We study the survival of super-currents in a system of impenetrable bosons
subject to a quantum quench from its critical superfluid phase to an insulating
phase. We show that the evolution of the current when the quench follows a
Rosen-Zener profile is exactly solvable. This allows us to analyze a quench of
arbitrary rate, from a sudden destruction of the superfluid to a slow opening
of a gap. The decay and oscillations of the current are analytically derived,
and studied numerically along with the momentum distribution after the quench.
In the case of small supercurrent boosts $\nu$, we find that the current
surviving at long times is proportional to $\nu^3$

Klich, I.

Lannert, C.

Refael, G.

English

arXiv

We study the survival of supercurrents in a system of impenetrable bosons on a lattice, subject to a quantum quench from its critical superfluid phase to an insulating phase. We show that the evolution of the current when the quench follows a Rosen-Zener profile is exactly solvable. This allows us to analyze a quench of arbitrary rate, from a sudden destruction of the superfluid to a slow opening of a gap. The decay and oscillations of the current are analytically derived and studied numerically along with the momentum distribution after the quench. In the case of small supercurrent boosts nu, we find that the current surviving at long times is proportional to nu^3

Caltech Authors - Main

Supercurrent Survival under a Rosen-Zener Quench of Hard-Core Bosons
I. Klich,1 C. Lannert,2 and G. Refael1
1Department of Physics, California Institute of Technology, MC 114-36 Pasadena, California 91125, USA
2Department of Physics, Wellesley College, Wellesley, Massachusetts 02481, USA
(Received 29 June 2007; published 16 November 2007)
We study the survival of supercurrents in a system of impenetrable bosons on a lattice, subject to a
quantum quench from its critical superfluid phase to an insulating phase. We show that the evolution of the
current when the quench follows a Rosen-Zener profile is exactly solvable. This allows us to analyze a
quench of arbitrary rate, from a sudden destruction of the superfluid to a slow opening of a gap. The decay
and oscillations of the current are analytically derived and studied numerically along with the momentum
distribution after the quench. In the case of small supercurrent boosts , we find that the current surviving
at long times is proportional to 3.
DOI: 10.1103/PhysRevLett.99.205303 PACS numbers: 67.40.Fd, 03.75.Kk, 05.70.Ln
Progress in the field of experimental cold atom systems
allows a controlled and direct access to the nonequilibrium
physics of quantum many body systems. This becomes
particularly exciting when the system is in the vicinity of
a phase transition and is driven through it as in the case of
the superfluid(SF)-insulator transition [1], or the magnetic
ordering transition [2]. In such situations, complex physics
is often exhibited; a prominent example is defect formation
in the ordered phase due to a fast quench [3], which is
qualitatively understood using the Kibble-Zurek mecha-
nism [4] originally proposed in cosmology to describe
domain formation during cooling of the universe.
Especially interesting are low-dimensional quantum sys-
tems under a ‘‘quantum quench,’’ with a Hamiltonian
driven through a quantum-critical point. This can be
achieved in quantum gases confined in highly anisotropic
traps and in optical lattices [5,6]. For instance, it was
recently demonstrated that when a system is driven from
an ordered to a disordered phase, the order-parameter
correlations experience ’’revival’’[1,7–10]. In the special
case of driving an off-critical 1d system into criticality,
Calabrese and Cardy [11] showed that correlation func-
tions can be expressed using correlations of the final criti-
cal state. Despite these advances, as well as qualitative
understanding of ‘‘revival‘‘ phenomena and the Kibble-
Zurek mechanism, analytical and exact results in this field
are scarce.
Here, we focus on a system driven out of criticality by
varying an external field. In practice, this is a generic case,
but elegant general results as in Ref. [11] for the opposing
case are mostly absent. Special cases that have previously
been studied analytically are the behavior of a dipole
model of a Mott insulator in an external electric field
when the field is suddenly changed from a ‘‘no dipole’’
state to a maximally polarized state [12], and the dynamics
of traversing spin chains between two phases [7]. A related
work, Ref. [8], describes bosons driven abruptly from a
Mott to superfluid phase and showed collective oscillations
of the superfluid order parameter with period proportional
to the gap in the initial Mott state. In [13], an SF to Mott
quench was analyzed within a mean field approach. In [10],
the evolution of hard-core bosons (HCBs) undergoing a
quantum quench was studied numerically.
A natural question arising in the SF-insulator transition
regards the fate of supercurrents in the system. In this
Letter, we describe the evolution of supercurrents under
quenching of a 1-d lattice gas of HCBs, the so called
Tonks-Girardeu gas [14,15], from SF to insulator. Such
systems may be formed by bosons at low temperatures and
densities [16]. Recently, a system of HCBs in an optical
lattice has been realized in an ultracold dilute gas [6]. We
calculate the current as a function of time, while concen-
trating on the long-time current survival rate. Our analysis
reveals fast, Bloch-like, oscillations that are superimposed
on a decay and a slower envelope function. The surviving
current is found to be proportional to 3, where  is the
initial supercurrent. Note that the decay of supercurrents
was considered before in Ref. [9] close to the Mott-
superfluid transition. In addition, we present numerics for
the evolution of the momentum distribution.
Our study relies on a generalization of the Rosen-Zener
problem [17] to the context of HCBs. The Rosen-Zener
problem describes a spin evolving in a time-dependent
magnetic field with a particular profile, where an x-y
magnetic field (analogous to a gap) is turned on while a
z-field remains constant. In contrast, Landau-Zener dy-
namics describe a spin in a constant x-y field, with Bz
swept through zero; thus, it is useful for describing travers-
ing the system through a quantum critical point [7]. The
integrability of the Rosen-Zener evolution allows us to
probe sudden quench dynamics as well as the response to
a finite quenching time and so goes beyond previous treat-
ments, which have dealt with an abrupt quench.
The system we consider has the Hamiltonian
 H  w
X
byi bi1  H:c:  Vt
X
1ibyi bi (1)
with byi a boson creation operator at site i which obeys
byi 
2  0, thus imposing the impenetrability of the HCBs.
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S
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Throughout, we work in units such that the lattice spacing
is 1. Vt is the amplitude of an externally applied poten-
tial, which tunes the system from its superfluid phase at
V  0, to an insulating phase at V  0. When Vt has the
Rosen-Zener shape, this evolution can be solved exactly.
The Rosen-Zener profile is given by
 Vt 

