Noise Mitigation in Quantum-Enhanced Fiber Optic Gyroscopes

Abstract

We analyze noise in a quantum-enhanced fiber optic gyroscope (FOG), focusing on one of the leading sources of phase uncertainty—uncorrelated photon saturation. Taking a squeezed state input as a source for N00N states, we compute the uncorrelated false coincidence counts at the optimal phase bias and determine an upper limit to the squeezed amplitude $\xi$ which allows for sub-shot noise precision. As examples, we apply parameters of present-day quantum FOG experiments and determine the maximum possible precision enhancement based on their respective $\xi$ and optimal phase bias points. With the aim of supporting future FOG setups with higher N00N state fluxes, our result highlights the need to transition to multimode states to bypass the $\xi$ limitation, such as photon pairs generated by the dynamical Casimir effect.

1. Introduction

The super-resolution of entangled photonic N00N states [1] offers an improved phase sensitivity over classical optical instruments. Originally posed as a photonic analogy to the de Broglie wavelength [2], in which the effective wavelength governing interference is reduced by the entangled photon order, the N00N state interference exhibits resolutions which are inaccessible from the classical lens. A notable example is the photon deposition resolution exploited for lithography [3].

Our focus is the fiber optic gyroscope (FOG) [4,5]. Building upon approaches to a quantum-enhanced Sagnac interferometer [6,7], recent experiments applied polarization and path-entangled $N=2$ photon order N00N states to FOGs, demonstrating the sought-after Sagnac phase super-resolution. The enhanced resolution permits sub-shot noise precision, which improves with the entangled photon number.

Sub-shot noise precision means that quantum FOGs surpass their classical counterparts when operating at the same photon flux. However, there remains a sizable gap between the quantum and classical FOG fluxes (MHz vs. ${10}^{13}$/s). Among the obstacles that prevent higher flux $N=2$ N00N states from being reached is the saturation of the signal with uncorrelated photons. Alternatively, higher $N>2$ N00N state orders as high as $N=6$ [8,9] have been accomplished, although here too there remains the complication of low fluxes in such post-selected combinations of quantum and classical light.

In this work, we analyze the noise arising from uncorrelated photon saturation, extracting a nontrivial dependence on the squeezed amplitude. This allows us to improve the phase uncertainty estimation and refine the optimal domains of phase bias for which sub-shot noise precision is possible.

We give a brief overview of the N00N state FOG in Section 2. In Section 3.1, we look at the phase uncertainty and insert a squeezed state to determine false coincidence counts and the resulting upper limit to the squeezed amplitude in Section 3.2. In Section 3.3, we apply the parameters of today’s experimental quantum FOGs [10,11]. As an outlook, we discuss an upcoming quantum FOG experiment, along with prospects for higher-order and flux photonic N00N states.

2. Sub-Shot Noise Resolution

We recall the Sagnac–Laue phase phase acquired between two counter-propagating beams in a fiber coil [5]

$$\begin{array}{c}\hfill {\varphi }_{\mathrm{SL}}={\displaystyle \frac{4\pi \mathsf{\Omega }Lr}{\lambda c}},\end{array}$$

(1)

with L representing fiber length, r representing the coil radius, and $\mathsf{\Omega }$ representing the angular velocity. Classical FOG measurements of Equation (1) are limited to (Poissonian) shot noise precision

$$\begin{array}{c}\hfill \mathsf{\Delta }{\varphi }_{\mathrm{SN}}=1/\sqrt{M},\end{array}$$

(2)

where M is the number of photons measured.

Recently, a polarization-entangled N00N state FOG experiment surpassed the shot noise limit by exploiting photon-number dependent phase resolution [10]:

$$\begin{array}{c}\hfill \left|{\psi }_{\mathrm{N}00\mathrm{N}}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}(\left|N0\right.〉-e^{iN({\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}})}\left|0N\right.〉).\end{array}$$

(3)

The photon order N enhancement of the Sagnac phase sensitivity enters the coincidence count as [10,11]

$$\begin{array}{c}\hfill M_{\mathrm{cc}}({\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}})={\displaystyle \frac{M_{\mathrm{N}00\mathrm{N}}}{2}}\left(1+cos\left(N({\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}})\right)\right),\end{array}$$

(4)

where $M_{\mathrm{cc}}$ is the number of coincident counts and $M_{\mathrm{N}00\mathrm{N}}$ is the number of N00N state pairs delivered to detectors within a time window to be specified below. Throughout this work, total photon number M and coincidence counts $M_{\mathrm{cc}}$ and $M_{\mathrm{N}00\mathrm{N}}$ label unitless quantities i.e., flux times measurement time.

