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Journal of the Optical Society of America B

Quantum theory of the nonlinear interferometer

Masataka Shirasaki, H. A. Haus, D. Liu Wong

Published Jan 1, 1989• 24 citations
Publisher

Abstract

A full quantum theory of the Mach–Zehnder interferometer with nonlinear Kerr media in both arms is presented. The nonlinear operator equations can be integrated, and the result can be normal-ordered. This permits evaluation of the factorial moments of the photon number to all orders of the nonlinearity. Various excitations and phase biases are studied. The degree of squeezing by the interferometer is determined.

DOI
10.1364/josab.6.000082

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A full quantum theory of the Mach–Zehnder interferometer with nonlinear Kerr media in both arms is presented. The nonlinear operator equations can be integrated, and the result can be normal-ordered. This permits evaluation of the factorial moments of the photon number to all orders of the nonlinearity. Various excitations and phase biases are studied. The degree of squeezing by the interferometer is determined.
A symmetric nonlinear Mach–Zehnder interferometer containing a Kerr medium performs the operation of squeezing. The operation is performed by the interference of cotraveling waves, and the pump wave is removed. This mechanism is broadband. We analyze the operation of the interferometer and determine the degree of shot-noise reduction achieved in a balanced detector. A modification of the interferometer into a fiber ring reflector is described that accomplishes the squeezing and the pump separation.
A Mach–Zehnder interferometer with a nonlinear medium in each arm is treated as an all-optical switching device. In a classical treatment, the interferometer can, in principle, operate as an ideal switch, but quantization of the electromagnetic fields limits the switching capability. In this treatment we illustrate the fundamental quantum limits that apply to switching in all-optical devices by studying the nonlinear Mach–Zehnder interferometer. In particular the input states are assumed to be coherent states and the reduction in the quality of the switching device is discussed.
The output state of a nonlinear Mach-Zehnder interferometer is shown to be an effective number-phase minimum-uncertainty state with reduced photon-number uncertainty. This interferometer includes an optical Kerr medium in one arm with a coherent-state input. Unusual ``crescent''-shaped squeezing which preserves photon number is revealed in the unitary evolution associated with a self-phase-modulation in the Kerr medium. Photon-number uncertainty 〈\ensuremath{\Delta}n${^}^{2}$〉 can be reduced by interference with a coherent-state reference wave. It can be minimized to 〈n^〉1/3{〉}^{1/3}〉1/3, far below the limit 〈n^〉2/3{〉}^{2/3}〉2/3 achieved by an ordinary squeezed state. The increased phase uncertainty due to self-phase-modulation and the reduced photon-number uncertainty still preserve the minimum-uncertainty product 〈\ensuremath{\Delta}n${^}^{2}$〉〈\ensuremath{\Delta}\ensuremath{\Phi}${^}^{2}$〉\ensuremath{\sim}(1/4). .AE
Nonlinear interferometers are Mach-Zehnder interferometers with Kerr media in either one or both arms. We refer to these devices, respectively, as the asymmetric and symmetric nonlinear interferometers. In the asymmetric case, with one input mode in the vacuum, it is possible to generate maximally entangled photonic states or superpositions of such states. We consider the device as a resource of entangled states for applications to Heisenberg-limited interferometry. Interferometry with the maximally entangled states cannot be performed by simply subtracting the output photocounts as in standard interferometry. Instead, one must perform parity measurements on only one of the output beams. We show that the symmetric nonlinear interferometer, with one input mode in the vacuum, may be used to perform such parity measurements. The same device is shown to produce, with an input coherent state and upon projective measurements, even or odd coherent states, examples of the Schr\"odinger-cat states.