V0
1
coshtT
; t < 0;
V0; t  0;
(2)
where T is the turning-on, or quenching time.
The HCB problem is equivalent to an infinite system
of spins precessing in a time-dependent magnetic field.
The evolution of a spin in the ẑ direction under Bx 
Vt given in Eq. (2) (Fig. 1) was solved exactly by
Rosen and Zener [17]. This exact nonequilibrium evolution
allows us to establish the state of the system at t  0, from
which the system is evolved by the final Hamiltonian. We
make use of this evolution to analyze the fate of a super-
current, introduced at t!1 [when the system is in the
gapless SF phase and V1  0]. At t! 1, it is
convenient to regard a current carrying state as the ground
state of the HCB Hamiltonian observed from a moving
frame, which corresponds to the boosted Hamiltonian
w
P
eibyi bi1  H:c:. Here,  is the ‘‘boost’’ wave
vector, which shifts modes with momentum k to k . 
is related to the current at t! 1 by jt  1 
2w= sin.
The first step in our analysis applies the Wigner-Jordan
transformation to the HCB operators [18], byi 
e
P
j<i
ayj ajayi , with aj being a fermion operator associated
with site j. In the ‘‘fermion’’ picture, the presence of the
supercurrent is expressed by choosing the ‘‘Fermi sea’’ of
the a fermions to be shifted in momentum. That is, the k
occupation of the fermions is
 nsk  kF   < k < kF  ; (3)
where kF is the Fermi momentum. Let us now assume half
filling, so kF  =2. The current density is given by
 j 
iw
L
X
l
hbyl1bl  H:c:i (4)
where L is the total number of sites.
To find the evolution of the current, we first address the
evolution of the fermion operators aj introduced above. By
rewriting the Hamiltonian as
 
H0  
X
k<jj
2w coskayk ak
Hd 
Vt
2
X
k<jj
ayk ak  a
y
kak;
(5)
we note that k couples to k  (or equivalently k 
since k is the same up to multiples of 2). Thus, we can
write the Hamiltonian as
 
Ht  jkj<=2Hkt
with Hkt  2w coskz  Vtx
(6)
acting in the fayk ; a
y
kg mode space, and explicitly break-
ing the problem into a product of noninteracting spin
systems. Thus, solving the evolution of the HCB system
is equivalent to solving for the time evolution of an infinite
series of time-dependent two-level systems.
For a given k, the Schrödinger equation generated by
Eq. (6) for a fermion operator  yt  stayk  pta
y
k
can be rewritten as the second order differential equation:
 
S  
V20
coshtT 
2 S

4iw cosk 

T
tanh

t
T

_S (7)
for St  e2iw cosktst, and the same equation for Pt 
e2iw cosktpt, with w! w. This equation, via a change
of variable z  12 tanh
t
T   1	, can be recast into a hyper-
geometric differential equation. The solution, satisfying
the initial condition jS1j  1, jP1  0j (recall
jkj< 2 ), is given in terms of the hyper-geometric function:
 
Sz  2F1;; c; z
Pz  i

z1 z
p 22e4iw coskt
cT

 2F11 ; 1 ; 1 c; z (8)
where [19]   V0T ; c 
1
2
2iwT cosk
 . We now use the
solution at t  0 [setting z! 12 in (8)] as a boundary
condition for the dynamics under the final Hamiltonian.
The evolution of the aj’s at times t > 0 is obtained by
diagonalizing Eq. (6) at fixed V  V0. This results in
 
ayk t  Aka
y
k 0  Bka
y
k0
aykt  Aka
y
k0  Bka
y
k 0
(9)
where A;B are given explicitly by the relations
 Ak  cosEkt  i cos sinEk Bk  i sin sinEkt
(10)
with
  < k< ; Ek 

