Hydrodynamic theory of motion of quantized vortex rings in trapped superfluid gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov Hydrodynamic theory of motion of quantized vortex rings in trapped superfluid gases.
Lev P. Pitaevskii , (Dated: October 17, 2018)I study vortex ring oscillations in a superfluid, trapped in an elongated trap, under the conditionsof the Local Density Approximation. On the basis of the Hamiltonian formalism I develop a hydro-dynamic theory, which is valid for an arbitrary superfluid and depends only on the equation of state.The problem is reduced to an ordinary differential equation for the ring radius. The cases of thedilute BEC and the Fermi gas at unitarity are investigated in detail. Simple analytical equations forthe periods of small oscillations are obtained and the equations of non-linear dynamics are solvedin quadratures. The results agree with available numerical calculations. Experimental possibilitiesto check the predictions are discussed. PACS numbers: 03.75.Lm, 3.75.Kk, 67.85.De
Introduction.
The quantized vortex ring in a superfluidis one of the most unusual objects of modern physics. Itcan be quite macroscopic in size but still keeps its quan-tum nature, carrying one quanta of circulation. Ringsof large size have small phase velocity and play a crucialrole in the phenomenon of critical velocity. The existenceof such rings was predicted by Feynman [1]. They werediscovered, in an indirect way, in experiments on ion mo-tion in liquid He [2]. Vortex rings were also observed ingaseous Bose-Einstein Condensates (BEC) in traps [3].There is every reason to believe that the development ofexperimental technique will permit the detailed investi-gation of rings, both in trapped BEC and Fermi gasesnear unitarity. The dynamics of rings under such condi-tions should be quite peculiar. In a uniform fluid a ringalways moves with an anomalous energy-velocity relation- its velocity decreases with increasing energy. Under thenon uniform conditions in a trap the situation is different.It was discovered in [4] that in a spherical trap a config-uration of maximum energy exists, where a circular ringis at rest. It was checked in [5] that the same situationtakes place also in an elongated trap. An initial deviationfrom this equilibrium configuration will result in oscilla-tions of the ring. These oscillations in a spherical trapwere investigated numerically in [4].The problem of oscillation of a ring along a symme-try axis of a superfluid sample in an elongated harmonictrap is one of the most natural subjects of experimentalinvestigation. Numerical simulations have been recentlyperformed in [6] for a Fermi gas at unitarity using a time-dependent density functional theory and in [7] for a BECusing the Gross-Pitaevskii (GP) equation. The main goalof these papers is the interpretation of experimental re-sults [8]. An approximate theory for ring motion in acylindrical trap in the presence of dissipation was devel-oped in [9].
General theory.
In this letter I will present the Hamil-tonian theory of the motion of a ring of radius R alongthe axis of a superfluid gas in a harmonic trap. I assumethat the trap is elongated, ω ⊥ ≫ ω z . I also assume theLocal Density Approximation (LDA) conditions, that is that the transverse radius of the gas is much larger thanthe healing length, R ⊥ ≫ ξ . Moreover I will assume thatthese conditions are satisfied strongly in the sense thatalso the logarithmic factor L ≡ log( R/ξ ), which enters inthe theory, is large and can be considered as a constantin all calculations. We will see that in this “logarith-mic”approximation one can solve the problem analyti-cally in a simple way for a superfluid of any nature. Iwill finally assume, that R ∼ R ⊥ , excluding very smallrings.The LDA conditions permit us to use hydrodynamics.The energy of a ring, i. e. the Hamiltonian, can bewritten directly. The key point is that the energy canbe obtained by integrating the kinetic energy of the flow ρ v / ρ is the density of the fluid).In the logarithmic