一次元量子気体の二重極振動の減衰における量子位相滑り
段下 一平
京大基研
I. Danshita, Phys. Rev. Lett. 111, 025303 (2013)
熱場の量子論とその応用
京都大学基礎物理学研究所
2013年8月27日

Outline:Damping rate ⇔ Nucleation rate

1. Introduction

2. Hand-waving picture
3. Numerical corroboration in the hardcore boson limit
4. Mechanism of the damping in softcore bosons

5. Finite temperature effects
6. Conclusions

Special thanks to Nikolay Prokof’ev (UMass Amherst)

1. Introduction

1.1. Experiments on transport of 1D lattice bosons
A 2D array of 1D Bose gases
H. Moritz et al., PRL (2003)

BEC

Strongthermal &quantum
fluctuations
Dipole oscillations:Moving optical lattices:
Bose gas
Velocity: v

ETH: T. Stöferle et al., PRL (2004)
NIST: C. D. Fertig et al., PRL (2005)
Innsbruck: E. Haller et al., Nature (2010)
Stony Brook: B. Gadway et al., PRL (2011)

MIT: J. Mun et al., PRL (2007)

NIST: C. D. Fertig et al., PRL (2005)

Center of mass velocity (mm/s)

1.2. Damped dipole-oscillation

Lattice depth:

V0 = 0 ER
V0 = 0.25 ER

V0 = 0.5 ER
V0 = 2.0 ER
Time (ms)・ They observed a dissipative flow (significant damping) even though
the flow velocity is much smaller than the critical value predictedby the Gutzwiller mean-field theory.

Maximum velocity

MF critical velocity for V0 = 2.0 ER

・ The damping rate rapidly increases when deepening the lattice.
Interpretation by Polkovnikov et al. PRA (2005):

This breakdown of superfluidity is due to phase slips via
thermal activation or quantum tunneling.

1.3. Superflow decay via phase slips

N. Giordano, PRL (1988)

Free energy: F(p)A supercurrent is flowing

A ring lattice

The flow momentum
is quantized asn: winding number
L: # of lattice sites,d: lattice spacing.

pA state with finite flow velocity can be metastable.
The momentum at which δF reaches zero.
Strong thermal or quantum fluctuations

Persistent current

Superfluid critical momentum
Superflow decay via phase slips

Phase-slip nucleation rate:
ThermalCritical momentum: pcQuantum

1.4. Other 1D superfluids (superconductors)
Dissipation

Liquid 4He in nanopores
R. Toda et al., PRL (2007)
J. Taniguchi et al., PRB (2010)
T. Eggel et al., PRL (2011)
etc.

Superconducting nanowires
and nanotubes

TcResistence

A. Bezryadin et al., Nature (2000)
F. Altomare et al., PRL (2006)
K. Yu. Arutyunov et al., Phys. Rep. (2008)
M. Kociak et al., PRL (2001)
Z. Wang et al., Nanoscience (2012)etc.

The concept of phase slip is central also to the understanding of 1D
superfluids (superconductors) in these condensed matter systems.
QPS in ultracold atoms

・ Unified view of 1D superfluids
・ Study of QPS in a highly controllable manner1.5. What we do in this work
A ring latticeDamping rate：G？？？

Supercurrent
is flowing

Phase-slip nucleation rate: Γ・ We find the relation between G and Γ:・Using the relation, we elucidate the mechanism of the damping at T=0.
Damping rate

・ Universal damping behavior:

where vc is the MF
critical velocity

Flow velocity

2. Hand-waving picture
・ Relation between the nucleation rate Γ and the damping rate GThe definition of the damping rate:

The maximum velocity:

Potential energy:(a) The lost potential energy

At t=0Ax0
At t=1

Ax1
Potential energy:2. Hand-waving picture
・ Relation between the nucleation rate Γ and the damping rate G
The definition of the damping rate:

The maximum velocity:(a) The lost potential energy(a) & (b)lead toDamping rate:(b) The Joule heat

Nucleation rate
Flow velocity

3. Numerical corroboration of the relation

3.1. 1D hardcore Bose-Hubbard model with a single barrier potential: Hopping energy,: Nearest neighbor interaction,: Curvature of the trapping potential,: Strength of the barrier potential: Displacement of the trap center,Advantages:・ Nucleation rate of a quantum phase slip at T=0:&(λ<<J) Yu. Kagan et al., PRA (2000)(λ>>J) H. P. Büchler et al., PRL (2001)

TL parameter: K

・ The Tomonaga-Luttinger (TL) parameter Kat ν=0.5 is related to V/J ase.g. M. Cazalilla et al., RMP (2011)

when・ This model is numerically solvable with TEBD
[G. Vidal, PRL (2004)], which can precisely
capture quantum phase slips.
I. Danshita & A. Polkovnikov, PRB (2010); PRA (2012)

SuperfluidV/J

3.2. Dipole oscillation and damping rate
Time evolution of the COM position:
0.3

potential

0.2

t=0
0.1

density0,,,※ The trap curvature is set such that
Assuming the under-damped oscillation,
we define the damping rate asSingle impurity potential
Florence: J. Catani et al., PRA (2012)
We expect∵3.3. The damping rate vs the velocity (hardcore BHM)
・ In the fitting region,
the damping rate obeys

・ The power η
hardly changes
for different λ .

