Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
From GPE to KPZ:
Finite temperature dynamical structure factor
of the 1D Bose gas
Aspen Winter Conference 2012
Austen Lamacraft (Virginia)
Manas Kulkarni (Toronto)
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Dynamical structure factor and inelastic scattering
S(k, ω) ∝ σ(k, ω) energy (~ω) and momentum (~k) resolved
e.g. Brillouin scattering
S(k, ω) measures dynamical density fluctuations in a system
Z
S(k, ω) = dx dt e i(ωt−kx) hρ(x, t)ρ(0, 0)i
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
S(k, ω) measurements in ultracold physics
http://quantumgases.lens.unifi.it
Present status - collective excitations exist and disperse
What else?
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
S(k, ω) of a classical liquid
ω = ck
S(k, ω)
Rayleigh ∝ Dk 2
Brillouin ∝ Γk 2
ω
effects influence the a phonon could decay int
xcitations Phonon nonlinearities and decay
ty of fluid
A
B
pply a unitransforExact
hf1
d the new
generates
Hydrodynamic picture
hf0
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Detection probability
Outline
gent quanhf2
unique to
Elementary decay process
y encount the interPicture fine in higher dimensions (e.g. Beliaev damping)
Following
theinfate
of phonons.
of interactions
1D spoils
GR calculationOne measu
anicsResonant
and nature
they move through it. (A) A diagram illustrati
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Recent experiments
Fabbri et al. Phys. Rev. A 79, 043623 (2009)
No measurements of S(k, ω) in phonon regime k .
√
2µm/~?
Hydrodynamic
picture(SiN). The
Relation
(noisy)
Burgers
equation
KPZ
Numerical simulations
nitride
wafertowas
epoxy
sealed
on and
a sample
P isOutline
provided across
the
holder [see Fig. 1(a) middle piece] that was subsequently
ent relationship is analoinserted into the body of the experimental cell. Both the lid
wing in a resistor with a
[top part of cell in Fig. 1(a)] and the sample holder were
ge drop is supplied across
sealed with soft indium o-rings to protect against leaks to
al differences in the nathe outside and leaks around the holder, respectively. The
e reservoir,
i.e., electrons
Helium
in nanopores
drain pressure below the membrane (PD ) is kept at vacuum
in a fluidic system are
sport processes are very
ces to the measurement
ductance of the channel.
ment of the gas flow connometric size. A direct
1] is made whose crosshen the mean free path of
meter, as expected.
ctance remains well delarge as a macroscopic
nanofabricated channel.
microfluidics [2] and the
for molecular detection
f the transport properties
es more accessible. With
ents involving quantum
[10], most experimental
stricted to either a single
o porous membranes with
the very large number of
FIG. 1 (color). (a) Experimental cell for the mass flow conchannels=cm2 ) [1,12,13].
ductance experiment. Gas inlet and outlet are connected through
conductance, for a single
Savard
etwith
al. conductance
Phys. Rev.
Lett.
GD . (b)103,
TEM 104502 (2009)
stainless steel
capillaries
GS and
t out between these two
image of the nanopore with diameter 101 $ 2 nm. (c) Cartoon of
Recent experiments
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
From Gross–Pitaevskii to fluid dynamics
i∂t Ψ = −
Writing Ψ =
√
1 2
∂ Ψ + g |Ψ|2 Ψ
2m x
ρe iθ gives
∂x (ρ∂x θ)
=0
m
√
1 ∂x2 ρ
1
2
(∂x θ) = −g ρ +
θ̇ +
√
2m
2m
ρ
∂t ρ +
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
From Gross–Pitaevskii to fluid dynamics
i∂t Ψ = −
Writing Ψ =
√
1 2
∂ Ψ + g |Ψ|2 Ψ
2m x
ρe iθ gives
∂t ρ + ∂x (ρv ) = 0
‘Quantum Pressure’
mρ (∂t v + v ∂x v ) = −∂x
}|
z
2 √ {
2
∂x ρ
gρ
1
+
ρ∂x
√
2
2m
ρ
v = ∂x θ/m
Continuity and Euler equations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Linear oscillations
Write ρ = ρ0 + % and linearize in %, v
ρ0 2
∂ θ=0
m x
∂2%
θ̇ = −g % + x
4ρ0
%̇ +
Oscillations with ω = ±Ek
s
Ek =
k2
2m
k2
+ 2ρ0 g
2m
Ek = ck + O(k 3 ),
Bogoliubov dispersion
r
g ρ0
c=
m
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Hydrodynamic equations
Throw away quantum pressure term, but retain nonlinearity
∂t ρ + ∂x (ρv ) = 0
mρ (v̇ + v ∂x v ) = −∂x
g ρ2
2
Can be put in Riemann form
∂t (v ± 2cρ ) + v± ∂x (v ± 2cρ ) = 0,
p
v± = v ± cρ , and cρ = c ρ/ρ0 is local speed of sound
Meaning: v ± 2cρ constant along characteristics X+ (t)
(Ẋ+ (t) = v+ (X+ (t), t))
Characteristic = trajectory of phonon wavepacket
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Motion of a phonon wavepacket
Alternate form
1
∂t v± + v± ∂x v± = (∂t + v± ∂x )v∓
3
Characteristics curved due to (random) counterpropagating waves
Origin of phonon lifetime!
