dynamics of trapped bose and fermi gases sandro stringari university of trento ihp paris, june 2007...
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DYNAMICS OF TRAPPED BOSE AND FERMI GASES
Sandro Stringari
University of Trento
IHP Paris, June 2007
CNR-INFM
Lecture 1
Lecture 1:- Hydrodynamic theory of superfluid gases - expansion - collective oscillations and equation of state - collective vs s.p. excitations :Landau critical velocity - Fermi superfluids with unequal masses - example of non HD behavior: normal phase of spin polarized Fermi gas
Lecture 2:- Dynamics in rotating superfluid gases - Scissors mode - Expansion of rotating gas - vortex lattices and rotational hydrodynamics - Tkachenko oscillations
Understanding superfluid features requires theory for transport phenomena
(crucial interplay between dynamics and superfluidity)
Macroscopic dynamic phenomena in superfluids(expansion, collective oscillations, moment of inertia)are described by theory of irrotational hydrodynamics
More microscopic theories required to describe other superfluid phenomena (vortices, Landau critical velocity, pairing gap)
HYDRODYNAMIC THEORY OF SUPERFLUIDS
Basic assumptions:
- Irrotationality constraint (follows from the phase of order parameter)
- Conservation laws (equation of continuity, equation for the current)
Basic ingredient:
- Equation of state
- Equation for velocity field involves gradient of chemical potential; - Includes smoothly varying external field in local density approximation
0))(2
1(
0)(
2
extVnmvvt
m
vnnt
Hydrodynamic equationsof superfluids (T=0)Closed equations fordensity and superfluidvelocity field
)(rVext
HYDRODYNAMIC EQUATIONS AT ZERO TEMPERATURE
is atomic mass, is superfluid density is superfluid velocity, is chemical potential (equation of state) is external potential
mSnnSvv
extV
T=0
irrotationality
- Have classical form (do not depend on Planck constant) - Velocity field is irrotational- Should be distinguished from rotational hydrodynamics.- Applicable to low energy, macroscopic, phenomena - Hold for both Bose and Fermi superfluids- Depend on equation of state (sensitive to quantum correlations, statistics, dimensionality, ...) - Equilibrium solutions (v=0) consistent with LDA
)(n
KEY FEATURES OF HD EQUATIONS OF SUPERFLUIDS
What do we mean by macroscopic, low energy phenomena ?
BEC superfluids
gn
mgn
2
BCS Fermi superfluids
Fv more restrictive than in BEC
0)()( rVn ext
superfluid gap
healing length
size of Cooper pairs
WHAT ARE THE HYDRODYNAMIC EQUATIONS USEFUL FOR ?
They provide quantitative predictions for
- Expansion of the gas follwowing sudden release of the trap - Collective oscillations excited by modulating harmonic trap
Quantities of highest interest from boththeoretical and experimental point of view
- Expansion provides information on release energy, sensitive to anisotropy
- Collective frequencies are measurable with highest precision and can provide accurate test of equation of state
-Initially the gas is confined in anisotropic trap in situ density profile is anisotropic too
- What happens after release of the trap ?
2/)( 2222 rzmV zext
Non interacting gas expands isotropically
- For large times density distribution become isotropic
- Consequence of isotropy of momentum distribution
- holds for ideal Fermi gas and ideal Bose gas above
- Ideal BEC gas expands anysotropically because momentum distribution of condensate is anysotropic
CT
Expansion from anysotropic trap
- Expansion inverts deformation of density distribution, being faster in the direction of larger density gradients (radial direction in cigar traps)- expansion transforms cigar into pancake, and viceversa.
Superfluids expand anysotropically
Hydrodynamic equations can be solved after switching off the external potential
For polytropic equation of state (holding for unitary Fermi gas ( ) and in BEC gases ( )), and harmonic trapping, scaling solutions are available in the form(Castin and Dum 1996; Kagan et al. 1996)
222
01
2
1),(
)/,/,/()(),,,(
zyxtrv
bzbybxnbbbtzyxn
zyx
zyxzyx
with the scaling factors satisfying the equation
and
)(
12
zyxiii bbbb
b
n3/2 1
ib
iiii bbtb /,1)0(
Experiments probe HD nature of theexpansion with high accuracy. Aspect ratio
HDHD
ideal gas ideal gas
ZR /
Expansion of BEC gases
First experimental evidence for hydrodynamic anisotropic expansion inultra cold Fermi gas (O’Hara et al, 2002)
Hydrodynamics predicts anisotropic expansion inFermi superfluids (Menotti et al,2002)
EXPANSION OF ULTRACOLD FERMI GAS
collisionless
HD theory
Measured aspect ratio after expansion along the BCS-BEC crossover of a Fermi gas
(R. Grimm et al., 2007)
prediction of HD at unitarity
prediction of HD for BEC
- Expansion follows HD behavior on BEC side of the resonance and at unitarity. - on BCS side it behaves more and more like in non interacting gas (asymptotic isotropy)
Explanation:
- on BCS side superfluid gap becomes soon exponentially small during the expansion and superfluidity is lost.
