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Physical Review A

Doppler-induced spatially uniform atom-cavity coupling in a standing-wave cavity and the unidirectional momentum transfer on a moving atom

Kyungwon An, Young-Tak Chough, Sun-Hyun Youn

Published Jul 20, 2000• 10 citations
Publisher

Abstract

Deflection of an inverted two-state atom by a standing-wave cavity vacuum is analyzed. It is shown that clean vacuum Rabi oscillations occur at a frequency of ${g}_{0}/2$ between the initial atom-field state and the state with the atomic transverse momentum displaced by $\ensuremath{\Elzxh}k$ if the cavity is tuned to the Doppler shift associated with the atomic motion along the cavity axis, given that the shift is much larger than the optimum atom-cavity coupling constant ${g}_{0}.$ Consequently, the atom can be deflected by the cavity, the direction of which is precisely controlled by the atom-cavity detuning.

DOI
10.1103/physreva.62.023819

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Deflection of an inverted two-state atom by a standing-wave cavity vacuum is analyzed. It is shown that clean vacuum Rabi oscillations occur at a frequency of g0/2{g}_{0}/2g0​/2 between the initial atom-field state and the state with the atomic transverse momentum displaced by \ensuremath\Elzxhk\ensuremath{\Elzxh}k\ensuremath\Elzxhk if the cavity is tuned to the Doppler shift associated with the atomic motion along the cavity axis, given that the shift is much larger than the optimum atom-cavity coupling constant g0.{g}_{0}.g0​. Consequently, the atom can be deflected by the cavity, the direction of which is precisely controlled by the atom-cavity detuning.
We analyze the interaction of an two-state atom with two dimensional cavity modes in the strong coupling regime, with the quantized atomic center of mass motion taken into account. By choosing proper initial condition for the atomic transverse momentum and the cavity detuning, we can control the atomic transverse momentum distribution in two dimensions. We have analyzed the two-dimensional momentum distribution of the excited atom fully quantum mechanically, which interacts with two independent pairs of counterpropagating traveling-wave modes, one pair of folded traveling-wave modes, two independent crossed standing-wave modes and one folded standing-wave mode, respectively.
We address a somewhat naive but intriguing question of momentum transfer between a matter particle and a standing-wave mode of electromagnetic radiation. A beam of two-state atoms prepared in the upper-state enters at a slightly tilted angle from the normal direction to the axis of a single mode cavity. A similar setup with a relatively short atom-field interaction time was considered by the MIT group in their microlaser experiment, where the microlaser output was measured as a function of the atom-cavity detuning, revealing two-peak gain spectra. In our present consideration, however, the atom-cavity interaction time is appropriately set that by the time an atom leaves the interaction region, it has undergone a Rabi flip. The cavity is slightly detuned from the atomic resonance frequency that each radiation quantum released from the Rabi flip of the atoms is expected to be launched into only one of the two possible directions in which the resonance condition is met by the Doppler shift due to the transverse velocity component of the atom.
Collapse and revival of the Doppler-Rabi oscillations of a two-level atom moving in a cavity electromagnetic field are analyzed. The coupled atom-field dynamics are predicted accurately by numerical calculation and approximately by using the stationary phase approximation combined with the Poisson summation formula. The collapse and revival patterns are shown to be qualitatively different in the cases of moving atom and atom at rest. In particular, quantum revivals of Doppler-Rabi oscillations occur with a period determined by the Doppler shift of the atomic transition frequency. This regime of Doppler-Rabi oscillations requires that the Rabi frequency and the Doppler shift satisfy the condition Ω R ≪ Ω D . Under the inverse relation, the collapse- revival phenomenon generally does not occur. It is shown that even a small amount of atom-cavity detuning eliminates collapse-revival behavior. The analysis is performed for both coherent and thermal cavity fields.
The interaction of a two-level atom uniformly moving along a classical trajectory with a high-Q cavity quantum mode is analyzed. The dressed-state method is used to derive a recurrence formula for the transition probability of the atom with photon emission; the temporal dynamics of this probability qualitatively depends on the Doppler shift of the atomic transition frequency, on the Rabi frequency of the atom-field system, and on the detuning of the atomic transition frequency from the field mode frequency. The emission dynamics of a moving atom is very sensitive to the detuning. Rabi-type oscillations with a frequency equal to the Doppler shift can arise under certain conditions. At resonance, the emission probability of a moving atom can considerably exceed the emission probability of an atom at rest. A plane-parallel-mirror cavity and a confocal spherical-mirror cavity are considered. It is shown that the peculiarities of Doppler-Rabi oscillations must be taken into account in micromaser theory.
The dynamics of energy exchange between an atom moving in a high-Q cavity and the cavity field is analyzed by taking into account the Roentgen interaction. A two-level atom coupled to a Fock or coherent state of an optical or microwave cavity is considered. The mean cavity photon number required for high-frequency Doppler-Rabi oscillations to occur is relatively high for both Fock-and coherent-state cavity fields and increases with the atomic transition frequency. Conditions are found when the Roentgen interaction plays a key role in the Doppler-Rabi oscillations and must be taken into account, in addition to the conventional electric field-dipole interaction.
We use a basis of Bloch wave functions to solve the problem of spontaneous emission from an atom inside a standing-wave cavity. In this way we incorporate the effects of the atom’s quantized center-mass-motion in the periodic light-force potential of the one-quantum dressed states during the spontaneous emission process. We calculate emission spectra and the momentum distribution of the atom. Under strong-coupling conditions, “vacuum” Rabi spectra become asymmetric, with an increased relative weight given to the red side of the spectrum because of the transfer of energy to the center-of-mass motion of the atom. Because of the nonperturbative interaction between the atom and the cavity mode, many momentum quanta are transferred to the atom from the cavity mirrors during the irreversible emission of just one photon. Thus, the atom is diffracted by its own radiation reaction field. We analyze these effects using the Bloch-state basis. Use of the secular approximation provides a simple rate equation description of transitions from the Bloch states excited in the initial state to the atom-cavity ground state. We compare this approach with other ways of doing the calculations and discuss the accuracy of the secular approximation. We consider the extension of these calculations to treat the effects of photon statistics on the momentum distribution of atoms traversing a standing-wave cavity.