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Vortices in Bose-Einstein condensates with anharmonic confinement

We examine an effectively repulsive Bose-Einstein condensate of atoms, that rotates in a quadratic-plus-quartic trapping potential. We investigate the phase diagram of the system as a function of the angular frequency of rotation and of the coupling constant, demonstrating that there are phase transitions between multiply- and singly-quantized vortex states. The derived phase diagram is shown to be universal and exact in the limits of small anharmonicity and weak coupling constant.

The behavior of Bose-Einstein condensates of alkalimetal atoms under rotation has become a very interesting topic in recent years, since these systems are superfluid and the only way to rotate them is by creating quantized vortex states. The experimental observation of vortex states in a two-component system was reported by Matthews et al. [1], while Madison et al. [2] observed vortex states in a stirred one-component Bose-Einstein condensate. Many more experimental studies on rotating condensates have been performed recently [3][4][5].
If, in a harmonic trapping potential, the frequency of rotation becomes equal to the trap frequency, the centrifugal force exactly cancels the restoring force and the atoms fly apart. In other words, the trap frequency sets an upper limit for the frequency of rotation of the cloud. To overcome this difficulty and also to examine the various phases which have been predicted theoretically [6][7][8][9][10][11][12][13], Bretin et al. [14] recently examined experimentally a rotating Bose-Einstein condensate in an anharmonic potential having the form Here, ρ is the cylindrical polar coordinate, M is the atomic mass, ω is the frequency of the harmonic trapping potential, a 0 = (h/M ω) 1/2 is the oscillator length, and λ is a dimensionless constant which in the experiment of Ref. [14] was very small, on the order of 10 −3 .
In what follows we investigate the behavior of a gas that is trapped in a potential of the form of V and is under rotation. In reality, the gas is also trapped in the z direction. However since it rotates around this axis, we neglect the trapping in this direction and assume that the gas has a constant density per unit length σ = N/Z, where N is the number of atoms and Z is the width of the condensate. In addition we assume that the degrees of freedom along the z axis are frozen so that our problem becomes two-dimensional. In this case and for a purely harmonic potential, λ = 0, the single-particle energy levels are given by where n r is the radial quantum number, and m is the quantum number corresponding to the angular momentum. Since ǫ grows linearly with the angular momentum, any superposition of states with different values of m (and total angular momentum m 0 ) will have the same energy as the pure state with m = m 0 . Interactions split this degeneracy. As shown in Refs. [15,16], repulsive interactions always favor a superposition of states. The argument is very much like that in the tight-binding model, where delocalization of the wavefunction lowers the energy. In practice, this implies that a singly-quantized vortex state always has a lower energy than a multiplyquantized vortex in a harmonic potential. This picture changes drastically when λ > 0. The single-particle energy levels grow faster than linearly with m in this case. As a result, for sufficiently weak interactions, a multiply-quantized vortex state has lower energy than a singly-quantized vortex with the same circulation. As the coupling constant increases, however, it eventually becomes energetically favorable for the system to spread its angular momentum to additional states, and multiplyquantized vortex states break into singly-quantized vortices. We demonstrate this phase transition explicitly below.
In the following, we assume that the interaction is described by a contact potential of the form where U 0 = 4πh 2 a/M is the strength of the effective twobody interaction with a equal to the scattering length for atom-atom collisions. We assume that the interaction is repulsive, a > 0. Attractive interactions in anharmonic potentials have been studied in Ref. [11]. We can estimate the interaction strength required at the transition for a given value of the anharmonicity coefficient, λ, by noting that the characteristic energy scale associated with the anharmonicity must be comparable to the interaction energy. (This argument, valid when the number of vortices is of order 1, will be generalized below.) The first scale is of order λhω, and the second is of order αnU 0 , where n is the characteristic atom density, and α is a dimensionless constant of order 10 −1 . (See, e.g., Fig. 1 of Ref. [16].) If λhω is to be ≈ αnU 0 , the effective coupling constant, σa, must be ≈ λ/α. For λ ∼ 10 −3 , we conclude that σa must be less than 1.
Therefore, provided that λ and σa are smaller than unity, we consider the eigenstates of the harmonic potential with zero radial excitations and angular momentum mh (m is positive for simplicity) as our unperturbed states, where φ is the angle in cylindrical polar coordinates. The measured with respect to the zero-point energy,hω. (All energies will henceforth be measured with respect tohω.) Equation (5) shows explicitly the faster-than-linear (i.e., quadratic) dependence of the energy on m.