????

Four conditions forthe fitting region:i)iii)(thin solid),(dashed),ii)(thick solid)iv)(dotted)3.4. The exponent vs V/J (hardcore BHM)

Using the fitting function,

we extract the exponent η.

The damping rate obeys
the following scaling formula:∴ The relation has been corroborated.

4. Mechanism of the damping in a 1D Bose gas
in an optical lattice4.1. The softcore Bose-Hubbard model with no barrier potential
which corresponds to actual experiments.

0.75
0.5

0.25

0The density profile is smooth.
The trap curvature Ω is set such that・ The damping rate G obeys the power-law
with respect to the momentum p.
※ Previous works have studied dipole oscillations of the same model with different parameters:
I. Danshita and C. W. Clark, PRL (2009); S. Montangero et al., PRA (2009)

4.2. The exponent vs U/J (softcore BHM)
The TL parameter is extracted fromthe one-body density matrix
numerically calculated with TEBD.

(periodic potential)(single barrier)

The damping rate obeys the scaling formulafor the quantum-phase slips in the presence ofa single impurity :Yu. Kagan et al., PRA (2000)
H. P. Büchler et al., PRL (2001)

rather than that for a periodic potential
at a commensurate filling:
I. Danshita & A. Polkovnikov, PRA (2012)

4.3. Effective impurities

Unit filling points
Transport in the regions near the unit filling points is much more suppressedthan in the other regions.
The unit-filling regions act as barrier potentials for the other partsof the gas.
The damping rate obeys the scaling formula for a single impurity.

5. Finite temperature effects

QPS rate is taken from
Kagan et al. PRA (2000)

B.
Thermallyassisted
quantumtunnelingCrossoverA.
Quantumtunneling

Crossover

5.1. QPS at finite temperatures

C.
Thermalactivation

where

Regime A

Tunneling fromthe lowest state

QPS rate is taken from
Kagan et al. PRA (2000)

B.
Thermallyassisted
quantumtunneling

CrossoverA.
Quantumtunneling

Crossover

5.1. QPS at finite temperatures

C.
Thermalactivation

where

Regime B

Tunneling frommany states areaveraged.

QPS rate is taken from
Kagan et al. PRA (2000)

B.
Thermallyassisted
quantumtunneling

CrossoverA.
Quantumtunneling

Crossover

5.1. QPS at finite temperatures

C.
Thermalactivation

whereThe crossover can be achievedby changing the velocity.

Regime B

In the experiment of Fertig et al., PRL (2005),
andTunneling frommany states areaveraged.

whilein typical experiments.

Damping rate

5.2. Universal damping behavior
B.
Thermally
assisted
quantumtunnelingA.
Purelyquantumtunneling

Flow velocity
This allows for the determination of K.

Mott transition point (K=2)

The same behavior may be seen universally in systems whose low-energy
physics can be described as the single-component TL liquid.
Note: vc is the mean-field critical velocity and EJ is the Josephson plasma energy.

5.3. Implication to a disorder potential
Disorder potential
Stony Brook: B. Gadway et al., PRL (2011)

In the case of a weak disorder,
Regime A

The same universal damping behavior

Regime B→ Localization transition of theGiamarchi-Schultz type (K=3/2)

S. Khlebnikov & L. P. Pryadko, PRL (2005)

In the case of a strong disorder,

The TEBD-based analyses may answer this question.
→ may also address the localization transition
in a strong disorder.

6. Conclusions
We have studied the transport of 1D Bose gases in strong connection
with quantum nucleation of phase slips.

・ We have found the relation between the damping rate G andthe phase-slip nucleation rate Γ:・ This relation allows to analyze QPS in cold atom experiments(and in the exact TEBD or tDMRG simulations).
・ We corroborate that the damping of the dipole oscillation of
1D lattice bosons is due to the nucleation of QPS.
・ We suggest that the damping rate vs the flow velocity exhibits
the universal behavior, which can be tested in future experiments.
・ Such experiments could be interpreted as a quantum simulationof 1D superfluids (or superconductors).

I. Danshita, Phys. Rev. Lett. 111, 025303 (2013)

3.2. Isn’t it due to trivial finite temperature effects ??
Well-known fact:
In 1D, the superfluid fraction ρs → 0 in the thermodynamic limit at any T > 0.
Naive guess: The system is not in the SF phase.
Large fraction
is condensed !!

, which is wrong.

MIT: J. Mun et al.,PRL (2007)

Bose gas

Condensate fraction

Momentum: p
Dissipationless (superfluid) flow
if the velocity is sufficiently small.
The system
is in the SF state.

1D Bose gases at finite temperatures exhibit superfluidity as long as
the “lifetime” of superflow is longer than the time scale in experiment.

Yu. Kagan et al.
PRA (2000)

Also in the context of liquid 4He: T. Eggel et al., PRL (2011)