Effects of boundaries and density inhomogeneity on states of vortex matter in Bose-Einstein condensates at finite temperature

Most of the literature on quantum vortices predicting various states of vortex matter in three dimensions at finite temperatures in quantum fluids is based on the assumption of an extended and homogeneous system. This is well known not to be the case in actual Bose-Einstein condensates in traps, which are finite systems with nonuniform density. This raises the question to what extent one can speak of different aggregate states of vortex matter (vortex lattices, liquids, and tensionless vortex tangles) in these systems. To address this point, in the present work we focus on the finite-size, boundaries and density inhomogeneity effects on thermal vortex matter in a Bose-Einstein condensate. To this end we perform Monte Carlo simulations on a model system describing trapped Bose-Einstein condensates. Throughout the paper, we draw on analogies with results for vortex matter obtained for extended systems. We also consider, for comparison, the cylindrical confinement geometry with uniform density profile from the center out to the edge of the trap. The trapping potential is taken to be generically of an anharmonic form in such a way as to interpolate between a harmonic trap and a cylindrical confinement geometry. It also allows for a dip in the density profile at the center. We find distinct thermal equilibrium properties of the vortex system as the qualitative characteristics of the trapping potential are varied. For a uniform cylindrical confinement geometry, we find a vortex lattice at the center of the trap as well as ringlike thermal fluctuations of the vortex system as the trap edge is approached. For a harmonic trap, we find a distinct region at the edge of the trap where the vortex lines appear to have lost their line tension. Due to the steep density gradient, this crosses directly over to a vortex-line lattice at the center of the trap at low temperatures. At higher temperatures, an intermediate tensionful vortex liquid may exist. For an anharmonic trap where the ground state condensate density features a local minimum at the center of the trap, we find a corresponding region which appears to feature a tensionless vortex-line liquid phase. This work suggests that, finiteness and intrinsic inhomogeneity of the system notwithstanding, one nonetheless can approximately invoke the notion of distinct aggregate states of vortex matter realized at certain length scales. This might be helpful, in particular, in the search for possible new states of vortex matter in Bose-Einstein condensates with multiple components and different symmetries.

Figure 1

(Color online) Monte Carlo results for a uniform system with periodic boundary conditions. The helicity modulus (superfluid density) ${{\rm Y}}_z$ along the $z$ direction is plotted $(+)$, but ${{\rm Y}}_x$ and ${{\rm Y}}_y$ give equal results. The specific heat $C_V/L^3$ $(\times )$ has a peak at $T_C$, where ${{\rm Y}}_z$ vanishes. In the second row, random snapshots of the vortex configurations are shown for three temperatures. These are sections with ${16}^3$ lattice sites cut out from the ${72}^3$ system.

Figure 2

(Color online) For each temperature, given below the columns, the structure function $S\left({\mathbf{k}}_{\perp }\right)$ is shown in the upper row [$S\left(0\right)$ is removed, and in the two rightmost images the color scale is magnified by a factor 25]. There are sharp peaks for the characteristic Bragg vectors at the lowest temperatures, before a transition to a ring structure corresponding to a vortex liquid phase at $T=1.67$. In the second row we show the real-space equivalent to the structure function, the $xy$ positions of the vortices, integrated over $z$ direction. The averages of the first two rows are calculated from 100 000 Monte Carlo sweeps. In the third row, we show sections of size ${16}^3$ from snapshots of the vortex configurations.

Figure 3

(Color online) Helicity modulus ${{\rm Y}}_z$ $(+)$ along the axis of rotation is cut off and vanishes when the vortex lattice melts. In the transverse direction, a zero ${{\rm Y}}_{\perp }={{\rm Y}}_x,{{\rm Y}}_y$ $(\square )$ indicates that there is no pinning of the vortices to the numerical grid. The rounded peak in $C_V/L^3$ $(\times )$ is a remnant of the vortex-loop blowout transition in the nonrotating system.

Figure 4

(Color online) Radial density profile $P_{i\mu }$ for the case of a uniform cylinder. The density is uniform in the $z$ direction.

