‘Integrability and its Breaking’ in two problems from
ultracold atomic physics
Austen Lamacraft
University of Virginia
May 27th, 2011
Trieste Integrability Workshop
arXiv:1104.0655
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
1 / 31
What is integrability?
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
2 / 31
What is integrability?
For a recent discussion see J.-S. Caux, J. Mossel, J. Stat. Mech. (2011)
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
2 / 31
What is integrability?
For a recent discussion see J.-S. Caux, J. Mossel, J. Stat. Mech. (2011)
ed student access to
ite part of the book,
alogero–Sutherland
eract via long-range
not as heavily studied
re fascinating in their
neralizations of the
welcome introduction
Physics Today
ed in presentation. It
by graduate students.
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
2 / 31
Integrability ≡ Non-diffractive scattering
Scattering of two particles p1 , p2 −→ p10 , p20 satisfies
p1 + p2 = p10 + p20
p2
p 02
p 02
p12
+ 2 = 1 + 2
2m1 2m2
2m1 2m2
For equal masses two solutions are
p10 = p1 , p20 = p2 (retain) or p10 = p2 , p20 = p1 (swap)
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
3 / 31
Integrability ≡ Non-diffractive scattering
Scattering of three particles (equal masses)
p1 + p2 + p3 = p10 + p20 + p30
p12 + p22 + p32 = p102 + p202 + p302
Solution not unique!
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
3 / 31
Three body scattering
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
4 / 31
Three body scattering
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
4 / 31
Diffractive vs. Non-diffractive scattering
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
5 / 31
Diffractive vs. Non-diffractive scattering
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
5 / 31
Outline
1
Breaking integrability in a simple model
Integrability and transport
Consequences for mobility of an impurity in a 1D Fermi gas
2
Noise correlations in an expanding 1D Bose gas
The Hanbury Brown Twiss effect
Tracy–Widom form of the N-particle propagator
Effect of interactions on the Hanbury Brown and Twiss effect
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
6 / 31
Outline
1
Breaking integrability in a simple model
Integrability and transport
Consequences for mobility of an impurity in a 1D Fermi gas
2
Noise correlations in an expanding 1D Bose gas
The Hanbury Brown Twiss effect
Tracy–Widom form of the N-particle propagator
Effect of interactions on the Hanbury Brown and Twiss effect
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
7 / 31
Integrability and transport
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
8 / 31
Integrability and transport
Within Boltzmann picture
f (p, x, t) is not altered by
2-body collisions
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
8 / 31
Integrability and transport
Within Boltzmann picture
f (p, x, t) is not altered by
2-body collisions
3-body collisions can lead to
relaxation (thermalization) if
non-diffractive
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
8 / 31
Formalizing the relationship...
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
9 / 31
Formalizing the relationship...
Integrability conjecture (Castella, Zotos, and Prelovšek, 1995)
A finite Drude weight at T 6= 0 is a generic property of integrable systems
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
9 / 31
Formalizing the relationship...
