Departament de Fìsica i Enginyeria Nuclear,
Campus Nord B4-B5, Universitat Politècnica de Catalunya,
Barcelona, Spain
Lieb-Liniger,
Tonks-Girardeau and
super Tonks-Girardeau
gases
G.E. Astrakharchik
Summer school, Trier, August 18 (2012)
1
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogoliubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
ONE-DIMENSIONAL QUANTUM WIRES
• One-dimensional quantum systems have been experimentally
realized and studied already in 80's in GaAs/AlGaAs
semiconductor structures.
• Systems with free electrons
confined to one dimension
(quantum wires) has been
realized.
• Conductance properties,
elementary excitations, etc
have been measured.
Figure with dispersion of excitations in a quantum wire taken from 1991
3
experiment by A. R. Goñi et al. in AT&T Bell Laboratories, New Jersey
1D BOSE GAS IN A MICROCHIP
• Atoms can be confined to onedimensional geometry
• heavier and larger
• easier to detect
• can be bosons or fermions
• microchip experiments:
cold atoms trapped in a
combination of two magnetic
fields: external magnetic field
and field generated by an
electric current in wire.
taken from “BEC on a microchip”,
4
J. Reichel et al. MPI, München
COLD GASES IN OPTICAL LATTICES
Advanced and unique features of dilute ultracold gas experiments
with optical lattices:
•
Possibility to fine-tune the interaction strength by using Feshbach
resonances
- contrary interactions between electrons in a quantum wire can be
hardly changed
•
highly controllable geometry of the confinement parameters
- for example spacing and height of an optical lattice
- can be changed dynamically
•
extremely pure systems
- no defects as in quantum wires
- no condensate fragmentation as might happen in micro chip
5
traps
CONDITION FOR ONE-DIMENSIONALITY
The gas behaves dynamically as onedimensional when the excitations of
the levels of the transverse
confinement are not possible:
• Condition for the energy
• Condition for the temperature
Figure is taken from
T.Esslinger et al./Zurich
Comparison of the frequencies of
the dipole and breathing modes
confirms the achievement of the
6
quasi-one-dimensional regime.
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogoliubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
WHY LOW-DIMENSIONAL SYSTEMS ARE SPECIAL?
From theoretical point of view low-D systems are interesting as:
• The role of quantum effects is increased as the dimensionality is
lowered.
•
In particular, in one dimension, quantum fluctuations destroy
- crystalline order (no solid in 1D)
- Bose-Einstein condensation (no BEC in 1D)
•
Quantum statitstics are topologically interconnected
- Bose-Fermi mapping from Tonks-Girardeau gas to an ideal
Fermi gas.
•
There is a number of exactly-solvable many-body Hamiltonians
•
Integrability in Hamiltonian might lead to absence of
thermalization, as shown in recent experiments
8
EXACTLY SOLVABLE 1D MODELS
9
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogoliubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
POTENTIAL ENERGY: TYPICAL SCALING WITH DENSITY
1.0
(r)
0.5
0.0
0.0
0.5
r/l
1.0
11
KINETIC ENERGY: TYPICAL SCALING WITH DENSITY
12
POTENTIAL vs KINETIC ENERGY
2
E/N
10
1
10
kinetic energy
potential energy
1D
2D
3D
0
10
-1
D
10
1
,
D
3
-2
10
-2
10
na
2D
-1
10
0
10
1
10
D
2
10
13
THREE-DIMENSIONAL GEOMETRY
2
E/N
10
1
10
kinetic energy
potential energy
0
10
-1
10
na
-2
10
-2
10
-1
10
0
10
1
10
3
2
10
14
TWO-DIMENSIONAL GEOMETRY
2
E/N
10
1
10
kinetic energy
potential energy
0
10
-1
10
na
-2
10
-2
10
-1
10
0
10
1
10
2
2
10
15
ONE-DIMENSIONAL GEOMETRY
2
E/N
10
1
10
kinetic energy
potential energy
0
10
-1
10
na
-2
10
-2
10
-1
10
0
10
1
10
2
10
16
MEAN-FIELD vs QUANTUM REGIMES
17
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogoliubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
LIEB-LINIGER HAMILTONIAN
N bosonic particles of mass m interacting with contact δ-function
pseudopotential in a one dimensional system are described by the
Lieb-Liniger Hamiltonian:
The effective quasi-one-dimensional coupling constant is
inversely proportional to the one-dimensional s-wave scattering
length
The model solved exactly by
E. H. Lieb and W. Liniger in (1963)
Phys. Rev. 130, 1605 (1963)
Elliott H. Lieb
19
BETHE-ANSATZ SOLUTION
20
ENERGY OF LIEB-LINIGER GAS
E/N
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
Lieb-Liniger gas
Gross-Pitaevskii gas
Tonks-Girardeau gas
-4
10
-5
10
-6
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
n1D |a1D|
21
CORRELATION FUNCTIONS
22
ANALYTIC RESUTLS
LIEB-LINIGER GAS (arbitrary value of n|a1D|).
