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Physical Review A

Amplification of superposition states in phase-sensitive amplifiers

M. S. Kim, K. S. Lee, Vladimír Bužek

Published May 1, 1993• 28 citations
Publisher

Abstract

We study statistical properties of quantum superposition states (Schr\"odinger-cat states) amplified by phase-sensitive (squeezed) amplifiers. We show that the phase-sensitive amplifier with a properly chosen phase can preserve quantum coherences and nonclassical behavior of the Schr\"odinger-cat-state input even for a gain factor G larger than 2. In particular, we show that for an even coherent state (CS) phase-sensitive amplifiers can preserve squeezing for G>2 but simultaneously in the process of amplification the noise added by the amplifier leads to a rapid increase of fluctuations in the photon number. Because of the finite maximum degree of squeezing obtainable for the even CS the maximum gain factor ${\mathit{G}}_{\mathit{m}}$ for which squeezing can still be observed in the output state is finite. The phase-sensitive amplifier with a properly chosen phase can also reduce fluctuations in the photon number of the initial even CS. Nevertheless, one cannot amplify the initial even CS with super-Poissonian photon statistics into the state with sub-Poissonian photon statistics.

DOI
10.1103/physreva.47.4302

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We study statistical properties of quantum superposition states (Schr"odinger-cat states) amplified by phase-sensitive (squeezed) amplifiers. We show that the phase-sensitive amplifier with a properly chosen phase can preserve quantum coherences and nonclassical behavior of the Schr"odinger-cat-state input even for a gain factor G larger than 2. In particular, we show that for an even coherent state (CS) phase-sensitive amplifiers can preserve squeezing for G>2 but simultaneously in the process of amplification the noise added by the amplifier leads to a rapid increase of fluctuations in the photon number. Because of the finite maximum degree of squeezing obtainable for the even CS the maximum gain factor Gm{\mathit{G}}_{\mathit{m}}Gm​ for which squeezing can still be observed in the output state is finite. The phase-sensitive amplifier with a properly chosen phase can also reduce fluctuations in the photon number of the initial even CS. Nevertheless, one cannot amplify the initial even CS with super-Poissonian photon statistics into the state with sub-Poissonian photon statistics.
We analyze statistical properties of multicomponent superpositions of coherent states of light amplified by quantum-mechanical amplifiers with reduced quantum fluctuations. We show that sub-Poissonian (i.e., amplitude-squeezed) amplifiers can transform any initial superposition of coherent states of light into a nonclassical state with a high degree of sub-Poissonian photon statistics. Simultaneously the shape of the photon-number distribution of the initial state of light is preserved but shifted toward higher photon numbers. On the other hand, we show that phase-sensitive (i.e., quadrature-squeezed) amplifiers may preserve the initial reduction of quadrature fluctuations (quadrature squeezing). We show that by using amplifiers with reduced quantum noise one can, in principle, control the influence of the noise transferred from the amplifier to the system.
It has long been known that amplification will degrade an optical signal and will rapidly destroy any intrinsically quantum features that may have been associated with the input. However, devices that exhibit squeezed fluctuations in the gain or loss medium can reduce the detrimental effects of the inevitable amplifier noise. This makes it possible to retain nonclassical features (in particular squeezing) at higher levels of gain than is possible in conventional devices. In this paper we study the effect of these amplifiers on the photon-number fluctuations. We find that, for suitable inputs, phase-sensitive amplifiers can reduce the tendency for the photon-number distribution to spread and allow us to retain sub-Poissonian statistics at a higher level of gain than is possible in conventional devices.
We examine the nonclassical property of a Schr\"odinger cat state, i.e., a superposition of two coherent states, when it is amplified by either a phase-sensitive or a phase-insensitive amplifier. In both cases the corresponding P representation remains nonpositive in some area of the complex plane, for any amplification factor. The contribution to the P function from these areas decreases exponentially with an amplification factor. With a phase-sensitive amplifier the onset of decay can be postponed to a larger amplification factor. Thus the nonclassical nature of the field can be preserved under a wide range of conditions, in spite of the noise introduced in the amplification process. \textcopyright{} 1996 The American Physical Society.
Using the Wigner function formalism in phase space, we analyze the decay of quantum coherences in phase-sensitive reservoirs. We show that the decay rate of quantum coherences in phase-sensitive reservoirs can be significantly modified compared to the decay rate in ordinary (phase-insensitive) thermal reservoirs. Depending on the phases of the quantum system (field mode) and the squeezed reservoir, the decay rate of the quantum coherence can be either enhanced or significantly suppressed, which is in agreement with the results obtained recently by other methods [T. A. B. Kennedy and D. F. Walls, Phys. Rev. A 37, 152 (1988)]. We show that in an ideally squeezed reservoir with a high degree of squeezing, the decay rate of the quantum coherence (i.e., the decay rate of off-diagonal terms of the density matrix in the coherent-state basis) can be equal to the decay rate of the energy of the system (i.e., the decay rate of diagonal terms of the density matrix). Suppression of the decay rate of the quantum coherence leads to preservation of nonclassical effects such as the oscillations in the photon number distribution. Moreover, we find that some initial superposition states of light exhibiting super-Poissonian photon statistics can be transformed into intermediate sub-Poissonian states under the influence of phase-sensitive reservoirs.
Amplification and cloning of the Schrödinger-cat state (even coherent-state superposition) in the optical nondegenerate parametric amplifier are analysed. In the long-time limit, distinguishability and interference in the marginal probability distributions of the Schrödinger-cat state cannot be preserved simultaneously during amplification. The mutual ratio between distinguishability and interference can be controlled by the squeezed vacuum state injected in the idler mode. Thus only one of either quantum interference of indistinguishable states or distinguished states without interference can be obtained on the output of the amplifier. The purity of the initial state in the signal mode disappears during the amplification for both cases, even if the interference is preserved. Thus the interference in the marginal probability distribution is connected with indistinguishability in the phase space rather than with purity of the state. On the other hand, the nonclassical oscillations in the photon-number distribution vanish simultaneously, as the state is more mixed, irrespective of the existence of the interference.