Impact of Non-Unitary Spin Squeezing on Atomic Clock Performance

Spin squeezing is a form of entanglement that can improve the stability of quantum sensors operating with multiple particles, by inducing inter-particle correlations that redistribute the quantum projection noise. Previous analyses of potential metrological gain when using spin squeezing were performed on theoretically ideal states, without incorporating experimental imperfections or inherent limitations which result in non-unitary quantum state evolution. Here, we show that potential gains in clock stability are substantially reduced when the spin squeezing is non-unitary, and derive analytic formulas for the clock performance as a function of squeezing, excess spin noise, and interferometer contrast. Our results highlight the importance of creating and employing nearly pure entangled states for improving atomic clocks.


I. INTRODUCTION
Spin squeezed states (SSSs) [1] offer a path toward entanglement-enhanced quantum sensors by reducing the variance of one spin quadrature. Typically, this potential improvement is quantified in terms of the metrological Ramsey squeezing parameter ξ R [2], defined as min is the smallest variance of any spin quadrature of the state, S is the maximum possible length of the spin vector, and C is the contrast of the complete Ramsey sequence. In this picture of quantum-enhanced Ramsey spectroscopy, spin squeezing reduces the measurement noise variance by a factor of ∆S 2 min S/2 compared to the standard quantum limit (SQL) that can be attained in the absence of entanglement using a coherent spin state (CSS), while the 1 C 2 term accounts for the reduction in squared signal due to interferometer contrast loss. However, the expression for ξ R does not account for other downsides in using SSSs in Ramsey spectroscopy, such as the increase in quantum noise (antisqueezing) in the conjugate spin direction [3].
Single-particle decoherence, i.e. uncorrelated noise between the particles, due to atom loss or spontaneous emission, typically has a more deleterious effect on SSSs than on uncorrelated collective atomic states. In particular, if the coherence properties of an atomic clock are limited by single-particle decoherence, squeezing is found to offer at best a small, and constant with atom number, improvement in ultimate clock stability [4,5]. However, in state-of-the-art optical atomic clocks [6][7][8][9], the dominant noise is not single-particle decoherence but rather phase noise in the local oscillator (LO) laser used to interrogate the narrow atomic transition. Even as LO laser technology improves [10], there are many increasingly narrow atomic clock transitions [11][12][13][14][15] that would still leave LO stability as the primary limit to Ramsey time, and hence precision, in atomic clocks.
In an atomic clock, the atomic phase is used to stabilize the LO phase. While dephasing of the LO does not destroy the quantum correlations between the atoms, it invalidates the assumption that the antisqueezing does not affect the measurement precision [3]. The mechanism by which this happens is illustrated in Figure 1(a). Due to the curvature of the Bloch sphere, part of the antisqueezed quadrature enters into the final S z measurement when the phase deviation φ between the atoms and the LO is non-zero, as in this case the final π 2 rotation of the Ramsey sequence places the SSS away from the equator of the Bloch sphere. Therefore, to avoid this leakage of antisqueezing into the final S z measurement, a squeezed pointing along x is prepared with a quadrature reduced variance ∆S 2 y = S 2 ξ 2 and an increased variance ∆S 2 z = S 2 χ 2 . If the phase deviation φ is non-zero, the final readout of S z becomes sensitive to the antisqueezing χ. (b) and (c) S z distributions as a function of φ for a clock with N = 10 3 spins, using a coherent state and a pure spin squeezed state with ξ 2 = χ −2 = −20 dB, respectively. The S z distribution is narrowed by squeezing only when |φ| is small. clock must operate in a reduced range of |φ|, limiting the Ramsey time τ in the presence of LO dephasing to a smaller value than what would be necessary to merely avoid 2π phase errors. On the other hand, the frequency stability of atomic clocks improves with longer Ramsey time as τ −1 for a single measurement, and as τ −1/2 for repeated measurements over a fixed total duration. Therefore, a conflict exists between obtaining the greatest phase noise suppression by using a short Ramsey time, and maximizing clock frequency stability with a long Ramsey time. Note that this effect does not harm sensors that operate in a fixed bandwidth, such as atomic magnetometers, accelerometers, and gyroscopes.
Here, we extend the model of [3] to derive analytical expressions including non-unitary squeezing, and determine the impact on potential gains in the stability of atomic clocks.