4w2cos2k V20
q
(11)
and  is defined through tan   V02w cosk .
Using Eq. (8) with z! 1=2 to find ayk 0, a
y
k0 and
substituting in Eq. (4), we find
t
V0
FIG. 1. The Rosen-Zener superlattice switch-on profile, Vt.
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S
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hjti  4w
Z =2
=2
dk
2
sink

js0Ak  p0B

kj
2 
1
2


 nk  nk (12)
where nk is the fermion momentum (Fermi-Dirac) occu-
pation: hayk ak0 i  kk0nk. The above Eqs. (8)–(12) consti-
tute a complete description of current evolution under a
Rosen-Zener quench.
We first concentrate on an initial state with a super-
current at half filling and with a short switching time,
wT 1, which allows nonadiabatic transitions. In this
case, to leading order near k =2, we may take c 1=2,
c  1=2 


p
, giving
 p0  i sin

V0T
2

s0  cos

V0T
2

: (13)
Assuming small supercurrents ( 1), the current evolu-
tion, Eq. (12), simplifies to
 hji 
2w

M2t TV0; 

t
p
	 (14)
where
 M x; y 
cosxCy  sinxSy
y
(15)
and S, C are the Fresnel functions Sz 
Rz
0 du sin

2 u
2,
Cz 
Rz
0 du cos

2 u
2, and  

8w2
V0
q
.
Figure 2 shows the current evolution for an abrupt
quench. The evolution is characterized by fast oscillations
superimposed on a slow envelope which oscillates and
decays. The time between successive maxima of the slow
envelope can be estimated by looking at the extrema of the
Fresnel integrals in Eq. (15). From @zCz  cosz2=2,
we see that the extrema are at z 

2n
p
, with n integer.
Thus, the condition

2n
p


t
p
 yields the envelope pe-
riod e:
 e 
4
22
: (16)
The fast current oscillations are a consequence of Bloch-
like oscillations in each of the 2-state systems (k-state
fermions and their k  backscattered partners) described
by Eq. (6). These arise from the cos2t TV0 terms in
Eq. (14), and thus have period B 

V0
. The Bloch oscil-
lations decohere over time due to the spread in frequencies
(11) of the 2-level systems. Indeed, the current in Eq. (14)
eventually decays, and no current is left in the system, to
lowest order in . This decay should not be confused with
decoherence due to coupling with an external bath. Rather,
it is due to interference between the different k modes of
the system, which oscillate with different frequencies.
Thus, for a system of finite size, where the k frequencies
are quantized, after long enough time, the current will be
revived. However, this time grows rapidly with system size
and is much longer than the time shown in our numerics.
Nevertheless, even in the infinite system limit, some
current survives to long times. This appears in our results
at higher orders in . By computing the current averaged
over long times, h ji  limT!1
1
T
R
T
0 hjidt, we show that
the surviving current is proportional to 3 for small super-
currents. From Eq. (12), we find that the averaged current
is given exactly by
 h ji 
2w sin cosV0T


