Root
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Conference Proceedings
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4th International Conference on Physics and Control (PhysCon 2009)
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Quasideterministic generation of maximally entangled states of two mesoscopic atomic ensembles by adiabatic quantum feedback.
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Quantum entanglement is one of the most fundamental aspects of quantum

mechanics, as well as an essential resource in quantum communication

and information processing. Although very difficult to realize, entangled

states of material particles have been thoroughly studied in recent years

both theoretically and experimentally, and some schemes for their generation

have been designed and partially realized. Some studies concentrated on how

to produce entanglement between groups of two or few atoms. More recently,

there has been a growing interest on how to create multipartite entanglement

between atoms belonging to a single atomic ensemble considered as amulti-party

quantum system, by exploiting the interaction with a light field, and the

subsequent detection process. Finally, and more ambitiously, various schemes

have been proposed for the entanglement of different (two or more) macroscopic

or mesoscopic atomic ensembles.

In the cases of several (at least and typically two) macroscopic atomic

ensembles, where collective atomic operators can be described

by some continuous-variable approximation, it is only possible to design

schemes for the realization of weak entangled states. Some of these schemes

exploit quantum non-demolition (QND) measurements on auxiliary electromagnetic

fields (usually assumed in some Gaussian state) interacting with the atoms to

prepare entangled states of atomic systems. However, the probabilistic nature

of the quantum measurement events makes the generation of atomic entangled

states conditioned by the measurement outcomes, usually yielding a low

probability of success. This is particularly true for the preparation of

maximally entangled states. Such a shortcoming should be in principle overcome

by exploiting the knowledge of the state vector of the atomic system

conditioned on the outcome of a measurement, and then by introducing a proper

feedback scheme to efficiently drive the system toward a maximally entangled

state. Actually, Stockton, van Handel, and Mabuchi showed how this strategy can

be properly used to deterministically prepare highly entangled Dicke states of

a single atomic ensemble.

We introduce a reliable feedback scheme to generate maximal entanglement

of two mesoscopic atomic ensembles. In this scheme, the discrete quantum nature

of the atomic systems is fully taken into account, without resorting to any

continuous variable or Gaussian approximation.

Our proposal is based on a model introduced by Di Lisi and Moelmer, where two

collections of atoms, probed by a sequence of single-photon scattering

processes, are conditionally entangled by QND measurements of the total atomic

population difference between the two atomic samples. This model has been also

shown to be robust against spontaneous scattering.

We exploit the results of the QND measurements obtained by photo-detections to

drive the system into the maximally entangled state by a suitable feedback

mechanism. The feedback scheme that we introduce is a proper modification to

the fully discrete case of the continuous feedback strategy originally designed

by Thomsen, Mancini and Wiseman to generate high spin squeezing of a single

atomic ensemble, whose experimental realization was obtained by Geremia,

Stockton, and Mabuchi. The same scheme was generalized to the case of two

atomic ensembles to produce two-mode spin squeezing by Barry and Sanders.

Our procedure is monitored by quantitative wave-function simulations which

show how the sequence of photo-detection events, followed by the feedback

signal, gradually modifies the state of the samples and post-selects the

maximally entangled states. We show that the feedback scheme enormously

increases the rate of success in producing maximally entangled states of the

two atomic ensembles compared with the scheme in which feedback is absent. We

also show that the efficiency is further improved by adiabatically switching

off the feedback signal; in this way one obtains a quasi-deterministic

generation of the maximally entangled state. Finally, we study the problem for

the more realistic case of imperfect detectors, and we show how the feedback

scheme guarantees a very high probability of success in this case as well,

making the mechanism quite reliable against losses.

The two atomic ensembles we consider are identical, and each one is constituted

by N identical atoms in a static magnetic field, whose level structure consists

of two metastable lower states (that correspond to Zeeman sublevels of the

electronic ground state of alkali atoms), and one excited state.

We then consider an optical beam passing through the atomic clouds which is

coupled (out of resonance) only to the transition from one of the metastable

lower state. We can introduce the atomic spin operators for an atom, and then

define the x-, y- and z-components of the collective angular momentum.

In particular, the eigenvalues of the z-component are proportional to the

population difference in the two stable states.

The two atomic ensembles are initially prepared, by optical pumping, so that

each ensemble is fully polarized along the x-axis, and the atoms are

distributed between the two lower states according to a binomial distribution

with probability 0.5 for each state. The maximally entangled state of the

composed system can be explicitely computed, and the main goal is to realize

this state with high reliability.

The QND measurement of the total population difference between the two lower

atomic states by photo-detection can be realized, for example, by using light

with two polarization components that interact differently with the two atomic

states, generating in this way different phase shifts that produce a

polarization rotation signal. Another method may consist in using modulated

light, with one frequency component closer to resonance than the other, so that

the interaction with the atoms yields a phase difference between the two

components.

