Thermodynamics of a Bose-Einstein Condensate with Free Magnetization

We study thermodynamic properties of a gas of spin $3{}^{52}\mathrm{Cr}$ atoms across Bose-Einstein condensation. Magnetization is free, due to dipole-dipole interactions. We show that the critical temperature for condensation is lowered at extremely low magnetic fields, when the spin degree of freedom is thermally activated. The depolarized gas condenses in only one spin component, unless the magnetic field is set below a critical value, below which a nonferromagnetic phase is favored. Finally, we present a spin thermometry efficient even below the degeneracy temperature.

Figure 1

Phase diagram of a spin 3 BEC. The solid lines delimitate the three phases predicted for a noninteracting gas of bosons. $A$ phase: thermal gas in each Zeeman component; $B$ phase: BEC only in $m_S=-3$; $C$ phase: BEC in all Zeeman components. The histograms represent typical experimental population distributions. Black triangles are experimental $T_{c1}$ measurements (see text). The dashed lines give the evolutions predicted by theories with no interactions for a free magnetization (right), and a fixed magnetization (left), and apply in our case, respectively, to regimes above $B_c$ and (at the lowest temperatures, for which a multicomponent BEC is obtained) below $B_c$.

Figure 2

Evolution of the magnetization with temperature. Triangles: above $B_c$ ($B=0.9\mathrm{mG}$), the magnetization $M$ slowly departs from zero as the temperature is lowered. When condensation is reached $M$ decreases faster, and reaches about $-3$ for the lowest temperatures; the data agree with a theory without interactions and free magnetization (solid line, which corresponds to the right dashed line in Fig. 1). Diamonds: below $B_c$ (here $B=B_{\min }$), the magnetization remains almost constant and significantly different from $-3$, even at extremely low temperature (the horizontal solid line shows its average value).

Figure 3

Evolution of the condensed fraction $\eta$ with temperature. Filled triangles: for a polarized sample ($B=20\mathrm{mG}$), the usual $1-(T/T_{c0}{)}^3$ (corresponding to the right solid line) is obtained. Open triangles: at $B_{\min }$ (below $B_c$), the condensed fraction gets smaller at every temperature, as the spin degree of freedom is unfrozen. The left solid line corresponds to a theory with no interactions but fixed magnetization.

Figure 4

Comparison of the cloud temperatures deduced from a time of flight analysis ($T_{\mathrm{TOF}}$) with those of a spin components analysis ($T_{\mathrm{spin}}$), for $B=0.9\mathrm{mG}$. Inset: spin components populations. The magnetic field is set above $B_c$ ($B=0.5\mathrm{mG}$), so that only the thermal fraction gets depolarized. The temperature is below $T_{c1}$ ($T=160\mathrm{nK}$), hence the large $m_S=-3$ component. The other $m_S$ components almost obey a Boltzmann statistics [18], as shown by the LogPlot, from which $T_{\mathrm{spin}}$ is deduced.