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arXiv:cond-mat/0407022v1 [cond-mat.other] 1 Jul 2004

Correlation functions of cold bosons in an optical lattice

Radka Bach and Kazimierz Rza֒˙ zewski∗

Center for Theoretical Physics, Polish Academy of Sciences, al. Lotnik´ ow 32/46, 02-668 Warsaw, Poland

(Dated: February 2, 2008)

We investigate the experiment of collapses and revivals of matter wave field in more detail. To this

end we calculate the lowest-order correlation functions of the Bose field. We compare predictions of

the total Fock state with the commonly used coherent state approximation. We also show how to

observe an interference pattern for the celebrated Mott state.

I. INTRODUCTION

Cold bosons in optical lattices have gained a lot of at-

tention recently because of the ease and precision with

which they can be manipulated [1].

was an optical lattice that enabled the observation of

collapses and revivals of a matter wave field [2]. Even

though the theoretical analysis of this experiment pro-

vided by authors is based on the notion of coherent states

and as such is—in principle, at least—inadequate to de-

scribe systems with total number of atoms fixed, it seems

to work quite well. It is our purpose to describe this ex-

periment more accurately, in particular to examine the

role of total number of atoms conservation and identify

situations in which it will clearly manifest itself.

This will be achieved with the aid of correlation func-

tions. Contrary to the common belief, there is no need to

build a separate experimental set-up to measure higher-

order correlation functions in atomic systems – it is

enough to analyze photographs of a cloud of atoms. Such

photographs, which are typically obtained in the final

stage of experiments with cold gases, are nothing else but

a simultaneous detection of many atoms and therefore

probe the correlation function of the order of the number

of atoms [3]. And since the observed system comprises of

a fixed number of atoms, it is described by a Fock state;

consequently it is possible to reconstruct low-order cor-

relation functions out of these measurements [4]. Having

the possibility of measuring correlation functions, it is

justified to investigate them theoretically as well.

The article is organized as follows. In Section II we

start with a brief description of the experiment in which

collapses and revivals of a matter wave field were ob-

served. On this basis two states of the system are in-

troduced: one that obeys the total number of atom con-

servation and one that violates it (the latter for com-

parison). Correlation functions corresponding to both

states are calculated and investigated in Section III. In-

ter alia, we identify situations in which predictions of

the two examined states differ.

the collapsed phase the system effectively behaves as if

there was no site-to-site coherence, as pointed out in Sec-

For example, it

We also note that in

∗Also at Cardinal Wyszy´ nski University, al. Lotnik´ ow 32/46, 02-

668 Warsaw, Poland

tion IV. Even then, though, the second-order correlation

function shows nontrivial structure and interference pat-

tern should be seen in a single photograph. Counter-

intuitively, this is also the case for Mott insulator, as

shown in Section V, and it is extremely difficult to tell

these states apart on the basis of a single or many mea-

surements. We conclude in Section VI.

II. THE EXPERIMENT

The experiment in which collapses and revivals of the

matter wave field were observed [2] was composed of a

few steps. After preparing a Bose-Einstein condensate

in a harmonic oscillator potential, an optical lattice po-

tential was slowly raised. The height of this lattice has

been chosen such that the system was still completely in

the superfluid regime. Then, the intensity of light cre-

ating the lattice was rapidly increased and the resulting

optical lattice was so high that the tunneling between

the sites was strongly suppressed. Was this raising done

adiabatically, the system would move to the Mott insu-

lating phase; but because it was done rapidly, the atom

number distribution at each well from a superfluid state

was preserved at the high lattice potential, thus produc-

ing a mixture of Fock states with different number of

atoms at each well. The system was then left to evolve

for some time, which was varied from experiment to ex-

periment (we will call it interaction time and denote T).

Finally, after switching off all potentials the atomic cloud

was allowed to expand freely for time t before shooting

a photograph. As a function of the interaction time T

a collapse and a revival of the interference pattern were

seen.

