Optical implementation of spin squeezing

Quantum metrology enables estimation of optical phase shifts with precision beyond the shot-noise limit. One way to exceed this limit is to use squeezed states, where the quantum noise of one observable is reduced at the expense of increased quantum noise for its complementary partner. Because shot-noise limits the phase sensitivity of all classical states, reduced noise in the average value for the observable being measured allows for improved phase sensitivity. However, additional phase sensitivity can be achieved using phase estimation strategies that account for the full distribution of measurement outcomes. Here we experimentally investigate a model of optical spin-squeezing, which uses post-selection and photon subtraction from the state generated using a parametric downconversion photon source, and we investigate the phase sensitivity of this model. The Fisher information for all photon-number outcomes shows it is possible to obtain a quantum advantage of 1.58 compared to the shot-noise value for five-photon events, even though due to experimental imperfection, the average noise for the relevant spin-observable does not achieve sub-shot-noise precision. Our demonstration implies improved performance of spin squeezing for applications to quantum metrology.


Introduction
Quantum metrology uses non-classical states to enable measurement of physical parameters with precision beyond the fundamental shot-noise limit [1]. This is subject to intense research effort for measurements at the single-photon level [2], with higher intensity quantum optics [3] and with matter [4]. In all these cases the central motivation is to understand how to extract more information per-unit of resource (such as probe power and interaction time) and this will naturally lead to applications in precision measurement [5][6][7][8]. An approach that dates back to the beginning of quantum optics [9] is to improve phase sensitivity using squeezed states [3]. In discrete quantum optics, one approach has been to use path-entangled states, such as NOON states [10], as a means to achieve supersensitivity since they exhibit interference patterns with increased frequency compared to classical light. So far, experiments have reached photon numbers of up to six [11][12][13] photons, and recent works aim to address weaknesses in these schemes due to loss [14] and state generation using non-deterministic processes [15]. Using probes multiple times can also enable a precision advantage, which varies according to the chosen notion of resource [16,17].
Spin squeezing has proven to be a useful approach thanks to developments in experiments manipulating atomic ensembles [18][19][20][21][22][23]. In these experiments ensemble measurements are typically used, which correspond to collective observables for all particles in the ensemble. However, experiments that utilise detections at the single-particle level, can in principle achieve sensitivity beyond that achievable using ensemble measurements [24]. The total statistical information that can be extracted from a measurement of an unknown phase shift is captured by the Fisher information [25,26], which is evaluated for all measurement outcomes. Because it is well known that squeezing can improve the phase sensitivity in many set-ups, it is important to quantify the Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. sensitivity improvement with squeezing, and how close this sensitivity is to the maximum phase sensitivity as quantified using Fisher information.
In this paper, we focus on measurements using spin squeezing [27,28], which has been shown to enable increased sensitivity in several experiments using ultracold atoms [29][30][31][32]. We report on an optical implementation of a spin-squeezing model which was originally considered by Yurke et al [33,34], and we investigate how sensitivity is improved in this model. Our setup generates five-photon Yurke states by postselecting on cases with five detection events from the state emitted by a parametric downconversion source after one photon subtraction [35], and we use spatially-multiplexed pseudo-number-resolving detection to reconstruct photon-number statistics at the output [15]. Our analysis demonstrates increased sensitivity from the observed optical Yurke state, using all five-photon coincidence outcomes. We investigate the role of spin squeezing in achieving this quantum enhancement by using our optical spin-squeezing model.
Consider first N uncorrelated single photons, where each photon is in a superposition of horizontal (H) and vertical (V) polarisations, ñ + ñ (| | ) 1, 0 0, 1 2 HV HV . When we measure this state in HV-polarisation basis, the probability that n photons are detected with H polarisation and N−n photons with V polarisation is given by the Binomial distribution = ( ) ( ) The noise obtained from this distribution is given by N , which is called shot noise for phase estimation. The state which we consider here, sometimes referred to as the Yurke state [33], is a superposition of the two states of + ( ) N 1 2photons are in one optical mode (e.g. horizontal polarisation) and + -( ) N 1 2 1photons are in an orthogonal mode (e. g. vertical polarisation) of the form - If we adopt polarisation encoding and measure in the HV basis, the outcomes take two values with photon-number difference ±1. The noise of Yurke state is therefore 1 which is smaller than that using N uncorrelated photons with a noise of N .
More generally, the photon statistics of any two-mode N-photon system can be described by the Stokes parameters describing the photon-number differences between H and V polarisation, diagonal (D) and antidiagonal (A) polarisation, and right-circular (R) and left-circular (L) polarisation, whereˆ † a i ,â i andn i are the creation, annihilation and number operators for the corresponding modes. The average of these parameters, = á ñ á ñ á ñ (ˆˆˆ) S S S S , , ( figure 1(a)). A large number of parameters have been devised to quantify spin squeezing for various applications [28]. To characterise this squeezing, we use the squeezing parameter, x S , which is defined to be the ratio between the minimum uncertainty for directions orthogonal to S [27], where x < 1 S indicates reduced quantum noise below the shot-noise. For uncorrelated photons, x = 1 S . While for the Yurke state, x S is minimised along the S 1 direction with x = N 1 S which indicates strong squeezing for this choice of squeezing parameter. The squeezing property of the Yurke state can be used for improving the phase sensitivity of an interferometer. The effect of a phase rotation by f can be described by the unitary operator . Specifically,Ŝ 1 after the phase rotation is expressed as for both N uncorrelated photons and the Yurke state, the average of f ( ) S 1 is expressed by For estimates ofŜ 1 , phase error is given by the ratio of DS 1 and the phase derivative of á ñ S 1 . Specifically, the phase error at f = 0 is given by for the Yurke state, the phase error by squeezing of Yurke state is To characterise the improvement of the phase sensitivity due to squeezing, we use another squeezing parameter x R which was introduced in [36,37], which is the ratio of the phase error for a general state and phase error due to shot noise df , the phase error is smaller than the shot-noise limit attained by uncorrelated photons R for the high-N limit. Although squeezing of the Stokes parameters can improve phase sensitivity beyond the shot-noise limit, additional phase sensitivity can be achieved by phase estimation which accounts for the full distribution of measurement outcomes. Statistical information about f can be extracted from the frequencies of every measurement outcome occurring in an experiment and quantified using Fisher information f ( ) F . In a twomode N-photon problem, Fisher information is calculated from the More specifically, the Cramér-Rao bound states that any unbiased statistical estimator of f has mean-square error which is lower bounded by f ( ) F 1 , and this bound can be saturated using a suitable statistical estimator [38]. The minimum phase error for the Yurke state is then given by . The improvement factor compared to the shot-noise limit is df df » N 2 opt. SNL for the high-N limit. Note that the phase sensitivity obtained from the Fisher information is greater than the sensitivity obtained from equation (2) by a factor of approximately 2 , indicating that maximum sensitivity is achieved not only due to squeezing but also other quantum effects captured by the full set of measurement outcomes. Note that a theoretical analysis with similar motivation is given in [24].