Integrability conjecture (Castella, Zotos, and Prelovšek, 1995)
A finite Drude weight at T 6= 0 is a generic property of integrable systems
Subtleties e.g. for Heisenberg model (Sirker, Pereira, and Affleck, 2009)
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
9 / 31
Breaking integrability: a simple model
‘Impurity’ of mass M in gas of fermions of mass m
P
Interaction Hint = V i δ(xi − X )
For M = m problem integrable by Bethe ansatz (Yang–Gaudin, 1967)
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
10 / 31
Breaking integrability: a simple model
‘Impurity’ of mass M in gas of fermions of mass m
P
Interaction Hint = V i δ(xi − X )
For M = m problem integrable by Bethe ansatz (Yang–Gaudin, 1967)
Solved by McGuire (1965) by ‘baby’ (nested) Bethe ansatz
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
10 / 31
Breaking integrability: a simple model
‘Impurity’ of mass M in gas of fermions of mass m
P
Interaction Hint = V i δ(xi − X )
For M = m problem integrable by Bethe ansatz (Yang–Gaudin, 1967)
Solved by McGuire (1965) by ‘baby’ (nested) Bethe ansatz
Finite Drude weight at T > 0 (Castella, Zotos, and Prelovšek, 1995)
Austen Lamacraft (University of Virginia)
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May 27th, 2011
10 / 31
Equal masses
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Two problems
May 27th, 2011
11 / 31
Unequal masses
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
12 / 31
Kinematics at low temperatures
Impurity + 1 particle processes frozen out for kB T < 2MvF
2MvF
E (p)
vF p
p2
2M
p
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
13 / 31
Kinematics at low temperatures
Impurity + 1 particle processes frozen out for kB T < 2MvF
2MvF
E (p)
vF p
p2
2M
p
Low temperature kinetics determined by two-particle processes
(Castro-Neto and Fisher, 1996)
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
13 / 31
Perturbative calculation – 3-body scattering
T
(2)
=i
Austen Lamacraft (University of Virginia)
V
L
2
ξk1 +q1
1
− ξk1 + K −q1 − K
Two problems
May 27th, 2011
14 / 31
Perturbative calculation – 3-body scattering
T
(2)
=i
+
V
L
ξk2 +q2
2 1
1
−
ξk1 +q1 − ξk1 + K −q1 − K
ξk2 +q2 − ξk1 + K −k2 −q2 +k1 − K
1
1
−
.
− ξk2 + K −q2 − K
ξk1 +q1 − ξk2 + K −k1 −q1 +k2 − K
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
14 / 31
Implications for impurity mobility
Momentum relaxation rate of the impurity
−1
τmom
=
2π
~MT
X
k1 ,k2 ,q1 ,q2
Austen Lamacraft (University of Virginia)
(q1 + q2 )2 |T (2) |2 δ(Ei − Ef )
× nk1 nk2 (1 − nk1 +q1 ) (1 − nk2 +q2 )
Two problems
May 27th, 2011
15 / 31
Implications for impurity mobility
Momentum relaxation rate of the impurity
−1
τmom
=
2π
~MT
X
k1 ,k2 ,q1 ,q2
(q1 + q2 )2 |T (2) |2 δ(Ei − Ef )
× nk1 nk2 (1 − nk1 +q1 ) (1 − nk2 +q2 )
vanishes as T 4 , leading to a low temperature mobility
µ = τmom /M ∝ T −4
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
15 / 31
Implications for impurity mobility
Momentum relaxation rate of the impurity
−1
τmom
=
2π
~MT
X
k1 ,k2 ,q1 ,q2
(q1 + q2 )2 |T (2) |2 δ(Ei − Ef )
× nk1 nk2 (1 − nk1 +q1 ) (1 − nk2 +q2 )
vanishes as T 4 , leading to a low temperature mobility
µ = τmom /M ∝ T −4
Matrix element vanishes for M = m −→ ballistic motion
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
15 / 31
Implications for impurity mobility
Momentum relaxation rate of the impurity
−1
τmom
=
2π
~MT
X
k1 ,k2 ,q1 ,q2
(q1 + q2 )2 |T (2) |2 δ(Ei − Ef )
× nk1 nk2 (1 − nk1 +q1 ) (1 − nk2 +q2 )
vanishes as T 4 , leading to a low temperature mobility
µ = τmom /M ∝ T −4
Matrix element vanishes for M = m −→ ballistic motion
Calculation to leading order in M − m for any V using exact solution.
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
15 / 31
where the effective two-phonon amplitudes are given by
ðXÞ þ ð#d =2Þ"2 ðXÞ; cf. Eq. (4)
$
#
#
$
m2 c2
$M 2
#
article with the superfluid ve( # d ( 1 þ $l ; (12)
!" ¼ 2 ) 1 ( $l þ
d [5] by noticing that in the
m
#
nM
¼ 0 the energy of the particle
#
$
1
$M
by the dispersion relation "ðPÞ.
