Bose-Einstein condensationconf.kias.re.kr/.../lec_note/Bose-Einstein...Gases.pdf · S. N. Bose. A....
Transcript of Bose-Einstein condensationconf.kias.re.kr/.../lec_note/Bose-Einstein...Gases.pdf · S. N. Bose. A....
Bose-Einstein condensation
Lecturer: Yong-il Shin (SNU)
2010 KIAS-SNU Physics camp
Outline
1. What is Bose-Einstein condensation?
2. BEC in ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Outline
1. What is Bose-Einstein condensation?
2. BEC in ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Three boarders on slopes
A
B C
Here, we can’t distinguish them.They all look same.
Classical counting Quantum counting
Total number of cases: 27Probability of having all of them
in the same slope: 1/9Total number of cases: 10Probability of having all of them
in the same slope: 3/10
Indistinguishability makes them more likely to be together.
Saturation of occupation
Example) N-particles in a two-level system
Classical counting Quantum countingdue to the indistinguishability of particles
2N N+1
1
2E
ppNN+
=12 1
1
2 1)1(
1 +
+
−+−
−= N
N
ppN
ppN
TkE Bep /−=
In a thermodynamic limit N→∞,the occupation number of the excited state is saturated
Bose statistics
Occupation number of a state with energy
Total number of particles
iε
( )CN T≤
Density of states
When , the remaining particles are put into the ground state with .
CN N>0µ ε=
ε
( )n ε
Bose-Einstein condensate: Macroscopic occupation of a single quantum state
Criterion of Bose-Einstein condensation
: density matrix of a given many-body state of N-bosons
: single-particle density matrix
: corresponding eigenfunctions and values
Macroscopic occupation of a single quantum state
: wavefunction of a condensate
Penrose & Onsager (1956)
Yang (1962): off-diagonal long-range order (ODLRO)
Birth of the BEC idea (1920’s)
S. N. Bose
A. Einstein
E. Schrodinger
de Broglie
• Bose derived Plank distribution of Black-body radiation with a new photon counting way, but failed to publish his results.
• Einstein immediately agreed with Bose, and they described the indistinguishabilityof photons and Bose-Einstein statistics.
• Einstein extended this idea to include systems with a conserved particle number, adopting de Broglie’s new idea of matter waves.
• Einstein pointed a peculiar feature of the distribution: at low temperature it saturates.
• Schrodinger first heard about de Broglie’s idea from reading Einstein’s paper and later he developed his wave equation.
Matter wave picture of BEC
TmkBdB
22 πλ =
612.23 ≅dBnλ
de Broglie’s wavelength
Critical condition of BEC
BEC systems Superfluid Helium Lasers and Masers (macroscopic occupation in the same state) Superconductors Ultracold atomic gases
Outline
1. What is Bose-Einstein condensation?
2. BEC in ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Ultracold Atom Cloud
Typical sample size
Atom number ~ 106
Spatial size ~ 100 umn=1011~1015 /cm3
T= 100 nK (~ 1 Hz)
Bose-Einstein condensate (BEC)
A Bose-Einstein condensate is the macroscopic occupation of the ground state of a system.
T > TBEC T = 0
BECTBEC ~ 100nK
How to cool down atoms ?
Laser cooling
ground state
excited state
• photon has momentum• atom absorbs and emits photons
• Doppler effect : Optical molasses
kv+ω
kv−ω
Laser cooling(1997 Nobel prize)
v
T ~ 100 µK
How to cool down atoms ?
Evaporative cooling
Removing the tail of thermal distribution leads to lower average energy, i.e., cooling the sample.
Method 1: transitions to untrapable states
ωRF
Method 2: Reducing the trap depth
How to cool down atoms ?
Evaporative cooling
Removing the tail of thermal distribution leads to lower average energy, i.e., cooling the sample.
T ~ 100 nK
Time-of-flight Imaging
The expanded cloud reveals the momentum distribution of the sample.