1
V0 arctan
2w sin
V0

2w sin

: (17)
The leading term in  is of order 3:
 h ji 
8w23
3V20
cosV0T; (18)
and thus absent from the treatment leading to Eq. (14).
To complement the current evolution analysis, we next
study the dynamics of the momentum distribution of the
HCBs. Although the analysis of the current evolution is the
same for free fermions and a Tonks-Girardeu gas, the
momentum distribution of the two systems is quite differ-
ent. The presence of a supercurrent is described by a shift
of the Fermi step function of the free Jordan-Wigner
fermions, as in Eq. (3); in terms of the bosons, the super-
current leads to a peak in the boson momentum occupation
nk 
P
le
iklhbyl tb0ti at the boost value, i.e., nk / jk
j1=2 [15]. Unfortunately, an analytic description of the
evolution of nk is a much harder task, requiring analysis of
the determinants arising in the boson correlation functions.
At equilibrium, the analysis is simpler: translational invari-
ance of the system allows application of mathematical
machinery such as Szego limit theorems. Our case is
much less accessible, due to the incoherent mixing be-
tween the different k modes.
Using exact-diagonalization techniques (see, e.g., [10])
to monitor the time evolution of the boson momentum
distribution and the current, we investigated lattice sizes
of up to 350 sites. The momentum distribution indeed
reflects the supercurrent Bloch oscillations, as can be
seen in the bosonic nk plotted in Fig. 3. The period of these
oscillations of the peak in nk between  and   agrees
very well with the analytical result V0 . We also confirmed
numerically the main results of this manuscript, i.e., the
current survival after a quench, Eqs. (17) and (18). Our
FIG. 2. Current evolution for an abrupt quench, T  0. We set
superlattice strength V0  0:7, hopping w  1, and the boost
  15 [i.e., initial supercurrent of j0  2=15].
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S
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results for the long-time averaged current (in a chain of 350
sites) vs supercurrent boost  and superlattice strength
V0=w for an abrupt quench are summarized in Fig. 4.
Perfect agreement is obtained between the analytical and
numerical results.
In this Letter, we studied the behavior of hard-core
bosons under a Rosen-Zener quench in the presence of a
supercurrent. We described the evolution of the supercur-
rent, and its long-time survival fraction, as well as the
corresponding momentum distribution evolution. Perhaps
the most readily accessible result is the ‘‘3 law’’: starting
with supercurrent , the surviving current in the insulating
phase is / 3 for  1. By using the Wigner-Jordan
transformation, we essentially mapped the HCB gas to a
Fermi system with backscattering at the Luther-Emery line
[21]. In light of our results, it is particularly interesting to
ask what happens when we consider a Luttinger liquid
(describing either fermions or bosons): is the 3-law uni-
versal, or does it depend on the Luttinger parameter, g 
2? This question, as well as a test of our predictions and the
level of their universality, can be taken on experimentally.
This could be done, for instance, by probing cold-atoms in
a ring-shaped trap, with a superimposed optical lattice.
Such a geometry was recently discussed in Ref. [22]. If
the additional optical lattice is made to rotate as it turns on,
a supercurrent will exist in the rotating frame. The insulat-
ing phase can then be accessed by introducing a corotating
superlattice. Such an experimental setup will also be able
to probe nonequilibrium quenches well beyond the regimes
which we considered here analytically.
We would like to thank R. Santachiara and S. Powell for
discussions.
[1] A. K. Tuchman, C. Orzel, A. Polkovnikov, and M. A.
Kasevich, Phys. Rev. A 74, 051601 (2006).
[2] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore,
and D. M. Stamper-Kurn, Nature (London) 443, 312
(2006).
[3] A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett. 91,
197001 (2003).
[4] T. W. B. Kibble, J. Phys. A 9, 1387 (1976); W. H. Zurek,
Nature (London) 317, 505 (1985).
[5] M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch and
T. Esslinger, Phys. Rev. Lett. 87, 160405 (2001);
T. Stöferle, H. Moritz, C. Schori, M. Khl, and T. Ess-
linger, Phys. Rev. Lett. 92, 130403 (2004).
[6] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Flling,
I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch,
Nature (London) 429, 277 (2004); T. Kinoshita,
T. Wenger, and D. S. Weiss, Science 305, 1125 (2004).
[7] R. W. Cherng and L. S. Levitov, Phys. Rev. A 73, 043614
(2006).
[8] E. Altman and A. Auerbach, Phys. Rev. Lett. 89, 250404
(2002); A. Polkovnikov, S. Sachdev, and S. M. Girvin,
Phys. Rev. A 66, 053607 (2002).
[9] A. Polkovnikov, E. Altman, E. Demler, B. Halperin, and
M. D. Lukin, Phys. Rev. A 71, 063613 (2005).
[10] M. Rigol and A. Muramatsu, Phys. Rev. A 70, 031603(R)
(2004); 72, 013604 (2005).
[11] P. Calabrese and J. Cardy, Phys. Rev. Lett. 96, 136801
(2006).
[12] K. Sengupta, S. Powell, and S. Sachdev, Phys. Rev. A 69,
053616 (2004).
[13] R. Schützhold, M. Uhlmann, Y. Zu, and U. R. Fischer,
Phys. Rev. Lett. 97, 200601 (2006).
[14] L. Tonks, Phys. Rev. 50, 955 (1936); M. Girardeau,
J. Math. Phys. (N.Y.) 1, 516 (1960).
[15] A. Lenard, J. Math. Phys. (N.Y.) 7, 1268 (1966).
[16] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
[17] N. Rozen and C. Zener, Phys. Rev. 40, 502 (1932).
[18] E. Lieb, T. Shultz, and D. Mattis, Ann. Phys. (N.Y.) 16,
407 (1961).
[19] One may also include dissipation, which amounts to
taking c! c 	T , where 	 is related to the decay of
the higher energy level, which we shall ignore here [20].
[20] R. T. Robiscoe, Phys. Rev. A 17, 247 (1978).
[21] A. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589 (1974).
[22] D. W. Hallwood, K. Burnett, J. A. Dunningham,
arXiv:quant-ph/0609077; A. M. Rey, K. Burnett, I. I.
Satija, and C. W. Clark, Phys. Rev. A 75, 063616 (2007).
FIG. 3 (color online). nk as a function of k and time for a boost
of   =5, superlattice strength V0  4, and hopping w  1.
The initial peak is at k  , and the peak transports back and
forth to 4=5 before disappearing due to decoherence. Results
obtained using an exact diagonalization study of chains 150 sites
long.
j
3π/4π/40
ν
π
0.4
0.5
0.1
0.2
0.3
0.005
0.01
0.015
0.02
(a) (b)
π/8π/160
ν
π/2
FIG. 4 (color online). Supercurrent surviving through a sudden
quench. The symbols are the result of exact diagonalization of a
chain 350 sites long. The hopping, w, is set to unity. (a) Current
survival over the full range of boosts. From top to bottom,
V0=w  0:1, 0.25, 0.5, 0.7, and the solid lines are Eq. (17).
(b) The 3 behavior of current survival at small currents. From
top to bottom, V0=w  2, 4, 8. The solid lines are Eq. (18).
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S
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Supercurrent Survival under a Rosen-Zener Quench of Hard-Core Bosons

