Breathing mode frequencies of a rotating Fermi gas in the BCS-BEC crossover region
aa r X i v : . [ c ond - m a t . o t h e r] J u l Breathing mode frequencies of a rotating Fermi gas in the BCS-BEC crossover region
Theja N. De Silva
Department of Physics, Applied Physics and Astronomy,The State University of New York at Binghamton, Binghamton, New York 13902, USA.
We study the breathing mode frequencies of a rotating Fermi gas trapped in a harmonic plusradial quartic potential. We find that as the radial anharmonicity increases, the lowest order radialmode frequency increases while the next lowest order radial mode frequency decreases. Then at acritical anharmonicity, these two modes merge and beyond this merge the cloud is unstable againstthe oscillations. The critical anharmonicity depends on both rotational frequency and the chemicalpotential. As a result of the large chemical potential in the BCS regime, even with a weak anhar-monicity the lowest order mode frequency increases with decreasing the attractive interaction. Forlarge enough anharmonicities in the weak coupling BCS limit, we find that the excitation of thebreathing mode frequencies make the atomic cloud unstable.
I. INTRODUCTION
The rapid progress of ultra-cold atomic gas experi-ments provides unique opportunities for well controlledstudies of quantum many body physics. For the caseof Fermi atomic systems, the possibility of controllingthe s-wave scattering length ( a ) between two differentspin components allows to control the interaction by us-ing a magnetically tuned Feshbach resonance [1]. Thisunique capability allows one to investigate the cross-overbetween the weakly interacting BCS regime (the regimewhere a → − ) and Bose-Einstein condensate of dimers(the regime where a → + ) [2]. These two regimes meetin strongly interacting limit where the scattering lengthis divergent and at this unitarity limit, the physics is ex-pected to be universal [3].The appearance of quantized vortices of a quantumfluid under rotation offers direct evidence of superfluid-ity. For example, the observation of quantized circula-tion in a rotating superfluid He [4] and the observationof vortex lattice in a rotating Fermi gas of Li [5] aretwo classic demonstrations of phase coherence in a su-perfluid. These are analog to the vortex lattice in type-II superconductors in the presence of a magnetic filed.These vortices melt as the magnetic field increases andthen the superconductors turn into normal at sufficientlylarge magnetic fields. For the case of rapidly rotatingFermi gasses, the force due to the trapping frequency al-most balances the centrifugal force and superfluid cloudspreads in the plane perpendicular to the rotation axis.At the limit of very large rotation, the theory predictsthat the atomic system enters into the fractional quan-tum Hall regime [6, 7]. However, the fractional quantumHall window is expected to be very small and inverselyproportional to the number of atoms in the trap. A possi-ble way of stabilizing the fractional quantum Hall regimeis to add a positive quartic trapping potential.In this paper, we study the collective breathing modefrequencies of a rotating Fermi gas in the presence of aquartic trapping potential by using a hydrodynamic ap-proach. A negative, but small quartic term is always present with the Gaussian optical potentials in currentexperimental setups while added positive quartic termensures the stability of the fast rotating regime. Ther-modynamic properties of a Bose gas confined in har-monic plus quartic potential trap can be found in ref. [8].As breathing mode frequencies are very sensitive to theequation of state, these dynamical quantities can be usedas tests for various theories. The breathing mode fre-quencies have been measured for non-rotating Fermi sys-tems in the BCS-BEC region [9, 10]. In most of the pa-rameter regions, experimental results agree well with thehydrodynamic approaches, variational approaches andsum rule approaches [11]. However, the measured finitetemperature axial and radial breathing mode frequen-cies show a striking increase in the intermediate BCSregime [9, 10], in contrast to zero temperature theoreti-cal calculations in a harmonic trap. Furthermore, exper-imentalists were unable to measure the breathing modefrequencies in the weak coupling BCS limit. The devia-tion of the experimental data from theoretical results andthe lack of experimental data in the weak coupling BCSlimit were believed to be due the large Landau dampingwhen the superfluid energy gap is much smaller than thecollective oscillation energies. With inclusion of a posi-tive quartic term in the trapping potential, we find some-what similar deviation of the breathing mode frequenciesin the intermediate regions of BCS regime. We find thatthe breathing mode frequencies deviate significantly fromthe modes frequencies calculated in a harmonic trap andthe atomic cloud is unstable against the breathing modeoscillations at larger chemical potentials. As the chemicalpotential is larger in the weak coupling BCS limit, exci-tation of breathing mode frequencies make the atomiccloud unstable. It should be noted that the Gaussian op-tical trap potential provides a negative quartic term inthe trapping potential in experimental setups. We inves-tigate the effect of negative quartic term and find thatthe breathing mode frequencies tend to decrease in theentire BCS-BEC crossover region.This paper is organized as follows. In section II, wepresent the derivation of breathing mode frequencies us-ing a hydrodynamic approach. In section III, we presentour results together with a discussion. Finally in sectionIV, we draw our conclusions.
