A general variational approach for equilibrium phase boundaries of trapped spin-1 Bose-Einstein condensates

Abstract

We develop a simple and general variational method to estimate the solutions of the Gross-Pitaevskii equations and obtain the corresponding density profiles for all spin states of a trapped spin-1 Bose-Einstein condensate. We further employ this approach to obtain the complete phase diagram of the system under quasi-one-dimensional harmonic confinement, with ferromagnetic or antiferromagnetic spin interactions. We identify a suitable scaling that collapses all phase diagrams for different system sizes (i.e., total particle number) into a universal (system size-independent) phase diagram. The complete phase diagram for a confined system shows some significant qualitative differences compared to that of a condensate with homogeneous density distribution. The phase diagrams reported here could help identify the important parameter regimes in which phase transitions in the confined system, in general, occur. This knowledge of the region of phase boundaries can enable a reliable investigation of the instabilities near the boundaries that drive phase transitions.

Figures (5)

• Figure 1: Phase diagram in linear ($p$) and quadratic ($q$) Zeeman terms for spin-1 condensate in the absence of any trapping potential, for (a) antiferromagnetic-type, (b) in-the-absence-of, and (c) ferromagnetic-type of spin interaction.
• Figure 2: The total density $u_{tot}$ and sub-component densities $u_1$ and $u_{-1}$ as a function of the distance $\zeta$ from the centre of the quasi-1D harmonic trapping, obtained from the (a) variational method, (b) Thomas-Fermi approximation (in green-solid, red-dash-dot and blue-dashed line respectively), are compared against the same obtained from numerical simulation (in green-circle, red-triangle and blue-square respectively) for the antiferromagnetic state at $p'=0.2$ and $q'=-0.5$.
• Figure 3: (a) Phase boundary between anti-ferromagnetic and polar state with varying condensate particles of a spin-1 condensate with anti-ferromagnetic type of spin interaction. (b) The same type of phase boundary with varying spin-interaction strength for a fixed condensate particle of $N=30000$. (c) The scaling factor $A(N,\lambda_1)$ scales the phase boundary of the anti-ferromagnetic state and the polar state as $p'^2 = A(N,\lambda_1) q'$. The variation of scaling factor $A(N,\lambda_1)$ with the number of condensate particles $N$ and (d) with the spin-dependent interaction $\lambda_1$. (e) The universal phase boundary obtained after scaling the linear and quadratic Zeeman terms with the scaling factor $\lambda_1 N^{2/3}$.
• Figure 4: (a) The phase boundary between Ferromagnetic (F1) and polar states for different $N$ values. (b) Complete phase diagram with $\lambda_1>0$ for trapped condensates with varying values of $N$ in $(q',\:p')$ parameter space. (c) Universal phase diagram for all $N$ and $\lambda_1$ after scaling the linear and quadratic Zeeman strengths.
• Figure 5: For the ferromagnetic type of spin interaction $\lambda_1<0$, (a) the phase boundary between the polar and PM phase for a trapped spin-1 condensate for different values of $N$ at $\lambda_1<0$. These phase boundaries are hyperbolic in nature and follow $p'^2=k q'^2-B(N,\lambda_1)$. (b) Variation of the scaling factor $B(N,\lambda_1)$ with $N$ for a fixed value of $\lambda_1$, shown in a log-log plot to get the power-law dependence. (c) Universal Polar-PM phase boundary obtained after scaling of $p'$ and $q'$ axes with $N^{2/3}$. The phase boundary between the (d) ferromagnetic and PM state, and (e) ferromagnetic and polar state, for different values of $N$. (f) The complete phase boundary for $\lambda_1<0$ for trapped condensates with varying values of $N$. (g) The universal phase diagram for $\lambda_1<0$, after scaling the $p'$ and $q'$ axes with $N^{2/3}$, shows the region of existence for different states.