Non-canonical quantum optics

Abstract

A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a (quantum) superposition of different frequencies. The idea is applied to the electromagnetic radiation field and nonrelativistic quantum optics. Employing a Dirac-type mode-quantization of the electromagnetic field and using a single oscillator we obtain several standard properties such as coherent states or spontaneous and stimulated emission. Extending the formalism to a greater number of oscillators we arrive at a structure analogous to the Fock space but without the standard cyclic "vacuum state". In the modified formalism the notion of the vacuum state is replaced by a vacuum subspace spanned by ground states of the oscillators. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure. Atom-light interactions are described by an appropriately modified minimal-coupling Hamiltonian (no normal ordering of the free-field Hamiltonian is necessary). The Hamiltonian does not change the number of oscillators which leads to an additional conservation law. Using the "−(e/m) A · p " interaction we discuss in second-order perturbation theory a two-photon spontaneous emission. The result essentially agrees with the ordinary formulas but the nontrivial vacuum structure is explicitly seen in the two-photon amplitude. The probability of the 2-photon emission resulting from the new formalism consists of a product of several terms, a part of them resembling those arising in the standard formulation from detector inefficiency, and the remaining one being the well known quantum optics formula. The presence of the additional conservation law shows that the theory contains two kinds of bosons (oscillators, whose number is conserved, and their excitations, whose number is not conserved). Taking this distinction into account we calculate an analog of the blackbody radiation Planck law. For temperatures lower than some T critical the result is indistinguishable from the Planck distribution. For T > T critical the distribution is Planck-like but with the maximum lowered and shifted towards higher frequencies.

States of light via reducible quantization

“…This is a consequence of the fact that the cases k = 0 and k = 0, k 2 = 0, correspond to representations of the Poincaré group induced from SL(2, C) and E(2), respectively. It is quite remarkable that the ultraviolet cut-offs discussed in [1] appeared automatically due to the same property of the formalism: The nontrivial structure of the vacuum state. In the case of ultraviolet and vacuum divergences the regularization is a consequence of square integrability of O(k).…”

“…The idea of reducible quantization is to take a(k, s) as an operator analogous to the operator one finds for a harmonic oscillator whose frequency is indefinite. In the nonrelativistic case there is only one parameter ω and one finds [1] the reducible representation of CCR a(ω) = |ω ω| ⊗ a where a comes from the irreducible representation [a, a † ] = 1 [5]. The electromagnetic field involves the 3-momentum k and two polarizations.…”

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

“…Recently, in [1], it was shown that the representation can be, in principle, directly tested in cavity QED. The N -representation of electromagnetic field operators was introduced in [10], and further analyzed and generalized in [11,12,13,14].…”

Cavity-QED Tests of Representations of Canonical Commutation Relations Employed in Field Quantization

“…Such a Hilbert space represents essentially a single harmonic oscillator of indefinite frequency (for physical motivation cf. [23,24] and the Appendix in [15]). An important property of the representation is that k I k = I is the identity operator in H. A vacuum of this representation is given by any state annihilated by all a k .…”