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Journal of Physics A Mathematical and General

Mean value and uncertainty of optical phase-a simple mechanical analogy

Tomáš Opatrný

Published Nov 7, 1994• 52 citations
Publisher

Abstract

There exist several definitions of the mean value and the phase uncertainty of an optical field. We show that one of these definitions is especially simple and intuitive because of its mechanical analogy. On this basis, we also derive the summation rule and the Tchebyshev inequality for random angular and phase variables and a number-phase uncertainty relation.

DOI
10.1088/0305-4470/27/21/034

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There exist several definitions of the mean value and the phase uncertainty of an optical field. We show that one of these definitions is especially simple and intuitive because of its mechanical analogy. On this basis, we also derive the summation rule and the Tchebyshev inequality for random angular and phase variables and a number-phase uncertainty relation.
When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
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.