Disorder effects on the quantum coherence of a many-boson system

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by  Zhi-Hai Zhang

Disorder effects on the quantum coherence of a many-boson system
Zhi-Hai Zhang,1 Pei Lu,1 Shiping Feng,1 and Shi-Jie Yang∗1
1Department of Physics, Beijing Normal University, Beijing 100875, China
The effects of disorders on the quantum coherence for many-bosons are studied in a double well
model. For the ground state, the disorder enhances the quantum coherence. In the deep Mott
regime, dynamical evolution reveals periodical collapses and revivals of the quantum coherence
which is robust against the disorder. The average over variations in both the on-site energy and
2
theinteraction revealsabeatphenomenonofthecoherence-decoherenceoscillation inthetemporal
1
evolution.
0
2
PACSnumbers: 03.75.Lm,03.75.Gg,03.75.Kk

n
a
J The ultracold atoms in an optical lattice have pro- richandcomplexarrangementofdifferentinsulatingand
6 vided an unprecedent opportunity to explore the corre- superfluid states[24–26].
1 lated many-body problems and nonlinear dynamics in In this paper, we explore the quantum coherence of a
condensedmatter physics[1–6]. The atomstunneling be- correlated Bose gas in a double well system[27–29]. We

] tween lattice sites play the role of electrons in the solid. show that the on-site energy disorder enhance the quan-
s
a Due to the controllable manipulation of the optical lat- tum coherence in the ground state. In the deep Mott
g tice, the inter-particle interaction as well as the well de- regime, dynamical evolution reveals periodical collapses
-
t veloped measurement technique, scientists can now pre- and revivals of the quantum coherence. The periodicity
n ciselyexaminethecoincidenceoftheoreticalmodels,e.g., is robust against the disorder. In addition, the disorder
a
the Hubbard model in strongly correlated systems, with in the interatomic interaction also plays an important
u
q experiments[7–9]. Ontheotherhand,experimentaltech- role. We find that the revival oscillations of quantum
. niquesalsoexploitedthecollapseandrevivaldynamicsof coherence exhibit the beat phenomenon when a moder-
t
a ultracold atoms in the optical lattice[10, 11]. In a recent ate interaction disorder is taken accounted of. The cor-
m experiment, S. Will et al observedthe beat phenomenon responding Fourier spectra of the coherence parameter
- of quantum coherence for Bose atoms in the dynamical exhibit a double-peak structure.
d evolution[1]. This phenomenon was interpreted as the We are concerned with N bosonic atoms confined in
n
multi-bodyinteractionsduetomulti-orbitalvirtualtran- a double well potential. By considering only the lowest
o
c sitions. energy band, the Bose-Hubbard Hamiltonian is written
[ The disorders usually play important roles in realistic as [30, 31]
materials. It profoundly affects the transport proper-
1
ties of a wide range of materials. The dilute atomic gas U
4v in disordered optical potentials is again an ideal candi- Hˆ =−t(aˆ†1aˆ2+aˆ†2aˆ1)+ 2 Xnˆi(nˆi−1)+ε(nˆ1−nˆ2), (1)
date model to simulate various disorder effects in con- i
4
1 densed matter physics[12–15]. Disorders may be created
3 in the laser potential in different ways. One of them is where aˆ†i and aˆi (i = 1,2) are the creation and anni-
1. to add a speckle pattern to the regular lattice[16]. The hilation operators in either side of the well. nˆi are the
0 disorder strength is continuously tunable by controlling number operators. U and t are the Hubbard energy and
2 the intensity of the speckle field. The localization ef- the atom hopping, respectively. ε is the on-site energy
1 fects due to correlation and disorder compete against difference between the two sites. The disorder is intro-
: each other, resulting in a partial delocalization of the duced into the model via the fluctuation of ε , which is
v
i particles in the Mott regime, which in turn lead to in- assumedtorandomlyanduniformlydistributewithinthe
X creasedphasecoherence[17]. However,thereissignificant interval ε ε ε.
