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Physical Review A

Minimum-correlation mixed quantum states

G. S. Agarwal, Sergey A. Ponomarenko

Published Mar 13, 2003• 12 citations
Publisher

Abstract

We consider states leading to the equality sign in the uncertainty inequalities associated with correlations in open quantum systems which have been recently derived by Ponomarenko and Wolf [Phys. Rev. A 63, 062106 (2001)]. The new inequalities involve fluctuations defined in terms of the square of the density operator that characterizes mixed states. We find the minimum-correlation states associated with the quadratures of single-mode and two-mode electromagnetic fields in a cavity and for the angular momentum operators which can describe atomic degrees of freedom. We show that while in the case of single-mode quadratures the functional form of the minimum-correlation state is uniquely specified, this is not so for the other pairs of noncommuting operators. In general, the states with the least amount of correlations are mixed and they exhibit squeezing.

DOI
10.1103/physreva.67.032103

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We consider states leading to the equality sign in the uncertainty inequalities associated with correlations in open quantum systems which have been recently derived by Ponomarenko and Wolf [Phys. Rev. A 63, 062106 (2001)]. The new inequalities involve fluctuations defined in terms of the square of the density operator that characterizes mixed states. We find the minimum-correlation states associated with the quadratures of single-mode and two-mode electromagnetic fields in a cavity and for the angular momentum operators which can describe atomic degrees of freedom. We show that while in the case of single-mode quadratures the functional form of the minimum-correlation state is uniquely specified, this is not so for the other pairs of noncommuting operators. In general, the states with the least amount of correlations are mixed and they exhibit squeezing.
We study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.
Quantum squeezing, a major resource for quantum information processing and quantum metrology, is best analyzed in terms of the field quadratures - the quantum optical analogues of position and momentum, which form the continuous-variable formalism of quantum light. Degenerate squeezing admits a very helpful and simple description in terms of the single-mode quadrature operators, but the non-degenerate case, i.e. when the squeezing involves pairs of modes, requires a more complicated treatment involving correlations between the quadratures of the different modes. We introduce a generalized set of complex quadrature operators that treats degenerate and non-degenerate squeezing on equal footing. We describe the mode-pairs (and photon-pairs) as a single entity, generalizing the concept of single-mode quadrature operators to two-mode fields of any bandwidth. These complex operators completely describe the SU(1,1) algebra of two-photon devices and directly relate to observable physical quantities, like power and visibility. Based on this formalism, we discuss the measurement of optically-broad squeezed signals with direct detection, and present a compact set of phase-dependent observables that completely and intuitively determine the two-mode squeezed state, and quantify the degree of inseparability and entanglement between the modes.
A displacement operator for a subset of the supercoherent states is found. It is also shown that a diagonal version of the squeezing operator reduces the fluctuations of one of the quadratures, but the reduction is not enough to go below the standard quantum noise level. New states are found, which are minimum uncertainty for the new operators X and P, and lead to coherence and squeezing, in the sense of equal fluctuations in the two quadrature and reduction in one of them, respectively. Moreover, these states are associated with the model of Jaynes and Cummings (1963) in the strong-coupling limit.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.
Normal squeezing is described in terms of variables which are linear in the field creation and annihilation operators. Amplitude-squared squeezing, on the other hand, is defined in terms of variables which are quadratic in these operators. As a result processes, such as second-harmonic generation, which couple one mode amplitude to the square of another convert this kind of squeezing into normal squeezing. In particular, the variables one considers are the hermitian part and one over i times the antihermitian part of the square of the annihilation operator. The variances of these observables obey an uncertainty relation. In order to better understand the properties of amplitude-squared squeezing it is useful to find the minimum uncertainty states corresponding to this uncertainty relation. These states are described by two parameters; the first,, gives the expectation of the square of the annihilation operator and the other λ, describes the amount of amplitude-squared squeezing. AH of these states have a mean amplitude of zero. These states for which =0 have unusual noise properties in that their Q functions have a four-fold rotational symmetry. We also find that a squeezed vacuum state is an amplitude-squared minimum uncertainty state.