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Physical Review A

Superpositions of coherent states: Squeezing and dissipation

Vladimír Bužek, A. Vidiella-Barranco, P. L. Knight

Published May 1, 1992• 426 citations
Publisher

Abstract

In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.

DOI
10.1103/physreva.45.6570

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In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.
We consider the quantum superposition of two coherent states shifted in phase by 3π/2, |α⟩ and |− i α⟩, as well as special states that are characterized solely with respect to amplitude α. According to analysis of nonclassical properties such as oscillatory and sub-Poissonian distribution of photon numbers, photon antibunching, quadrature squeezing, and negativity of Wigner function, it is concluded that the considered superposition state is a nonclassical state. Also, the special state is found to exhibit even stronger nonclassical features when relative phase ϕ of the superposition is equal to the average photon number |α| 2 .
The investigation of photon statistics and quadrature squeezing in superpositions of two coherent states under the influence of a continuous measurement of photon number shows that the photon statistics of the cavity mode oscillates between sub-Poissonian and super-Poissonian distributions and that the quadrature fluctuation oscillates between squeezing and nonsqueezing at each time when one photon of a cavity mode is registered by a counter. These oscillatory behaviors are caused by the influence of the continuous measurement of photon number and the quantum interference between the two coherent states in the superposition.
Abstract Superposition of two coherent states, the Schrodinger’s cat state, can exhibit different nonclassical properties having foundational applications in quantum information processing. We consider the ‘superposition of Schrodinger’s cat state with the vacuum state (SCVS)’ of the optical field. We discuss different witness of nonclassicality properties such as lower- and higher-order squeezing (viz., squeezing, Hong & Mandel’s fourth-order squeezing, amplitude-squared squeezing) and sub-Poissonian photon statistics. Further, we discuss the negativity of the Wigner function of SCVS indicating the nonclassicality of the state under investigation. We find that the vacuum state contribution in SCVS exhibits different nonclassicalities under some conditions stronger where the nonclassicalities exhibited by the state without vacuum state contribution are weaker.
It is shown that a particular combination of so-called even coherent state dubbed the orthogonal-even coherent state, while exhibiting no ``ordinary'' or second-order squeezing, can furnish near-optimal simultaneous-quadrature fourth-order squeezing [where the latter term is defined in the sense of Hong and Mandel, Phys. Rev. Lett. 54, 323 (1985)]. The Wigner function of the particular quantum-mechanical superposition of states is calculated and various plots compared to the Wigner function of a simple statistical mixture of the same states. Rather uninteresting behavior is found in the latter case, which illustrates that the higher-order squeezing effect is due to nonclassical, quantum-mechanical ``interference in phase space.''
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.