We study the survival of supercurrents in a system of impenetrable bosons on a lattice, subject to a quantum quench from its critical superfluid phase to an insulating phase. We show that the evolution of the current when the quench follows a Rosen-Zener profile is exactly solvable. This allows us to analyze a quench of arbitrary rate, from a sudden destruction of the superfluid to a slow opening of a gap. The decay and oscillations of the current are analytically derived and studied numerically along with the momentum distribution after the quench. In the case of small supercurrent boosts ν, we find that the current surviving at long times is proportional to ν3

Lannert, Courtney

Smith College: Smith ScholarWorks

Supercurrent Survival under a Rosen-Zener Quench of Hard-Core Bosons
I. Klich,1 C. Lannert,2 and G. Refael1
1Department of Physics, California Institute of Technology, MC 114-36 Pasadena, California 91125, USA
2Department of Physics, Wellesley College, Wellesley, Massachusetts 02481, USA
(Received 29 June 2007; published 16 November 2007)
We study the survival of supercurrents in a system of impenetrable bosons on a lattice, subject to a
quantum quench from its critical superfluid phase to an insulating phase. We show that the evolution of the
current when the quench follows a Rosen-Zener profile is exactly solvable. This allows us to analyze a
quench of arbitrary rate, from a sudden destruction of the superfluid to a slow opening of a gap. The decay
and oscillations of the current are analytically derived and studied numerically along with the momentum
distribution after the quench. In the case of small supercurrent boosts , we find that the current surviving
at long times is proportional to 3.
DOI: 10.1103/PhysRevLett.99.205303 PACS numbers: 67.40.Fd, 03.75.Kk, 05.70.Ln
Progress in the field of experimental cold atom systems
allows a controlled and direct access to the nonequilibrium
physics of quantum many body systems. This becomes
particularly exciting when the system is in the vicinity of
a phase transition and is driven through it as in the case of
the superfluid(SF)-insulator transition [1], or the magnetic
ordering transition [2]. In such situations, complex physics
is often exhibited; a prominent example is defect formation
in the ordered phase due to a fast quench [3], which is
qualitatively understood using the Kibble-Zurek mecha-
nism [4] originally proposed in cosmology to describe
domain formation during cooling of the universe.
Especially interesting are low-dimensional quantum sys-
tems under a ‘‘quantum quench,’’ with a Hamiltonian
driven through a quantum-critical point. This can be
achieved in quantum gases confined in highly anisotropic
traps and in optical lattices [5,6]. For instance, it was
recently demonstrated that when a system is driven from
an ordered to a disordered phase, the order-parameter
correlations experience ’’revival’’[1,7–10]. In the special
case of driving an off-critical 1d system into criticality,
Calabrese and Cardy [11] showed that correlation func-
tions can be expressed using correlations of the final criti-
cal state. Despite these advances, as well as qualitative
understanding of ‘‘revival‘‘ phenomena and the Kibble-
Zurek mechanism, analytical and exact results in this field
are scarce.
Here, we focus on a system driven out of criticality by
varying an external field. In practice, this is a generic case,
but elegant general results as in Ref. [11] for the opposing
case are mostly absent. Special cases that have previously
been studied analytically are the behavior of a dipole
model of a Mott insulator in an external electric field
when the field is suddenly changed from a ‘‘no dipole’’
state to a maximally polarized state [12], and the dynamics
of traversing spin chains between two phases [7]. A related
work, Ref. [8], describes bosons driven abruptly from a
Mott to superfluid phase and showed collective oscillations
of the superfluid order parameter with period proportional
to the gap in the initial Mott state. In [13], an SF to Mott
quench was analyzed within a mean field approach. In [10],
the evolution of hard-core bosons (HCBs) undergoing a
quantum quench was studied numerically.
A natural question arising in the SF-insulator transition
regards the fate of supercurrents in the system. In this
Letter, we describe the evolution of supercurrents under
quenching of a 1-d lattice gas of HCBs, the so called
Tonks-Girardeu gas [14,15], from SF to insulator. Such
systems may be formed by bosons at low temperatures and
densities [16]. Recently, a system of HCBs in an optical
lattice has been realized in an ultracold dilute gas [6]. We
calculate the current as a function of time, while concen-
trating on the long-time current survival rate. Our analysis
reveals fast, Bloch-like, oscillations that are superimposed
on a decay and a slower envelope function. The surviving
current is found to be proportional to 3, where  is the
initial supercurrent. Note that the decay of supercurrents
was considered before in Ref. [9] close to the Mott-
superfluid transition. In addition, we present numerics for
the evolution of the momentum distribution.
Our study relies on a generalization of the Rosen-Zener
problem [17] to the context of HCBs. The Rosen-Zener
problem describes a spin evolving in a time-dependent
magnetic field with a particular profile, where an x-y
magnetic field (analogous to a gap) is turned on while a
z-field remains constant. In contrast, Landau-Zener dy-
namics describe a spin in a constant x-y field, with Bz
swept through zero; thus, it is useful for describing travers-
ing the system through a quantum critical point [7]. The
integrability of the Rosen-Zener evolution allows us to
probe sudden quench dynamics as well as the response to
a finite quenching time and so goes beyond previous treat-
ments, which have dealt with an abrupt quench.
The system we consider has the Hamiltonian
 H  wXbyi bi1  H:c:  VtX1ibyi bi (1)
with byi a boson creation operator at site i which obeys
byi 2  0, thus imposing the impenetrability of the HCBs.
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending16 NOVEMBER 2007
0031-9007=07=99(20)=205303(4) 205303-1 © 2007 The American Physical Society
Throughout, we work in units such that the lattice spacing
is 1. Vt is the amplitude of an externally applied poten-
tial, which tunes the system from its superfluid phase at
V  0, to an insulating phase at V  0. When Vt has the
Rosen-Zener shape, this evolution can be solved exactly.
The Rosen-Zener profile is given by
 Vt 

