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Physical Review Letters

Quantized Phase Difference

Sixia Yu

Published Aug 4, 1997• 21 citations
Publisher

Abstract

A quantum description for the phase difference between two harmonic oscillators is proposed. Being represented by a Hermitian operator, phase difference displays two striking quantum features. First, its eigenvalues form a new type of quantized value which is countable and dense in a $2\ensuremath{\pi}$ window. Second, because quantized phase difference cannot be defined as a difference of phases, it obeys a quantum addition rule which may not be commutative.

DOI
10.1103/physrevlett.79.780

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A quantum description for the phase difference between two harmonic oscillators is proposed. Being represented by a Hermitian operator, phase difference displays two striking quantum features. First, its eigenvalues form a new type of quantized value which is countable and dense in a 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ window. Second, because quantized phase difference cannot be defined as a difference of phases, it obeys a quantum addition rule which may not be commutative.
We present a comprehensive and self-consistent theory of relative-phase measurements and the associated Hermitian relative-phase operator of two harmonic oscillators. We find that since Nature does not favor any particular initial condition of the two oscillators, the relative-phase operator is not unique. We show that the relative-phase eigenstates are maximally entangled. Therefore, most relative-phase operators lack a classical correspondence, even in the high-excitation limit. Furthermore, we find that the relative phase and the excitation number difference are noncommuting, noncanonical observables and we derive a commutation relation.
The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.
From the experimentalist’ spoint of view, it is natural to think of aphase measurement as a measurement that involves a reference time. As a part of the system, the reference time generating “clock” must be taken into account in the quantum treatment of the measuring process. This could be argued to be the reason why there does not exist any observable corresponding to the absolute phase in the unrestricted Hilbert space.Luis and Sánchez-Soto [Phys. Rev. A 48, 4702 (1993)] have investigated the case where the “clock” is another harmonic oscillator and defined a Hermitian phase-difference operator, which is valid in the infinite Hilbert space.In the present work we propose and experimentally demonstrate a setup for direct measurement of the phase-difference. We derive the phase-difference distributions for some two-mode states and compare them with the experimental results.
Second quantization is a powerful technique for describing quantum mechanical processes in which the number of excitations of a single particle is not conserved. A textbook example of second quantization is the presentation of the simple harmonic oscillator in terms of creation and annihilation operators, which, respectively, represent addition or removal of quanta of energy from the oscillator. Our aim in this article is to bolster this textbook example. Accordingly, we explore the physics of coupled second-quantized oscillators. These explorations are phrased as exactly solvable eigenvalue problems, the mathematical structure providing a framework for the physical understanding. The examples we present can be used to enhance the discussion of second-quantized harmonic oscillators in the classroom, to make a connection to the classical physics of coupled oscillators, and to acquaint students with systems employed at the frontiers of contemporary physics research.