Download 76, 023605 (2007).

PHYSICAL REVIEW A 76, 023605 共2007兲
Probing n-spin correlations in optical lattices
Chuanwei Zhang, V. W. Scarola, and S. Das Sarma
Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
共Received 27 November 2006; revised manuscript received 2 May 2007; published 13 August 2007兲
We propose a technique to measure multispin correlation functions of arbitrary range as determined by the
ground states of spinful cold atoms in optical lattices. We show that an observation of the atomic version of the
Stokes parameters, using focused lasers and microwave pulsing, can be related to n-spin correlators. We
discuss the possibility of detecting not only ground state static spin correlations, but also time-dependent
spin-wave dynamics as a demonstrative example using our proposed technique.
DOI: 10.1103/PhysRevA.76.023605
PACS number共s兲: 03.75.Lm, 03.75.Mn, 39.25.⫹k, 75.10.Pq
I. INTRODUCTION
The advent of optical lattice confinement of ultracold
atomic gases 关1–4兴 opens the possibility of observing a vast
array of phenomena in quantum condensed systems 关5兴. In
particular, optical lattice systems may turn out to be the ideal
tools for the analog simulation of various strongly correlated
interacting lattice models 共e.g., Hubbard model 关2,3兴, Kitaev
model 关6兴兲 studied in solid state physics. The great advantage
of optical lattices as analog simulators of strongly correlated
Hamiltonians lies in the ability of optical lattices to accurately implement lattice models without impurities, defects,
lattice phonons, and other complications which can obscure
the observation of quantum degenerate phenomena in the
solid state.
In this context optical lattices can support a variety of
interacting spin models which to date have been only approximately or indirectly observed in nature or remain as
rather deep but unobserved mathematical constructs. Three
exciting possibilities are currently the subject of active study
关5兴. The first 共and the most direct兲 envisions simulation of
conventional condensed matter spin-lattice models in optical
lattices. Quantum magnetism arising from strong correlation
leads to many-body spin ground states that can be characterized by spin-order parameters. Spin order can, in some cases,
show long-range behavior arising from spontaneous symmetry breaking, e.g., ferromagnetism and antiferromagnetism.
Such long-range spin-ordering phenomena are reasonably
well understood in most cases. Recent work also relates conventional spin-order parameters to entanglement measures
which yield scaling behavior near quantum phase transitions
关7,8兴. The second possibility, simulation of topological spin
states, arises from the surprising fact that optical lattices can
also 共at least in principle兲 host more complicated spin models
previously thought to be academic. The ground states of
these models do not fall within the conventional Landau
paradigm, i.e., there is no spontaneously broken symmetry,
but show topological ordering and, as a result, display nontrivial short-range behavior in spin correlation functions. Examples include the chiral spin-liquid model 关9兴 and the Kitaev model 关6,10–12兴. Finally, optical lattices are also
particularly well suited to realize coherent and collective
spin dynamics because dissipation can be kept to suitably
low levels 关13兴.
While optical lattices offer the possibility of realizing all
three of the above examples one glaring question remains.
1050-2947/2007/76共2兲/023605共7兲
Once a suitable spin Hamiltonian is realized, how do we
observe the vast array of predicted phenomena in spinoptical lattices? To date, time-of-flight measurements have
proven to yield detailed information related to two types of
important correlation functions of many-body ground states
of particles trapped in optical lattices. The first, a first-order
correlation function 共the momentum distribution兲, indicates
ordering in one-point correlation functions 关14兴. The second
is a second-order correlation function 共the noise distribution兲
which indicates ordering in two-point density-density correlators 关15–18兴. The former can, for example, detect longrange phase coherence while, as we will see below, the latter
is best suited to probe long-range order in two-point correlation functions, e.g., the lattice spin-spin correlation function.
We note that recent proposals suggest that time-of-flight imaging can in principle be used to extract other correlation
functions 关19,20兴.
In this paper we propose a technique to observe equal
time n-spin correlation functions characterizing both longand short-range spin ordering useful in studying all three
classes of spin-lattice phenomena mentioned above. Our proposal utilizes realistic experimental techniques involving focused lasers, microwave pulsing, and fluorescence detection
to effectively measure a general n-spin correlation function
defined by
n
␰兵␣ jk,k = 1, . . . ,n其 ⬅ 具⌿兩 兿 ␴ j jk兩⌿典,
k=1
␣
k
共1兲
where ⌿ is the many-body wave function of the atomic en␣
semble, 兵jk其 is a set of sites, and ␴ j jk 共␣ jk = 0 , 1 , 2 , 3兲 are
k
Pauli spin operators at sites jk with the notation ␴0 = I, ␴1
= ␴x, ␴2 = ␴y, and ␴3 = ␴z. Examples of order detectable with
one-, two-, and three-spin correlation functions are magnetization 共具␴zj 典 = 1兲, antiferromagnetic order 关具␴zj ␴zj⬘典 = 共−1兲 j−j⬘兴,
and chiral spin-liquid order 关具␴ j · 共␴ j⬘ ⫻ ␴ j⬙兲典 = 1兴, to name a
few.