Let us now incorporate the interactions. We use perturbation theory to determine the interaction energy per particle in the state Φ m , Combining Eqs. (5) and (6) we obtain the total energy per particle in Φ m , In a rotating frame of reference with some angular velocity Ω, the energy is The above expression gives the energy per particle in the rotating frame in terms of the parameters Ω/ω, λ, and σa, in a given state Φ m .
From Eq. (8) we see that E ′ 1 becomes lower than E ′ 0 for a critical frequency of rotation which is This is actually the critical angular frequency for inducing rotation in the cloud. For λ = 0, this result is in agreement with Refs. [15,16]. As Ω increases further, states with larger m become energetically favorable. The critical value of Ω m for a multiply-quantized vortex state to form with mh units of angular momentum is given by A similar result was derived using a variational approach in Ref. [11] for an attractive Bose gas trapped in all three directions. According to our model and for sufficiently weak interactions, the angular momentum mh of the system is quantized and increases stepwise at the frequencies given by Eq. (10), as shown in Fig. 1 for λ = 0.05 and σa = 0.4. For m ≫ 1, Ω m /ω ≈ 1 + λm. Figure 2 shows the phase diagram of multiply-quantized vortex states in the Ω/ω -σa plane, as given by Eq. (10), for λ = 0.05. We argued earlier that, as the strength of the interaction increases, a multiply-quantized vortex state will split into singly-quantized vortices. To attack this problem more generally, one can expand the order parameter [15,16], Here the coefficients c m are (complex) variational parameters subject to The first condition in simply the normalization, while the second implies that the state Ψ is assumed to have some angular momentum per particle, lh, where l is not necessarily quantized. After minimization of the energy in the rotating frame with respect to the variational parameters c m and with respect to l, one can determine Ψ and l = l(Ω).
The phase transition we consider first appears in the case with m = 2, and this is the example we consider below. However our method can be generalized to higher values of m. Let us therefore consider the trial wavefunction [15,16] The state Ψ reduces to Φ 2 when |c 2 | = 1, c 0 = c 4 = 0 (i.e., it is a doubly-quantized vortex state), and it describes two singly-quantized vortices when c 0 c 4 = 0 [17]. As we show below, there is a critical value of σa above which c 0 and c 4 become nonzero. It is important to note that, as a consequence of the spherical symmetry of the interaction, the phase boundary between single and multiple quantization is given exactly by the trial order parameter of Eq. (13) so long as the states with nonzero radial nodes, n r = 0, can be neglected. Equation (13) can be extended to larger values of m by considering the local stability of the multiply-quantized state, Φ m , with respect to the admixture of states with any angular momentum m 1 and m 2 = 2m − m 1 . For small m, the leading instability is given by m 1 = 0. These results will be addressed in a future publication. Calculating the energy per particle in the state Ψ in the rotating frame of reference we get, Using Eqs. (12), we can eliminate |c 0 | and |c 4 | in Eq. (14). For a fixed λ, we then vary c 2 and l to minimize E ′ for various values of Ω/ω and σa and thus investigate the phase diagram in the Ω/ω -σa plane. The result for λ = 0.05 is shown as the dashed line in Fig. 2 in the m = 2 phase. Above this line there are two singly-quantized vortices with |c 2 | < 1. Below this line there is a doublyquantized vortex state, where |c 2 | = 1. The actual phase boundary is expected to lie lower than the dashed line because of the variational nature of the present calculation (i.e., the neglect of states with n r = 0). Extending the same approach to higher values of m, we derive the corresponding phase boundaries for m = 3, 4 and 5, which are shown as dashed lines in Fig. 2. The overall phase diagram of Fig. 2 resembles that obtained numerically by Lundh [6] for traps of some power law.
It is important to note that the transition seen here is continuous and of second order. Thus, it is sufficient to consider the local stability of the states Φ m in determining the phase boundary. As σa increases, the separation between the two vortices increases continuously in agreement with numerical simulations [6]. Note also that the form of the phase diagram is unaltered by changes in the strength of the anharmonicity since (1 − Ω/ω) and σa at the phase boundary simply scale linearly with λ. The phase diagram shown in Fig. 2 is thus universal and exact in the limits of small anharmonicity and weak interactions.
