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Journal of Mathematical Physics

Relative number state representation and phase operator for physical systems

Masashi Ban

Published Nov 1, 1991• 51 citations
Publisher

Abstract

A relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed. A phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the RNS representation space. The phase operator is expressed as a unitary exponential operator. The properties of the relative number operator, the phase operator, and their eigenstates are investigated in detail. The phase variable has a maximum uncertainty in any stationary state. Also, a time operator as a dynamical variable can be defined in the RNS representation space. The RNS representation is closely related to the Liouville space formulation and to thermofield dynamics. The RNS representation is shown to be a suitable method for investigating the Josephson junction with ultrasmall capacitance. A basic formulation of number-phase quantization in the Josephson junction is given in terms of the RNS representation.

DOI
10.1063/1.529054

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Publications of the Research Institute for Mathematical Sciences•Jun 30, 1994•19 citations
A relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed. A phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the RNS representation space. The phase operator is expressed as a unitary exponential operator. The properties of the relative number operator, the phase operator, and their eigenstates are investigated in detail. The phase variable has a maximum uncertainty in any stationary state. Also, a time operator as a dynamical variable can be defined in the RNS representation space. The RNS representation is closely related to the Liouville space formulation and to thermofield dynamics. The RNS representation is shown to be a suitable method for investigating the Josephson junction with ultrasmall capacitance. A basic formulation of number-phase quantization in the Josephson junction is given in terms of the RNS representation.
A relative-state formulation of quantum systems is presented in terms of relative-coordinate states, relative-number states, and relative-energy states. The relative-coordinate states are used to describe quantum systems in position and momentum representations. The probability distribution is calculated in terms of the relative-coordinate states and is shown to be equivalent to the functional definition of the quantum probability in phase space. It is shown that a quantum-mechanical phase operator can be constructed in terms of the relative-number states without the well-known difficulties. The results are compared with those obtained by the Pegg-Barnett phase-operator formalism and the relations to various other phase-operator methods are also discussed. The energy-measurement and energy-probability distributions are discussed in terms of the relative-energy states. Furthermore, a relative-state formulation is developed in the Liouville space. A phase representation in the Liouville space is introduced to investigate the time evolution of quantum coherence. In the Liouville space a time operator is defined as a canonical conjugate of the time-evolution generator, but not the Hamiltonian energy operator. The relation to the internal time presented by Prigogine and Misra is discussed.
Phase-space representation of quantum state vectors is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phase-space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative-position states and the relative-momentum states. The absolute-square of the phase-space function represents the probability distribution of the phase-space variables. Time-evolution of a quantum system is investigated in terms of the phase-space functions. The relations to the phase-space representations formulated by the other methods are obtained.
The phase operator for a two-mode photon system is defined in terms of the relative-number states, and its property is investigated in detail. The phase operator thus defined has a unitary exponential form. It is shown that the average value and fluctuation of the phase operator, where one mode is in some physical state, such as a coherent or a squeezed state, and the other is in a vacuum state, are equivalent to those obtained by means of the Pegg–Barnett phase operator for a single-mode photon. The average value and fluctuation of the phase operator are calculated for coherent and squeezed states of a two-mode photon system.
The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum Lz{L}_{z}Lz​ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator NNN) exists is shown to be erroneous. However, cosine and sine operators CCC and SSS exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather JJJ, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce CCC and SSS. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.
We review some of the difficulties previously encountered in defining the phase operator for finite quantum systems. We then propose the Wigner–Weyl quantization of the angle function φ on phase space as the phase operator, and derive a closed expression for its matrix elements with respect to the Hermite functions. We also determine the quantization of e^{iφ} , which turns out to be a weighted shift operator, its spectrum and that of its adjoint. This is done in the framework of quantization of a certain symbol class of phase space distributions, specialized to those which depend on one variable only. After recalling some results for the position and momentum variables, we apply the scheme to functions of radius or angle. We give necessary and sufficient conditions for operators obtained by quantizing functions of the angle to be elements of \mathscr L^+ [\mathscr S(\mathbf R)] and \mathscr L^+ [\mathscr S (\mathbf R), L^2(\mathbf R)] , and a sufficient condition for boundedness. We then consider the associated questions of commutation relations and uncertainties for operators in \mathscr L^+ [\mathscr S(\mathbf R), L^2(\mathbf R)] , which we define as bilinear forms. As must be the case, the commutator between our phase operator and the number operator exhibits noncanonical terms. Not surprisingly, the Poisson bracket of their phase space symbols also exhibits a noncanonical term.