The N N00N state order gives an effective enhancement to the total photon number which reduces the shot noise phase uncertainty:

$$\begin{array}{c}\hfill \mathsf{\Delta }{\varphi }_{\mathrm{QM}}=\mathsf{\Delta }{\varphi }_{\mathrm{SN}}/\sqrt{N}=1/\sqrt{NM},\end{array}$$

(5)

where ‘QM’ labels the quantum-enhanced shot noise resolution. Equation (5) possesses a factor $\sqrt{N}$ smaller phase uncertainty compared to the classical case Equation (2).

Considering the angular velocity at which the phase uncertainty is equal to the Sagnac phase: ${\mathsf{\Omega }}_{\min }=\mathsf{\Omega }({\varphi }_{\mathrm{SL}}=\mathsf{\Delta }{\varphi }_{\mathrm{QM}})$ is

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\min }={\displaystyle \frac{\lambda c}{4\pi Lr\sqrt{NM}}.}\end{array}$$

(6)

This sets an order of magnitude regime for the lower limits to angular rotation sensitivity. Taking for example a 1 km fiber loop with 40 cm radius, a wavelength of $\lambda =1550$ nm, $M={10}^6$ photons measured in a given time interval, and an $N=2$ order N00N state, we have ${\mathsf{\Omega }}_{\min }=6.5\times {10}^{-5}$ rad/s. It is in this low-intensity regime that the N00N state enhancement makes a decisive difference—we recall e.g., Earth’s rotation rate of $7.29\times {10}^{-5}$ rad/s. Transitioning to higher intensities however requires that we revisit noise sources—in the next section we look at uncorrelated photon signals.

3. Uncorrelated Photon Noise

3.1. Phase Uncertainty

We consider the uncertainty in a N00N state-enhanced phase measurement. As an addition to the variance in the desired signal coincidence counts in Equation (4), we have spurious coincidences due to uncorrelated photons which produce a Sagnac + bias phase-dependent uncertainty. From here on, we write ${\varphi }_{\mathrm{SL}}$, implicitly labeling the full phase including the bias $({\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}})$.

We consider an experiment run time $t_{\mathrm{meas}}$ spanning many detection time windows (${\tau }_{\mathrm{detector}}$). The number of false counts scales with the probability $P_{\mathrm{cc}-\mathrm{false}}$ of a single spurious count within ${\tau }_{\mathrm{detector}}$:

$$\begin{array}{c}\hfill M_{\mathrm{cc}-\mathrm{false}}=P_{\mathrm{cc}-\mathrm{false}}{\displaystyle \frac{t_{\mathrm{meas}}}{{\tau }_{\mathrm{detector}}}},\end{array}$$

(7)

and similarly the signal N00N state pairs delivered

$$\begin{array}{c}\hfill M_{\mathrm{N}00\mathrm{N}}=P_{\mathrm{N}00\mathrm{N}}{\displaystyle \frac{t_{\mathrm{meas}}}{{\tau }_{\mathrm{detector}}}.}\end{array}$$

(8)

Combining the two, we have a spurious count which adds to Equation (4): $M_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)\to M_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)+M_{\mathrm{cc}-\mathrm{false}}$. The false count contribution can be extracted from the dark fringes ($N{\varphi }_{\mathrm{SL}}=\pm \pi ,\pm 3\pi ,\cdots$) of coincidence count fitting functions and modeled by a limited visibility [10,11,12].

