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Physical Review A

Displaced squeezed number states: Position space representation, inner product, and some applications

Klaus B. Mo ller, Thomas Godsk Jørgensen, Joachim Dahl

Published Dec 1, 1996• 60 citations
Publisher

Abstract

For some applications the overall phase of a quantum state is crucial. For the so-called displaced squeezed number state (DSN), which is a generalization of the well-known squeezed coherent state, we obtain the position space representation with the correct overall phase, from the dynamics in a harmonic potential. The importance of the overall phase is demonstrated in the context of characteristic or moment generating functions. For two special cases the characteristic function is shown to be computable from the inner product of two different DSNs. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.5378

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For some applications the overall phase of a quantum state is crucial. For the so-called displaced squeezed number state (DSN), which is a generalization of the well-known squeezed coherent state, we obtain the position space representation with the correct overall phase, from the dynamics in a harmonic potential. The importance of the overall phase is demonstrated in the context of characteristic or moment generating functions. For two special cases the characteristic function is shown to be computable from the inner product of two different DSNs. \textcopyright{} 1996 The American Physical Society.
The displaced parity operators are shown to have properties that bear a deep relationship with those of the Wigner functions. By exploiting these properties we show that these operators play an important role in linking together many of the aspects of the various exact phase-space mappings of quantum mechanics. These include the Wigner and Weyl representations, coherent states and the Bargmann representation, the P and Q representations, the Weyl correspondence, and the Moyal star product formalism. We also introduce corresponding displaced Fourier operators and show that their squares are just the displaced parity operators. The formalism is extended to squeezed and displaced parity operators, and their corresponding central role in the theory of squeezed coherent states and general squeezing is explained. We also elucidate the part played by the displaced parity operators in the Moyal star product and its extensions, as a first step towards a potential application of these operators in such modern developments as deformation theory and quantum groups. Finally, we indicate how the apparatus developed might also find applications in other recent exact classical mappings of many-particle quantum mechanics or quantum field theory, which are not special cases of deformation theory. Prime examples here include the powerful so-called independent-cluster method techniques, which incorporate the coupled-cluster method formalism with its inbuilt supercoherent states. Throughout the work we stress the central and unifying role played by the displaced parity operator and its generalizations.
We compute the photon-number distribution, the Q (α) distribution function and the wavefunctions in the momentum and position representation for a single mode squeezed number state using generating functions which allow one to obtain any matrix element in the squeezed number state representation from the matrix elements in the squeezed coherent state representation. For highly squeezed number states we discuss the previously unnoted oscillations which appear in the Q (α) function. We also note that these oscillations can be related to the photon-number distribution oscillations and to the momentum representation of the wavefunction.
The fundamental properties of recently introduced stretched coherent states are investigated. It has been shown that stretched coherent states retain the fundamental properties of standard coherent states and generalize the resolution of unity, or completeness condition, and the probability distribution that nnn photons are in a stretched coherent state. The stretched displacement and stretched squeezing operators are introduced and the multiplication law for stretched displacement operator is established. The results of the action of the stretched displacement and stretched squeezing operators on the vacuum and the Fock states are presented. Stretched squeezed stretched coherent states and stretched squeezed stretched displaced number states are introduced and their properties are studied. The inner product of two quantum mechanical vectors was defined in terms of their stretched coherent state representations, and functional Hilbert space was introduced.
We introduce, generate and study a family of supersymmetric displaced number states (SDNS) that can be considered generalized coherent states of the supersymmetric harmonic oscillator. The family is created from the seminal supersymmetric boson-fermion entangling annihilation operator introduced by Aragone and Zypman and later expanded by Kornbluth and Zypman. Using the momentum representation, the states are obtained analytically in compact form as displaced supersymmetric number states. We study their position-momentum uncertainties, and their bunchiness by classifying them according to their Mandel Q-parameter in phase space. We were also able to find closed form analytical representations in the space and number basis.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have different correlations with the squeezing and coherence components. Moreover, different from the properties of the position, the average value and variance of the particle number operator as well as the second-order correlation function are time independent.
We develop a new method of representation of quantum states in terms of the displaced number states. We call it representation, where is an amplitude of the base displaced states. In particular, representation was obtained for set of the displaced number states with different amplitude of the displacement, two-mode squeezed vacuum (TMSV) and superposition of vacuum and single photon. The treatment is employed for building of elementary linear operators, in particular, two-qubit control-sign gate, Hadamard matrix between two two-dimensional Hilbert spaces. Main idea underlying the method is extraction of the displaced number state from initial state. Coherent qubit with large amplitude taken simultaneously two modes shifts two-mode target qubit on phase plane and constructive interference appears after measurement of single photon. The results are exact, any approximations are not exploited. We show implementation of the gates with almost unity fidelity of the output states is possible in realistic scenario.