Enhanced Whispering Gallery Mode Phase Shift using Indistinguishable Photon Pairs

arXiv:2404.17409 · quant-ph, physics.optics · Submitted 2024-04-26 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Enhanced Whispering Gallery Mode Phase Shift using Indistinguishable Photon Pairs".

Mira: Here's the gist of the paper. The gist: The paper presents a theoretical investigation showing that an input state of an indistinguishable photon pair leads to a bimodal coincidence transmission spectrum when coupled to a whispering gallery mode resonator, resulting in at least a four-fold improvement in the signal-to-noise ratio (SNR) for WGM sensing Quantum Optical Model for WGM Coupling The model uses the operator valued phasor addition (OVPA) approach described in Ref.

12: for a quantum optical treatment of coupling an optical mode in a waveguide to a ring resonator.

Mira: The coupling to the resonator is characterized by the amplitude coupling coefficient κ and amplitude reflection coefficient r, where κ2 + r2 = 1.

Kai: The mode in the ring resonator has an amplitude transmission α and phase θ per round trip.

Title and authors: Kai: So let's talk about who wrote this—callum jones, antonio vidiella-barranco, jolly xavier, and frank vollmer. They’re bringing together the hardware side and the theory here.

Mira: I think it’s interesting because they are taking a fairly complex setup involving WGM resonators and coupling them to a Mach-Zehnder interferometer, which is the context for this kind of sensing.

Lev: From my side, I’m looking at how this relates to what we can actually build. It involves these coupling coefficients κ and r where κ squared plus r squared equals one, so that’s the basic physics of coupling you have to manage in real hardware.

Kai: Exactly, Lev. It shows the math behind what you have to engineer in your setup when you are trying to get a good signal out of it.

Mira: And they are using a specific model for this, the operator valued phasor addition approach from reference twelve which is how they treat the quantum optical part of coupling a mode into a resonator <ref:2404.17409#pg2>.

The paper's summary: Kai: Now let's get into what the paper actually says in its summary. It talks about taking that input state and seeing how it plays out inside the interferometer after coupling to the WGM.

Mira: Basically, they describe this two-photon N00N state that appears due to Hong-Ou-Mandel interference at the first beamsplitter when you feed in those indistinguishable photons.

Lev: That N00N state is key because it introduces a double phase shift, which means the phase difference between the two photons in modes five and six is twice what you would normally expect from just coupling to a WGM resonator.

Kai: So that doubled phase shift is what makes their results interesting; it changes how the final output state looks when you act on those modes with an output beamsplitter.

Mira: They show the final output state equation, which involves terms depending on A over t squared, and specifically highlights where that double phase shift comes into play in the modes five and six.

The paper's improvements: Kai: The main improvement they highlight is comparing three measurement cases: 'Classical WGM', 'Classical WGM-MZI', and then the 'Entangled WGM-MZI' case.

Mira: For the entangled case, they give us a coincidence probability formula that looks like PCoinc(ω) equals one over four A over t squared plus one.

Lev: What we see is that in the overcoupled regime, where alpha equals zero point nine nine nine seven and r equals zero point nine nine nine six, the entangled WGM-MZI has reduced noise in the wavelength shift compared to those two classical cases you mentioned earlier <ref:2404.17409#pg1>.

Kai: And they say that for this overcoupling region, the SNR enhancement factor for the entangled WGM-MZI relative to a classical WGM is consistently higher and it peaks at that critical coupling condition.

Mira: They also look at the uncertainty delta omega, which shows that for photon number N greater than one hundred five the resonance frequency noise dominates and stays constant even as the optical power increases.

Conclusion: Kai: So to wrap up this paper, they show that using an indistinguishable photon pair input state can lead to a four-fold improvement in SNR for WGM sensing.

Mira: And they found that this advantage is most robust in the overcoupled regime, especially when considering the spectral width of the input photons.

Lev: From a hardware standpoint, they suggest that the optimum enhancement is a trade-off between having noise much lower than your dynamic range and having a more overcoupled resonator.

Kai: It points toward needing to tune those parameters carefully based on what you want to measure, whether it’s temperature or pressure sensing.

Mira: This work suggests that using these quantum input states can really help improve the performance of WGM sensors in applications like biosensing or single molecule detection.

Lev: And they point out a limitation, though, that the most robust SNR advantage is obtained when you are in the overcoupling regime, which might not always be achievable depending on your experimental constraints.

Kai: So yeah, this paper shows a path forward for getting better sensitivity in these specific quantum sensing applications.