- At unitarity gap instead always remains of the order of Fermi energy and hence pairs are not easily broken during the expansion
Collective oscillations: unique tool to explore consequence of superfluidity and test the equation of state of interacting quantum gases (both Bose and Fermi)
Experimental data for collective frequencies are available with high precision
Collective oscillations in trapped gases
ST
ET
)1(4
)1(3
04
00
z
TkF Beq
1
1)(
6),,(
0
03
2220
czTTk
zTTF ESBzES
Asymptotic behavior of the force out of thermal equilibrium(Antezza et al. PRL 95 083202, 2005)
Lifschitz result (thermal equilibrium)
First measurement of thermal effect on the Casimir force(JILA, Obrecht et al. PRL 2007)
Recent example of high precision measurement
Relative frequency shift
Results for the frequency shift of the center of mass oscillation due to the Casimir-Polder force produced by a substrate at different temperatures
theory
Propagation of sound in elongated traps
-If wave length is larger than radial size of elongated trapped gas sound has 1D character
where and n is determined by TF eq.
one finds
In uniform medium HD theory gives trivial sound wave solution
with nmccq /; 2 )( tqzien
11121 / nnmc D
dxdyn
ndxdymc D 1
21
)/(
1)()( rVn ext
2/1 bulkD cc (Zaremba, 1998)For BEC gas ( )
5/31 bulkD cc
ndxdyn1
nFor unitary Fermi gas ( )
3/2n (Capuzzi et al, 2006)
Sound wave packets propagating in a BEC (Mit 97)
velocity of sound as a function of central density
Sound wave packets propagating in an Interacting Fermi gas (Duke, 2006)behavior along the crossover
BCS mean field
QMC
Difference bewteen BCS and QMC reflects: -at unitarity: different value of in eq. of state-On BEC side different dimer-dimer scattering length
3/2)1( n
Collective oscillations in harmonic trap
When wavelength is of the order of the size of the atomic cloud sound is no longer a useful concept. Solve linearized 3D HD equations
))((2
2
nn
nnt
-Surface modes are unaffected by equation of state
- For isotropic trap one finds where is angular momentum
- surface mode is driven by external potential, not by surface tension - Dispersion law differs from ideal gas value (interaction effect)
hol l
hol
Solutions of HD equations predictsurface and compression modes
Surface modes
well understood in atomic BEC Stringari,1996here we focus on Fermi gases
where is non uniform equilibrium Thomas Fermi profile
)(rn
Quadrupole mode measured on ultracold Fermi gas along the crossover (Altmeyer et al. 2007)
HD prediction
2
Ideal gas value
Enhancement of damping
Minimum damping near unitarity
- Experiments on collective oscillations show that on the BCS side of the resonance superfluidity is broken for relatively small values of (where gap is of the order of radial oscillator frequency)
- Deeper in BCS regime frequency takes collisionless value
- Damping is minimum near resonance
akF/1
Compression modes
- Sensitive to the equation of state
-analytic solutions for collective frequencies available for polytropic equation of state
- Example: radial compression mode in cigar trap
- At unitarity ( ) one predicts universal value
- For a BEC gas one finds
n
)1(2
83.13
10
2
3/2
Equation of state along BCS-BEC crossover
- Fixed Node Diffusion MC (Astrakharchick et al., 2004)- Comparison with mean field BCS theory ( - - - - - )
On BEC side of the resonance QMC confirms result for the dimer-dimer scattering length (Petrov et al., 2005)and points out occurrence of Lee-Huang-Yang correction in equation of state
aaM 6.0
])(3
321[2 2/132
MddMd anna
1 100.01
0.1
(E -
Eb)
/ (N
3/5
F)
-1 / kFa
Fixed Node Monte Carlo
Gross Pitaevskii (aM=0.6a)
Gross Pitaevskii (aM=2a)
Lee Huang Yang (aM=0.6a)
aaM 2
LHY
In harmonic trap LHY effect results in enhancement of radial compressionfrequency according to: Pitaevskii and Stringari (1998) Braaten and Pearson (1999)
])0(256
1051[2 3
dMna
w
Frequency of radialcompression mode in cigar trap
(Stringari 2004)
BEC beyond mean field unitarity
akF/1
2
0
))(094.01(2 5/60akF 3/10
- Collective frequency has non monotonic behaviour (effect missed by BCS theory, accounted for by MC)- can be evaluated numerically in whole regime starting from knowledge of equation of state
)/91.1( 6/10hoF aNk
Radial breathing mode at Innsbruck (Altmeyer et al., 2007)
MC equation of state (Astrakharchick et al., 2005)
BCS eq. of state(Hu et al., 2004)
83.13/10
includes beyond mf effects
does not includes beyond mf effects
universal value at unitarity
Measurement of collective frequencies provides accurate test of equation of state !!
akF0/1
- Accurate confirmation of the universal HD value predicted at unitarity.
- Accurate confirmation of QMC equation of state on the BEC side of the resonance.
- First evidence for Lee Huang Lee effect.