1 ( $l þ
:
(13)
!# ¼
week ending
mentum of such particle is P þ
m
m
PHYSICAL REVIEW LETTERS
20 FEBRUARY 2009
PRL 102, 070402 (2009)
vs ðP þ Mvs Þ ¼ "ðPÞ þ Pvs þ
The momentum relaxation rate may be evaluated in the
entum in the laboratory frame,
second order in the two-phonon amplitudes !";# , averaged
second power in vs , one finds Bloch
a One-Dimensional
Spinor
Gas
overOscillations
the Luttingerin (quadratic)
part of the
phonon
Þvs þ ðM2 "00 ðPÞ ( MÞv2s =2,
Hamiltonian (8). This does
not assume smallness
of the
1,*
2
is V ¼ "0 ðPÞ. At a sufficiently
Gangardt and A. Kamenev
amplitudesD.!M.
";# , but rather gives the leading low1 1=M ) one thus
and "00 ð0Þ ¼
School of Physics and Astronomy,
University
Edgbaston,
B15 2TT, United Kingdom
temperature,
T !of!,Birmingham,
contribution.
The Birmingham,
semiclassical
f the spin excitations2School of Physics
and Astronomy,
University
Minnesota,
Minneapolis,
55455, USA
equation
of motion
for the of
spin
excitation
acquires aMinnesota
form
v2s
The view from LuttingerLand
(Received 21 November 2008; revised manuscript received 19 January 2009; published 18 February 2009)
mc2
# applied to a spin-flipped particle in a one-dimensional spinor gas may lead to Bloch oscillations
A force
þ
ð1 ( $l Þ" þ d "2 ;
of the particle’s
position and velocity. The existence of Bloch oscillations crucially depends on the viscous
n
2
friction force(9)
exerted by the rest of the gas on the spin excitation. We evaluate the friction in terms of the
quantum fluid parameters. In particular, we show that the friction is absent for integrable cases, such as an
symmetric gas of bosons or fermions. For small deviations from the exact integrability the friction
the absence ofSUð2Þ
interactions
very weak, opening the possibility to observe Bloch oscillations.
amiltonian (9) isisindependent
rms containing "DOI:
¼ "ðXÞ
and
10.1103/PhysRevLett.102.070402
PACS numbers: 05.30.Jp, 03.75.Kk, 03.75.Mn
garded as effective interaction
e particle
X. ultracold
This
associated with an accelerated quantum particle in the
Thecoordinate
dynamics of
atomic gases with internal
me for
the moving
presence of a static periodic potential; see, e.g., Ref. [13]
(spinor)
degreesparticle.
of freedom has been a focus of a number
m canonical
of The observed collective phefor a recent experimental realization. However, the exisof recenttransformation
experiments [1].
þ $Mv
along
with interest
the
s !P
tence of Bloch oscillations in 1D spinor condensates does
nomena
have
revived
in earlier theoretical works
ð$M=mÞ"
ðxÞ !
"ðxÞ,inwhere
not rely on a presence of an external periodic potential. It is
[2] ondspin
waves
helium and opened the possibility to
density
particle. Thedynamics
FIG. 2.
Second-order
diagrams the
contributing
to two-phonon
1D quantum
liquid itself that provides a quasiperiodic
studyofthethenonequilibrium
of quantum
liquids.
transformation
fluid
amplitudes.
Spin
excitation is represented
a full
while
potentialby
with
theline,
lattice
spacing n'1 and thus 2"@n reBecause oftothetheunprecedented
degree of
experimental
bsorbed
in the
by wavy
Diagramsciprocal
(a),(b) and
(c) contribute to
control
it ismodified
possible imputo excite aphonons
few atoms
into alines.
different
vector.
(12),
diagram
!# , to
Eq.a (13).
