Quasideterministic generation of maximally entangled states of two mesoscopic atomic ensembles by adiabatic quantum feedback.
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.