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Hierarchies of nonclassical states in quantum optics

Narasimhaiengar Mukunda

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

The conventional separation of states of the quantised radiation field into “classical‿ and “nonclassical‿ types is expressed in a dual operator form and then refined. This is based on new features of the normal ordering rule for passage from classical to quantum dynamical variables. The cases of single and two-mode radiation fields are discussed

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Classical optics describes quite successfully the propagation of laser light, both in free-space and inside a transparent medium. On the other hand, neither classical optics nor the conventional formulation of quantum optics can address adequately the propagation of nonclassical light. Nonclassical states of light are usually described by second-quantization operators acting onto the vacuum state of the radiation field, and the corresponding expectation values of the electric and magnetic fields are often zero. Clearly, the description of propagation of nonclassical light requires a direct quantum mechanical treatment of the spatial progression of the field operators. This chapter outlines the main features of propagative quantum optics, and examines more particularly the treatment of propagation through a linear (refractive) medium, and across a refractive interface. In order to facilitate the discussion of the physical ideas, a plane-wave propagation geometry, which greatly simplifies the mathematical treatment, is used.
A new operator-based condition for distinguishing classical from nonclassical states of quantized radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum-dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalization of the notion of sub-Poissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalization is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair-coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted to one another, comparing the generalized sub-Poissonian statistics with extant signatures of nonclassical behaviour.
The problem of relating the semiclassical and quantum treatments of statistical states of an optical field is re-examined. The case where the rule of association between functions and operators is that of antinormal ordering is studied in detail. It is shown that the distribution function for each mode corresponding to this case is a continuous bounded function, and is also a boundary value of an entire analytic function of two variables. The nature of the distribution for the normal ordering rule of association and its relation to this entire function are discussed. It is shown that this distribution can be regarded as the limit of a sequence of tempered distributions in the following sense: One can find a sequence of density operators ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}$ which converges in the norm to the density operator $\stackrel{^}{\ensuremath{\rho}}$ of any given field (consisting of a single mode), such that each member of the sequence can be expressed in the form ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}=\ensuremath{\int}{\ensuremath{\varphi}}_{(\ensuremath{\nu})}(z)|z〉〈z|{d}^{2}z$, where ${\ensuremath{\varphi}}_{(\ensuremath{\nu})}$ is a tempered distribution.