!" , Eq.
hyperfine internal state [3]. This
leads
to while
effective
spin(d) represents
Contrary
static periodic potential, a 1D quantum
excitations which may be
regarded as impurities moving
liquid exhibits quantum and thermal fluctuations. Because
070402-3
Austen
Lamacraft
(University
Virginia)
Two problems
2011dispersion
16 / 31
through
the quantum
liquidofformed
by the majority spins.
of the latter, the spin excitations May
with 27th,
periodic
Outline
1
Breaking integrability in a simple model
Integrability and transport
Consequences for mobility of an impurity in a 1D Fermi gas
2
Noise correlations in an expanding 1D Bose gas
The Hanbury Brown Twiss effect
Tracy–Widom form of the N-particle propagator
Effect of interactions on the Hanbury Brown and Twiss effect
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
17 / 31
‘Traditional’ uses of integrability
N
H=−
X
1 X ∂2
+c
δ(xi − xj ).
2
2
∂xi
i=1
i<j
Lieb–Liniger gas (1963)
Two conceptual steps...
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
18 / 31
‘Traditional’ uses of integrability
N
H=−
X
1 X ∂2
+c
δ(xi − xj ).
2
2
∂xi
i=1
i<j
Lieb–Liniger gas (1963)
Two conceptual steps...
1 The ‘Bethe ansatz’
ΨN (x) =
X
P
Austen Lamacraft (University of Virginia)

aP exp i
Two problems
N
X
j=1

kP(j) xj 
x1 ≤ x2 ≤ · · · ≤ xN
May 27th, 2011
18 / 31
‘Traditional’ uses of integrability
N
H=−
X
1 X ∂2
+c
δ(xi − xj ).
2
2
∂xi
i=1
i<j
Lieb–Liniger gas (1963)
Two conceptual steps...
1 The ‘Bethe ansatz’
ΨN (x) =
X
P
2

aP exp i
Impose boundary conditions
kj L = 2πnj − 2 arctan
Austen Lamacraft (University of Virginia)
Two problems
N
X
j=1

kP(j) xj 
kj − ki
c
x1 ≤ x2 ≤ · · · ≤ xN
May 27th, 2011
18 / 31
‘Traditional’ uses of integrability
N
H=−
X
1 X ∂2
+c
δ(xi − xj ).
2
2
∂xi
i=1
i<j
Lieb–Liniger gas (1963)
Two conceptual steps...
1 The ‘Bethe ansatz’
ΨN (x) =
X
P
2

aP exp i
Impose boundary conditions
kj L = 2πnj − 2 arctan
N
X
j=1

kP(j) xj 
kj − ki
c
x1 ≤ x2 ≤ · · · ≤ xN
Thermodynamic limit, form factors...
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
18 / 31
he initial momentum spread of the particles.
The setup
∆
Time
�
FIG. 1. 1D expansion of atoms
from an optical May
lattice.
Two problems
27th, 2011 Noise
19 / 31
Austen Lamacraft (University of Virginia)
Ready availability of noise correlations in ultracold physics
PHYSICAL REVIEW A 70, 013603 (2004)
Probing many-body states of ultracold atoms via noise correlations
Ehud Altman, Eugene Demler, and Mikhail D. Lukin
Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
(Received 10 June 2003; published 6 July 2004)
We propose to utilize density-density correlations in the image of an expanding gas cloud to probe complex
many-body states of trapped ultracold atoms. In particular, we show how this technique can be used to detect
superfluidity of fermionic gases and to study spin correlations of multicomponent atoms in optical lattices. The
feasibility of the method is investigated by analysis of the relevant signal to noise ratio including experimental
imperfections.
DOI: 10.1103/PhysRevA.70.013603
PACS number(s): 03.75.Ss, 03.75.Mn, 42.50.Lc
of the excitement in the field of Bose-Einstein conn (BEC) is due to the clear demonstration it provides
wave character of matter. The condensed state of
Austen Lamacraft (University of Virginia)
tices. In particular, spin ordered Mott states prop
two-component bosons [7,8] can be detected. Fin
verify the experimental feasibility of the proposed
Two problems
May 27th, 2011
20 / 31
Hanbury Brown and Twiss
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
21 / 31
Hanbury Brown and Twiss
The two-particle wavefunction is
1
Ψ2 (x1 , x2 ; t) = √ [ϕi (x1 ; t)ϕj (x2 ; t) ± ϕi (x2 ; t)ϕj (x1 ; t)]
2
Austen Lamacraft (University of Virginia)
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May 27th, 2011
21 / 31
Hanbury Brown and Twiss
For separation ∆ between harmonic wells
1
(y − α∆)2
ϕα (y ) =
exp −
2`2
(π`2 )1/4
s
`
i
`2
2
2
√ exp
ϕα (x; t ` ) →
− 2 (x − α∆) .
