Adaptive quantum metrology with large dynamic range using short one-axis twists

Abstract

Phase estimation with potentially large phase values, i.e., with large dynamic range, has many applications in quantum metrology, for example to atomic clocks. A recently proposed phase estimation scheme approaches the Heisenberg scaling in this global setting using sequences of increasingly squeezed Gaussian states as probes and adaptively chosen, potentially mid-circuit, measurements. In this work, we first observe that the pattern of increase in the squeezing of the probes is applicable even to states with some non-Gaussian features. We then propose an experimentally feasible version of this phase estimation scheme, based on the alternating application of one-axis twist (OAT) operations and rotations. Our protocols are explicitly described in terms of multiple OAT angles whose durations decrease polynomially with system size and spin-squeezing parameters that decay as $N^{-μ}$, with $μ>2/3$ in most cases. Using numerical computation of the system-size dependence $N^{-ν}$ of the Bayesian mean-squared error of an estimator, we show that these states are suitable for use in the phase estimation scheme, and highlight the protocols to achieve $ν=17/9$ and $53/27$ using two and three OAT operations respectively in the last adaptation stage. We also analyze the limited non-Gaussianity of the resulting probe states and discuss the role of non-Gaussianity in this protocol more generally. Finally, we analyze how robust these protocols are with respect to imperfections such as particle number fluctuations and coherent control fluctuations.

Figures (9)

• Figure 1: In the adaptive protocols we consider, the spin-1/2 particles that make up the sensor are divided up into separate ensembles. The ensembles then undergo separate squeezing protocols where different ensembles undergo a different number of one-axis twisting operations, denoted by $T(\chi_{j}^{(k)})$, interspersed with $x$-rotations, denoted by $R_{x}(\theta_{j}^{(k)})$. After the squeezing, the ensembles undergo a unknown rotation about the $z$-axis, denoted by $R_{z}(\phi)$, by the angle to be estimated. Finally, the ensembles are measured one at a time from smallest, and least squeezed, to largest with larger ensembles being counter rotated by the estimate of the residual phase made by the previous ensembles, i.e. by $\hat{\phi}_{j}$'s. The quasi-probability distributions are the Q-distributions (on a logarithmic scale) associated with the fourth, largest, ensemble at each step of the protocol.
• Figure 2: Here we show the dependences of the kurtoses of $J_{y}$ and $J_{z}$ on the value of $\mathcal{C}$. In (a) the orange dots are the numerically obtained kurtosis for the $J_{y}$ observable as a function of $\mathcal{C}$, the factor by which we shrink the twisting angle for the single twist case. The blue squares display the numerically obtain value of $\textrm{Kurt}[J_{z}]$. The green dashed line is a fit of $\textrm{Kurt}[J_{y}]$ to a sigmoid-exponential function of the form Eq. \ref{['eq:sig_exp']} while the red dash-dotted line is a fit of $\textrm{Kurt}[J_{z}]$ to the same functional form. The fits are of a fairly high quality with $P_{4}\approx0.84$ for the $\textrm{Kurt}[J_{y}]$ indicating a turning-on behavior around this point. In (b) we plot for $N=750$ (orange circles and red hexagons), $N=1500$ (blue squares and purple crosses), and $N=2250$ (green diamonds and yellow $x$-crosses) the kurtosis of $J_{y}$ and $J_{z}$ respectively with respect to $\mathcal{C}$. These curves lie nearly on top of one another indicating that the turning-on is approximately independent of system size.
• Figure 3: (a) Example system size dependence of the two twisting parameters in our protocol targeting $\mu=8/9$ where the orange circles correspond to $\chi_{1}$ and the blue squares correspond to $\chi_{2}$. The magnitudes of both parameters approximately follow a simple algebraic decay. (b) Examples the squeezing protocols considered here yielding a spin-squeezing parameter that decreases faster as the system size is increased than is possible with a single twist. The blue circles result from a single twist and correspond to the protocols proposed in Kitagawa_1993. The red squares correspond to the a two-twist protocol which achieves $\mu\approx8/9$. and the green triangles correspond to the three-twist protocol that achieves $\mu\approx26/27$.
• Figure 4: Performance of adaptive phase estimation using one-axis twists. (a) Plot of the averaged estimation errors obtained by this protocol as a function of system size by the three-ensemble, two-twist protocol for various values of the prior standard deviation $\sigma$ where we target $\nu\approx17/9$. (b) Plot of the scaling exponents $\nu$ obtained as a function of the prior standard deviation $\sigma$. The dark orange circles correspond to the behavior exhibited by a protocol that uses one ensemble squeezed by a single twist. The scaling exponent rapidly decays in this case. Note that for this curve the exact values depend on the system size the fit is performed at but the qualitative behavior is consistent. The blue circles correspond to a protocol that uses two ensembles with the first being unsqueezed and the second being squeezed with a single twist. The green upward triangles correspond to a protocol that uses three ensembles where the first ensemble is unsqueezed and the second and third ensembles are squeezed with a single twist. The purple downward triangles correspond to a protocol that uses three ensembles but with the third ensemble undergoing two twists. The brown crosses correspond to a protocol that uses four ensembles where ensembles one through four are squeezed with zero to three twists respectively. The dashed lines correspond to the expected values for our two, three, and four ensemble protocols. The gold dashed line is at $\nu=5/3$, the orange dot-dashed line is at $\nu=17/9$, and the red dot-dot-dashed line is at $\nu=53/27$. The black circles around the purple downward triangles indicate the points associated with the curves in (a). indicate the points associated with the curves in subfigure (a). The numerical simulations give results that approximately agree with those suggested by the theory of the twisted Gaussian states. The "Three Ensembles (One Twist)" and "Four Ensembles" curves exhibit additional fluctuations due to the use of Monte Carlo integration to evaluate several of the sums in Eq. \ref{['eq:monte']} while for the other curves we are able to evaluate the sums exactly.
• Figure 5: The average squeezing parameter produced vs. the anticipated system size for various different particle number distributions in the case where $\mu\approx8/9$ is targeted. The orange circles correspond to the case where there is no particle number uncertainty, the green dashed curve corresponds to the case of a Poisson distribution, and the blue dash-dot curve corresponds to a binomial distribution with $p=0.98$. The vertical bars denote a single standard deviation from the mean for each distribution. We find that the noiseless squeezing parameter remains within the error bars for all system sizes.