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Aside: the 1D Fermi gas
What about a Fermi gas, with Hamiltonian density
Fermi pressure
2
HFermi
ρ (∂x θ)
=
+
2m
z }| {
π 2 ρ3
6m
?
Yields the uncoupled Burgers equations
∂t v± + v± ∂x v± = 0
v± = v ± πρ/m are the right and left moving Fermi velocities.
Reflects description in terms of noninteracting Fermions!
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Aside: the 1D Fermi gas
Characteristics
are straight
lines
CHAPTER
7. THE
MATHEMATICS OF REAL
t
?
P
x
waveswave
inevitable!
Figure 7.10: Shock
Simple
characteristics.
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
The Kardar–Parisi–Zhang (KPZ) equation
Describes a growing interface of height h(x, t)
Nonlinear growth
∂t h =
Noise
Smooths
z }| {
z }| {
z }| { √
λ www.nature.com/scientificreports
(∂x h)2 + ν∂x2 h + Dη
2
(a) and flat (b) interface. Binarised snapshots at successive times are shown with different colours.
Takeuchi
etx
the laser emission. The local height h(x, t) is defined in each case as a function of the lateral
coordinate
al. (2011)
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ and the Burgers equation
∂t h =
√
λ
(∂x h)2 + ν∂x2 h + Dη
2
Take λ = 1, v = −∂x h
∂t v + v ∂x v = ν∂x2 v +
√
D∂x η
Noisy Burgers equation
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ and the Burgers equation
∂t h =
√
λ
(∂x h)2 + ν∂x2 h + Dη
2
Take λ = 1, v = −∂x h
∂t v + v ∂x v = ν∂x2 v +
√
D∂x η
Noisy Burgers equation
Recall GPE hydrodynamics
1
∂t v± + v± ∂x v± = (∂t + v± ∂x )v∓
3
= (∂t + v± ∂x )Noise
Suggests close relation
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ scaling
KPZ-type problems have dynamic critical exponent z = 3/2
[Length] ∼ [Time]1/z ∼ [Time]2/3
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ scaling
KPZ-type problems have dynamic critical exponent z = 3/2
[Length] ∼ [Time]1/z ∼ [Time]2/3
L ∼ T 2/3
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ scaling
KPZ-type problems have dynamic critical exponent z = 3/2
[Length] ∼ [Time]1/z ∼ [Time]2/3
L ∼ T 2/3
Notice that this is faster than diffusive, a result of nonlinearity
∂t h =
√
λ
(∂x h)2 + ν∂x2 h + Dη
2
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Examples of KPZ scaling
Few but varied observations
• Bacterial colony growth
• Burning paper
• Liquid crystal interfaces
Wakita et al. (1997)
Maunuksela et al. (1997)
Takeuchi et al (2010)
(Nematic interface)
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
KPZ and phonon lifetime
KPZ describes phonon dynamics in comoving frame
Suggests scaling form for Brillouin peaks1
ω ± c|k|
1
(±)
fPS
Sphonon (k, ω) ∝
Γk
Γk
Γk ∝ |k|3/2
fPS (x) is scaling function for slope fluctuations in KPZ
Prähofer & Spohn (2004)
e.g. fPS (x) → |x|−7/3 ,
|x| → ∞
c.f. x −2 tail for Lorentzian peak
1
van Beijeren arXiv:1106.3298
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Origin of z = 3/2
• Consider noisy Burgers equation
∂t v + v ∂x v = ν∂x2 v +
√
D∂x η
• Scaling must preserve Galilean invariance, leading to identical
scaling of two terms on RHS
• Scale x → λx, t → λz t
• Scaling preserves equilibrium at fixed temperature:
R
ρ0
dx v 2 const. so v → λ−1/2 v , and
z = 1 + 1/2 = 3/2
Forster, Nelson, and Stephen (1977)
For
a finite tunnelRelation
coupling
Methods)
two
sysisolated
1D Bose gases
are
Outline
Hydrodynamic
picture
to (see
(noisy)
Burgersbetween
equationthe
and
KPZ
tems, we also observe an increase in the waviness of the interference.