We are going to describe this experiment via a sim-

ple model which nonetheless includes the most relevant

features of the system. For example, we will neglect the

non-uniformity of the optical lattice stemming from addi-

tional harmonic oscillator potentials (one used to create

the Bose-Einstein condensate and another one associated

with widths of laser beams), but consider a system com-

posed of N atoms distributed among K equivalent lattice

sites instead. The notion of atom density, ρ = N/K, will

be used interchangeably. The wells’ exact shape is also

not important since we are going to assume that atoms

are so cold that they do not have enough energy to occupy

states other than the ground state, which is justified by

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the fact that the most important part of the evolution—

the one from which collapses and revivals stem— takes

place when lattice potential is very high and when the

energy of an atom is far less than the energy gap to the

first excited state.

Prior to turning on the strong optical lattice the sys-

tem was in a superfluid state and so right after raising

the potential the distribution of atoms between wells is

multinomial. Hence the state of the system is:

|ψ(T = 0)? =

?

N!

KN

N

?

n1=0

?

N

?

n2=0

···

N

?

nK=0

???

1

√n1!n2!...nK!

|n1,n2,...,nK?

(1)

where the underbrace denotes total number of atoms con-

servation law, i.e. n1+ n2+ ··· + nK= N and we have

already assumed that each atom can occupy any of the

wells with equal probability.

Once prepared, the system is left to evolve for time T.

Since atoms are now imprisoned in a strong optical lat-

tice in which tunneling is highly suppressed, in each well

they evolve independently of others. Assuming contact

interactions between atoms, the Hamiltonian in each well

is effectively of the form:

ˆH =?g

2ˆ n(ˆ n − 1) (2)

where ˆ n is the operator of the number of atoms in this

well and g denotes the rescaled coupling constant:

g =4π?a

m

?

dx|ϕ(x)|4

(3)

(ϕ(x) is the wavefunction of the ground state of a well

and a is the s-wave scattering length). Then, after time

T the state of the system is:

|ψmnm(T)? =

?

N!

KN

N

?

n1=0

?

N

?

n2=0

···

N

?

nK=0

???

cn1(T)···cnK(T)|n1,n2,...,nK?

(4)

where the coefficients cn(T) are:

cn(T) =

1

√n!exp

?

−igT

2n(n − 1)

?

. (5)

To investigate the role of total number of atoms conser-

vation law, however, we are going to analyze a coherent

counterpart of this state as well:

|ψcoh(T)? = e−ρK/2

∞

?

n1=0

∞

?

n2=0

···

∞

?

nK=0

αn1+n2+···+nK

cn1(T)···cnK(T)|n1,n2,...,nK?

where the coefficients cn(T) are defined as before and

|α|2= ρ for consistency.

(6)

As far as a subsystem composed of a fixed number of

wells is concerned, the multinomial state approaches the

coherent one when the number of atoms and the number

of wells are increased in such a way that the density is

kept constant [7]. In this limit—let us call it thermo-

dynamic limit in analogy with statistical mechanics—the

remaining part of the system serves as a particle reservoir

and not surprisingly the distribution of atoms factorizes

and becomes poissonian in each well independently, as it

is for coherent states.

III.CORRELATION FUNCTIONS

Let us now calculate explicit analytical formulas for

correlation functions for the multinomial and coherent

state introduced above. The r-th order correlation func-

tion is defined as:

G(r)(x1,x2,...,xr;T,t) =

?

ψ(T)

???ˆΨ†(x1,t)···ˆΨ†(xr,t)ˆΨ(xr,t)···ˆΨ(x1,t)

???ψ(T)

?

(7)

In the above formula two distinct times appeared: the

interaction time T, denoting how long atoms were left

to evolve in the strong optical lattice potential, and the

measurement time, t, which is the time between switching

off all binding potentials and shooting the actual pho-

tograph. During the latter time there are no external

potentials and we also assume that the interaction does

not play any major role—consequently the only effect

this process has on the system is the free expansion of

wells’ wavefunctions. Although this way the interaction

phases that atoms might have acquired are neglected, the

assumption is not only justified (since during expansion

the system becomes extremely dilute), but it also clarifies

the overall picture. Each of the times is now responsi-

ble for different physical effects: the interaction time,

T, governs the phases of different Fock states and there-

fore is responsible for collapses and revivals, while the

measurement time, t, determines the behaviour of wells’

wavefunctions and as such does not influence the relation

between different Fock states. Note also that the time t

is treated here rather as a convenient control parameter

than a true argument of the correlation function.

