WO2024175705A1

WO2024175705A1 - Single-photon triggered optical switches and/or optical signal routers

Info

Publication number: WO2024175705A1
Application number: PCT/EP2024/054507
Authority: WO
Inventors: Anthony LAING; Dara MCCUTCHEON; Alberto POLITI; Lawrence ROSENFELD; Ross WAKEFIELD
Original assignee: Duality Quantum Photonics Ltd
Current assignee: Duality Quantum Photonics Ltd
Priority date: 2023-02-22
Filing date: 2024-02-22
Publication date: 2024-08-29
Anticipated expiration: 2025-08-22
Also published as: GB202302560D0; GB2627731A; AU2024226073A1

Abstract

Optical switching using an optical resonator into and out of which travelling optical modes (4), of a given optical signal frequency, to selectively couple by electro-optically modulating the resonance characteristics (e.g., resonance optical frequency) of the optical resonator (12) in response to a signal produced by the single-photon detector (20). The detection of a photon (27B) controls how (and when) the travelling optical modes couple into and/or out of the optical resonator. In this way, the route of travelling optical modes of light may be switched in response to (i.e., condition upon) the detection of a single photon.

Description

SINGLE-PHOTON TRIGGERED OPTICAL SWITCHES AND/OR OPTICAL SIGNAL ROUTERS

Field of the Invention

The present invention relates to optical switches and/or optical signal routers and particularly, although not exclusively, to single-photon triggered optical switches and/or optical signal routers.

Background

Integrated photonics as a platform for quantum technologies has many advantages, such as the potential to integrate thousands of components on a single chip, ease of integration with more traditional photonic technologies, and the relatively cheap, fast and robust nature of information carries in the form of single photons or other quantum states of light. However, the main drawback of photonics is the challenge in engineering some form of coupling between optical signals, since light signals do not mutually interact in the same way as, for example, two electrons do via the coulomb force. This means that constructing all- optical logic gates between optical signals is challenging.

One way to engineer effective interactions between optical signals is to electrically detect all or part of one of the signals, thereby converting it into an electronic signal, then to amplify this electronic signal so that it can be used to drive or modulate the target optical signal. However, this approach has the drawback that there is some latency associated with the detection, amplification and optical-to-electronic conversion process. The total latency has multiple contributions, including the response time of the single photon detector, the bandwidth of the amplifier which converts the voltage signal produced from the detector to drive the optical modulator, the bandwidth of the modulator, and the delay induced by electronic connections between the various elements.

This latency in turn means that the target signal typically must pass through a delay line of order metres in length. Such delay lines can take up considerable and valuable physical space on an integrated chip. Moreover, propagation losses will also be incurred. This is particularly damaging for single photons in quantum information applications, since information encoded onto the quantum states of single photons cannot be amplified or copied. The result of these factors means that we do not currently have an effective way to have any kind of switching or controllable coupling involving single photons or other quantum states of light. This is a major challenge for optical quantum technologies, where fast switching is needed for, e.g., multiplexing probabilistic sources, providing the necessary feedforward for quantum teleportation, or conditional operations for quantum gates and ultimately optical quantum information processing.

The present invention has been devised in light of the above considerations. Summary of the Invention With the issues above in mind, the inventors have recognised that optical quantum technologies as they stand are lacking in the fundamental capability to do fast switching conditioned on single photon detection. The key to achieving this is to remove the need for bulky and/or off-chip amplification of the electrical signal generated from a single photon detector, since the amplification typically incurs the greatest latency penalty. In order to remove the amplification step and modulate the target signal directly from the voltage pulse produced by a single photon detector, the inventors have realised that an advantage may be gained by the use of a sensitive structure such as a resonator that will respond to a relatively weak signal. Preferably, the use of a non-standard material platform such as thin-film lithium niobate which has a high electro-optic coefficient (or to use carrier injection in CMOS materials) may be used in this regard. The inventors have realised that a photon may be detected and an optical switch or re-router may be operated in response to that so as to re-route or switch a target optical signal without the need for off-chip amplification and the associated long delay lines and incurred losses. In a general sense the invention provides a method to re-route or switch a target optical signal conditional on the detection of one or more photons in a manner that does not require a bulky and/or off-chip electronic amplifier. This innovation may be implemented preferably by using the small (tens to hundreds of mV) voltage output from e.g., known and available single photon detectors to modulate the resonance of a resonator. For example, the resonator may in general couple multiple optical modes (e.g., waveguide modes). Optical signals propagating in one of the waveguide modes may be distributed via the resonator across the modes (e.g., the waveguide modes), in a manner that depends on the resonance of the resonator and the coupling between the resonator and optical modes (e.g., the waveguides). Thus, an initial optical signal may be distributed differently depending on the detection of a single photon. This may provide a fast single-photon transducer which may enable performance of optical transformations conditional on the detection of a single photon, and thus removes the need for long delay lines and the requirement for real time feedforward logic with long electronic connections. The fast single photon transducer may produce a laser pulse output (e.g., a strong laser pulse) when a single photon is detected, with the laser pulse then used to directly drive the optical transformations (e.g., using a non- linear optical element). This allows an optical transformation on a target mode of light (e.g., a waveguide mode), thus giving rise to a single-photon conditioned optical operation. At its most general, the invention is to implement optical switching using an optical resonator into and out of which travelling optical modes, of a given optical signal frequency, can selectively couple by electro- optically modulating the resonance characteristics (e.g., resonance optical frequency) of the optical resonator in response to a signal produced by the single-photon detector (SPD). The detection of a photon can thereby control how (and when) the travelling optical modes couple into and/or out of the optical resonator. In this way, the route of travelling optical modes of light may be switched in response to (i.e., condition upon) the detection of a single photon. That single photon can be prepared by any desired optical or photonic process occurring within a part of an optical or photonic circuit of which the optical switch forms a part or is optically coupled to. Examples of optical or photonic circuits include photonic information processor circuits, photonic quantum computer circuits, photonic communications circuits and components and the like. The quantum states of travelling optical modes conditionally routed by the optical switching may then be subject to further optical processing, such as quantum-optical processing, as desired. Here, the term ‘quantum-optical processing’ is intended to include a reference to the processing of an optical signal by a process defined by treating the optical signal as possessing a quantum state which may be manipulated according to defined quantum processes to achieve a desired quantum state. The ‘quantum-optical processing’ may comprise applying a process corresponding to the application of a quantum operator to the optical signal (e.g., a ‘displacement operator’, a ‘phase shift operator’ etc.). The ‘quantum-optical processing’ may require the optical signal to be defined in terms of a quantum state or states e.g., representing a photon or a stream of photons rather than a classical electromagnetic wave. For example, ‘quantum-optical processing’ may comprise entanglement. Noting that quantum states of travelling optical modes can be entangled, e.g., by mixing the modes at an appropriately chosen multiport. The simplest example is the superposition of two travelling modes by an optical beam splitter. A combination of beam splitters with measuring instruments in certain output channels may therefore be a method for engineering quantum states of travelling optical fields. For example, ‘quantum-optical processing’ may comprise generation of ‘squeezed’ states from the optical signal. For example, ‘quantum-optical processing’ may comprise generation of phase-shifted quantum states from the optical signal. For example, ‘quantum-optical processing’ may comprise generation of ‘displaced’ quantum states from the optical signal (e.g., to displace a state in phase space by a desired magnitude). The optical resonator may comprise a ring resonator, a racetrack resonator, a disk resonator, a bow-tie resonator, a photonic crystal ring resonator, a photonic crystal cavity resonator (2D or 1D cavities), a semiconductor micro-pillar cavity resonator. A ring resonator structure, racetrack resonator structure has been found to be particularly beneficial as it can blend high quantity-factor ( ^^), relatively easier integration into an optical circuit, and greater efficiency of modulation. The invention may implement optical switching by conditionally routing an optical signal, of a given optical signal frequency, from an optical signal source according to the following conditions: Route 1: When the Resonator is Resonant with the Light from the Light Source This condition may hold when no photon is detected by the single-photon detector, such that light input to the system is strongly coupled into the resonator with the effect that an optical output via a specified optical output route of the system is low or negligible. Route 2: When the Resonator is Non-resonant with the Light from the Light Source This condition may hold when a photon is detected by the single-photon detector, such that light input to the system is weakly, or negligibly, coupled into the non-resonant resonator with the effect that the optical output via the specified optical output route of the system is high. In this sense, the absence of a single-photon detection event by the single-photon detector may cause the optical resonator to be able to accommodate an input optical signal (i.e., the resonator becomes at least a part of the route), whereas the occurrence of a single-photon detection event may cause the optical resonator to be unable to accommodate the input optical signal (i.e., the resonator is not a part of the route). Alternatively, the re-router may be configured in the opposite sense: i.e., when no single- photon is detected, an input optical signal is not coupled strongly with the resonator such that the optical resonator is unable to accommodate the input optical signal (i.e., the resonator is not a part of the route), and the input optical signal may exit a particular waveguide. Whereas when a single-photon is detected, the input optical signal is made resonant thereby causing the optical resonator to be able to accommodate an input optical signal (i.e., the resonator becomes at least a part of the route). The optical signal may not exit or, in some examples, may exit via a different waveguide. Thus, in the alternative, the invention may implement optical switching by conditionally routing an optical signal, of a given optical signal frequency, from an optical signal source according to the following conditions: Route 1: When the Resonator is Resonant with the Light from the Light Source This condition may hold when a photon is detected by the single-photon detector, such that light input to the system is strongly coupled into the resonator with the effect that an optical output via a specified optical output route of the system is low or negligible. Route 2: When the Resonator is Non-resonant with the Light from the Light Source This condition may hold when no photon is detected by the single-photon detector, such that light input to the system is weakly, or negligibly, coupled into the non-resonant resonator with the effect that the optical output via the specified optical output route of the system is high. In a first aspect, the invention may provide an optical switch for receiving an optical signal of a given optical signal frequency from an optical signal source and for outputting the optical signal, the optical switch comprising; an optical router configured for routing the optical signal conditional on the detection of a single photon, the optical router comprising: an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switch further comprises: a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; wherein the optical modulator is configured to modulate the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to either: (a) change from being a resonance bandwidth that includes the optical signal frequency to being a resonance bandwidth that excludes the optical signal frequency; or, (b) change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency. It is to be understood that a resonance bandwidth of an optical resonator may be set by factors determined during fabrication, such as physical dimensions and other characteristics, it is also determined – after it has been fabricated – by the refractive index of the optical resonator. The refractive index of the optical resonator may be controllably adjusted after fabrication of the resonator, as is discussed in more detail below. References herein to the terms “…resonance bandwidth…” may be considered to include a reference to a continuous range of frequencies including not only the exact resonance frequency, ω_res, but also including closely neighbouring frequencies that differ from the exact resonance frequency by not more than one half of the value of the resonance line-width (e.g., FWHM),

of the spectral resonance profile of the optical resonator. For example, the terms “…resonance bandwidth…” may be considered to include a reference to a continuous range of frequencies, ω , which satisfy the following condition:

Accordingly, the optical resonator may be considered to resonate at any optical frequency within the resonance bandwidth, and all such optical frequencies may be considered to be “resonant optical frequencies” for practical purposes. For example, light of frequency ^^ may be considered to reside within the “…resonance bandwidth…” of a resonator possessing a quality factor, ^^, within the meaning of the term used herein, if the following condition is met:

Here,

It will be understood that the terms “…modulate the resonance bandwidth…” may be considered to include a reference to a change, shift or translation of the spectral location or position of the resonance bandwidth (e.g., the position of its centre). Noting that the resonance bandwidth surrounds the exact resonance frequency, ω_res, which may be at or close to the bandwidth centre, this means that a change, shift or translation of the spectral location or position of the resonance bandwidth corresponds to a change, shift or translation of the spectral location or position of the resonance frequency within it. Thus, a change (modulation) of the frequency position of the resonance bandwidth may comprise a change/shift (modulation) in the resonance frequency. It is also noted that a modulation of the refractive index of the optical resonator may result in a change in the size of the resonance line-width (e.g., FWHM), , of the spectral resonance profile of the optical resonator. Thus, a change in the size of the resonance bandwidth (due to a change in line-width of the spectral resonance profile) alone or together with a change in the spectral position of the resonance bandwidth (and of the resonance frequency) may occur in response to a modulation of the refractive index of the optical modulator. Either or both effects may contribute to the end result of the resonance bandwidth either including or excluding a given optical signal frequency, as desired. The configuration (a) stated above whereby the resonance bandwidth is changed to exclude the optical signal frequency may thereby permit the optical input signal to be transmitted to the output optical waveguide part as a non-resonant optical output signal for output from the optical switch. The configuration (b) stated above whereby the resonance bandwidth is changed to include the optical signal frequency may thereby: (b1) permit the optical input signal to be transmitted to the output optical waveguide part as a resonant optical output signal for output from the optical switch, or (b2) permit suppression (e.g., prevention) of the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch. These two alternative possibilities (b1 and b2) under configuration (b) may be determined according to how the output optical waveguide part is optically coupled to the input optical waveguide part. For example, under configuration (b1), the optical coupling may be indirect via the optical resonator as an intermediate part if the output optical waveguide part and the input optical waveguide part are two physically separate waveguides. For example, under configuration (b2), the optical coupling may be direct if the output optical waveguide part and the input optical waveguide part are two separate parts of one continuous waveguide. Accordingly, the optical input signal to be transmitted to the output optical waveguide part either directly from the input optical waveguide part substantially without resonating within the optical resonator, or indirectly from the input optical waveguide part via the optical resonator, having first been resonantly ‘loaded’ into the optical resonator from the input optical waveguide part. In this way, in the quiescent state when the single-photon detector produces no electrical detection signal, the optical switch is configured to implement a signal routing (e.g., Route 1) which includes the resonator as part of the route or path of the travelling optical mode of any input optical signal, but which substantially excludes or neglects another available, specified optical output route from the path of the travelling optical mode. Conversely, in the active state when the single-photon detector produces an electrical detection signal, the optical switch is configured to implement a signal routing (e.g., Route 2) which substantially excludes or neglects the resonator as part of the route or path of the travelling optical mode of any input optical signal, but which includes the other available, specified optical output route as part of the path of the travelling optical mode. In other words, in the quiescent state the travelling optical mode is substantially resonant with the resonator and travels into the resonator. Once within the optical resonator, the resonant travelling optical mode may subsequently remain within the optical resonator (e.g., be trapped there) or may pass through the resonator and onto an onward route travelling away from the resonator. The onward route may be via an optical waveguide that is optically coupled (e.g., critically coupled) to the optical resonator. In any case, the result of implementing this signal routing (Route 1) is that the optical signal output at the other available, specified optical output route of the system is low or negligible. In the active state, the optical resonator is rendered non-participating in the route or path of the travelling optical mode of an input optical signal and this permits the route or path of the input optical signal to include the other available, specified optical output route of the system. The refractive index may be changed along only a limited section of the length/circumference of the optical resonator, or preferably along the entire length/circumference of the optical resonator. Changing the refractive index along only a limited section of the length/circumference of the optical resonator may induce more optical backscattering and may be less efficient (i.e., have a lower resonant shift for the same electric signal) than if the refractive index were changed along the entire length/circumference of the optical resonator. For example, the optical modulator may be operable to implement local refractive index changes to the material of the optical resonator at local parts (i.e., not the whole of) the optical pathway defined by the optical resonator. Alternatively, the optical modulator may be operable to implement global refractive index changes to the material of the optical resonator at substantially all parts (i.e., the whole of) the optical pathway defined by the optical resonator. The optical modulator may be operable to implement refractive index changes induced by effects including, but not limited to, thermo-optic, electro-optic, carrier-injection, piezo-electric, birefringent, micro-electro-mechanical, strain-inducing, or acousto-optic. The input optical waveguide part and/or the output optical waveguide part may be evanescently optically coupled to the optical resonator (e.g., physically and materially separated from the optical resonator but in sufficient proximity to permit respective evanescent electromagnetic fields, or quantum modes/states, to couple across the separation). Alternatively, the input optical waveguide part and/or the output optical waveguide part may be physically optically coupled to the optical resonator (e.g., in physical contact with, or integrally formed with, or optically bonded to the optical resonator). The input optical waveguide part and/or the output optical waveguide part may be optically coupled to the optical resonator via a tunable directional coupler or via a Mach-Zehnder Interferometer. The coupling may be tuned prior to use in order to vary the ‘at-rest’ state of coupling to the optical resonator (i.e., when there is no photon detected). This would allow for fine-tuning to get better critical coupling to the optical resonator when no photon is detected. The input optical waveguide part and the output optical waveguide part may each be a respective part of one continuous optical waveguide that is optically coupled to the optical resonator at an optical coupling region of the optical resonator, wherein the input optical waveguide part extends to the optical coupling region and the output optical waveguide part extends from the optical coupling region. For example, the optical resonator may comprise a looped waveguide such as a ring optical resonator (e.g., a micro-ring optical resonator), a racetrack resonator, or a disk resonator, that is physically spaced from the continuous optical waveguide by a spacing (which may be occupied with a material or substance as appropriate) so as to be evanescently optically coupled to the continuous optical waveguide. Alternatively, the continuous optical waveguide may comprise the optical resonator, which may be formed integrally with (e.g., within) the continuous optical waveguide as an optical waveguide structure containing an optical resonator cavity (e.g., a photonic crystal cavity resonator (2D or 1D cavities), or a semiconductor micro-pillar cavity resonator). The optical resonator may comprise a looped optical resonator (e.g., a ring optical resonator, a optical racetrack resonator, a photonic crystal ring optical resonator, a disk optical resonator), a bow-tie optical resonator, a photonic crystal cavity optical resonator (2D or 1D cavities), or a semiconductor micro-pillar cavity optical resonator. A looped optical resonator structure (e.g., especially a ring, racetrack, or photonic crystal ring optical resonator, and also a disk optical resonator) has been found to be particularly beneficial as it can blend high quantity-factor ( ^^), relatively easier integration into an optical circuit, and greater efficiency of modulation. Desirably, the optical resonator comprises a ^^-factor of not less than 1,000,000. This has the benefit of permitting a relatively small modulation in the resonance frequency of the optical resonator to produce the effect of a very large relative change (e.g., % change) on the optical coupling strength between the optical resonator and the output optical waveguide part. This relative change is reflected in a corresponding relative change in the optical output signal intensity from the switch. These ^^-factor may be less than 1,000,000 in the case of using a SPAD that can output a larger voltage, providing a bigger frequency shift. Thus, scenarios exist where it is beneficial to employ a ^^-factor of less than 1,000,000, such as when rerouting a spectrally broader optical signal. The optical resonator is preferably a single-mode waveguide, but may be a multi-mode waveguide. The continuous optical waveguide is preferably critically coupled to the optical resonator. Accordingly, by critically coupling the continuous optical waveguide to the optical resonator, this permits that as light couples into the waveguide and back out again, it can accumulate the necessary amount of phase to generate destructive interference in the output of the continuous optical waveguide to filter out light travelling in the onward direction of the waveguide beyond the resonator. This renders the optical output there very suppressed or negligible, or substantially non-existent in the onward direction of the waveguide beyond the resonator. The optical switch may comprise a further output optical waveguide wherein the continuous optical waveguide and the further output optical waveguide are separate optical waveguides each separately optically coupled to the optical resonator at separate respective optical coupling regions of the optical resonator. The further output optical waveguide is preferably critically coupled to the optical resonator. Accordingly, by critically coupling the further output optical waveguide to the optical resonator, this permits that by the time light within the resonator couples into the further output optical waveguide it has accumulated the necessary amount of phase to generate destructive interference onward parts of the optical resonator whilst accumulating the necessary amount of phase to generate constructive interference onward parts of the further output optical waveguide. This has the effect of suppressing light travelling in the onward direction of the optical resonator while simultaneously enhancing and promoting light travelling in the onward direction of the further output optical waveguide beyond the resonator. This renders the optical output to the further output optical waveguide enhanced and routed to propagate along the further output optical waveguide beyond the resonator. The optical modulator is preferably configured to modulate the refractive index of the optical resonator by the electrical modulation signal during a pre-set modulation time interval, ^^, starting from a time of detection of a single photon by the single-photon detector unit thereby to modulate the resonance bandwidth during the pre-set modulation time interval. Preferably the pre-set modulation time interval, ^^, is not greater than twice the decay time, ^^ = 1/Γ^ത, of the optical resonator, or more preferably not greater than the decay time of the optical resonator, wherein the decay time of the optical resonator corresponds to the inverse of the optical loss rate, Γ^ത, of the optical resonator. The optical loss rate, Γ^ത, may correspond to a rate of the optical resonator when an electrical detection signal is output by the single-photon detector unit, and a modulation of the refractive index of the optical resonator occurs. In other words, preferably, ^^ ≤ 2 ^^, or more preferably ^^ ≤ ^^. Note that an optical signal travelling within the input optical waveguide is not resonantly coupled into the resonator if it is travelling there during the pre-set modulation time interval, ^^, but may be resonantly coupled into the resonator if it is travelling there after (e.g., immediately after) the end of the pre-set modulation time interval, ^^. During the pre-set modulation time interval, ^^, a consequence of modulating the refractive index of the optical resonator is to permit light that may already be present and resonating within the optical resonator (e.g., ‘trapped’ within the resonator) to begin to couple out of the resonator and into the output optical waveguide or the further output optical waveguide. This ‘leaked’ light emanating from the resonator may undesirably combine with the optical signal that has travelled from the input optical waveguide directly to the output optical waveguide or the further output optical waveguide without having couped into the optical resonator. To mitigate against this undesirable combination of light, the inventors have found that by limiting the duration of the pre-set modulation time interval to be no greater than twice the decay time,