Living Systems Institute, University of Exeter · Gleb Wataghin Institute of Physics - University of Campinas · SeNSE, Indian Institute of Technology Delhi

quant-ph, physics.optics

Submitted: 2024-04-26

Updated: 2026-10-22

Comments: Withdrawn due to inconsistencies identified during verification of the theoretical framework. Specifically, the treatment of the environment modes in Eq. 6 and an incorrectly normalised state in Eq. 8. Correcting this significantly changes the results and the reported sensing advantage no longer occurs for measuring WGM resonance shifts

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist: The paper presents a theoretical investigation showing that an input state of an indistinguishable photon pair leads to a bimodal coincidence transmission spectrum when coupled to a

Key concepts

Indistinguishable Photon Pair Input State: This input state is created via Hong-Ou-Mandel interference, meaning two photons entering a beamsplitter interfere perfectly. This quantum property leads to a two-photon N00N state inside the interferometer, which doubles the phase shift compared to classical measurements.
Overcoupled Regime: This refers to operating the resonator where its coupling strength is high (e.g., $\alpha = 0.9997$). In this regime, the system exhibits a more robust signal-to-noise ratio enhancement for sensing applications compared to other coupling conditions.
Signal-to-Noise Ratio (SNR) Enhancement: The quantum measurement using entangled photon pairs significantly reduces noise in wavelength shift measurements. For specific conditions, such as when the input photon pair spectral width is 0.1 times the WGM linewidth, an SNR enhancement factor of up to four can be achieved.
WGM Coupling Parameters ($\alpha$ and $r$): These parameters describe how strongly the waveguide mode couples to the ring resonator. $\alpha$ relates to amplitude transmission, while $r$ is the amplitude reflection coefficient. The relationship between these parameters dictates whether the system operates near critical coupling or in an overcoupled regime.

Terminology

Signal-to-Noise Ratio (SNR) — 1× in this paper · explained in 6 episodes

Summary

The gist: The paper presents a theoretical investigation showing that an input state of an indistinguishable photon pair leads to a bimodal coincidence transmission spectrum when coupled to a whispering gallery mode resonator, resulting in at least a four-fold improvement in the signal-to-noise ratio (SNR) for WGM sensing

Quantum Optical Model for WGM Coupling

The model uses the operator valued phasor addition (OVPA) approach described in Ref. [12] for a quantum optical treatment of coupling an optical mode in a waveguide to a ring resonator. The coupling to the resonator is characterized by the amplitude coupling coefficient κ and amplitude reflection coefficient r, where κ2 + r2 = 1. The mode in the ring resonator has an amplitude transmission α and phase θ per round trip. This phase θ is dependent on the detuning δ between the frequency of the waveguide mode and a WGM, via θ = 2πRnδ/c. The input-output relation for the waveguide mode coupled to the WGM resonator is given by aˆout = t(ω)ˆain + Fˆ(ω). The complex amplitude transmission past the WGM resonator at frequency ω, t(ω), is defined as t(ω) = r − αe(iθ)/(1 − r∗αe(iθ)).

Indistinguishable Photon Pair Input State

For an indistinguishable photon pair input state, the state inside the interferometer is a two-photon N00N state due to Hong-Ou-Mandel (HOM) interference at the first beamsplitter. The output state in modes 5 and 6 after coupling to the WGM is given by ψ5,6⟩ = √2i/2 (t∗(ω))2 A(ω)2, 0⟩5,6+0, 2⟩5,6⊗0⟩env − i (t∗(ω))2 A(ω)1, 0⟩5,6⊗Fˆ† (ω)0⟩env+ i/2 (t∗(ω))2 A(ω)0, 0⟩5,6⊗Fˆ†2 (ω)0⟩env. The expectation values of noise operators Fˆ are evaluated using the commutation relation for the output mode amplitudes and using Equation 6, as described by Alsing et al. [12]. The double phase shift due to the two-photon N00N state comes in because now due to the factor (t∗(ω))2, the phase difference between photons in modes 5 and 6 is twice the usual phase shift from coupling to a WGM resonator.

Enhanced Signal-to-Noise Ratio

The paper compares three measurement cases: 'Classical WGM', 'Classical WGM-MZI', and 'Entangled WGM-MZI'. For the entangled WGM-MZI case, the coincidence probability is given by PCoinc(ω) = 1/4 A(ω) (t∗(ω))2 + 1. The change we would like to detect for sensing experiments is a shift in the WGM resonance frequency δω0. In the overcoupled regime, where α = 0.9997 and r = 0.9996, the entangled WGM-MZI has reduced noise in the wavelength shift compared to the two classical cases. The SNR enhancement factor for entangled WGM-MZI relative to classical WGM is consistently higher in the overcoupling region and has a peak at the critical coupling condition.