The field operatorˆΨ(x,t) can be decomposed in any

complete basis of annihilation operators. In the system

we are considering it is convenient to introduce annihila-

tion operators of atoms at individual sites’ ground states:

ˆΨ(x,t) =

?K

though straightforward calculations one obtains:

k=1ϕk(x,t)ˆ ak. Then after some lengthy

G(r)(x1,...,xr;T,t) = ρr

?

l1,...,lr

?

k1,...,kr

ω({l,k},T)×

× ϕ∗

l1(x1,t)···ϕ∗

lr(xr,t)ϕkr(xr,t)···ϕk1(x1,t)

(8)

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where:

ωcoh({l,k},T) = R({l,k},T)exp{ρQ({l,k},T)}

for coherent states and

(9)

ωmnm({l,k},T) =

N!

Nr(N − r)!R({l,k},T)

for multinomial states. The functions R and Q are:

R({l,k},T) =

Q({l,k},T) =

i=1

?

1 +Q({l,k},T)

K

?N−r

(10)

(11a)

2

(11b)

= exp

igT

2

K

?

i=1

?

r

?

j=1

δlj,i

2

−

r

?

j=1

δkj,i

?

K

?

eigT?r

j=1(δlj,i−δkj,i)− 1

First of all the expression for the correlation does not de-

pend on the interaction time T nor the coupling strength

g alone, but on the product of the two. It means that

at T = 0 the interacting system effectively mimics the

behaviour of an ideal gas, for which, as can be easily

checked, all correlation functions factorize (for multino-

mial states there appears an additional proportionality

constant). And because the functions R and Q are peri-

odic functions of gT, this also happens for any time such

that gT = 2kπ, where k is an integer. This re-appearance

of the interference pattern is nothing else but the revival

phenomenon.

A. The first-order correlation function

Density—the first-order correlation function—is ob-

tained from above formulas via setting r = 1 and can

be written as:

?

k

G(1)(x) = ρ

|ϕk(x)|2+ κρ

?

l

?

k?=l

ϕ∗

l(x)ϕk(x) (12)

where:

κ =

exp

?

−4ρsin2

?gT

?gT

2

2

??N−1

??

for coherent states

?

1 −4

Ksin2

for multinomial states

(13)

First of all, the density is a sum of two kinds of terms:

the background, which is a simple consequence of having

on average ρ atoms at each lattice site, and interference

terms. The presence and relative amplitude of the latter

is governed by the coefficient κ.

The Fig. 1(a) depicts typical density profiles versus the

interaction time. To observe interference pattern two

conditions must be simultaneously fulfilled: (i) the in-

teraction time gT is such that the coefficient κ is not

00 0.250.250.5 0.5 0.750.7511

gT/2π

gT/2π

-5-5

00

55

xx

0

1

2

3

4

5

6

7

(a)

x

x’

1

2

0

4

(b)

FIG. 1: First- and second-order correlation functions for one-

dimensional system composed of K = 10 wells and N = 30

atoms.The wells’ wavefunctions were Gaussians with ini-

tial width, σ = 0.12 (this particular value has been cho-

sen such that the resulting nonorthogonality of neighbouring

wells’ wavefunctions is still negligible—of order of 10−31—

but there are only few fringes in the second-order correlation

function) and the measurement time was t = 3σ. Arrows

denote the positions of wells’ centers. (a) Density profiles

versus the interaction time, gT. Both the collapse and the re-

vival of interference pattern is clearly visible. (b): Close-up

for a certain value of interaction time, gT = 0.3 × 2π. The

curve at the top is the density profiles (this is a cut along

the dotted line from the figure above). The 2-dimensional

density plot denotes the normalized second-order correlation

function, g(2)(x,x′), where x and x′are plotted at the vertical

and horizontal axis, respectively.

vanishing and (ii) the measurement time, t, is such that

the wavefunctions ϕi(x) substantially overlap. The lat-

ter condition controls the exact shape of interference pat-

tern observed while the first one influences its visibility.