/ , of the optical resonator, or more preferably not greater than the decay time of the optical resonator, this denies sufficient time for the leaked light to accumulate to significant levels in the output optical waveguide or the further output optical waveguide. Similarly, this mitigation may be implemented in terms of a controlled relationship between the pre-set modulation time interval, ^^, and a coupling rate of the resonator. For example, the output optical waveguide part may be optically coupled to the optical resonator according to a coupling rate, Γ₁, which is less than the inverse of the pre-set modulation time interval (i.e., Γ₁ < 1/ T) such that, in use, optical input light resonating within the optical resonator is rendered non-resonant during the pre-set modulation time interval, ^^, and is transferred by the optical resonator to the output optical waveguide part at the coupling rate, Γ_^, during the pre-set modulation time interval, T. The optical coupling rate, Γ_^, may correspond to a rate of coupling when an electrical detection signal is output by the single-photon detector unit, and a modulation of the refractive index of the optical resonator occurs. It is possible that the modulation of the refractive index of the resonator may also have a side-effect of modulating the coupling rates between the optical resonator and the input/output waveguides. This effect is expected to be small compared with the detuning of the resonator and input optical signals. The output optical waveguide part may be optically coupled to the optical resonator according to a coupling rate, Γ₁, and, the single-photon detector unit may be configured to output the electrical detection signal in the form of a voltage pulse such that the voltage value of the electrical detection signal reduces from a voltage pulse peak value according to a pre-set voltage decay rate, Γ_d , that is greater than the coupling rate: Γ_d > Γ₁. Here, Γ_d = 1/τ_d where τ_d is the decay time of the voltage pulse. In this way, the electrical detection signal pulse duration may be arranged to occur more swiftly than the time frame of the coupling time (i.e., the inverse of the coupling rate) for coupling light out from the optical resonator and into the output optical waveguide part. That is, the coupling of light the optical resonator can be delayed, or suppressed, for the time-scale of the voltage signal, but light already within the optical resonator may continue to couple out (i.e., leak out). The amount of leaked-out light may be reduced to suitable levels by imposing the above constraint on the electrical detection signal pulse duration. In addition, it has been found that by using a suitably short voltage decay rate, T_d , suppresses oscillations in the optical output signal intensity. The optical coupling rate, Γ₁, may correspond to a rate of coupling when an electrical detection signal is output by the single-photon detector unit, and a modulation of the refractive index of the optical resonator occurs. Alternatively, in other implementations, the optical resonator may be pre-loaded with resonant optical signal light ready for release from the optical resonator to the output optical waveguide part or the further output optical waveguide part as a non-resonant optical output signal for output from the optical switch, in response to a single-photon detection event by the single-photon detector. For example, the output optical waveguide part may be optically coupled to the optical resonator according to a coupling rate, Γ₁, which is greater than the inverse of the pre-set modulation time interval (i.e., Γ₁ > 1/ T). The optical coupling rate, Γ₁, may correspond to a rate of coupling when an electrical detection signal is output by the single-photon detector unit, and a modulation of the refractive index of the optical resonator occurs. In use, an optical input signal when resonating within the optical resonator may be rendered non-resonant during the pre-set modulation time interval, T, and may be transferred by the optical resonator to the output optical waveguide part at the coupling rate, Γ₁, as a non-resonant optical output signal for output from the optical switch. In other words, the response time of the optical resonator may be such that it is less than the pre-set modulation time interval such that the optical resonator has ‘enough time’ to respond to the modulation of its resonance bandwidth by releasing the pre-loaded resonant light within it, during the pre-set modulation time interval. The output optical waveguide part may be optically coupled to the optical resonator according to a coupling rate, Γ₁. The optical resonator may comprise an optical loss rate,

Γ. The optical modulator may be configured to modulate the refractive index of the optical resonator to modulate the resonance bandwidth so to change the frequency position thereof by a resonance frequency shift, ω₁, that has a value exceeding the optical loss rate (e.g., total loss rate), Γ. For example, the resonance frequency shift, ω₁, may have a value of at least five times the loss (e.g., total loss) rate,

or more preferably a value of at least ten times the optical loss rate,

or more preferably a value of at least 50 times the optical loss rate,

or at least 100 times the optical loss rate,

ω ≥ 100Γ. The inventors have found that these constraints provide better routing efficiency. The optical loss rate,

, may correspond to a rate of coupling when an electrical detection signal is output by single-photon detector unit, and a modulation of the refractive index of the optical resonator occurs. The optical loss rate (e.g., total loss rate) of the ring, Γ defines the bandwidth/FWHM of the transmission spectrum. When the frequency shift is more than (approximately) twice this amount this corresponds to circumstances when the optical signal has largely uncoupled from the optical resonator (e.g., ring). When the frequency shift is below (approximately) twice the total loss rate of the optical resonator, the optical signal remains significantly coupled into the optical resonator. Desirably, when the input optical waveguide part and the output optical waveguide part form parts of one continuous optical waveguide, the continuous optical waveguide is optically coupled to the optical resonator according to a coupling rate, Γ_^, and the optical resonator may comprise an optical loss rate (e.g., total loss rate),

Γ, that has a value which is substantially equal to twice the coupling rate, Γ₁, such that

. This may enable critical coupling to occur, such as when the continuous optical waveguide provides the only optical input and output channels to/from the optical resonator. The optical coupling rate, Γ₁, may correspond to a rate of coupling when an electrical detection signal is output by the single- photon detector unit, and a modulation of the refractive index of the optical resonator occurs. The optical switch may further comprise the optical signal source configured to output an optical signal comprising a given optical signal frequency. The optical signal source is configured to output an optical signal in the form of an optical pulse comprising either a plurality of photons, or not more than a single photon. Desirably, the output optical waveguide part is optically coupled to the optical resonator according to a coupling rate, , and the optical signal source may be configured to output an optical signal in the form of an optical pulse comprising a pulse temporal width, σ, which exceeds the inverse of the coupling rate,

. The optical coupling rate,

, may correspond to a rate of coupling when no electrical detection signal is output by the single-photon detector unit, and no modulation of the refractive index of the optical resonator occurs. The further output optical waveguide part is preferably optically coupled to the optical resonator according to a coupling rate, Γ₂, which is substantially equal to a coupling rate, Γ₃ with which the input optical waveguide part is optically coupled to the optical resonator, Γ₂ = Γ₃. This may enable that the input optical waveguide part and the further output optical waveguide part to be collectively critically coupled to the optical resonator. Desirably, the optical resonator may comprise an optical loss rate,

Γ, that has a value which is substantially equal to twice the coupling rate, Γ₂, of the further output optical waveguide part (and of the input optical waveguide part, Γ₃) such that

This may enable critical coupling to occur. The optical coupling rates, Γ₂, Γ₃, may correspond to a rate of coupling when no electrical detection signal is output by the single-photon detector unit, and no modulation of the refractive index of the optical resonator occurs. As a result, the efficiency with which an input optical signal is routed may be greatly improved. In other words, critical coupling provides that resonant travelling optical modes of an input optical signal may be substantially fully coupled to the further output optical waveguide part via the optical resonator. When the output optical waveguide part and the further output optical waveguide part collectively form one continuous waveguide, then preferably the one continuous waveguide is optically critically coupled to the optical resonator. The critical coupling may be provided at least when no electrical detection signal is output by single-photon detector unit, and no modulation of the refractive index of the optical resonator occurs. The optical signal source may be configured to output an optical signal in the form of an optical pulse. Preferably, the spectral width of the optical signal is smaller than the resonance bandwidth of the resonator. As a result, the spectral components of the travelling modes of the optical input signal may substantially fully fit within the resonance bandwidth of the optical resonator when no electrical detection signal is output by single-photon detector unit and may be fully excluded from the resonance bandwidth of the optical resonator when an electrical detection signal is output by single-photon detector unit. The optical signal source may be configured to output an optical signal in the form of a continuous optical output. The optical signal may be substantially monochromatic. In a second aspect, the invention may provide an optical switching assembly comprising an optical switch as disclosed herein according to any aspect of the invention and further comprising an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal. Preferably, the output monitoring unit is configured to determine the presence or the absence of the optical output signal according to whether a detected optical output power exceeds a pre-set detection power threshold value wherein the pre-set detection power threshold value exceeds the optical output power detected by the output monitoring unit in the absence of an electrical detection signal by the single-photon detector unit. In a third aspect, the invention may provide an optical switching assembly comprising an optical switch as disclosed herein according to any aspect of the invention and further comprising an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal, wherein: the optical signal source is configured to output an optical signal in the form of an optical pulse comprising a pulse temporal width, σ; and, the output monitoring unit is configured to determine the presence of the optical output signal within a monitoring time interval, Δt , which has a duration not exceeding the pulse temporal width, ∆ t ≤ σ. This has been found to be an effective and reliable choice of time interval encompassing the peak of an optical output signal associated with the optical pulse and consequential to an electrical detection signal being output by single-photon detector unit. The monitoring time interval corresponds to a binning time interval. In some instances, “no detection” and “detection” cases can give either a single peak or a double peak in the output optical signal, so if the binning time interval is less than the separation between these peaks, then having a signal or not within a particular bin may be dependent on a detection event. In a fourth aspect, the invention may provide an optical switching assembly comprising an optical switch as disclosed herein according to any aspect of the invention wherein the optical resonator comprises an optical loss rate (e.g., total loss rate),

Γ, and the optical switching assembly further comprises an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal, wherein the output monitoring unit is configured to determine the presence of the optical output signal within a monitoring time interval, Δ t, which has a duration not exceeding the inverse of the optical loss rate,

