Superfluid-supersolid phase transition of elongated dipolar Bose-Einstein Condensates at finite temperatures

We analyse the finite-temperature phase diagram of a dipolar Bose Einstein Condensate confined in a tubular geometry. The effect of thermal fluctuations is accounted for by means of Bogoliubov theory employing the local density approximation. In the considered geometry, the superfluid-supersolid phase transition can be of first- and second-order. We discuss how the corresponding transition point is affected by the finite temperature of the system.

For the same reasons, thermal fluctuations can also have substantial effects on the phases of dipolar quantum gases, even well below the condensation temperature [38][39][40][41].In particular, recent calculations showed how heating a dipolar superfluid can induce a transition to a solid phase with a periodically modulated condensate density [41].
In this work, we use Bogoliubov theory [38,39] to numerically study finite-temperature effects on a dipolar BEC with strong transverse confinement in the thermodynamic limit (see Fig. 1).Recent studies of the zerotemperature phase diagram showed that the superfluidsupersolid phase-transition of this system can of first-as well as of second-order [17,18,42,43].Here, we explore Schematic representation of the dipolar system in the tubular geometry.Dipoles are polarized along the z axis, while the system extends infinitely along the x axis.By keeping the number of condensed atoms fixed and increasing the temperature, the system transitions from an unmodulated gas (b) to a supersolid (a).The trapping strengths in the y − z plane are given by ω ⊥ = 0.165ϵ d /ℏ, with ) and a d denoting the dipolar length.
the effects of thermal fluctuations on the nature of the phase transition.

II. FINITE TEMPERATURE THEORY
Detailed discussions of finite-temperature effects in dilute Bose Einstein condensates can be found, e.g., in [44][45][46][47][48].In order to account for the effect of thermal fluctuations in dipolar BECs, we apply Bogoliubov theory and use local density approximation [38,41].This yields a temperature-dependent extended Gross-Pitaevskii equation (TeGPE) given by µψ for the condensate wave function ψ(r).Here, µ is the chemical potential, m is the mass, U describes de trapping potential, V dd denotes the dipolar interactions and a corresponds to the s-wave scattering length.Furthermore, H qu and H th describe effective nonlinear potentials that arise from quantum fluctuations and thermal fluctuations, respectively.They are given by where and T denotes temperature.Ṽ (k) represents the Fourier transform of the sum of the dipole-dipole interaction (DDI) and the contact interaction, given by The parameter a d = mC dd /(12πℏ 2 ) corresponds to the dipolar length, C dd is the strength of the dipolar interaction, and the auxiliary function The term in Eq. 2 that accounts for quantum fluctuations acts as a defocusing nonlinearity whose strength increases with the condensed density and is responsible for arresting collapse as discussed above [22,49,50].Thermal fluctuations, as described by Eq. 3, on the other hand, generate a focusing nonlinearity that decreases with increasing density ρ [38,41].Fig. 2 illustrates this density dependence and shows that the total fluctuation energy, H fl = H qu + H th , features a minimum that shifts towards higher densities with increasing temperature.
The evaluation of Eq. 3 for values of the scattering length that are lower than the dipole length requires special attention since the integrand can in this case become complex.This reflects the instability of a homogeneous condensate, as the excitation spectrum, ε k , turns imaginary for small momenta and a < a d .The finite transverse size of the partially confined condensate, however, introduces a natural low-momentum cut-off for the considered system.Due to the symmetry of the dipole-dipole interaction, the contribution to the fluctuation energies depends only on k z and k ρ = k 2 x + k 2 y .Considering radial confinement as shown in Fig. 1 with typical system sizes l y and l z along the y-axis and the z-axis, respectively, one obtains lower bounds, k z > 2π/l z and k ρ > 2π/l y , for both momenta.Here, we use k z > 0.007/a d and k ρ > 0.017/a d and have checked that a 30% increase of these values does not significantly affect the numerical results.

