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Physical Review A

Number-phase minimum-uncertainty state with reduced number uncertainty in a Kerr nonlinear interferometer

Masahiro Kitagawa, Y. Yamamoto

Published Nov 1, 1986• 404 citations
Publisher

Abstract

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^${〉}^{1/3}$, far below the limit 〈n^${〉}^{2/3}$ 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

DOI
10.1103/physreva.34.3974

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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
The evolution of a quadrature-squeezed state through a Kerr medium is studied from the point of view of amplitude squeezing. We propose a Mach-Zehnder interferometer configuration, which incorporates a squeeze- state generator and a nonlinear Kerr medium, to generate and subsequently displace these states. The displaced states show amplitude squeezing for a proper choice of the initial quadrature squeezed state, and we study two such cases here in detail. In the limit where the evolution through a Kerr medium results in a macroscopic superposition, an initial phase-squeezed coherent state shows significant amplitude squeezing, and we have shown that the photon-number uncertainty 〈(\ensuremath{\Delta}n^)2{)}^{2})2〉 can be minimized to a value smaller than 〈n^〉2/3{\mathrm{〉}}^{2/3}〉2/3. On the other hand, an initial quadrature-squeezed state, whose direction of squeezing is along the direction of excitation, upon evolving through a Kerr medium for a suitable small duration of time shows substantial amplitude squeezing for an appropriate displacement, and we show that the photon-number uncertainty in this case can be minimized to 〈n^〉1/5{\mathrm{〉}}^{1/5}〉1/5, which is far below the previously achieved limits. \textcopyright{} 1996 The American Physical Society.
We study the statistical and phase properties of the output states of a Mach-Zehnder interferometer with a nonlinear Kerr medium in one of its arms. The combination of nonlinearity and displacement produced by the interferometer generates output states (displaced Kerr states) that, unlike those derived from a Kerr medium alone (Kerr states), do not necessarily retain the photon statistics of the initially coherent field. As the parameter characterizing the nonlinear medium is varied, the statistics of the output field vary from sub-Poissonian to super-Poissonian. The photon-number distributions of these states are rationalized by examining their quasiprobability distributions, which vary from banana-shaped for sub-Poissonian statistics to complicated interference structures for highly super-Poissonian statistics. The number-phase properties of the Kerr and displaced Kerr states are studied for the case of small photon numbers (relevant to cavity quantum electrodynamics) using a methodology based on the Pegg-Barnett formalism. In particular, we determine the range of parameters for which the states are minimum-uncertainty states and whether they are squeezed with respect to photon number and phase, drawing a distinction between a sub-Poissonian nature and number squeezing. Finally, we find that sub-Poissonian statistics can coexist with quadrature squeezing for a range of parameters.
Photon number of coherent light obeys Poisson distribution, and its variance is proportional to the intensity of the light. Photon-number squeezed state is the light whose variance is reduced below the standard quantum limit. In general, photon-number squeezed state has been generated with femtosecond pulses and asymmetric fiber interferometer [1]. However, due to difference of nonlinearity in the two arms, mismatch of spectrum shape and phase is occurred for two pulses. As a result, the squeezing level becomes lower. It is especially true for pulses in the normal dispersion regime.
Minimum-uncertainty states in variables which are quadratic in mode creation and annihilation operators can be used to increase the accuracy of interferometric measurements. For a general interferometer we show how to find the relevant uncertainty relation. We consider two specific examples, a Mach-Zehnder interferometer and a Mach-Zehnder interferometer in which the first beam splitter has been replaced by a four-wave mixer. For both devices su(2) squeezed minimum-uncertainty states can be used to achieve phase-measurement accuracies of 1/N, where N is the total photon number at the input. We also describe a method of producing approximate versions of these states.
Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.