In general our proposed technique can be used to experimentally characterize a broad class of spin-lattice models of
the form
冉
M
冊
H共J;A兲 = J共t兲 兺 A兵jk其 兿 ␴ j jk ,
兵jk其
k=1
␣
k
共2兲
where J has dimensions of energy and can vary adiabatically
as a function of time, t, while the dimensionless parameters
023605-1
©2007 The American Physical Society
PHYSICAL REVIEW A 76, 023605 共2007兲
ZHANG, SCAROLA, AND DAS SARMA
A兵jk其 are kept fixed. For example, M = 2 represents the usual
two-body Heisenberg model. Several proposals now exist for
simulating two-body Heisenberg models 关5,10兴. In the following we, as an example, consider optical lattice implementations of the Heisenberg XXZ model,
冉兺
具j,j⬘典
x
y
z
冊
共␴xj ␴ j⬘ + ␴yj ␴ j⬘兲 + ⌬␴zj ␴ j⬘ ,
共3兲
where 具j , j⬘典 denotes nearest neighbors and ⌬ and J are
model parameters that can be adjusted by, for example, varying the intensity of lattice laser beams 关10兴.
The paper is organized as follows: in Sec. II we show that,
in practice, short-range spin correlations are difficult to detect in noise correlation measurements. Section III lays out a
general procedure for detecting n-spin correlations with local
probes. An experimental scheme for realizing such a procedure is proposed and a quantitative feasibility analysis is
presented. Section IV is devoted to an investigation of the
time dependence of correlation functions using the technique. In particular, we show how the technique can be used
to engineer and probe spin-wave dynamics. We conclude in
Sec. V.
II. LOCAL CORRELATIONS IN TIME OF FLIGHT
We first discuss the measurement of spin-spin correlation
functions by analyzing spatial noise correlations 共two-point
density-density correlations兲, i.e., 关具n共rជ兲n共rជ⬘兲典 − 具n共rជ兲典
⫻具n共rជ⬘兲典兴 / 具n共rជ兲典具n共rជ⬘兲典 in time-of-flight images from atoms
confined to an optical lattice modeled by the XXZ Hamiltonian. The ground states of this and a variety of spin models
can be characterized by the spin-spin correlation function
between different sites. For instance, the spin-spin correlation function in a one-dimensional XXZ spin chain
共with J ⬎ 0兲, shows power-law decay
具␴zj ␴ j⬘典 ⬃ 共− 1兲 j−j⬘/兩j − j⬘兩␩
z
共4兲
in
the
critical
regime
共−1 ⬍ ⌬ ⱕ 1兲,
where
␩ = 1 / 共1 − ␲1 cos−1 ⌬兲. In principle this correlation function
can be probed by spatial noise correlation in time of flight.
We argue that, in practice, short-range correlations 共e.g.,
␩ ⬎ 1 in the XXZ model兲 are difficult to detect in time-offlight noise correlation measurements. To see this note that
the noise correlation signal is proportional to 关15兴
G关Q共r − r⬘兲兴 = 兺 eiQ共j⬘−j兲a具␴zj ␴ j⬘典,
z
Correlation Amplitude
HXXZ共J;⌬兲 = J
1 (a)
共5兲
j,j⬘
where Q is the lattice wave vector which gets mapped into
coordinates r and r⬘ in time of flight on the detection screen,
and a is the lattice spacing. Including normalization the noise
correlation signal is proportional to N−1 for systems with
long-range order 共e.g., antiferromagnetic order giving ␩ = 0
in the above XXZ model兲 but shows a much weaker scaling
for short-range correlations. In fact the ratio between correlators in a ground state with ␩ ⬎ 1 共short-range power-law
order兲 and ␩ = 0 共long-range antiferromagnetic order兲 scales
as N−1 making the state with power-law correlations rela-
N = 20
0.5
0
1 (b)
N = 200
0.5
0
−
3π
a
−
π
π
a
a
Q (r − r')
3π
a
FIG. 1. 共Color online兲 Noise correlation plotted as a function of
wave vector of the one-dimensional XXZ model. The solid 共dashed兲
line corresponds to a ground state with ␩ = 0, long-range
共␩ = 2, short-range兲 spin correlator. The amplitudes are normalized
by the maximum for antiferromagnetic order 共␩ = 0兲. The number of
atoms in panels 共a兲 and 共b兲 are N = 20 and N = 200, respectively.
tively difficult to detect in large systems. To illustrate this we
compare the calculated noise correlation amplitude, G, in
Fig. 1 for two cases ␩ = 0 共solid line兲 and ␩ = 2 共dashed line兲
with N = 20 关Fig. 1共a兲兴 and N = 200 关Fig. 1共b兲兴 for the 1D XXZ
model. We see that the correlation amplitude for short-range
共power-law兲 order is extremely small in comparison to longrange antiferromagnetic order for large N.