The object of interest is the deviation in the number of photons delivered during $t_{\mathrm{meas}}$, which was shown in [12] to produce a nontrivial optimization of the phase bias angle. The deviation combines the signal photons in Equation (4) and false coincidences: for $P_{\mathrm{N}00\mathrm{N}},P_{\mathrm{cc}-\mathrm{false}}\ll 1$,

$$\begin{array}{c}\hfill \mathsf{\Delta }{M}_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)=\sqrt{M_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)+M_{\mathrm{cc}-\mathrm{false}}}.\end{array}$$

(9)

$\mathsf{\Delta }{M}_{\mathrm{cc}}$ can be recast as a phase uncertainty $\mathsf{\Delta }{\varphi }_{cc}$ accompanying the Sagnac–Laue phase in the coincidence count expression Equation (4):

$$\begin{array}{c}\mathsf{\Delta }{M}_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)=M_{\mathrm{cc}}({\varphi }_{\mathrm{SL}}+\mathsf{\Delta }{\varphi }_{cc})-M_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right).\end{array}$$

(10)

We solve for $\mathsf{\Delta }{\varphi }_{cc}$ exactly without expanding, as is usually done to first-order derivatives:

$$\begin{array}{c}\mathsf{\Delta }{\varphi }_{cc\mp }=\pm {\displaystyle \frac{1}{N}}\mathrm{acos}\left({\displaystyle \frac{2\mathsf{\Delta }{M}_{\mathrm{cc}}}{M_{\mathrm{N}00\mathrm{N}}}}+cos\left(N{\varphi }_{\mathrm{SL}}\right)\right)+{\displaystyle \frac{2\pi n}{N}}-{\varphi }_{\mathrm{SL}},\end{array}$$

(11)

where $n=0,\pm 1,\pm 2,\cdots$ corresponds to the range $N{\varphi }_{\mathrm{SL}}=[(2n-1)\pi ,(2n+1)\pi ]$. Accounting for symmetry, the $\mathsf{\Delta }{\varphi }_{cc+}$ solutions are positive valued and valid within $N{\varphi }_{\mathrm{SL}}=[(2n-1)\pi ,2n\pi ]$, while $\mathsf{\Delta }{\varphi }_{cc-}$ are negative and valid within $N{\varphi }_{\mathrm{SL}}=[2n\pi ,(2n+1)\pi ]$. We also note that $\mathsf{\Delta }{\varphi }_{\mathrm{cc}}^{-2}$ provides the Fischer information [12].

We recall that $\mathsf{\Delta }{M}_{\mathrm{cc}}$ describes the distribution of coincidence counts which one needs to average over. For completeness, we note that deviations in counts on the order of $\mathsf{\Delta }{M}_{\mathrm{cc}}$ take on both signs. We thus consider absolute values $\mathsf{\Delta }{\varphi }_{cc}=\left|\mathsf{\Delta }{\varphi }_{cc\mp }\right|$, accounting for the $\mathrm{sgn}\left(\mathsf{\Delta }{M}_{\mathrm{cc}}\right)$ dependence in $\mathsf{\Delta }{\varphi }_{cc}$ by using the sign which produces the larger noise contribution at the optimal bias point.

We evaluate phase uncertainty in the well-defined phase regions where $\mathsf{\Delta }{\varphi }_{cc}$ as defined in Equation (11) is real: $|{\displaystyle \frac{2\mathsf{\Delta }{M}_{\mathrm{cc}}}{M_{\mathrm{N}00\mathrm{N}}}}+cos\left(N{\varphi }_{\mathrm{SL}}\right)|<1$. Within the range $(0,\pi /N)$, this condition implies

$$\begin{array}{c}\hfill {\varphi }_{\max }<{\varphi }_{\mathrm{SL}}<\pi /N-{\varphi }_{\min },\end{array}$$

(12)

where ${\varphi }_{\max }$ is the offset from the maxima ($N{\varphi }_{\mathrm{SL}}=2\pi n$) and ${\varphi }_{\min }$ from the minima ($N{\varphi }_{\mathrm{SL}}=\pi (2n+1)$) at which the phase uncertainty becomes well defined. For ${\varphi }_{\min }\ll 1$ and ${\varphi }_{\max }\ll 1$, the nested solutions reduce to $N{\varphi }_{\min }\approx \sqrt{(1/M_{\mathrm{N}00\mathrm{N}})\left(\sqrt{4M_{\mathrm{cc}-\mathrm{false}}+1}+1\right)}$ and $N{\varphi }_{\max }=2/\sqrt{M_{\mathrm{N}00\mathrm{N}}}$. The undefined regions repeat periodically at the maxima ($|N{\varphi }_{\mathrm{SL}}-2\pi n|<{\varphi }_{\max }$) and minima ($|N{\varphi }_{\mathrm{SL}}-\pi (2n+1)|<{\varphi }_{\min }$) of the coincidence count expression Equation (4), corresponding to the divergent uncertainties in the first-order derivative expansion of Equation (10) ([12]). With an exact solution, however, a conveniently chosen fitting where the fringe visibility is slightly exaggerated can provide a well-defined phase uncertainty—we do not consider this here, since it comes at the cost of phase measurement precision.

