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American Journal of Physics

A basis-independent approach to quantum optics

Ivan Deutsch

Published Sep 1, 1991• 29 citations
Publisher

Abstract

States of the radiation field can be defined without reference to a given basis of normal modes. In this way optical systems for which there is no natural set of orthogonal modes can be handled quantum mechanically in a more rigorous manner. In particular, coherent states and associated phase-space methods can easily be generalized to this basis-independent representation.

DOI
10.1119/1.16731

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States of the radiation field can be defined without reference to a given basis of normal modes. In this way optical systems for which there is no natural set of orthogonal modes can be handled quantum mechanically in a more rigorous manner. In particular, coherent states and associated phase-space methods can easily be generalized to this basis-independent representation.
For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A mathematically rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. All other phase quantities are defined in terms of P and P†. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It is easily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and the set of phase eigenstates is not complete. Mathematically precise operator relations, including the rigorous statement of commutation rules, are developed. For each of the phase operators, a complete spectral analysis is given, with the nature of the spectrum, spectral decomposition (if any), and eigenstates shown explicitly. The values of the various phase operators are compared between phase states and coherent states.
Glauber-Sudarshan diagonal coherent state P-representation has been used to determine geometric phase for non-classical states of light. For a given density operator ρ1^\hat{ρ_1}ρ1​^​ of two mode optical beam, we evolve it in complex projective ray space (R)(\cal R)(R) to ρ2^\hat{ρ_2}ρ2​^​ and to ρ3^\hat{ρ_3}ρ3​^​ by changing its state of polarisation using unitary operator U^p(θ)\mathrm{\hat{U}_p}(θ)U^p​(θ). The diagonal coherent state basis has been utilized to represent the density operators instead of fock state basis as in the fock state basis the state vector in present work evolve under unitary operator produe infinitely numerous terms which make the density operators very messy to handle. This cumbersome situation can be easily avoided by using the approach proposed above. The trace of the product of ρ1^\hat{ρ_1}ρ1​^​, ρ2^\hat{ρ_2}ρ2​^​ and ρ3^\hat{ρ_3}ρ3​^​ is taken to get three vertex Bargmann invariant. Argument of this gives the geometric phase which is represented in terms of phase space variables containing a notion of symplectic area.
The Hilbert space of a single mode quantum system is spanned by a countable set of orthonormal basis states denoted |n〉 for n = 0, 1, 2, … Such a basis can be used to define quasi-probability distributions over a two-dimensional quantum phase plane. Multi-mode quantum systems are spanned by orthonormal product states |n1,…nk〉 = |n1〉 ⊗ … ⊗ |nk〉 which can be used to define quasiprobability distributions over a 2k-dimensional phase volume. However, this multi-mode basis set is also countable and can be uniquely specified by a single integer label. This labeling allows us to define new collective operators and generalized coherent states to construct probability distributions over a new two-dimensional phase plane. These distributions represent the system's 2k-dimensional phase volume with perfect fidelity, and allow us to apply phase space interference (PSI) approaches to multi-mode quantum systems.