II. FORMALISM
We consider a rotating Fermi atomic system trapped ina harmonic plus radial quartic potential in the BCS-BECcrossover region. The trapping potential is V ex ( r, z ) = 12 M ω r r + 12 M ω z z + K r (1)where M is the atom mass, ω i ’s are the harmonic trap-ping frequencies, and r = x + y . The Fermi atomicsystem rotates about the z -axis at frequency Ω.We restrict ourselves to the case of large number of vor-tices in the system where the wavelength of the oscillationfrequencies is much larger than the inter-vortex distance.This condition always satisfies as the typical wavelengthof the lowest mode oscillations is of the order of the sys-tem size. Further, we assume that the vortices are uni-formly distributed in the superfluid so that we do nothave to consider microscopic details of the single vortices.These assumptions are valid at the limit of large rotationswhere the atomic cloud spreads in the plane perpendicu-lar to the rotation. Within this diffused vorticity approx-imation [12], diffuse vorticity is given by ∇ × v = 2 Ω ,where superfluid velocity is given by v = (¯ h/ M ) ∇ θ .The local superfluid density n and the local phase θ arerelated through the wave function ψ = √ ne iθ . The uni-form vortex density is given by n v = 2 M Ω / ¯ h .Assuming local equilibrium, we start with the continu-ity and Euler equations of rotational hydrodynamics, ∂n∂t = −∇ · [ n ( r ) v ] (2)and M ∂ v ∂t = −∇ [ 12 M v + V ex ( r, z ) −
12 Ω r + µ ( n )]+2 M v × Ω + M v × ∇ × v (3)This hydrodynamic description is valid as long as thecollisional relaxation time τ is much smaller than the in-verse of the oscillation frequencies; ωτ <<
1. The equa-tion of state enters through the density dependent localchemical potential µ ( n ). We fix the local chemical po-tential by introducing the equation of state in the formof µ ( n ) ∝ n γ . As we will discuss in the next subsec-tion, the polytropic index γ is calculated by the methodproposed by Manini and Salasnich [13] in the entire BCS-BEC crossover region. Linearizing the density n and thesuperfluid velocity v around their equilibrium values as n = n ( r ) + δn , v = v + δ v and µ ( n ) = µ ( n ) + δµ with δµ = ( ∂µ/∂n ) | n = n δn , we obtain the linearized versionof the hydrodynamic equations. ∂δn∂t = −∇ · [ n ( r ) δ v ] (4)and M ∂δ v ∂t = −∇ δµ + 2 M δ v × Ω (5)Starting from these two linearized equations, collec-tive breathing mode frequencies have been calculated inref.[14] and ref.[15] for a harmonic trap. As the authorshave used two different ansatz for the velocity fluctua-tion, they produce two different results for the breathingmode frequencies. In this paper we closely follow the ap-proach adopted in ref.[15], generalizing the theory to ananharmonic trap. The ansatz used in ref.