David Roberts- Casimir-like drag in slow-moving Bose-Einstein condensates

8/3/2019 David Roberts- Casimir-like drag in slow-moving Bose-Einstein condensates

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Casimir-like drag

in slow-moving

Bose-Einstein condensates

David RobertsENS, Paris

August 2005



Is there dissipationless flow around

a weak perturbation as 0 ?

Object in a moving superfluid

(dilute BEC) flow at T=0



Is there dissipationless flow arounda weak perturbation as 0 ?

• Why do we care?

Superfluidity

• What is known?

Results from experiments & mean field theory

• What is missing?

Drag arising from quantum fluctuations



Why do we care?

Superfluidity

• etc.

• Dissipationless flow for at T=0 Non-zero critical velocity

Experimental manifestations

• Second sound, collective excitations

• Persistent currents, Hess-Fairbank effect• Quantized vortices

Theory

• BEC (London, 1938)

– Macroscopically occupied mode

– Long range order

– Order parameter =>

• However, 2-fluid model by Landau



Landau theory predictsdissipationless flow if

Landau criterion:

Dissipationless flow at T = 0 ??

Excitation spectrum

Non-zero critical velocity for dilute BEC(measured by

ion mobility)

Crucial Point::Assuming dissipation is caused

by the creation of excitations

at T = 0



Are alkali condensates superfluid?

Image: Mike Matthews, JILA research team

• Coherence (MIT, 1997 )

• Collective oscillations (phonons at low ) ( Boulder, MIT, 1996 )

• Non-classical moment of inertia (e.g. scissor modes) (Oxford, 1999)

• Quantized vortices ( ENS, Boulder, etc.)

• Dissipationless flow ( MIT, 1999, 2000)



What is known experimentally?Experimental evidence for dissipationless flow in dilute BECs

Impurity collisions:

A.P. Chikkatur et. al. PRL 85, 483 (2000)

Moving macroscopicobject (blue-detunedlaser) throughcondensate:

asymmetry(drag force)

heatingrate

R. Onofrio et. al. PRL 85, 2228 (2000)



So… Object in a slow-movingsuperfluid flow at T=0

Is there dissipationless flow around

a weak perturbation as

0 ?



Theory

Drag force:

zero populationof q.p’s

describes

BEC flow

describesstationary

object

breakstranslational

symmetry



Mean field calculation

order parameter symmetric

=> density asymmetry in

leads to drag force

where

scaled by healing length

Galilean transformationfor moving flow

Note:

scattering length

(repulsive interactions)

Notes: - Flow is in x-direction- is flow velocity far from potential

GPE:



Numerical simulations of macroscopic objects• Vortex shedding

–in 2-d, ( Frisch, Pomeau, Rica, 1992;Winiecki, McCann, Adams, 1999; Huepe, Brachet, 1997 )

–in 3-d, ( Adams et al )

–Complications:• edge effects ( Fedichev & Schlyapnikov)

• vortex stretching ( Brachet et al )• etc.

Note: all velocities at infinity

• Drag on weak repulsive impurity (linear analysis)3-d, 2-d; ( Astrakharchik and Pitaevskii, 2004)

• Repulsive potential1-d; (creation of gray solitons – Hakim, 1997; Pavloff, 2002

What is known theoretically?Some drag results from GPE



Lesson from GPE

If max. local fluid velocity > dissipation/drag(non-linear effects: vortex shedding, etc.)

Local adjustment:

Recall d’Alembert’s paradox in a potential flow

GPE (mean field)

Non-zero modified critical velocity

Dissipationless flow exists

(at level of mean field approx.)



q.f’s scale as [<< 1 for (typical) alkali condensates]

can be ignored for many purposes

e.g. density profile, collective oscillations,

interference, quantized vortices, etc.

order parameter modified by fluctuations

What is missing?GPE (mean field) ignores quantum fluctuations

But q.fs have indirect experimental consequences –shift of collective frequencies (Pitaevskii, Stringari, 1999)

–suppression of density fluctuations in the phonon regime (Ketterle, 1999)

–quantum phase transition in an optical lattice (Bloch, 2002)

–source for entangled atoms (Ketterle, 2002) …



Can quantum fluctuations produce a force?

EM vacuum(Casimir, 1948)

EM: Perfect

conductors

A

Putting boundary conditions on (static) EM vacuum => force



Analogy: French fleet, 1700s

• Destructive interference between boats - calm• Force from differences in energy densities



Static Casimir force in a BEC

Quasiparticle

vacuum in BEC

Putting boundary conditions on (static) q.p. vacuum => force

BEC: Infinitely thin

and repulsive walls

A

1010 smaller than for EM

C i i f i EM d BEC



Casimir force in EM and BEC:Physical manifestation of q.fs

Putting boundary conditions on (static) vacuum => force

dominated by low k part of energy spectrum

speed of light polarizations speed of

sound polarizations

EM vacuum

(Casimir, 1948)

Quasiparticle vacuum

in BEC

C t fl t ti



Can quantum fluctuations

produce a drag force at ?Boundary conditions

on vacuum

( )( )

non-local perturbation

=> drag force?



