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Acta Sinica Quantum Optica

Calculation of the Quadrature-amplitude Measurement in the Two-mode Squeezed Vacuum State via the Entangled State Representation

Xuefen Xu

Published Jan 1, 2006• 0 citations
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Abstract

Based on the subject proposed in Ref.[4](Acta Sinica Quantum Optica V.8(2002) 47) this paper makes some new profound discussions.It calculates the result of the quadrature-amplitude measurement on the first-mode field of a two-mode squeezed vacuum state via the entangled state representation,and it is shown that the second-mode field collapses to a single-mode squeezed state,whose explicit form is derived.

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Based on the subject proposed in Ref.[4](Acta Sinica Quantum Optica V.8(2002) 47) this paper makes some new profound discussions.It calculates the result of the quadrature-amplitude measurement on the first-mode field of a two-mode squeezed vacuum state via the entangled state representation,and it is shown that the second-mode field collapses to a single-mode squeezed state,whose explicit form is derived.
Based on the fact that a two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l -photon measurement for the two-mode squeezed number state S 2 | m , n ⟩ is discussed. It is found that a remaining field-mode simultaneously collapses into a number state | n – m + l ⟩ with the coefficient being a Jacobi polynomial of n , m and l , which manifestly exhibits the entanglement between the two modes, i.e. it depends on the number-difference between the two modes. The second mode collapses into an excited coherent state when the first mode is measured as a coherent state.
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.
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.