The small correlation signal originates from the fact that
the noise correlation method is in practice a conditional
probability measuring collective properties of the whole system, while short-range spin correlations describe local properties and are therefore best detected via local operations. In
the following we propose a local probe technique to measure
local correlations thus providing an experimental scheme
which compliments the time-of-flight–noise correlation technique, best suited for detecting long-range order.
III. DETECTING n-SPIN CORRELATION WITH LOCAL
PROBES
A. General procedure
We
find
that
general
n-spin
correlators,
␰兵␣ jk , k = 1 , . . . , n其, can be related to the Stokes parameters
broadly defined in terms of the local reduced density matrix
␳ = Tr兵jk,k=1,. . .,n其 兩 ⌿典具⌿兩 on sites 兵jk , k = 1 , . . . , n其, where the
trace is taken on all sites except the set 兵jk其. The Stokes
parameters 关21兴 for the density matrix ␳ are S␣ j . . ..␣ j
1
␣
n
= Tr共␳兿k=1
␴ j jk兲 leading to the decomposition
k
3
␳ = 2−n
␣j
1
兺
,¯,␣
jn=0
冉
n
S ␣ j . . .␣ j
1
n
␣
␴j
兿
k=1
jk
k
冊
.
n
共6兲
Using the theory of quantum state tomography 关21兴, we find
the n-spin correlators
023605-2
PHYSICAL REVIEW A 76, 023605 共2007兲
PROBING n-SPIN CORRELATIONS IN OPTICAL LATTICES
n
␰兵␣ jk,k = 1, . . . ,n其 = 兿 共P兵兩␾␣ j 典其 ± P兵兩␾␣⬜j 典其兲,
k
k=1
(a)
共7兲
k
A
B
↑
A
B Microwave
↓
A B
where the plus 共minus兲 sign indicates a 0 共nonzero兲 index
and 兵兩␾␣ j 典 , 兩␾␣⬜ 典其 denote the measurement basis for the atom
k
Focused lasers
B
(b)
F=2
jk
↑
at jk. We define the measurement basis to be 兩␾1典 = 共兩 ↓ 典
⬜
冑
冑
+ 兩 ↑ 典兲 / 冑2, 兩␾⬜
1 典 = 共兩 ↓ 典 − 兩 ↑ 典兲 / 2, 兩␾2典 = 共兩 ↓ 典 + i 兩 ↑ 典兲 / 2, 兩␾2 典
= 共兩 ↓ 典 − i 兩 ↑ 典兲 / 冑2, 兩␾3典 = 兩 ↓ 典, 兩␾⬜
3 典 = 兩 ↑ 典. Finally, P兵兩␾␣ j 典其 is
k
the probability of finding an atom in the state 兩␾␣ j 典.
k
The expansion of the product defining ␰ then yields a
quantity central to our proposal,
-2
0
-1
1
(c)
2
↑
Microwave
F=1 ↓
-1 0
1
(d)
3
B
A
B
A
↓
A B
2
B
Fluorescence Signals
Detection
laser
A
B
↑
F =1
n
␰兵␣ jk,k = 1, . . . ,n其 = 兺 共− 1兲l Pl ,
共8兲
l=1
where Pl is the probability of finding l sites in the states 兩␾⬜
jk 典
and n − l sites in 兩␾ jk典. Equation 共8兲 shows that the n-spin
correlation function can be written in terms of experimental
observables. We can now write a specific example of the
two-spin correlation function 共discussed in the preceding
section兲 in terms of observables: ␰兵3 , 3其 = P兩↓典 j 兩↓典 j + P兩↑典 j 兩↑典 j
1
2
1
2
− 共 P兩 ↓ 典 j1兩 ↑ 典 j2 + P兩↑典 j 兩↓典 j 兲 . In the following subsection we dis1
2
cuss a specific experimental procedure designed to extract
precisely this quantity using local probes of cold atoms confined to optical lattices.
