Download Quantum phase transitions in atomic gases and

Quantum phase transitions of
ultracold atoms
Subir Sachdev
Quantum Phase Transitions
Cambridge University Press (1999)
Transparencies online at
http://pantheon.yale.edu/~subir
What is a quantum phase transition ?
Non-analyticity in ground state properties as a function of
some control parameter g
E
E
g
True level crossing:
Usually a first-order transition
g
Avoided level crossing which
becomes sharp in the infinite
volume limit:
second-order transition
Why study quantum phase transitions ?
T
Quantum-critical
gc
• Theory for a quantum system with strong correlations:
describe phases on either side of gc by expanding in
deviation from the quantum critical point.
g
• Critical point is a novel state of matter without
quasiparticle excitations
• Critical excitations control dynamics in the wide
quantum-critical region at non-zero temperatures.
Important property of ground state at g=gc :
temporal and spatial scale invariance;
characteristic energy scale at other values of g:  ~ g  g c
z
Outline
I. The
The superfluid—Mott-insulator
superfluid—Mott-insulator transition
transition
II. Mott insulator in a strong electric field.
S. Sachdev, K. Sengupta, and S. M. Girvin,
Physical Review B 66, 075128 (2002).
III. Conclusions
I. The Superfluid-Insulator transition
Boson Hubbard model
Degrees of freedom: Bosons, b†j , hopping between the
sites, j , of a lattice, with short-range repulsive interactions.
U
†
H  t  bi b j -  n j   n j ( n j  1) 
2 j
ij
j
nj  b b
†
j j
M.PA. Fisher, P.B. Weichmann, G.
Grinstein, and D.S. Fisher Phys.
Rev. B 40, 546 (1989).
For small U/t, ground state is a superfluid BEC with
superfluid density

density of bosons
What is the ground state for large U/t ?
Typically, the ground state remains a superfluid, but with
superfluid density
density of bosons
The superfluid density evolves smoothly from large values at
small U/t, to small values at large U/t, and there is no quantum
phase transition at any intermediate value of U/t.
(In systems with Galilean invariance and at zero temperature,
superfluid density=density of bosons always, independent of the
strength of the interactions)
What is the ground state for large U/t ?
Incompressible, insulating ground states, with zero
superfluid density, appear at special commensurate densities
nj  3
nj  7 / 2
t

U
Ground state has “density wave” order, which
spontaneously breaks lattice symmetries
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Boson Green's function G ( p, ) :
Insulating ground state
Cross-section to add a boson
while transferring energy  and momentum p
G  p, 
Z      p  
Quasiparticle pole
~3
Continuum of
two quasiparticles +
one quasihole

Similar result for quasi-hole excitations obtained by removing a boson
Entangled states at g  t / U of order unity
Z ~  gc  g 

Quasiparticle
weight Z
A.V. Chubukov, S. Sachdev, and J.Ye,
Phys. Rev. B 49, 11919 (1994)
gc
g
 p ,h ~  gc  g  for g  gc

Excitation
energy gap 
 p ,h =0 for g  gc
g
rs ~  g  gc 
Superfluid
density rs
ggcc
g
( d  z  2)
Crossovers at nonzero temperature
Quasiclassical
dynamics
Quasiclassical
dynamics
Relaxational dynamics ("Bose molasses") with
phase coherence/relaxation time   given by
1

  Universal number 
k BT
Q2   
Conductivity (in d=2) =


h  kBT 
1 K  20.9kHz 
S. Sachdev and J. Ye,
Phys. Rev. Lett. 69, 2411 (1992).
K. Damle and S. Sachdev
Phys. Rev. B 56, 8714 (1997).
  universal function
M.P.A. Fisher, G. Girvin, and G. Grinstein, Phys. Rev. Lett. 64, 587 (1990).
K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997).
Outline
I.
The superfluid—Mott-insulator transition
II. Mott
insulatorininaastrong
strongelectric
electric
field
II.
Mott insulator
field.
S. Sachdev, K. Sengupta, and S. M. Girvin,
Physical Review B 66, 075128 (2002).
III. Conclusions
Superfluid-insulator transition of 87Rb atoms in a magnetic trap
and an optical lattice potential
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch,
Nature 415, 39 (2002).
Related earlier work by C. Orzel, A.K. Tuchman, M. L. Fenselau,
M. Yasuda, and M. A. Kasevich, Science 291, 2386 (2001).
Detection method
Trap is released and atoms expand to a distance far larger than
original trap dimension
 mR02
 mR2 
mR0  r 
  R, T   exp  i
i
  0, 0 
  0, 0   exp  i
T 
 2 T
 2 T
where R = R0 + r, with R0 = the expansion distance, and r  position within trap
In tight-binding model of lattice bosons bi ,
 
detection probability   b b exp iq  ri  rj
†
i j
i, j

mR0
with q 
T
Measurement of momentum distribution function
Superfluid state
Schematic three-dimensional interference pattern with measured absorption images taken
along two orthogonal directions. The absorption images were obtained after ballistic
expansion from a lattice with a potential depth of V0 = 10 Er and a time of flight of 15 ms.
Superfluid-insulator transition
V0=0Er
V0=13Er
V0=3Er
V0=7Er
V0=10Er
V0=14Er
V0=16Er
V0=20Er
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Applying an “electric” field to the Mott insulator
V0=10 Erecoil perturb = 2 ms
V0= 16 Erecoil perturb = 9 ms
V0= 13 Erecoil perturb = 4 ms
V0= 20 Erecoil perturb = 20 ms
What is the
quantum state
here ?
H  t 
ij