approximation the main contributionto the integral is due to a small region near the vortexline. Thus one can use the expression for the energy ina uniform fluid, taking the density ρ ( r, z ) to be its valuenear the ring: E R ( R, Z ) = 2 π ~ M Rρ ( R, Z ) log (cid:18) Rξ (cid:19) , (1)where M is the atomic mass m for the Bose superfluidand the pair mass 2 m for the Fermi one, and Z is the z -coordinate of the ring. Notice that the coordinate de-pendence of the density in the LDA regime can be writtenin the form ρ ( r, z ) = ρ (cid:2) µ (cid:0) − r /R ⊥ − z /R z (cid:1)(cid:3) , (2)where µ is the chemical potential in the center of thetrap and R ⊥ = (cid:0) µ/mω ⊥ (cid:1) / , R z = (cid:0) µ/mω z (cid:1) / arethe Thomas-Fermi (TF) radii of the fluid.The momentum of the ring can be calculated as P R ( R, Z ) = ( ~ /M ) R ρ ( r, z ) ∂ z φd r , where φ is the phaseof the order parameter (condensate wave function in BECcase). However, for an elongated trap with R z ≫ R ⊥ one can substitute z ≈ Z. Then the integration will bereduced to an integration on the surface, stretched on thering aperture, where the phase φ undergoes a 2 π jump: P R ≈ ~ M Z ρ ( r, Z ) ∂ z φd r = 2 π ~ M Z R ρ ( r, Z ) 2 πrdr. (3)The velocity of the ring can be calculated with the Hamil-ton equation as V ≡ V z = (cid:18) ∂E R ∂P R (cid:19) Z = ( ∂E R /∂R ) Z ( ∂P R /∂R ) Z . (4)Such a method was used in [10] to calculate o the ve-locity for a uniform fluid. The possibility to use thisequation in a non-uniform fluid, if the energy and mo-mentum are calculated properly, is a central point ofthe theory. Taking into account that, according to (3),( ∂P R /∂R ) Z = (cid:0) π ~ /M (cid:1) Rρ ( R, Z ), we finally obtain anelegant equation ( ρ = ρ ( R, Z )): V ( R, Z ) = ~ M LRρ (cid:18) ∂ ( Rρ ) ∂R (cid:19) Z . (5)Notice that the equation (5) admits a transparent inter-pretation. The quantity F = πR (cid:0) ∂E R ∂R (cid:1) is the force,acting on a unit of length of the ring. According to theMagnus equation, the ring drifts then with the velocity F M/ (2 π ~ ρ ) in agreement with (5).Equations (1) and (5) give a full description of themotion of the ring in a trap. Notice that the theoryis completely hydrodynamic in its nature. Properties ofthe fluid enter only through the equation of state ρ ( µ ).It is convenient to introduce the dimensionless variables X = Z/R z and Y = R/R ⊥ . Energy can be presented as E R ( R, Z ) = (cid:0) π ~ R ⊥ ρ L/M (cid:1) f ( Y, X ), where ρ is thedensity in the center of the trap and f is a dimensionlessfunction. The trajectory of the ring on the X, Y plane isgiven by the energy conservation equation f ( Y, X ) = f , (6)where f fixes the energy of the ring. The plane wherethe ring can be at rest is always at X = 0. The equi-librium radius R EQ = R ⊥ Y EQ is defined by the equa-tion ∂f ( Y, /∂Y = 0 . The energy E R has a maximumat this point. The equation f ( Y,
0) = f has two pos-itive solutions, Y and Y , where 0 < Y < Y . Then R min = R ⊥ Y is the minimal radius of the ring on thegiven trajectory and R max = R ⊥ Y is the maximal one.Below I will consider Y as an ”initial point” of the tra-jectory. The equation ( ∂f /∂Y ) X = 0 defines the lineof ”turning points” on Y, X plane, where the velocitychanges sign. Together with (6) it gives the turning point Y A , X A for a trajectory of given energy. It follows thata ring with Y < Y A moves in the same direction as inan uniform fluid and a ring with Y > Y A in the op-posite direction. The amplitude of the oscillations in z -direction is Z A = R z | X A | . It is worth noting, thatthe ring at rest has zero velocity, but finite momentum,in analogy with rotons in superfluid He. The quantity,
Z/R z R / R ⊥ Z/R z FIG. 1: (Color online) Trajectories of a vortex ring in the