1/Γ, such that

/ This has been found to be an effective and reliable choice of time interval encompassing the peak of an optical output signal consequential to an electrical detection signal being output by single-photon detector unit. The optical switching assembly may comprise a laser unit configured to generate either a continuous wave (CW) laser light output, or a pulsed laser light output for input into the input optical waveguide part. For example, the laser unit may comprise an optical output port that is optically coupled to the input optical waveguide part (e.g., to an input port at a terminal end thereof, or at a position along the waveguide via any suitable optical coupler) for propagation towards the optical resonator. In this way, the laser light output from the laser unit may provide the optical input signal to be transmitted along the input optical waveguide part to the output optical waveguide part either directly from the input optical waveguide part substantially without resonating within the optical resonator, or indirectly from the input optical waveguide part via the optical resonator, having first been resonantly ‘loaded’ into the optical resonator from the input optical waveguide. The laser unit may produce a laser pulse output (e.g., a strong laser pulse) or a CW laser output when a single photon is detected, with the laser pulse or the CW laser output then used to directly drive an optical transformation (e.g., using a non-linear optical element). For example, the output monitoring unit may comprise: a further optical signal input port for receiving a further optical input signal; an optical transformer unit optically coupled to the further optical signal input port and configured to apply a pre-set optical transformation to the further optical input signal conditional on generation of a detection signal thereby to generate a transformed optical signal; an optical signal output port for outputting the transformed optical signal as an optical output signal. In a fifth aspect, the invention may provide an optical switching assembly comprising: an optical switch for receiving an optical signal of a given optical signal frequency from an optical signal source and for outputting the optical signal, the optical switch comprising; an optical router configured for routing the optical signal conditional on the detection of a single photon, the optical router comprising: an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switch further comprises: a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal, wherein the optical modulator is configured to modulate the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to suppress the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch; an output monitoring unit responsive to the presence of said optical output signal output from the optical switch to generate a detection signal, the output monitoring unit further comprising: a further optical signal input port for receiving a further optical input signal; an optical transformer unit optically coupled to the further optical signal input port and configured to apply a pre-set optical transformation to the further optical input signal conditional on generation of a detection signal thereby to generate a transformed optical signal; and, an optical signal output port for outputting the transformed optical signal as an optical output signal. The pre-set optical transformation, according to any aspect of the invention, preferably comprises one or more of: a transformation to a single-mode optically squeezed state; a transformation to a two-mode optically squeezed state; a transformation to an optically phase-shifted quantum state; a transformation to a displaced quantum state. In a sixth aspect, the invention may provide an integrated photonic circuit comprising an optical switch or an optical switching assembly as disclosed herein according to any aspect of the invention. The invention may provide a photonic processing network or circuit which is part of a broader optical circuit or system. In some aspects, the invention may comprise (or be comprised within) an integrated photonic chip. The photonic circuit and/or chip may be formed in materials including, but not limited to: Silicon (Si), Silicon nitride (SiN), Silica (SiO2), Gallium Arsenide (GaAs), Indium Phosphide (InP), Polymer, Lithium Niobate (LiNbO) or Aluminium Nitride (AlN). In a seventh aspect, the invention may provide an optical switching method for switching an optical signal of a given optical signal frequency from an optical signal source by routing the optical signal conditional on the detection of a single photon, the method comprising: providing an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; providing an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; providing an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switching method further comprises: providing a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, providing an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; and, by the optical modulator, modulating the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to either: (a) change from being a resonance bandwidth that includes the optical signal frequency to being a resonance bandwidth that excludes the optical signal frequency; or, (b) change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency. The configuration (a) of the method stated above, in which the resonance bandwidth is changed to exclude the optical signal frequency, may permit the optical input signal to be transmitted to the output optical waveguide part as a non-resonant optical output signal for output from the optical switch. The configuration (b) of the method stated above, in which the resonance bandwidth is changed to include the optical signal frequency may permit either of the following outcomes/methods: (b1) to permit the optical input signal to be transmitted to the output optical waveguide part as a resonant optical output signal for output from the optical switch, or (b2) to permit suppression (e.g., prevention) of the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch. The method may include providing an optical transformer unit for receiving a further optical input signal, and therewith applying a pre-set optical transformation to the further optical input signal. The method may include providing an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal. The method may include, by the output transformer unit, applying the pre-set optical transformation to the further optical input signal conditional on generation of the detection signal thereby to generate a transformed optical signal. The method may include outputting the transformed optical signal as an optical output signal. In an eighth aspect, the invention may provide an optical switching method for switching an optical signal of a given optical signal frequency from an optical signal source by routing the optical signal conditional on the detection of a single photon, the method comprising: providing an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; providing an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; providing an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switching method further comprises: providing a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, providing an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; and, by the optical modulator, modulating the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to suppress (e.g., prevent) the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch; wherein the optical switching method further comprises: generating a detection signal in response to the presence of said optical output signal from the optical switch; providing a transformation unit and thereat receiving a further optical input signal; by the transformation unit, applying a pre-set optical transformation to the further optical input signal conditional on generation of said detection signal thereby to generate a transformed optical signal; and, outputting the transformed optical signal. The pre-set optical transformation preferably comprises one or more of: a transformation to a single-mode optically squeezed state; a transformation to a two-mode optically squeezed state; a transformation to an optically phase-shifted quantum state; a transformation to a displaced quantum state. The invention includes the combination of the aspects and preferred features described except where such a combination is clearly impermissible or expressly avoided. Summary of the Figures Embodiments and experiments illustrating the principles of the invention will now be discussed with reference to the accompanying figures in which: Figure 1 shows a ring optical resonator device. Figure 2 shows a ring optical resonator device. Figure 3A shows an optical switching/routing device according to a first embodiment of the invention in a first switching/routing state. Figure 3B shows the optical switching/routing device of Figure 3A in a second switching/routing state. Figure 3C shows a schematic diagram of an optical transmission spectrum of the optical switching/routing device within the optical transformation device of Figure 3A and 3B according to a first embodiment. Figure 4A shows an optical switching/routing device according to a second embodiment of the invention in a first switching/routing state. Figure 4B shows the optical switching/routing device of Figure 4A in a second switching/routing state. Figure 4C shows a schematic diagram of an optical transmission spectrum of the optical switching/routing device within the optical transformation device of Figure 4A and 4B according to a first embodiment. Figure 4D shows an optical switching/routing device according to a third embodiment of the invention in a first switching/routing state. Figure 4E shows the optical switching/routing device of Figure 4D in a second switching/routing state. Figure 4F shows an optical switching/routing device according to a fourth embodiment of the invention in a first switching/routing state. Figure 4G shows the optical switching/routing device of Figure 4F in a second switching/routing state. Figure 5 shows a schematic view of parameters of the optical switching/routing device according to a first embodiment of the invention. Figure 6 shows a schematic view of parameters of the optical switching/routing device according to a second embodiment of the invention. Figure 7 graphically shows optical power output spectra for an optical switching/routing device according to an embodiment of the invention. Figures 8A to 8C show an optical power spectrum of a laser pulse (Figure 8A) and corresponding optical power spectra of an optical output for an optical switching/routing device according to an embodiment of the invention. Figures 9A to 9C show an optical power temporal profile of a laser pulse (Figure 9A) and corresponding optical power temporal profiles of an optical output for an optical switching/routing device according to an embodiment of the invention. Figures 10A to 10E show an optical power temporal profile of a laser pulse (Figure 10A) and corresponding optical power temporal profiles of an optical output for an optical switching/routing device according to an embodiment of the invention. Figures 11A and 11B each show the time profile of the electrical signal generated by a single-photon detector. Figures 12A to 12E show an optical power temporal profile of a laser pulse (Figure 12A) and corresponding optical power temporal profiles of an optical output for an optical switching/routing device according to an embodiment of the invention. Figures 13A to 13B show optical power temporal profiles of an optical output for an optical switching/routing device according to an embodiment of the invention. Figures 14A to 14C show optical power temporal profiles of an optical output for an optical switching/routing device according to an embodiment of the invention. Figure 15 shows a schematic diagram of an optical transformation device. Figures 16A to 16D show schematic diagrams of different implementations of the optical transformation device of Figure 15. Figures 17A to 17C show schematic diagrams of different implementations of an optical resonator according to embodiments of the invention. Figures 18A to 18E show schematic diagrams of different implementations of an optical waveguide according to embodiments of the invention. Detailed Description of the Invention Aspects and embodiments of the present invention will now be discussed with reference to the accompanying figures. Further aspects and embodiments will be apparent to those skilled in the art. All documents mentioned in this text are incorporated herein by reference. In the following examples, a photonic transducer is described where the voltage output from a single photon detector creates a light signal at a designated optical output port. The optical output signal could be continuous wave (CW), a coherent pulse, or a single photon. This may form the basis of a ‘photonic transistor’, whereby a pulsed optical input is switched (or not) by the detection (or lack of detection) of a single photon. A conditioned single photon gate or switch may be provided, whereby the detection of a single photon causes an operation to happen on a second photon (e.g., a quantum optical operation defined according to the application of quantum operators to a quantum state of light). Consider three types of optical inputs: Continuous Wave (CW), pulsed light and single photons (e.g., with a Gaussian wavepacket). These optical inputs may be coupled into a waveguide which is then coupled into an optical resonator (e.g., an optical ring resonator) and coupled out of the resonator either into the same optical waveguide or a different optical waveguide. The following disclosure presents detailed quantum mechanical modelling which contains both optical loss and optical backscattering effects. The time-dependent voltage output of a single photon detector is used to modulate (i.e., shift) a resonant frequency of an optical resonator which changes how the light couples into and out of the optical resonator. The inventors have found that a large enough voltage output from the single photon detector can cause a large enough shift in the resonant frequency of the optical resonator such that an initially resonant input light may be caused to cease coupling into the optical resonator. The rate at which the cessation of in-coupling has been found to be determined by the time scale of the time-dependent voltage output of a single photon detector. This contrasts with the out- coupling of light coupling out of the optical resonator, which happens at the time scale of the optical resonator (i.e., the inverse to the total loss rate of the resonator, which is related to the Q-factor of the resonator). However, it has been found that if the input light is initially non-resonant with the optical resonator and the resonant frequency of the optical resonator is subsequently shifted so that the input light becomes resonant, the light also couples into the optical resonator on the time scale of the ring rather than on the time scale of the time-dependent voltage output of a single photon detector. It has been found that performance is greatly enhanced when using optical resonators with high Q-factor values, which result in low loss rates from the resonator and, consequently, long time scales for the light to couple into the optical resonator. As a result of this, the inventors have found that a far greater degree of sensitivity and control may be achieved when the input light is constrained to be initially resonant with the optical resonator and the resonance frequency of the optical resonator is then modulated in the much shorter time scale of the time-dependent voltage output of a single photon detector. Going from on- resonance to off-resonance is easier to implement. Simulations show that the system is more efficient this way. Nevertheless, the opposite process may be implemented according to the invention, whereby initially the optical resonator is non-resonant with input light and is then modulated to become resonant. The inventors investigate the use of input light which takes the form of a continuous wave (CW). The inventors have found that when CW input light is used then good performance can be achieved when there are two optical waveguides coupled to the optical resonator. The two optical waveguides may include a first output optical waveguide arranged such that when there is no detection event from the single photon detector, the optical output of that waveguide is as close to zero as possible while the light within the optical resonator is found to couple out into the further (second) output optical waveguide. As such, when the single photon is detected, the input CW light stops coupling into the optical resonator thereby allowing an optical output from the first waveguide that is easily distinguishable from the optical output when no photon is detected by the single photon detector. In this way an optical signal is created where there was effectively (i.e., nearly, or practically) none before. This is a good characteristic for an optical transducer. The inventors also investigate the use of pulsed input light. This is equivalent to a single photon input in as far as the proportion of the input signal that is coupled is concerned, provided the bandwidth of the photon matches that of the corresponding pulsed input light. The inventors have discovered conditions that enable one to switch as much of the light as possible, conditioned on the detection of a single photon by the single photon detector. Good performance can be achieved when using a single optical output waveguide coupled to the optical resonator. This has been found to be effective in allowing as much light as possible to travel out of the system (i.e., optical output waveguide) when the optical resonator is made non-resonant with the pulsed input light. A procedure of time-binning the output has also been found to be advantageous. As noted above, detection/no-detection can result in multiple peaks or a single peak in the output optical signal. Time binning to isolate a single peak may then results in a detection-dependent signal. In order to switch as much of the light as possible, it has been found to be advantageous that the time scale of the optical pulse/photon is of the same order of magnitude as the decay time of the optical resonator. This has been found to be advantageous because, otherwise one may require a long voltage signal decay time in the detection signal generated by the single photon detector, which is an undesirable use of time because it is hard to engineer a long detector response and hard to get respective timescales of the various other components to match. In addition, it has also been found that without this careful tuning of time scales, insufficient light from the input optical signal will couple into an optical resonator to begin with, in which case it has been found that there is then insufficient difference between the output with and without single photon detector. The research and modelling of the inventors have found that careful tuning of the parameters of the system allows for a regime where, when no photon is detected by the single photon detector, the output optical signal may comprise a significant and reliably identifiable pulse/peak output. When a photon is detected by the single photon detector, the output optical signal may also comprise a significant and reliably identifiable different pulse/peak output. This allows the system to provide the function of a photonic transducer or transistor, and/or a conditioned gate. The disclosures below aim to model the properties and highlight a range of preferable configurations of implementations of the invention in terms of a ring optical resonator coupled to one or two waveguides, and Single Photon Detector (SPD), such as a Single Photon Avalanche Diode (SPAD) or Superconducting Nanowire Single Photon Detector (SNSPD) arranged to generate a voltage signal after detecting a single photon. This voltage signal is used to modulate the resonant frequency of the ring optical resonator with time. A light signal is input to this system, which could either be CW in nature or pulsed (e.g., a single photon wavepacket). The model includes the effects of the optical power, the wavefunctions, and the phases etc. of the input optical signal vs. the output optical signal from the system, as well as the effects of optical backscattering within the system. This modelling is presented for a classical (CW and Gaussian) coherent state (both with/without backscattering effects) and including time dependence due to the single photon detector voltage signal input (both with/without backscattering effects). This modelling is also presented for the conditions noted above in terms of an arbitrary quantum state. A Ring Resonator: The Classical Description An optical ring resonator typically consists of a straight optical waveguide section 3 coupled to a circular optical waveguide section 2, as shown in ring resonator 1 of Figure 1, or two separate straight waveguide sections (3, 9) each separately optically coupled to one common circular waveguide 2 and, therefore, optically coupled to each other via the circular waveguide section, as shown in ring resonator 11 of Figure 2. In the first case, shown in Figure 1, input light 4 (e.g., a continuous beam or a pulse) input to the ring resonator via a waveguide input port of the straight optical waveguide section, is coupled (i.e., transferred) into the circular optical waveguide section according to the strength of an optical coupling present at a coupling point, or short coupling region, 7 of the straight optical waveguide section where it is in close proximity to (or in contact with) the circular optical waveguide section. Due to the presence of this optical coupling, a proportion of light 5 is transferred into the circular optical waveguide section from the input light 4 within the straight optical waveguide section. The quantity so transferred depends upon the strength of optical coupling provided by the coupling point, or short coupling region, as well as the wavelength of the light undergoing the coupling in question. By contrast, the proportion of light 6 fully transmitted through the straight optical waveguide section, from its waveguide input to its waveguide output, depends sensitively on the proportion of light that is coupled into the circular optical waveguide section and, therefore, is not passed to the waveguide output of the straight optical waveguide section. The proportion of fully transmitted light 6 relative to the amount of input light 4, is defined as the transmittance, T, of the optical ring resonator, and is given by the formula:

Here, the parameter r is the self-coupling coefficient of light between the straight waveguide 3 and the circular waveguide 2, and the parameter a is a loss parameter related to the optical power attenuation coefficient α through the equation

( ) with L being the circumferential length of the circular waveguide. Here, φ is the single-pass phase shift, defined as:

Here, ^^ is the effective refractive index of the propagating mode in the circular waveguide, and λ is the wavelength of the light in free space. The transmittance is minimized when

( ) 10, which implies that the condition for minimum transmittance, as far as the phase shift is concerned, is:

Here, ^^ is an integer. It follows that the values of the optical wavelengths minimizing the transmittance (i.e., the resonant wavelengths) are given by the formula: Consequently, the values of the op

tical frequencies minimizing the transmittance (i.e., the resonant frequencies) are given by the formula:

Here, ^^ is the radius of the circular waveguide. Consequently, the transmittance takes the form:

When ^^ = ^^ , the transmittance is zero. This condition is known as “critical coupling”. Referring to Figure 2, in this arrangement the transmittance of light 6 through the lower one (item 3) of the two straight waveguide sections (3, 9) to the lower waveguide output port B, is given by:

Here, ^^_^ is the self-coupling coefficient of light between the lower straight waveguide 3 and the circular waveguide 2 at the coupling region 7, and ^^ is the self-coupling coefficient of light between the upper straight waveguide 9 and the circular waveguide 2 at the coupling region 8 where the upper straight waveguide is in close proximity to (or in contact with) the circular waveguide 2. The transmittance through the upper straight waveguide 9, to produce an optical output 10A at the upper waveguide output port A, is given by:

It can be shown that the condition for the minimum transmittance through the lower waveguide 3, and simultaneously the maximum transmittance through the upper waveguide 9, is described again by:

In this case one can see that a resonance condition means that the minimum transmittance through the lower waveguide 3 takes the form:

Similarly, the resonance condition means that the minimum transmittance through the upper waveguide 3 takes the form:

The equation defining

implies that if:

then

. In this case the transmittance through the upper straight waveguide 6 takes the form:

This means that when a = 1, then

and

Thus, in the case of a symmetrical optical ring resonator, such as shown in Figure 2, input light 4 is perfectly coupled to the upper waveguide 9, when the wavelength is given by

(i ) and the loss parameter, a, is equal to one (i.e., a = 1 when optical power attenuation coefficient α = 0), and the coupling coefficients,

are equal to each other (i.e.,

). This condition is known as “critical coupling”. The theoretical condition of α = 0 is impossible but α ≪ r is a good approximation. An important result of the above analysis is that, in both configurations, the resonant wavelength is given by

, which practically implies that it is proportional to the effective refractive index, n, of the propagating mode in the circular waveguide. In more detail, supposing that the refractive index, n, changes by Δ n, then:

Here, ^^_^^^^ ( Δω_res^) is the resonant wavelength (frequency) corresponding to the refractive index, ^^. It follows that: and

Consequently, the relative change in the resonant wavelength (frequency) is equal to the relative change in the effective refractive index. Accordingly, an important parameter regarding the ability to vary the resonance wavelength (frequency) of an optical ring resonator is the dependence of the refractive index experienced by a propagating mode of light with the resonator. This is a measure of the sensitivity of the system when the refractive index, n, is modulated (Δ n) deliberately as a means to modulate (Δ λ_res) the resonant wavelength of the system, or modulate (Δω_res ) the resonant frequency of the system. Figures 3A and 3B each illustrate an optical switch according to an embodiment of the invention employing the optical ring resonator illustrated in Figure 1. The optical switch is a single photon triggered optical signal switch comprising the ring resonator acting as an optical router configured for routing an optical signal 4 (either a CW signal or an optical pulse) output by a laser unit 140 into an input optical waveguide 14 to which the optical output port of the laser unit 140 is optically coupled, for onward propagation along the input optical waveguide 14 towards the ring resonator 22. The routing is put into effect by the action of the ring resonator, as described herein, conditional on the detection of a single photon (27A/27B). The optical router comprises an optical ring resonator 22 configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index, n, of the optical resonator. An input optical waveguide part 14 optically coupled to the optical ring resonator 22 by the physical proximity of the input optical waveguide part to the ring resonator 22 at a coupling region 7. The input optical waveguide part 14 is configured to receive an optical input signal 4 from the optical signal source (not shown) as an input to the optical router. The optical switch includes an output optical waveguide part 18 which is a continuation of the input optical waveguide part 14, and which is therefore also optically coupled to the optical resonator. The output optical waveguide part 18 is configured for receiving an optical signal 4 from the optical signal source, via the input optical waveguide part 14, for output from the optical router. The optical switch further comprises a single-photon detector unit 20 configured to output an electrical detection signal in response to detection of a single photon. An optical input port of the single-photon detector is coupled to the optical output end of a feed waveguide 21 for conveying photons to the single photon detector. An optical input end of the feed waveguide 21 is optically coupled to a source (not shown) of single photons, for receiving single photons therefrom and for guiding the single photons to the optical input port of the single-photon detector for detection. This source of single photons may be any suitable source readily apparent and available to the person skilled in the art. The single-photon detector unit 20 is electrically coupled to an electro-optical modulator 26 which is configured to receive the electrical detection signal and to output an electrical modulation signal in response to the electrical detection signal. An optional electrical contact part 24 electrically links the single-photon detector 20 and the electro-optical modulator 26. This electrical contact part 24 serves to provide signal processing or amplification of the electrical detection signal produced by single-photon detector unit 20 and may comprise signal filtering elements or capacitive elements. It may be used to generate a larger output voltage pulse, such as by passing the signal through a cascading array of nanowire detectors to amplify the output pulse from a single detector. Alternatively, or in addition, the electrical contact part 24 may comprise an impedance-matching taper, or/and may comprise a compact low power signal amplifier. The electro-optical modulator 26 is configured to modulate the refractive index, n, of the material forming the waveguide defining the optical ring resonator 22 by the electrical modulation signal, in response to detection of a single photon by the single-photon detector unit. The effect of the electrical modulation signal is to modulate (Δω_res ) the spectral position ω_res0 , of the centre of the resonance bandwidth of the optical ring resonator to change from being a resonance bandwidth that includes the optical signal frequency (Figure 3A) to being a resonance bandwidth that excludes the optical signal frequency (Figure 3B). The effect of the modulation in spectral position is to permit an optical input signal 4 to be transmitted from the input optical waveguide part 14 to the output optical waveguide part 18 (Figure 3B) as a non- resonant optical output signal for output from the optical switch without resonating within the optical resonator 22. As shown in Figure 3A, in the absence 27A of a photon in the feed waveguide 21 to the single-photon detector 20, no detection of a photon takes place and, consequently no electrical detection signal is generated. This means that the electro-optical modulator is not driven to modulate, Δ n, the refractive index of the material of the waveguide forming the ring resonator (i.e., the modulator is “OFF”). Consequently, no modulation,Δω_res , of the resonance frequency (or resonance wavelength modulation,Δ λ_res) takes place. This means that, because the resonance bandwidth of the ring resonator is configured to include the frequency (wavelength) of the input signal 4 upon the input waveguide part 14, and because the ring resonator is critically coupled to the input waveguide part 14, that input signal 4 is resonantly coupled into the ring resonator and is not routed to the output waveguide part 18. Critical coupling of a ring with a single waveguide input means either high loss or low coupling rate. For a pulsed input, the input signal will be routed into the output waveguide, but with some additional phase and time delay By contrast, as shown in Figure 3B, upon detection of one or more photons 27B via the feed waveguide 21, the single-photon detector 20 produces a photon detection electrical signal which drives the electro- optical modulator 26 (i.e., switches the modulator “ON”) to change the resonant frequency (wavelength) of the ring resonator 22, and the bandwidth of the resonance profile, to a resonant frequency (wavelength) position at which the bandwidth of the resonance profile excludes the frequency (wavelength) of the input signal 4 upon the input waveguide part 14. The result is that the input signal 4 is not coupled into the ring resonator and is, instead, routed to the output waveguide part 18. In this way, detection of a single photon by the single photon detector 20 changes the function of the optical router 12 from being a router that prevents the routing of an input optical signal 4 to the output waveguide part, to being a router that permits the routing of an input optical signal 4 to the output waveguide part as an output signal 10. In this way, single-photon detection affects the how impinging optical signal is redistributed amongst the ring resonator 22 and the output waveguide part 18. Figures 4A and 4B each illustrate an optical switch according to another embodiment of the invention employing the optical ring resonator illustrated in Figure 2. The optical switch comprises the structure of the optical switch described above with reference to Figures 3A and 3B, but further comprises a further output optical waveguide in the form of the upper output optical waveguide 12 (corresponding to the upper straight waveguide 9 of Figure 2) which is separately optically coupled to the optical resonator 22 at a separate respective optical coupling region 8 of the optical resonator. As shown in Figure 4A, in the absence 27A of a photon in the feed waveguide 21 to the single-photon detector 20, no detection of a photon takes place and, consequently no electrical detection signal is generated. This means that the electro-optical modulator is not driven to modulate, Δ n, the refractive index of the material of the waveguide forming the ring resonator (i.e., the modulator is “OFF”). Consequently, no modulation,Δω_res , of the resonance frequency (or resonance wavelength modulation,Δ λ_res) takes place. This means that, because the resonance bandwidth of the ring resonator is configured to include the frequency (wavelength) of the input signal 4 upon the input waveguide part 14, and because the ring resonator is critically coupled to the input waveguide part 14, that input signal 4 is resonantly coupled into the ring resonator and is routed to the waveguide output A of the upper waveguide part 12. It is not routed to the waveguide output B of the lower waveguide part 18, as an optical output signal 10A. By contrast, as shown in Figure 4B, upon detection of one or more photons 27B via the feed waveguide 21, the single-photon detector 20 produces a photon detection electrical signal which drives the electro- optical modulator 26 (i.e., switches the modulator “ON”) to change the resonant frequency (wavelength) of the ring resonator 22, and the bandwidth of the resonance profile, to a bandwidth that excludes the frequency (wavelength) of the input signal 4 upon the input waveguide part 14. The result is that the input signal 4 is not coupled into the ring resonator 22 and is, instead, routed to the waveguide output B of the lower waveguide part 18, as an optical output signal 10B. In this way, detection of a single photon by the single photon detector 20 changes the function of the optical router 12 from being a router that routs an input optical signal 4 to the upper waveguide 12 for output form its waveguide output A, to being a router that routs the input optical signal 4 to the lower waveguide part 18 for output form its waveguide output B as an output signal 10B. In this way, single- photon detection affects the how impinging optical signal is redistributed amongst the upper waveguide part 12 and the lower waveguide part 18. Quality Factor ( Q) Optical resonators temporarily confine light in all three dimensions, yielding dramatic enhancement of the electric field strength in the resonator with respect to freely propagating beams in continuous media. The decay lifetime, τ, of a mode of light within an optical resonator may be thought of in terms of a rate of change of the energy, ℇ( t), of the light mode stored within the resonator as follows:

This means that the stored energy ℇ( t) changes over time as follows: ( ) ( ) ⁄ ( ) ⁄

This means that the quality factor, ^^, is related to the ratio of the stored energy and its rate of change:

Thus, if the loss per round-trip ‘pass’ of the light mode (e.g., pass of a photon) within the resonator is denoted as ℒ and the path length of each ‘pass’ is ℓ, then the fractional loss per unit of time is given bycℒ/ nℓ, where ^^ is the refractive index of the resonator material and ^^ is the speed of light in a vacuum, then we may write that:

For example, the steady state mean photon number N in a low loss ring optical resonator driven coherently on resonance is given by

where ^^_^^ is the mean number of incoming photons per optical period. This expression is for a CW input and does not include nonlinear effects or backscattering. The quality factor Q may be regarded as the figure of merit quantifying the degree of field enhancement. The Q in a ring optical resonator can range to well over 1,000,000 and therefore the increase in N is not merely a perturbation to the optical dynamics involved. The quality factor associated with an optical resonator is also related to ratio of the resonance frequency,Δω_res, at which the resonator resonates, and the resonance line-width (e.g., FWHM),

of the spectral resonance profile of the optical resonator as:

Thus, a spectral resonance profile having a narrow full width at half maximum (FWHM) will be a property of an optical resonator having a large quality factor, Q. This means that only small shifts in the spectral position of the resonance frequency, Δω_res, of such a resonator will have the effect of rendering wholly non-resonant (i.e., significantly off-resonance) any light that was initially resonant light with the resonator. Thus, more generally, light of frequency ^^ may be considered to reside within the “…resonance bandwidth…” of a resonator possessing a quality factor, ^^, within the meaning of the term used herein, if the following condition is met:

Thus, given that this condition translates to:

This means that the “…resonance bandwidth…” generally includes not only the exact resonance frequency, ω_res, but also includes closely neighbouring frequencies that differ from the exact resonance frequency by not more than one half of the value of the resonance line-width (e.g., FWHM),

Γ^ത of the spectral resonance profile of the optical resonator. One can see that as the quality factor, Q, of the resonator increases in value, the “…resonance bandwidth…” becomes increasingly narrowly centred upon the exact resonance frequency position. It is to be noted that the definition of the quality factor, Q , is not limited to a specific structure and configuration of an optical resonator but is concerned with energy storage and the change/loss of that stored energy. A Ring Resonator: The Quantum Description The above description includes a description of the conditions necessary for critical coupling according to a classical (i.e., non-quantum) explanation of the optical processes taking place. This is adequate as an explanation when the input optical signal 4 to be routed is of sufficient light intensity to be adequately described by classical physics whereby light is described accurately as an electromagnetic wave, which may be either a continuous optical signal or a sufficiently slowly changing (i.e., long duration) optical pulse. However, a more accurate description is given by quantum mechanics whereby the input optical signal 4 is described as a quantum state. This may be more appropriate if the input optical signal 4 comprises either only a single photon, or a few photons, or a rapidly changing optical pulse, when the optical processes taking place are inadequately described by classical physics and are accurately described by quantum mechanics. It is to be noted that although the following discussion will be framed in terms of a ring optical resonator formed using waveguides, it is to be understood that the following analysis and conclusions are not limited to that specific structure and configuration of an optical resonator. In fact, the following analysis and conclusions are readily applicable to other structures and configurations of an optical resonator, as is noted above. The benefits obtained by using optical resonators with a large quality factor may have to be balanced against the consequences of manufacturing imperfections which lead to light being lost by scattering from the resonator. Although manufacturing imperfections are also present devices with a smaller quality factor, their consequences are magnified in resonators having a large quality factor, Q, due to the longer dwelling time of photons in those resonators. In other words, the enhanced confinement of the light also enhances the interactions with defects that lead to scattering. These consequences are especially critical for quantum technology applications, which may be required to operate using few photons. As an example, a photon within an optical resonator may either escape the resonator via a dedicated output channel, with resonator-to-channel coupling rate Γ, where it can be used for a dedicated purpose, or may be uselessly lost/escape due to scattering from the resonator at a loss rate M . The probability, P, that the photon escapes to the dedicated output channel may be written as:

Only if Γ exceeds M by a sufficient extent will ^^ be large enough for the optical resonator to be useful for practical purposes. Close attention is therefore paid to the role of scattering losses in the implementations of the present invention. Figure 5 schematically shows a diagram of the ring resonator model used to describe quantum mechanically the processes occurring in the optical switch described above with reference to Figure 3A and Figure 3B. Figure 6 schematically shows a diagram of the ring resonator model used to describe quantum mechanically the processes occurring in the optical switch described above with reference to Figure 4A and Figure 4B. Each ring resonator model consists of a micro-ring resonator 22 with a resonant frequency of and loss

that optically couples to one continuous optical waveguide (14, 18 as per Figure 3A and Figure 3B), or optically couples to two separate waveguides (e.g., 14, 18, 12 as per Figure 4A and Figure 4B), with the input light 4 in the input (lower) waveguide 14, travelling from left to right in the figures. This input light 4 couples into the anticlockwise mode 22A of the ring resonator 22. In the model of Figure 6, corresponding to the embodiment per Figure 4A and Figure 4B, the input light 4 leaves from the upper waveguide 12 travelling right-to-left or leaves the lower waveguide 18 travelling left-to-right. Backscattering of light causes coupling to additional modes of the ring/waveguide system, so that light also couples to the clockwise mode 22C of the ring as well as right-to-left in the lower waveguide 14 and left-to-right in the upper waveguide 12. A voltage output from a single-photon detector, such as a superconducting nanowire single-photon detector (SNSPD) causes a time-dependent shift,

of the resonant frequency, of the ring whereby

( ) h ( ) is a time-varying function and ω₁ is the amplitude, or maximum, of the frequency shift, such that

Heisenberg-picture calculations are constructed below, to describe the optical output of the system in response to an initial input state. The analysis below is in terms of linear dynamics which characterise the relevant travelling optical modes involved in the pertinent phenomena of the system. These modes include narrowband resonant modes of the optical resonator and a continuum of modes of channel fields into which light is injected into the resonator and from which light is extracted from the resonator. In addition, modes that arise from scattering losses are represented. Suitable couplings between these modes are introduced so as to provide appropriate spectral, spatial, and temporal features of the dynamics of the light within the system. The methods, analysis and conclusions set out herein can be used to describe any structure supporting a set of optical modes that are weakly coupled to an input/output channel over a spatial region that is smaller than the wavelength of light. This is applicable to a wide class of resonator structures. Effective Hamiltonian The following methods use a known general formulation of the Hamiltonian that has been used in the context of Maxwell’s equations when used in the context of a periodic medium, such as a notional 3D photonic crystal, to generate equations of motion. Sipe et al. [ref. [1]: Sipe et al. “Effective field theory for the nonlinear optical properties of photonic crystals”: PHYSICAL REVIEW E 69, 016604 (2004)] have shown that the displacement field, corresponding to an electromagnetic field can be written as:

Here, the terms aα ( t) are mode amplitudes complying with the following commutation relations:

The mode index α then consists of a crystal wave vector ^^ in a Brillouin zone and a crystal band mode m . Sipe et al. show that the stationary form of the displacement field can be written in terms of the mode amplitudes

shown below, where:

This formulation starts with the general expression for the displacement field, ,as:

Here, “c.c.” refers to the complex conjugate of the preceding expression in the equation, as will be readily understood by the person skilled in the art. The terms ( ) are amplitudes associated with a notional

3D photonic crystal with a periodically varying refractive index with lattice vector R such that:

This means that:

For field mode propagation far from a Brillouin band gap, or within a band gap but close to the band edge, then it has been found that one of the amplitudes, will dominate over others. This dominant

amplitude is known as the “principal component” and a Schroedinger equation for the “principal component” may be used. In this regard, an “effective field”,

, may be formulated for modes as:

Here the “reference” crystal wavenumber

^̅ is used, and the contributions to the above integral from the coefficients ( ) are significant over only the small range of the parameter The term

is a

selected “reference” carrier wave vector with which the set of amplitudes

( ) is associated whereby the underlying spatial variation in the optical properties (e.g., the refractive index) of the medium through which the modes propagate, is contained within the term

above. The

of interest are slowly varying in space over distances on the order of the lattice spacing. These “effective fields” satisfy:

It can be shown that the displacement field

can be written in terms of this effective field as follows:

Here, the terms

( ) ( ) have been expanded by a Taylor expansion about the point and a partial integral performed, and noting that:

This can be shown to result in a Hamiltonian for describing the linear dynamics of the system of the form:

and,

This discussion has been in terms of a medium that has a periodicity in all three dimensions of space. Of course, the discussion is equally applicable to media having a periodicity in two dimensions or in only one dimension of space. Examples of media that are periodic in only two dimensions of space include a looped optical resonator structure (e.g., a ring optical resonator, racetrack optical resonator, disk optical resonator or the like). Examples of media that are periodic in only one dimension of space include a dielectric stack structure, a distributed Bragg reflector, a fibre Bragg grating structure. More generally, an optical resonator may be considered to be periodic in the sense that modes of light travelling within it repeatedly pass through, along or within the structure of the resonator in a periodic manner (e.g., reflections back-and-forth in a linear resonator, or cyclical loos in a looped resonator structure). Such structures as these are considered herein. Channel Modes An optical channel outside the resonator is assumed to accommodate a continuum of modes confined in the transverse yz plane, and freely propagating in the longitudinal ^^ direction. The quantized electric displacement field operator, D( r), can then be expressed as follows:

Here, and throughout this disclosure, the term “ H . c” refers to the Hermitian conjugate of the terms preceding it, as will be readily understood by the person skilled in the art. The term ω_lk is the angular frequency of an optical mode with polarization I and wavevector k, the sum over “ I” in the above expression is taken over all polarizations, and the term d_lk( y, z) is transverse mode profile. However, henceforward we consider only one polarization such that the sum over “ I” in the above expression is no longer necessary. In addition, the following analysis concerns only forward-propagating modes, so that the integral extends only from k = 0 to k = +∞. Photon field operators ψ_j ( x), will be used in the following analysis in place of the displacement field operator, D ( r), and these are defined as:

Here, the index J indexes the different frequency ranges of interest corresponding to reference wavevectors k_j, and ( ) The k_j

correspond to resonances of the optical resonator to which the channel couples. Because these resonances are narrow and well-separated in frequency, the integral in the above expression may merely span a range of k which wholly captures the spectral features near resonance at k_j, but does not overlap with any other resonance. The operators are annihilation operators which annihilate photons in these modes, and satisfy the following comm

utation relations:

Similarly, the photon field operators,

ψ_j ( x), satisfy the following commutation relations:

The photon field operators, , are used to formulate the Hamiltonian H_ch

which describes light propagating in an isolated channel, as follows:

The Hamiltonian neglects zero-point energy and includes only the optical modes of interest. Those modes of interest have the same one polarisation and have frequencies in ranges defined by a set of reference wavevectors ^^_^. Consequently, we may write:

Here, the photon field operators , are defined above. Thus, applying this to the Hamiltonian gives:

Here, the term represents the frequency associated with

, and the group velocity term is defined as:

Resonator Modes Initially ignoring any optical coupling to the channel or scattering modes, the resonator can be assumed to support a set of discrete resonant modes

with frequencies The electric displacement field operator inside the resonator may be expressed as:

The resonant mode profiles

for a ring optical resonator of radius and circumference

, may be written in the form:

Here, the quantity

and is the angular coordinate around the ring circumference. The term

represents the cross-sectional field profile on the coordinates

which are transverse to the tangent line at position The wavevectors must satisfy the resonance condition:

Here, is an integer. The resonant modes defined by this condition comprise evenly spaced values of each one of which corresponds to a respective one of the different mode orders . It is assumed that

the waveguide from which the ring is constructed has the same dimensions and material properties as the waveguide defining the channel such that group velocities

in the ring are the same as those defined above for the channel. However, it is to be understood that this assumption is not essential and different dimensions and material properties may be assumed and accounted as would be readily apparent to the skilled person. For each resonant mode

of the ring resonator, a quantized harmonic oscillator is used which is described by the discrete annihilation operator noted above in the expression for . The,

Hamiltonian for these modes, ignoring the contribution from zero-point energy, can be written as:

The annihilation operators satisfy the following commutation relations:

Resonator-Channel Coupling

The following describes the transfer of energy between the resonator and the channel structures. A coupling term is derived for the Hamiltonian which defines a coupling between the ring operators to the channel fields. The coupling is defined as occurring at the single point located at in the channel.