Bose-Einstein condensation in a dilute gas
BEC @ JILA, 1995(2001 Nobel prize)
MOT
Many-body Hamiltonian in cold atom gases
∑∑<=
−+
+=
jiji
N
iiext
i rrVrVm
pH )()(21
2
∑∑<=
−+
+=
jiji
N
iiext
i rrUrVm
pH )()(2 0
1
2
δ
Model system
Scattering problem
)(rVFor a given Interparticle potential
refe
ikrikz )(θψ +=
)(θf : Scattering amplitude
Partial wave description
)(cos)()12()(0
θθ ll
l Pkflf ∑∞
=
+=
ikkikekf
l
i
l
l
−=
−=
δ
δ
cot1
21)(
2
∑∞
=
+=0
22 sin)12(4
lltot l
kδπσ
Phase shift
Cold atom collisions
2
2
2)1()()(
mrllhrVrVeff+
+=For non-zero l,
van der Waals attractionrc ac
V(r)
r
4/1
262
=
Cma rcCharacteristic length
Centrifugal barrier mK 12
22
≈≈cr
c amlE
For gases in the sub-milikelvin regime, only s-wave collisions are relevant.
s-wave scattering length
Physical meaning of scattering length
rc acV(r)
r
krkrei
out)sin( 0
0 δψδ +
≅
At r >>ac
a<0
a<<0
a>0
Sign of scattering length and energy shift
Corresponding energy shift?
Positive a : repulsive Negative a : attractive
L
a
2 22 2 2 201
3
2( )2 2 2
kk a naEm m m L m
πδ = −
0kLπ
=
1kL aπ
=−
Effective potential
In the regime of ultracold collision, kac<<1The two-body collision is completely specified by a single parameter, a
Effective pseudopotential
∑∑<=
−+
+=
jiji
N
iiext
i rrm
arVm
pH )(4)(2
2
1
2
δπ
Realizing the toy-model Hamiltonian,
Mean-field description of a dilute Bose gas
),(),()(2
),( 20
22
trtrUrVm
trt
i ext Φ
Φ++∇−=Φ
∂∂
Gross-Pitaevskii (GP) equation
A simplest approximation for many-body states a product of a single-particle wavefunction:
2
01
( ) ( )2
Ni
ext i i ji i j
pH V r U r rm
δ= <
= + + −
∑ ∑
Wave function of condensate
Outline
1. What is Bose-Einstein condensation?
2. Ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Laser lightOrdinary light
diffraction limited (directional)coherentone big wavesingle mode (monochromatic)
divergentincoherentmany small wavesmany modes
Interference @ MIT, 1997(2001 Nobel Prize)
Interference of two BECs
Hanbury Brown – Twiss Effect
Hanbury Brown & Twiss, Nature 177 (1956)
Photon bunching in light emitted by a chaotic source Highlight the importance of two-photon correlations Modern quantum optics
Quantum theory of optical coherenceGlauber, PRL 10 (1963)
)()();( 2,2)(
1,1)(
2,21,1)1( trEtrEtrtrG +−=
First-order coherence function
Laser light
Chaotic light
a a+
How to describe the state of light
Correlations in many-body systems First-order coherence function
)(ˆ)(ˆ),( 2121)1( xxxxG ψψ += one-particle density matrix
0),(lim 021)1(
21
≠=∞→−
nxxGxx
: condensate fraction
1),(),(
),(),(22
)1(11
)1(21
)1(
21)1( ≤=
xxGxxGxxGxxg Normalized first-order coherence function
For a translational invariant system
−=
Tcl
rrgλπ 2
)1( exp)(
Tmkp
kBen 2
2
~−
For a classical gas
nnrgr
/)(lim 0)1( =
∞→With a BEC
At T~Tc
−
)(exp~)()1(
Trrg
ξ
r
Spatial coherence of a trapped Bose gas
Bloch et al., Nature 403 (2000)
T<<Tc T~Tc T>>Tc
