Transcript Josephson Devices with Cold Atoms Andrea Trombettoni (SISSA, Trieste) Perugia, 18 July 2007 Outlook -) Bosonic equivalent of superconducting devices: Josephson junction (JJ) arrays of JJ SQUID analogies/differences -) Phase.

Josephson Devices with Cold Atoms
Andrea Trombettoni
(SISSA, Trieste)
Perugia, 18 July 2007
Outlook
-) Bosonic equivalent of superconducting devices:
Josephson junction (JJ)
arrays of JJ
SQUID
analogies/differences
-) Phase transitions of two-dimensional bosonic arrays:
the Berezinskii-Kosterlitz-Thouless transition
Superconducting weak links: a
Josephson junction
Josephson current at T<TBCS
Bosonic Josephson junctions
A Bose-Einstein condensate in a double well (T<TBEC) is a bosonic
Josephson junction:
[Oberthaler et al. (2005)]
T=0 Dynamics in a Double Well
i
2
 (r , t ) 
2
2
 
  Vext (r )  g 0   
t
 2m

Two-mode ansatz:
Gross-Pitaevskii equation
(r , t )  left t left  r    right t right  r 
2
  left
i
  K right  U  left  left


t

2
i  right   K  U 
left
right  right

interaction term / charging energy
t

2


2
4
K    d r left  
  Vext  right ; U  g 0  d r left
 2m

tunneling rate / Josephson energy
Phases and numbers:
 left  N left eileft


i right

 right  N right e
Josephson Hamiltonian
Relative phase:
Current:
 left right
Fractional imbalance:
z
Nleft  N right
.
z   2 K 1  z 2 sin 
H eff
U 2
 z  2 K 1  z 2 cos 
2
[Smerzi et al., 1997]
[Oberthaler et al., 1997]
NT
Ultracold bosons in an optical lattice:
an array of bosonic Josephson junctions
Vext (r )  V  sin 2 (k x)  sin 2 (k y )  sin 2 (k z ) 
a 3D lattice
It is possible to control:
- height barrier (i.e., the Josephson energy)
- interaction term (i.e., the charging energy)
- the shape of the network
- the dimensionality (1D, 2D, …)
…
Dynamics in Bosonic Arrays


 (r , t )   2 2
2
i
 
  Vext (r )  g 0   
t
 2m

When V0>>m:


 r , t    j t   j r 
j
 i
i
  t  i 1  i 1   U | i |2  i
t
[Cataliotti et al. (2001)]
Discrete Non Linear
Schroedinger Equation
Quantum Effects in Bosonic Arrays
1
ˆ
ˆi i , j nˆ j   m   i   nˆi  K   aˆ iaˆ j +aˆ j aˆ i
H   nU
2 i, j
i
i , j >

ˆ
ˆ
a
,
a
 i j  ij

nˆi  aˆ iaˆ i
Increasing V, one passes from a

superfluid to a Mott insulator
[Greiner et al. (2001)]
Similar phase transitions studied in superconducting arrays
[see e.g. Fazio and van der Zant 2001]:
ˆ , nˆ  i
1
ˆ
H   Qˆ i Ci,1j Qˆ j  EJ  cos(ˆi  ˆ j )
2 i, j
i, j
interaction term
Josephson term
Connection among the two models
i
j
i, j
ˆ  2e nˆ
Q
j
j