Calculation of Casimir-like drag

Fluctuations with non-uniform medium (object)

f cond: Force from condensate

modified by fluctuations

measure of density asymmetry symmetric

f fluc: Force

from fluctuations

Reminder:



Expand in terms of q.p. operators

Weakly interacting particles

Ignoring quasiparticleinteractions (Beliaev/Landau terms)

Non-interacting q.p.:

Quantum depletion/fluctuations

for uniform gases

Definition of T=0:(zero population of q.p.)

Note: Easily modified for finite T

B-E distribution function:

Equations governing quantum fluctuations



Equations governing quantum fluctuations(non-uniform medium)

Bogoliubov-de Gennes equations:

Normalization:

moving BEC

q.p’s obey bosoniccommutation relations

Generalized GPE (Castin and Dum, 1998):

UV divergent because of contactapproximation (need to renormalize)

Ensures orthogonality between excited statesand condensate

In general acts as effective complex potential

=> mass transfer between condensate and fluctuations

(T=0)

2 l f d f i



2 examples of drag force in a

3-D condensate, Boundary Conditions specified

Force well defined

1) Localized and varying only in flow direction:

2) Point impurity:

1) Potential locali ed in flo direction



1) Potential localized in flow direction

Restriction on : weak and 1-d

where (1st Born approx.)

Assume assures convergence of perturbation series inBogoliubov equations

Boundary conditions atOnly outgoing finite scattered waves exist far from potential

Determined by group velocityof reflected wave:

wave number of

reflected wave

Note: Flow is in x-direction



Expression for drag force

all q.fs in 3-d

zeroth order interaction pressure:(mean field density)

cross-sectional area

A l ti i f f( )



Analytic expressions for f( )

Reminder:

Similarly for from BCs at

Ch f ll i



Choose following

Localized in x-directionRestrictions:Limits:

Non-localized limit(scattering problem ceases to make sense)

( )(normalization and convergenceof perturbation series)

Localized limit

(Grand canonical) energy



(Grand canonical) energy

of the flow in the presence of

relative to uniform flowdecreases as increases

=> negative kinetic mass

An analogy gray soliton behavior



An analogy - gray soliton behavior

Rest frame of soliton

• Width is amplified by• Density modulation corresponds to hole

GPE with

=> (Tsuzuki, 1971)

Grand canonical energy(relative to uniform flow)

disappearsat

⇒ In presence of dissipation soliton accelerates to cs and disappears

⇒ negative kinetic mass ( Fedichev, Muryshev and Shlyapnikov, PRA, 1999)

h i f f( )



Behavior of f( )

Dynamical modulation instability at long wavelengthsGPE + lattice => (perfect lattice)

Note:

References

Theory: Smerzi et al, PRL, 2002

Wu and Niu, PRA, 2001Experiment: Inguscio et al, 2004

Experimentally measured parameters



Experimentally measured parameters

densityasymmetry:

Typical experimental parameters:

Heating rate (energy transfer per atom):

typical condensate length

Resolution achieved in MIT experiment is 10nK/s

to detect Casimir drag effect

Interpretation



InterpretationStates at -∞, ∞:

In principle, assuming local thermal equilibrium (higher order processes), can define by calculating

using distribution function:

normal fluid

(in rest frame of superfluid)

unperturbed reflectioncoefficient

Scattered fluctuations

– create ρn far from potential

=>

2) Point impurity in a 3 D condensate



2) Point impurity in a 3-D condensate

where

=>

uv cutoff:

scattering length characterizing

particle-impurity interactions

• Exists at all velocities

• Dominant effect at

Summary



SummaryIs BEC always dissipationless as v 0 ?

⇒ NO• Showed 2 examples:

Weakly interacting 3-D BEC where vc = 0

• Consistent with experiments - semblance of vc as the dominantmean field effect

• However, Casimir-like drag dominant term v < vc and potentially

measurable• Outlook:

– in principle, any moving density perturbation (e.g. vortices,solitons, impurities, laser, etc.) will also experience Casimir-likedrag

– Amplified with (quasi) condensates of lower D

– Liquid Helium (rough surface momentum transfer between q.f.

and walls)

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David Roberts- Casimir-like drag in slow-moving Bose-Einstein condensates