V0
1
coshtT
; t < 0;
V0; t  0;
(2)
where T is the turning-on, or quenching time.
The HCB problem is equivalent to an infinite system
of spins precessing in a time-dependent magnetic field.
The evolution of a spin in the z^ direction under Bx 
Vt given in Eq. (2) (Fig. 1) was solved exactly by
Rosen and Zener [17]. This exact nonequilibrium evolution
allows us to establish the state of the system at t  0, from
which the system is evolved by the final Hamiltonian. We
make use of this evolution to analyze the fate of a super-
current, introduced at t! 1 [when the system is in the
gapless SF phase and V1  0]. At t! 1, it is
convenient to regard a current carrying state as the ground
state of the HCB Hamiltonian observed from a moving
frame, which corresponds to the boosted Hamiltonian
wPeibyi bi1  H:c:. Here,  is the ‘‘boost’’ wave
vector, which shifts modes with momentum k to k . 
is related to the current at t! 1 by jt  1 
2w= sin.
The first step in our analysis applies the Wigner-Jordan
transformation to the HCB operators [18], byi 
e
P
j<i
ayj ajayi , with aj being a fermion operator associated
with site j. In the ‘‘fermion’’ picture, the presence of the
supercurrent is expressed by choosing the ‘‘Fermi sea’’ of
the a fermions to be shifted in momentum. That is, the k
occupation of the fermions is
 nsk  kF   < k < kF  ; (3)
where kF is the Fermi momentum. Let us now assume half
filling, so kF  =2. The current density is given by
 j  iw
L
X
l
hbyl1bl  H:c:i (4)
where L is the total number of sites.
To find the evolution of the current, we first address the
evolution of the fermion operators aj introduced above. By
rewriting the Hamiltonian as
 
H0  
X
k<jj
2w coskayk ak
Hd  Vt2
X
k<jj
ayk ak  aykak;
(5)
we note that k couples to k  (or equivalently k 
since k is the same up to multiples of 2). Thus, we can
write the Hamiltonian as
 
Ht  jkj<=2Hkt
with Hkt  2w coskz  Vtx
(6)
acting in the fayk ; aykg mode space, and explicitly break-
ing the problem into a product of noninteracting spin
systems. Thus, solving the evolution of the HCB system
is equivalent to solving for the time evolution of an infinite
series of time-dependent two-level systems.
For a given k, the Schro¨dinger equation generated by
Eq. (6) for a fermion operator  yt  stayk  ptayk
can be rewritten as the second order differential equation:
 
S   V
2
0
coshtT 2
S

4iw cosk  
T
tanh

t
T

_S (7)
for St  e2iw cosktst, and the same equation for Pt 
e2iw cosktpt, with w! w. This equation, via a change
of variable z  12 tanhtT   1	, can be recast into a hyper-
geometric differential equation. The solution, satisfying
the initial condition jS1j  1, jP1  0j (recall
jkj< 2 ), is given in terms of the hyper-geometric function:
 
Sz  2F1;; c; z
Pz  i

z1 z
p 22e4iw coskt
cT

 2F11 ; 1 ; 1 c; z (8)
where [19]   V0T ; c  12  2iwT cosk . We now use the
solution at t  0 [setting z! 12 in (8)] as a boundary
condition for the dynamics under the final Hamiltonian.
The evolution of the aj’s at times t > 0 is obtained by
diagonalizing Eq. (6) at fixed V  V0. This results in
 
ayk t  Akayk 0  Bkayk0
aykt  Akayk0  Bkayk 0
(9)
where A;B are given explicitly by the relations
 Ak  cosEkt  i cos sinEk Bk  i sin sinEkt
(10)
with
  < k< ; Ek 