Many properties of the system can be obtained from the Bethe ansatz.
1) The ground-state energy  / N = e(n|a1D|) 2n2 / 2m
[Lieb, Liniger Phys. Rev. 130, 1605 (1963)]
2) The expansion of the OBDM g1(z) at zero
g1(z) = 1 - 1/2 (e + (n|a1D|) e’(n|a1D|)) |nz|2 + e’(n|a1D|)/6 |nz|3
[Olshanii, Dunjko PRL. 91, 090401 (2003)]
3) The value of the pair distribution function g2(z) at zero
g2(0) = - (n|a1D|) e’(n|a1D|) /2
[Gangardt, Shlyapnikov PRL 90, 010401 (2003)]
4) Relation between short-distance(2-body) physics and equation of state
for δ-interaction: analog of Tan’s contact M. Barth, W. Zwerger (2011)
5) Large-range asymptotic g1(z) are predicted from hydrodynamic theory
[Haldane PRL 47, 1840 (1981)] g1(z) = C / |n z|α with the coefficient
α=
23
mc / 2πn = c / 2cF related to the speed of sound mc2 = μ / n
PAIR DISTRIBUTION FUNCTION
Pair distribution function function for different values of the gas parameter.
Arrows indicate the value of g2(0) as obtained from the equation of state. At24
n|a1D|=10-3 the g2(z) function is similar to the one of the a Tonks-Girardeau gas.
LOCAL PAIR CORRELATION FUNCTION
25
ONE-BODY DENSITY MATRIX
One-body density matrix g1(z) (solid lines), power-law fits (dashed lines).
The long-range asymptotic value of OBDM gives the condensate fraction.
• i.e. condensate is absent in all cases.
26
TRAPPED TG GAS: EXPERIMENTS
27
ENERGY EXPANSION
In the mean-field description the
chemical potential is linear with the density:
Beyond mean-field terms can be found perturbatively within Bogliubov
theory.
• for 3D Bose gas the correction was found by K.W. Huang and Nobel
prize winners C. N. Yang and T. D. Lee.
• for 1D Bose gas the correction can be found from Bogoliubov theory
and it coincides with expansion of the Bethe ansatz result:
28
This coincidence is not trivial as condensate fraction is zero!
THREE-BODY CORRELATION FUNCTION
DEFINITION
The three-body correlation function (its value at zero) is defined as
N ( N  1)( N  2 )  |  0 ( 0 , 0 , 0 , z 4 ,.., z N ) | dz 4 .. dz N
2
g 3 (0) 
1
n
3



ˆ ( z ) ˆ ( z ) ˆ ( z ) ˆ ( z ) ˆ ( z ) ˆ ( z ) 
n
3
|
2
0
( z1 ,.., z N ) | dz 1 .. dz N
This function is related to the probability of three-body collisions
LIMITING EXPRESSIONS
1) In the Tonks-Girardeau regime (n|a1D|  1) the three-body collisions are highly
suppressed
6
g 3 (0) 
( n a 1 D )
60
2) In the limit of weakly interacting gas (Bogoliubov theory), (n|a1D|  1)
g 3 (0)  1 
6 2

na 1 D
3) Approximation (arbitrary density n|a1D|)
g 3 ( 0 )  ( g 2 ( 0 ))
3
29
THREE-BODY CORRELATION FUNCTION
Value at zero of the three-body correlation function g3(0) (black squares) on the
log-log scale, TG limit (blue dashed line), Bogoliubov limit (red dashed line),
30
3
(g2(0)) (solid black line), experimental result of Tolra et al.’04 (green diamond).
THREE-BODY CORRELATION FUNCTION
Value at zero of the three-body correlation function g3(0) on the log-log scale, as a
function of γ). Solid line V.V. Cheianov et al, JSTAT 8, P08015. (2006).
31
Experimental data: H.-C. Nägerl group, PRL 107, 230404 (2011).
STATIC STRUCTURE FACTOR: LIEB LINIGER
1.0
S (k )
10
3
30
0.8
1
0.3
0.6
10
-3
0.4
0.2
0.0
0
2
4
6
8
10
k / n1D
Static structure factor S(k) for different values of the gas parameter (solid lines).
The dashed lines are the corresponding long-wavelength asymptotics.At n|a321D|=10-3
the static structure factor is similar to the one of the a Tonks-Girardeau gas.
