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Physical Review A

Three-level atom in a broadband squeezed vacuum field. I. General theory

Z. Ficek, P. D. Drummond

Published Jun 1, 1991• 103 citations
Publisher

Abstract

A complete treatment of a three-level ladder system interacting with a broadband squeezed vacuum field is presented. It is assumed that the ground state and the second excited state are decoupled in the electric-dipole approximation, and are tuned close to the sum frequency of the incident squeezed vacuum field. Using Zwanzig's projection-operator techniques, we derive the master equation, assuming that the system interacts with a broadband squeezed vacuum field in one or more dimensions. It is shown that, in the first case, the squeezed vacuum introduces new decay constants and frequency-shift parameters. These have the same dependence on the atom-radiation coupling parameter as the ordinary vacuum decay rate and frequency shift, the only major difference being that they are multiplied by the squeezing parameters M and N. In more dimensions, the decay constants and frequency-shift parameters depend on the solid angle \ensuremath{\Omega} over which the squeezing is propagated. For \ensuremath{\Omega}=0 these correspond to the usual Einstein A coefficients and Lamb shifts of the atomic levels, while for large \ensuremath{\Omega} they are similar to those for the one-dimensional squeezed vacuum.

DOI
10.1103/physreva.43.6247

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Journal of Physics B Atomic Molecular and Optical Physics•Jun 8, 2004•1 citation
A complete treatment of a three-level ladder system interacting with a broadband squeezed vacuum field is presented. It is assumed that the ground state and the second excited state are decoupled in the electric-dipole approximation, and are tuned close to the sum frequency of the incident squeezed vacuum field. Using Zwanzig's projection-operator techniques, we derive the master equation, assuming that the system interacts with a broadband squeezed vacuum field in one or more dimensions. It is shown that, in the first case, the squeezed vacuum introduces new decay constants and frequency-shift parameters. These have the same dependence on the atom-radiation coupling parameter as the ordinary vacuum decay rate and frequency shift, the only major difference being that they are multiplied by the squeezing parameters M and N. In more dimensions, the decay constants and frequency-shift parameters depend on the solid angle \ensuremath{\Omega} over which the squeezing is propagated. For \ensuremath{\Omega}=0 these correspond to the usual Einstein A coefficients and Lamb shifts of the atomic levels, while for large \ensuremath{\Omega} they are similar to those for the one-dimensional squeezed vacuum.
Using the formalism developed earlier, we treat spontaneous emission from a three-level atom (ladder system) interacting with a broadband squeezed vacuum field. We obtain expressions for the transient and steady-state populations of the atomic levels with the conditions that the atom interacts with either a multimode perfect squeezed vacuum field, or a three-dimensional vacuum field in which the squeezed modes lie within a solid angle over which squeezing is propagated. The results are compared with those obtained for the atom interacting with a thermal field. We show that in the perfect case the first excited state is not populated when the squeezed vacuum field is in a minimum-uncertainty squeezed state. Moreover, the second excited state can have a steady-state population larger than 1/2. These features are completely absent when the atom interacts with the thermal field. In addition, for a low-intensity squeezed vacuum field the population in the second excited state exhibits a linear rather than quadratic dependence on the intensity of the squeezed vacuum field. In the three-dimensional case the presence of unsqueezed modes considerably reduces the effect of squeezing on spontaneous emission. However, a significant reduction in a population of the first excited state and a population larger than 1/2 in the second excited state can be achieved provided the squeezing is propagated over a large solid angle. We also discuss the effect of the two-photon detuning between the double carrier frequency of the squeezing and atomic transition frequency ${\mathrm{\ensuremath{\omega}}}_{3}$ on the steady-state atomic population.
We review the use of operator reaction field theory in quantum optics by considering the problem of one atom in an isotropic three-dimensional broad-band squeezed vacuum. In these terms we give a quantum electrodynamical analysis of this system exact up to all contributions of order one in the ratio γω 0 -1 , the ratio of the A -coefficient to the atomic resonance frequency. These terms include all of the counter-rotating terms at this order. At zero order, namely in the usual rotating wave approximation, we find the usual Einstein rate equation while we confirm that the fluorescence spectrum consists of two Lorentzian peaks. However, at order one we find an additional oscillating term in the rate equation and additional resonant dispersive terms correct for each of the two Lorentzian peaks in the fluorescence spectrum. Similar results are expected for broad-band correlated squeezed vacua of arbitrary, rather than isotropic, geometry.
