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Bulletin of the American Physical Society

Master equation with quantized atomic motion including dipole-dipole interactions

François Damanet, Daniel Braun, John Martin

Published May 26, 2016• 0 citations
OpenAlex

Abstract

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion [1]. Our equation provides a unifying picture of the consequences of recoil and in- distinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimen- tally observable with Rydberg atoms [2]. [1] F. Damanet, D. Braun, and J. Martin, arXiv:1512.06676v2. [2] K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. D. Martin, Phys. Rev. Lett. 93, 233001 (2004)

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We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion [1]. Our equation provides a unifying picture of the consequences of recoil and in- distinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimen- tally observable with Rydberg atoms [2]. [1] F. Damanet, D. Braun, and J. Martin, arXiv:1512.06676v2. [2] K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. D. Martin, Phys. Rev. Lett. 93, 233001 (2004)
We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and applies equally well to distinguishable and indistinguishable atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find closed-form formulas for a number of relevant states (Gaussian states, Fock states, and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion.
We investigate superradiance and subradiance of indistinguishable atoms with quantized motional states, starting with an initial total state that factorizes over the internal and external degrees of freedom of the atoms. Due to the permutational symmetry of the motional state, the cooperative spontaneous emission, governed by a recently derived master equation [F. Damanet et al., Phys. Rev. A 93, 022124 (2016)], depends only on two decay rates $\ensuremath{\gamma}$ and ${\ensuremath{\gamma}}_{0}$ and a single parameter ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{dd}}$ describing the dipole-dipole shifts. We solve the dynamics exactly for N=2N=2N=2 atoms, numerically for up to 30 atoms, and obtain the large-NNN limit by a mean-field approach. We find that there is a critical difference ${\ensuremath{\gamma}}_{0}\ensuremath{-}\ensuremath{\gamma}$ that depends on NNN beyond which superradiance is lost. We show that exact nontrivial dark states (i.e., states other than the ground state with vanishing spontaneous emission) only exist for $\ensuremath{\gamma}={\ensuremath{\gamma}}_{0}$ and that those states (dark when $\ensuremath{\gamma}={\ensuremath{\gamma}}_{0}$) are subradiant when $\ensuremath{\gamma}<{\ensuremath{\gamma}}_{0}$.
We investigate the evolution of interacting Rydberg gases in the limit of strong noise and dissipation. Starting from a description in terms of a Markovian quantum master equation we derive effective equations of motion that govern the dynamics on a 'coarse-grained' timescale where fast dissipative degrees of freedom have been adiabatically eliminated. Specifically, we consider two scenarios which are of relevance for current theoretical and experimental studies—Rydberg atoms in a two-level (spin) approximation subject to strong dephasing noise as well as Rydberg atoms under so-called electromagnetically induced transparency (EIT) conditions and fast radiative decay. In the former case we find that the effective dynamics is described by classical rate equations up to second order in an appropriate perturbative expansion. This drastically reduces the computational complexity of numerical simulations in comparison to the full quantum master equation. When accounting for the fourth order correction in this expansion, however, we find that the resulting equation breaks the preservation of positivity and thus cannot be interpreted as a proper classical master rate equation. In the EIT system we find that the expansion up to second order retains information not only on the 'classical' observables, but also on some quantum coherences. Nevertheless, this perturbative treatment still achieves a non-trivial reduction of complexity with respect to the original problem.
Master equations are derived for the time evolution of density-matrix elements characterizing atoms interacting with classical radiation fields that drive transitions between electronic-state manifolds of levels. The atomic center-of-mass motion is quantized and recoil accompanying absorption, stimulated emission, and spontaneous emission is included. An adiabatic elimination of the optical coherences is used to obtain quantum rate equations for ground- and excited-state density-matrix elements. Subsequent adiabatic elimination of excited-state density-matrix elements results in rate equations for the slow evolution of ground-state density-matrix elements. It is shown that recoil during spontaneous emission mixes spatial and internal atomic degrees of freedom. \textcopyright{} 1996 The American Physical Society.
Decoherence, known as the consequence of the coupling of any quantum system to its environment, causes information loss in the system and represents a major problem in the physical realization of quantum computers [1]. Decoherence-Free States (DFS) are considered as a possible solution to this problem. A set of trapped cold atoms placed in a DFS state will be immune against decoherence due to spontaneous emission. However, because of dipole-dipole interactions between atoms, induced dephasing effects are likely to destroy the coherence and drive the system out of its DFS [1, 2]. In this work, we study numerically the dynamics of a set of two-level atoms initially in a DFS with respect to dissipative processes by solving the master equation including both dissipative dynamics and dipole dipole interactions. We fo- cus our attention on the infuence of dipolar coupling on the radiated energy rate and coherence of the system as in [3]. In particular, by averaging over many realizations of close randomly distributed atomic positions, we show the formation of a superradiant-like pulse and we study its properties as a function of the dipolar coupling strength. [1] D. A. Lidar & K. B. Whaley, Lectures Notes in Phys., Vol. 622, p83-120, Springer (2003). [2] M. Gross & S. Haroche, Physics reports 93, 301-396 (1982). [3] W. Feng, Y. Li & S-Y. Zhu, Phys. Rev. A 88, 033856 (2013).
We derive the stochastic equations and consider the non-Markovian dynamics of a system of multiple two-level atoms in a common quantum field. We make only the dipole approximation for the atoms and assume weak atom–field interactions. From these assumptions, we use a combination of non-secular open- and closed-system perturbation theory, and we abstain from any additional approximation schemes. These more accurate solutions are necessary to explore several regimes: in particular, near-resonance dynamics and low-temperature behavior. In detuned atomic systems, small variations in the system energy levels engender timescales which, in general, cannot be safely ignored, as would be the case in the rotating-wave approximation (RWA). More problematic are the second-order solutions, which, as has been recently pointed out (Fleming and Cummings 2011 Phys. Rev. E 83 031117), cannot be accurately calculated using any second-order perturbative master equation, such as RWA, Born–Markov, Redfield, etc. The latter problem, which applies to all perturbative open-system master equations, has a profound effect upon calculation of entanglement at low temperatures. We find that even at zero temperature all initial states will undergo finite-time disentanglement (sometimes termed ‘sudden death’), in contrast to the previous work. We also use our solution, without invoking the RWA, to characterize the necessary conditions for Dicke subradiance at finite temperature. We find that the sub-radiant states fall into two categories at finite temperature: one that is temperature independent and one that acquires temperature dependence. With the RWA, there is no temperature dependence in any case.