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arXiv:quant-ph/0601204 · Physical Review A

Cooperative spin decoherence and population transfer

Claudiu Genes, P. R. Berman

Published May 10, 2006• 1 citation
arXivHTMLPDF

Abstract

An ensemble of multilevel atoms is a good candidate for a quantum information storage device. The information is encrypted in the collective ground state atomic coherence, which, in the absence of external excitation, is decoupled from the vacuum and therefore decoherence free. However, in the process of manipulation of atoms with light pulses (writing, reading), one inadvertently introduces a coupling to the environment, i.e., a source of decoherence. The dissipation process is often treated as an independent process for each atom in the ensemble, an approach which fails at large atomic optical depths where cooperative effects must be taken into account. In this paper, the cooperative behavior of spin decoherence and population transfer for a system of two, driven multilevel atoms is studied. Not surprisingly, an enhancement in the decoherence rate is found, when the atoms are separated by a distance that is small compared to an optical wavelength; however, it is found that this rate increases even further for somewhat larger separations for atoms aligned along the direction of the driving field's propagation vector. A treatment of the cooperative modification of optical pumping rates and an effect of polarization swapping between atoms is also discussed, lending additional insight into the origin of the collective decay.

DOI
10.1103/physreva.73.053809

Related papers

Atomic entanglement and decoherence.

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The generation of entanglement in atomic systems plays a central topic in the fields of quantum information storage and processing. Moreover, a special category of entangled states of multi-atom ensembles, spin squeezed states, have been proven to lead to considerable improvement in the sensitivity of precision measurements compared to systems involving uncorrelated atoms. A treatment of entanglement in open systems is, however, incomplete without a precise description of the process of decoherence which necessarily accompanies it. The theory of entanglement and decoherence are the two main topics of this thesis. Methods are described for the generation of strong correlations in large atomic ensembles using either cavity quantum electrodynamics or measurement outcome conditioned quantum dynamics. Moreover, the description of loss of entanglement resulting from the coupling to a noise reservoir (electromagnetic vacuum) is explored. A spin squeezing parameter is used throughout this thesis as both a measure of entanglement strength and as an indication of the sensitivity improvement above the so-called standard quantum limit (sensitivity obtained with uncorrelated particles) in metrology. The first scheme considered consists of a single mode cavity field interacting with a collection of atoms for which spin squeezing is produced in both resonant and off-resonant regimes. In the resonant case, transfer of squeezing from a field state to the atoms is analyzed, while in the off-resonant regime squeezing is produced via an effective nonlinear interaction (one-axis twisting Hamiltonian). A second, more experimentally realistic case, is one involving the interaction of free space atoms with laser pulses; a projective measurement of a source field originating from atomic fluctuations provides a means of preparing atomic collective states such as spin squeezed and Schrodinger cat states. A new unravelling is proposed, that employs the detection of photon number in a single output channel and is capable of producing mesoscopic Schrodinger cat states in a single step. As a first step to understanding the role of cooperative decoherence, of importance in the case of dense pencil-shaped atomic ensembles, the collective spin decoherence of a two multilevel atom system is derived. This calculation is also relevant to entanglement loss for two qubits manipulated using with reading/writing pulses. Finally, a scheme is proposed in which arbitrarily strong entanglement can be produced in a four-wave mixing setup, where the preparation of the atoms in a dark state limits the decoherence to negligible values.

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Microscopic models based on multilevel atoms are central to optimizing nonlinear optical responses and the coherent control of light. These models are traditionally based on single-atom effects that are parametrically extrapolated to include collective effects, such as an enhanced response or propagation within atomic media. In this work, we present a systematic analysis of the cooperative effects arising in driven systems composed of multilevel atoms coupled via a common electromagnetic environment. The analysis is based on an interplay between dressed states induced by the driving field and photon exchanges, and collective decay channels. This theory is applied to the case of four-wave mixing induced by a pair of lasers acting on an atomic pair with internal levels in the diamond configuration. The effect of interatomic correlations and collective decay over the photons created in this nonlinear process is then explored. We identify three regions of operation: (i) laser-dominated, (ii) intermediate, and (iii) dipole-dominated. The dependence of single- and two-photon correlations is studied in detail for each region by varying atomic orientations and laser parameters consistent with current experiments involving atomic gases. Photonic correlation functions are shown to exhibit a transition from a Lorentz-like dependence on the two-photon detuning—with general features that can be obtained in an isolated atom scheme—to a two-peaked distribution when the dipole-dipole interactions become relevant. For weak Rabi frequencies whose value is smaller than the highest collective decay rate, the atoms are trapped inside their ground state as they approach each other. It is found that the anisotropy of the dipole-dipole interaction and its wave nature are essential to understand the behavior of the photon correlations. Signatures of these processes are identified for existing experimental realizations. The intuition obtained from this connection helps to uncover relevant parameters that could be exploited for quantum control protocols based on dispersive and dissipative cooperative effects in multilevel systems.