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogolubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
TONKS-GIRARDEAU GAS
34
BOSE-EINSTEIN STATISTICS
35
FERMI-DIRAC STATISTICS
36
ONE-DIMENSIONAL FERMI GAS
37
MAPPING OF THE WAVE FUNCTION
38
TONKS-GIRARDEAU GAS: WF
39
TG GAS: PDF and STATIC STRUCTURE FACTOR
1.0
1.0
g2
Sk
2
n
0.5
0.0
0.5
0
1
2
nz
3
0.0
0
1
2
3
k / kF
4
40
TG GAS: MOMENTUM DISTRIBUTION
3
nk
fermions
(ideal Fermi gas)
bosons
(Tonks-Girardeau gas)
2
1
0
0
1
2
k / kF
3
41
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogolubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
UNITARY AND TG GASES: 3D
43
UNITARY AND TG GASES: BOSONS IN 3D
44
UNITARY AND TG GASES: BOSONS IN 1D
2
E/N
10
1
10
kinetic energy
potential energy
1D
2D
3D
0
10
-1
1D
,
3D
2D
10
-2
10
-2
10
-1
10
na
0
10
1
10
D
2
10
45
GOING BEYOND TONKS-GIRARDEAU LIMIT
1 .0
0 .8
g (z )
m e an f ie ld G ro s s- P ita e vs k ii
2
na
0 .6
na
1D
1D
=10
-3
= 0 .3
n a1 D=1
0 .4
na
na
0 .2
1D
1D
=30
= 1 00
To n k s- G ira rd e au
0 .0
0
1
2
3
4
5
n z
46
TWO-BODY SCATTERING SOLUTION
(x)
1.0
0.8
0.6
Tonks-Girardeau
g1D + 
0.4
0.2
super Tonks-Girardeau
g1D < 0
0.0
-0.2
-0.4
a1D < 0
-0.1
hard rods
a1D > 0
0.0
0.1
0.2
0.3
0.4
0.5
x, [a.u.]
47
6
4
0
a1D
MF
-GP
2
resonance
TG,
g1D /(aoscosc); a1D / aosc
8
g1D  +
CONFINEMENT INDUCED RESONANCE
g1D
-2
g 
sTG, 1D
one
collapse / solit
-
-4
-6
-3
-2
-1
0
1
2
3
a3D / aosc
48
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogolubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
6
4
0
a1D
MF
-GP
2
resonance
TG,
g1D /(aoscosc); a1D / aosc
8
g1D  +
SUPER TONKS-GIRARDEAU SYSTEM
g1D
-2
g 
sTG, 1D
one
collapse / solit
-
-4
-6
-3
-2
-1
0
1
3
2
a3D / aosc
50
STATIC STRUCTURE FACTOR: HARD-RODS
Static structure factor S(k) of a gas of hard-rods for different values of the gas
parameter na1D (color lines) and static structure factor of a Tonks-Girardeau gas
51
(black solid line).
PAIR DISTRIBUTION FUNCTION
Pair distribution function, n1Da1D = 0.1
1.2
g2(x)
1.0
0.8
0.6
0.4
VMC
HR
0.2
0.0
0
5
10
15
20
25
x / a1D
52
PAIR DISTRIBUTION FUNCTION
Pair distribution function, n1Da1D = 0.2
1.2
g2(x)
1.0
0.8
0.6
0.4
VMC
HR
0.2
0.0
0
5
10
15
20
25
x / a1D
53
PAIR DISTRIBUTION FUNCTION
Pair distribution function, n1Da1D = 0.3
1.2
g2(x)
1.0
0.8
0.6
0.4
VMC
HR
0.2
0.0
0
5
10
15
20
25
x / a1D
54
4
10
2
2
E / N, [/ m a1D ]
GROUND STATE ENERGY: COMPARISON
3
10
2
10
1
10
Lieb-Liniger
Super-Tonks-Girardeau
Hard rods
Gross-Pitaevskii
Tonks-Girardeau
0
10
-1
10
-2
10
-3
10
-4
10
0.01
0.1
1
10
n1D |a1D|
100
• At small n LL and ST energy correction to TG gas has same absolute value.
55
• There is an instability of sTG state for a certain value of the gas parameter.
COLLECTIVE OSCILLATIONS
56
SUPER TONKS-GIRARDEAU SYSTEM
57
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogolubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
DYNAMIC FORM FACTOR
59
DYNAMIC FORM FACTOR OF LIEB-LINGER GAS
60
LANDAU’S CRITERIA OF SUPERFLUIDITY
61
DYNAMIC FORM FACTOR AT 2kF
S(2kF,)
10
sTG
8
6
GP
4
TG
2
0
0
1


2
62
CONTENTS
• Introduction
• Why low-dimensional systems are special?
• Density scaling: potential vs kinetic energy
• Lieb-Liniger gas
• Energy expansion in Bogolubov theory
• Tonks-Girardreu gas
• Unitary gas
• Super Tonks-Girardeau gas
• Dynamic form factor
• Conclusions
CONCLUSIONS 1/2
64
CONCLUSIONS 2/2
65
DANKE SCHÖN
FÜR IHRE
AUFMERKSAMKEIT!
66