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Journal of Mathematical Physics

Symmetries of two-mode squeezed states

D. Han, Y. S. Kim, Marilyn E. Noz, Leehwa Yeh

Published Dec 1, 1993• 28 citations
Publisher

Abstract

It is known that the symmetry of two-mode squeezed states is governed by the group Sp(4) which is locally isomorphic to the O(3,2) de Sitter group. It is shown that this complicated ten-parameter group can be regarded as a product of two three-parameter Sp(2) groups. It is shown also that two coupled harmonic oscillators serve as a physical basis for the symmetry decomposition. It is shown further that the concept of entropy is needed when one of the two modes is not observed. The entropy is zero when the system is uncoupled. The system reaches thermal equilibrium when the entropy becomes maximal.

DOI
10.1063/1.530318

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It is known that the symmetry of two-mode squeezed states is governed by the group Sp(4) which is locally isomorphic to the O(3,2) de Sitter group. It is shown that this complicated ten-parameter group can be regarded as a product of two three-parameter Sp(2) groups. It is shown also that two coupled harmonic oscillators serve as a physical basis for the symmetry decomposition. It is shown further that the concept of entropy is needed when one of the two modes is not observed. The entropy is zero when the system is uncoupled. The system reaches thermal equilibrium when the entropy becomes maximal.
It is shown that the basic symmetry of two-mode squeezed states is governed by the group Sp(4) in the Wigner phase space which is locally isomorphic to the (3+2)-dimensional Lorentz group. This symmetry, in the Schr\"odinger picture, appears as Dirac's two-oscillator representation of O(3,2). It is shown that the SU(2) and SU(1,1) interferometers exhibit the symmetry of this higher-dimensional Lorentz group. The mathematics of two-mode squeezed states is shown to be applicable to other branches of physics including thermally excited states in statistical mechanics and relativistic extended hadrons in the quark model.
It is noted that the symmetry of two-mode squeezed states of light is governed by the group Sp(4) that is locally isomorphic to O(3,2). This group has subgroups that are locally isomorphic to the two-dimensional Euclidean group. Two-mode states having the E(2) symmetry are constructed. The translation-like transformations of this symmetry group shear the Wigner distribution function defined over the four-dimensional phase space consisting of two pairs of canonical variables. Sheared states are constructed in the Schrödinger picture of quantum mechanics and in the Fock space for photon numbers. It is shown that the Wigner phase-space picture is a convenient representation of quantum mechanics for calculating measurable quantities of the sheared states.
Mode entanglement in a two-mode squeezed state is studied. The entanglement spectrum is derived and interpreted as Boltzmann coefficients of harmonic oscillator eigenstates at thermal equilibrium. The effective temperature is expressed in terms of squeezing parameters and is equal to zero when the state is unsqueezed and increases without bound as the state is infinitely squeezed. It is demonstrated that the reduced density operator can be identified as a canonical ensemble of single harmonic oscillator at some effective temperature Te. The Renyi entanglement entropy is calculated and found to increase without bound as the squeezing parameter increases. The single-copy entanglement, which is a limiting value of the Renyi entropy, is interpreted as a partition function.
A general anaylsis of squeezing transformations for two-mode systems is given based on the four-dimensional real symplectic group Sp(4,openR). Within the framework of the unitary (metaplectic) representation of this group, a distinction between compact photon-number-conserving and noncompact photon-number-nonconserving squeezing transformations is made. We exploit the U(2) invariant squeezing criterion to divide the set of all squeezing transformations into a two-parameter family of distinct equivalence classes with representative elements chosen for each class. Familiar two-mode squeezing transformations in the literature are recognized in our framework and seen to form a set of measure zero. Examples of squeezed coherent and thermal states are worked out. The need to extend the heterodyne detection scheme to encompass all of U(2) is emphasized, and known experimental situations where all U(2) elements can be reproduced are briefly described.
In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) applicable to a six-dimensional space consisting of three space-like and three time-like coordinates, or SL(4,r) in the four-dimensional phase space consisting of two position and two momentum variables. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r) respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics. In quantum mechanics, there is the lower bound in the size of phase space for each mode determined by the uncertainty principle while there are no restriction on the phase-space size in classical mechanics. This is the reason why the symmetry is smaller in quantum mechanics.
The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the O ( 3 , 2 ) de Sitter group containing two O ( 3 , 1 ) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the S p ( 4 ) group, which serves as the basic language for two-mode squeezed states. Since the S p ( 4 ) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.