The phase uncertainties near the divergent points are well above the shot noise limit. Near the minima, where uncorrelated counts $M_{\mathrm{cc}-\mathrm{false}}$ provide the dominant contribution, $\mathsf{\Delta }{\varphi }_{cc}(\pi /N-{\varphi }_{\min })=(\sqrt{2}-1){\varphi }_{\min }$. This is a substantially larger phase uncertainty than the shot noise Equation (2): ${\varphi }_{\min }/{\varphi }_{\mathrm{SN}}=\sqrt{2/N}{\left(M_{\mathrm{cc}-\mathrm{false}}\right)}^{1/4}$ (unless $N\gg 2$, a point we return to in Appendix B). We are thus interested in the uncertainty at the optimal phase bias between the singular points, which we now determine as a function of squeezed amplitude and experimental parameters.

3.2. Squeezed State Input

Squeezed states are the prevalent source of N00N states in quantum sensing experiments. We focus here on $N=2$ N00N state interferometry, evaluating the noise due to random in-phase false coincidence counts arising from 4 and higher photon number states generated in the down-conversion process. Extensions to $N>2$-order N00N states are discussed in Appendix B.

We consider for simplicity the squeezed vacuum state

$$\begin{array}{cc}\hfill \left|\xi \right.〉=& e^{({\xi }^{*}{\widehat{a}}^2-\xi {\widehat{a}}^{†2})/2}\left|0\right.〉\hfill \\ \hfill =& {\displaystyle \mathsf{\sum }_{m=0}^{\infty }}{\displaystyle \frac{\sqrt{\left(2m\right)!}}{2^{m}m!}}{\displaystyle \frac{(-e^{i \arg \left(\xi \right)}{tanh\left|\xi \right|)}^m}{\sqrt{cosh\left|\xi \right|}}}\left|2m\right.〉,\hfill \end{array}$$

(13)

where the photon number probability

$$\begin{array}{c}\hfill P_{2m}^{\xi }={\left|〈2m|\xi 〉\right|}^2={\displaystyle \frac{\left(2m\right)!}{{(2^{m}m!)}^2}}{\displaystyle \frac{{(tanh|\xi \left|\right)}^{2m}}{cosh\left|\xi \right|}},\end{array}$$

(14)

and the average photon number

$$\begin{array}{c}\overline{n}=\left.〈\xi \right|{\widehat{a}}^{†}\widehat{a}\left|\xi \right.〉={sinh}^2\left|\xi \right|.\end{array}$$

(15)

Since today’s SPDC sources produce two-photon-state fluxes on the order of ${10}^6$ pairs per second, we consider the regime where two-photon states dominate the SPDC output. The resulting average photon number is $\overline{n}\sim {10}^{-4}$ ($\left|\xi \right|\sim 0.01$) over a typical single-photon counter time resolution window of ${\tau }_{\mathrm{detector}}=\mathcal{O}\left(100\mathrm{ps}\right)$, where we consider coherence times $\lesssim {\tau }_{\mathrm{detector}}$. Comparing the four-photon and two-photon probabilities,

$$\begin{array}{c}\hfill {\displaystyle \frac{P_4^{\xi }}{P_2^{\xi }}}={\displaystyle \frac{3}{4}}{tanh}^2\left|\xi \right|\sim 7.5\times {10}^{-5},\end{array}$$

(16)

and similarly $P_6^{\xi }/P_4^{\xi }=(5/6){tanh}^2\left|\xi \right|$. Given that, in general, $P_{2m}^{\xi }\gg P_{2m+2}^{\xi }$ for this perturbative in the $\xi$ regime, we focus on $P_4^{\xi }$ and neglect $P_6^{\xi }$ and higher orders.

The $|4\rangle$ state produces predominantly spurious coincident counts with random phase at the detection. Due to loss, a negligible amount of four-photon N00N states with well-defined phase relations survive. The majority of loss occurs after the optical elements used to generate the N00N states, e.g., in the Sagnac interferometer, producing uncorrelated two- and three-photon states responsible for false coincidence counts. For a discussion of loss at earlier stages in the experiment, see Appendix C.

