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Coherence conditions for groups of Rydberg atoms I
NSTITUTE OF
P
HYSICS
P
UBLISHING
J
OURNAL OF
P
HYSICS
B: A
TOMIC,
M
OLECULAR AND
O
PTICAL
P
HYSICS
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 48834893
doi:10.1088/0953-4075/39/23/007
Coherence conditions for groups of Rydberg atoms
J V Hern´andez and F Robicheaux
Department of Physics, Auburn University, AL 36849-5311, USA
E-mail:
hernajv@physics.auburn.edu
Received 11 August 2006, in nal form 21 September 2006
Published 10 November 2006
Online at
stacks.iop.org/JPhysB/39/4883
Abstract
We investigate the excitation of a collection of cold atoms to Rydberg states. By
a direct numerical solution of Schr¨odingers equation, we are able to compute
various interesting properties of the many-body wavefunction.
The high
polarizability of Rydberg atoms allows them to support large dipole moments
which in turn can interact with each other over long ranges. If the interaction
energy between excited atoms is large enough, the resultant energy shift will
move the two excitation states out of resonance, thus effectively blocking a
two excitation state from occurring. One particular topic investigated is the
quantum phase gate, where both groups of atoms are within a blockade radius
R
b
and subjected to a
2 sequence of pulses. We examine the regime
where the groups are neither totally within nor totally outside the blockade
radius. Our results explore the tolerance in variation of intergroup lattice
distances for a series of quantum gates.
1. Introduction
Atoms excited into Rydberg states are large in size and thus able to support large dipole
moments. The long-range interactions between these large dipoles have been a popular topic
of study over the last several years. With recent advancements in cooling and trapping, it
has been experimentally shown that the laser excitation of a frozen gas from an initial state
to a Rydberg state is suppressed when driving on resonance [
1 5
]. There have also been
various numerical simulations investigating this system [
1
,
6 8
]. If the interaction energy
between two excited atoms is large enough, the shift will move the two excitation state out
of resonance, thus effectively blocking this state from occurring. The number of particles
able to be excited is now suppressed, exhibiting a dipole blockade [
9
]. This allows for the
coherent manipulation of a large collection of atoms, enabling careful macroscopic control
over microscopic systems. The ability to precisely interact with quantum systems is critical
in the development of quantum computing.
The original proposals for creating a dipole blockade [
10
] consisted of exciting Rydberg
states at F¨orster resonance and using the interaction between these transition dipole moments
to create energy shifts. Evidence of a dipole blockade has been successfully observed via this
0953-4075/06/234883+11$30.00
© 2006 IOP Publishing Ltd
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4883 4884
J V Hern´andez and F Robichcaux
process [
2
]. A dipole blockade could also be created by exciting a group of cold Rydberg
atoms in a static electric eld (along the z-axis). The resultant Stark states have relatively
large static dipole moments, d, along the direction of the electric eld. For two Rydberg atoms
within the n manifold in a static electric eld, separated by a distance R, the dominant long-
range interaction is proportional to 1/R
3
and scales like n
4
. This is similar to the proposal for
a blockade that utilizes the second-order interaction (van der Waals) between two np Rydberg
atoms [
1
]. It is proportional to 1/R
6
and scales like n
11
. In this paper we will investigate
both the rst-order dipoledipole and second-order van der Waals (vdW) interactions. For
the vdW case, we will only be looking at the nsns interaction in order to eliminate any
directional dependence. In both cases, the initial state is coherent; we do not include decays
or repopulations from our ground level. If a narrow bandwidth laser is used to drive from
the ground to a high Rydberg state (n > 50) on resonance, the main transition will be
the dipole-allowed one [
11
]. We can then treat the entire collection as a group of two-level
systems.
