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arXiv:1203.2099 · Physical Review A

Quantum metrology for nonlinear phase shifts with entangled coherent states

Jaewoo Joo, Ki-min Park B, Hyunseok Jeong, William J. Munro, Kae Nemoto, Timothy P. Spiller

Published Oct 18, 2012• 95 citations
arXivHTMLPDF

Abstract

We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent $k$. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.

DOI
10.1103/physreva.86.043828

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We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number \ensuremath⟨n\ensuremath⟩\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath⟨n\ensuremath⟩ and nonlinearity exponent kkk. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large \ensuremath⟨n\ensuremath⟩\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath⟨n\ensuremath⟩. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number \ensuremath⟨n\ensuremath⟩\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath⟨n\ensuremath⟩.
We present an improved phase estimation scheme employing entangled SU(1,1) coherent states in comparison to NOON and entangled coherent states under perfect and lossy conditions for a fixed mean photon number. The study is also devoted to the phase enhancement of the quantum states resulting from a generalized nonlinearity of the phase shifts, both without and with losses. Furthermore, we show that these states give the smallest variance in the phase parameter for a large number of photons in a different order of nonlinearity. Finally, the phase sensitivity of this interferometric setting with parity detection is discussed.
Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision (NNN is the mean photon number of the light field). But when NNN is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the 1/N1/21/{N}^{1/2}1/N1/2 (1/N3/21/{N}^{3/2}1/N3/2) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the 1/N1/N1/N scaling for the linear phase and 1/N21/{N}^{2}1/N2 scaling for the nonlinear phase.
Quantum entanglement offers the possibility of making measurements beyond the classical limit, however some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has used the quantum Fisher information (QFI) to show that entangled coherent states (ECSs) may be useful for this purpose as they combine sub-classical phase precision capabilities with robustness (Joo et al., 2011). However, to date no effective scheme for measuring a phase in lossy systems using an ECS has been devised. Here we present a scheme that does just this. We show how one could measure a phase to a precision significantly better than that attainable by both unentangled "classical" states and highly-entangled NOON states over a wide range of different losses. This brings quantum metrology closer to being a realistic and practical technology.
Abstract The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system’s robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.
We study the precise phase estimation using squeezed states in the presence of photon loss. Our exact quantum Fisher information calculation shows significant enhancement even though the loss is very large and sets a benchmark for experimental realization. We show that if we blindly use the existing parity measurement scheme [Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)] for the ideal case, the result will be even worse than the classical case given very small loss. Using our formulas, we can optimize the measurement result by making loss-dependent phase shift and choosing appropriate squeezed states (average photon number of squeezed states).
There has been much interest in developing phase estimation schemes which beat the so-called Heisenberg limit, i.e., for which the phase resolution scales better than 1/n, where n is a measure of resources such as the average photon number or the number of atomic qubits. In particular, a number of nonlinear schemes have been proposed for which the resolution appears to scale as 1/n^k or even exp(-n), based on optimising the quantum Cramer-Rao bound. Such schemes include the use of entangled coherent states. However, it may be shown that the average root mean square errors of the proposed schemes (averaged over any prior distribution of phase shifts), cannot beat the Heisenberg limit, and that simple estimation schemes based on entangled coherent states cannot scale better than 1/n^{1/4}. This paradox is related to the role of 'bias' in Cramer-Rao bounds, and is only partially ameliorated via iterative implementations of the proposed schemes. The results are based on new information-theoretic bounds for the average information gain and error of any phase estimation scheme, and generalise to estimates of shifts generated by any operator having discrete eigenvalues.