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The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied ($c$-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective radiation time ${\ensuremath{\tau}}_{R}$, and the pump pulse duration ${\ensuremath{\tau}}_{P}$ satisfy the inequalities, ${\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}$ and ${\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}$ ($\ensuremath{\gamma}$ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area $\ensuremath{\pi}$ and that ${\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}$. Our results show that when the former conditions are satisfied the delay time ${\ensuremath{\tau}}_{D}$ for collective free pulse evolution is a function of both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{R}$. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.

Physical Review A•Oct 1, 1978•58 citations

Coherent control of collective atomic spins

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In this thesis I explore the use of collective spin angular momentum as a platform for quantum information processing. Such systems have several nice features that make them excellent choices for such protocols, especially ones where information is also stored in photonic variables. The longer coherence times of atoms makes it possible to store information from light in atoms for future use, and it is generally easier to couple atomic variables than to create nonlinear interactions of light. Above and beyond the advantages of atomic systems, collective atomic systems have additional strengths. In the limit of a large number of atoms, the collective variables of atomic systems have a natural connection to the bosonic algebra of light (known as the Holstein-Primakoff or HP approximation) where components of the collective spin angular momentum effectively act as quadratures, making them natural systems for coupling to light. 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Squeezing and loss is initially treated assuming ensembles of spin-1/2 atoms, but this work is further extended to higher dimensional subsystems. For a higher spin case, preparing the atoms in spin coherent states is not optimal. One can engineer a stronger interaction by preparing the atoms in an atomic "cat state", i.e., a superposition of the two stretch states along the direction of propagation of the light beam. The fluctuations of such a state are more strongly coupled to the light, resulting in a stronger nonlinearity. This leads to strong correlations between the atoms, but they are not immediately useful for squeezing; the cat state must be coherently mapped to a coherent state to achieve atomic spin squeezing. The state created in this manner is ultimately more squeezed than that achieved with the same interaction but prepared initially in a coherent state.

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C. P. Sun, S. Yi, Li You

We investigate the decoherence of a superposition of symmetric collective internal states of an atomic ensemble due to inhomogeneous coupling to external control fields. For asymptotically large system, we find $\sqrt{N}$ as the characteristic decoherence rate scale with N being the total number of atoms. Our results shed new light on attempts for realizing quantum information processing and storage with atomic ensembles.