When a collection of cold atoms is in a blockade conguration (i.e. the physical parameters
are such that the interaction energy between two Rydberg atoms is large enough to shift the
pair out of the two excitation resonance), the number of atoms that can be excited (N
e
)
is
suppressed. By taking repeated measurements of N
e
, we can nd the relationship between
the mean N
e
and the variance N
2
e
N
e 2
. The Mandel Q parameter, Q N
2
e
N
e 2
N
e
1,
is a useful quantity to compare the atom counting statistics to a Poissonian distribution [
12
].
For a Poissonian distribution the mean is equal to the variance, so Q
= 0. In the case of
blockaded atoms, Q should be less than 0; this corresponds to a sub-Poissonian distribution.
The Q parameter reects the measure of how efciently the system is blockaded. Recent
experiments have been able to measure Q values [
4
] and in this paper we will present the
results of our simulations of Q.
We also investigate the situation where there are two spherical, localized groups of cold
atoms. The radii of both spheres are chosen so that all pairs of atoms are in a blockade
conguration when the two spheres are just touching. The maximum number of atoms able
to be excited in this case should be 1, creating a two-level system. We then subject the system
to the following sequence of pulses: group 1 is excited by a pulse, then group 2 is excited
by a 2 pulse and nally group 1 is deexcited by another pulse. If all pairs of atoms
are in a blockade conguration, the sequence of pulses will maintain all the atoms in their
ground level, but this nal state will now have a phase shift relative to the initial state.
This sequence of pulses acts as a phase gate [
9
]. If the radii of each sphere are held constant
and the centre-to-centre distance D is increased, pairs of intergroup atoms will no longer be
blockaded. There is no longer just a two-level system, but if D is increased to the point where
no intergroup atoms are blockaded, then each group is now effectively independent of the
other leaving two two-level systems. In such a conguration, the
2 pulse will leave
the nal state exactly in phase with the initial one. Our interest is in the regime in between
all intergroup atoms being blockaded and none of them being so. The behaviour of the phase
gate as D is increased will determine how far apart or close together the groups of atoms must
be in order to minimize errors.
The relatively large and long-range interaction between a pair of Rydberg atoms implies
that a collection of N atoms must be treated as a many-body system. By a direct numerical
solution of Schr¨odingers equation, we are able to compute and retain various interesting
properties of the many-body wavefunction. Even if every atom was strictly treated as a two-
level system, the number of basis states needed for a direct solution would still be 2
N
. This
severe limitation in size can be overcome by utilizing the simplications described in [
7
]. Coherence conditions for groups of Rydberg atoms
4885
Although the method used in this paper utilizes the full wavefunction for calculations, other
techniques have been used such as the Monte Carlo procedure used in [
6
] which requires much
less computational effort. This is appropriate in the case of an incoherent ground state such as
in [
4
], but it cannot be used to probe quantities such as phase shifts or the amplitudes of pieces
of the full wavefunction useful to quantum computing. The mean eld approach developed
in [
1
] works well for high laser power, but it is unable to give spatial correlation functions or
phase shifts needed to check the error of phase gates. Unless otherwise noted, atomic units
will be used throughout this paper.
2. Theory
In this section we will describe the techniques involved in the direct numerical solution to
the many-atom wavefunction. We will also discuss how we solved for the wavefunction in a
manner that allowed us to check for convergence. Once the wavefunction has been solved for,
it is possible to compute the number of excited atoms and various correlation functions.
We begin by treating each atom as a purely two-level system with one level being the
initial tightly bound state
|g and the other being a highly excited Rydberg state |e . For the
purposes of this paper, the locations of the atoms will be xed in space. This is a reasonable
approximation if the temperature of the gas is low enough and the time duration of the exciting
laser pulse is short enough. For conditions similar to recent experiments, the laser pulse must
be
200 ns. We expanded the wavefunction
| (t) = a
gg...g
(t )
|gg . . . g + a
eg...g
(t )
|eg . . . g
· · · + a
ee...g
(t )
|ee . . . g + a
ee...e
(t )
|ee . . . e
(1)
= a (t )
| .
(2)
We do not use all of the states in the expansion, but recursively eliminate them as described in
[
7
]. At time t
= 0, all atoms ar