Experimental setup
In order to demonstrate experimentally the phase sensitivity obtained from squeezing and maximum phase sensitivity obtained from Fisher information, we have implemented a five-photon Yurke state model by using a post-selection technique. Figure 2 shows the experimental setup for generating the Yurke state. Downconverted photon pairs are generated from biaxial Type-I bismuth borate (BiBO) crystal in a non-collinear configuration. A half wave plate (HWP) is placed on each path so that one path is horizontally polarised and the other is vertically polarized. Each beam is then combined into a single spatial mode at the polarisation beam splitter (PBS1). The state after the PBS1 is a superposition of photon number states with equal photon number in the horizontal and vertical polarisation, where the sum is taken over even values of N. If we postselect N photons from this state, the state is equivalent to the Holland-Burnett state ñ |N N 2, 2 HV [13]. To generate the Yurke state [35], one photon is subtracted from the down-converted photon source, by detection of a single-photon in the D/A basis. After the one-photon subtraction, the conditional output is the five-photon Yurke state. In the setup, we put a beam splitter after PBS1 so that each of the N photons in the beam is transmitted with probability 10%. The transmitted one-photon state was measured in the D/A basis using a HWP set at 22.5°and PBS3. After the one-photon detection, the reflected -N 1 photons are analysed by the polarisation interferometer.
Note that the conditional output state after the one photon subtraction is the superposition of arbitrary (odd) photon-number Yurke states, since the down-converted photon state is the superposition of even-photon number states described in equation (4). Typical count rates for two, four, six-fold coincidence are7 10 5 , 40 and 10 −2 Hz, respectively. In this experiment, we only focus on cases with five detection events after the onephoton subtraction.
To demonstrate the sub-shot noise phase measurement, we measured all possible coincidence outcomes, of which there are six, at the output as f is varied. We used a pseudo-number-resolving multiplexed detection system using 1×7 fibre beam splitters, 14 avalanche photodiodes (APDs) and a multi-channel photon correlator (DPC-230, Becker and Hickl GmbH) [15]. The phase shift was measured by using a HWP and PBS3 which were placed on the reflected path of the beam splitter.
Details of measurement times and efficiency for our experiment are as follows. For our six-photon measurements, data collection took eight hours per point in the interference fringe. Quantum efficiencies of our APDs are roughly 55% at 808 nm. Transmittance through the all the optical components including coupling efficiency from free space to fibre is estimated as 20%. The reflectivity of the non-polarising beamsplitter is 90%. Assuming equal splitting probability amongst the seven detectors at each interferometer output, the efficiency for detecting five-photon outcomes ranges from 15% to 52%. Hence, overall efficiency can be estimated as´´´» 0.55 0.2 0.9 0.15 0.014. and +5), and results in the reduced quantum noise at the phase where the average is nearly zero (figure 3(c)). On the other hand, for the classical uncorrelated case, the probability distribution of a To extract the maximum phase sensitivity, we calculated the phase sensitivity by using Fisher information obtained from equation (3) for the Yurke state. Figure 4 shows the bias phase dependence of Fisher information. The maximum of Fisher information is F = 7.89 at bias phase of f = 0.21 which is slightly different from the phase where the squeezing is maximum. Thus the obtained state can actually achieve sensitivity that is a factor of 1.58 smaller than the shot-noise limit. Note that maximum Fisher information is obtained at a slightly-different bias phase from where the squeezing is maximum. We can conclude that the improvement in the phase sensitivity is not only due to squeezing but also additional information reflected in the higher moments of the distributions [41]. In particular, Fisher information can extract the full information for changes in the phase parameter from the interference fringes at the output.