The two atomic ensembles are placed in one arm of an interferometric setup.

The incoming field is a highly collimated single photon pulse, which is

decomposed in two components by means of a 50-50 beam splitter: the reflected

component follows the free, path, while the transmitted component goes through

the atomic samples and interacts with them.

Since only one lower state is off-resonantly coupled with the excited level by

the interacting field, the phase shift between the two field components is

proportional to the population difference between the atomic ground states, and

can be resolved by the intensities measured in the two output ports of the

interferometerby suitably engineered photo-detectors. The sequence of

measurements of the field phase shift yields a nondestructive evolution of

the global state of the two atomic ensembles.

Photo-detection losses are accounted for by considering a finite efficiency of

the measurement process, i.e. assuming that only a fraction of the probe

photons is actually detected. In this non-ideal situation the evolution of the

density matrix is timed by the rate at which the single photon enters the

interferometric set-up, and is conditioned by the possibility of

photo-detection.

The feedback scheme is suggested by the fact that the maximally entangled state

is the only simultaneous eigenvector of three collective atomic spin operators

with zero eigenvalues. This in a natural way leads to impose a quantum feedback

that, at each step, properly counter-rotates the atomic spin operators.

This quantum feedback is realized acting with a unitary feedback operator.

The feedback operator contains a feedback (angle) parameter, whose value can

be computed for a generic step.

We perform numerical simulations of the controlled evolutions, and quantify the

efficiency of the scheme by determining the fraction of simulations for which

the value of the overlap between the final state of the two atomic ensembles

after the fotodetection sequence and the maximally entangled state is larger

then 0.99.

The results of the simulations show a substantial efficiency of the fedback

scheme (about 54 per cent of efficiency in the ideal case, and about 24 per

cent of efficiency in presence of losses) compared with those obtained by pure

probabilistic schemes without feedback (about 6 per cent in both ideal and

realistic cases).

We have however further improved the feedback mechanism by observing that

the not completely satisfying performance in the case of simple feedback

is due to the emergence of "anomalous" feedback angles, driving the system

toward a state orthogonal to the maximally entangled one. Then, we have

suitably modifyied the feedback algorithm in order to force the controlling

parameter to go uniformly to zero as the maximally entangled state is

approached. The improved feedback allows remarkably high successful rates

(about 92 per cent of efficiency in the ideal case, and about 78 per cent of

efficiency in presence of losses). Moreover, the reliability of the scheme

can also be appreciated by considering stronger constrained data.

mechanics, as well as an essential resource in quantum communication

and information processing. Although very difficult to realize, entangled

states of material particles have been thoroughly studied in recent years

both theoretically and experimentally, and some schemes for their generation

have been designed and partially realized. Some studies concentrated on how

to produce entanglement between groups of two or few atoms. More recently,

there has been a growing interest on how to create multipartite entanglement

between atoms belonging to a single atomic ensemble considered as amulti-party

quantum system, by exploiting the interaction with a light field, and the

subsequent detection process. Finally, and more ambitiously, various schemes

have been proposed for the entanglement of different (two or more) macroscopic

or mesoscopic atomic ensembles.

In the cases of several (at least and typically two) macroscopic atomic

ensembles, where collective atomic operators can be described

by some continuous-variable approximation, it is only possible to design

schemes for the realization of weak entangled states. Some of these schemes

exploit quantum non-demolition (QND) measurements on auxiliary electromagnetic

fields (usually assumed in some Gaussian state) interacting with the atoms to

prepare entangled states of atomic systems. However, the probabilistic nature

of the quantum measurement events makes the generation of atomic entangled

states conditioned by the measurement outcomes, usually yielding a low

probability of success. This is particularly true for the preparation of

maximally entangled states. Such a shortcoming should be in principle overcome

by exploiting the knowledge of the state vector of the atomic system

conditioned on the outcome of a measurement, and then by introducing a proper

feedback scheme to efficiently drive the system toward a maximally entangled

state. Actually, Stockton, van Handel, and Mabuchi showed how this strategy can

be properly used to deterministically prepare highly entangled Dicke states of

a single atomic ensemble.

We introduce a reliable feedback scheme to generate maximal entanglement

of two mesoscopic atomic ensembles. In this scheme, the discrete quantum nature

of the atomic systems is fully taken into account, without resorting to any

continuous variable or Gaussian approximation.

Our proposal is based on a model introduced by Di Lisi and Moelmer, where two

collections of atoms, probed by a sequence of single-photon scattering

processes, are conditionally entangled by QND measurements of the total atomic

population difference between the two atomic samples. This model has been also

shown to be robust against spontaneous scattering.