III. FINITE-TEMPERATURE PHASE-DIAGRAM
Equation 1 can be solved numerically by imaginary time evolution to obtain the condensate wave function ψ for a finite temperature, T , and a fixed condensate density or chemical potential, µ. Figure 3 shows the contrast where ρ max and ρ min denote the maximum and minimum of the axial density ρ(x) = dydz|Ψ(r)| 2 along the xaxis.The latter is also used to define the overall axial density ρ = L −1 L 0 dxρ(x) for a given value of the length L of the simulation box along the x-direction.
The axial density contrast vanishes in the superfluid phase, for large ratios a/a d in Fig. 3, and takes on finite values below a critical value of a/a d as one enters the solid phase with finite density modulations.For the chosen density of ρa d = 4.77, the zero temperature simulation yields a discontinuous increase of the contrast, characteristic for a first order phase transition.For a finite temperature of k B T /ϵ d = 2, however, one finds a second order phase transition with a continuous rise of the density contrast.Apart from shifting the transition point, thermal fluctuations, therefore, may also qualitatively affect the fluid-solid phase transition.
Thermal effects are further illustrated in Fig. 4, where we show the contrast across the phase transition as a function of temperature for two different values of the condensate density.In all depicted cases, heating the system induces a transition to a density-modulated state, irrespective of the order of the transition.Around the second order phase transition, one finds moderately modulated states with a density contrast that remains significantly below unity.Concurrently, such states are expected to feature a substantial superfluid fraction [51][52][53] and should, therefore, realise a supersolid.On the other hand, a direct first order transition from a superfluid to a solid with near unit modulation contrast and no global superfluidity should eventually occur with decreasing density.
Figure 5 provides a more complete picture of the fluidsolid transition, showing the phase diagram at zero temperature and k B T /ϵ d = 2 as a function of the density and the competing interaction strengths.The chosen parameters lie in typical regimes of current experiments, e.g., whereby the temperature corresponds to T ≃ 87nK for a quantum gas of 164 Dy atoms.The calculations show that such low temperatures do not qualitatively alter the phase diagram compared to the ground state behaviour of dipolar condensates, discussed recently in [18,42].
As the temperature is increased, the solid-fluid transition line shifts towards larger values of the scattering length a. Starting from the superfluid phase close to the quantum phase transition (T = 0) and increasing the temperature, therefore, leads to the emergence of a solid phase upon heating the system regardless of the precise values of the otherwise fixed parameters (i.e., a, a d , and ρ).This effect, which has been reported in recent experiments with 164 Dy atoms [41,54], can be understood from the characteristic density dependence of the energy, H th , of thermal fluctuations shown in Fig. 2. A decreasing energy with increasing density, |ψ| 2 , implies  that H th [|ψ(r)|] acts as a focusing nonlinearity in the generalized GPE [41] and, therefore, tends to support the density-modulated phase.
One observes in Fig. 5 a convergence of the phase boundaries for the two different temperatures for large densities, which shows again that thermal effects on the phase boundary weaken as the density increases.This can again be readily understood from Fig. 2, which shows that quantum fluctuations yield the dominant contribution to the energy correction H fl at higher densities.While thermal fluctuations always shift the phase boundary towards larger scattering lengths, a, their effect on the critical density depends on the density itself.At lower densities, where the energy corrections from quantum fluctuations and thermal fluctuations are comparable, the phase boundary is shifted towards lower densities and thereby facilitates the formation of the solid phase.On the contrary, at higher densities, where quan- tum fluctuations dominate the energy correction, H fl , a larger temperature requires an increased density to form a modulated state.Yet, heating still facilitates the solid phase, since the critical scattering length decreases with density in this regime (see Fig. 2).This effect is illustrated in Fig. 7, where we show the contrast as a function of ρ in the two different density regimes.
We finally discuss the order of the phase transition and how it is affected by thermal fluctuations.At very low densities, the transition is of first-order type but turns into a continuous second order phase transition with increasing density.Eventually, the phase transition becomes once again discontinuous in the high density regime.This general phenomenology of the quantum phase transition (T = 0) [18,42] prevails at finite temperatures, while thermal fluctuations can shift the critical points at which the order of the phase transition changes.
At the low density critical point (cf.Fig 5) the change of the critical value of the scattering length ((a/a dd ) = 0.655 ± 0.005) is small compared to the experimental resolution for the values considered [cf.Fig. 7(a)].In contrast to that, one finds a substantial effect on the critical density, which decreases significantly with increasing temperature of the BEC [cf.Fig. 7(b)].On the other hand, the shift in the critical point is less pronounced in the high density regime, as can be seen in Fig. 5.This is a consequence of the aforementioned weakening of thermal effects for increasing condensate density.

IV. CONCLUSIONS
In conclusion, we have characterized finitetemperature effects in the phase diagram of an elongated dipolar BEC.We have shown that an increase of temperature at constant condensed density yields a transition from an unmodulated superfluid to a supersolid, thus pushing the solid-fluid boundaries of the zero temperature diagram to higher values of the scattering length.For a sufficiently large condensed density we enter a regime where quantum fluctuations dominate and thermal effects become small in comparison.We have also seen how the low density critical point where the fluid-solid transition shifts from first to second order (or vice versa) changes with temperature and have shown that it moves to lower values of the condensed density, thus yielding a range of parameters for which temperature effectively changes the order of the phase transition.We have observed that the high density critical point experiments a less significant shift than the low density one, due to the weakening of thermal effects as the axial condensed density increases.
The role of temperature in dipolar systems still remains a relatively unexplored subject.It remains unclear how temperature will affect the different geometrical phases both in infinite and trapped quasi-2D dipolar systems [16,19,55] as well as their superfluid properties [56].Furthermore, improved theoretical calculations for specific geometries where the local density approximation is not necessary could present avenues towards more accurate quantitative predictions that can be compared with future experiments.Similarly, the realization of ab-initio calculations [25] that are able to fully account for the effect of temperature could help extending the formalism presented in this work to the high temperature regime, where the fraction of condensed atoms is small.

FIG. 2 .
FIG. 2. Density dependence of the energy contributions to the TeGPE from quantum fluctuations, Hqu, and thermal fluctuations, H th , along with the total energy H fl = H th + Hqu.The results are shown for a/a d = 0.7 and different indicated temperatures kBT /ϵ d = 1, 2, 3.

FIG. 3 .
FIG. 3. Contrast of the wave function versus scattering length in a (a) first and (b) second order phase transition for ρa d = 4.77 and two different temperatures: kBT /ϵ d = 0 (a) and kBT /ϵ d = 2 (b).

FIG. 4 .
FIG. 4. Contrast of the wave function as a function of temperature for ρa d = 6.89, a/a d = 0.676 (a) and ρa d = 4, a/a d = 0.64 (b).The lines correspond to eye-guides.

FIG. 6 .
FIG. 6. Contrast C of the wave function versus axial density in the (a) low density and (b) high density regimes as function of the density.

FIG. 7 .
FIG. 7. Scattering length (a) and density (b) of the low density critical point as a function of the temperature.