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Journal of Optics B Quantum and Semiclassical Optics

Amplification of Schrödinger-cat state: distinguishability and interference in phase space

Radim Filip, Jan Peřina

Published Mar 14, 2001• 8 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/3/2/301

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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.
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 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.
Amplification of the Schrödinger-cat state (even coherent-state superposition) can be accompanied by its non-exponential decoherence and strong squeezing effect in the degenerate optical parametric amplifier. In the classical pump approximation and small thermal noise limit, the Wigner function behaviour illustrates that this non-exponential effect is induced by the phase-selective character of amplification. The non-exponential decoherence is much slower, in contrast with phase-unselective amplification exhibiting faster exponential decay. Thus, if one is able to produce the Schrödinger-cat state with at least a small amplitude, then it can be partially amplified with only a weak decoherence effect.
Quantum fluctuations in optical amplification are investigated with a nondegenerate optical parametric amplifier whose internal idler mode is coupled to a squeezed vacuum. Reductions of the inherent quantum noise of the amplifier are observed with a minimum noise level 0.7 dB below the usual noise level of the amplifier with its internal idler mode in a vacuum state. With a small coherent field as the signal input, the amplified output exhibits an improvement in signal-to-noise ratio of 0.5 dB for the case of a squeezed-vacuum as compared to a vacuum state for the amplifier's internal mode.
Coherent state superpositions, also known as Schrödinger cat states, are widely recognized as promising resources in quantum information, quantum metrology, as well as fundamental tests. These states are hard to produce deterministically and most schemes for their probabilistic generation can only attain amplitudes too small for practical use. This is for example the case for photon-subtracted squeezed vacuum (PSSV), which can be used to approximate cat states of amplitude no larger than y = 1.5 if the fidelity is to be maintained above 95%. One way to reach larger amplitudes is to start with pairs of small cats and then to interfere them on a balanced beam splitter. The projective measurement of one of the outputs is used to herald a larger cat resulting from the constructive interference of the initial states. The scheme proposed here uses the projection |x = 0〉〈x = 0| as the heralding condition. Homodyning is proposed, as opposed to photon counting, because homodyne detection has high a quantum efficiency, and - as demonstrated in the paper - can be tuned to increase the success probability of the amplification without heavily compromising the output's fidelity.
Quantum fluctuations in an optical amplifier are of fundamental as well as technological significance and lead to basic limitations on noise performance. It is well-known that the signal-to-noise ratio (SNR) of an injected signal is degraded in the amplification process due to added noise from thermal or (at best) vacuum fluctuations of the internal modes of the amplifier.[1] However, if the internal modes are coupled to a squeezed vacuum, the added noise in amplification can be reduced.[2] With this motivation, we have constructed a nondegenerate parametric amplifier whose idler mode is itself coupled to a squeezed vacuum, which is generated by a second parametric amplifier. Phase-sensitive noise amplification of a vacuum input to the signal mode is observed with the minimum amplified noise level 0.5dB below the usual phase-insensitive amplified noise level for the amplified signal of the same amplifier but with the idler mode coupled to a vacuum state. Prospects for improvement by increasing the squeezing that is coupled into the idler mode of the amplifier beyond the current level of 2dB as well as surprising results from our theoretical analysis which includes losses will be discussed.