[15] ensures theconservation of angular momentum properly. In orderto solve the linearized equations for the breathing modefrequencies, we take the equilibrium density in the lo-cal density approximation as n ( r ) ∝ [ µ − (1 / M ( ω r − Ω ) r − (1 / M ω z z − ( K/ r ] /γ and use following vari-ational ansatz for the density fluctuations and velocityfluctuations. δ v = { δ Ω × r + δ Ω × r r + ∇ [ α ⊥ r + α z z + βr ] }× exp[ − iωt ](6)and δn = n − γ { a + a ⊥ r + a z z + br } exp[ − iωt ] (7)The first two terms δ Ω and δ Ω in Eq. (6) are parallel tothe axis of rotation and guarantee that angular momen-tum is conserved during the oscillations. Substitutingthese two ansatz into Eq. (4) and Eq. (5), we derive fourlinear equations for the variational parameters a , a ⊥ , a z , and b . These linear equations yield three non-zerosolutions for the breathing mode frequencies ω m ≡ ω/ω r as roots of the following equation. A + Bω m + Cω m + ω m = 0 (8)with, A ≡ − (1 − ζ ) δ [64 dγ ( γ + 1) + 48 γ + 56 γ +16] − (1 − ζ ) ζ δ (32 γ + 104 γ + 48) − ζ δ ( γ + 2), B ≡ (1 − ζ ) [8( γ +1)(2 γ +1)+16 dγ ( γ +2)]+(1 − ζ )[ δ (8 γ +26 γ + 12) + ζ (40 γ + 24)] + 8 ζ δ ( γ + 2) + 16 ζ and C ≡ (1 − ζ )(2 − γ ) − δ ( γ +2) −
8. The constants, ζ ≡ Ω /ω r , δ ≡ ω z /ω r , and d ≡ [1 / (1 − ζ ) ]( K ¯ h/M ω r )( µ / ¯ hω r ) area set of dimensionless parameters. The three solutions ofEq. (8) are the lowest order axial breathing mode fre-quency ω and the lowest and next lowest order radialbreathing mode frequencies ω and ω . The Effective polytropic index and the chemicalpotential in the BCS-BEC crossover region
We use the proposal made by Manini and Salasnich [13]to calculate the effective polytropic index γ and thechemical potential µ in the BCS-BEC crossover region.In the weak coupling BCS limit ( a −→ − ) and the uni-tarity limit ( a −→ ∞ ), the polytropic index is γ = 2 / a −→ + ), the polytropic index is γ = 1. In the BCS-BEC crossover regime, the scatteringlength’s dependence on γ is given by [13] γ = 2 / − yǫ ′ ( y ) / y ǫ ′′ ( y ) / ǫ ( y ) − yǫ ′ ( y ) / y = 1 / ( k f a ) is the interaction pa-rameter with k f being the Fermi wave vector. The func-tion ǫ ( y ) is related to the energy per atom given by E = (3 / E f ǫ ( y ), where E f = ¯ h k f / M is the Fermiatomic energy of a non-interacting Fermi system in thetrap. Above ǫ ′ ( y ) = ∂ǫ ( y ) /∂y and the double prime in-dicates the second derivative of the function on its argu-ment. Using the data presented in reference [16], Maniniand Salasnich [13] used a data fitting scheme to derivean analytical form of the function ǫ ( y ) in the entire BCS-BEC region, ǫ ( y ) = α − α arctan (cid:20) α y β + | y | β + | y | (cid:21) (10)Two different sets of parameters are proposed for α i ’s and β i ’s in the BCS regime ( y <
0) and the BEC regime ( y > α = 0 . α = 0 . α = 1 . β = 1 . β = 0 . α = 0 . α =0 . α = 5 . β = 0 . β = 0 . µ is given by [13] µ = E f [ ǫ ( y ) − yǫ ′ ( y ) /
5] (11)We determine the Fermi energy E f of a non-interactingFermi system in a harmonic plus radial quartic potentialthrough the number equation. N = 115 δ s M ω r ¯ h | K | (cid:18) E f ¯ hω r (cid:19) / f ( E f ) (12)We define the function f ( E f ) as f ( E f ) = ± (cid:18) ± − ζ ) | ˜ K | ˜ E f (cid:19) ∓ s (1 − ζ ) | ˜ K | ˜ E f × (cid:20)
15 + 5(1 − ζ ) K ˜ E f + 12 (1 − ζ )
1( ˜ K ˜ E f ) (cid:21) (13) - - - (cid:144)H k f a L Μ (cid:144) Ñ Ω r FIG. 1: Central chemical potential ( µ ) of a non-rotatingFermi gas as a function of interaction parameter 1 / ( k f a ) forvarious values of ˜ K . From top to bottom ˜ K = 0 .
