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Journal of Optics B Quantum and Semiclassical Optics

Wavefunctions of two-mode states in entangled-state representation

Hong-Yi Fan, Alfred Wünsche

Published May 16, 2005• 8 citations

Abstract

Two-mode entangled-state wavefunctions and with complex variables z and y are introduced and investigated. They are direct analogues of the wavefunctions and in the single-mode case or are linearly related to each other by two-dimensional Fourier transformation. The states and are eigenstates of two pairs of commuting operators ( Z , Z † ) (or ( Q + , P − )) and ( Y , Y † ) (or ( P + , Q − )) to eigenvalues ( z , z * ) and ( y , y * ), respectively, and are normalized by means of the two-dimensional delta function. The entangled states and are represented as two limiting cases of two-mode squeezed coherent states. Different representations of these states, in particular, the analogue of the Agarwal–Simon representation of squeezed vacuum states, are derived and the properties of these states are discussed. The transition from the single-mode to the two-mode case is made using the common property of the squeezing operators to belong to different realizations of the abstract SU (1,1) group. The Wigner quasiprobability in representation of the states and is discussed and is explicitly calculated for two-mode squeezed vacuum (and squeezed coherent) states.

DOI: 10.1088/1464-4266/7/6/r02

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Two-mode entangled-state wavefunctions and with complex variables z and y are introduced and investigated. They are direct analogues of the wavefunctions and in the single-mode case or are linearly related to each other by two-dimensional Fourier transformation. The states and are eigenstates of two pairs of commuting operators ( Z , Z † ) (or ( Q + , P − )) and ( Y , Y † ) (or ( P + , Q − )) to eigenvalues ( z , z * ) and ( y , y * ), respectively, and are normalized by means of the two-dimensional delta function. The entangled states and are represented as two limiting cases of two-mode squeezed coherent states. Different representations of these states, in particular, the analogue of the Agarwal–Simon representation of squeezed vacuum states, are derived and the properties of these states are discussed. The transition from the single-mode to the two-mode case is made using the common property of the squeezing operators to belong to different realizations of the abstract SU (1,1) group. The Wigner quasiprobability in representation of the states and is discussed and is explicitly calculated for two-mode squeezed vacuum (and squeezed coherent) states.

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We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n -pair entangled state representation . The explicit form of the 2 n -mode squeezing operator which squeezes the n -pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2 n -mode squeezed state for M being Hermitian, and obtain the variances of the 2 n -mode quadrature operators in the 2 n -mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.

The properties of states generated by the application of the two-mode squeeze operator to the pair coherent states are studied. These states are the two-mode analogues of the single-mode squeezed states generated by the application of the single-mode squeeze operator to an ordinary coherent state. In the present case there are correlations between the modes and strong non-classical properties are to be expected. We study the statistical properties of the photon number distributions, squeezing, violations of the Cauchy-Schwartz inequality, quasiprobability distributions and the phase distributions.