Quantum Phase Transitions and Continuous Observation of Spinor Dynamics in an Antiferromagnetic Condensate

Condensates of spin-1 sodium display rich spin dynamics due to the antiferromagnetic nature of the interactions in this system. We use Faraday rotation spectroscopy to make a continuous and minimally destructive measurement of the dynamics over multiple spin oscillations on a single evolving condensate. This method provides a sharp signature to locate a magnetically tuned separatrix in phase space which depends on the net magnetization. We also observe a phase transition from a two- to a three-component condensate at a low but finite temperature using a Stern-Gerlach imaging technique. This transition should be preserved as a zero-temperature quantum phase transition.

Figure 1

Faraday signal (proportional to $⟨F_x{⟩}^2$) taken from a single measurement for $m=0$ at two magnetic fields 26 and $40\mu \mathrm{T}$ starting with ${\rho }_0=0.5$, $\theta =0$. The solid line is a fit with a damped sinusoid. The signals show (a) an oscillating phase and (b) a running phase at $B$ below and above $B_c$, respectively, as evidenced by the signal reaching zero or not.

Figure 2

Equal-energy contour plots generated from Eq. (1) at two magnetic fields 26 (left) and $40\mu \mathrm{T}$ (right), with $m=0$ and $c/h=33\mathrm{Hz}$. Dashed (red) lines represent the energy of a state with ${\rho }_0(t=0)=0.5$. Heavy (blue) solid lines represent the energy of the separatrix ($E_{\mathrm{sep}}$) between oscillating and running phase solutions. Darker colors represent lower energies.

Figure 3

The minimum of the Faraday signal as a function of magnetic field for $m=0$ (red dots) and $m=0.3$ (blue bowties). A scale factor is applied to the Faraday signal to correct for the PD response at different $f_L$. The lines are fits based on Eq. (2), which yield the fit parameters ${\rho }_0=0.42$ and $N=1.50\times {10}^5$ for $m=0$ and ${\rho }_0=0.54$ and $N=1.32\times {10}^5$ for $m=0.3$. The fit parameters are within the 3% uncertainty of those derived from absorption images.

Figure 4

Evidence of phase transitions in the mean-field ground state of the antiferromagnetic spinor BEC. (a) Points indicate ${\rho }_0$ as a function of $B$ at $m=0.71(2)$. The intersection of two linear fits defines $B_{2-3}$, the critical magnetic field for the phase transition. (b) Inset: ${\rho }_0$ versus $m$ for $B=21.2\mu \mathrm{T}$. The intersection of two linear fits defines $m_{2-3}$, the critical magnetization. The main figure shows $m_{2-3}$ versus $B$. The solid line is the prediction from the SMA.