Modelling Signal-to-Noise Ratio Enhancement

The uncertainty ∆omega for the entangled WGM-MZI case is compared against classical cases, showing that for photon number N > 105, the resonance frequency noise dominates and is constant as optical power increases. In the regime where shot noise begins to dominate for N < 104, the uncertainty in wavelength shift increases as √N. The SNR enhancement factor is up to a factor of 4 when the input photon pair spectral width is 0.1 times the WGM linewidth. The most robust SNR advantage is obtained in the overcoupled regime, where there is an enhancement of at least a factor of 4 (for linewidth ratio 0.1).

Experimental Considerations

To realize this experiment, it requires a source of indistinguishable photon pairs showing high HOM interference visibility. The setup might be built with an all-fibre MZI, using a tapered optical fibre to couple photons to e.g. a microsphere, or alternatively by fabricating the MZI with integrated photonics and using a microring resonator. To observe the SNR enhancement, the photon pairs must be spectrally narrow: ideally at least ∼0.1 of the WGM linewidth (typically ∼100 MHz for WGM microsphere sensors). Photon pairs generated by cavity-assisted SPDC seem the most promising technique. The optimum enhancement is achieved when the measurement is in the shot noiselimited regime, and the WGM resonator is overcoupled, and the input photon pair state is spectrally narrow: ∼0.1 times the WGM linewidth.

Discussion

A SNR enhancement in measurements of the WGM resonance shift promises to improve the performance of WGM sensors which are applied to precise measurements in, e.g., temperature [16–18], pressure [19, 20], and biosensing [3, 4] including single molecule detection when combined with optoplasmonic enhancements [5, 6, 21]. The authors showed an advantage to using indistinguishable photon pairs in the shot noise limited sensing regime, with an achievable factor of 4 enhancement in the SNR, taking into account the photon spectral width. The optimum enhancement is a trade-off between having noise much lower than the dynamic range for a more overcoupled resonator, and higher SNR enhancement closer to critical coupling. The limits on r will be determined by experimental constraints.

Appendix A

The dimensionless parameters α, r can be related to these quantities through the following expressions: ∆λ = -λo2 / (4π2/2R ln(αr)) and Q = λo / ∆λ = -4π2/2R λo ln(αr). The WGM resonator radius is R and the resonance wavelength λo.

Appendix B

To meet all these requirements, photon pairs generated by cavity-assisted SPDC seem the most promising, which is a technique already being used for coupling entangled photons to atomic transitions [15, 25] (typically for quantum memories [26]). Refs. [15, 25] report photon pair bandwidths of 7 MHz and 2 MHz, respectively, therefore a bandwidth of < 0.1 of the WGM linewidth is realistic.

Appendix C

For a fair comparison between measurement cases, the average photon number resources⟨N⟩ reaching the WGM resonator should be equal in all three cases. The maximum detected rate for the classical WGM-MZI transmission difference measurement is 2R. A photon number per time bin N was used to calculate the shot noise level in the classical WGM and entangled WGM-MZI cases, and 2N in the classical WGM-MZI case.

Appendix D

The most robust SNR enhancement is obtained in the overcoupling regime. The dynamic range condition also rules out the large SNR enhancements predicted near critical coupling, as did considering the spectral width of the input photons in the main text. The factor of 4 SNR enhancement at linewidth ratio 0.1 is still achieved with values of r for which the 3σ noise level is 40-60% of the dynamic range for the entangled WGMMZI measurement. If we allow the photon spectral width to be arbitrarily narrow we see that in principle a factor of 10 enhancement in the SNR is possible when the 3σ noise level is 60-80% of the dynamic range.

Appendix A (Wavelength Shift)

Appendix B (Experimental)

To realize this experiment, we require a source of indistinguishable photon pairs showing high HOM interference visibility (e.g. from SPDC in a nonlinear crystal such as PPKTP [23, 24]). The setup might be built with an all-fibre MZI, using a tapered optical fibre to couple photons to e.g. a microsphere, or alternatively by fabricating the MZI with integrated photonics and using a microring resonator.

Improvements for AI systems

Improved WGM Sensing for High Precision Measurements

An AI system utilizing this technique can perform wavelength shift measurements with a four-fold improvement in signal-to-noise ratio (SNR) compared to classical methods. This enhancement is achieved by using an indistinguishable photon pair input state, which doubles the WGM phase shift, allowing interference fringes to oscillate at double the frequency of the classical MZI.

Enhanced Single Molecule Detection Sensitivity

The system can detect even smaller conformational changes in biomolecules by leveraging quantum-enhanced WGM sensing. This is possible because the entangled WGM-MZI measurement shows a reduced noise in wavelength shift compared to classical cases when operating in the shot noise limited regime, particularly when the resonator is overcoupled.

Optimized Sensing Conditions Determination

The AI can dynamically determine optimal experimental parameters for WGM sensing by analyzing the SNR enhancement factor across varying coupling conditions (r) and input photon spectral widths. The paper shows that the most robust SNR advantage is obtained in the overcoupled regime, which guides experimental setup choices to maximize performance.