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Practical Quantum Metrology with Large Precision Gains in the Low Photon Number Regime

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Practical Quantum Metrology with Large Precision Gains in the Low Photon Number Regime Practical quantum metrology with large precision gains in the low photon number regime P. A. Knott,1, ∗ T. J. Proctor,2, 3 A. J. Hayes,1 J. P. Cooling,1 and J. A. Dunningham1 1Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 3Berkeley Quantum Information and Computation Center, University of California, Berkeley, CA 94720, USA (Dated: April 21, 2016) Quantum metrology exploits quantum correlations to make precise measurements with limited particle numbers. By utilizing inter- and intra- mode correlations in an optical interferometer, we find a state that combines entanglement and squeezing to give a 7-fold enhancement in the quantum Fisher information (QFI) { a metric related to the precision { over the shot noise limit, for low photon numbers. Motivated by practicality we then look at the squeezed cat-state, which has recently been made experimentally, and shows further precision gains over the shot noise limit and a 3-fold improvement in the QFI over the optimal Gaussian state. We present a conceptually simple measurement scheme that saturates the QFI, and we demonstrate a robustness to loss for small photon numbers. The squeezed cat-state can therefore give a significant precision enhancement in optical quantum metrology in practical and realistic conditions. PACS numbers: 42.50.St,42.50.Dv,03.65.Ud,03.65.Ta,06.20.Dk INTRODUCTION It is this small photon number regime that is considered herein, and whilst in this case theoretical lower bounds on precision do exist [23], it is an open question as to Optical quantum metrology utilizes quantum mechan- which practical states can give significant improvements ical correlations to make high precision phase measure- over the SNL. In this paper we make significant progress ments with a significantly lower particle flux than would towards this question by introducing an experimentally be required by classical systems. This is a crucial re- realisable scheme that can measure to a precision with a quirement for many applications such as biological sens- p p 7 factor improvement over the SNL, and a 3 improve- ing, where disturbing the system can damage the sample ment over the commonly used quantum states, including [1,2], or gravitational wave detection, which suffers from Caves's scheme [5]. the effects of radiation pressure and mirror distortion if the photon flux is too high [3,4]. Squeezed states of light The general setting of optical quantum metrology can have shown much promise for quantum-enhanced metrol- be understood in terms of a two-mode (two-path) in- ogy beyond the classical shot noise limit (SNL), and since terferometer. The enhancement gained from employing the seminal proposal of Caves [5] significant progress has quantum states for phase estimation can be then framed been made in exploiting the potential of such states [6{ in terms of different types of correlations: those be- 9]. As a result the effectiveness of squeezing in quantum tween photons on each mode of the interferometer (intra- metrology has been demonstrated experimentally [10], mode), as well as the correlations between the paths and a squeezed vacuum is now routinely injected into (inter-mode). Both types of correlations can contribute the dark port of gravitational wave detectors to improve to improvements in precision, and hence it is natural to their measurements [11{13]. consider states in which both are present. Observing Remarkably, in the large photon-number limit in which that the squeezed vacuum exhibits high intra-mode cor- arXiv:1505.04011v3 [quant-ph] 20 Apr 2016 gravitational wave detectors operate, it has been shown relations due to non-classical photon statistics [23], and that when photon losses are present the original scheme that inter-mode correlations may be provided by mode- of Caves is optimal [17]. However, in many applications it entanglement, this naturally leads us to introduce the `squeezed-entangled state' jΨ i / jz; 0i + j0; zi where is not this regime that is of interest and it is instead nec- SES essary to consider metrology with low photon numbers. jzi represents the squeezed vacuum which will be defined Measurements on fragile systems are of much interest, below. It will be shown that the fundamental bound on with examples including measurements of spin ensem- the phase precision possible with this state is a substan- bles [1], biological systems [2, 18], atoms [19, 20] and tial improvement over the states normally considered in single molecules [21], and in all these applications it is the literature, including the state proposed by Caves [5], of utmost importance to minimize the probe's interac- the NOON state [24], and the optimal Gaussian state tion with the sample to avoid damage. Examples of such (created from only Gaussian transformations) [23, 25]. damage are the scattering induced depolarisation of spin The squeezed-entangled state (SES) has clear poten- ensembles [1], or direct degradation of living cells [22]. tial for precision phase estimation but has the signifi- 2 Loss 60 NOON Photon SES Count SSV State preparation: 50 SVCS BS SCS 40 Photon Count Loss 30 FIG. 2. A quantum state jΨi is prepared as an input into the arms of an interferometer which contains an unknown QFI 20 relative phase shift φ ≡ φa −φb, generated by the linear phase shift unitary operator U^ = exp(i(φan^a + φbn^b)). For the states introduced herein the optimal measurement scheme is mixing the modes on a balanced (50:50) beam splitter (BS), 10 followed by photon number counting. When photon losses are considered these can be modelled by ‘fictitious’ variable transmissivity beam splitters after the phase shift. 0 0 0.5 1 1.5 2 2.5 3 3.5 Average number phase estimation problem of measuring a phase differ- ence φ between two optical modes containing unknown FIG. 1. The QFI (plotted against average photon numbern ¯) linear phase shifts, as shown in Fig.2. This is appli- for both the squeezed-entangled state (SES) and the squeezed cable to a wide range of physical scenarios and is the cat-state (SCS) shows dramatic improvements over the com- monly used states for optical quantum metrology, includ- canonical approach to a very broad range of metrology ing Caves's state (SVCS), the optimal Gaussian state (SSV), schemes. The fundamental limit to the precision with and the NOON state. Furthermore, the squeezed cat-state which a state ρ can measure the phase φ is given by the has been made experimentally [14{16], and in this paper we quantum Cram´er-Rao bound (CRB) [26, 27]: present a measurement scheme that can be employed to read out the phase. 1 ∆φ ≥ p ; (1) µFQ(ρ) cant disadvantage that it is not clear if a simple high- where µ is the number of independent repeats of the fidelity preparation procedure can be found. Hence we experiment and FQ(ρ) is the QFI of ρ. For pure and introduce a practical alternative, the `squeezed cat-state' path-symmetric states (only path-symmetric states will (SCS), which has been demonstrated experimentally [14{ be considered herein) it is shown in Appendix A that the 16]. The quantum Fisher information (QFI) is a useful relevant QFI is simply given by and commonly used measure which quantifies the phase precision obtainable using a given probe state, and us- FQ(Ψ) = 2 (VarΨ − CovΨ) ; (2) ing this metric the potential for phase estimation of both 2 2 states proposed herein is shown in Fig.1 (the requisite where VarΨ = hn^ai − hn^ai is the variance of the photon QFI formalism will be provided in the next section). In- number in mode a (or mode b) and CovΨ = hn^a ⊗ n^bi − triguingly, as well as being more practical, the SCS also hn^aihn^bi is the covariance of the two modes (the expec- outperforms the SES, showing that this state is of great tation values are taken with respect to the state jΨi). interest from both a practical and theoretical perspec- This explicitly highlights the roles played by inter- and tive. Furthermore, it will be seen that the SCS is robust intra-mode correlations. enough to exhibit a precision advantage with up to 27% We now introduce the relevant states in the quantum photon loss. Finally, it is shown that high-precision phase metrology literature. In the following we denote a co- herent state and a squeezed vacuum by jαi ≡ D^(α)j0i measurements can be obtained both in the ideal and lossy ^ cases using a photon-number counting measurement. and jzi ≡ S(z)j0i respectively (α; z 2 C) where the displacement operator is D^(α) = exp (αa^y − α∗a^) and h 2 2 i ^ 1 ∗ y the squeezing operator is S(z) = exp 2 (z a^ − za^ ) . CORRELATIONS IN OPTICAL METROLOGY Caves [5] proposed the use of squeezing to enhance the phase precision via a probe state obtained from mix- We begin by reviewing the relevant background ma- ing a squeezed vacuum and a coherent state (SVCS) terial. In this work we consider the standard optical on a balanced (50:50) beam splitter, which is given by 3 ^ ^ jΨSVCS i = UBS (jαia ⊗ jzib), where UBS denotes the beam is even more significant, with FQ(ΨSES ) ≈ 7FQ(ΨNOON ) splitter unitary operator. This state has been studied ex- forn ¯ = 1. In Fig.1 we compare the QFI of the SES, tensively and has an asymptotic phase precision of 1=n¯ the NOON state, the SVCS and the optimal Gaussian [6] (wheren ¯ = hn^a +n ^bi is the average number of pho- state (SSV). Fig.1 clearly shows the great potential of tons in the interferometer).
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