05 (long blackdashed line), and 0 .
01 (short black dashed line), 0 harmonictrap (black solid line), − .
005 (short gray dashed line), and − .
01 (long gray dashed line). For the calculation, we use N = 2 . × and δ = 0 . - - - (cid:144)H k f a L Ω (cid:144) Ω r FIG. 2: The lowest order radial breathing mode frequenciesof a rotating Fermi gas in a harmonic trap with aspect ratio δ = 0 . ζ = 0 (solid line), 0 . . where the scaled parameters are ˜ K ≡ ¯ hK/ ( M ω r ) and˜ E f ≡ E f / (¯ hω r ). The upper and lower signs are cor-responding to K >
K < E f =¯ hω r [3 N δ (1 − ζ )] / . III. RESULTS AND DISCUSSION
In FIG. 1, we plot the central chemical potential ofa non-rotating Fermi system calculated from Eq. (11)as a function of inverse scattering length for two differ-ent representative values of anharmonicity. We use N =2 . × number of atoms in the trap with δ = 0 . ǫ ( y ) proposed by Manini andSalasnich [13] (This kink appears in all the calculatedmacroscopic quantities).Solving Eq. (8) for the case of harmonic potential trap - - - (cid:144)H k f a L Ω (cid:144) Ω z FIG. 3: The lowest order axial breathing mode frequenciesof a rotating Fermi gas in a harmonic trap with aspect ratio δ = 0 . ζ = 0 (solid line), 0 . . ( K = 0) in a rotating ( ζ = 0) Fermi system, the lowestaxial and radial breathing modes frequencies are givenby ω m = 12 { γ (2 + δ − ζ ) + 2(1 + δ + ζ ) ±{ [ γ (2 + δ − ζ ) + 2(1 + δ + ζ )] − δ [ γ ( − ζ ) − ζ )] } / } . (14)For the isotropic trap ( δ = 1), at non-interacting limitand at unitarity limit ( γ = 2 / ω m = 2 and ω m = p ζ /
3. In the deepBEC limit ( γ = 1), the mode frequencies are ω m = q (1 / ± p − ζ ). For the case of harmonic poten-tial ( K = 0) in a non-rotating ( ζ = 0) limit, Eq. (14)reduces to ω m = 12 { δ ) + γ (2 + δ ) ± p [2(1 + δ ) + γ (2 + δ )] − γ ) δ } . (15)For the case of highly anisotropic limit ( δ << γ = 2 / ω /ω z = p / ω /ω r = p / γ = 1 are ω /ω z = p / ω /ω r = 2. In the BCS-BEC crossover region, the lowestorder breathing modes frequencies are calculated fromEq. (15) by using the γ from Eq. (9). The results forseveral representative values of ζ are given in FIG. 2 andFIG. 3.We solve Eq. (8) for the breathing mode frequen-cies for various values of K in both rotating and non-rotating Fermi systems. We calculate the central chem-ical potential µ for fixed number of atoms N =2 . × in the trap. We find that as d ≡ [1 / (1 − ζ ) ]( K ¯ h/M ω r )( µ / ¯ hω r ) = [1 / (1 − ζ ) ] ˜ K ˜ µ increases, - - K Ž ΜŽ (cid:144)H -Ζ L Ω (cid:144) Ω r a nd Ω (cid:144) Ω r FIG. 4: The two lowest radial breathing mode frequencies asa function of d = [1 / (1 − ζ ) ] ˜ K ˜ µ . Black (lowest mode) andgray (second lowest) solid lines are the mode frequencies forweakly interacting limit and unitarity limit( γ = 2 / γ = 1). The value of δ = ω z /ω r = 0 . d and beyond this critical value, the atomic cloud is unstableagainst the breathing mode oscillations. K Ž Ω (cid:144) Ω z FIG. 5: Axial breathing mode frequencies as a function of ˜ K for ζ = 0 (black), 0.3 (long dashed), 0.6 (short dashed) and,0.9 (dotted) at unitarity. We use the values δ = 0 .