R, Z plane. The left panel corresponds to BEC, the rightone - to a Fermi gas at unitarity. Different curves correspondto different values of minimal radius Y = R min /R ⊥ , whichcan be seen on the Y axis. The bold lines (black online),connecting the points (0,1) and (1,0), are the TF boundary ofthe fluid. The dashed line (blue online) connects the turningpoints of the trajectories. The stars on the Y axises show theequilibrium values of the radius. which can be easily measured, is a period of oscillationof the ring. One can calculate the period of small oscil-lations T in a general form by writing the energy nearthe point R = R EQ and Z = 0 in the oscillator form E R ( R, Z ) − E R ( R EQ , ∝ − h(cid:0) T π (cid:1) V + Z i . Directcalculation gives T = 2 √ T z M µmL ~ ω ⊥ (cid:18) f ∂ X f ∂ Y f (cid:19) / , (7)where T z = 2 π/ω z is the trap period and quantities in theparenthesis should be taken at X = 0 , Y = Y EQ . In LDAregime ( µ/L ~ ω ⊥ ) ≫ T ≫ T z , as revealed in nu-merical calculations in [6],[7]. Notice that the prefactor,fixed by the parameters of the system, can be presentedas ( T z M µ/mL ~ ω ⊥ ) = ( πM R ⊥ R z /L ~ ), demonstrating asimple dependence on pure geometric factors.One can find the time dependence of Z and R from theequation t = R dZV , where the integral should be takenalong the trajectory. It is more convenient to go to vari-able R . Using Eq. (5) we get t ( Y ) = − T z M µπ mL ~ ω ⊥ πf Z YY (cid:18) ∂X∂f (cid:19) Y dY . (8)The period for oscillations for an arbitrary amplitude canbe found as T = 2 t ( Y ). This quantity is interesting,because it reflects the peculiarity of the dynamics of therings, and also important, because it is difficult to observeoscillations of a small amplitude in an experiment. Vortex ring in a trapped dilute BEC.
In a dilute BECthe chemical potential is µ = gn , where g is the couplingconstant and n is the atom density. This means that theenergy function f is f ( Y, X ) = Y (cid:0) − Y − X (cid:1) . (9) Y =R min /R ⊥ T / T FIG. 2: (Color online) The period of oscillations in the unitsof the period of small oscillations as a function of the minimalradius Y for a ring in BEC. Then the equilibrium radius is R EQ = R ⊥ Y EQ = R ⊥ / √ . This value coincides with one obtained in [4] inthe logarithmic approximation for a spherical trap. Theline of the turning points is 3 Y A + X A = 1. The mini-mal radius of the ring is related to the energy as f = Y (cid:0) − Y (cid:1) and the maximal radius Y = p − Y − Y . The equation of the trajectory with initial radius Y can be written as Y (cid:0) − Y − X (cid:1) = Y (cid:0) − Y (cid:1) andthe amplitude of oscillation can be expressed through en-ergy as X A = q − / f / . Trajectories for differentvalues of Y are shown in the left panel in Fig. 1. (Onecan see values of Y as initial values of Y on the y − axis,)The period of small oscillations can be calculated ac-cording to Eq. (7). A simple calculation gives T ( B )0 T z = 43 √ µL ~ ω ⊥ ≈ . µL ~ ω ⊥ . (10)Scaling of T as gn/ log ( gn ) was predicted in [7] on thebasis of qualitative considerations. For oscillations of anarbitrary amplitude let us present the period as T ( B ) = T z (cid:0) µ/π L ~ ω ⊥ (cid:1) τ ( B ) . One obtains the expression for thedimensionless period ττ ( B ) = Z Y Y πf dY p Y ( Y − Y )( Y − Y )( Y − Y ) , (11)where Y = − p − Y − Y is a negative root of theequation f ( Y,
0) = f . Equation (11) can be expressedin terms of the complete elliptic integral of the first order: τ ( B ) = 4 πf p Y ( Y − Y ) K ( k ) , k = s ( Y − Y ) ( − Y ) Y ( Y − Y ) . (12)For small oscillations Y → Y → / √ , f =2 / (3 √ , Y = − / √ . Then τ ( B ) → π / √ Y one gets formally τ ( B ) ≈ πY ln(16 /Y ). However, this limit violates theapplicability of the approximation. The dependence ofthe period on the minimal radius of a ring Y is shownin Fig. 2. τ F Y = R / R ⊥ FIG. 3: (Color online) The time dependence of the radiusof a ring, oscillating in the Fermi gas at unitarity. Differentcurves correspond to different values of minimal radius Y = R min /R ⊥ , which can be seen on the Y axis. X A T / T FIG. 4: (Color online) The period of oscillations in the units ofthe period of small oscillations as a function of the amplitudeof oscillations X A for a ring in the Fermi gas at unitarity. Vortex ring in a trapped Fermi gas at unitarity.