B. Proposed experimental realization
We now describe and critically analyze an experimental
procedure designed to find the probabilities, Pl, from which
we can determine the spin-correlation function
␰兵␣ jk , k = 1 , . . . , n其 through Eq. 共8兲. To illustrate our technique
we consider a specific experimental system: 87Rb atoms confined on a single two-dimensional 共xy plane兲 optical lattice
with two hyperfine ground states 兩 ↓ 典 ⬅ 兩F = 1 , mF = −1典 and
兩 ↑ 典 ⬅ 兩F = 2 , mF = −2典 chosen as the spin of each atom. Here
the atomic dynamics in the z direction are frozen out by high
frequency optical traps 关22兴. However, the scheme can be
directly applied to three-dimensional optical lattices by using
one additional focused laser which propagates along the xy
plane and plays the same role as the focused laser 共propagating along the z axis兲 discussed in the following step 共II兲.
In the Mott insulator regime with one atom per lattice site
spin Hamiltonians, defined in Eq. 共2兲, may be implemented
using spin-dependent lattice potentials in the superexchange
limit 关10兴. The spin coupling J共t兲, i.e., the overall prefactor in
Eq. 共2兲, can be controlled by varying the lattice depth. Our
proposed experimental procedure will build on such spin
systems, although it can be generalized to other implementations where H is generated by other means. In the spin
systems, the ground states are strongly correlated many-body
spin states and their properties can be characterized by the
spin correlations between atoms at different lattice sites.
Step 共I兲. We start with a many-body spin state ⌽0 and turn
off the spin-spin interactions generated by superexchange between lattice sites. We achieve this by ramping up the spindependent lattice depth to ⬃50ER adiabatically with respect
FIG. 2. 共Color online兲 Schematic plot of the experimental procedure used to measure n-spin correlation functions. A indicates one
of the target atoms. B indicates all other atoms. 共a兲 Target atoms A
are transferred to a suitable measurement basis using a combination
of focused lasers and microwave pulses 关see step 共II兲兴. 共b兲 Atoms at
state 兩 ↓ 典 are transferred to state 兩2典 by two microwave ␲ pulses,
then atoms at state 兩 ↑ 典 are transferred to state 兩 ↓ 典 by another microwave ␲ pulse 关see step 共III兲兴. 共c兲 Target atoms A are transferred
back to state 兩 ↑ 典 from 兩 ↓ 典 关see step 共IV兲兴. 共d兲 A detection laser is
applied to detect the probability of finding target atoms at 兩 ↑ 典
关see step 共IV兲兴.
to the band splitting, where ER = h2 / 2m␭2 is the photon recoil
energy and ␭ is the wavelength of the optical lattice. The
ramp up time for each lattice is chosen properly so that only
the overall energy scale J共t兲 in the spin Hamiltonian 共2兲 is
modified, which preserves the highly correlated spin state
⌽0. In the deep lattice, the time scale for the spin-spin interactions 共⬃ប / J ⬎ 10s兲 becomes much longer than the time
共⬃1 ms兲 taken to perform the spin-correlation measurement.
The spin-spin interactions play no role in the measurement
process and the following detection steps are quickly performed on this “frozen” many-body spin state ⌽0.
Step 共II兲. In this step, we selectively transfer target atoms
A at site共s兲 jk 共chosen a priori兲 to a suitable measurement
basis 兵兩␾ jk典 , 兩␾⬜
jk 典其 from initial states 兵兩 ↓ 典 jk , 兩 ↑ 典 jk其, without affecting nontarget atoms B at other sites 关Fig. 2共a兲兴. Selective
manipulation of quantum states of single atoms in optical
lattices is currently an outstanding challenge for investigating physics in an optical lattice, because spatial periods of
typical optical lattices are shorter than the optical resolution.
In Ref. 关23兴, a scheme for single atom manipulation using
microwave pulses and focused lasers is proposed and analyzed in detail. Here we apply the scheme to selectively
transfer atoms between different measurement bases.
To manipulate a target atom A, we adiabatically turn on a
focused laser that propagates along the ẑ axis having the
maximal intensity located at A. The spatially varying laser
intensity I共r兲 induces position-dependent energy shifts
⌬Ei共r兲 = ␤iI共r兲
共9兲
for two spin states 兩 ↓ 典 and 兩 ↑ 典, where the parameter ␤i for
state 兩i典 共i = ↓ or ↑兲 is determined by the focused laser parameters. Different polarizations and detunings of the focused
laser lead to different ␤i and thus yield different shifts of the
023605-3
PHYSICAL REVIEW A 76, 023605 共2007兲
ZHANG, SCAROLA, AND DAS SARMA
⍀共t兲 = ⍀0 exp共− ␻20t2兲 共− t f ⱕ t ⱕ t f 兲
and parameters ␻0 = 14.8ER / ប, ⍀0 = 13.1ER / ប, and t f = 5 / ␻0.