2t
2t
i πt
Probability density is then
|Ψ2 (x1 , x2 ; t)|2 →
`2 −`2 (ξ12 +ξ22 )
e
[1 ± cos ([ξ1 − ξ2 ] ∆)]
πt 2
ξ1,2 ≡ x1,2 /t and e −∆
Austen Lamacraft (University of Virginia)
Two problems
2 /4`2
neglected
May 27th, 2011
21 / 31
Hanbury Brown and Twiss
Array of N particles −→ higher harmonics arising from pairs of particles
separated by multiples of ∆
Z
C(x1 , x2 ; t) ≡ dx3 · · · dxN |ΨN (x1 , x2 , . . . , xN ; t)|2 .
#
"
∞
`2 −`2 (ξ12 +ξ22 )
2π X
→ 2e
δ(∆ [ξ1 − ξ2 ] − 2πn)
1±
πt
N n=−∞
Austen Lamacraft (University of Virginia)
Two problems
!
May 27th, 2011
21 / 31
Acknowledgements!I%!-0.3*B1%9'%!9&)06))&*3)!B&($!J7!81(2-3
Expansion of Mott state (Fölling et al., 2005)
@?!($%!LMK;!8MNOP!-39!($%!JQ!639%/!-!A-/&%RS6/&%!M%11*B)$&,
Austen Lamacraft (University of Virginia)
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May 27th, 2011
22 / 31
'+&+2#$(E#)(%&!)%!#!2+'7$#2!#22#:!%,!*(F!*%725+*!4()-!)-+!*#8+!*0#5(&'!(*!*-%4&!(&!'2++&.!#.7.3!
Expansion
of Mott state (Fölling et al., 2005)
#2=()2#2:!7&()*.!
Austen Figure
Lamacraft2!G(&'$+!*-%)!#=*%20)(%&!(8#'+!(&5$71(&'!67#&)78!,$75)7#)(%&*!#&1!)-+!#**%5(#)+1!
(University of Virginia)
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May 27th, 2011
22 / 31
1D case: interactions must be important!
Trajectories must cross
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
23 / 31
1D case: interactions must be important!
Trajectories must cross
For infinite repulsion have mapping to free fermions
Austen Lamacraft (University of Virginia)
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May 27th, 2011
23 / 31
1D case: interactions must be important!
Trajectories must cross
For infinite repulsion have mapping to free fermions
Crossover from bosonic to fermionic HBT effect
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
23 / 31
Correlations in the crossover
Correlations in the crossover
N
�
C(x1 ,x2 ;t)
C(0,0;t)
�
−1
ξ2 ∆
ξ1 ∆
�
�
−1
Austen Lamacraft (University
of Virginia)
C(x,−x;t)
N
C(0,0;t)
Austen Lamacraft (University of Virginia)
Two problems
Two problems
c∆ = 2,May`/∆
= 0.224 / 26
27th, 2011
May 27th, 2011
24 / 31
ξ2 ∆
ξ1 ∆
Correlations in the crossover
(4)
N
ms of
sityplus
n in
1, is
reladips
�
C(x,−x;t)
C(0,0;t)
−1
�
(5)
ξ∆
rom
ssed
FIG.
�
2.
C(x1 ,xof2Virginia)
;t)
Austen Lamacraft (University
c∆
= 2, `/∆ = 0.2.