i
aˆ i  ni ei
The route for the bosonic SQUID
Ring trap: difficult to experimentally obtain, but significant progress in the
groups of Phillips and Stamper-Kurn – as well as in atom chip setups.
Or adding two or more barriers and
measure rotation…
[Anderson, Dholakia, and Wright (2003)]
Outlook
-) Bosonic equivalent of superconducting devices:
Josephson junction (JJ)
arrays of JJ
SQUID
analogies/differences
-) Phase transitions of two-dimensional bosonic arrays:
the Berezinskii-Kosterlitz-Thouless transition (with
P. Sodano and A. Smerzi - very recent experimental
results in Cornell’s group at JILA, Boulder)
BKT in a nutshell
In two dimensions:
T=0
T<TBKT
Few, tightly bound
vortex antivortex pairs
superfluid state
T>TBKT
Free vortices
normal state
Experiments:
liquid He films (e.g. Reppy and coworkers),
superconducting Josephson junction arrays (e.g. Tinkham and coworkers),
2D trapped Bose gas: Z. Hadzibabic et al. Nature 441, 1118 (2006),
recent work in W. Phillips group,
2D Hydrogen (Turku group, Finland)
Signature of the BKT in a 2D array
central peak of the momentum distribution
effects of vortex-antivortex pairs
on the momentum distribution
[A. Trombettoni, A. Smerzi and P. Sodano, New J. Phys. (2005)]
A two-dimensional Josephson junction array
Initial
condensate
. .
.
. T.
. .
.
.
. .
. .. .
. .. .
....
. .. . .
....... ......
..... .....
Tunneling energy
 J Cos( )
J
Nwell
VOL

“Charging energy“
Due to on-site interactions
1
EC N 2
2
Period 4.7mm
No quantum fluctuations
of relative phases !
TJ
Thermal fluctuations
of relative phases!
[Schweikhard, Tung, Cornell, arXiv: 0704.0289]
Observation of phase defects
Lattice strength
VOL
Image
J
10s
2s
tr
t
No lattice
J/T=15
J/T=2.2
J/T=1.4
J/T=1.1
J/T=0.8
J/T=0.3
Thermally driven vortex proliferation
cold: 30<T<40nK
hot: 55<T<70nK
0.06
0.10
0.04
vortex density
0.08
0.02
0.06
10
100
J [nK]
1000
0.04
0.02
0.0
10
100
J [nK]
1000
4
10
Conclusions
Ultracold atoms in trapping periodic potentials provide new
experimentally realizable quantum devices on which to test wellknown paradigms of the statistical mechanics / field theory:
-) varying the height potential  quantum phase transitions
-) 2D physics
-) dilute fermions and boson/fermion mixtures
-) interaction can be enhanced/tuned through Feshbach resonances
-) inhomogeneity can be tailored – defects/impurities can be added
-) effects of the nonlinear interactions on the dynamics
-) strong analogies with superconducting and superfluid systems
-) quantum coherence / superfluidity on a mesoscopic scale
-) quantum vs finite temperature physics
-) long-range interactions can be controlled (dipolar atomic gases)
…
Trapped ultracold atoms: Fermions
A non-interacting
Fermi gas
Tuning the interactions…
… and inducing a fermionic
“condensate”
Probing the superfluidity for fermionic gases
Close to the crossover …
… and in a lattice
More on the BCS-BEC crossover (I)
More on the BCS-BEC crossover (II)
Typycal phase-diagram of hole-doped
high-Tc superconductors
Inhomogeneous network = non-translationally invariant network
Inhomogeneity due to topology = how the lattice sites are
connected and/or to external fields
- Dream
To induce desired macroscopic coherent behaviors by acting
on the topology of networks:
Effects induced by the topology (i.e.: not observable on a regular lattice)
on bosonic systems: ultracold bosonic gases and Josephson networks
Rather new area:
A. Kitaev, quant-ph/9707021
R. Burioni et al., Europh. Lett. 52, 251 (2000)
L. B. Ioffe et al., Nature 415, 503 (2002)
B. Doucout et al., Phys. Rev. Lett. 90, 107003 (2003)
P. Sodano et al., cond-mat/0609639 (to appear on New J. Phys.)
Some examples
backbone
Bosons undergo Bose-Einstein
condensation:
they localize on the comb’s backbone
[R. Burioni et al., EPL 52, 251 (2000)]
Ground states with high
degeneracy
[B. Doucout et al., PRL 90, 107003 (2003)]
Creating a comb-shaped network
with superconducting Josephson junctions
-Nb trilayer technology
-Josephson critical currents I C 10 m A
- capacitance
C 2 pF
- classical regime
P. Silvestrini et al., cond-mat/0512478;
P. Sodano et al., cond-mat/0609372
EJ 
IC
2e2
e
Ec 
2C
Ec / EJ  0.001
Creating a star-shaped network
with ultracold bosons
Corresponding
network
V (x)  V0 cos2 (kx)
Temperatures ~ 0-500 nK
Number of particles ~ 1000-10000
Number of wells ~ 100
k  2
ER 
  800 nm