U
b b  b b   ni  ni  1   E  ri ni
2 i
i
†
i j
†
j i
ni  bi†bi
U  E ,t
E,U
Describe spectrum in subspace of states resonantly
coupled to the Mott insulator
Effective Hamiltonian for a quasiparticle in one dimension (similar for a quasihole):


H eff   3t b†j b j 1  b†j 1b j  Ejb†j b j 
j
Exact eigenvalues  m  Em ; m  

Exact eigenvectors  m  j   J j  m  6t / E 
All charged excitations are strongly localized in the plane perpendicular electric field.
Wavefunction is periodic in time, with period h/E (Bloch oscillations)
Quasiparticles and quasiholes are not accelerated out to infinity
Semiclassical picture
dk
E
dt
k
Freesituation:
particle
isStrong
accelerated
outpotential
to occurs
infinity
periodic
invia Zener
In aExperimental
weak periodic
potential,
escape
to infinity
which there istunneling
negligibleacross
Zenerband
tunneling,
gaps and the
particle undergoes Bloch oscillations
Important neutral excitations (in one dimension)
Nearest-neighbor
dipole
Creating dipoles
Nearest-neighbor
Nearest
on neighbor
nearest neighbor
dipole
dipoles links creates a
state with relative energy U-2E ; such states are not
part of the resonant manifold
Dipoles can appear resonantly on non-nearest-neighbor links.
Within resonant manifold, dipoles have infinite on-link
and nearest-link repulsion
A non-dipole state
State has energy 3(U-E) but is connected to resonant
state by a matrix element smaller than t2/U
State is not part of resonant manifold
Hamiltonian for resonant dipole states (in one dimension)
d †  Creates dipole on link


H d   6t  d †  d  (U  E ) d †d
Constraints: d †d  1 ; d †1d 1d †d  0
Determine phase diagram of Hd as a function of (U-E)/t
Note: there is no explicit dipole hopping term.
However, dipole hopping is generated by the
interplay of terms in Hd and the constraints.
Weak electric fields: (U-E)
t
Ground state is dipole vacuum (Mott insulator)
First excited levels: single dipole states
0
d† 0
d †dm† 0
t
Effective hopping between dipole states
d† 0
t
t
t
dm† 0
0
If both processes are permitted, they exactly cancel each other.
The top processes is blocked when , m are nearest neighbors
t2
 A nearest-neighbor dipole hopping term ~
is generated
U E
Strong electric fields: (E-U)
t
Ground state has maximal dipole number.
Two-fold degeneracy associated with Ising density wave order:
d1† d3† d5† d7† d9† d11†
0
or
d2† d4† d6† d8† d10† d12†
0
Ising quantum critical point at E-U=1.08 t
0.220
Equal-time structure
factor for Ising order
parameter
N=8
N=10
N=12
N=14
N=16
0.216
0.212
3/4
S /N
0.208
0.204
0.200
-1.90
-1.88
-1.86

-1.84
-1.82
-1.80
Hamiltonian for resonant states in higher dimensions
p†, n  Creates quasiparticle in column and transverse position n
h†, n  Creates quasihole in column and transverse position n
H ph   6t   p 1,n h ,n  p †1,n h†,n 
,n
Terms as in one dimension
(U  E )
†
†

p
p

h


,n
,n
,n h ,n 
2
,n
t

†
†
2
h
h

3
p
 ,n ,m
,n p ,m  H.c.
Transverse hopping
, nm
p †,n p ,n  1 ; h†,n h ,n  1 ;
p †,n p ,n h†,n h ,n  0
Constraints
New possibility: superfluidity in transverse direction (a smectic)
Resonant states in higher dimensions
Quasiparticles
Dipole states in one
dimension
Quasiholes
Quasiparticles and
quasiholes can move
resonantly in the transverse
directions in higher
dimensions.
Constraint: number of
quasiparticles in any
column = number of
quasiholes in column
to its left.
Possible phase diagrams in higher dimensions
 (U  E ) / t
Ising density wave order
Transverse superfluidity
 (U  E ) / t
Implications for experiments
•Observed resonant response is due to gapless spectrum near
quantum critical point(s).
•Transverse superfluidity (smectic order) can be detected by
looking for “Bragg lines” in momentum distribution function--bosons are phase coherent in the transverse direction.
•Present experiments are insensitive to Ising density wave order.
Future experiments could introduce a phase-locked subharmonic
standing wave at half the wave vector of the optical lattice---this
would couple linearly to the Ising order parameter.
Conclusions
I.
Study of quantum phase transitions offers a controlled and systematic method of
understanding many-body systems in a region of strong entanglement.
II.
Atomic gases offer many exciting opportunities to study quantum phase
transitions because of ease by which system parameters can be continuously
tuned.