Dynamical probing of superfluidity and shear rigidity in different phases of a dipolar Bose-Einstein condensate

TL;DR

The paper addresses how to dynamically diagnose superfluidity and rigidity in dipolar Bose-Einstein condensates across superfluid, supersolid, and droplet phases. By applying a sudden tilt to the polarization direction and tracking the resulting scissors-mode dynamics via $C_{xz}(t)$ and its spectrum, the authors show that $\Omega_{\mathrm{sc}}$ in the SF phase is largely trap-dominated and undamped, while SS and droplets exhibit damped, multi-frequency responses whose width $\Gamma$ tracks the system's rigidity. Large-angle quenches reveal elastic-to-plastic transitions at critical angles $\Delta \theta_c$, with SS and SD undergoing irreversible structural changes and the SD fragmenting into MD; the SF phase remains reversible. The results establish a practical dynamical benchmark for mapping elasticity and phase properties in dipolar BECs and motivate extensions to finite temperature and sensing applications, including vector magnetometry.

Abstract

We show that a sudden change in the polarization direction of the magnetic dipole moments of the atoms in a dipolar Bose-Einstein condensate (BEC) can serve as a useful probe to sense its superfluid and solid-like properties. We find that for small angular deviation of the polarization direction, actuated for instance by modifying an external magnetic field, the superfluid state undergoes an undamped scissors mode oscillation, a characteristic signature of superfluidity. In contrast, both the droplet and supersolid states exhibit a scissors-mode oscillation, which is effectively damped due to multiple closely spaced frequency components. Notably, we find that this damping rate provides a direct quantitative measure for the rigidity of different phases of a dipolar BEC. Furthermore, there exists a maximum angular deviation of the polarization direction, beyond which the droplet and the supersolid states undergo a permanent deformation i.e., we find an analog of the usual elastic to plastic phase transition of solids. We characterize this transition numerically using the fidelity of the condensate wavefunction with the ground state as well as the droplet width and periodicity of the supersolid density of the condensate which are experimentally accessible. Thus, the technique introduced here can be an important experimental benchmark to identify and characterize the superfluid and solid properties of different phases of dipolar BECs.

Figures (9)

• Figure 1: The panels $\rm{(a-c)}$ demonstrate the temporal evolution of the scissors mode in the $x$-$z$ plane for (a) the SF ($a_s=120a_0$), (b) SS ($a_s=90a_0$) and (c) the SD ($a_s=70a_0$) state, respectively, following a sudden change in polarization direction by $\Delta \theta=2^{\circ}$. The insets in $\rm{(a- c)}$ display an enlarged view of $C_{xz}(t)$ within the time interval from $t =0-50$ ms. Panel (d) depicts the corresponding frequency spectrum $\tilde{C}_{xz}(\omega)$ obtained by the Fourier transformation of the signal $C_{xz}(t)$ shown in $\rm{(a-c)}$. Panel (e) shows the scissors mode frequency as a function of $a_s$. Brown star-shaped markers correspond to the prominent frequency of the signal $C_{xz} (t)$ and the cyan colored circles correspond to the frequency estimated by the sum rule (see Appendix \ref{['A']}). Panel (f) illustrates the variation of the superfluid fraction $f_s$ and the HWHM $\Gamma$ of the dominant peak in the Fourier spectrum $\tilde{C}_{xz}(\omega)$, which quantifies the damping rate of the corresponding state. The background color shading in $\rm{(e)–(f)}$ delineates the different phase domains obtained from the imaginary-time evolutions of the eGPE.
• Figure 2: Time evolution of the fidelity of (a) SF, (b) SS and (c) SD state under the influence of the sudden change in polarization direction. The different colored lines represent different angular deviations, as shown in the legend. $\rm{(d i-d iv)}$ Represent the density profiles $n(x,y,z=0,t)$ of the SF state under a $\Delta \theta = 20^\circ$ angular deviation, resulting in $\mathcal{F} = 1$, thereby no structural change occurs. $\rm{(e i-e iv)}$ Illustrate the density profile of the SS state at different times after the sudden $\Delta \theta = 20^\circ$ change in polarization direction, resulting in a structural deformation of the crystalline structure, with $0<F < 1$. $\rm{(f i-f iv)}$ Depict the density profile of the SD state at different time intervals following a sudden change in the polarization direction of $\Delta \theta = 15^\circ$, causing permanent deformation ($\mathcal{F} = 0$) and transition to the MD state. Insets in $\rm{(d-f)}$ demonstrate the corresponding density distribution in the $y=0$ plane.
• Figure 3: Variation of the average rms width $\bar{\sigma}_x$ along the $x$-axis (solid dark blue line with circular markers) and the average rms width $\bar{\sigma}_z$ along the axial direction of the condensate initially in the SD phase (brown dashed line with star shape markers) as a function of the $\Delta \theta$ following the sudden change in polarization direction. The horizontal black dashed line indicates the value of axial width of the ground state in the presence of a static magnetic field along the $z$-axis. The shaded regions indicate the elastic ($\bar{\sigma}_x <\bar{\sigma}_z$) and plastic ($\bar{\sigma}_x >\bar{\sigma}_z$) regimes of the SD state, with the vertical dashed line marking the transition point from elastic to plastic phase domain. The insets demonstrate the temporal evolution of the integrated density profile $n (x)/n_{\rm max}$ corresponding to the highlighted points in the main plot.
• Figure 4: (a) Temporal evolution of the integrated density profile $n (x)/n_{\rm max}$ for the SS subjected to a sudden angular deviation of $\Delta \theta=30^{\circ}$ in the polarization direction. Panel (b) displays an enlarged view of the temporal evolution of the density profile within the interval $t=0-50\rm{ms}$. Panel (c) demonstrates the spatial density distribution $n (x)/n_{\rm max}$ at time $t=0 \rm{ms}$ and $t=199.8 \rm{ms}$. The corresponding correlated density peaks are indicated by green and red circular markers, consistent with those shown in panel (a). (d) Variation of the density peak separation, measured relative to its initial value, as a function of the angular deviation $\Delta \theta$.
• Figure 5: (a) Temporal evolution of the rms width $\sigma_x$ for the SD state subjected to a linear change in the polarization angle with $\Delta \theta = 15^{\circ}$ for different quench times. The black dashed line represents the axial width $\sigma_z$ of the SD ground state. (b) The corresponding fidelity dynamics for the same set of linear ramps, highlighting the effect of the quench time on the elastic to plastic phase transition. The dynamics are shown for five distinct linear ramps associated with different quench times, as indicated in the legend of panel (b).