2
k2
V0  s  E R
2m
s  10  30

A theoretical model for
bosons in optical networks
(Bose-Hubbard model)
1
t
ˆ
H   nˆiUi , j nˆ j  m   i  nˆi   aˆ iaˆ j + aˆ j aˆ i
2 i , j >
2 i , j >
i

ˆ
ˆ
a
,
a
 i j  ij
aˆ i  ni e

nˆ j  aˆ j aˆ j
ii
nˆ j
large
Quantum Phase model for
superconducting Josephson networks

In the
following: JJN on discrete structures which
are not necessarily regular lattices:
Graphs
B.E.C. of non interacting bosons on inhomogeneous
low-dimensional structures: d<2.
Adjacency matrix
1 if i - j is a link
Aij  
otherwise
0
Coordination number zi 
A
ij
j
Chemical distance dij
(shortest path from i to j)
Star lattice
arms
13
14
12
1
5
16
17
O
8
i  (x, y)
11
21
x 1,...,L distance from the center
1
y  1,...,p labels arms

z i coordination number of a given site: 2
Total number of sites: N
S  pL  1 z coordination number of the center: p
O

Spatial Bose-Einstein condensation
in the center at T  TC


ˆ

ˆ
H  t A a aˆ  A  (   )(1  )   

i,j
ij i
j
i, j
x',x1
x,x'1
t  Aij ( j)  E (i)
j
x,0
y,y'
x,0 x,1
Energy spectrum
Formed by NS states and divided in 3 parts: { E0 , s0 , E+}
s0
pL-1 delocalized states with
E  2t,2t 
E  
Density of states: E   E  E n 
 4t 2  E 2
n

E0 <-2t
two bound states
(E0 : localized ground-state)
and
E+>2t


1
E 0  t
E 0  2t
p=2
p≠2
p
p 1

p  E 0  2t
Linear chain
Gapped Spectrum
Ground-state wavefunction
p  2  x 0
 E0 ( x) 
e
2p  2
2
0 
, p> 2
log( p 1)
Exponentially localized around
the center, i.e. around

the topological defect
(~Anderson localization on
inhomogeneous media)
Adding arms
enhances
localization
Thermodynamics for bosons hopping
on a star lattice
NT  N E0 
s
E
0
N S  E 
 N E
1   E  E0 
z e
1
Delocalized
Ground-State
T  TC 
Excited
N E0
NS
0
p  2 EJ
TC 
2 p 1 kB
Topology effect
N E0  T 
 TC   1  T
NT
TC
L
I. Brunelli et al., J. Phys. B 37, S275 (2004)
NT
f 
NS
E J  2tf
with the interwell
 barrier V  2π
0

50 KHz and filling
f  200 then:
 50 nK
 EJ 
Typical of one-dimensional
condensate [see W. Ketterle and N.
J. van Druten,
 Phys. Rev. A 54, 656
(1996)]

Boson distribution
T


T
NBx >>1; T  f

C 
TC
x far away form the center

Signature of the spatial
Bose-Einstein condensation:
decrease of the Josephson
critical currents
I CB x >> 1; T TC
I. Brunelli et al., J. Phys. B 37, S275 (2004)
I CA

T

TC
Bose-Einstein condensation on a comb lattice
Mapping with a 1-dimensional system with an
“external field” in the centre of the finger
Ground-state
eigenfunction
EJ
TC 
kB
I CB x >>1; T TC
I CA

