From Classical to Quantum Glasses with Ultracold Polar Molecules

We study the dynamics of a bilayer system of ultracold polar molecules, which exhibits classical and quantum glassy behavior, characterized by long tails in the relaxation time and dynamical heterogeneity. In the proposed setup, quantum fluctuations are of the order of thermal fluctuations and the degree of frustration can be tuned by the interlayer distance. We discuss the possible observation of a glassy anomalous diffusion and dynamical heterogeneity in experiment using internal degrees of freedom of the molecules in combination with optical detection.

Figure 1

Bilayer setup of trapped polar molecules with parallel dipole moments (a) and antiparallel dipole moments (b). The lower layer represents a high density dipolar crystal, the upper layer is loaded with low density (see text). A typical route to enter the glass phase is to quench the system from the liquid to a supercooled phase. In contrast, here the glass phase is reached by decreasing the layer separation $s$. (c) Sketch of the phase diagram of a single layer of bosonic polar molecules with the ratio between kinetic energy and interaction $K$ and $r_d$ from Ref. 10. Note that the limit $m\to \infty$ corresponds to the classical limit as the extension of the wave function vanishes with ${\lambda }_{dB}=h/(2\pi m{k}_{B}T{)}^{1/2}$. The arrow indicates the parameter region of interest. (d) For parallel dipoles [(a)], the effective volume and symmetry of the defects depends on the interlayer separation. The ground state from a classical simulated annealing calculation at $T=0$ includes patterns of triangular and cubic symmetries where particles in the crystal layer (red) are displaced by the molecules in the defect layer (blue). (e) For antiparallel dipoles [(b)], defect patterns can range from triangular, cubic to fivefold symmetries.

Figure 2

Glass order parameters in the classical (top) and semiclassical regime from path integral calculations (bottom). (a) In the glass phase, the average mean squared displacement of the particles deviates from the constant diffusion. In the antiparallel setup the plateau extends over the whole sampling time below $K<0.02$ of the combined system. (b) The relaxation of the time dependent structure factor $F(k^{*},t)$ for $s=0.5a$ with $d_B=4d_A$ (black) and $d_B=-d_A$ (red) diverges when approaching the glass phase [symbols as in (a)]. (c) Relaxation dynamics with initial temperature $K=0.01$ in the antiparallel setup for various choices for $r_d=Dm/[{\hslash }^2⟨a⟩]$ of the combined system, with $⟨a⟩$ the average distance in the $xy$ plane. Parameters are chosen such that the system enters the glass phase in the classical limit $m\to \infty$ and melts with smaller $r_d$ due to quantum fluctuations (blue). This allows one to study the crossover from a classical glass to a quantum phase. (d) Relaxation dynamics as a function of the layer separation with effective $r_d\approx 20–25$ and $K=0.02$ in the antiparallel setup. While for distance $s=0.5a$ the system is a liquid due to large quantum fluctuations (blue), one reaches the glass phase when approaching $s=0.2a$ (black).

Figure 3

Dynamical heterogeneity of a quantum glass in the proposed setup. Molecules in an excited internal state act as marker molecules to identify glassy dynamics. (a) Typical initial configuration of layer $A$ (red) and layer $B$ (blue) depicted as the projection of the paths onto real space. (b) The parameter ${\psi }_6^i$ measures the relative order in the vicinity of particle $i$ (see Supplemental Material [38]). Here, regions of large (small) order are regions of small (large) classical diffusion and a small (large) extension of the particle wave function. (c) Snapshots of two possible initial choices of marker molecule positions (green and orange) at $\Delta t=0$ with $R_g=R_0$. The particle positions indicated by spheres represent the maximum of the path probability in layer $A$ (gray) and layer $B$ (light blue). (d) Indicating dynamical heterogeneity, the radius of gyration of marker molecules after time $\Delta t=1.5\times {10}^2$ differs depending on the initial position. The cloud of marker molecules placed in an inactive region (green) of the glass follows the dynamics of an amorphous crystal $R_g=R_0$ while molecules in a mobile region (orange) show a fast increase in size (inset). Note that dynamical heterogeneity is a general feature of glasses, which is also present in systems without apparent partial crystallinity (see Supplemental Material [38]).