arXiv:quant-ph/0203072•Physical Review A•Jun 26, 2003•28 citations
An ensemble of multilevel atoms is a good candidate for a quantum information storage device. The information is encrypted in the collective ground state atomic coherence, which, in the absence of external excitation, is decoupled from the vacuum and therefore decoherence free. However, in the process of manipulation of atoms with light pulses (writing, reading), one inadvertently introduces a coupling to the environment, i.e., a source of decoherence. The dissipation process is often treated as an independent process for each atom in the ensemble, an approach which fails at large atomic optical depths where cooperative effects must be taken into account. In this paper, the cooperative behavior of spin decoherence and population transfer for a system of two, driven multilevel atoms is studied. Not surprisingly, an enhancement in the decoherence rate is found, when the atoms are separated by a distance that is small compared to an optical wavelength; however, it is found that this rate increases even further for somewhat larger separations for atoms aligned along the direction of the driving field's propagation vector. A treatment of the cooperative modification of optical pumping rates and an effect of polarization swapping between atoms is also discussed, lending additional insight into the origin of the collective decay.
The generation of entanglement in atomic systems plays a central topic in the fields of quantum information storage and processing. Moreover, a special category of entangled states of multi-atom ensembles, spin squeezed states, have been proven to lead to considerable improvement in the sensitivity of precision measurements compared to systems involving uncorrelated atoms. A treatment of entanglement in open systems is, however, incomplete without a precise description of the process of decoherence which necessarily accompanies it. The theory of entanglement and decoherence are the two main topics of this thesis. Methods are described for the generation of strong correlations in large atomic ensembles using either cavity quantum electrodynamics or measurement outcome conditioned quantum dynamics. Moreover, the description of loss of entanglement resulting from the coupling to a noise reservoir (electromagnetic vacuum) is explored. A spin squeezing parameter is used throughout this thesis as both a measure of entanglement strength and as an indication of the sensitivity improvement above the so-called standard quantum limit (sensitivity obtained with uncorrelated particles) in metrology. The first scheme considered consists of a single mode cavity field interacting with a collection of atoms for which spin squeezing is produced in both resonant and off-resonant regimes. In the resonant case, transfer of squeezing from a field state to the atoms is analyzed, while in the off-resonant regime squeezing is produced via an effective nonlinear interaction (one-axis twisting Hamiltonian). A second, more experimentally realistic case, is one involving the interaction of free space atoms with laser pulses; a projective measurement of a source field originating from atomic fluctuations provides a means of preparing atomic collective states such as spin squeezed and Schrodinger cat states. A new unravelling is proposed, that employs the detection of photon number in a single output channel and is capable of producing mesoscopic Schrodinger cat states in a single step. As a first step to understanding the role of cooperative decoherence, of importance in the case of dense pencil-shaped atomic ensembles, the collective spin decoherence of a two multilevel atom system is derived. This calculation is also relevant to entanglement loss for two qubits manipulated using with reading/writing pulses. Finally, a scheme is proposed in which arbitrarily strong entanglement can be produced in a four-wave mixing setup, where the preparation of the atoms in a dark state limits the decoherence to negligible values.
Microscopic models based on multilevel atoms are central to optimizing nonlinear optical responses and the coherent control of light. These models are traditionally based on single-atom effects that are parametrically extrapolated to include collective effects, such as an enhanced response or propagation within atomic media. In this work, we present a systematic analysis of the cooperative effects arising in driven systems composed of multilevel atoms coupled via a common electromagnetic environment. The analysis is based on an interplay between dressed states induced by the driving field and photon exchanges, and collective decay channels. This theory is applied to the case of four-wave mixing induced by a pair of lasers acting on an atomic pair with internal levels in the diamond configuration. The effect of interatomic correlations and collective decay over the photons created in this nonlinear process is then explored. We identify three regions of operation: (i) laser-dominated, (ii) intermediate, and (iii) dipole-dominated. The dependence of single- and two-photon correlations is studied in detail for each region by varying atomic orientations and laser parameters consistent with current experiments involving atomic gases. Photonic correlation functions are shown to exhibit a transition from a Lorentz-like dependence on the two-photon detuning—with general features that can be obtained in an isolated atom scheme—to a two-peaked distribution when the dipole-dipole interactions become relevant. For weak Rabi frequencies whose value is smaller than the highest collective decay rate, the atoms are trapped inside their ground state as they approach each other. It is found that the anisotropy of the dipole-dipole interaction and its wave nature are essential to understand the behavior of the photon correlations. Signatures of these processes are identified for existing experimental realizations. The intuition obtained from this connection helps to uncover relevant parameters that could be exploited for quantum control protocols based on dispersive and dissipative cooperative effects in multilevel systems.
The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied (ccc-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective radiation time ${\ensuremath{\tau}}_{R}$, and the pump pulse duration ${\ensuremath{\tau}}_{P}$ satisfy the inequalities, ${\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}$ and ${\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}$ ($\ensuremath{\gamma}$ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area $\ensuremath{\pi}$ and that ${\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}$. Our results show that when the former conditions are satisfied the delay time ${\ensuremath{\tau}}_{D}$ for collective free pulse evolution is a function of both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{R}$. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.