In aFermi gas at unitarity ρ ∝ µ / . Correspondingly, theenergy function is f ( Y, X ) = Y (1 − Y − X ) / . (13)Equation (6) then defines the trajectories in the X, Y plane, which are presented in the right panel of Fig. 1.The equilibrium radius is now Y EQ = 1 / T ( F )0 T z = √ √ µL ~ ω ⊥ = 1 . µL ~ ω ⊥ . (14)The calculations for arbitrary amplitudes are more cum-bersome that in the BEC case and I will present them inshort. The equation for the line of the turning pointsis 4 X A + Y A = 1. The maximal value of X A , i. e.the amplitude of oscillations, is X A = q − / f / and the corresponding Y A = / f / . Minimal andmaximal values of the radius are given by the equation f ( Y,
0) = f , which, introducing the variable y = Y / ,can be transformed to y (1 − y ) = f . For a trajectorywith the minimal radius Y = y / this equation can bepresented as ( y − y ) Q ( y ) = 0, where Q ( y ) = y + y y + yy + y − . (15)The equation Q ( y ) = 0 has 3 roots. The root y is realand defines the maximum value of the radius, Y = y / .Roots y and y are complex conjugated. One can findthe roots analytically or numerically.Changing the variable of integration in (8) from Y to y , I present the time of motion as t ( F ) = T z (cid:0) µ/π L ~ ω ⊥ (cid:1) τ ( F ) , where τ ( F ) ( y ) = Z yy πf / dy r ( y − y ) ( y − y ) (cid:16) y − y ) y + | y | (cid:17) . (16)The integral can be expressed in terms of an elliptic in-tegral (see [12], Eq. 3.145) τ ( F ) = 4 πf / √ pq F ( ϕ, k ) , (17)where the parameters are ϕ = 2arccot s q ( y − y ) p ( y − y ) , k = s ( y − y ) − ( p − q ) pq (18)and p = h | y | + | y | − y Re( y ) i / ,q = h | y | + | y | − y Re( y ) i / . (19)Inverse dependence y ( τ F ) can be expressed in terms ofthe Jacobi cn( t, k ) function. The period of oscillationsis 2 t ( F ) ( Y ). For small amplitudes the result coincideswith (14). For a ring of small radius, Y →
0, one gets τ ( F ) → . Y / . Notice that the authors of [6] used aslightly different dependence of period on radius for rela-tively small rings: T ( F ) ∝ R as opposite to T ( F ) ∝ R / here. In Fig. 3 I show the time dependence of the radiusof a ring, oscillating in the Fermi gas at unitarity. Onecan see that the oscillations are almost harmonic. Onlyfor the small initial radius Y = 0 . T z and increaseswhen the radius of the ring decreases or the interactionincreases. I tried to compare quantitatively my resultswith numerical calculations [7]. These calculations were produced for a ring in a BEC with R ⊥ /ξ = 2 µ/ ~ ω ⊥ ≈ L ∼ . ω ⊥ /ω z = 4, that, of course, doesnot ensure applicability of my asymptotic theory. How-ever, formal use of the theory gives T ( B ) /T z ≈ R min /a z ≈ .
2. The data presented in Fig. 5 of [7] give T ( B ) /T z ≈ . R ⊥ /ξ ≈
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