Conclusions
In conclusion, we have demonstrated, using our set-up, suppression of quantum noise by a factor of 2.56 with the effects of the squeezing being clearly shown by the measured interference fringes. Spin squeezing is often characterised using parameters x S and x R , where values <1 correspond to supra-classical performance. Our measurements show clear spin-squeezing using the parameter x = 0.63 s , while our measurements of x R , which is traditionally used to quantify sub-shot noise phase-noise error in spin-squeezing experiments, is >1. Nonethe-less, the extracted Fisher information was 1.58 times better than shot-noise-limit demonstrating that quantum enhanced precision is possible even with  x 1 R . As an alternative to the multiplexed pseudo-numbercounting detectors we used, recently-developed high-efficiency number-resolving detectors [42,43] could be used to improve detection efficiency and therefore reduce measurement time. Our experimental demonstration is important not only for optical sub-shot-noise measurement but also other applications demonstrating subshot-noise spin-squeezed states [44][45][46]. , on the bias phase at the output for the classical state, respectively.
We emphasise that since we only focus on the post-selected five photon events for this experiment, there are contributions to the output from lower photon-number states that are ignored, and the actual sensitivity is much lower if all these contributions are considered. To obtain actual quantum-enhanced sensitivity using Yurke states, we would need a deterministic or heralded source with fixed photon number. We note that the sensitivity using a setup for generating heralded two-photon Holland-Burnett states was reported and analysed in [47]. It is outstanding challenge to achieve heralded generation of Holland-Burnett states with high photon number, and furthermore this would need to be combined with single-photon subtraction to create Yurke states.
Note also that we did not make phase estimates using our model, but showed the possibility of improving the sensitivity by looking at the obtained probability distributions (as was done in [12,13] for example). If the standard maximum-likelihood procedure were to be used to obtain phase estimates using our setup, individual estimates would each require tens or more counts to be accumulated. A comprehensive analysis of the statistics of these estimates, and therefore their sensitivity, would require thousands of such estimates to be obtained, which was not practical in our experiment due to coincidence counts of roughly 300 counts/8 h for each bias phase. We also remark that for the phase-estimation experiment reported in [15], it was shown that methods for measuring sensitivity using Fisher information derived from probability distributions, and maximumlikelihood estimation from simulated data (sampled from the same probability distributions) achieve close agreement.

Reconstruction of interference fringes for the Yurke state
Photon-number counts at our multiplexed detectors are analysed as follows. Single photons are detected at each APD with probabilities of s a i (i=1, 2, ... 7) in mode a and s b j ( j=1, 2, ... 7) in mode b, which account for propagation loss and detector efficiency. In our analysis, we assume that five-fold coincidence detections arise only due to the generation of three photon pairs at the source (and neglect higher-order contributions). We define efficiency parameters for coincidence events at our multiplexed detectors as follows, where we assume that m clicks in path a andm

Derivation of probability distributions for the Yurke state including temporal mode mismatch
To derive f ( ) P m I , , we start from theoretical model in [48]. The quantum state generated before the BS in figure 2 is given by can be written as a superposition of one indistinguishable and one distinguishable component: where I is the indistinguishability given by á ñ , and the symbolV denotes the orthogonal mode to H and V. In the following, we assume that modesˆ( ) a H V and^( ) a H V do not interact so that we can consider reduced density matrix, r yñ | where offdiagonal terms can be neglected as follows, where C d is given by Replacing the annihilation operators for indistinguishable and distinguishable modes as where f ( ) U is unitary transformation due to a half-wave plate, which is expressed as s was determined by fitting and the average value of s over 200 simulations was 0.085. F in figure 4 is computed using these modified distributions, which have lower values compared to the unmodified distributions around f = 0.
Note that the effect of imperfect indistinguishability derived in section 5.2 cannot explain the effect of this phase insensitive noise, which is roughly 8%. The phase-insensitive noise could arise from several factors, perhaps the most important being six-fold coincidence counts arising from components of the downconversion state with eight or more photons (and which have lost photons due to inefficiencies in the setup). However, the six-fold coincidence count rate (10 −2 ) in our current setup is too low to enable analysis of the effect of these higher-order contributions. This analysis would be done by repeating measurements over a range of pump power (or other gain parameter for the downconversion source) which we leave for future work. We also note that the probability distributions derived by considering the effect of imperfect indistinguishability given by equation (15) can explain most of the features of experimental data as shown in figure 5.