If a broadband squeezed vacuum is treated as a reservoir with respect to a two-level atom, the non-zero phase-dependent reservoir correlation functions characterizing the squeezed vacuum introduce `squeezing terms' to the master equation. These terms are responsible, for example, for the well known narrowing of the spectral lines in the resonance fluorescence spectrum of the atom. For a squeezed vacuum reservoir with finite bandwidth it is possible to derive the master equation that is consistent with the Born-Markov approximation by a two-step procedure which consists in dressing the atom first and next coupling it to the finite-bandwidth reservoir. The master equation is valid whenever the bandwidth of the reservoir is much broader than the atomic linewidth but not necessarily broader than the Rabi frequency. This procedure can be applied not only for the squeezed vacuum reservoir but also for the ordinary vacuum with the structure of the mode density that is not flat, e.g., modelled by a Lorentzian function as in a cavity. The master equation for this case, in the operator form, shows some similarities to the master equation for the squeezed vacuum reservoir due to the presence of the `squeezing-like terms'. The similarities and differences of the two master equations are discussed in this paper.
A cavity QED system is analysed which duplicates the dynamics of a two-level atom in free space interacting exclusively with broadband squeezed light. We consider atoms in a three-or four-level Λ configuration coupled to a high-finesse optical cavity which is driven by a squeezed light field. Raman transitions are induced between a pair of stable atomic ground states via the squeezed cavity mode and coherent driving fields. An analysis of the reduced master equation for the atomic ground states shows that a three-level atomic system has insufficient parameter flexibility to act as an effective two-level atom interacting exclusively with a squeezed reservoir. However, the inclusion of a fourth atomic level, coupled dispersively to one of the two ground states by an auxiliary laser field, introduces an extra degree of freedom and enables the desired interaction to be realized. As a means of detecting the reduced quadrature decay rate of the effective two-level system, we examine the transmission spectrum of a weak coherent probe field incident upon the cavity. The extension to multiple atoms is also considered and we show that such a cavity QED system may also duplicate the dynamics of an ensemble of two-level atoms collectively coupled to broadband squeezed light.
The relaxation process of a single two-level atom driven by an intense resonant coherent field is studied in the presence of a broadband squeezed vacuum field. Generalized forms for the self-field operator and the field-dependent damping coefficients are derived. In the steady state, positive atomic inversion (~5%) is shown for some range of the phase of the squeezed vacuum field. The squeezing-induced enhanced and asymmetric coherence (i.e.?non-zero dispersive atomic polarization) induces profound asymmetry in the side-bands of the Mollow fluorescent spectrum and the absorptive?dispersive spectra near the Rabi side-band frequencies. The case of two and three cooperative atoms, instead of a single atom, shows (i)?positive atomic inversion in the steady state for larger interval of the squeeze phase parameter and more enhanced extrema for the atomic polarization components and (ii) the extra side peaks in the fluorescent spectrum become asymmetric and of dispersive-like profile. Dressed-state analysis in the single-atom case with the field-dependent decay process in the presence of the squeezed vacuum field is presented and shows that the inequality of the (field-dependent) decay rates of the dressed states leads to more positive inversion in the dressed states as compared with the normal vacuum case.
An expression for dipolar and dissipative forces is obtained for a two-level atom with squeezed ground state. The atom is exposed to a coherent beam of frequency ωL with the spatial cross-sectional profile and Guoy phase . In our model, we assume that the broadband squeezed vacuum has a line width larger than that of a transition from a state |2 to a state |1 of the two-level atom. Moreover, we assume that the squeezed band is centred around ωL. Our expressions for the optical forces reduce to the standard expressions when the squeezing parameter goes to zero. We use an appropriate master equation for the reduced density operator with the squeezed vacuum bath. A steady-state solution of the density operator ρ is obtained to evaluate π12 and π21 which are required to find optical forces. We also obtain the time evolution of the dipolar force and the dissipative force in an intense field limit. We have shown that the dipolar force in a squeezed vacuum is non-conservative and non-zero even if the detuning parameter is zero.