To estimate the false coincidence probability, we consider the four-photon state undergoing loss via beamsplitting with the vacuum environment. The probability of a two- or three-photon byproduct is

$$\begin{array}{c}\hfill P_{\mathrm{cc}-\mathrm{false}}=\left(6T^2{(1-T)}^2+4T^3(1-T)\right)P_4^{\xi },\end{array}$$

(17)

where T labels the transmission coefficient. The two-photon contribution (the first term in parenthesis) dominates for a transmission of $T\sim 0.1$ relevant to FOG setups. In comparison, we show the probability of the desired $N=2$ N00N state reaching the detector scales with transmission squared:

$$\begin{array}{c}\hfill P_{\mathrm{N}00\mathrm{N}}=T^2{P}_2^{\xi }.\end{array}$$

(18)

Applying the above probabilities to Equations (7) and (8) gives the uncorrelated and signal coincidence counts.

Following the steps in Section 3.1, we evaluate the phase uncertainty in Equation (11) for different squeezed amplitudes, transmission coefficients, and measurement times. We find in Figure 1 that the minimum uncertainties at the optimal phase bias points grow with the squeezed amplitude. Note that we consider transmission coefficients stemming from optical loss, and not single-photon detector quantum efficiencies, which we assume are large (e.g., >95%, as is the case in superconducting nanowires).

We find an upper limit to squeezed amplitude for which sub-shot noise precision is possible. To show this in more detail, we plot the minima in uncertainty as a function of $\left|\xi \right|$ in Figure 2. This limit grows with transmission ($\left|\xi \right|=0.181$ at $T=0.1$, and $\left|\xi \right|=0.406$ at $T=0.75$), yet is largely independent of the measurement time and detection time window. Note that at larger transmissions we approach values of $\left|\xi \right|$, where higher photon number contributions (>4) to the false coincidence counts start to matter. These contributions further decrease the permissible $\left|\xi \right|$ range, making this result a conservative noise saturation estimate. A similar estimate can be derived for coherent state setups which rely on post-selection schemes; see Appendix A.

As an alternative to solving for optimal bias points numerically as in the above examples, one may also obtain an analytic expression via perturbative expansion of Equation (11); see [12]. We plug Equations (4) and (9) into the phase uncertainty $\mathsf{\Delta }{\varphi }_{cc}^2\approx \mathsf{\Delta }{M}_{\mathrm{cc}}^2\left({\varphi }_{\mathrm{SL}}\right)/{\left({M^{\prime }}_{\mathrm{cc}}\left({\varphi }_{\mathrm{SL}}\right)\right)}^2$, and differentiate with respect to ${\varphi }_{\mathrm{SL}}$ to find the minima: within $(0,\pi /N)$ we have ${\varphi }_{\mathrm{SL}-\min }\equiv N^{-1}\mathrm{acos}(-B/2+\sqrt{B^2/4-1})$, where $B=4M_{\mathrm{cc}-\mathrm{false}}/M_{\mathrm{N}00\mathrm{N}}+2$. This is nearly identical to the numerical result at $T=0.1$, and accurate on the percent order at $T=0.75$.

3.3. Experimental Implications

We apply the above phase uncertainty considerations to recent quantum FOG experiments. We consider first the $N=2$ N00N state FOG setup in [11], the first path entanglement-based FOG to resolve angular velocities below Earth’s rotation rate. This setup reported a transmission of $T=0.1$, 4000 N00N state coincidence counts per second amounting to $M_{\mathrm{N}00\mathrm{N}}=7.2\times {10}^6$ over $t_{\mathrm{meas}}=1800$ s, and a detector timing resolution ${\tau }_{\mathrm{detector}}=156$ ps.

Following the steps in Section 3, these experimental parameters amount to a squeezed amplitude of $\left|\xi \right|=0.011$. Applying the results of Figure 1, this is within the range of squeezed amplitudes for which sub-shot noise phase uncertainty is possible. Additional dark photon counts beyond the higher-photon-number sourced flux considered above (e.g., sub kHz count per second thermal background) produce a negligible number of false coincidence counts and are easily mitigated by edgepass or bandpass filters.