T

TC
Superconducting Josephson junctions on a
comb lattice
On a comb of superconducting Josephson networks, one expects
that critical currents along the backbone increase and along the
fingers decrease:
a) 4.2 K
b) 1.2 K
backbone finger
vs. chain vs. chain
P. Silvestrini et al., cond-mat/0512478,
cond-mat/0609372
Comparison for the critical currents
with the experimental results
Using a discretized version of the Bogoliubov-de Gennes equation, one finally gets
~  /2
 bV
b  c 
 dk

0
 
tanh   k 
2
2 
1  cos k
cos k
k
Contribution of the localized
eigenstates of the adjacency matrix
P. Sodano, A. Trombettoni, P. Silvestrini, R. Russo, and B. Ruggiero, cond-mat/0609639
Anderson localization for bosons
Random Impurities
Single/Many Impurities
(with or without an optical lattice)
Very active field of research:
J. Lye et al., PRL 95, 070401 (2005)
C. Fort et al., PRL 95, 170410 (2005)
D. Clement et al., PRL 95, 170409 (2005)
T. Schulte et al., PRL 95, 170411 (2005)
H. Gimperlein et al., PRL 95, 170401 (2005)
M. Modugno, cond-mat/0509807
General idea: topological defect ~ single impurity
Anderson localization around
topological defects
Analogy with the single impurity problem:
Chain
Chain + single defect
Star
E  2t cos(k )
Delocalized states
Exponentially localized states
N.B. Also differences: ordering on the chain – no ordering
on the star …
Localization vs. interactions I:
single impurity
 i    i 0
H  t   i  i*1  c.c.  U | i |    i i
4
i
i
Variational ansatz:
(impurity in 0)
i
 i  K s  e|i|/s
Minimization of the energy for U=0 gives the correct result
With fixed ,the minimization of the energy
for finite U gives a critical value for the interaction:
U  Uc
U > Uc
e1/ s
(UN )c  4
2
1    
 

  4 
2  t 
t


Localization vs. interactions II:
star lattice
H  t  Aij  i  *j  c.c.  U | i |
4
ij
Aij adjacency matrix of the star
i
Variational ansatz:
 i  K s  e|x|/s
(x distance from the star’s center)
Similarly to before, the minimization
of the energy for finite U gives a critical
value for the interaction:
(UN )c  4 t  p  2
(p number of star’s arms)
From the point of view of the ground-state localization, the
star’s topology corresponds to an equivalent defect
= 4 t (p-2)
A. Smerzi, P. Sodano, and A. Trombettoni, in preparation
Conclusions and perspectives
- Topology role in inducing and control new coeherent states at
for bosons on inhomogenoeus networks
- Observed enhancement of critical currents in inhomogeneous
networks of superconducting Josephson junctions: good agreement
with Bogoliubov-de Gennes results
- Localization around topological defects:
Analogies with single impurity’s localization
Competition between topologically induced disorder and interactions
- Propagation of wavepackets and solitons in inhomogeneous networks:
Controlling soliton dynamics through the topology
- Statistical mechanics models (both classical and quantum)
on inhomogenous networks: a lot to do ...
Theoretical models I:
superconducting Josephson networks
(Quantum Phase model)
1
ˆ
ˆ  E cos(
ˆ i  
ˆ j)
H  Qˆ iC1
Q
i, j j
J
2 i, j
i, j
ˆ
ˆ
Q
ˆ
ˆ
j  2e n j
i, n j  ii, j
C i, j




1
C
Ec  e2 i ,i
2
EJ
E J >> E c
E J  E c
Capacitance
matrix
Charging energy

Josephson coupling
Classical XY regime
Quantum XY regime
Trapped ultracold atoms: Bosons
Bose-Einstein condensation
of a dilute bosonic gas
Probe of superfluidity:
vortices
Bogoliubov-de Gennes Theory for the Critical Current Enhancement in Comb Shaped Josephson Networks
Inhomogeneous Comb
i  ( x, y )
x  position on the backbone
y  position on the finger
Homogeneous Chain
i  position on the chain
Aij   xx´ ( y , y´1   y , y´1 ) 
  y 0 y´0 ( x , x´1   x , x´1 )
Aij   i , j 1   i , j 1
Spectrum
 t  Aij  ( j )  e  (i)
j
Ground state localized around the backbone –
“Hidden” spectrum of localized states
R. Burioni et al., Europhys. Lett. 52, 251 (2000)
Planewave solutions
Retrieving the standard BCS theory
In the homogeneous limit, the quantum number  is the momentum k:
 k  2  Ek2