Another important point is that in the phases of single quantization, the function l = l(Ω) is not necessarily quantized. Therefore, the graph of Fig. 1 for higher values of σa will be partly continuous and partly discontinuous. This is similar to the case of harmonic confinememt [15] except for the fact that, in the present problem l does not diverge when Ω → ω because of the presence of the quartic term in the trapping potential.
To understand better the mechanism which gives rise to the phase transition between single and multiple quantization, we return to the trial order parameter of Eq. (13) and consider the extra constraint l = 2. Eliminating |c 0 | and |c 4 | in the energy, we find in the laboratory frame that The right side of the above equation must be minimized with respect to |c 2 |. However, it is clear that the energy that results from the anharmonicity is minimized for |c 2 | = 1, while the interaction energy is minimized for |c 2 | 2 = (64 √ 6 − 106)/2(64 √ 6 − 109) ≈ 0.53, which is less than 1. It is essentially the balance between these two terms that gives rise to the phase transition between multiple and single quantization. Some trivial algebra shows that the critical coupling constant above which the vortices separate is given by The dashed curve in Fig. 2 (for m = 2) saturates for larger Ω to the value of σa given by Eq. (16). (In fact, it has essentially reached this point at the highest value of σa shown.) Since the solid line separating the m = 2 and m = 3 phases in Fig. 2 cuts this dashed line at some σa ≤ (σa) c , Eq. (16) provides an upper limit for σa for which there is a doubly-quantized vortex state. In Ref. [6] the function (σa) c = (σa) c (λ) was calculated numerically (in Fig. 4 of [6]), and (σa) c of Eq. (16) indeed lies above this curve. The discrepancy is less than 10 percent for the smallest λ shown, λ = 0.01, and grows as λ increases. We note that Ref. [10] has predicted the existence of a triple point in the Ω/ω -σa plane, at which a multiplyquantized vortex state, a vortex lattice, and a vortex lattice with a hole coexist. The present study is far from this point. However, such a triple point would be indicated in our approach when the leading instability is due to the admixture of states with m 1 = 0, since Φ 0 is the only component in the wavefunction that does not vanish at ρ = 0. Since the effective trapping potential, V − M Ω 2 ρ 2 /2, has the familiar "Mexican hat" shape for Ω/ω > 1, the contribution of Φ 0 to Ψ will decrease with increasing Ω/ω.
Turning to the experiment of Ref. [14], for λ = 10 −3 Eq. (16) implies a value of (σa) c ≈ 0.02, which is very small. Actually this experiment is closer to the Thomas-Fermi limit of strong interactions [18]. For a number of atoms N = 3 × 10 5 , ω = 2π × 65.6 Hz, ω z = 2π × 11.0 Hz, and a = 53Å, we find a transverse radius of the cloud of ≈ 6.6 µm, and a width in the z direction of ≈ 39.4 µm. With these values σa turns out to be ≈ 20, which is much greater than (σa) c . This fact explains why no multiply-quantized vortices were observed. On the other hand, tens of vortices were created in this experiment, and (σa) c increases with the number of vortices [10]. We can obtain a more general estimate for (σa) c for the m-th multiply-quantized vortex state as by equating the energy due to the quartic term, Eq. (5), with the interaction energy, Eq. (6). Equation (17) is indeed an increasing function of m. In Eq. (17) λ(m + 1)(m + 2)/2 must be smaller than m, and m should thus not exceed 2/λ. Self-consistency requires that (σa) c should not exceed unity. To consider a specific example with λ = 10 −3 for a multiply-quantized vortex state with m = 15, Eq. (17) implies that (σa) c ∼ 0.94, while Eq. (10) gives a rotation frequency Ω 15 ≈ 1.01ω. In summary, we have examined a weakly-repulsive Bose-Einstein condensate of atoms that is confined in a quadratic-plus-quartic trapping potential and is under rotation. We have investigated the two phases that the system exhibits -multiply-and singly-quantized vortices -as functions of the frequency of rotation, the coupling constant and the anharmonicity. We have also demonstrated that, by varying the coupling constant, the cloud undergoes a phase transition from multiply-to singlyquantized vortex states.
Author GMK wishes to thank Gordon Baym, Emil Lundh, Ben Mottelson, and Chris Pethick for useful discussions. He also acknowledges financial support from the Swedish Research Council (VR), and from the Swedish Foundation for Strategic Research (SSF).