In this thesis I explore the use of collective spin angular momentum as a platform for quantum information processing. Such systems have several nice features that make them excellent choices for such protocols, especially ones where information is also stored in photonic variables. The longer coherence times of atoms makes it possible to store information from light in atoms for future use, and it is generally easier to couple atomic variables than to create nonlinear interactions of light. Above and beyond the advantages of atomic systems, collective atomic systems have additional strengths. In the limit of a large number of atoms, the collective variables of atomic systems have a natural connection to the bosonic algebra of light (known as the Holstein-Primakoff or HP approximation) where components of the collective spin angular momentum effectively act as quadratures, making them natural systems for coupling to light. Also, collectively addressing the atoms allows one to access a large state space without individual addressing of the atoms. A natural first step towards control of these collective variables is generating large amounts of atomic spin squeezing. In the HP approximation going from rotations to squeezing corresponds to going from linear to quadratic interactions in the atomic quadratures; for extremely large squeezing one should see the breakdown of the HP approximation and the ability to generate arbitrary collective atomic states. I have sought to improve previous schemes for the spin squeezing of atomic ensembles, such as the proposal of Takeuchi et. al. based on coherent quantum feedback. In this scheme a beam of linearly polarized light passes through the atomic ensemble (prepared in a coherent state), coupling to the atoms through a state-dependent index of refraction (the Faraday effect). The light is then passed through a wave-plate and reflected back through the atoms for a second pass. This double-pass scheme leads to an effective nonlinearity as the atomic fluctuations are mapped onto the light on the first pass and then back on to the atoms in the second pass. The light acts as a bus coupling each atom to each of the others. This nonlinear interaction forms a shearing of the atomic coherent state that results in squeezing. The light is entangled to the atoms through these interactions, and remains entangled as it escapes the system. This leads to decoherence of the atoms as the light is lost to the environment, reducing the amount of spin squeezing achieved. The first step towards improving the double-pass scheme was to add a quantum eraser step in which the light is disentangled from the squeezed atoms. By first measuring one quadrature of the light, and then performing a measurement-dependent rotation on the atomic ensemble, it is possible to decouple the atoms and light so that the loss of the light does not reduce the atomic squeezing. This results in an improvement of the rate of atomic spin squeezing. The nonlinear shearing interaction that remains still falls short of the exponential squeezing seen in optical parametric amplification. The reason for this can be seen by decomposing the shearing interaction into squeezing along a given quadrature followed by a rotation. The rotation leads to a constantly changing axis along which the squeezing occurs, and thus the squeezing builds up more slowly. This can be corrected for by breaking up the light pulse into a sequence of small pulses, and performing a small phase matching rotation after each pulse. The squeezing then adds up along a constant direction, rather than along a constantly varying direction. This results in still further improvement of the rate of atomic spin squeezing. A complete model includes the effects of photon-atom scattering and other noise and loss effects on the overall rate of squeezing. Previous derivations of noise due to photon-atom scattering have started with the unjustified assumptions that the atomic and photonic decoherence channels were both Gaussian and independent of correlations between the two subsystems. They then proceed with very general statistical arguments that rely upon these simplifications. My work begins with the more fundamental master equation picture in which I find the Linblad jump operators. I find that in general the photonic and atomic loss channels are not independent, with the intensity of the light dictating the details of this dependence, and find the conditions under which the Gaussian approximation holds. Squeezing and loss is initially treated assuming ensembles of spin-1/2 atoms, but this work is further extended to higher dimensional subsystems. For a higher spin case, preparing the atoms in spin coherent states is not optimal. One can engineer a stronger interaction by preparing the atoms in an atomic "cat state", i.e., a superposition of the two stretch states along the direction of propagation of the light beam. The fluctuations of such a state are more strongly coupled to the light, resulting in a stronger nonlinearity. This leads to strong correlations between the atoms, but they are not immediately useful for squeezing; the cat state must be coherently mapped to a coherent state to achieve atomic spin squeezing. The state created in this manner is ultimately more squeezed than that achieved with the same interaction but prepared initially in a coherent state.
We report an investigation to establish the physical mechanisms responsible for decoherence in the generation of photon pairs from atomic ensembles, via the protocol of Duan et al. for long-distance quantum communication [Nature (London) 414, 413 (2001)] and present the experimental techniques necessary to properly control the process. We develop a theory to model in detail the decoherence process in experiments with magneto-optical traps. The inhomogeneous broadening of the ground state by the trap magnetic field is identified as the principal mechanism for decoherence. The theory includes the Zeeman structure of the atomic hyperfine levels used in the experiment, and the polarization of both excitation fields and detected photons. In conjunction with our theoretical analysis, we report a series of measurements to characterize and control the coherence time in our experimental setup. We use copropagating stimulated Raman spectroscopy to access directly the ground-state energy distribution of the ensemble. These spectroscopic measurements allow us to switch off the trap magnetic field in a controlled way, optimizing the repetition rate for single-photon measurements. With the magnetic field off, we then measure nonclassical correlations for pairs of photons generated by the ensemble as a function of the storage time of the single collective atomic excitation. We report coherence times longer than $10\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, corresponding to an increase of two orders of magnitude compared to previous results in cold ensembles. The coherence time is now two orders of magnitude longer than the duration of the excitation pulses. The comparison between these experimental results and the theory shows good agreement. Finally, we employ our theory to devise ways to improve the experiment by optical pumping to specific initial states.
We investigate the decoherence of a superposition of symmetric collective internal states of an atomic ensemble due to inhomogeneous coupling to external control fields. For asymptotically large system, we find N\sqrt{N}N​ as the characteristic decoherence rate scale with N being the total number of atoms. Our results shed new light on attempts for realizing quantum information processing and storage with atomic ensembles.