xponential coherence decay,
12
However, in contrast to the completely separated case, the final equiredictions . For two coupled
librium state shows a non-random phase distribution (Fig. 2c, d).
observed to approach a non13
This is caused by the phase randomization being counterbalanced by
by a Bogoliubov approach .
the coherent particle exchange between the two fractions. The final
coherence is the matter wave
width of the observed phase spread depends on the strength of the
sers by injection. The nontunnel coupling22,23.
s has an important role in a
as superconductors, quantum
spin systems14–16. Our experishow that 1D Bose gases are
lass of phenomena.
nts is a 1D Bose gas of a few
d
ongated, cylindrical magnetic
typical transverse and longit< 4.0 kHz and nz < 5 Hz. Our
condensate regime1, which is
e T and chemical potential m
Numerical simulations
Application of KPZ scaling: condensate dephasing
ingle 1D system, we perform a
nsverse direction by means of a
atic potential11. As shown in
wo 1D quasi-condensates in a
ntial20. They are separated by
of which is controlled by the
ess initializes the system in a
splitting, the phase fluctuation
ndensates are identical, resulte. This is a highly non-equilibwill relax to equilibrium over
phase coherence, the two 1D
ll configuration for a varying
recombined during the timepattern is recorded using
y
L
x
z
Figure 1 | Schematic of the experiment. A single 1D quasi-condensate is
phase coherently split into two parts separated by distance d using r.f.
potentials on an atom chip (top). A combination of two r.f. fields allows
balanced splitting in the vertical direction20, as indicated in the figure. After
the separation, the system is held in the double-well potential for a variable
time t and is then released from the trap. The resulting interference pattern
(centre) is imaged along the transverse direction of the system onto a CCD
camera (right). Thermal phase fluctuations in the two quasi-condensates can
Hofferberth et al. (2007)
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Application of KPZ scaling: condensate dephasing
Measure coherence
1
C(t) ≡ Re
L
Z
dx he i[θ1 (x,t)−θ2 (x,t)] i
Phase analogous to height h(x, t) in KPZ problem
θ1 (x, t) − θ2 (x, t) ∼ t 1/3 χx
χx random variable
Thus
h
i
C(t) ∼ exp − (t/t0 )2/3
Burkov, Lukin, Demler (2007)
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
GPE simulations
Classical treatment OK for ρ0 ξ −1 = (gmρ0 )1/2
i.e. Luttinger parameter K ≡
Populate modes according to equipartition2
r
ρ0 X −κk e
%(x) =
bk e ikx + c.c
2L
k6=0
X
i
θ(x) = √
e κk bk e ikx − c.c .
2ρ0 L k6=0
bk complex Gaussian random variables with h|bk |2 i =
2
Assumes nonlinearity sufficiently weak
T
Ek
πρ0
mc
1
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Numerical simulations
GPE simulations
At each time step find Fourier components of density ρ(x, t)
ρk (t) =
N−1
X
|Ψ(na, t)|2 e −2πikna
k = 0,
n=0
2π
π
,..., .
L
a
Resulting time series is Fourier transformed to give S(k, ω)
S(k, ω) = h|ρk,ω |2 i
Average over ∼ 128 runs
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Results for S(k, ω)
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Results for S(k, ω)
(±)
Sphonon (k, ω)
1
∝
fPS
Γk
ω ± c|k|
Γk
Numerical simulations
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Results for S(k, ω)
z = 1.510 ± 0.018
Numerical simulations
19
Outline
Hydrodynamic picture
Relation to (noisy) Burgers equation and KPZ
Conclusions
Cottrell Schola
Numerical simulations
26
Additional Pro
28
Advisory Comm
30
Financials
32
Officers, Board
• Real systems always finite T and (often) weakly interacting!
Let’s make a virtue of these features
• In 1D quantum fluids phonon nonlinearities play a vital role
• These systems offer potential to study KPZ scaling