3 / 2
ik  r
uk (r )  L U k e
1
U 
2
2

k
 Ek 
1 

  
k 

Ek   2 k 2 / 2 m  µ  U


3 / 2
ik r
vk (r )  L Vk e
1
V 
2
2

k
 Ek 
1  
  
k 

Putting U=0 and µ=EF and assuming a BCS point-like interaction,
one gets the BCS equation for the gap:
1
n(0)VBCS
2

 D
 D

tanh
2
E 2  2
dE

E 2  2 

Relation between the chemical potential
and the Fermi energy
Using the equation for the number of particles


 2

 2
 2
N  2  dr | v (r ) |  2  dr f | u (r ) |  | v (r ) |



it follows at T=0 when <<EF
m  U  EF
Since U<0, then µ<EF : increasing the attraction, the Hartree-Fock term U
increases and the chemical potential µ decreases.
Bogoliubov-de Gennes equations:
continuous case
For an inhomogeneous fermionic systems with attractive
interactions
H  H 0  H1



H 0   dr   r s  h 0  r s 
s

h0   2 2 / 2m U0 (r )  m
H1  



V
 
 







d
r

r
s

r
s
´
h

r
s
´

r
s

0

2
ss ´





 u r   h0  U r u r   r  v r 




* 
 v r    h0  U r v r    r  u r 



 
(r )  V  u (r ) v* (r ) tanh  
2 




U (r )  V  | u (r ) |2 f  | v* (r ) |2 (1  f )


f  (e    1) 1

BdG Equations
Self-consistency conditions
Bogoliubov-de Gennes Equations:
lattice case




u (r )   u (i) i (r ); v (r )   v (i) i (r )
Discretization:


 u i    iju  j   i  v i 
j
 v i     ij v  j   * i  u i 
Lattice BdG Equations
j
~
ij   t Aij  U (i)  ij  µ
ij
Encoding the network´s
connectivity (=topology)
 


t    dr  (r )    / 2m  U (r )  (r )
~  µ  dr  (r )    / 2m  (r )
µ

2
2
i
0
2
i
 
~
V  V i2 (r  ri )
j
2
i
~
 
(i)  V  u (i) v* (i) tanh  
2 

Self-consistency condition
Hopping parameter
Lattice chemical potential
Lattice Bogoliubov-de Gennes equations
for the comb
Away from the backbone, the fingers may be regarded as a linear chain
(U(i)=Uc and i)=c). Setting on the backbone U(i)=Ub and i)=b,
~ U
one gets with µ
b
~  /2
 bV
b  c 
 dk

0
 
tanh   k 
2
2 
1  cos k
cos k
k
Contribution of the localized
eigenstates of the adjacency matrix
At low temperature:
 b (T  0)

 c (T  0)
1
~
V
1
2t


1


log 1  2 
 
2


Lattice Bogoliubov-de Gennes
equations for the chain
i  1,...,N S
 k
 k  2  E 2k
~ U ;
Ek  2t cos k  µ
c
~  2t
V
1
4  t 2t
dE
E2
1 2
4t
~  U 2
2c  E  µ
c

~  U 2 
tanh
2c  E  µ
c
2

We have to set
 k  Ek
~ U  0
i.e., µ
c
~ U  0
µ
c
One gets the „bulk“ BCS results with <<t
~
 c (T  0)  8t e 2 t / V
~
k BTc  Ct e  2 t / V
 c (T  0)
 1.76
k BTc
(C  4.54)