The pulse transfers the measurement basis of the target atoms A in 16.9 ␮s, while the change in the quantum state of
nontarget atoms is found to be below 3 ⫻ 10−4 by numerically integrating the Rabi equation that describes the coupling between two spin states by the microwave pulse. The
focused lasers are adiabatically turned off after the microwave pulse. During the step 共II兲, the probability for spontaneous scattering of one photon from target atoms inside the
⌫
focused laser can be roughly estimated as P ⬇ 兰␶␶ f ប␽ V共t兲dt
i
−4
⬃ 2 ⫻ 10 , where ␶i and ␶ f represent the times when the
focused laser is turned on and off, ⌫ is the decay rate of 6P
state, ␽ is the detuning, and V共t兲 is the potential depth of the
focused laser.
The distance between any two target atoms can be as
short as a lattice spacing. This is because the basis transfer
processes for different target atoms are preformed sequentially in time, i.e., the process for an atom at site j2 starts
after the process for an atom at site j1 is accomplished. In the
special case that sites j1 and j2 are spatially well separated
and the final basis 兵兩␾ jk典 , 兩␾⬜
jk 典其 at two sites are the same, the
transfer process can be done simultaneously for two sites
with one microwave pulse.
Step 共III兲. In this step, we transfer all atoms to the
兩F = 1典 hyperfine level 关Fig. 2共b兲兴 to avoid stray signal in the
detection step 共IV兲. We apply two ␲ microwave pulses to
20
(a)
(d)
n pA
0.2
n pA
ρ 33A
0.4
10
0
20 (b)
0
10
1×10 −4 (e)
n Bp
0
1×10 −5 (c)
5 ×10 −5
n Bp
hyperfine splittings 兩⌬E共r兲 兩 = 兩⌬E↑共r兲 − ⌬E↓共r兲兩 between two
spin states. We choose a ␴+-polarized laser that drives the
5S → 6P transition to obtain a small diffraction limit. The
wavelength ␭ f ⬇ 421 nm 共corresponding to a detuning
⌬0 = −2␲ ⫻ 1209 GHz from the 6 2 P3/22 state兲 is optimized to
obtain the maximal ratio between energy splittings of two
spin states and the spontaneous scattering rate 关23兴.
Because of the inhomogeneity of the focused laser intensity I共r兲, 兩⌬E共r兲兩 reaches a maximum at the target atom A
and decreases dramatically at neighboring sites. Therefore
the degeneracy of hyperfine splittings between different atoms is lifted. By adjusting the focused laser intensity, the
differences of the energy shifts ␦ = 兩⌬E共0兲 兩 −兩⌬E共␭ / 2兲兩 between target atom A and nontarget neighboring atom B can
be varied and calculated through Eq. 共9兲. Here we choose
␦ = 74ER because it balances the speed and fidelity of single
spin manipulation, leading to less spontaneously scattered
photons from the target atoms in the spin-dependent focused
lasers. To avoid excitations of atoms to higher bands of the
optical lattice, the rise speed of the focused laser intensity
should satisfy the adiabatic condition. We use the adiabatic
condition to estimate the ramp up time of the focused laser to
be 35 ␮s to give a 10−4 probability for excitation to higher
bands.
We then change the measurement basis by applying a microwave ␲ / 2 pulse that drives a suitable rotation to target
atoms A 关Fig. 2共a兲兴. The microwave is resonant with the
hyperfine splitting between two spin states of the target atoms A, but has a detuning larger than ␦ for nontarget atoms
B. Consider a pulse with Rabi frequency
5 ×10−6
0
0 10 20 30 40 50
Γt
0
0
1
2
Ω/Γ
3
FIG. 3. 共a兲 Time evolution of the probability for the target atoms
A to be in the excited state 兩3典. 共b兲 The number of scattering photons
nAp versus time for atoms A at state 兩 ↑ 典. 共c兲 The number of scattering
photons nBp versus time for atoms 共A or B兲 at states 兩 ↓ 典 and 兩2典. 共d兲
The number of scattering photons nAp versus the Rabi frequency of
the resonant laser for atoms A at state 兩 ↑ 典. 共e兲 The number of scattering photons nBp versus the Rabi frequency of the resonant laser
for atoms 共A or B兲 at states 兩 ↓ 典 and 兩2典. The time period for 共d兲 and
共e兲 is 50/ ⌫.
transfer all atoms at 兩 ↓ 典 first to 兩F = 2 , mF = 0典 then to 兩2典
⬅ 兩F = 1 , mF = 1典. Another ␲ microwave pulse is then applied
to transfer all atoms at 兩 ↑ 典 to 兩 ↓ 典. The ␲ microwave pulse
can be implemented within 12.5 ␮s for a microwave Rabi
frequency ⍀ = 2␲ ⫻ 40 kHz.
