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arXiv:0910.3364 · Chinese Physics B

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

Fan Hong-Yi, Yuan Hong-Chun

Published Jul 1, 2010• 3 citations
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Abstract

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ (μ, ν) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a † 1 – a 2 ) and ( a † 1 – a 2 ) can be solved and the contents of phase space quantum mechanics can be enriched, where a i , a † i are bosonic creation and annihilation operators, respectively.

DOI
10.1088/1674-1056/19/7/070301

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As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ (μ, ν) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a † 1 – a 2 ) and ( a † 1 – a 2 ) can be solved and the contents of phase space quantum mechanics can be enriched, where a i , a † i are bosonic creation and annihilation operators, respectively.
Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys. A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator Δ ( q',p' ) (q-number transform) in phase space quantum mechanics, and its inverse where Q , P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q–P ordering, P–Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.
Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator Δ(q′,p′)Δ(q',p')Δ(q′,p′) (qqq-number transform) in phase space quantum mechanics, ∬−∞∞dp′dq′/πΔ(q′,p′)e−2i(p−p′)(q−q′)=δ(p−P)δ(q−Q), \iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q), ∬−∞∞​dp′dq′/πΔ(q′,p′)e−2i(p−p′)(q−q′)=δ(p−P)δ(q−Q), and its inverse ∬−∞∞dqdpδ(p−P)δ(q−Q)e2i(p−p′)(q−q′)=Δ(q′,p′),\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'), ∬−∞∞​dqdpδ(p−P)δ(q−Q)e2i(p−p′)(q−q′)=Δ(q′,p′), where Q,Q,Q, PPP are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among Q−PQ-PQ−P ordering, P−QP-QP−Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.
Based on the two mutually conjugate entangled state representations |ξ⟩ and |η⟩, we propose an integration transformation in ξ — η phase space , and its inverse transformation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to ( Q 1 – Q 2 ) ↔ ( P 1 – P 2 ) ordered and simultaneously ( P 1 + P 2 ) ↔ ( Q 1 + Q 2 ) ordered operators.
We find a new complex integration-transform which can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.