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Journal of Modern Optics

Photon statistics, parity measurements, and Heisenberg-limited interferometry: example of the two-mode SU(1,1) SU(1,1) coherent states

Christopher C. Gerry, Richard J. Birrittella, Andrae Raymond, Raul Carranza

Published Aug 22, 2011• 7 citations
Publisher

Abstract

Parity measurement-based quantum optical interferometry and metrology is known to be sensitive to the joint photon number distribution, , of the two-mode quantum states inside the interferometer. In cases where the statistics display a clustering of the populations around the borders (i.e. along the axes ) with a plateau over most of the interior, Heisenberg-limited (or super-sensitive) measurements of phase shifts are enabled. This follows from the heuristic number-phase uncertainty relation because of the uncertainty in the location of the photons being ΔN ∼ . As an example of the connection between the photon statistics and Heisenberg-limited phase uncertainty, we study parity measurement-based interferometry of the two-mode Barut–Girardello coherent states for SU(1,1)SU(1,1) obtained from the coupling together of two representations of SU(1,1) each associated with single-mode boson realizations with Bargmann indices as discussed by Gerry and Benmoussa [Phys. Rev. A 2000, 62, 033812]. These states generally have distributions with clustered populations along the borders. We demonstrate that these states do, indeed, lead to super-sensitive measurements and also lead to super-resolved interference fringes.

DOI
10.1080/09500340.2011.607927

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Parity measurement-based quantum optical interferometry and metrology is known to be sensitive to the joint photon number distribution, , of the two-mode quantum states inside the interferometer. In cases where the statistics display a clustering of the populations around the borders (i.e. along the axes ) with a plateau over most of the interior, Heisenberg-limited (or super-sensitive) measurements of phase shifts are enabled. This follows from the heuristic number-phase uncertainty relation because of the uncertainty in the location of the photons being ΔN ∼ . As an example of the connection between the photon statistics and Heisenberg-limited phase uncertainty, we study parity measurement-based interferometry of the two-mode Barut–Girardello coherent states for SU(1,1)SU(1,1) obtained from the coupling together of two representations of SU(1,1) each associated with single-mode boson realizations with Bargmann indices as discussed by Gerry and Benmoussa [Phys. Rev. A 2000, 62, 033812]. These states generally have distributions with clustered populations along the borders. We demonstrate that these states do, indeed, lead to super-sensitive measurements and also lead to super-resolved interference fringes.
We study the application of photon-subtracted two-mode squeezed vacuum states (TMSVS), where identical photon numbers are subtracted from each beam, to the problem of quantum optical interferometry with the photon-number parity measurements scheme. Previously, Anisimov et al. [Phys. Rev. Lett.104, 103602 (2010)] have studied the prospect of using the TMSVS for parity measurements based interferometry. However, the joint photon number distribution of this state before beam splitting is thermal-like in each mode, meaning that its statistics are super-Poissonian and that the most probable state of the field is the double vacuum and that the average photon number is low. Furthermore, with these states the sensitivity of the phase-shift measurement is extremely sensitive to the size of the phase shift to be measured for large average photon numbers. The simultaneous subtraction of 1, 2, or 3 photons from each mode has the effects of increasing the average photon numbers of each mode and rendering the statistics nearly Poissonian and in some cases sub-Poissonian. We show that phase uncertainties for such states are less sensitive to the phase shift itself at large average photon numbers. We also show that the photon-subtracted TMSVS also lead to increased resolution.
Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number NNN injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ However, the method of phase detection suggested by those authors, obtaining the difference of the photocurrents at the output ports of the interferometer, is not sensitive to the phase difference between the two interferometer paths; in fact, the photocurrent vanishes. In this paper we show that the use of parity measurements on just one of the output modes not only is sensitive to the phase difference but that the sensitivity approaches the Heisenberg limit for large N.N.N.
After reviewing parity-measurement-based interferometry with twin Fock states, which allows for supersensitivity (Heisenberg limited) and super-resolution, we consider interferometry with two different superpositions of twin Fock states, namely, two-mode squeezed vacuum states and pair coherent states. This study is motivated by the experimental challenge of producing twin Fock states on opposite sides of a beam splitter. We find that input two-mode squeezed states, while allowing for Heisenberg-limited sensitivity, do not yield super-resolutions, whereas both are possible with input pair coherent states.
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
We present a new operator method in the Heisenberg representation to obtain the signal of parity measurement within a lossless SU(1,1) interferometer. Based on this method, it is convenient to derive the parity signal directly in terms of input states, including general Gaussian and non-Gaussian states. As applications, we revisit the signal of parity measurement within an SU(1,1) interferometer when a coherent or thermal state and a squeezed vacuum state are considered as input states. In addition, we obtain the parity signal of an arbitrary single-mode state when it passes through an SU(1,1) interferometer, which is also a new result. Then, we analytically prove that parity measurement can saturate the quantum Cramér–Rao bound when the estimated phase approaches zero. Therefore, the operator method proposed in this work may bring convenience to the study of quantum metrology, particularly the phase estimation based on an SU(1,1) interferometer.
We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.