4w2cos2k V20
q
(11)
and  is defined through tan   V02w cosk .
Using Eq. (8) with z! 1=2 to find ayk 0, ayk0 and
substituting in Eq. (4), we find
t
V0
FIG. 1. The Rosen-Zener superlattice switch-on profile, Vt.
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending16 NOVEMBER 2007
205303-2
 hjti  4w
Z =2
=2
dk
2
sink

js0Ak  p0Bkj2 
1
2


 nk  nk (12)
where nk is the fermion momentum (Fermi-Dirac) occu-
pation: hayk ak0 i  kk0nk. The above Eqs. (8)–(12) consti-
tute a complete description of current evolution under a
Rosen-Zener quench.
We first concentrate on an initial state with a super-
current at half filling and with a short switching time,
wT 1, which allows nonadiabatic transitions. In this
case, to leading order near k =2, we may take c 1=2,
c  1=2  p , giving
 p0  i sin

V0T
2

s0  cos

V0T
2

: (13)
Assuming small supercurrents ( 1), the current evolu-
tion, Eq. (12), simplifies to
 hji  2w

M2t TV0; 

t
p 	 (14)
where
 M x; y  cosxCy  sinxSy
y
(15)
and S, C are the Fresnel functions Sz  Rz0 du sin2 u2,
Cz  Rz0 du cos2 u2, and  

8w2
V0
q
.
Figure 2 shows the current evolution for an abrupt
quench. The evolution is characterized by fast oscillations
superimposed on a slow envelope which oscillates and
decays. The time between successive maxima of the slow
envelope can be estimated by looking at the extrema of the
Fresnel integrals in Eq. (15). From @zCz  cosz2=2,
we see that the extrema are at z  2np , with n integer.
Thus, the condition

2n
p  tp  yields the envelope pe-
riod e:
 e  422 : (16)
The fast current oscillations are a consequence of Bloch-
like oscillations in each of the 2-state systems (k-state
fermions and their k  backscattered partners) described
by Eq. (6). These arise from the cos2t TV0 terms in
Eq. (14), and thus have period B  V0 . The Bloch oscil-
lations decohere over time due to the spread in frequencies
(11) of the 2-level systems. Indeed, the current in Eq. (14)
eventually decays, and no current is left in the system, to
lowest order in . This decay should not be confused with
decoherence due to coupling with an external bath. Rather,
it is due to interference between the different k modes of
the system, which oscillate with different frequencies.
Thus, for a system of finite size, where the k frequencies
are quantized, after long enough time, the current will be
revived. However, this time grows rapidly with system size
and is much longer than the time shown in our numerics.
Nevertheless, even in the infinite system limit, some
current survives to long times. This appears in our results
at higher orders in . By computing the current averaged
over long times, h ji  limT!1 1T
RT
0 hjidt, we show that
the surviving current is proportional to 3 for small super-
currents. From Eq. (12), we find that the averaged current
is given exactly by
 h ji  2w sin cosV0T


