Magnetic resonance spectroscopy and characterization of magnetic phases for spinor Bose-Einstein condensates

The response of spinor Bose-Einstein condensates to dynamical modulation of magnetic fields is discussed with linear response theory. As an experimentally measurable quantity, the energy absorption rate (EAR) is considered, and the response function is found to access quadratic spin correlations which come from the perturbation of the quadratic Zeeman term. By applying our formalism to spin-1 condensates, we demonstrate that the EAR spectrum as a function of the modulation frequency is able to characterize the different magnetically ordered phases.

Figure 1

The MF phase diagram of spin-1 BECs with respect to the effective spin-exchange interaction $n{c}_1$ and QZ coupling $q$, which are here restricted to being weak compared with the density interaction, i.e., $\left|q\right|,n|c_1|\ll n{c}_0$. The thick-solid and dashed boundary lines denote first- and second-order phase transitions, respectively. We note that the phase diagram was first given in [8], which covers a wider parameter regime than shown here. The phase boundary between the BA and LP phase is given by $q=2n|c_1|$.

Figure 2

The EAR spectra as a function of modulation frequency for the different $\tilde{q}=q/2n\left|c_1\right|$: (a) the LP phase for $c_1,q>0$ or for $q>0$ and $q+2n{c}_1>0$, (b) the TP phase for $c_1>0$ and $q<0$, and (c) the BA phase for $c_1<0$ and $0<q<2n|c_1|$. Spectra (a) and (b) are qualitatively the same, but the intensity differs due to the different number of accessible gapful-spin modes. Spectra (c) apart from $\tilde{q}=1$ show an abrupt enhancement of the EAR around $\hslash \omega \approx 2n|c_1|$, which indicates the appearance of the contribution from $r_{{\mathrm{BA}}_d}\left(\omega \right)$.