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To compare predictions of multinomial versus coherent

states it is therefore enough to investigate the coefficient

κ only. It is astonishing how quickly—when increasing

the number of wells—predictions of the two states coin-

cide: practically from 3 wells upwards there is no way

to distinguish multinomial from coherent states through

density profiles. Only for two wells, K = 2, is there a

substantial discrepancy: binomial distribution predicts

an additional revival at gT = π. Interesting enough, the

interference pattern observed at this additional revival is

in-phase with the pattern at gT = 0 for odd number of

atoms and out-of-phase for even number of atoms. One

might think that investigating the situation with only

two wells is pushing the analysis beyond the limits set

by experimental reality. This is not true, however. Only

slight modification of the experiment is required to go

into this interesting regime [8]. Imagine that before the

first optical lattice (the one, in which the atomic system

is in a superfluid state) is turned on, another optical lat-

tice is raised: a lattice deep enough to force the system

into the Mott insulator state, with a well defined number

of atoms per site. Let us now turn on a lattice with half

the spatial period of the one already existing: this will

produce a binomial distribution of atoms in each deep

potential well between two shallow wells.

The vanishing of the coefficient κ for almost all inter-

action times apart from the ones close to revival times is

the origin of the collapse phenomenon. To estimate the

time of collapse T∗one can require the coefficient κ to

reach a pre-set small value, ǫ. Then:

cos(gT∗

mnm) ≤ 1 −K

coh) ≤ 1 +lnǫ

2

?

1 − ǫ

1

N−1

?

(14a)

cos(gT∗

2ρ

(14b)

Note that the collapse is (apart from K = 2,3 cases)

never exact, i.e. when demanding the interference fringes

to vanish completely, which corresponds to setting ǫ = 0,

the above inequalities cannot be satisfied. At least not

without simultaneously increasing the density ρ. How-

ever, for any practical purposes it is enough to have the

visibility of fringes smaller than the sensitivity of detec-

tors to speak about complete collapse of interference pat-

tern. For small number of wells, on the other hand, there

exist interaction times for which fringes vanish identi-

cally: these are gT∗=

K = 2 case, and gT∗=

K = 3 case.

1

4× 2π and gT∗=

1

3× 2π and gT∗=

3

4× 2π for

2

3× 2π for

B. The second-order correlation function

Let us concentrate on the normalized second order cor-

relation function, defined as:

g(2)(x,x′) =

G(2)(x,x′)

G(1)(x)G(1)(x′)

(15)

because this way information about the density is set

aside and only correlations are being investigated. For in-

teraction time such that gT = 2kπ, where k = 0,1,2,...,

i.e. exactly the revival time, the normalized second-order

correlation function is constant, because then the ideal

gas case is recovered and consequently:

?1

1 −

g(2)

gT=2kπ(x,x′) =

for coherent state

for multinomial state

1

N

(16)

Already, there is a difference between the two states ex-

amined: multinomial states predict a small antibunching

effect, g(2)(x,x′) < 1, which is due to the total number of

atoms conservation law: the chance of detecting an atom

at position x is smaller if another atom was already de-

tected somewhere else (than if it was not) because there

are less atoms altogether. Note that this result will hold

also for other interaction times, provided x and x′are

distant enough.

However, if one moves away from the exact revival

time, so that gT is no longer an integer multiple of 2π,

an additional structure in the second order correlation

function appears. Typically, these are diagonal stripes

in the x − x′plane, such that the normalized second or-

der correlation function depends only on the difference

of its spatial arguments: g(2)(x,x′) ≈ g(2)(x − x′), as

seen at Fig. 1(b) (at least far from the boundaries of the

system). It means that in every realization interference

fringes are going to be observed, but the whole pattern

will move from shot to shot and will vanish after aver-

aging, producing a smooth density profile. The range of

g(2)is closely related to the size of expanding wavepack-

ets. Speaking roughly, to have a nontrivial correlation,

wavepackets must spread over the distance equal to the

separation between the points at which g(2)is calculated

or measured. The separation between the fringes, on the

other hand, is determined by the initial size of expand-

ing wavepackets, or, in other words, by the momenta

involved: the steeper the optical lattice potential, the

more localized the wavefunction and the more fringes in

the second-order correlation function structure.