Slice with function
x1 = −x2
(Top)
Normalized
correlation
�
Two problems
May 27th, 2011
24 / 31
Correlations in the crossover
Series of Fano lineshapes
[qn Γn /2 + (ε − ηn )]2
Γ2n /4 + (ε − ηn )2
ε = ∆ (ξ1 − ξ2 ) − 2πn is deviation from the nth peak.
The asymmetry parameter qn is
arg S(2πn/∆) =
2qn
.
qn2 − 1
expressed in terms of two particle scattering matrix
S(k) = −
c − ik
,
c + ik
Evolution from qn = ∞ for free bosons (resonance lineshape) to
qn → 0 as c → ∞ (antiresonance).
Austen Lamacraft (University of Virginia)
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May 27th, 2011
24 / 31
Tracy–Widom propagator (2008)
GN (x|y; t) =
X Z
σ∈SN
Austen Lamacraft (University of Virginia)
···
Z
Aσ
N
Y
it
e ikσ(j) (xj −yσ(j) ) e − 2
j=1
Two problems
P
j
kj2 dk1
2π
···
dkN
2π
May 27th, 2011
25 / 31
Tracy–Widom propagator (2008)
GN (x|y; t) =
Aσ =
X Z
σ∈SN
Y
···
Z
Aσ
N
Y
it
e ikσ(j) (xj −yσ(j) ) e − 2
j=1
P
j
kj2 dk1
2π
···
dkN
2π
S(kσ(α) − kσ(β) ) : xα < xβ but yσ(α) > yσ(β) .
Austen Lamacraft (University of Virginia)
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May 27th, 2011
25 / 31
Tracy–Widom propagator (2008)
GN (x|y; t) =
Aσ =
X Z
σ∈SN
Y
···
Z
Aσ
N
Y
it
e ikσ(j) (xj −yσ(j) ) e − 2
P
j
kj2 dk1
j=1
2π
···
dkN
2π
S(kσ(α) − kσ(β) ) : xα < xβ but yσ(α) > yσ(β) .
Convolve with initial (product) state
Z
Y
1
ΨN (x; t) = √
GN (x, y; t)
ϕj (yj )dy
N!
j
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
25 / 31
Tracy–Widom propagator (2008)
GN (x|y; t) =
Aσ =
X Z
σ∈SN
Y
···
Z
Aσ
N
Y
it
e ikσ(j) (xj −yσ(j) ) e − 2
P
j
kj2 dk1
j=1
2π
···
dkN
2π
S(kσ(α) − kσ(β) ) : xα < xβ but yσ(α) > yσ(β) .
Convolve with initial (product) state
Z
Y
1
ΨN (x; t) = √
GN (x, y; t)
ϕj (yj )dy
N!
j
Still have to
1
Do the integrals
2
Sum over permutations
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
25 / 31
Going to the ‘far field’
Stationary phase approximation at long times
N
1 N/2 X 0 Y i ( 2t ξj2 −ξj yσ(j) )
Aσ
e
GN (x|y; t) →
2πit
σ∈SN
j=1
ξj = xj /t and A0σ denotes
Y
A0σ =
S(ξα − ξβ ) : xα < xβ but yσ(α) > yσ(β) .
Austen Lamacraft (University of Virginia)
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May 27th, 2011
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Going to the ‘far field’
Stationary phase approximation at long times
N
1 N/2 X 0 Y i ( 2t ξj2 −ξj yσ(j) )
Aσ
e
GN (x|y; t) →
2πit
σ∈SN
j=1
ξj = xj /t and A0σ denotes
Y
A0σ =
S(ξα − ξβ ) : xα < xβ but yσ(α) > yσ(β) .
Austen Lamacraft (University of Virginia)
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May 27th, 2011
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Density correlations
Require the ‘forward and back’ propagator
Y
1 N X
∗
GN (x|y; t)GN (x|ỹ; t) →
A0σ1 A0∗
e −iξj (yσ1 (j) −ỹσ2 (j) ) .