Recalling that phase super-resolution with a factor $N=2$ enhancement was achieved in [11], sub-shot noise precision is also possible for these experimental parameters. We find $\mathsf{\Delta }{\varphi }_{\mathrm{cc}}/\mathsf{\Delta }{\varphi }_{\mathrm{SN}}<1$ to be accessible within the (periodically repeating) range $\pi /4<{\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}}<0.493\pi$. At the optimal point ${\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}}=0.453\pi$, we have $\mathsf{\Delta }{\varphi }_{\mathrm{cc}}/\mathsf{\Delta }{\varphi }_{\mathrm{SN}}=0.723$, close to the ideal ($N=2$ N00N state-enhanced) shot noise reduction factor of $1/\sqrt{2}=0.707$.

We turn to an earlier experiment [10] which pursued a similar $N=2$ N00N state FOG measurement. Here, $t_{\mathrm{meas}}$ = 20 ms data points were reported, each collecting 1956 photons, and a similar transmission of $T\sim 0.1$. We assume that we have the same 156 ps detection timing window as in the above example.

These experimental parameters give $\left|\xi \right|=0.039$, also within the range of squeezed state amplitude for which sub-shot noise precision is possible. In this case, $\mathsf{\Delta }{\varphi }_{\mathrm{cc}}/\mathsf{\Delta }{\varphi }_{\mathrm{SN}}<1$ is accessible within $\pi /4<{\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}}<0.467\pi$. At the optimal point ${\varphi }_{\mathrm{SL}}+{\varphi }_{\mathrm{bias}}=0.403\pi$, we have $\mathsf{\Delta }{\varphi }_{\mathrm{cc}}/\mathsf{\Delta }{\varphi }_{\mathrm{SN}}=0.786$. Impressively, a factor 0.87 reduction in the shot noise was achieved in [10] at the optimal phase bias point, close to the smallest achievable uncertainty ratio of 0.786 at this $\left|\xi \right|$.

4. Conclusions

We have computed the N00N state-based phase measurement uncertainty as a function of an input squeezed state amplitude $\xi$. We found an upper limit to $\left|\xi \right|$ allowing for sub-shot noise precision. The crucial component is the uncorrelated photon flux as a function of $\xi$—without this contribution, an optimal phase bias point and achievable precision cannot be established theoretically, potentially leading one to overestimate the accessible quantum enhancement to the shot noise resolution.

We applied this analysis to two recent $N=2$ N00N state FOG experiments. In one case [10], we determined the minimum phase uncertainty to be larger than the expected (ideal) factor $\sqrt{2}$ shot noise reduction. We found this corrected form to be close to the measured sub-shot noise resolution achieved in [10]. In the second experiment [11], a similar bound to precision was obtained. Our result thus serves to refine the analysis of both prior and upcoming experimental measurements, allowing one to quantify the achievable sub-shot noise precision and select the suitable bias angle to exploit it.

We recall that the analysis here is based on the assumption that all loss byproducts of the higher (than 2) photon number states have uncorrelated phase, since most loss occurs after the N00N states are generated. However, a fraction of loss still occurs in between the SPDC squeezed state source and the N00N state generation, in which case some residual phase bias dependence may remain—this is discussed in Appendix C.

Our analysis will be applied to an upcoming quantum FOG experiment detailed in [13], building on the work of [10,11]. The projected 6.65 dB loss ($T=0.216$, noting a typo in Table 3 of [13]), provides a competitive upper bound on $\left|\xi \right|$, allowing for sub-shot noise precision angular velocity measurements below Earth’s rotation rate.

In future experiments one will in principle need higher $N=2$ N00N state fluxes than the bounds on the $\left|\xi \right|$ permit for an SPDC source. A candidate for bypassing this limit is a multi-mode squeezed state [14]. Overcoming system complexity, e.g., due to coupling losses with multiple fibers, the continuous variable entanglement over multiple spatial modes can reduce shot noise beyond the $\sqrt{2}$ enhancement via $N=2$ N00N states.

One may also consider the dynamical Casimir effect [15,16]. In this case the 4-photon and higher order states, which are otherwise detrimental in the SPDC-based setup, possess frequencies which sum to the medium modulation rate (or mechanical motion frequency). The resulting false coincidences can be easily filtered out, since there is negligible overlap with the two-photon sector frequencies. Overcoming the challenge of achieving high photon pair fluxes in the optical or infrared domain at room temperature, a path to quantum advantage is possible.