Step 共IV兲. We transfer target atoms A at jk from 兩 ↓ 典 back
to 兩 ↑ 典 with the assistance of the focused lasers and microwave pulses 关Fig. 2共c兲兴, using the same atom manipulation
procedure as that described in step 共II兲. We then apply a
detection laser resonant with
兩↑典 → 兩3典 ⬅ 兩5 2 P3/2:F = 3,m = − 3典
to detect the probability of finding all target atoms at 兩 ↑ 典
关corresponding to the basis state 兩␾⬜
jk 典 because we transferred
atoms to the measurement basis in step 共II兲兴 关Fig. 2共d兲兴. The
beam size of the detection laser should be large enough so
that target atoms at different sites 兵jk其 experience the same
laser intensity, and scatter the same number of photons if
they are in the 兩 ↑ 典 state. The fluorescence signal 共the number
of scattered photons兲 is from one of the n + 1 quantized levels, where the lth level 共l = 0 , . . . , n兲 corresponds to states
with l sites of target atoms on state 兩 ↑ 典 共兩␾⬜
jk 典兲. By repeating
the whole process many times, we obtain the probability distribution Pl, and thus the spin correlation function
␰兵␣ jk , k = 1 , . . . , n其 via Eq. 共8兲.
The scattering photons of the fluorescence signals come
mostly from the target atoms A at state 兩 ↑ 典 共Fig. 3兲. Signal
from atoms at any 兩F = 1典 state is suppressed because of the
large hyperfine splitting 共␯ ⬇ 2␲ ⫻ 6.8 GHz兲 between 兩F = 1典
and 兩F = 2典 states. Atoms B do not contribute to the fluorescence signal because they are already transferred to the
兩F = 1典 state in step 共III兲. The dynamics of photon scattering
is described by the optical Bloch equation 关24兴,
023605-4
d
i
␳33 = − ⌫␳33 + ⍀共␳13 − ␳31兲,
dt
2
PHYSICAL REVIEW A 76, 023605 共2007兲
PROBING n-SPIN CORRELATIONS IN OPTICAL LATTICES
冉
冊
冉
冊
d
⌫
i
␳13 = −
+ i␯0 ␳13 + ⍀共2␳33 − 1兲,
dt
2
2
⌫
d
i
␳31 = −
− i␯0 ␳31 − ⍀共2␳33 − 1兲,
dt
2
2
共10兲
where ␳33共t兲 = 兩c3共t兲兩2 is the probability for the atom to be in
the excited state 兩3典, 兩1典 represents state 兩 ↑ 典 for target atom A
at state 兩 ↑ 典 and 兩F = 1典 for other cases, ␳13共t兲 = c1共t兲c*3共t兲ei␯0t,
*
␳31共t兲 = ␳13
共t兲. The detuning of the laser ␯0 is zero for target
atoms A at state 兩 ↑ 典 and ⬃␯ for all nontarget atoms B and
part of the target atoms A at hyperfine state 兩2典. ⌫ = 2␲
⫻ 6.07 MHz is the decay rate of the excited state 兩3典, ⍀ is
the Rabi frequency of the resonant laser that is related to the
on-resonance saturation parameter by s0 = 2 兩 ⍀兩2 / ⌫2.
We numerically integrate the optical Bloch equation and
calculate the number of scattering photons
n p共t兲 = ⌫
冕
t
␳33共t⬘兲dt⬘
共11兲
0
for both target and nontarget atoms. In Fig. 3共a兲, we plot the
A
for target atoms A at state 兩3典 with respect to
probability ␳33
A
increases initially and reaches the saturation
time. We see ␳33
value s0 / 2共1 + s0兲. The number of scattering photons reaches
20 in a short period, 1.3 ␮s for atoms A at 兩 ↑ 典 关Fig. 3共b兲兴, but
it is only 10−5 for nontarget atoms B and target atoms A at
state 兩2典 关Fig. 3共c兲兴. Therefore the impact of the resonant
laser on the nontarget atoms B can be neglected. In Figs. 3共d兲
and 3共e兲, we see that for a wide range of Rabi frequencies
共laser intensities兲, the scattering photon number for the nontarget atoms B is suppressed to undetectable levels, below
10−4.
