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Physical Review A

Quantum statistics of a two-mode SU(1,1) interferometer

Ulf Leonhardt

Published Feb 1, 1994• 36 citations
Publisher

Abstract

A comprehensive formalism for a two-mode SU(1,1) interferometer (a nondegenerate parametric amplifier, a degenerate four-wave mixer, or a phase-conjugating mirror) is presented. The device changes the quantum state of the two incident modes by an SU(1,1) transformation. Apart from phase shiftings, the quadrature wave functions are Lorentz transformed. A transformation formula in the Fock space of the two modes is derived. Hybrids of s and -s quasiprobability distributions are changed by the inverse-mode transformation. It is related to quantum noise in phase conjugation. Linear amplification caused by Gaussian reservoirs corresponds exactly to a heuristic nondegenerate-parametric-amplifier model. It allows a simple geometrical explanation for the sensitivity of squeezing or quantum superposition with respect to phase-sensitive amplification.

DOI
10.1103/physreva.49.1231

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A comprehensive formalism for a two-mode SU(1,1) interferometer (a nondegenerate parametric amplifier, a degenerate four-wave mixer, or a phase-conjugating mirror) is presented. The device changes the quantum state of the two incident modes by an SU(1,1) transformation. Apart from phase shiftings, the quadrature wave functions are Lorentz transformed. A transformation formula in the Fock space of the two modes is derived. Hybrids of s and -s quasiprobability distributions are changed by the inverse-mode transformation. It is related to quantum noise in phase conjugation. Linear amplification caused by Gaussian reservoirs corresponds exactly to a heuristic nondegenerate-parametric-amplifier model. It allows a simple geometrical explanation for the sensitivity of squeezing or quantum superposition with respect to phase-sensitive amplification.
A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.
A new formalism for analyzing two-photon devices, such as parametric amplifiers and phase-conjugate mirrors, is proposed in part I, focusing on the properties and the significance of the quadrature-phase amplitudes and two-mode squeezed states. Time-stationary quasi-probability noise is also detailed for the case of Gaussian noise, and uncertainty principles for the quadrature-phase amplitudes are outlined, as well as some important properties of the two-mode states. Part II establishes a mathematical foundation for the formalism, with introduction of a vector notation for compact representation of two-mode properties. Fundamental unitary operators and special quantum states are also examined with an emphasis on the two-mode squeezed states. The results are applied to a previously studied degenerate limit (epsilon = 0).
The theory of a two-mode nondegenerate parametric amplifier in a cavity is reformulated in terms of quadrature-phase-amplitude variables. The corrrespondence with a genuine classical stochastic linear process is found (non-negative-definite diffusion matrices) for the case of a cavity device immersed in thermal or ordinary (nonsqueezed) vacuum sources. A special kind of squeezing, i.e., quadrature squeezing [B. L. Schumaker, Phys. Rep. 135, 318 (1986)], is found to be characteristic of the internal steady state for the case of a cavity model subjected to an additional phase-sensitive noise source coupled to one of the two internal modes. Finally, the usual squeezing spectrum of the output field is calculated in both cases by means of an input-output formalism based upon a symmetric ordering scheme of noncommuting operators, well adapted for the quadrature-phase description of the fields involved in the interaction.
In the framework of Hamiltonian formalism the quantum statistical properties of an optical field propagating inside a directional coupler operating by nondegenerate parametric amplification are studied. We investigate the effect of switching between the input modes and the outgoing fields from the coupler. Particular attention is paid to two-mode squeezing, the second-order correlation function, quasiprobability distribution functions, and photon-number distribution. Incident number and coherent states are considered. It has been shown that when one of the modes enters the coupler in the Fock state |1 and the other modes are in vacuum states, the coupler can serve as a generator of a coherent state. Furthermore, regimes for the generation and transmission of squeezed and/or sub-Poissonian light are found.
An SU(1,1) interferometer uses a sequence of two optical parametric amplifiers for achieving sub-shot-noise sensitivity to a phase shift introduced in between. We present the first realization of a wide-field SU(1,1) interferometer, where the use of a focusing element enables spatially multimode operation within a broad angle. Over this angle, the interference phase is found to be flat. This property is important for the high sensitivity to the phase front disturbance. Further, -4.3 +/- 0.7 dB quadrature squeezing, an essential requirement to the high sensitivity, is experimentally demonstrated for plane-wave modes inside the interferometer. Such an interferometer is useful not only for quantum metrology, but also in remote sensing, enhanced sub-shot-noise imaging, and quantum information processing. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
A Lie-group-theoretical approach to the analysis of interferometers is presented. Conventional interferometers such as the Mach-Zehnder and Fabry-Perot can be characterized by SU(2). We introduce a class of interferometers characterized by SU(1,1). These interferometers employ active elements such as four-wave mixers or degenerate-parametric amplifiers in their construction. Both the SU(2) and SU(1,1) interferometers can in principle achieve a phase sensitivity \ensuremath{\Delta}\ensuremath{\varphi} approaching 1/N, where N is the total number of quanta entering the interferometer, provided that the light entering the input ports is prepared in a suitable quantum state. SU(1,1) interferometers can achieve this sensitivity with fewer optical elements.