We find that for typical states realized so far in experiments with dilute atomic ensembles amenable to clock applications [26][27][28][29][30], the impact can be severe, potentially wiping out any metrological gain from squeezing. We also analyze the effect of contrast loss, and find that the system is more robust to contrast loss during the squeezing than to contrast loss during the Ramsey time. We conclude that experimental efforts should be in part directed toward reducing excess antisqueezing, since moderate near-unitary squeezing can yield better clock stability than larger non-unitary squeezing.

II. ANTISQUEEZING AND MEASUREMENT PRECISION
We consider an atomic clock based on a SSS consisting of N = 2S spin-1/2 atoms, with phase variance squeezed by a factor of ξ 2 ≤ 1 and population variance increased by a factor χ 2 ≥ ξ −2 . In terms of spin operator variances as shown in Figure 1, this is equivalent to an initial phase squeezed state pointing along S x , with (∆S y ) 2 = S 2 ξ 2 and (∆S z ) 2 = S 2 χ 2 . We quantify the imperfection of the squeezing process through the excess antisqueezing factor A 2 = ξ 2 χ 2 ≥ 1, with equality corresponding to unitary squeezing (at contrast C = 1).
After the Ramsey time τ , the atomic state is displaced in phase relative to the LO by the phase deviation angle φ due to LO noise. Contrast loss during state preparation or the Ramsey sequence arising from single-particle decoherence is neglected for the time being (but is discussed in Figure 5 and Appendix C). The final π/2 rotation in Figure 1(a) results in a distribution for S z that depends on the accumulated phase φ, as shown in Figures   1(b) and 1(c). In a clock, the final measurement of S z is used to estimate φ, and thereby apply feedback to stabilize the LO phase. For squeezed states, we note that for φ = 0, the banana-like shape of the state wrapping around the Bloch sphere causes a leakage of the antisqueezed variance S 2 χ 2 into the final measurement of S z , and therefore a deterioration in the ability to extract the true value of φ [3]. To determine the clock stability, we first calculate the variance (∆φ) 2 of the phase measurement as a function of the squeezing ξ, antisqueezing χ, and phase deviation φ. We then use the derived analytical relation to determine the optimum Ramsey time τ and best attainable clock frequency stability.
The process by which φ is estimated is as follows. Given our knowledge of the initial quantum state, we calculate the conditional probability distribution P (S z |φ) for each value of the true phase deviation φ. With no prior knowledge about φ, the Bayesian estimate of φ, given a particular measured value S z,f of the S z operator at the conclusion of the Ramsey sequence, will be given by the conditional distribution P (φ|S z,f ). In the limit where P is Gaussian with respect to S z and φ, the two distributions P (S z |φ) and P (φ|S z ) are directly related to one another through the signal slope ∂ S z,f ∂(φ) , found from the mean signal produced by the clock sequence, S z,f (φ) = S sin(φ). Note that this last expression is only approximate for squeezed states, but remains sufficiently accurate for all φ of interest, as can be seen by comparing Figures 1(b) and (c).
For the initial state in the Ramsey sequence, we already defined (∆S y ) 2 = S 2 ξ 2 and (∆S z ) 2 = S 2 χ 2 . The only remaining variance to compute is (∆S x ) 2 . Using the Holstein-Primakoff approximation (see Appendix A), we find The variance of the final S z projection after the Ramsey sequence has two components: the initial S y variance, with weight cos 2 (φ), and the leaking in of (∆S x ) 2 into S z , with weight sin 2 (φ). Putting these together gives The expected variance (∆φ) 2 of the estimate for φ at the end of the Ramsey sequence is simply the variance of the final S z measurement given by (2), normalized by the slope of φ as a function of S z,f : (2) gives Note that (3) is only valid when |φ| < π/2, since that is the phase range in which the clock signal S z,f (φ) is invertible. Moreover, (3) predicts a divergence of the phase error when |φ| = π/2, which is an unphysical artifact of using a locally linearized model.
To estimate the maximum possible phase error φ max near |φ| = π/2, we note that if we measure a value of S z near the top (or bottom) of the Ramsey fringe, there is a finite range of φ which could have produced this value of S z . The variance of S z at |φ| = π/2 is given . Therefore, the largest possible phase error ∆φ max must To complete the picture, we need to estimate the phase error for |φ| > π/2. In this case, the inversion function φ(S z,f ) will give a result in the range |φ| < π/2, with an error approximately equal to 4(|φ| − π/2) 2 . To make the result continuous with the error in the vicinity of |φ| = π/2, we approximate the phase error as The complete dependence of the phase estimation error (∆φ) 2 on φ can be obtained by stitching together (3), (4), and (5) for different values of φ. To validate this approximate analytical formula, we perform a complete numerical simulation of the process of estimating a phase in a Ramsey sequence using states with different levels of squeezing, described in Appendix B. We find excellent agreement between these analytical approximations and the results of the full numerical simulation, especially around φ ≈ 0 where the phase estimation error is low, the crucial region for predicting optimal clock performance.