The formulas for correlation functions are quite dis-

tinct for both kinds of states, it is thus somewhat sur-

prising that the final plots are practically indistinguish-

able. To spot possible discrepancies we have investigated

coefficients ω({l,k}) defined by Eq. (8) in more detail,

and found that even though these coefficients depend

on four indices, they can take only one of six values

(this happens—technically speaking—because functions

R and Q depend not on the values of the indices them-

selves, but on their mutual relation, i.e. on the way they

split into equivalence classes with respect to the number

of identical values). In the limit K → ∞ and N → ∞

such that ρ = N/K = const, the coefficients correspond-

ing to multinomial states become identical with the ones

stemming from coherent states, which is a manifestation

of the equivalence of the two states in the thermodynamic

limit. This limit is achieved so quickly that from K = 5

upwards there is practically no way of distinguishing be-

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tween the two states [9]. Only for K = 2,3,4 wells case is

there a room for discrepancies and indeed such situations

are found and depicted in Fig. 2.

The conclusions presented above—stemming from one-

dimensional simulations—are not altered when extending

the analysis to higher dimensions. With one important

note, however. The structure in the second-order corre-

lation function implies that interference pattern should

be seen in every single realization and smooth out when

averaged over many experiments, producing a relevant

density profile. This is true for a measurement which

detects atoms in the full, 3-dimensional physical space

and is in general false for its 2-dimensional projection

(think about holograms versus photographs). The latter

is governed by a column-averaged correlation function, in

which information about correlations in the direction of

the illuminating light pulse is lost, and such a situation

corresponds to performing many 2-dimensional experi-

ments and averaging over them. This explains why the

collapse was seen at all in a single experimental run. Nu-

merical simulations of a lattice composed of 5 × 5 sites

show that the visibility of the interference pattern drops

from 44% when averaged over one layer to 19% when av-

eraged over all 5 layers, and the effect is expected to be

even more profound for larger systems.

IV. INCOHERENT SYSTEMS

The system examined herein before was implicitly as-

sumed to be perfectly coherent, in a sense that in nu-

merical simulations wavefunctions of all optical lattice

sites had identical phases. This was because in the ex-

perimental situation the system prior to the turning on

the strong optical lattice potential was in a superfluid

phase, which exhibits long-range coherence. However, in

principle, the same formalism is also applicable to the

situation in which there is no long-range coherence in

the initial state, i.e. the wavefunctions’ phases are com-

pletely random. Even though the physical relevance of

such a state might be questionable, it is worth to exam-

ine this limit as well, because then one could separate

effects originating from having a mixture of Fock states

at each site from the ones necessarily requiring the coher-

ence between sites. The state with completely random

phases is also interesting because—at least when large

measurement times are concerned—it can be to a large

extent investigated analytically independently of its size

and dimension.

Let us therefore assume that (i) the wavefunctions

of individual wells have random phases:

eiφkϕk(x), where φkare random numbers between 0 and

2π, and (ii) the measurement time t is so large that at

each point in space wavefunctions of many sites overlap.

Then, in the expressions for the first- and second-order

correlation functions all terms that explicitly depend on

ϕk(x) →

gT= 0.25 × 2 π

0

3

coherent

0

1

2

0

3

multinomial

gT= 0.50 × 2 π

0

5

coherent

0

1

2

0

5

multinomial

gT= 0.33 × 2 π

0

4

coherent

0

1

2

0

4

multinomial

gT= 0.50 × 2 π

0

5

coherent

0

1

2

0

5

multinomial

FIG. 2: Situations, at which coherent and multinomial states

give distinctively different predictions. Density plots denote

the normalized second-order correlation function, g(2)(x,x′),

where x and x′are on the horizontal and vertical axis, re-

spectively. Arrows denote the position of the wells’ centers.

One-dimensional graph on top of the density plot is the den-

sity, G(1)(x). The parameters of physical systems, apart from

the number of wells and the interaction time, are identical for

all plots: atom density, ρ = 5, measurement time, t = 2σ,

and initial width of expanding wavepackets, σ = 0.12.