This is an accurate approximation given that the spatial extent of the coupling region is small in practical applications. The coupling Hamiltonian has the form:

Here, the ^^_^ are channel-ring coupling constants. The model makes it possible that the coupling strength between different resonances may vary, as occurs in real systems. Scattering Losses As noted above, the effects of imperfections in an optical resonator of high value are magnified relative

to the same effects that may be present in an optical channel outside of the resonator. For example, sidewall roughness in the resonator, or other manufacturing defects, will lead to a loss of photons from the resonator due to scattering. In general, scattering losses from the resonator for a given resonance, maybe described by including a term

in the Hamiltonian. This term describes a continuum of scattering modes with annihilation operators , and also describes the couplings of these scattering modes to the optical modes of the resonator:

Here, represent the scattering mode frequencies, and

represent their couplings to the modes of the

resonator. In addition, field operators

are introduced and are defined as:

Assuming that the couplings

vary slowly over the extent of the resonator resonance in question, such that ( ) one may approximate as follows:

Here is assumed to be a constant. The result is that the coupling term in in the Hamiltonian then becomes:

By expanding the term about the reference wavevector , giving one may write:

Here, the group velocity is u_j is given by:

The result is that the coupling term in H_sc in the Hamiltonian can then be written as:

It will be noted that this Hamiltonian has the same form as a Hamiltonian describing a notional additional channel possessing field operators

and group velocities

u_j coupled to the resonator with coupling constants Scattering losses are taken account of in the following model as is explained in more detail below. Note that in the above disclosures, the index

has been explicitly applied to index the different frequency ranges of interest corresponding to reference wavevectors

The quantities

correspond to resonances of the optical resonator to which the channel couples. In the following discussion, for simplicity, we confine the discussion to one frequency range of interest corresponding to one reference wavevector , and assume one resonance of the optical resonator to which the channel couples. Initially, we ignore photon backscattering effects, as this corresponds to a linear model with a single (‘pump’) input (which could be, for example, a single frequency continuous-wave (CW) input or a light pulse of Gaussian pulse profile). We start with a single mode operator

(in the ring),

When considering a ring resonator coupled to a single waveguide, then the waveguide number n, is only 1 (one), and when considering a ring resonator coupled to two separate waveguides, then the waveguide number n, counts over two values (n = 1, 2) with n = 1 denoting a lower (input, conditional output) waveguide n = 2 denoting an upper (conditional output) waveguide. The field,

represents a notional mode travelling in a ‘phantom waveguide’ and is used to model optical losses from the system. The Hamiltonian taking account of linear effects (i.e., non-linear effects not included) is then given by:

which has the pump frequency propagation speed In the following discussion, merely for ease of

presentation, the speed is

assumed to be the same in both waveguides,

but a generalisation to different speeds,

is clearly possible and applicable in the above Hamiltonian. The propagation speed in the ‘phantom waveguide’ is given by

and the ring- waveguide coupling is given by whereas the ring-‘phantom waveguide’ coupling is given by

(directly giving optical loss). All couplings, including that to the ‘phantom waveguide’, are taken to be point couplings. It is to be understood that the above Hamiltonian, the analysis which follows, and the conclusions herein are applicable more generally to a wide range of optical resonator structures and is not limited to (or intended to be limited to) the ring optical resonator structure disclosed in the drawings. The exemplification of an application of the invention in terms of a ring optical resonator structure has been selected to be useful to aid the reader in gaining a better understanding of the invention. Probability Density and Probability Current Density It is instructive at this point to note that Hamiltonian terms

each describes a channel (either physical or notional) possessing a field operator,

respectively, and a group velocity and

respectively, coupled to the resonator with associated coupling constants,

and respectively. Each of these Hamiltonian terms includes a component of the form of a probability density:

which is multiplied by a photon energy

. In addition, each of these Hamiltonian terms includes a component of the form of a probability current density:

These probability current density terms are equivalent to the product of the relevant group velocity of the mode and the expectation value of its momentum, noting that the expectation value of the momentum of a field is given in terms of the momentum operator and the probability current density,

, as follows:

Here, the probability current density is defined by the Wronskian in the well-known expression:

Thus, the Hamiltonian terms ^^ and ^^ each describes an energy contribution in the following

terms: Energy = (Photon Energy)x(Probability Density) + (Velocity)x(Momentum) The Model in Detail The following discussions and analysis are presented in terms of a ring optical resonator structure optically coupled to two separate optical waveguides (n = 1 and n = 2). Of course, it will be readily understood that where coupling to only a single optical waveguide is to be considered, then this is simply modelled by taking

As an illustrative example, and referring to Figure 5 and Figure 6, consider the following conditions:

As a simple way of accounting for any dispersion effects within waveguide, and ring, material etc., in the first instance one may take the waveguide operators immediately before and immediately after the point of coupling to the ring (i.e., at x = 0) by splitting them into ‘input’ (for x < 0) and ‘output’ (for x > 0) operators, as follows for the first (lower) waveguide and the ‘phantom waveguide’:

and ‘output’ (for x < 0) and ‘input’ (for x > 0) operators for the second (upper) waveguide

and looking at

( ) ( ) ( ) ( ) ( ) ( ) d ( ) Any effects due to the waveguide can then be considered by propagating these operators from

x = 0 to whichever onward point coupling in/out of the system (e.g., a downstream optical port, or the next optical element etc, of the system comprising the present switch/router). This gives the relation:

and using the rotating basis

the Heisenberg equations of motion are:

The following expectation values apply:

A single point coupling model of loss is equivalent to having some amount of loss at each point of the ring optical resonator. Two main approaches are presented for analysing this: the first is to consider a continuous wave optical input, CW, solved in the steady state; the second is to consider a pulsed optical input, e.g., solved in frequency space using Fourier Transforms. Continuous Wave Regime In the CW regime, the input can be given by a single frequency input with some detuning from the ring ^^ so that:

where is a constant. This gives a further rotated basis:

As with the phantom channel, having a single input gives exactly the same results as having multiple inputs by defining:

when those inputs have the same frequency. In the CW regime, this therefore gives the (ring) solution as:

which gives the output as:

That is:

These expressions show the extra ‘-’ sign in the phase of the output of the ring that occurs as a direct result of coupling to and from the ring. Setting

and

gives the graphs shown in Figure 7. Referring to Figure 7, there is shown (left-hand graphs) the CW optical power,

^{| |} output (10, 10B) from the output part 18 of the lower waveguide (14, 18), and (right-hand graphs) the CW optical power,

output (10A) from the output part of the upper waveguide (12). Here, the optical power output is scaled to the input power, with relative frequency in units o

f assuming no intrinsic loss (top graphs) and input into the lower waveguide 14. It can be seen that the swapping of coupling rates between waveguides has no effect on the optical output power of that waveguide. Adding optical losses can be considered in two different ways: firstly, by keeping the coupling rates between the waveguides and ring the same as when there is no loss, and changing the linewidth

and the relative rates

and

/ (bottom graphs). Note that in Figure 7, the rate

is denoted as

Γ indicating the lower waveguide. The lines marked ‘critical’ indicate parameters corresponding to critical coupling (note that the CW optical power, , falls to zero on resonance) and no single-photon detection event. Here the ratio of the

spectral width of the ring resonator to the optical input pulse is 10. In particular, at critical coupling for zero detuning

( ) and zero additional loss

is the point at which complete destructive interference occurs and no optical output (10, 10B) emanates from the output end 18 input waveguide 14. For

the optical output (10, 10B) has the same phase and for , the optical output (10, 10B) has the opposite phase as the optical input 4.

Pulsed Regime The pulsed solution can be found by Fourier Transforming the operators to get:

Similar to the CW case, this gives the output as:

which is (unsurprisingly, due to linearity) the same as the CW solution, except with and

frequency dependence. Assuming a gaussian optical input such that:

and further setting

0 gives the solutions

with the input pulse shown in Figure 8A and the optical output powers shown in Figure 8B and 8C. Turning to Figure 8A, there is shown a spectral profile of a normalised pulsed power optical input 4 to the input (lower) waveguide 14. Here the optical frequency,

is expressed relative to the resonance frequency, of the ring resonator in units of the spectral width (standard deviation,

Hz) of the optical input pulse (i.e.,

( ). The inventors have found that there are 3 different regimes arising, depending on whether the spectral width of the optical pulse is: (1) considerably less than, (2) roughly equal to, or (3) considerably greater than the linewidth of the resonance spectral profile of the ring resonator. The spectral width of the wavepacket used here is different to the spectral width of the power input/output by a factor of √2 due to squaring of the wavepacket function to obtain the optical power. When the spectral width of the optical pulse is much less than the linewidth of the resonance spectral profile of the ring resonator, the ring resonator effectively ‘sees’ what appears to it to be effectively a CW input, with the result that the optical output is approximately Gaussian. When the spectral width of the optical pulse is much greater than the linewidth, the ring ‘carves out’ a section of the optical pulse corresponding to the linewidth and resonant frequency of the ring. Figures 8B and 8C show plots of the calculated output intensities as a function of normalised relative frequency (i.e.,

( ) in the lower output waveguide 18 (Figure 8B) and the upper output waveguide 12 (Figure 8C) for a broadband input pulse. The lines marked with the work “critical” correspond to parameters resulting in critical coupling and no photon detection event. The coupling rate of the lower (input) optical waveguide to the resonator ring is indicated in the graphs as This rate is

indicated for each curve on each graph in units of the linewidth

of the resonance spectral profile of the ring resonator, such as

Here, the coupling rate of

corresponds to the “critical coupling” rate. Here the ratio of the width of the optical pulse optical input pulse to the linewidth of the resonance spectral profile of the ring resonator is 10 (i.e.,

). Figure 9A, shows the temporal profile of the normalised pulsed power optical input 4 to the input (lower) waveguide 14. The units of time are expressed in units of the decay time,

of the ring resonator. Figures 9B and 9C show plots of the calculated output intensities as a function of normalised relative time (i.e., ^^/ ^^) in the lower output waveguide 18 (Figure 9B) and the upper output waveguide 12 (Figure 9C) for a broadband input pulse. The lines marked with the work “critical” correspond to parameters resulting in critical coupling and no photon detection event. The coupling rate of the lower (input) optical waveguide to the resonator ring is indicated in the graphs as

This rate is indicated for each curve on each graph in units of the linewidth of the resonance spectral profile of the ring resonator, such as

. Here, the coupling rate of corresponds to the “critical coupling” rate.

In figures 8A, 8B, 9A and 9B, one can see how “critical coupling” leads to a very effective transfer of the power 10A of the input optical pulse (Figs 8A and 9A) to the upper output waveguide 12 (cf. Fig.6) giving very little output power 10B at the lower optical waveguide 18 output. Including Back-Scattering Back-scattering can then be considered using a counter-propagating mode, modelled as an additional ring resonator. The amount (i.e., rate) of back-scattering then gives the amount (i.e., rate) of coupling between these modes. Incoherent scattering can be modelled using ‘phantom channels’ as noted above. This introduces three couplings in total: • back-scattering within the ring = ring-ring coupling • back-scattering within the waveguide = waveguide-waveguide coupling • back-coupling of waveguide-ring (

) These are modelled using the parameters (ring-ring coupling), g (waveguide-waveguide coupling) and

(proportion of ring-waveguide coupling), giving ring-waveguide coupling as

and

. Back-scattering within the ring is introduced by including an additional ring mode. Back-scattering and back-coupling of the waveguides can be introduced by doubling the number of waveguides, with and

. In total, this gives four waveguide modes, two phantom waveguide modes and two ring modes that, in the absence of backscattering, can be considered as

where the group velocities are propagates anti-clockwise within the ring. Back-scattering is then

modelled with the additional coupling Hamiltonian as follows:

Here, back-scattering within the ‘phantom waveguide’ is ignored, as it is being used to model incoherent loss. This gives the evolution of the ring operators as:

The equation of motion of the channel operator for the lower (input) waveguide 14 are given by

To solve this, the general solution is denoted by:

The particular solution can then be found by first solving:

which has the solution:

This can be solved using the convolution:

Note that when solving a wave equation, for channels with waves travelling in the opposite direction, the velocity is negative, but

is now only non-zero for

. and introduces an additional

‘ ’ sign that gives all the

equivalent equations despite different directions of travel etc. This solution is considerably simplified for either

0 (i.e., no back-scattering from waveguide-waveguide) or

(i.e., back-scattering only at interaction with the ring). Alternatively, the full solution could be found perturbatively. Note that

0 reduces to that of Vernon and Sipe [ref. [2]: Z. Vernon and J. E. Sipe, “Spontaneous four-wave mixing in lossy microring resonators": Physical Review A, vol.91, no.5, p.053802, 2015. (arXiv:1502.05900v2 [quant-ph] 5 May 2015); ref. [3]: Z. Vernon and J. Sipe, “Strongly driven nonlinear quantum optics in microring resonators" Physical Review A, vol.92, no.3, p.033840, 2015. (arXiv:1508.03741v1 [quant-ph] 15 Aug 2015)], while

is that of McCutcheon [ref. [4]: W. McCutcheon, “Gaussian nonlinear optics in coupled cavity systems: Back-scattering in micro-ring resonators," arXiv preprint arXiv:2010.09038, 2020]. As the latter reduces to the former at ^^ = 0, we look at the latter case here, which is:

At this point, it is convenient to write these equations in matrix form, as follows:

The equations of motion of the ring operators can be written in matrix form as:

The ‘phantom waveguide’ is analogous to the real waveguide(s), with the following conditions:

In the rotated basis, and in terms of the optical inputs, this gives:

In this equation, the term

given by:

Similarly, the matrices

are given by:

This allows us to write the following relation:

Here contains only off-diagonal elements defined as:

Similarly, we have:

For a continuous wave (CW) optical input in the further rotated basis and semiclassical approximation, this gives the following relation:

Changing

as before then gives the Fourier Transformed version for a pulsed optical input. Setting

gives the solutions discussed above (e.g., see Figure 8B and 8C, for example). Time Dependence (no back-scattering) Going back to the equations of motion, for a time-independent Hamiltonian, we have that:

The Heisenberg equation of motion for an operator A (with no explicit time dependence) is

However, this is no longer necessarily true when the Hamiltonian, H , is time-dependent. In this specific case, we are making the resonant frequency of the ring optical resonator time-dependent, so there is still no explicit time-dependence in the creation/annihilation operators. Applying the time-dependent impulse ( ) to the resonant frequency of the ring, as discussed above, the relevant part of the Hamiltonian that

becomes time dependent is as follows:

The commutation relation is then defined by:

However, for

if the coupling terms in the Hamiltonian are changed to the following form:

we find that the Hamiltonian will commute. Alternatively, this can also be considered through a Heisenberg perturbation theory approach. For example, we may expand the unitary evolution as a Dyson series, as follows:

We may also write the Hamiltonian as follows:

We wish to transform this Hamiltonian into the effective Hamiltonian, as follows:

To do so we may use the unitary, expressed as e.g., a Dyson or Magnus series:

To do this, we take the following definition:

We then match terms of equal order in λ:

However, instead of taking the perturbative term to be:

we take it to be (using double perturbation theory) as follows:

This is valid because

This means that H₀ commutes with H_λ1 and the full (non-perturbative) solution to H₀ + H_Λ1 is known. As such, we aim for a solution of the form:

Note that when λ = 0 or when Λ = 0, we have that H' = H . The terms in the Magnus expansion are then as follows:

The integrand in the last term is then given by the expression:

There are further higher-order terms, although this first correction term gives the effective Hamiltonian as follows:

This effectively corresponds to the following changes:

Here higher order terms can be ignored because γ_n ≪ ω₀, and ω₁ ≪ ω₀ (by comparison with the H^'2 term in the matrix exponential expansion of H'). The expression:

is solved by multiplying both sides by an integrating factor of the following form:

This gives the following result:

further giving the solution:

In the case where the input is only to a single waveguide ^^, this gives (NB. this is now similar to Lagrangians and propagators):

One particular case of this is when ^^ is time-dependent, such as when the resonant frequency of the ring optical resonator is modulated/driven by an electrical signal (e.g., voltage) output from e.g., a superconducting nano-wire single-photon detector (SNSPD) upon the detection of a single photon. When given in terms of a voltage, this electrical signal may be expressed as:

Here, τ_d is the decay time, τ_r is the rise time, Z₀ is the circuit impedance, G is the amplifier gain in dB, and I_b is the bias current. This expression may be accurately modelled, for the purposes of calculations, by an analytical approximation as follows:

Figure 11A shows the temporal shape of the electrical signal output by the conducting nano-wire single- photon detector (SNSPD) upon the detection of a single photon. Figure 11B shows the temporal shape of the analytical approximation noted above. In this case, the equation of motion becomes:

Here, the resonance frequency of the ring optical resonator is given by:

This is expressed in terms of a response function f( t), which may be taken to be

. This gives an integrating factor as follows:

This gives:

However, integrating the experimentally corrected voltage output (e.g., by substituting u = e^-t) gives a hypergeometric function which may be unnecessarily computationally burdensome when integrated, and therefore, for simplicity, we use the following analytical formula:

This allows the integration to be split into two parts, with:

Then, for t > t₀, we may write:

For t < t₀, we may write:

As the wavefunction input to the waveguide is a Gaussian, this process could also be considered by instead taking the Fourier Transform. That is, by using:

This gives:

This may result in a convolution of the response function and the wavefunction of the ring mode, which would require e.g., a perturbative solution. However, it is possible that the input wavefunction has a different frequency to the resonant frequency of the optical ring resonator. Accordingly, we provide the following analysis which is not in the rotated basis and we denote the (unperturbed) resonant frequency of the ring by ω₀ and the frequency of the input wavefunction (assuming no dispersion) by ω . Now, the input is given by either:

Here, the CW input can effectively considered to have been input at t = −∞, when the time difference between the input and measurement being looked at is

. The output is then given by:

This gives, for a CW input before any detection event from the single-photon detector (SPD):

After the detection of a single photon by the SPD, setting t₀ = 0 wlog ,we have:

The output power of this is then given by:

An example is shown in Figures 13A, 13B and Figures 14A, 14B and 14C. For a pulsed optical input, such as is shown in Figure 10A, the outputs in time before any photon detection are given by the following expressions:

Examples of these optical outputs are shown in figures 10B, 10C, 10D and 10E. Figure 10A shows the temporal profile of the normalised pulsed power optical input 4 to the input (lower) waveguide 14. The units of time are expressed in units of the decay time,

of the ring resonator. Figures 10B and 10C show plots of the calculated output intensities as a function of normalised relative time (i.e., ^^/ ^^) in the lower output waveguide 18 (Figure 10B) and the upper output waveguide 12 (Figure 10C) for the input pulse of Figure 10A. Figures 10D and 10E show plots of the calculated output intensities as a function of normalised relative time (i.e., t/ τ) in the lower output waveguide 18 (Figure 10D) and the upper output waveguide 12 (Figure 10E) for the input pulse of Figure 10A. The lines marked with the word “critical” correspond to parameters resulting in critical coupling. The coupling rate of the lower (input) optical waveguide to the resonator ring is indicated in the graphs as Γ_↓. This rate is indicated for each curve on each graph in units of the linewidth

of the resonance spectral profile of the ring resonator, such as

Here, the coupling rate of

corresponds to the “critical coupling” rate. The graphs of Figures 10D and 10E show the pulse ‘width’ (standard deviation, σ) of σ = τ⁄ 10. When the pulse ‘width’ is much smaller (i.e., the pulse changes much faster) than the temporal response time

of the ring resonator, the pulse largely has passed the ring before it has had a chance to interact with it, as the required interaction time, τ, is much longer than σ. The small amount of pulse that does couple into the ring then stays in the ring for a much longer time than the pulse ‘width’ of the pulse, slowly leaking out with the decay time of the ring. The coupling rate into the ring has an effect as well (best shown in Figure 10B), as the larger the coupling rate,