III. IMPACT ON CLOCK STABILITY
Armed with an analytical model for the dependence of the phase estimation error ∆φ on the phase difference φ between the LO and atomic ensemble given by (3), we can now tackle the main question: How much can the frequency stability of a clock be improved by spin squeezing, in the limit where LO dephasing is the dominant noise source?
To quantify the precision of our clock, we need to evaluate the RMS difference between the true LO phase and the estimate of the LO phase we obtain using Ramsey spectroscopy of the atomic ensemble. We divide the total measurement time T into T /τ intervals of duration τ , corresponding to the individual Ramsey sequences of the experiment. The overall variance of the LO phase estimate after a time T is T /τ multiplied by the expected phase error variance after a single measurement, with corresponding frequency stability (or Allan deviation) given by σ 2 ω = T −2 σ 2 φ . The probability distribution P (φ, τ ) of the LO having a phase deviation φ after Ramsey time τ depends on the linewidth and lineshape of the LO. As a simple example, we approximate the phase evolution of the LO as a Gaussian process, following the model of [3].
For a free-running LO linewidth (or equivalently a dephasing rate) of γ, this model gives a Gaussian distribution for the phase deviation φ, with variance σ LO (τ ) = γτ : Using this probability distribution, we calculate the phase estimation error (∆φ) 2 and corresponding clock stability in several instructive cases, as shown in Figure 2.
First, we consider a clock operating without excess antisqueezing. The corresponding phase estimation errors are shown in Figure 2(a). We see that as the squeezing increases, the ability to resolve small rotations around φ = 0 improves: (∆φ(0)) 2 = ξ 2 /N. However, this comes at a price: the range of phase deviations φ for which the measurement is better than the SQL is reduced. In terms of clock precision, as shown in Figure 2(b), this effect manifests itself in a reduction of the optimal Ramsey time. As a result, the best possible clock stability increases with squeezing until an optimum is reached near ξ 2 = N −1/3 , worsening again for larger values of squeezing, as found in [3] for unitary squeezing.
Next, we consider the effect of non-unitary squeezing, where the excess antisqueezing is a factor of A 2 = ξ 2 χ 2 in variance. We fix ξ 2 = −15 dB and vary the antisqueezing The error in the phase measurement is shown in Figure 2(c). As expected, excess antisqueezing has no effect on the measurement precision near φ = 0, but makes the error ∆φ significantly worse for larger |φ|. Turning to the clock stability in Figure 2(d), we see that squeezed states with excess antisqueezing lead to clocks with the same stability for short Ramsey times, but the stability saturates at much smaller values of γτ . As a result, when the state area exceeds A 2 = 15 dB (for N = 10 4 atoms), the best attainable long-term stability of the squeezed clock is worse than if we had simply used a coherent spin state.  shorter, with τ opt = γ −1 √ N ξ 3 A −2 , yielding a clock stability of σ 2 φ (γ −1 ) = A 2 ξ −1 N −3/2 . Note that in this regime, the clock performance deteriorates with increasing squeezing.
Numerically, we find the boundary of the two regions to lie near A 4 ξ −6 N −1 = 5, in agreement with the result ξ 2 ∝ N −1/3 for optimum unitary squeezing (A = 1) found in [3].
Using this approach, we can analyze the potential gains in clock stability that could be obtained using spin squeezed states that have been experimentally realized. As shown in where this squeezing cannot be used in such an efficient manner. It is interesting to note that [20][21][22]27] all employ collective interactions to generate squeezing, while [29,30] use QND measurements for entangling the atoms. In measurement-based squeezing, any undetected photons produce excess antisqueezing; to minimize the latter, one needs to maximize the quantum efficiency of light detection and optimally use the available information in the probe light. This suggests that even though measurement-based squeezing has been used to create smaller spin variances, collective atom-atom interactions [33][34][35] may offer better performance for spin squeezed clocks.