Unlike the noise correlation method, the accuracy of our
detection scheme does not scale with the number of total
atoms N, but is determined only by manipulation errors in
the above steps. We roughly estimate that n-spin correlations
can be probed with a cumulative error ⬍n ⫻ 10−2, which is
sufficient to measure both long- and short-range spincorrelation functions. Our estimate takes into account possible experimental errors in all four steps, and the fact that
the same experimental procedure is repeated many times to
determine the probability Pl. The total failure probability of
the four-step detection process is p共n兲 ⬃ n ⫻ 10−3 for n target
atoms to give an incorrect measurement of the target atom
quantum state for determining the quantity Pl. Assuming that
the experimental procedure is repeated ␯ times, the total
probability for obtaining one incorrect measurement result is
about ␯ p共n兲. Since one incorrect measurement result leads to
an error ⬃1 / ␯ in Pl, the expectation for the error is ␯ p共n兲
⫻ 1 / ␯ = p共n兲, which should be chosen to be 1 / ␯ to minimize
the total error. In addition, the uncertainty in measurements
of probability Pl itself in repeated experiments is also about
1 / ␯. Taking into account all of these errors, we find that
n-spin correlations can be probed with an error that scales as
Cp共n兲, where C ⬃ 3 in our rough estimate. As a conservative
estimate, we take C = 10 to give an overall error in measuring
n-spin correlation function ␰兵␣ jk , k = 1 , . . . , n其 to be less than
10p共n兲 ⬃ n ⫻ 10−2.
We note that the scheme requires repeated production of
nearly the same condensate and repeated measurements, two
standard techniques which have been realized in many experiments. We have proposed a powerful technique for investigating strongly correlated spin models in optical lattices
and now consider one of several possible applications.
IV. SPIN-WAVE DYNAMICS
Our technique can be used to investigate time dependence
of correlation functions. In the following, we show how our
scheme can be used to engineer and probe spin-wave dynamics in a straightforward example, the Heisenberg XX model
realized in optical lattices with a slightly different implementation scheme than the one discussed in the preceding section. Consider a Mott insulator state with one boson per lattice site prepared in the state 兩0典 ⬅ 兩F = 1 , mF = −1典 in a threedimensional optical lattice. By varying the trap parameters or
with a Feshbach resonance, the interaction between atoms
can be tuned to the hard-core limit. With large optical lattice
depths in the y and z directions, the system becomes a series
of one-dimensional tubes with dynamics described by the
Bose-Hubbard Hamiltonian,
Hs = − ␹共t兲 兺 共a†j a j+1 + a†j+1a j兲.
共12兲
j
This Bose-Hubbard model can be solved exactly. It offers a
testbed for spin-wave dynamics by mapping the Hamiltonian
共12兲 onto the XX spin model, HXXZ共−2␹ ; ⌬ = 0兲, with the
Holstein-Primakoff transformation 关25兴.
We now study the time-dependent behavior of the XX
model using our proposed scheme. In the Heisenberg picture,
the time evolution of the annihilation operator can be written
as a j共t兲 = 兺 j⬘a j⬘共0兲i j⬘−jJ j⬘−j共␣兲, where J j⬘−j共␣兲 is the Bessel
function of the interaction parameter ␣共t兲 = 2兰t0␹共t⬘兲dt⬘. To
observe spin-wave dynamics, we first flip the spin at one site
from ↑ to ↓, which, in the bosonic degrees of freedom, corresponds to removing an atom at that site. Because of the
spin-spin interactions, initial ferromagnetic order gives way
to a reorientation of spins at neighboring sites which propagates along the spin chain in the form of spin waves. This
corresponds to a time-dependent oscillation of atom number
at each site. Therefore, spin-wave dynamics can be studied in
one- and two-point spin-correlation functions by detecting
the oscillation of the occupation probability at certain sites
and the density-density correlator between different sites,
respectively.
Single atom removal at specific sites can be accomplished
with the assistance of focused lasers. With a combination of
microwave radiation and two focused lasers 共propagating
along y and z directions, respectively兲, we can selectively
transfer an atom at a certain site from the state 兩0典 to the state
兩4典 ⬅ 兩F = 2 , mF = −2典. A laser resonant with the transition
兩4典 → 兩3典 ⬅ 兩5 2 P3/2 : F = 3 , m = −3典 is then applied to remove
an atom at that site. Following an analysis similar to the one
023605-5
PHYSICAL REVIEW A 76, 023605 共2007兲
V共t兲 = V0/共1 + 4冑2PexeV0/ER␻rt兲,
共13兲
where Pexe is the probability of making an excitation to
higher bands and ␻R = ER / ប. For Pexe = 4 ⫻ 10−4, we find the
interaction parameter to be ␣共thold兲 = 0.0146+ 0.0228␻Rthold,
with the tunneling parameter
␹共t兲 = 共4/冑␲兲ER1/4V3/4共t兲exp关− 2冑V共t兲/ER兴.