, the faster the interaction. However, having a larger coupling Γ_↓ between the lower waveguide and ring while keeping the overall loss rate or decay time ( τ) the same requires that the coupling rate to the upper waveguide to decrease so that m

ore light remains in the ring to couple back out to the lower waveguide after a time

In figures 10D and 10E one can see how a photon detection event under “critical coupling” conditions leads to a very effective transfer of the power 10B of the input optical pulse to the lower output waveguide 18 (cf. Fig.6) giving very little output power 10A at the upper optical waveguide 12 output. For a pulsed optical input, such as is shown in Figure 12A, the outputs after a single photon is detected are then shown in figures 12B, 12C, and 12D. These are discussed in more detail below. Alternatively, we may approximate the SPD voltage output modulation f( t) as the top hat function, as follows:

Where is related to τ_d by a numerical approximation. This approximation involves first recognising that the frequency shift of the ring has to be at least approximately twice the linewidth

for any light originally coupled to appreciably uncouple from the ring. As such, the second part of this approximation is that the time for which the above is true defines the effective time

′ That is ^^′ _^^is defined as the time interval over which

This is when the detuning has the most noticeable effect. This may be useful to approximate the output of the system, is (numerical) integration using more exact expressions for the SPD voltage output modulation f ( t) becomes undesirable. For parameters typically used in examples disclosed here,

, this gives 16 The integral of this top hat function is then given by:

This can be exponentiated to give:

In doing this, we can approximate the output without relying on numerical integration in both temporal and frequency space. Using this approximation gives: Before any SPD detection event, as above:

During the SPD detection event

Time dependence with Back-scattering We start with the equation:

and take the expectation value as before to get:

or, the matrix differential equation:

This commutes, so the solution can be given by the Green’s function solution:

where

^^ is then the matrix exponential of:

Alternatively, we can do the same without taking the expectation value to get:

As the input is taken to be only in the input (lower) waveguide 14, denoted in equation parameters by the downward arrow symbol (↓) herein, this simplifies to:

which gives the anticlockwise mode b₁ and clockwise mode b₂ as:

The correlation function of mode n at time (t₁, t₂) is then given by:

For a coherent state, we also have:

which gives, for the input (lower) waveguide 14:

For the upper (output) waveguide 12, we have:

The correlation functions, in shown in Figures 12A, 12B, 12C and 12D for a pulsed optical input, and shown in Figures 13A and 13B for a continuous wave (CW) optical input, are:

Referring to figures 12A, 12B, 12C, 12D and 12E, Figure 12A shows the temporal intensity profile of the pulsed optical input 4 upon the input end of the lower waveguide 14 of the system illustrated in Figure 5 and Figure 3A, or Figure 6 and Figure 4A. The time is measured in units of the decay time of the optical ring resonator. Figures 12B and 12C correspond to the response of the system schematically shown in Figure 6 and Figure 4A. Figure 12B shows the concurrent optical output power in an optical output 10B within the lower optical waveguide 18 in response to the pulsed optical input 4 upon the input end of the lower waveguide 14 of the system schematically shown in Figure 6 and Figure 4A. The time is measured in units of the decay time

of the optical ring resonator, for consistency with Figure 12A. Figure 12C shows the concurrent optical output power in an optical output 10A within the upper waveguide 12. Figure 12D corresponds to the response of the system schematically shown in Figure 5 and Figure 3A. This figure shows the optical output power in an optical output 10 within the lower optical waveguide 18 in response to the pulsed optical input 4 upon the input end of the lower waveguide 14 of the system of Figure 5 and Figure 3A. Figure 12E corresponds to the response of the system schematically shown in Figure 5 and Figure 3A. This figure shows the back-scattered optical output power in an optical output 10C within the lower optical waveguide 14 in response to the pulsed optical input 4 upon the input end of the lower waveguide 14 of the system of Figure 5 and Figure 3A. Note that the intensity scale (vertical axis) in this graph is about one tenth (1/10) the scale of the corresponding intensity scale in the graph of Figure 12D, indicating low back-scattering levels. Figures 12B, 12C, 12D and 12E each show the optical output power for two separate conditions: (1) when a single-photon detection event takes place (SPD voltage output) at time t = τ/10, in the graphs shown in figures 12B and 12C, and at time t = − τ in the graph shown in Figure 12D; and, (2) when no single-photon detection event takes place. The results correspond to a ‘critical coupling’ of the waveguides and ring resonator, and this corresponds to the coupling condition

The graphs correspond to the following conditions: a resonance frequency shift amplitude of

, in a resonator ring 22 with a quantity factor of

Backscattering parameters were chosen to be

, for simplicity. From these results, one can conclude that when the modulation (frequency shift) in the resonance frequency of the ring resonator is large enough, the detection of a single photon by the SPD causes the non-resonant input light to stop coupling into the ring for an interval of time corresponding to the duration of the frequency shift modulation. This non-resonant light may then interact with the light already in the ring which has also been rendered non-resonant and which is in the process of being coupled out of the ring at the decay rate of the ring. That is to say, the coupling of light into the ring can be delayed by the time-scale of the SPD electrical (e.g., voltage) signal, but pre-resonant light already within the ring continues to couple out. Figures 12D and 12E correspond to the output optical signal caused by a pulsed input light to the ring resonator when that ring resonator is only coupled to a single waveguide, as shown in Figure 5 and Figure 3A, and this appears to indicate an advantageous configuration in that the output pulse intensity is relatively high and relatively narrow/sharp (at least with respect to the initial rise and fall of the output pulse). As a result, the configuration exemplified in Figure 3A and Figure 5 may be better able to produce an output pulse that is easier to distinguish between when there is a single photon detection event and when there is not a single photon detected. It gives a ‘cleaner’ signal, as well as coupling all (except loss) of the light back into the single waveguide. This may be advantageous when used as a single-photon gate. Figure 13A shows the optical output power in an optical output 10B within the lower waveguide 18 in response to a CW optical input 4 upon the input end of the lower waveguide 14 when a single-photon detection event takes place at time ^^ = 0, in the graph. Figure 13B shows the concurrent optical output power in an optical output 10A within the upper waveguide 12 in response to a CW optical input 4 upon the input end of the lower waveguide 14 when a single-photon detection event takes place at time t = 0, in the graph. The result corresponding to a ‘critical coupling’ of the waveguides and ring resonator is indicated in the graphs and corresponds to the coupling condition

. The time is measured in units of the decay time

/ of the optical ring resonator. The graphs correspond to the following conditions: a resonance frequency shift amplitude of

, in a resonator ring 22 with a quantity factor of Q = 10⁶. Backscattering parameters were chosen to be

, for simplicity. Figures 14A, 14B and 14C each show the optical output power in an optical output 10B within the lower waveguide 18 in response to a CW optical input 4 upon the input end of the lower waveguide 14 when a single-photon detection event takes place at time ^^ = 0, in the graphs. The time is measured in units of the decay time

of the optical ring resonator. The result corresponding to a range of different coupling regimes as between the optical waveguides of the system and the ring resonator they are coupled to. These coupling regimes include a ‘critical coupling’ of the waveguides and ring resonator, and this is indicated in the graphs and corresponds to the coupling condition

The temporal pulse shape of the SPD electrical pulse is also indicated in each graph. The graphs correspond to the following conditions: Figures 14A and 14B, correspond to a resonance frequency shift amplitude of

10Γ in a resonator ring 22 with a quantity factor of . Figure 14C corresponds to a resonance frequency shift amplitude of

100Γ in a resonator ring 22 with a quantity factor of

. In all cases, backscattering parameters were chosen to be

for simplicity. Figure 14A shows a relatively long decay time of the temporal pulse shape of the SPD electrical pulse, with the result that the output optical power has oscillations with multiple peaks. Note that these are actual oscillations in the output optical power, and not merely an oscillating optical phase. Figure 14B shows a relatively reduced (i.e., relative to that of Fig.14A) decay time of the temporal pulse shape of the SPD electrical pulse may have the effect of removing the oscillations in the output optical power seen in Fig.14A, but at the cost of generating a relatively lower optical output power (note the vertical scale in the graphs, indicating approximately a five-fold drop in output power) for a shorter period of time. Figure 14C shows that the output optical power can be increased again (e.g., to a value comparable to that seen in Fig.14A) by increasing the amplitude, of the shift in resonant frequency of the ring (e.g.,

100Γ, a ten-fold increase in this example). This increasing in the amplitude,

, does not increase the duration of time, i.e., the width of the modulation function , that the frequency is shifted.

One can see that critical coupling of the waveguides and the resonator ring has the significant advantage that the optical output power in the optical output 10B within the lower waveguide 18 rises sharply from a substantially negligible value in response to the occurrence of a single-photon detection event and the rise of the SPD electrical pulse, and subsequently decays rapidly back to a substantially negligible value soon after the SPD electrical pulse has subsided. The width (Δt_width) of the optical output pulse 10B within the lower waveguide 18, as shown in Figure 12B, Figure 12D and Figure 13A, may be defined as the full-width at half-maximum (FWHM). In the present example, the width (Δt_width) of the optical output pulse is approximately equal to the decay time of the optical ring resonator:

This gives a brief, clear and distinct optical output pulse intensity as the optical output 10B of the switch of Figure 5 or Figure 6, which is well suited to serve as an optical switching signal or pulse for use in implementing an onward switching operation within a photonic component (e.g., an optical pulse generator, a laser pulse generator, an electrical pulse generator, a non-linear optical element, an optical phase shifter, or transmissive optical beam-splitter, as shown in Figure 15 and Figures 16A to 16D) downstream from the output end of the lower waveguide 18. For example, an optical detector (not shown) may be provided as a part of the photonic component downstream from the output end of the lower waveguide 18 which is responsive to detection of the optical switching signal or pulse so as to issue a control signal to a component of a photonic circuit, or the like. The optical detector may be configured to deem a detection of the optical switching signal or pulse to have occurred once when the detection event occurs anywhere within a given one of a succession of detection time slots (e.g., detection time bins). The time of the occurrence of such a detection event may be determined as being the time associated with the detection time slot in which the detection event occurred (e.g., the temporal beginning, middle or end of the detection time slot). Preferably, each of the time slots has a duration of not less than the width (e.g., FWHM) of the optical switching signal or pulse (e.g., not less than the decay time ^^ of the optical resonator). This means that the majority of the optical switching signal or pulse may reside within one detection time slot, and it mitigates against the unwanted occurrence of the same one optical switching signal or pulse spanning multiple detection time slots thereby erroneously triggering multiple ‘detection’ events. Preferably, each of the time slots has a duration of not more than twice (x2) or three-times (x3) the width (e.g., FWHM) of the optical switching signal or pulse (e.g., not less than twice (x2) or three-times (x3) the decay time ^^ of the optical resonator). This enhances the likelihood of encompassing substantially the whole of one optical switching signal or pulse within one detection time slot yet constrains the duration of the detection time slots to be similar to the duration of the optical switching signal or pulse. Alternatively, or in addition, the optical detector (not shown) provided as a part of the photonic component downstream from the output end of the lower waveguide 18 may be responsive to detection of the optical switching signal or pulse so as to issue a control signal to a component of a photonic circuit, or the like, if the detected optical output power rises to a value that is greater than the output power detected when no photon is detected by the single-photon detector. The optical detector may comprise a signal thresholder configured to compare the detected optical output power against a pre-set optical power threshold value and to issue a detection signal on condition that the detected optical output power exceeds the pre-set optical power threshold value. The clear and distinct nature of the switching signal or pulse shown in Figure 12B, Figure 12D, Figure 13A and Figures 14A, 14B and 14C under conditions of critical coupling has the particular advantage of making this thresholding process efficient. Notably, the inventors have found that any light already within the ring resonator when resonant is coupled out of the ring when rendered non-resonant. The out-coupling time-frame is preferably made to be slower than the voltage modulation time frame so that an optical input pulse can pass by the ring (without coupling into it) before a significant amount of non-resonant extant light (within the ring) is released from the ring and contributes to the optical output of the device. The use of a single waveguide coupled to the optical ring resonator may reduce the effects of leakage from ring resonator appearing in the optical output. Using one single waveguide means greater optical output, and back-scattered optical output is only relatively small (e.g., about 10%) compared to the optical output. By using a relatively short SPD voltage decay time, one may to some extent suppress oscillations in the optical output power at the waveguide output. The SPD voltage decay time can be considered to be relatively short when the effective decay time τ_d (defined above) should be less than (or approx. equal to) the decay time, τ, of the resonator. The coupling rate for coupling of light into the ring resonator from the input waveguide does not significantly change the height of the oscillations in the optical power of the output signal, at least for long SPD voltage decay times. These results suggest that an advantageous implementation of the optical switch/router may be achieved when implementing a down-stream detection scheme for detecting a pulsed optical output generated by the switch/router in response to a pulsed optical input. Detection schemes that may be well suited to such an optical output may comprise the detection of an output optical pulse (created using a pulsed optical input) within one of a series of detection time bins, each time bin being of a duration of the order of one or a few pulse widths (temporal) of the optical input pulse, and/or implement a thresholding process/device that counts an output optical pulse only if the detected optical output power is greater than the detected peak optical output power generated by the optical switch/router when no photon is detected by the SPD. As a further illustration of the ‘time binning/thresholding’ methods discussed above, for example consider a series of time-bins which may (or may not) be of the order of the decay time of the resonator. For each of these time-bins, we can then define a threshold power output for which any output above this threshold is considered a ‘1’ and any output below this threshold is then a ‘0’ (one may also have multiple thresholds with multiple values). In the case where a single-photon is not detected, this then results in a particular power distribution which gives a particular binary (or otherwise) output string after applying the above thresholding rules. When a single-photon is detected, the output power distribution is considerably different (i.e., there are multiple peaks in at least one of the photon being detected/not detected cases, with at least one peak in a different temporal location, although some overlap is expected) and as such, after the thresholding rule (or similar) above is applied, the output binary string will be different. Figure 15 shows a schematic diagram of an optical transformation device 99 for performing single photon conditional optical transformations. Figures 16A, 16B, 16C and 16D each show different implementations of the device of Figure 15. The optical transformation device 99 is configured to detect single photons 27B propagating into the device along a first waveguide 21 (‘waveguide A’) which is in optical communication with the single- photon detector of a single-photon transducer (switch/router) unit 100 according to the aspects of the invention described above. A laser pulse 131 is output by the single-photon transducer (switch/router) unit 100 to propagate along second waveguide 130 (‘waveguide B’) conditional upon detection of a single photon thereby. Optionally, the laser pulse 131 is input, via the waveguide 131 (‘waveguide B’) to a laser pulse re-shaper unit 132 which is arranged to reshape the temporal profile and/or spectral profile of the input laser pulse 131 to a desired new profile and to output the re-shaped laser pulse 133 on an output waveguide of the pulse re-shaper 138. The laser pulse 131 (or optionally the reshaped laser pulse 132) is input to an optical transformer 136 which comprises a first optical input port that is optically coupled to the output optical waveguide 130 (‘waveguide output B’) of the single-photon transducer (switch/router) unit 100 (or optionally the output waveguide 138 of the pulse re-shaper, if used). The laser pulse 131 (or 132 if reshaped) is then input to the optical transformer where it is used to drive a nonlinear optical process which enacts a transformation on optical signals input to the optical transformer 136 on a third optical waveguide (waveguides C) of the device. The third optical waveguide (waveguides C) is optically coupled to a second optical input port configured for receiving optical signals for optical transformation. A schematic layout of two embodiments of the optical switch/router is shown in figures 3A, 3B and 5, or figures 4A, 4B and 6 herein and each is discussed in detail above. It is to be understood that either of these two embodiments may be employed as the single-photon transducer (switch/router) unit 100 of the optical transformation device 99. The embodiments of the optical switch/router as shown in figures 3A to 3C, 4A to 4C and in figures 5 and 6 will be referred to below as separate examples of the single-photon transducer (switch/router) unit 100. The structure shown in figure 3A and 3B, or as shown in figures 4A and 4B, may be implemented as an integrated photonic chip containing the single-photon transducer (switch/router) unit 100 whose various different components comprise: - A collection of waveguides (12, 14, 16, 18) used to carry optical signals (4, 10, 10A, 10B). These could be realised in any material transparent as the wavelengths of the optical signals, e.g. silicon, silicon nitride, aluminium nitride or thin film lithium niobate. In general, the waveguides may support modes propagating in both directions (i.e. towards the left or the right, in the diagram), continuous wave or pulsed signals, and of either high intensity (i.e. laser light) or low intensity (i.e. single photons). The waveguides may be partly off-chip, or could be fed by off-chip optical fibres. - A high quality factor ring resonator 22 which couples to the collection of waveguides (12, 14, 16, 18), and which could be made from a different material or manufacturing process to collection of waveguides (12, 14, 16, 18). The ring resonator 22 may be constructed by looping a waveguide structure in any of the materials listed above, or may be another resonator structure (e.g., a disk resonator, a linear resonator, a photonic crystal structure etc.) as discussed above. In general, the resonator structure may be any one of a variety of shapes (not necessarily circular). Preferably, it supports optical modes propagating in both clockwise and anticlockwise directions. Coupling between the resonator and collection of waveguides (12, 14, 16, 18) may be achieved and controlled by bringing the structures into physical/spatial proximity. The resonance bandwidth of the resonator may be controlled by tuning either the coupling to collection of waveguides (12, 14, 16, 18) or coupling into ambient loss modes, as desired. - An on-chip integrated single photon detector 20 placed at, and optically coupled to, the terminal end of the first waveguide 21. This single-photon detector may be realised using either a superconducting nanowire detector or a single photon avalanche photo diode. It preferably is configured to absorb photons propagating in the first waveguide 21 in a direction towards the single-photon detector, which detector is responsive to receipt of the single-photon by producing an electrical signal. The single-photon detector may be optimised to produce an optimum output voltage, for example utilising multiple superconducting nanowires in parallel or by increasing the load resistance used in the detector drive circuitry. - An electrical contact part 24 (optional) is configured to electrically couple the single-photon detector 20 to an electro-optic modulator unit 26. This electrical contact part 24 preferably serves to provide on-chip processing and/or amplification of the electrical signal produced by the single-photon detector for use in driving the electro-optic modulator unit 26. The processing may include voltage signal filtering functions, and the electrical contact part may comprise suitable capacitive elements, as appropriate for filtering and/or signal amplification. It may be used to generate a larger output voltage pulse, such as by cascading a plurality of nanowire single-photon detectors to amplify the output pulse from a single detector, using an impedance-matching taper, or using a compact low power on-chip amplifier. - A high speed electro-optic modulator 26 is configured to modulate the refractive index of the ring resonator thereby to produce corresponding changes the resonant frequency of the ring resonator 22. This electro-optic modulator may be configured to implement a process such as DC Kerr effect, or Pockels effect, in the material of the ring resonator, or carrier injection into the material of the ring resonator, to achieve the desired modulation of refractive index. Methods and techniques such as are readily available to the skilled person may be employed to this end. The modulator may be optimised to provide a strong effect on the ring resonator (shift of the frequency resonance per voltage input) using a combination of engineered electrodes and high impedance. With the various elements described above in place, operation of the device as a whole proceeds as follows. Consider first the case schematically illustrated in Figure 3A for which the single-photon detector 20 produces no electrical signal and the electro-optic modulator 26 has its “rest” default effect on the ring resonator, which in turn will have a resonance frequency at its default frequency value (fres) given by the total coupling strength to all external optical modes, as determined by the manufacturing process used to create the optical switch/router. Figure 3C (upper graph) shows a schematic diagram of the transmission spectrum 140 of the input and output waveguide structure (14, 18) via the ring resonator when in the state illustrated in Figure 3A. That is to say, the frequency (ωsignal) of the optical signal 4 within the input optical waveguide part 14 corresponds to (or at least falls within the resonance bandwidth 142) of the resonance frequency of the ring resonator (i.e., ωres = ωsignal). This means that the optical signal 4 within the input optical waveguide 14 is strongly, resonantly, coupled into the ring resonator 22, and substantially is not transmitted to the output optical waveguide part 18. In this way, optical signals within optical modes of the waveguides, which impinge on the ring resonator, will be coupled into and out of the resonator as determined by their frequency distribution and the strength of the coupling between the optical modes of the waveguide and resonator. At “critical coupling” a signal propagating to the right in waveguide part 14 when resonant with the ring resonator will not exit the resonator 12. Similarly, for a broadband signal propagating to the right in waveguide 14, those frequency components closest to the resonance of the ring will enter the resonator, while off-resonant components will not enter the resonator and will exit the device via waveguide 18 propagating to the right to ‘waveguide output’. As is shown in Figure 3B, when a photon is present in waveguide 21 propagating to the right, the single- photon detector will absorb this photon thereby creating an output voltage signal which is used to drive the electro-optical modulator 26, which in turn changes the resonant frequency of the ring resonator 22. The result of the modulation is to shift the resonance frequency of the ring resonator from its default frequency value (ωres) to a shifted resonance frequency (ω_res → ω_res + Δω) which is shifted by a frequency shift 144 (Δω) determined by the modulator 26. Figure 3C (lower graph) shows a schematic diagram of the transmission spectrum 146 of the ring resonator 22 when in the state illustrated in Figure 3B. That is to say, the frequency (ωsignal) of the optical signal 4 within the input optical waveguide 14 is excluded from the resonance bandwidth 148 of the ring resonator. This means that the optical signal 4 within the input optical waveguide 14 is no longer resonantly coupled into the ring resonator 22, and is transmitted to the output optical waveguide part 18. This controls how incoming optical signals in the input waveguide 14 is redistributed either to the ring resonator to supress a corresponding outgoing signal or to the output waveguide 18 to provide a corresponding outgoing signal, thus enacting a single-photon triggered rerouting of optical signals. Consider next the case schematically illustrated in Figure 4A for which the single-photon detector 20 produces no electrical signal and the electro-optic modulator 26 has its “rest” default effect on the ring resonator. Once more, the ring resonator will have a resonance frequency at its default frequency value (ω_res) given by the total coupling strength to all external optical modes, as determined by the manufacturing process used to create the optical switch/router. Figure 3C (upper graph) applies equally to this arrangement and shows a schematic diagram of the transmission spectrum 140 of the input and output waveguide structure (14, 18) via the ring resonator when in the state illustrated in Figure 4A. Optical signals within optical modes of the waveguides, which impinge on the ring resonator, will be coupled into and out of the resonator as determined by their frequency distribution and the strength of the coupling between the optical modes of the waveguide and resonator. The process amounts to a redistribution of the impinging optical signals into outgoing modes in the same waveguides. For example, when the coupling strength between modes propagating to the right in waveguides 14 and 18 and the anticlockwise mode in the ring resonator is equal to the coupling strength between modes propagating to the left in waveguides 12 and 16 and the anticlockwise mode in the ring resonator, the system is said to be at “critical coupling”. At “critical coupling” a signal propagating to the right in waveguide 14 resonant with the ring resonator will exit the resonator only via waveguide 12 and propagating to the left, to ‘waveguide output A’. Similarly, for a broadband signal propagating to the right in waveguide 14, those frequency components closest to the resonance of the ring will exit via waveguide 12 propagating to the left, while off resonant components will exit via waveguide 18 propagating to the right to ‘waveguide output B’. The waveguide part 16 (waveguide 3) does not receive optical signals by direct output from the optical resonator 22, but may receive back-scattered components of the optical output 10 of the ring as that output is coupled into the waveguide 12 towards ‘waveguide output A’. As is shown in Figure 4B, when a photon is present in waveguide 21 propagating to the right, the single- photon detector will absorb this photon thereby creating an output voltage signal which is used to drive the electro-optical modulator 26, which in turn changes the resonant frequency of the ring resonator 22. This affects how incoming optical signals in waveguides (12, 14, 16, 18) are redistributed to their corresponding outgoing signals thus enacting a single-photon triggered rerouting of optical signals. For example, at “critical coupling”, frequency components of a broadband optical signal propagating to the right in waveguide 14 that were previously resonant with the ring resonator and which exited at ‘waveguide output A’ via waveguide 12, will now exit at ‘waveguide output B’ via waveguide 18 propagating to the right. Figure 3C (lower graph) shows a schematic diagram of the transmission spectrum 146 of the ring resonator 22 when in the state illustrated in Figure 4B. Alternative embodiments are shown schematically in figure 4D, 4E, 4F and 4G. These embodiments correspond to a configuration in which when the single-photon detector 20 detects no photon, the modulator 26 is in the “OFF” state and therefore produces no electrical signal. Thus, the electro-optic modulator 26 has a “rest” (or “OFF”) default effect on the ring resonator such that the resonance frequency of the resonator is at its default frequency value (ωres). Figure 4C (upper graph) shows a schematic diagram of the transmission spectrum 147 of the input and output waveguide structure (14, 18) coupled to the ring resonator when in the state illustrated in Figure 4D. That is to say, the frequency (ωsignal) of the optical signal 4 within the input optical waveguide part 14 is excluded from the resonance bandwidth 149 of the ring resonator (i.e., ωres = ωsignal + Δω). This means that the optical signal 4 within the input optical waveguide 14 is not resonantly coupled into the ring resonator 22 and is not transmitted to the output optical waveguide part 18. Figure 4E shows the optical switching/routing device of Figure 4D in a second switching/routing state when the single-photon detector 20 does detect a photon. The modulator 26 is in the “ON” state and therefore produces an electrical modulation signal. Thus, the electro-optic modulator 26 has an “active” (or “ON”) effect on the ring resonator such that the resonance frequency of the resonator is shifted from its default frequency value by a frequency shift 145 (-Δω) determined by the modulator 26, to a shifted resonance frequency (ω_res → ω_res - Δω = ω_signal + Δω – Δω = ω_signal) which places the ring resonator in resonance with the input optical signal 4. Figure 4C (lower graph) shows a schematic diagram of the transmission spectrum 144 of the input and output waveguide structure (14, 18) coupled to the ring resonator when in the state illustrated in Figure 4E. That is to say, the frequency (ωsignal) of the optical signal 4 within the input optical waveguide part 14 is included in the resonance bandwidth 143 of the ring resonator (i.e., ωres = ωsignal). This means that the optical signal 4 within the input optical waveguide 14 is resonantly coupled into the ring resonator 22 and is not transmitted to the output optical waveguide part 18. Figures 4F and 4G each show an optical switching/routing device according to a configuration of the invention similar to that shown in figure 4D but with the addition of a second output waveguide 12. Input optical signals 4 that do not couple resonantly into the ring resonator 22 when the modulator is “OFF” are not passed for output at waveguide output A (see Fig.4F), and are instead passed along output waveguide 18 for output at waveguide output B. Conversely, input optical signals 4 that do couple resonantly into the ring resonator 22 when the modulator is “ON” are passed to the output waveguide part 12 for output at waveguide output A (see Fig.4G) and are not passed along output waveguide 18 for output at waveguide output B. Single photon transducer The single-photon transducer (switch/router) unit 100 of the optical transformation device 99 may be realised as follows. With the arrangement of components shown in Figure 4A and 4B, one may configure/manufacture the ring resonator and waveguides such that the system is at “critical coupling”. Then, one may inject a broadband optical signal, or an optical pulse corresponding to an individual photon, propagating to the right into waveguide 14 (‘waveguide input’), such that its spectral width in frequency is small compared to the spectral frequency width of the resonator 22. The single-photon detector 20, the electrical contact part 24 (optional) 24 and the electro-optic modulator 26 are configured such that the duration of the modulator’s effect (modulation) on the ring resonator is sufficiently large compared to the temporal duration of the optical signal in waveguide 14. With these conditions met, when no photon is present in waveguide 21, such that no single-photon detection event occurs, the optical signal in 14 will exit the device via waveguide 12 propagating to the left towards waveguide output A. When a single photon propagates to the right in waveguide 21 such that a single-photon detection event occurs, and the subsequent modulation of the position of the resonance bandwidth of the ring resonator 22 coincides with the optical signal in waveguide 14 impinging on the resonator, then the optical signal will exit the device via waveguide 18 propagating to the right towards waveguide output B. The optical transformation device 99 provides a means to condition a laser pulse (e.g., a strong pulse) on the detection of a single photon, as described above. Consequently, a scheme to generate single-photon conditioned optical transformations becomes possible. An example of an arrangement for the optical transformation device 99 is shown schematically in Figure 15 and in Figures 16A to 16D. This example comprises: - Waveguides used to carry optical signals (110, 130, 133 and 134). As above, these waveguides could be realised in a number of integrated photonics platforms e.g., silicon, silicon nitride, aluminium nitride or thin film lithium niobate. The waveguides in general support multiple spectral, transverse and polarisation modes, and the optical fields can be continuous wave or pulsed. The quantum states of light confined to these waveguides are considered to be completely general, i.e., may be both Gaussian (e.g., squeezed light) or non-Gaussian (e.g., single photons). Furthermore, these quantum states may be entangled and/or correlated with the quantum states of any other physical systems not explicitly described or shown. - A single-photon transducer (switch/router) unit 100 as described above, which is configured such that the optical input end at the terminal of the waveguide 21 is optically coupled to the optical output end of the waveguide 110 (‘waveguide A’) of the of the optical transformation device 99. Accordingly, the single-photon detector 20 may receive and detect single photons 27B input to the waveguide 110 (‘waveguide A’). On detecting such a single-photon, the single-photon transducer (switch/router) unit 100 is responsive to generate a strong laser pulse 131 upon the output waveguide 18 of the unit 100 for output at ‘waveguide output B’ thereof, and for delivery to a waveguide 130 (‘waveguide B’) of the optical transformation device 99, with which it is optically coupled for this purpose. The photon in waveguide 110 (‘waveguide A’) of the optical transformation device 99 may be detected using a single-photon detector device 20 comprising any one of a number of configurations, selected from: a superconducting nanowire single-photon detectors; or a single photon avalanche diode. In this example this transduction is performed by a device as described above. - A pulse re-shaper (132) modifies the optical signal 131 delivered via waveguide 130 (‘waveguide B’). Such modifications could be achieved by linear passive elements such as optical filtering, or active elements such as electro-optic or acousto-optic modulators, as would be readily apparent to the skilled person. - An optical transformer unit 136, such as a multimode optical transformer e.g., a nonlinear optical element, configured to use the laser pulse (131, or 133) received from the single-photon transducer unit 100 (or from the pulse re-shaper 132) to perform an optical transformation on those optical modes of the received the laser pulse on waveguide B and optical modes of an optical signal 135 concurrently received via a second input waveguide 134 (‘waveguide C’), to produce an optically transformed optical output result 137. Depending on the configuration and settings of the optical transformer unit 136, transformations performed on optical signals received via a second input waveguide 134 (‘waveguide C’) may include, but are not limited to, displacements, squeezing or rotations in phase space. These may be realised by a combination of available integrated photonics components including optical resonators, waveguides and modulators, e.g., as exemplified below with reference to Figures 16A to 16D. Squeezing operations In this configuration, the optical transformer unit 136 is configured to perform single mode optical squeezing (e.g., photon-pair generation) in the second input waveguide 134 (‘waveguide C’). This may be achieved, for example, with a periodically polled thin-film lithium niobate waveguide, where quasi-phase matching is optimised for type-0 pair-generation i.e., both photons of the generate pair of photons are produced in the same polarisation and spatial mode as the pump, and are frequency degenerate, leading to a single-mode squeezed output state. In a different embodiment, the laser pulse 133 may be configured to pump a spiral or ring resonator in a highly-nonlinear third order material, such as silicon, silicon nitride or gallium arsenide, to generate squeezing via self-phase modulation. The phase and energy matching conditions can be adjusted so that the nonlinear waveguide produces frequency non- degenerate photon pairs, giving rise to a two-mode squeezing transformation on two of the modes in the second input waveguide 134 (‘waveguide C’). Passive operations: Phase-shifting In this configuration, the optical transformer unit 136 is configured to comprise a passive phase-shifting optical element to implement a phase-shift in the quantum state of the light received in the second input waveguide 134 (‘waveguide C’). The passive phase-shifting optical element may employ an electro-optic waveguide modulator acting on the second input waveguide 134 (‘waveguide C’) within the optical transformer unit 136. As such, the optical waveguide 130 (‘waveguide B’), the pulse re-shaper unit 132, and the first optical input waveguide 138 of the optical transformer 136 may be replaced, in the device 99, by an electrical transmission line (Figure 16C) arranged for transmitting an electrical pulse signal from the single-photon transducer unit 100 to the optical transformer unit 136, for use in driving the electro-optic waveguide modulator. Passive operations: displacement Alternatively, the pulse re-shaper unit 132 may be configured to generate a bright coherent state output pulse 133 which is mode-matched to the modes in the second input waveguide 134 (‘waveguide C’). The passive phase-shifting optical element may implement optical mixing on a highly transmissive beam- splitter to achieve the result of an optical displacement of the quantum state of the target optical mode. The examples of Figs.16A (1) and 16B (2) can be achieved with e.g. Spontaneous Parametric Down Conversion (SPDC) or Four Wave Mixing (FWM) processes, which can be implemented using resonators/spirals/waveguides etc. and are done by X⁽²⁾ and X⁽³⁾ processes respectively, with corresponding Hamiltonians:

and

When S (signal photon) = I (Idler photon), this gives degenerate processes (case (1)/fig.16A) and non- degenerate processes otherwise (case (2)/fig.16B). The example of Fig.16C (3) can e.g., be implemented by using the electrical output of the SPD to directly adjust the refractive index of a given length of material (as disclosed per the rest of this disclosure) to directly cause a phase shift on input state

to give the output state:

The example of Fig.16D (4) is then given by implementing the displacement operator:

on the input state |ψ> to give the output state

Figures 17A to 17C show schematic diagrams of different implementations of an optical resonator according to embodiments of the invention. Note that the embodiments shown in Figure 3A, Figure 3B and Figure 5 each employ a linear waveguide coupled to a looping optical resonator structure (e.g., a circular ring structure in these examples). However, the linear waveguide may, in other embodiments, be coupled to a linear optical resonator structure. Examples, which are not intended to be exhaustive, are shown schematically in figures 17A to 17C. For example, Figure 17A shows a linear optical resonator structure 150A formed within the material of the core of an optical waveguide (or at least a section of one). A resonator optical cavity 153 is formed in the linear space between two fibre Bragg grating structures 152 (e.g., distributed Bragg reflectors) formed within the core of the waveguide and separated by linear separation along the axis of the waveguide to define a linear grating-free region extending along the core of the waveguide between the two separated fibre Bragg grating structures. Light 162 input to the linear optical resonator structure 150A is coupled into the resonator optical cavity 153 via a first one of the two separated Bragg grating structures 152 and, depending upon the parameters of the resonator, may resonate 164 within the resonator optical cavity 153 or may pass through the cavity 153 as output light 162. The electro-optical modulator 26 (serving the same purpose as item 26 of Fig.3A, 3B) is arranged to apply an electrical signal generated by the single- photon detector (item 20 of Fig.3A) with which to modulate the refractive index of the material of the core of the optical waveguide defining the optical cavity 153 thereby to cause a shift in the frequency at which the resonance spectral profile of the linear optical resonator structure 150A is centred or positioned, as discussed above. The linear optical resonator structure 150A may be formed within the input optical waveguide (14, 18) of the optical switch/router described above with reference to Figure 3A and Figure 3B, with the optical ring resonator 22 omitted. Figure 17A shows an alternative linear optical resonator structure 150B formed within a dielectric stack defining an optical waveguide (or at least a section of one). A resonator optical cavity 154 is formed in the linear space between two distributed Bragg reflectors 156 formed by the dielectric stack and separated by linear separation along the axis of the stack to define a linear grating-free region extending along the axis of the stack between the two separated distributed Bragg reflectors 156. Light 162 input to the linear optical resonator structure 150B is coupled into the resonator optical cavity 154 via a first one of the two separated distributed Bragg reflectors 156 and, depending upon the parameters of the resonator, may resonate 164 within the resonator optical cavity 154 or may pass through the cavity 154 as output light 162. The electro-optical modulator 26 (serving the same purpose as item 26 of Fig.3A, 3B) is arranged to apply an electrical signal generated by the single-photon detector (item 20 of Fig.3A) with which to modulate the refractive index of the material of the core of the dielectric stack defining the optical cavity 154 thereby to cause a shift in the frequency at which the resonance spectral profile of the linear optical resonator structure 150B is centred or positioned, as discussed above. The linear optical resonator structure 150B may be formed within the input optical waveguide (14, 18) of the optical switch/router described above with reference to Figure 3A and Figure 3B, with the optical ring resonator 22 omitted. Figure 17C shows an alternative linear optical resonator structure 150C formed by a photonic crystal defect structure within a dielectric material of an optical waveguide (or at least a section of one). The photonic crystal is formed as a periodic linear array of cylinders of a first dielectric material (e.g., a solid substance, or an airgap, or void) embedded within a surrounding material of the waveguide which is a second dielectric material different the first dielectric material. A defect on the periodicity of the periodic linear array of cylinders is provided by the absence of one or more such cylinders where there would otherwise be such a cylinder(s) according to the periodicity of the array. This defect, namely the absence of one of more such cylinders from the periodic array, defines a resonator optical cavity 160 in the linear space between two separate sub-sections of the linear array of cylinders 158 separated by linear separation along the axis of the waveguide to define a linear cylinder-free region extending along the axis of the photonic crystal between the two separated sub-sections of the linear array of cylinders 158. Light 162 input to the linear optical resonator structure 150C is coupled into the resonator optical cavity 160 via a first one of the two separated sub-sections of the linear array of cylinders 158 and, depending upon the parameters of the resonator, may resonate 164 within the resonator optical cavity 160 or may pass through the cavity 160 as output light 162. The electro-optical modulator 26 (serving the same purpose as item 26 of Fig.3A, 3B) is arranged to apply an electrical signal generated by the single-photon detector (item 20 of Fig.3A) with which to modulate the refractive index of the dielectric material of waveguide defining the optical cavity 160 thereby to cause a shift in the frequency at which the resonance spectral profile of the linear optical resonator structure 150C is centred or positioned, as discussed above. The linear optical resonator structure 150C may be formed within the input optical waveguide (14, 18) of the optical switch/router described above with reference to Figure 3A and Figure 3B, with the optical ring resonator 22 omitted. It is to be noted that even though the detailed analysis provided above is given in terms of a looped optical resonator structure (e.g., a ring), the analysis has general applicability to other structures of optical resonator such as a linear waveguide. Figures 18A to 18E each show a respective example of a waveguide structure suitable for use in forming the optical waveguides and resonator structures according to the invention. Figure 18A shows a buried channel waveguide which is formed with a high-refractive index (n₁) waveguiding core 171, of width “w” and depth “d”, buried in a low-refractive index (n₂) surrounding medium 170. The waveguiding core can have any cross-sectional geometry though it is often a rectangular shape. Figure 18B shows a strip-loaded waveguide 172 formed by loading a planar waveguide comprising a low- refractive index (n2) layer 174 beneath a high-refractive index (n1) slab 173, which already provides optical confinement in the depth “d” direction, with a dielectric strip 175 of intermediate refractive index n3<n1 or a metal strip to facilitate optical confinement in the width “w” direction. The waveguiding core of a strip waveguide is the high-refractive index (n₁) region under the loading strip 175, with its thickness determined by the thickness “d” of the high-refractive index (n1) region, and its width “w” defined by the width of the loading strip 175. Figure 18C shows a ridge waveguide which has a structure similar to that of a strip waveguide, but in which a strip, or ridge 177 of width “w”, depth “d” and high-refractive index (n1), is disposed on top of a planar structure 176 of low refractive index (n₂), and this acts as the waveguiding core. A ridge waveguide has strong optical confinement because it is surrounded on three sides by low refractive index air (or cladding material). Figure 18D shows a rib waveguide has a structure similar to that of a strip or ridge waveguide, but differs in that the strip 181, of width “w” and height “h”, has the same high-refractive index (n1) as the high index planar layer 180 beneath it and is part of the waveguiding core. The combined thickness of the planar layer 180 and the strip (of height “h”) is denoted as “d”. The planar layer is disposed on top of a planar structure 179 of low refractive index (n₂). These four types of waveguides, shown in Figure 18A to 18D, are usually termed rectangular waveguides with a thickness “d” (or height “h”) in the x direction and a width “w” in the y direction, though their shapes are normally not exactly rectangular. Figure 18E shows a diffused waveguide formed by creating a high-index region 183, of depth “d” and width “w”, in a substrate 182 through diffusion of dopants. An example is a LiNbO3 waveguide with a core of high-refractive index (n₁) formed by Ti diffusion into the substrate material of low refractive index (n₂). Because of the diffusion process, the core boundaries in the substrate are not sharply defined. A diffused waveguide also has a thickness “d” defined by the diffusion depth of the dopant in the x direction and a width “w” defined by the distribution of the dopant in the y direction. Any of the waveguide structures noted above can be used in the invention. Preferable waveguide structures are rib, ridge and strip-loaded waveguides. Suitable dimensions are a depth or thickness of waveguide core in the range: d = about 200nm to about d = 800nm and a width of waveguide core in the range: w = about 300 to w = about 2000nm. The preferable wavelength of the optical signal 4 to be guided by the optical waveguides of the invention may be a wavelength in the range of: about 700nm to about 1600nm. The bend radius of a looped resonator structure, such as the ring resonator 22, may be a bend radius (i.e., radius of curvature) in the range of: about 10μm to about 200μm. The features disclosed in the foregoing description, or in the following claims, or in the accompanying drawings, expressed in their specific forms or in terms of a means for performing the disclosed function, or a method or process for obtaining the disclosed results, as appropriate, may, separately, or in any combination of such features, be utilised for realising the invention in diverse forms thereof. While the invention has been described in conjunction with the exemplary embodiments described above, many equivalent modifications and variations will be apparent to those skilled in the art when given this disclosure. Accordingly, the exemplary embodiments of the invention set forth above are considered to be illustrative and not limiting. Various changes to the described embodiments may be made without departing from the spirit and scope of the invention. For the avoidance of any doubt, any theoretical explanations provided herein are provided for the purposes of improving the understanding of a reader. The inventors do not wish to be bound by any of these theoretical explanations. Any section headings used herein are for organizational purposes only and are not to be construed as limiting the subject matter described. Throughout this specification, including the claims which follow, unless the context requires otherwise, the word “comprise” and “include”, and variations such as “comprises”, “comprising”, and “including” will be understood to imply the inclusion of a stated integer or step or group of integers or steps but not the exclusion of any other integer or step or group of integers or steps. It must be noted that, as used in the specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Ranges may be expressed herein as from “about” one particular value, and/or to “about” another particular value. When such a range is expressed, another embodiment includes from the one particular value and/or to the other particular value. Similarly, when values are expressed as approximations, by the use of the antecedent “about,” it will be understood that the particular value forms another embodiment. The term “about” in relation to a numerical value is optional and means for example +/- 10%. References A number of publications are cited above in order to more fully describe and disclose the invention and the state of the art to which the invention pertains. Full citations for these references are provided below. The entirety of each of these references is incorporated herein. [1] Sipe et al. “Effective field theory for the nonlinear optical properties of photonic crystals”: PHYSICAL REVIEW E 69, 016604 (2004) [2] Z. Vernon and J. E. Sipe, “Spontaneous four-wave mixing in lossy microring resonators": Physical Review A, vol.91, no.5, p.053802, 2015. (arXiv:1502.05900v2 [quant-ph] 5 May 2015) [3] Z. Vernon and J. Sipe, “Strongly driven nonlinear quantum optics in microring resonators" Physical Review A, vol.92, no.3, p.033840, 2015. (arXiv:1508.03741v1 [quant-ph] 15 Aug 2015) [4] W. McCutcheon, “Gaussian nonlinear optics in coupled cavity systems: Back-scattering in micro-ring resonators" arXiv preprint arXiv:2010.09038, 2020.

Claims

Claims: 1. An optical switch for receiving an optical signal of a given optical signal frequency from an optical signal source and for outputting the optical signal, the optical switch comprising; an optical router configured for routing the optical signal conditional on the detection of a single photon, the optical router comprising: an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switch further comprises: a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; wherein the optical modulator is configured to modulate the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to either: (a) change from being a resonance bandwidth that includes the optical signal frequency to being a resonance bandwidth that excludes the optical signal frequency thereby to permit the optical input signal to be transmitted to the output optical waveguide part as a non-resonant optical output signal for output from the optical switch; or, (b) change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to permit the optical input signal to be transmitted to the output optical waveguide part as a resonant optical output signal for output from the optical switch. 2. An optical switch according to any preceding claim wherein the input optical waveguide part and the output optical waveguide part are each a respective part of one continuous optical waveguide that is optically coupled to the optical resonator at an optical coupling region of the optical resonator, wherein the input optical waveguide part extends to the optical coupling region and the output optical waveguide part extends from the optical coupling region. 3. An optical switch according to claim 2 wherein the continuous optical waveguide is critically coupled to the optical resonator. 4. An optical switch according to claim 2 or claim 3 comprising a further output optical waveguide wherein the continuous optical waveguide and the further output optical waveguide are separate optical waveguides each separately optically coupled to the optical resonator at separate respective optical coupling regions of the optical resonator. 5. An optical switch according to claim 4 wherein the further output optical waveguide is critically coupled to the optical resonator. 6. An optical switch according to claim 4 wherein the further output optical waveguide part is optically coupled to the optical resonator according to a coupling rate which is substantially equal to the coupling rate with which the output optical waveguide part is optically coupled to the optical resonator. 7. An optical switch according to any preceding claim wherein: the optical modulator is configured to modulate the refractive index of the optical resonator by the electrical modulation signal during a pre-set modulation time interval, T, starting from detection of a single photon by the single-photon detector unit thereby to modulate the resonance bandwidth during the pre-set modulation time interval. 8. An optical switch according to claim 7 wherein: the output optical waveguide part is optically coupled to the optical resonator according to a coupling rate, Γ₁, which is less than the inverse of the modulation time interval, Γ₁ < 1/ T, such that, in use, an optical input signal when resonating within the optical resonator is rendered non-resonant during the pre-set modulation time interval, T, and is transferred by the optical resonator to the output optical waveguide part at the coupling rate, Γ₁, as a non-resonant optical output signal for output from the optical switch. 9. An optical switch according to any preceding claim wherein: the output optical waveguide part is optically coupled to the optical resonator according to a coupling rate, Γ₁; and, the single-photon detector unit is configured to output the electrical detection signal in the form of a voltage pulse such that the voltage value of the electrical detection signal reduces from a voltage pulse peak value according to a pre-set voltage decay rate, T_d , that is greater than the coupling rate: Γ_d > Γ₁. 10. An optical switch according to any preceding claim wherein: the optical resonator comprises an optical loss rate,

; and, the optical modulator is configured to modulate the refractive index of the optical resonator to modulate the resonance bandwidth so to change the frequency position thereof by a resonance frequency shift, ω₁, that has a value exceeding the optical loss rate,

. 11. An optical switch according to any preceding claim wherein: the output optical waveguide part is optically coupled to the optical resonator according to a coupling rate, Γ₁; and, the optical resonator comprises an optical loss rate, , that has a value exceeding the coupling rate, Γ₁, such that

12. An optical switch according to any preceding claim wherein the optical resonator comprises a ring optical resonator. 13. An optical switch according to any preceding claim wherein the optical resonator comprises a Q- factor of not less than 1,000,000. 14. An optical switch according to any preceding claim further comprising the optical signal source wherein: the output optical waveguide part is optically coupled to the optical resonator according to a coupling rate; and, the optical signal source is configured to output an optical signal in the form of an optical pulse comprising a pulse temporal width which exceeds the inverse of the coupling rate. 15. An optical switch according to any preceding claim comprising the optical signal source configured to output an optical signal comprising a given optical signal frequency. 16. An optical switch according to claim 15 wherein the optical signal source is configured to output an optical signal in the form of a continuous optical output. 17. An optical switch according to claim 15 wherein the optical signal source is configured to output an optical signal in the form of an optical pulse comprising either a plurality of photons, or not more than a single photon. 18. An optical switching assembly comprising an optical switch according to any preceding claim and further comprising an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal. 19. An optical switching assembly according to claim 18 wherein: the output monitoring unit is configured to determine the presence or the absence of the optical output signal according to whether a detected optical output power exceeds a pre-set detection power threshold value wherein the pre-set detection power threshold value exceeds the optical output power detected by the output monitoring unit in the absence of an electrical detection signal by the single-photon detector unit. 20. An optical switching assembly according to any preceding claim when dependent on claims 17 and 18, wherein: the optical signal source is configured to output an optical signal in the form of an optical pulse comprising a pulse temporal width, σ; and, the output monitoring unit is configured to determine the presence of the optical output signal within a monitoring time interval, Δt , which has a duration not exceeding the pulse temporal width, ∆ t ≤ σ. 21. An optical switching assembly according to any preceding claim when dependent on claim 18, wherein the optical resonator comprises an optical loss rate,

, and the optical switching assembly further comprises an output monitoring unit responsive to the presence of an optical output signal from the optical switch to generate a detection signal, wherein the output monitoring unit is configured to determine the presence of the optical output signal within a monitoring time interval, Δt , which has a duration not exceeding the inverse of the optical loss rate, , such that

.

22. An optical switching assembly according to any of claims 18 to 21 wherein the output monitoring unit comprises: a further optical signal input port for receiving a further optical input signal; an optical transformer unit optically coupled to the further optical signal input port and configured to apply a pre-set optical transformation to the further optical input signal conditional on generation of a detection signal thereby to generate a transformed optical signal; an optical signal output port for outputting the transformed optical signal as an optical output signal. 23. An optical switching assembly comprising: an optical switch for receiving an optical signal of a given optical signal frequency from an optical signal source and for outputting the optical signal, the optical switch comprising; an optical router configured for routing the optical signal conditional on the detection of a single photon, the optical router comprising: an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switch further comprises: a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; wherein the optical modulator is configured to modulate the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to suppress the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch; an output monitoring unit responsive to the presence of said optical output signal output from the optical switch to generate a detection signal, the output monitoring unit further comprising: a further optical signal input port for receiving a further optical input signal; an optical transformer unit optically coupled to the further optical signal input port and configured to apply a pre-set optical transformation to the further optical input signal conditional on generation of a detection signal thereby to generate a transformed optical signal; and, an optical signal output port for outputting the transformed optical signal as an optical output signal. 24. An optical switching assembly according to claim 22 or 23 wherein the pre-set optical transformation comprises one or more of: a transformation to a single-mode optically squeezed state; a transformation to a two-mode optically squeezed state; a transformation to an optically phase-shifted quantum state; a transformation to a displaced quantum state. 25. An integrated photonic circuit comprising an optical switch or an optical switching assembly according to any preceding claim. 26. An optical switching method for switching an optical signal of a given optical signal frequency from an optical signal source by routing the optical signal conditional on the detection of a single photon, the method comprising: providing an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; providing an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; providing an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switching method further comprises: providing a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, providing an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; and, by the optical modulator, modulating the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to either: (a) change from being a resonance bandwidth that includes the optical signal frequency to being a resonance bandwidth that excludes the optical signal frequency thereby to permit the optical input signal to be transmitted to the output optical waveguide part as a non-resonant optical output signal for output from the optical switch; or, (b) change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to permit the optical input signal to be transmitted to the output optical waveguide part as a resonant optical output signal for output from the optical switch. 27. An optical switching method for switching an optical signal of a given optical signal frequency from an optical signal source by routing the optical signal conditional on the detection of a single photon, the method comprising: providing an optical resonator configured to resonate at resonant optical frequencies within a resonance bandwidth determined by a refractive index of the optical resonator; providing an input optical waveguide part optically coupled to the optical resonator and operable to receive the optical input signal from the optical signal source as an input to the optical router; providing an output optical waveguide part optically coupled to the input optical waveguide part and to the optical resonator, and operable to receive the optical signal for output from the optical router; wherein the optical switching method further comprises: providing a single-photon detector unit configured to output an electrical detection signal in response to detection of a single photon; and, providing an optical modulator coupled to the single-photon detector and configured to output an electrical modulation signal in response to the electrical detection signal; and, by the optical modulator, modulating the refractive index of the optical resonator using the electrical modulation signal so as to modulate the resonance bandwidth to change from being a resonance bandwidth that excludes the optical signal frequency to being a resonance bandwidth that includes the optical signal frequency thereby to suppress (e.g., prevent) the optical input signal from being transmitted to the output optical waveguide part as an optical output signal for output from the optical switch; wherein the optical switching method further comprises: generating a detection signal in response to the presence of said optical output signal from the optical switch; providing a transformation unit and thereat receiving a further optical input signal; by the transformation unit, applying a pre-set optical transformation to the further optical input signal conditional on generation of said detection signal thereby to generate a transformed optical signal; and, outputting the transformed optical signal. 28. An optical switching method according to claim 26 or 27 wherein the pre-set optical transformation comprises one or more of: a transformation to a single-mode optically squeezed state; a transformation to a two-mode optically squeezed state; a transformation to an optically phase-shifted quantum state; a transformation to a displaced quantum state.