045 and N = 2 . × . the lowest order radial breathing mode frequency in-creases while the next lowest order breathing mode fre-quency decreases. Then at a critical value of d = d c ,these two modes merge and beyond this critical d c , theatomic cloud is unstable against the oscillations. FIG. 4shows the lowest and the next lowest order radial modefrequencies at γ = 2 / γ = 1 (deep BEC limit).As evidenced by FIG. 5, the lowest order axial breath-ing mode frequencies are almost insensitive to the radialanharmonicities. FIG. 5 shows the axial breathing modefrequencies for various values of rotational frequencies asa function radial anharmonicities at unitarity.In the BEC limit where γ = 1, we calculate the lowestand next lowest radial breathing mode frequencies as a K Ž Ω (cid:144) Ω r a nd Ω (cid:144) Ω r FIG. 6: Radial breathing mode frequencies at BEC limit( γ = 1) for ζ = 0 (solid line), 0 . . δ = ω z /ω r = 0 .
045 and theatom number is N = 2 . × . - - (cid:144)H k f a L Ω (cid:144) Ω r a nd Ω (cid:144) Ω r FIG. 7: The two lowest radial breathing mode frequencies atfor ˜ K = 0 .
005 (solid line), 0 .
01 (long dashed line) and 0 . K = − . δ = ω z /ω r = 0 .
045 and the atom number is N = 2 . × . function of radial anharmonicity ˜ K for three represen-tative values of rotational frequencies ζ . We fixed thenumber of atoms to be 2 . × and δ = ω Z /ω r = 0 . K the lowest or-der radial breathing mode frequency increases, while thenext lowest order radial breathing mode frequency de-creases. Further increase of ˜ K merges these two modesand beyond this merging point the atomic cloud is un-stable against the oscillations.The lowest order and next lowest order radial breath-ing mode frequencies as a function of the interaction pa-rameter [1 / ( k f a )] are shown in FIG. 7. In the presenceof radial anharmonicity, the lowest order mode frequencytends to increase in the BCS regime while the next lowestorder breathing mode frequency tends to decrease. Thisdeviation becomes large as the anharmonicity increases. - - (cid:144)H k f a L Ω (cid:144) Ω r FIG. 8: The lowest order radial breathing mode frequenciesat for ˜ K = 0 . .
025 and 0 .
001 (Top to bottom). Theblack lines is the lowest order breathing mode in a harmonictrap. The value of δ = ω z /ω r = 0 .
045 and the atom numberis N = 2 . × . The dots are the experimental data forcomparison [10]. As we have discussed before, at larger Kµ values thetwo lowest order modes merge and the cloud is unsta-ble against the breathing mode oscillations beyond thispoint. The data in the FIG. 8 shows the same informa-tion as FIG. 7, but we plot only the lowest order radialbreathing mode frequencies for small quartic potentialstogether with experimental data from ref. [10]. IV. CONCLUSIONS
We have discussed the breathing mode frequencies ofa rotating Fermi gas trapped in a harmonic plus ra-dial quartic potential. We find that the radial breath-ing mode frequencies strongly depend on the rotationand anharmonicity through parameter d = [1 / (1 − ζ ) ]( K ¯ h/M ω r )( µ / ¯ hω r ). As d increases, the lowest or-der radial breathing mode’s frequency increases and thenext lowest order mode decreases. Beyond some criti-cal d c , these two modes merge and the cloud is unstableagainst the oscillations.As the chemical potential is large in the intermediateBCS regime, even with a very weak quartic potential theparameter d is large. As a result, the lowest order breath-ing mode frequency increases in the intermediate BCSregime. Even though the Gaussian optical trap potentialprovides a negative anharmonic term in the trapping po-tential, this positive anharmonic behavior has been seenin recent experiments [9, 10]. In the weak coupling BCSlimit, the chemical potential is even larger so that wefind the atomic cloud is unstable against the oscillationsat large positive anharmonicities. For negative quarticpotentials, the breathing mode frequencies tend to de-crease in the BCS-BEC crossover region. V. ACKNOWLEDGEMENTS
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