σ2
2πt
σ1 ,σ2 ∈SN
j
Y
A0σ1 A0∗
{S(ξα − ξβ ) : σ1 (α) > σ1 (β) but σ2 (α) < σ2 (β)}
σ2 =
Austen Lamacraft (University of Virginia)
Two problems
May 27th, 2011
27 / 31
Density correlations
Require the ‘forward and back’ propagator
Y
1 N X
∗
GN (x|y; t)GN (x|ỹ; t) →
A0σ1 A0∗
e −iξj (yσ1 (j) −ỹσ2 (j) ) .
σ2
2πt
σ1 ,σ2 ∈SN
j
Y
A0σ1 A0∗
{S(ξα − ξβ ) : σ1 (α) > σ1 (β) but σ2 (α) < σ2 (β)}
σ2 =
Form of the product does not depend upon the ordering of the {xj }.
yσ1 (j)
Forward
=
xj
Back
yσ2 (j)
Austen Lamacraft (University of Virginia)
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May 27th, 2011
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The Golden Rule
How to calculate the integrals
Z
C(x1 , x2 ; t) ≡ dx3 · · · dxN |ΨN (x1 , x2 , . . . , xN ; t)|2
Use the Golden Rule!
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The Golden Rule
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The Golden Rule
‘Never impose on others what you would not choose for yourself.’
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The Golden Rule
Golden Rule (from analyticity of S matrix)
For those xj we integrate over:
A particle moving to the left (right) must be overtaken by another particle
moving to the left (right)
yσ1 (1)
yσ1 (α)
yσ1 (2)
ξ1
ξ2
ỹσ2 (2)
ỹσ2 (α)
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ỹσ2 (1)
May 27th, 2011
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Usual HBT trajectories
yσ1 (1)
yσ1 (α)
ξ1
ỹσ2 (2)
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yσ1 (2)
ξ2
ỹσ2 (α)
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ỹσ2 (1)
May 27th, 2011
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Usual HBT trajectories
yσ1 (1)
yσ1 (α)
ξ1
ỹσ2 (2)
yσ1 (2)
ξ2
ỹσ2 (α)
ỹσ2 (1)
Integrate over {xα : α 6= 1, 2}, convolve with Gaussians and sum series
"
`2 −`2 (ξ12 +ξ22 )
2
C(x1 , x2 : t) → 2 e
1 + Re
πt
N
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S(ξ2 − ξ1 )e i∆(ξ1 −ξ2 )
1 − e i∆(ξ1 −ξ2 ) ζ(ξ1 , ξ2 )
May 27th, 2011
!#
29 / 31
Usual HBT trajectories
yσ1 (1)
yσ1 (α)
ξ1
ỹσ2 (2)
yσ1 (2)
ξ2
ỹσ2 (α)
ỹσ2 (1)
Integrate over {xα : α 6= 1, 2}, convolve with Gaussians and sum series
"
`2 −`2 (ξ12 +ξ22 )
2
C(x1 , x2 : t) → 2 e
1 + Re
πt
N
S(ξ2 − ξ1 )e i∆(ξ1 −ξ2 )
1 − e i∆(ξ1 −ξ2 ) ζ(ξ1 , ξ2 )
!#
Fano with width Γn = 2(1 − Re ζ) > 0
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What about ‘non HBT’ trajectories?
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What about ‘non HBT’ trajectories?
Small in e −2c∆ , where c∆ = γ of equivalent LL gas
Austen Lamacraft (University of Virginia)
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May 27th, 2011
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What about ‘non HBT’ trajectories?
Small in e −2c∆ , where c∆ = γ of equivalent LL gas
In the same way, show that the power of e −c∆ is at least twice the
total number of moves to the right (or to the left).
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Conclusions
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Conclusions
1
Weak breaking of integrability can lead to anomalously large transport
coefficients, calculable in a simple model
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Conclusions
1
Weak breaking of integrability can lead to anomalously large transport
coefficients, calculable in a simple model
2
Integrability is useful simply at the level of the scattering problem in
certain natural situations in ultracold physics
Example of HBT effect in interacting 1D Bose gas
Crossover from bosonic to fermionic HBT with increasing interaction
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