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Physical Review A

Quantum phase superoperator

Masashi Ban

Published Mar 1, 1995• 11 citations
Publisher

Abstract

A unitary superoperator that represents a quantum phase is introduced by means of a density-matrix formalism of quantum mechanics. In this formalism, a quantum phase can be treated without using any of the additional techniques that are indispensable for defining a unitary phase operator when conventional phase operator methods are used. Furthermore, a phase-space distribution function for number and phase variables is introduced by means of the eigenoperator of the phase superoperator, and the basic properties of the distribution function are investigated.

DOI
10.1103/physreva.51.2469

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A unitary superoperator that represents a quantum phase is introduced by means of a density-matrix formalism of quantum mechanics. In this formalism, a quantum phase can be treated without using any of the additional techniques that are indispensable for defining a unitary phase operator when conventional phase operator methods are used. Furthermore, a phase-space distribution function for number and phase variables is introduced by means of the eigenoperator of the phase superoperator, and the basic properties of the distribution function are investigated.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
Abstract Operators in quantum mechanics—either observables, density or evolution operators, unitary or not—can be represented by c-numbers in operator bases. The position and momentum bases are in one-to-one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner–Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi–Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a ‘double’ Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.
The Wigner-Weyl (WW) phase-space formulation of quantum mechanics is discussed within the Liouville-space formalism, where quantum operators A^ are viewed as vectors, represented by L kets \ensuremath{\Vert}A^>>, on which act ``superoperators''; the scalar product is A^\ensuremath{\Vert}B^>>=TrA${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$B^=${\mathrm{\ensuremath{\Elzxh}}}^{\mathrm{\ensuremath{-}}1}$[A^,B^], A${\mathrm{^}}^{+}$B^=1/2(A^B^+B^A^). The WW representation corresponds to the choice of a special basis in Liouville space, namely, the eigenbasis of the position and momentum anticommutation superoperators q${\mathrm{^}}^{+}$ and p${\mathrm{^}}^{+}$ (where [q^,p^]=i\ensuremath{\Elzxh}). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$]=[q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$,p${\mathrm{^}}^{+}$]=i (the other commutators vanishing), as functions of which all other super- operators can be expressed. Weyl ordering is expressed as f(q^,p^)Weyl{)}_{\mathrm{Weyl}})Weyl​ ordering=f(q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{+}$)1^. A generalization of Ehrenfest's theorem is obtained.
A phase space for the single-mode light field is constructed by using optical phase variables. This phase-space formalism is directly connected to optical-phase-related operators. The phase-state basis \ensuremath{\Vert}\ensuremath{\theta}〉 is used to represent various quantum operators and states. Properties of the density operator \ensuremath{\rho}^ phase-state matrix elements P(\ensuremath{\theta},\ensuremath{\theta}')=〈\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\theta}'〉 and their relations to phase-related quasiprobability distributions are discussed. By using the phase representation, different types of number-phase intelligent states are analyzed.
The choice of mathematical representation when describing physical systems is of great consequence, and this choice is usually determined by the properties of the problem at hand. Here we examine the little-known wave operator representation of quantum dynamics and explore its connection to standard methods of quantum dynamics, such as the Wigner phase-space function. This method takes as its central object the square root of the density matrix and consequently enjoys several unusual advantages over standard representations. By combining this with purification techniques imported from quantum information, we are able to obtain a number of results. Not only is this formalism able to provide a natural bridge between phase- and Hilbert-space representations of both quantum and classical dynamics, we also find the wave operator representation leads to semiclassical approximations of both real and imaginary time dynamics, as well as a transparent correspondence to the classical limit. It is then demonstrated that there exist a number of scenarios (such as thermalization) in which the wave operator representation possesses an equivalent unitary evolution, which corresponds to nonlinear real-time dynamics for the density matrix. We argue that the wave operator provides a new perspective that links previously unrelated representations and is a natural candidate model for scenarios (such as hybrids) in which positivity cannot be otherwise guaranteed.