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arXiv:1904.04950

Universal density matrix for the phase space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Evgenii V. Burlakov

Published Apr 10, 2019• 2 citations
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Abstract

In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form $W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]$, where $ρ$ is the density matrix, and $\mathcal{W}$ is the universal density matrix. The density matrix $ρ$ for each quantum system is different, and the universal matrix $\mathcal{W}$ is the same for any quantum system. Thus, the matrix $\mathcal{W}$ has a fundamental character. In the work, the elements of the universal matrix $\mathcal{W}$ were found explicitly and their properties were investigated. The diagonal elements of the matrix $\mathcal{W}$ are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix $\mathcal{W}$ contain frequency oscillations responsible for dissipations in the quantum systems.

DOI
10.48550/arxiv.1904.04950

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In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form W=Sp⁡[ρW]W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]W=Sp[ρW], where ρρρ is the density matrix, and W\mathcal{W}W is the universal density matrix. The density matrix ρρρ for each quantum system is different, and the universal matrix W\mathcal{W}W is the same for any quantum system. Thus, the matrix W\mathcal{W}W has a fundamental character. In the work, the elements of the universal matrix W\mathcal{W}W were found explicitly and their properties were investigated. The diagonal elements of the matrix W\mathcal{W}W are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix W\mathcal{W}W contain frequency oscillations responsible for dissipations in the quantum systems.
Abstract In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems. Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.
A general approximation scheme for solving the quantum Liouville equation has been developed and is presented here specifically within the Hartree approximation. This method uses the connection between the density matrix and the Wigner phase-space representations. By use of the Wigner representation the off-diagonal elements of the density matrix are expanded in powers of deviations from the mean momentum value. The time evolution of these deviations is governed by a closed self-consistent set of equations.
In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$). With the implicit $\ensuremath{\hbar}$-dependence of the wave function $\ensuremath{\psi}$ ignored in the integral representation, this limit differs from ${|\ensuremath{\psi}(q)|}^{2}\ensuremath{\delta}(p)$ by terms $\frac{o(\ensuremath{\hbar})}{\ensuremath{\hbar}}$. With the implicit $\ensuremath{\hbar}$ dependence of $\ensuremath{\psi}$ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of qqq, multiplied by an energy-conserving $\ensuremath{\delta}$ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schr\"odinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble---in general, and in the particular cases of high temperature and of $\ensuremath{\hbar}\ensuremath{\rightarrow}0$; where $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the asymptotic behavior of the Wigner function is explicitly manifest.