−△ ≤ ≤△
r disagreement regarding features of the disordered Bose- The Hamiltonian (1) is explicitly represented in the
a
Hubbard model[18–23]. Experiments are still some way Fock basis set N,0 , N 1,1 , , 0,N . The gen-
{| i | − i ··· | i}
from the appropriate regime to observe, e.g., Anderson eral eigenstates are expressed as linear combinations of
localization,duetothelargespecklesizeandthedelocal- the occupation bases, ψ = N c N k,k (j =
| ji Pk=0 jk| − i
izingeffectoftheinteratomicinteractions. Theinterplay 0,1,2,...,N), which correspond to the eigenvalues ω .
j
between the interaction and the disorder also lead to a The coefficients c satisfy the recursive relation
∗Correspondingauthor: [email protected] jk
U
−tp(N −k)(k+1)cj(k+1)−tp(N −k+1)kcj(k−1)+[2(N2−2Nk−N +2k2)+ε(N −2k)−ωj]cjk =0. (2)
2
The temporal evolution of the state is governed by the
Schr¨odinger equation for a given initial state ψ(0) :
| i 1
∆ε/t=0
N N 0.8 (a) ∆ε/t=5
ψ(τ) =Xfj(τ)ψj =Xgk(τ)N k,k , (3) ∆ε/t=11
| i | i | − i ∆ε/t=20
j=0 k=0 0.6
α
where gk(τ) = PNj=0fj(0)cjke−iωjτ, with fi(0) = 0.4
ψ ψ(0) . 0.2
j
h | i
To depict the coherence degree of the system, we in-
0
troduce a characteristic parameter[31]: 0 10 20 30 40 50 60
U/t
1
λ λ
1 2
α(τ)= λ −+λ (4) 0.8 (b)
1 2
0.6
where λ and λ are the two eigenvalues of the single-
1 2 α
particle density ρµν(τ) = hψ(τ)|aˆ†µaˆν|ψ(τ)i (µ,ν = 0.4 U/t=2
1,2)[32, 33]. When α 1, the system is in the coherent
U/t=10
(quasicoherent) state s→ince in this case there is only one 0.2 U/t=20
large eigenvalue of the matrix ρ . Accordingly, α 0 U/t=50
indicates the system is in the deµcνoherent or fragmen→ted 00 5 10 15 20 25 30
∆ε/t
state because there are two densely populated natural
orbits. In the weak-interaction or strong-tunneling limit
(U/t 1), each atom is in a coherent superposition of
≪
the left-well and the right-well states. The ground state
FIG.1: (a) αversusU/t for several on-site disorderstrength
of the system is a state with a mean number N/2 of
(from lower to upper) ∆ε/t= 0,5,11,20. (b) α versus ∆ε/t
atoms in each well. In the opposite limit of the strong-
for several values of Hubbard energies. From upperto lower:
interaction or weak tunneling (U/t 1), the tunneling U/t=2,10,20,50.
≫
term is negligible. In this case, the Hamiltonian is the
product of number operators for the left and right wells.
The eigenstates are products of Fock states and are re-
ferred as decoherent states. This regime is analogous to Thelatticeparameterisnon-adiabaticallychangedtothe
the Mott insulator (MI) phase in a lattice system. deep MI regime. Figures 2 are the temporal evolution
In the following we consider the typicalcases in which of the coherence degree α for two typical interactions
the initial state is a coherent state. We choose N = 10 U/t=10,50, respectively. It is clearly shown that α os-
and set t = 1 as the energy units. All results are taken cillatesbetweenoneandzero,implying the systemexpe-
50repetitionsoftherandomchoiceofdisorders,whichis riences revivals and collapses of the coherence. Fig.2 (a)
enough to average out the fluctuations. and(b)arefor the systemwithoutthe disorder. Clearly,
Figures 1 display the coherence degree α versus the the periodicity is more distinct for larger U/t. The pe-
disorder in the ground states by taking average over the riod of the oscillation is T = π/U. Fig.2(c) and (d)
on-site energy within the range of ∆ε/t = 0,5,11,20, present temporal evolution of α with the on-site disor-
respectively. From Fig.1 (a), it can be seen that in all der ∆ε/U = 20% for two different values of U/t = 10
range of U/t the disorder raises α, in agreement with andU/t=50,respectively. ForsmallU/t (Fig.2(c)), the
previous studies. The coherence increaseby the disorder amplitude of the oscillation is small. The large ampli-
ismostprominentintheregimeofintermediatestrength tude of periodical oscillation of α(τ) in Fig.2(d) implies
oftheHubbardinteractions. Fig.1(b)showsthedisorder the system experiences revivals and collapses of the co-
dependence of the coherence degree α at several values herence. In contrary to one’s intuition that the disorder
of the Hubbard energies U/t=2,10,20,50,respectively. may destroy the periodicity of the revival and collapse
ForsmallerU/t,thegroundstatehaslargerα,indicating oscillation,they enhance the periodicity by damping the
that it is in a coherent (superfluid) state. For all U, the irregular fluctuations of the coherence degree α in the
on-site energy disorder helps to enhance the quantum temporal evolution.
