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arXiv:1310.7421

Repeatable two-mode phase measurement

Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi

Published Oct 28, 2013• 0 citations
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Abstract

A measurement scheme to perform a repeatable phase detection on a two-mode field is presented. The interaction with the probe state (the output state of a phase-insensitive high-gain amplifier) is described by a Hamiltonian which is physically realizable in the rotating wave approximation. Information on the system is obtained through unconventional heterodyne measurements performed on the probe field after the interaction with the system. The expressions for the probability distribution and the state reduction are given.

DOI
10.48550/arxiv.1310.7421

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A measurement scheme to perform a repeatable phase detection on a two-mode field is presented. The interaction with the probe state (the output state of a phase-insensitive high-gain amplifier) is described by a Hamiltonian which is physically realizable in the rotating wave approximation. Information on the system is obtained through unconventional heterodyne measurements performed on the probe field after the interaction with the system. The expressions for the probability distribution and the state reduction are given.
We present an experimental scheme that achieves ideal phase detection on a two-mode field. The two modes a and b are the signal and image band modes of a heterodyne detector, with the field approaching an eigenstate of the photocurrent Z^=a+${\mathit{b}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$. The field is obtained by means of a high-gain phase-insensitive amplifier followed by a high-transmissivity beam splitter with a strong local oscillator at the frequency of one of the two modes.
We consider the simultaneous measurement of two conjugate variables by coupling the system of interest to two independent probe modes. Our model consists of linearly coupled boson modes that can be realized by quantum optical fields in the rotating-wave approximation. We approach the setup both as a device to extract observable information and to prepare an emerging quantum state. The initial states of the probe modes and the coupling are utilized to optimize the operation in the various regimes. In contrast to the Arthurs and Kelly ideal scheme [Bell. Syst. Tech. J. 44, 725 (1965)], our more realistic coupling does not allow perfect operation but the ideal situations can be approximated closely. We discuss the conditions for maximum information transfer to the probe modes, information extraction with minimum disturbance of the system mode, and optimal state preparation for subsequent measurements. The minimum disturbance operation can be made to approximate a nondemolition measurement, especially when the information is carried in one quadrature component only. In the preparation mode, we find that the recording accuracy of the probe signals plays an essential role. We restrict the discussion to the first and second moments only, but the method can easily be generalized to any situation. Choosing all modes to be in squeezed coherent states originally, we can carry out analytical considerations; other cases can be treated numerically. The results are presented and discussed in detail as the paradigm of a class of realizable measurements.
The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.
Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number N and number of input modes M of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on the input modes to the amplifier is diagonal in the multimode number basis are proven to be quantum optimal under the same gain-independent measurement. We compare the best precision achievable using classical probes to the performance of an explicit photon-counting-based estimator on quantum probes and show that an advantage exists even for single-photon probes and inefficient photodetection. A closed-form expression for the energy-constrained Bures distance between two product amplifier channels is also derived.
The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.