In practice, Ramsey contrast in clocks is usually below unity, due to atom loss or technical imperfections. By combining the effects of antisqueezing and Ramsey contrast decay, we can readily determine parameter regions of SSSs that can enhance atomic clocks, as discussed in Appendix C. The total Ramsey contrast C = C 1 × C 2 has two components. The first, C 1 , is the reduction in contrast acquired during the preparation of the SSS, which will be present in optically induced spin squeezing due to the inevitable scattering of probe photons by the atoms during state preparation. The second factor, C 2 , is contrast loss during the Ramsey time itself.
An example of the effect of contrast loss is shown in Figure 5, where we analyze the experiment of [30], which reported the creation of a state with ξ 2 = −20.1 dB, A 2 = 19 dB, and C = 0.962 for N = 5 × 10 5 atoms. For these parameters, a change in clock stability ranging between an improvement by 2.7 dB and a deterioration by 0.6 dB could be expected, as compared to a clock operating with a CSS, depending on where in the Ramsey sequence the contrast loss originates. We see that contrast loss during spin squeezing, shown in Figure 5(a), is relatively benign in terms of clock stability compared to contrast loss during the Ramsey time, as shown in Figure 5(b); this is also noted in [26] and further discussed in Appendix C. For the SSS of [30], if the excess antisqueezing could be reduced to A 2 < 7 dB while maintaining C 1 > 0.7, C 2 > 0.96, then the clock could be operated 10 dB below the SQL. Thus, practical atomic clocks must control both excess antisqueezing and contrast loss at a tight level to profit from spin squeezing.

IV. CONCLUSION
We have analyzed the effect of non-unitary squeezing and derived simple expressions for potential improvements in clock stability in the presence of LO noise, as a function of the squeezing and anti-squeezing. We find that for a state with N atoms the squeezed state offers no metrological gain over a coherent spin state if the excess antisqueezing variance exceeds N −1/2 , and that highly squeezed states with large excess antisqueezing can lead to worse clock performance than moderately squeezed states with less antisqueezing.
We emphasize that these results apply to atomic clocks but not to sensors that require signal readout after an externally-imposed Ramsey interrogation time, such as those used to measure a time-dependent signal. In this case, for short interrogation times, it is the spin squeezing alone that determines the sensor performance.
We would also like to note that schemes to extract the full metrological gain from squeezed states in the presence of LO noise have been proposed, by using ensembles of clocks [31], or by measurement and active feedback onto the atomic state [32]. It remains to be analyzed In the Holstein-Primakoff approximation [36] of an N-particle Bloch sphere as a tangent plane perpendicular to the +x direction, we have the following mapping between harmonic oscillator operators a, a † and spin operators: The harmonic oscillator ground state |0 corresponds to an atomic coherent state pointing in the +x direction: The squeezing and displacement operators are defined in the usual way: When λ is real, the state S(λ) |0 is squeezed in the S y quadrature with variance S 2 e −λ and antisqueezed in the S z quadrature with variance S 2 e λ . A non-unitary squeezed state with antisqueezing variance χ 2 and squeezing variance ξ 2 can be decomposed as a statistical mixture of unitary squeezed states with squeezing ξ 2 0 = χ −2 , displaced in the S y direction with a Gaussian probability density of variance ξ 2 − ξ 2 0 . For a squeezed state |ψ = D(α)S(λ) |0 , we find the following expectation values for the number operator n = a † a: n = sinh 2 (λ) + |α| 2 n 2 = n 2 + 2 sinh 2 (λ) cosh 2 (λ)+ + |α| 2 sinh 2 (λ) + cosh 2 (λ) + + α 2 + α * 2 sinh(λ) cosh(λ) A displacement operator in the S y direction corresponds to a purely imaginary value for α. Therefore, (∆S x ) 2 = n 2 − n 2 equals Evaluating the expectation value over the α distribution and substituting χ and ξ gives the final result: For any squeezed state, we have 0 ≤ ξ 2 − χ −2 < 1, which will have negligible impact on the phase estimate precision ∆φ. Therefore, we can simplify (A4) to give

Appendix B: Numerical Validation of Phase Variance Formulas
Here, we describe the procedure for validating the analytical formulas (3), (4), and (5) through a numerical simulation and generating the results in Figure 6. For each value of unitary squeezing ξ and LO-atom phase deviation φ, we calculate the probability distribution P (S z , φ, ξ) of obtaining a particular value of S z during the projective measurement at the end of the Ramsey sequence. From this probability distribution, we determine the estimate for φ with the smallest root-mean-square error, given a particular observed value for S z , assuming a uniform Bayesian prior for φ in the interval (−π/2, π/2). This estimate is given by −π/2 dφ P (S z , φ, ξ) Next, for each true value of φ (which can lie outside the (−π/2, π/2) range), we evaluate the expected error in the φ estimate: This is the function plotted in Figure 6. The effects of contrast loss and excess antisqueezing can be introduced by the appropriate modification of the function P (S z , φ, ξ) to yield similarly good agreement with the analytical estimates of (∆φ) 2 derived above.