1 V0 arctan
2w sin
V0

2w sin

: (17)
The leading term in  is of order 3:
 h ji  8w
23
3V20
cosV0T; (18)
and thus absent from the treatment leading to Eq. (14).
To complement the current evolution analysis, we next
study the dynamics of the momentum distribution of the
HCBs. Although the analysis of the current evolution is the
same for free fermions and a Tonks-Girardeu gas, the
momentum distribution of the two systems is quite differ-
ent. The presence of a supercurrent is described by a shift
of the Fermi step function of the free Jordan-Wigner
fermions, as in Eq. (3); in terms of the bosons, the super-
current leads to a peak in the boson momentum occupation
nk 
P
le
iklhbyl tb0ti at the boost value, i.e., nk / jk
j1=2 [15]. Unfortunately, an analytic description of the
evolution of nk is a much harder task, requiring analysis of
the determinants arising in the boson correlation functions.
At equilibrium, the analysis is simpler: translational invari-
ance of the system allows application of mathematical
machinery such as Szego limit theorems. Our case is
much less accessible, due to the incoherent mixing be-
tween the different k modes.
Using exact-diagonalization techniques (see, e.g., [10])
to monitor the time evolution of the boson momentum
distribution and the current, we investigated lattice sizes
of up to 350 sites. The momentum distribution indeed
reflects the supercurrent Bloch oscillations, as can be
seen in the bosonic nk plotted in Fig. 3. The period of these
oscillations of the peak in nk between  and   agrees
very well with the analytical result V0 . We also confirmed
numerically the main results of this manuscript, i.e., the
current survival after a quench, Eqs. (17) and (18). Our
FIG. 2. Current evolution for an abrupt quench, T  0. We set
superlattice strength V0  0:7, hopping w  1, and the boost
  15 [i.e., initial supercurrent of j0  2=15].
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending16 NOVEMBER 2007
205303-3
results for the long-time averaged current (in a chain of 350
sites) vs supercurrent boost  and superlattice strength
V0=w for an abrupt quench are summarized in Fig. 4.
Perfect agreement is obtained between the analytical and
numerical results.
In this Letter, we studied the behavior of hard-core
bosons under a Rosen-Zener quench in the presence of a
supercurrent. We described the evolution of the supercur-
rent, and its long-time survival fraction, as well as the
corresponding momentum distribution evolution. Perhaps
the most readily accessible result is the ‘‘3 law’’: starting
with supercurrent , the surviving current in the insulating
phase is / 3 for  1. By using the Wigner-Jordan
transformation, we essentially mapped the HCB gas to a
Fermi system with backscattering at the Luther-Emery line
[21]. In light of our results, it is particularly interesting to
ask what happens when we consider a Luttinger liquid
(describing either fermions or bosons): is the 3-law uni-
versal, or does it depend on the Luttinger parameter, g 
2? This question, as well as a test of our predictions and the
level of their universality, can be taken on experimentally.
This could be done, for instance, by probing cold-atoms in
a ring-shaped trap, with a superimposed optical lattice.
Such a geometry was recently discussed in Ref. [22]. If
the additional optical lattice is made to rotate as it turns on,
a supercurrent will exist in the rotating frame. The insulat-
ing phase can then be accessed by introducing a corotating
superlattice. Such an experimental setup will also be able
to probe nonequilibrium quenches well beyond the regimes
which we considered here analytically.
We would like to thank R. Santachiara and S. Powell for
discussions.
[1] A. K. Tuchman, C. Orzel, A. Polkovnikov, and M. A.
Kasevich, Phys. Rev. A 74, 051601 (2006).
[2] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore,
and D. M. Stamper-Kurn, Nature (London) 443, 312
(2006).
[3] A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett. 91,
197001 (2003).
[4] T. W. B. Kibble, J. Phys. A 9, 1387 (1976); W. H. Zurek,
Nature (London) 317, 505 (1985).
[5] M. Greiner, I. Bloch, O. Mandel, T. W. Ha¨nsch and
T. Esslinger, Phys. Rev. Lett. 87, 160405 (2001);
T. Sto¨ferle, H. Moritz, C. Schori, M. Khl, and T. Ess-
linger, Phys. Rev. Lett. 92, 130403 (2004).
[6] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Flling,
I. Cirac, G. V. Shlyapnikov, T. W. Ha¨nsch, and I. Bloch,
Nature (London) 429, 277 (2004); T. Kinoshita,
T. Wenger, and D. S. Weiss, Science 305, 1125 (2004).
[7] R. W. Cherng and L. S. Levitov, Phys. Rev. A 73, 043614
(2006).
[8] E. Altman and A. Auerbach, Phys. Rev. Lett. 89, 250404
(2002); A. Polkovnikov, S. Sachdev, and S. M. Girvin,
Phys. Rev. A 66, 053607 (2002).
[9] A. Polkovnikov, E. Altman, E. Demler, B. Halperin, and
M. D. Lukin, Phys. Rev. A 71, 063613 (2005).
[10] M. Rigol and A. Muramatsu, Phys. Rev. A 70, 031603(R)
(2004); 72, 013604 (2005).
[11] P. Calabrese and J. Cardy, Phys. Rev. Lett. 96, 136801
(2006).
[12] K. Sengupta, S. Powell, and S. Sachdev, Phys. Rev. A 69,
053616 (2004).
[13] R. Schu¨tzhold, M. Uhlmann, Y. Zu, and U. R. Fischer,
Phys. Rev. Lett. 97, 200601 (2006).
[14] L. Tonks, Phys. Rev. 50, 955 (1936); M. Girardeau,
J. Math. Phys. (N.Y.) 1, 516 (1960).
[15] A. Lenard, J. Math. Phys. (N.Y.) 7, 1268 (1966).
[16] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
[17] N. Rozen and C. Zener, Phys. Rev. 40, 502 (1932).
[18] E. Lieb, T. Shultz, and D. Mattis, Ann. Phys. (N.Y.) 16,
407 (1961).
[19] One may also include dissipation, which amounts to
taking c! c 	T , where 	 is related to the decay of
the higher energy level, which we shall ignore here [20].
[20] R. T. Robiscoe, Phys. Rev. A 17, 247 (1978).
[21] A. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589 (1974).
[22] D. W. Hallwood, K. Burnett, J. A. Dunningham,
arXiv:quant-ph/0609077; A. M. Rey, K. Burnett, I. I.
Satija, and C. W. Clark, Phys. Rev. A 75, 063616 (2007).
FIG. 3 (color online). nk as a function of k and time for a boost
of   =5, superlattice strength V0  4, and hopping w  1.
The initial peak is at k  , and the peak transports back and
forth to 4=5 before disappearing due to decoherence. Results
obtained using an exact diagonalization study of chains 150 sites
long.
j
3π/4π/40
ν
π
0.4
0.5
0.1
0.2
0.3
0.005
0.01
0.015
0.02
(a) (b)
π/8π/160
ν
π/2
FIG. 4 (color online). Supercurrent surviving through a sudden
quench. The symbols are the result of exact diagonalization of a
chain 350 sites long. The hopping, w, is set to unity. (a) Current
survival over the full range of boosts. From top to bottom,
V0=w  0:1, 0.25, 0.5, 0.7, and the solid lines are Eq. (17).
(b) The 3 behavior of current survival at small currents. From
top to bottom, V0=w  2, 4, 8. The solid lines are Eq. (18).
PRL 99, 205303 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending16 NOVEMBER 2007
205303-4


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Supercurrent survival under Rosen-Zener quench of hard core bosons

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