coherence. Generally,thestrongerdisorderhasthelarger This phenomenon can be understood as follows.
coherence degree. The initial state can be expanded as ψ(0)
| i ∼
Next we study the disorder effect on the quantum dy- N Ck(aˆ†)N−k(aˆ†)k 0 . When U/t 1, the Fock
namics of the N-bosons system. Suppose the system is Pstakt=e0s (aˆN†)k10 in ea2ch|wiell are the eigen≫states, and the
initially in a coherent state, j | i
corresponding eigenenergies are Uk(k 1)/2. The sys-
−
aˆ† +aˆ† temisthenasuperpositionofproductsoftheFockstates,
ψ(0) =( 1 2)N 0 . (5) which evolves independently as,
| i √2 | i
3
1 1
(a) U/t=10 (b) U/t=50
0.8 0.8
0.6 0.6
α α
0.4 0.4
0.2 0.2
0 0
0 1 2 3 4 5 0 0.1 0.2 0.3 0.4
τ τ
1 1
(c) U/t=10 (d) U/t=50
0.8 0.8
0.6 0.6
α α
0.4 0.4
0.2 0.2
0 0
0 1 2 3 4 5 0 0.1 0.2 0.3 0.4
τ τ
FIG.2: (coloronline)Thetemporalevolutionofthecoherence
degree α(τ) without the disorder for (a) U/t = 10 and (b)
U/t=50, and with the on-site disorder ∆ε/U =20% for (c)
U/t=10 and (d) U/t=50. The dashed line in (d) is for the
independentevolution as explained in the text.
N
|ψ(τ)i ∼ XCNke−i[12U(N−k)(N−k−1)+ε1(N−k)]τ ×(aˆ†1)(N−k)e−i[21Uk(k−1)+ε2k]τ(aˆ†2)k|0i
k=0
N
= e−i[12UN(N−1)+ε1N]τ XCNkei[U(N−k)k+εk]τ(aˆ†1)N−k(aˆ†2)k|0i, (6)
k=0
where a time-dependent phase factor is attached in each vivalsandcollapsesofthecoherenceisrobustagainstthe
term. If there are no disorders (∆ε = 0), the system disorder[5].
will recover its initial state when the time evolves an
We explore the system by further taking account of
integer multiples of π/U. In between this period, super-
the interaction disorders with the Hubbard energy U in
position form various terms cancels and the coherence
arangofuniformlydistributedintervalU ∆U. Theon-
is destroyed[31]. In the presence of the disorder, there ±
sitedisorderisfixedat∆ε/U =20%withU/t=50. Fig-
are additional phases associating to the ∆ε in eachterm
ures3showthetemporalevolutionofαforvariousinter-
that seem destroy the coherence of the quantum state.
action disorders (left panels). The right panels (d-f) are
However,numericalcalculationsshowthat itis nottrue.
the corresponding Fourier spectra which exhibit double-
TheoriginisthatinthedeepMottregimes,particletun-
peak structures as signals of beat effects. In the absence
neling is suppressed. All Fock states have almost the
of the interaction disorders (∆U/U =0), Fig.3(a) shows
same oscillating periods which only depends on the in-
that the beat effect is smeared by additional oscillations
teraction strength T = π/U. Hence average over onsite
in the temporal evolutionof α, which is evidently shown
energy ǫ does not reduce the amplitude. The dashed
in Fig.2(b). These oscillations give rise to an additional
line in Fig.2(d) reveals that the periodicity are perfectly
peak at two times of the double-peak frequency. When
reserved. We conclude that the phenomenon of the re-
interaction disorders are counted in (Fig.3(b) and (c)),
4
not clear yet if there are inherent relations between the
beat phenomena observed by S. Will et al and our theo-
1 0.8 retical model.