Figure 5

(Color online) $xy$ positions of vortices in a cylindrical container integrated over $z$ direction, and averaged over every tenth of a total of $5\times {10}^5$ MC sweeps. Top and bottom rows have $L=36$ and 72, respectively. At $T=0.5,1.0$, we discern circular ordering close to the cylinder wall combined with a hexagonally ordered state closer to the center. At $T=1.25,1.67$ we observe dominance of angular fluctuations closest to the edge.

Figure 6

(Color online) Results for ${\widetilde{{\rm Y}}}_z\left(R_1,R_2\right)$ in a cylindrical container. In (b)–(e) are shown the helicity moduli ${\widetilde{{\rm Y}}}_z\left(R_1,R_2\right)$ for radii $R_1,R_2=0,R/4,2R/4,3R/4,R$, as indicated by the white circles in the images on the left, and compared to the results for the extended uniform system (a). The upper curve $(+)$ is the helicity modulus without rotation, while the lower curve $(\times )$ is the result with rotation-induced vortices present (filling fraction $f=1/36$). All regions in the cylindrical container behave almost as the uniform system.

Figure 7

(Color online) Radial density profile $P_{i\mu }$ for the case of a harmonic trap.

Figure 8

(Color online) Snapshots of vortex configurations in a rotating trapped BEC at $T=0.5$ (left figures) and 1.0 (right figures). The top row shows a selection of 16 out of 72 layers in $z$ direction. The bottom row shows smaller selections in the $xy$ plane, but 32 out of 72 layers in the $z$ direction. Fluctuations are minimal in the trap center, and increase toward the edge of the trap. A distinct front separating regions of ordered and disordered vortices is easily identified.

Figure 9

(Color online) $xy$ positions of vortices in a trapped BEC integrated over the $z$ direction. In the top row, we show snapshots, while the middle and bottom rows show averages of ${10}^5$ and $5\times {10}^5$ MC sweeps, respectively. Every tenth configuration has been sampled. This provides information on the stability of the ordered region and the evolution of the disordered region as $T$ varies.

Figure 10

(Color online) Results for ${\widetilde{{\rm Y}}}_z\left(R_1,R_2\right)$. The two top panels show computation results for the thermal depletion of the superfluid density in a harmonically trapped condensate ($r_{\perp }$ is the distance from the center of the trap) at the temperatures $T=2.50$ (the lowermost curve), 2.00, 1.67, 1.25, 1.00, and 0.50. The uppermost curve is the pure ground state density profile $P_{i\mu }$. In (c)–(g), the upper curve $(+)$ is the helicity modulus without rotation, while the lower curve $(\times )$ is the helicity modulus with rotation-induced vortices with filling fraction $f=1/36$ as functions of temperature. (c) shows ${{\rm Y}}_z$ for a cubic uniform system with periodic boundary conditions. The remaining panels (d)–(g) show ${\widetilde{{\rm Y}}}_z\left(0,R/4\right)$, ${\widetilde{{\rm Y}}}_z\left(R/4,2R/4\right)$, ${\widetilde{{\rm Y}}}_z\left(2R/4,3R/4\right)$, and ${\widetilde{{\rm Y}}}_z\left(3R/4,R\right)$, respectively. The vortex plots on the left (obtained at $T=0.50$) define the radii $R_1$ and $R_2$ as white circles.

Figure 11

(Color online) Radial density profile $P_{i\mu }$ for the case of an anharmonic trap with a local minimum of the ground state condensate density at the center of the trap.

Figure 12

(Color online) $xy$ positions of the vortices in a pure quartic trapping potential (top row) with filling fractions $f=1/36$ (left) and 1/18 (right). In the second row, the vortices are trapped by a harmonic plus quartic potential, with $\alpha =2$ in Eq. (19). Due to lower density, the visibility of the vortices close to the rotation axis is reduced for the highest temperatures. All images are averages over 100 000 Monte Carlo sweeps.

Figure 13

(Color online) Results for ${\widetilde{{\rm Y}}}_z\left(R_1,R_2\right)$, comparing the low- and high-density regions in the top and bottom panels, respectively. The lower curve $(\times )$ in both panels, as well as the images on the left, are calculated for a system with filling fraction $f=1/18$ and anharmonic density profile defined by $\alpha =2$ [see Eq. (19)]. Also included, are results for the same density profile, but with no rotation-induced vortices (upper curve, $(+)$).