共14兲
Two physical quantities that can be measured in experiments are the single atom occupation probability
D j共␣兲 = 具␸兩a†j a j兩␸典 =
J2l−j共␣兲
兺
l⫽␬
共15兲
at the site j, and the density-density correlator
†
G jj⬘共␣兲 = 具␸兩a†j a ja j⬘a j⬘兩␸典 =
−
2
J2l−j共␣兲J␥−j⬘共␣兲
兺
l⫽␬,␥⫽␬
2
关J2l−j共␣兲Jl−j⬘共␣兲
兺
l⫽␬
+ Jl−j共␣兲J␬−j共␣兲J␬−j⬘共␣兲Jl−j⬘共␣兲兴
共16兲
between sites j and j⬘, where ␸ is the initial wave function
with one removed atom at site ␬. The former is related to the
local transverse magnetization through
具␸兩szj 共␣兲兩␸典 = D j共␣兲 − 1/2,
共17兲
and the latter is related to the spin-spin correlator via
z
G jj⬘共␣兲 = 具␸兩szj 共␣兲s j⬘共␣兲兩␸典 + 关D j共␣兲 + D j⬘共␣兲兴/2 + 1/4.
共18兲
In Fig. 4, we plot D j共␣兲 and G jj⬘共␣兲 with respect to the
interaction parameters ␣ 共which scales linearly with holding
time兲. Here the initial empty site at j = 0 is located at the
center of the spin lattice. We see different oscillation behavior at different sites. Initially, all sites are occupied except
j = 0, i.e., D j⫽0共0兲 = 1, D j=0共0兲 = 0 关Fig. 4共a兲兴. The initial spinspin correlation G jj⬘共0兲 between different sites is zero if either j or j⬘ is zero, and one if both of them are nonzero 关Fig.
4共b兲兴. As ␣ increases, atoms start to tunnel between neighboring sites and the spin wave propagates along the onedimensional optical lattice, which is clearly indicated by the
1
0.8 (a)
0.6
0.4
0.2
0
0 1
Dj (α)
above, we see that the impact on other atoms can be neglected. To observe fast dynamics of spin-wave propagation,
we may adiabatically ramp down the optical lattice depth
共and therefore increase ␹兲 from the initial depth V0 = 50ER to
a final depth 13ER, with a hold time, thold, to let the spin
wave propagate. Finally, the lattice depth is adiabatically
ramped back up to V0 for measurement. The time dependence of the lattice depth in the ramping down process is
chosen to be
1
0.8 (b)
0.6
0.4
0.2
0
0 1
6
Gjj ' (α)
ZHANG, SCAROLA, AND DAS SARMA
2
3
α4
5
2
3
α
4
5
6
FIG. 4. Site occupation probability 共a兲 and density-density correlation 共b兲 with respect to scaled spin interaction parameter ␣⬀
time for N = 30 and j = 0 关共a兲 solid line兴; j = 1 关共a兲 dashed line兴; j
= 0, j⬘ = 3 关共b兲 dashed line兴; and j = 2, j⬘ = 3 关共b兲 solid line兴. The site
index j ranges from −N to N.
increase 共decrease兲 of the site occupation at j = 0 共j ⫽ 0兲. In
Fig. 4, the oscillation of D j共␣兲 and G jj⬘共␣兲 in a long time
period and the decay of the oscillation amplitudes originate
from the finite size of the spin lattice, which yields the reflection of the spin waves at the boundaries.
To probe the single site occupation probability D j共␣兲, we
use two focused lasers and one microwave pulse to transfer
the atom at site j to 兩4典. A laser resonant with the transition
兩4典 → 兩3典 is again applied to detect the probability to have an
atom at 兩4典, which is exactly the occupation probability
D j共␣兲. To detect G jj⬘共␣兲, we transfer atoms at both sites j
and j⬘ to the state 兩4典 and use the same resonant laser to
detect the joint probability for atoms at 兩4典. The fluorescence
signal has three levels, which correspond to both atoms
G jj⬘共␣兲, one atom D j共t兲 + D j⬘共t兲, and no atoms at state 兩4典. A
combination of these measurement results gives the spin-spin
correlator 具␸ 兩 szj 共␣兲szj⬘共␣兲 兩 ␸典.
V. CONCLUSION
We find that a relation between general spin-correlation
functions and observable state occupation probabilities in optical lattices allows for quantitative measurements of a variety of spin correlators with the help of local probes, specifically focused lasers and microwave pulsing. Our proposal
includes a realistic and practical quantitative analysis suggesting that n-spin correlations can be probed with an error
⬍n ⫻ 10−2, which is sufficient to measure both long- and
short-range spin-correlation functions. Our work establishes
a practical and workable method for detecting n-spin correlations for cold atoms in one-, two- or three-dimensional
optical lattices. Applications to a broad class of spin physics
including topological phases of matter 关6,12兴 realized in
spin-optical lattices are also possible with our proposed technique.
ACKNOWLEDGMENT
This work is supported by ARO-DTO, ARO-LPS, and
LPS-NSA.
023605-6
PHYSICAL REVIEW A 76, 023605 共2007兲
PROBING n-SPIN CORRELATIONS IN OPTICAL LATTICES
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