α 0.05 ∆U/U=0 (a) Spectral density (a.u.)0.04 (d) In summ1ary, we have investigated the diso u.)rd0e.2r effect
αα 00..015015000 111∆∆UU//UU==13%%222 τττ 333 444((cb)) 555Spectral density (a.u.)Spectral density (a.u.)0000....012101211000FFFrrr555eee00qq0quuueee(nnn(999efccc)00)0yyy111222000 ααα 000...150150500 11UUU///ttt===543000 22 ττ 33 44((c(ba))) 55 density (a.u.)Spectral density (a.Spectral density (a.u.)00000.....120101211 00FFrree55qq00uu(ee((genf99n)))c0c0yy1122 00
0 pectral 0
0 1 2 3 4 5S 10 50 90120
τ Frequency
1 u.)0.4
FcoIhGe.re3n:c(ecodleogrroeenlαinew)itLhefbtoptahnoenls-:siTteemdipsoorradleresvo∆luεt/iUon=of2t0h%e α 0.5 U/t=50 (d) density (a.0.2 (h)
aanndd i(nct)er∆acUti/oUn d=iso3r%de,rsw.it(ha)U∆/tU/=U5=0.0,R(ibg)ht∆pUa/nUels=: 1t%he, 0 pectral 0
corresponding Fourier spectra. 0 1 2 3 4 5 S 10 50 90120
τ Frequency
FIG.4: (color online)Left panels: Temporalevolution ofthe
the additional oscillations in Fig.2(b) are damped, as
coherence degree α(τ) with both on-site disorders ∆ε/U =
shown in Fig.2(d). Consequently, the interaction disor-
20% and interaction disorders ∆U/U =2% for (a) U/t=30,
derenhancesthebeateffectofthecoherence-decoherence
(b) U/t = 40, and (c) U/t = 50. For comparison, (d) is
osccilations. independentevolution (6) with both disorders. Rightpanels:
Figures 4 show the temporal evolution of α for var- thecorresponding Fourier spectra.
ious interactions U/t = 30,40,50, respectively, with
∆ε/U = 20% and ∆U/U = 2%. As a comparison,
Fig.4(d) displays the result from independent evolution
described by equation (6). Figs.4 (e)-(h) are the corre-
on the quantum coherence of a many-boson system. We
sponding Fourier spectra. It shows that in deeper MI
findthatthe disorderenhancequantumcoherenceinthe
regime, the beat effects become more prominent.
ground states. Dynamical evolution of the cold Bose
In a recent paper, it has been observed the beat phe-
atoms exhibits collapses and revivals of the coherence
nomenonoftherevivalsandcollapsesofthequantumco-
which is robust against the disorder. In particular, the
herence in an optical lattice in a longer time interval[1].
interaction disorders generate the beat phenomenon of
The beat signal of the revival and collapse oscillation
coherence in the temporal evolution. It may provide a
was attributed to the multi-body interactions, emerging
signal to observe the effects of the interaction disorders
through virtual transitions of particles from the lowest
experimentally.
energy band to the higher energy bands. In our model,
the atoms are assumed to occupy a single spatial orbital ThisworkissupportedbythefundsfromtheMinistry
and only the two-body interaction Hubbard energy U is of Science and Technology of China under Grant Nos.
independentofthefillingatthelatticesite. Thedisorder 2012CB821403 and 2011CB921700 and by the National
canenhancetheperiodicityoftherevivalandcollapseos- Natural Science Foundation of China under grant No.
cillation by damping the higher-order frequencies. It is 10874018.
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