Appendix C: Effect of Contrast Loss
The model we use can also be extended to include the effects of contrast loss, as shown in Figure 7. We model contrast loss during the SSS preparation by combining a SSS comprised of NC 1 atoms with two equally populated sub-ensembles, one with N 2 (1 − C 1 ) atoms in the ground state, and the other with the same number of atoms in the excited state, giving a total Ramsey contrast of C 1 . The orientation of the SSS relative to the S z = 0 plane depends on the method by which squeezing is produced -measurement-based squeezing [28][29][30] gives θ = 0 while feedback-based squeezing [27,33,34] has θ = arcsin(1/χ).
During the Ramsey sequence, additional contrast loss by a factor C 2 may occur when some of the atoms decay from the excited clock state. We model this effect by a population transfer to a CSS containing N(1 − C 2 ) atoms in the ground state. The net Ramsey contrast of the entire sequence equals C = C 1 × C 2 , as only the atoms remaining in the SSS are still contributing to the signal.
Contrast loss adversely affects the clock performance by reducing the magnitude of the signal by a factor C, and by adding additional noise in the final S z measurement: with the first term arising from contrast loss during squeezing preparation, and the second term from contrast loss during the Ramsey sequence. In typical situations where θ and φ are small, the second term is greater, reflecting the fact that contrast loss during the Ramsey sequence is a more severe problem than contrast loss during the squeezing preparation.
These noise terms are added to the noise arising from the spin squeezed state itself, giving The mean signal produced by the clock sequence is S z,f (φ) = C × S sin(φ), giving the full expression for (∆φ) 2 : Similarly, the maximum phase estimation error becomes By replacing (3) and (4) with (C3) and (C4), we can evaluate the performance of clocks in the presence of contrast loss and potentially non-unitary squeezing.
As an example, consider a clock operating without any squeezing, for different amounts of remaining contrast C after the Ramsey time τ . This contrast loss may occur during the state preparation (C 1 < 1, C 2 = 1) or during the Ramsey sequence itself (C 1 = 1, C 2 < 1).
The errors in estimating the LO phase for these two situations are shown in Figure 8(a) and (c). When C = 1, the error in phase estimation is always at the SQL for N = 10 4 atoms: (∆φ) 2 = 10 −4 = 1/N. As the contrast becomes worse, two mechanisms cause (∆φ) 2 to grow. For φ = 0 and C 2 = 1, the imperfect contrast C 1 leads to a smaller signal with measurement noise corresponding to only NC 1 atoms, so (∆φ(0)) 2 = 1/(NC 1 ) in this case. However, when C 2 < 1, all N atoms contribute to the S z measurement noise, giving (∆φ(0)) 2 = 1/(NC 2 2 ). As |φ| increases, ∆φ becomes equal for both situations because the N(1 − C 1 ) atoms decohered during state preparation are rotated toward the equator and contribute their projection noise to the final S z measurement. The corresponding clock stabilities are shown in Figure 8 Finally, we focus on the potential gain in clock stability, optimizing over the Ramsey time τ . Figure 9 shows the effect of contrast loss on the performance of a squeezed clock with N = 10 4 atoms. The colors and contours are in units of dB, normalized to the SQL: a clock with no squeezing and perfect contrast (top-right corner of these plots). As expected, contrast loss C 2 during the Ramsey time has a much more severe effect on ultimate clock stability than imperfections in the initial state C 1 . We can see that if the former contrast falls below C 2 = 0.7, the clock performance will always be worse than a clock without squeezing and unity contrast. More interestingly, as C 2 becomes smaller, the potential improvement from squeezing also decreases -for C 2 = 1, we can gain as much as −13.7 dB in clock stability by squeezing, but for C 2 = 0.4, this decreases to only −1.7 dB. This finding is consistent with the results of [4,5]: when single-particle loss limits the Ramsey time, squeezing is unable to significantly improve clock stability.