We exploit the results of the QND measurements obtained by photo-detections to

drive the system into the maximally entangled state by a suitable feedback

mechanism. The feedback scheme that we introduce is a proper modification to

the fully discrete case of the continuous feedback strategy originally designed

by Thomsen, Mancini and Wiseman to generate high spin squeezing of a single

atomic ensemble, whose experimental realization was obtained by Geremia,

Stockton, and Mabuchi. The same scheme was generalized to the case of two

atomic ensembles to produce two-mode spin squeezing by Barry and Sanders.

Our procedure is monitored by quantitative wave-function simulations which

show how the sequence of photo-detection events, followed by the feedback

signal, gradually modifies the state of the samples and post-selects the

maximally entangled states. We show that the feedback scheme enormously

increases the rate of success in producing maximally entangled states of the

two atomic ensembles compared with the scheme in which feedback is absent. We

also show that the efficiency is further improved by adiabatically switching

off the feedback signal; in this way one obtains a quasi-deterministic

generation of the maximally entangled state. Finally, we study the problem for

the more realistic case of imperfect detectors, and we show how the feedback

scheme guarantees a very high probability of success in this case as well,

making the mechanism quite reliable against losses.

The two atomic ensembles we consider are identical, and each one is constituted

by N identical atoms in a static magnetic field, whose level structure consists

of two metastable lower states (that correspond to Zeeman sublevels of the

electronic ground state of alkali atoms), and one excited state.

We then consider an optical beam passing through the atomic clouds which is

coupled (out of resonance) only to the transition from one of the metastable

lower state. We can introduce the atomic spin operators for an atom, and then

define the x-, y- and z-components of the collective angular momentum.

In particular, the eigenvalues of the z-component are proportional to the

population difference in the two stable states.

The two atomic ensembles are initially prepared, by optical pumping, so that

each ensemble is fully polarized along the x-axis, and the atoms are

distributed between the two lower states according to a binomial distribution

with probability 0.5 for each state. The maximally entangled state of the

composed system can be explicitely computed, and the main goal is to realize

this state with high reliability.

The QND measurement of the total population difference between the two lower

atomic states by photo-detection can be realized, for example, by using light

with two polarization components that interact differently with the two atomic

states, generating in this way different phase shifts that produce a

polarization rotation signal. Another method may consist in using modulated

light, with one frequency component closer to resonance than the other, so that

the interaction with the atoms yields a phase difference between the two

components.

The two atomic ensembles are placed in one arm of an interferometric setup.

The incoming field is a highly collimated single photon pulse, which is

decomposed in two components by means of a 50-50 beam splitter: the reflected

component follows the free, path, while the transmitted component goes through

the atomic samples and interacts with them.

Since only one lower state is off-resonantly coupled with the excited level by

the interacting field, the phase shift between the two field components is

proportional to the population difference between the atomic ground states, and

can be resolved by the intensities measured in the two output ports of the

interferometerby suitably engineered photo-detectors. The sequence of

measurements of the field phase shift yields a nondestructive evolution of

the global state of the two atomic ensembles.

Photo-detection losses are accounted for by considering a finite efficiency of

the measurement process, i.e. assuming that only a fraction of the probe

photons is actually detected. In this non-ideal situation the evolution of the

density matrix is timed by the rate at which the single photon enters the

interferometric set-up, and is conditioned by the possibility of

photo-detection.

The feedback scheme is suggested by the fact that the maximally entangled state

is the only simultaneous eigenvector of three collective atomic spin operators

with zero eigenvalues. This in a natural way leads to impose a quantum feedback

that, at each step, properly counter-rotates the atomic spin operators.

This quantum feedback is realized acting with a unitary feedback operator.

The feedback operator contains a feedback (angle) parameter, whose value can

be computed for a generic step.

We perform numerical simulations of the controlled evolutions, and quantify the

efficiency of the scheme by determining the fraction of simulations for which

the value of the overlap between the final state of the two atomic ensembles

after the fotodetection sequence and the maximally entangled state is larger

then 0.99.

The results of the simulations show a substantial efficiency of the fedback

scheme (about 54 per cent of efficiency in the ideal case, and about 24 per

cent of efficiency in presence of losses) compared with those obtained by pure

probabilistic schemes without feedback (about 6 per cent in both ideal and

realistic cases).

We have however further improved the feedback mechanism by observing that

the not completely satisfying performance in the case of simple feedback

is due to the emergence of "anomalous" feedback angles, driving the system

toward a state orthogonal to the maximally entangled one. Then, we have

suitably modifyied the feedback algorithm in order to force the controlling

parameter to go uniformly to zero as the maximally entangled state is

approached. The improved feedback allows remarkably high successful rates

(about 92 per cent of efficiency in the ideal case, and about 78 per cent of

efficiency in presence of losses). Moreover, the reliability of the scheme

can also be appreciated by considering stronger constrained data.