- LHY effect (and in general enhancement of compression frequency above the BEC value) very sensitive to thermal effects (thermal fluctuations prevail on quantum fluctuations except at very low temperature)
Main conclusions concerning the m=0 radial compression mode
3/10
Landau’s critical velocity
While in BEC gas sound velocity provides critical velocity,in a Fermi BCS superfluid critical velocity is fixed by pair breaking mechanisms (role of the gap)
Landau’s critical velocity
p
pv pcr
)(min
- Landau’s criterion for superfluidity (metastability): fluid moving with velocity smaller than critical velocity cannot decay (persistent current)
- Hydrodynamic theory cannot predict value of critical velocity Needed full description of dispersion law (beyond phonon regime)
- Ideal Bose gas and ideal Fermi gas one has
- In interacting Fermi gas one predicts two limiting cases:
Dispersion law of elementary excitations
BEC (Bogoliubov dispersion)
cvcr
BCS (role of the gap)
Fcr pv /(sound velocity)
0crv
Landau’s critical velocity ishighest near unitarity !!
BEC
resonance
Sound velocity
BCS BEC
B
F
cr
v
v
BCSBEC
Critical velocity
unitarity
gap gapgap
(Combescot, Kagan and Stringari 2006)
Dispersion law along BCS-BEC crossover of Fermi gas
Healing length is smallest !!(Marini et al 1997)
crmv
Lecture 1:- Hydrodynamic theory of superfluid gases - expansion - collective oscillations and equation of state - collective vs s.p. excitations :Landau critical velocity - Fermi superfluids with unequal masses - example of non HD behavior: normal phase of spin polarized Fermi gas
Fermi superfluids with unequal masses ( )(G.Orso, L.Pitaevskii, S.S, in preparation)
NN
Hydrodynamic eqs. keep standard form:
0)~
)(2
1(
0)(
2
extVnmvvt
m
vnnt
with
mm
mm
rmV
mmm
iii
iiext
222
22
~~~
~2
1~
2/)(
- Relevant frequencies for HD oscillations
-If oscillator lengths are equal (natural conditionto cool down into superfluid phase) one has
ii mm
iii ~
What happens if one suddenly switches off ?
- Sudden change in trapping conditions:
- In the absence of interactions atoms down will fly away
- Superfluidity keeps component down confined even if - obvious in molecular regime - non trivial many-body effect at unitarity - weak effect in BCS regime
- Condition for energetic stability of superfluid with respect to separation of two components
- Gas starts oscillating with frequency
mm
mnew
22 ~~
0
1)1(
m
mm easily satisfied if mm
new~
)()~( NnewS EE(unitarity)
Example: mixture (aspect ratio 0.2)
KLi 406
Oscillation of radial sizeif one removes K-trapping
Equilibrium of superfluidin new trapping conditions
Oscillation of radial sizeif one removes Li-trapping
Will superfluid survive in conditionsof strong energetic instability?
Collective oscillations in spin polarized Fermi gas (normal phase)
Lobo, Recati, Giorgini and S.S
- Recent experiments on spin polarized Fermi gas show phase separation- between superfluid and spin polarized normal gas if
- Density profiles at unitarity well agree with theory (Lobo et al. 2006) based on QMC equation of state of superfluid and normal phases -Equation of state in normal phase well approximated by expansion holding for small concentrations
-QMC simulation yields A = 0.97, m/m* = 1.04
75.0
NN
NNP
..]1[5
3 3/5*
xm
mAxE
N
EF
N
n
nx
Axial densitiesin spin polarizedFermi gas
Exp. Shin et al. 2006Theory ideal gas Lobo et al. 2006
Critical value for normal-superfluidphase separationP = 0.75
P=0.80
P=0.58
Clogston-Chandrasekharlimit at unitarity (and in trap)
normal
normal plussuperfluid
At zero temperature oscillations in normal spin polarized phase are not governed by hydrodynamics (like in superfluid), but by Landau’s theory of Fermi liquids (collisionless regime + mean field)
Spin down feel potential generated by spin upresulting in renormalizationof oscillator frequency
If spin down atomsoscillate with frequencies
hoho
hoho
mAm
mAm
46.2/)5/31(2
23.1/)5/31(
*
*
dipole
quadrupole
Spin dipole mode difficult to excite because shift of harmonic trap resultsin excitation of center of mass motion (Kohn’s theorem)
Spin quadrupole is instead excited by quadrupole modulation of trap.
In general one expect to excite two modes (sum rule approach)
2/)(
22
/)5/31(2
21
2
*1
QQQQQ
QQQ
mmAQQ
ho
ho
Exciting Q one excites also spin mode !!
QUESTION: Will the spin mode be strongly damped? Pauli principle is expected to quench damping at small T(collaboration with Ch. Pethick)
Quadrupolemodes
NN
Lecture 2:- Dynamics in rotating superfluid gases - Scissors mode - Expansion of rotating gas - vortex lattices and rotational hydrodynamics - Tkachenko oscillations
Main conclusions
- Collective oscillations in trapped quantum gases can be measured with high precision.- They provide unique information on the consequences of superfluidity and accurate test of equation of state in Fermi superfluids
Can their measurement be considered a proof of superfluidity?
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