Two-slit experiment to measure spatial coherenceUsing two rf waves, outcouple two atomic beams in different positionsVisibility of the interference pattern indicates spatial coherence
Spatial coherence of a trapped Bose gasBloch et al., Nature 403 (2000)
1),(),(),(
),(21
)1(
22)1(
11)1(
21)1(
≤== xxgxxGxxG
xxGV
Quantum theory of optical coherence (2)
)()()()();( 1,1)(
2,2)(
2,2)(
1,1)(
2,21,1)2( trEtrEtrEtrEtrtrG ++−−=
Second-order coherence function
Fluorescence From a single atom
Laser light
Chaotic light
How to describe the state of light Glauber, PRL 10 (1963)
Correlations in many-body systems (2) Second-order coherence function
)(ˆ)(ˆ)(ˆ)(ˆ),( 122121)2( xxxxxxG ψψψψ ++=
),()()()()()(ˆ)(ˆ 21)2(
2121121 xxgxnxnxxxnxnxn
+−= δ
)()(),(),(
21
21)2(
21)2(
xnxnxxGxxg =
Normalized second-order coherence function
Density-density correlation function
Prob. To have another particle in a shell [r, r+dr]drrgnr )(4 )2(22π
Higher order phase coherenceOttl et al., PRL 95 (2005)
Higher order phase coherenceOttl et al., PRL 95 (2005)
Coherent outcoupling Incoherent outcoupling
Bunching and anti-bunching
Using 3He* (fermion) and 4He* (boson)
Schellekens et al., Science 310 (2005) / Jeltes et al., Nature 445 (2007)
Bunching and anti-bunching
Using 3He* (fermion) and 4He* (boson)
Schellekens et al., Science 310 (2005) / Jeltes et al., Nature 445 (2007)
Higher order phase coherence
6!3)0()3( ==thg
1)0()3( =BECg
Three-body decay rate is six-times smaller for condensates.
Burt et al., PRL 79 (1997)
Outline
1. What is Bose-Einstein condensation?
2. Ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Superfluid
Superfluid, having a phenomenological definition, can flow without dissipation.
Q) Can this particle excite this fluid, or give its kinetic energy to this fluid?
Landau Criterion of Superfluidity
L.D. Landau, J. Phys. (USSR) 5, 71 (1941).
min
)(
=
pp
cευ
Critical velocity
2 21 1 ( )2 2
( )
mv m v v mv v
p mv m v v m v
ε δ δ
δ δ
= − − ≈
= − − =
If the particle excite the fluid, ( )pvp
ε=
Excitation energy for momentum p
Excitation spectrum of Superfluid Helium
Phonon
Roton
Maxon
∆pC1=ε
r
ppµ
ε2
)( 20−
+∆=
Excitation spectrum of Superconductor
Normal
Super
Excitation spectrum of a non-interacting Bose gas
2( )p pε ∝
0Cv =p
( )pε
2
1 2
N
i
pHm=
=∑
Excitation spectrum of a non-interacting Bose gas
p
( )pε
2
01
( )2
N
i ji i j
pH U r rm
δ= <
= + −∑ ∑
Microscopic theory of a Bose gas at T=0
20000 )(2 qqq Un εεε +=
Bogoliubov approximation: replacing a0 with c-number N01/2
Diagonalize with canonical transformation:
Elementary excitation of an interacting Bose gas
20000 )(2 qqq Un εεε +=
2
01
( )2
N
i ji i j
pH U r rm
δ= <
= + −∑ ∑
mUns
qsUn qq
/
2
00
000
=
=≈ εε
000 Unqq +≈ εεPhonon regime
Free particle regime
Many-body ground state
Dominant scattering processes at T~0
)()()0()0( pp −++⇔+Two atoms in condensate collide into +p and –p atoms.
02 ≠== +pppp vaan
Non-condensed atom number
Quantum depletion
Outline
1. What is Bose-Einstein condensation?
2. Ultracold atomic gases
3. Phase coherence of BEC
4. Superfluidity and BEC
5. BEC in an optical lattice
Optical dipole trap
R. Grimm, et al, Adv. At., Mol., Opt. Phys. 42, 95 (2000)
Complex polarizability
Far detuning limit (∆ << Γ)
Optical lattice
When two laser beams overlap, they interfere, leading to a periodic pattern of the intensity, i.e. a periodic potential for atoms.
Standing potential
Moving lattice potential21
21
ωωωω
≠=
Lattice period is controlled by the angle between the two beams.
Optical lattice2D optical lattice / quantum wire, 1D physics
3D optical lattice
Superlattice potentials
Optical lattice
Atoms moving in an optical lattice have the same basic physics as electrons in a crystal lattice in solids.
Lattice constant
Solid crystal ~10-10mOptical lattice ~10-7m
Lattice barrier height
Solid crystal ~105 KOptical lattice ~10-5 K
Optical lattice: Magnifying laboratory for condensed matter physics.
Band structure
The presence of an optical lattice modifies the single-particle energy spectrum to a band structure.
Energy band structure
Bloch wave function for nth band with momentum q-distributed over all lattice sites
Bloch wavefunction
)()()()(
xudxuxuex ikx
=+=ψ
Time-of-flight image of a BEC in an optical lattice
BEC from a harmonic trap
BEC from a lattice
Sudden release from a trap Revealing the in-trap momentum distribution. Diffraction from an optical grating
Adiabatic mapping of quasimomentumUnder adiabatic transformation of the lattice depth the quasimomentum q is preserved during slow turn-off process
PRL 87, 160405 (2001).
Brillouin zones
BEC in a double-well potential
BEC1 BEC2
Relative phase of two condensates
Tunneling of particles between the wells
Time evolution of the phase and the atom number: Josephson dynamics
Simple description
++−= ++
=
++∑ )(21
21122,1
aaaaJaaaaUHi
iiii
Interaction term Tunneling term
Ground state for the non-interacting case U=0
−+−+
++ +−= aJaaJaH 2/)(
2/)(
21
21
aaa
aaa
−=
+=
−
+Symmetric state
Anti-symmetric state
Symmetric ground state If we start with a BEC in one well, it will oscillate at hJ /2
Two-mode approximation
Coherent state and number state
Non-interacting case U=0
0)( 21Ni
coh aea ++ +∝ φψ : Coherent state with a well-defined relative phase
Strongly-interacting case U>>J
0)()( 22
21
NN
num aa ++∝ψ : Number (Fock) statewith well-defined atom numbers
++−= ++
=
++∑ )(21
21122,1
aaaaJaaaaUHi
iiii
Particle number uncertainty ~ N
Bose-Hubbard model
Kinetic energyHopping to nearest neighbors
On-site interactions
Both the thermal and mean interaction energies at a single site are much smaller than the separation to the first excited band. Only the lowest band is involved.
Wannier functions decay essentially within a singel lattice constant. Only the hopping to nearest neighbors are counted.
In the limit of a sufficient deep optical lattice.
Superfluid-Mott-insulator transition
The many-body ground state is determined via the competition between the kinetic energy and the interaction energy.
Superfluid phasenn
NNn L
=∆
= /
Mott-insulating phase0
1=∆=n
n
can be controlled by the lattice intensity.
Quantum phase transition from SF to Mott-Insulating phase
Superfluid-Mott-insulator transitionNature 415, 39 (2001).
V=0 Er 3 7 10
13 14 16 20
Interference peaks disappear Loosing superfluidity
Phase coherence in SF-to-MI transition
Existence of BECSuperfluid phase
Perfect Mott regime, J=0
vanishes exponentially beyond R=0.
The momentum distribution is a structureless Guassian.
With non-zero J, a coherent admixture of particle-hole pairs
Short range coherence
Summary
Ultracold atom gases
: Model system for many-body physics
BEC has laser-like properties
BEC with interactions is a superfluid
Optical lattice systems simulate solid state physics.