Thermalization near Integrability in a Dipolar Quantum Newton’s Cradle

Isolated quantum many-body systems with integrable dynamics generically do not thermalize when taken far from equilibrium. As one perturbs such systems away from the integrable point, thermalization sets in, but the nature of the crossover from integrable to thermalizing behavior is an unresolved and actively discussed question. We explore this question by studying the dynamics of the momentum distribution function in a dipolar quantum Newton’s cradle consisting of highly magnetic dysprosium atoms. This is accomplished by creating the first one-dimensional Bose gas with strong magnetic dipole-dipole interactions. These interactions provide tunability of both the strength of the integrability-breaking perturbation and the nature of the near-integrable dynamics. We provide the first experimental evidence that thermalization close to a strongly interacting integrable point occurs in two steps: prethermalization followed by near-exponential thermalization. Exact numerical calculations on a two-rung lattice model yield a similar two-timescale process, suggesting that this is generic in strongly interacting near-integrable models. Moreover, the measured thermalization rate is consistent with a parameter-free theoretical estimate, based on identifying the types of collisions that dominate thermalization. By providing tunability between regimes of integrable and nonintegrable dynamics, our work sheds light on the mechanisms by which isolated quantum many-body systems thermalize and on the temporal structure of the onset of thermalization.

Popular Summary

Novel quantum technologies will likely operate in regimes where the thermalization of particles must be controlled, and it behooves us to learn how interactions cause their motion to become more chaotic. Weakly interacting systems, such as planets in our Solar System, remain stable despite long-range gravitational interactions. But systems with stronger long-range interactions can exhibit irregular, chaotic motion, which thermalizes the particles. The transition between the two regimes—nonthermalizing and thermalizing—is understood for classical systems but not for the quantum world. Our work is the first to experimentally explore this question. We find that when chaotic motion is enhanced through long-range interactions between quantum particles, then the systems thermalize differently from their classical counterparts and from what some simulations had predicted.

We create the first quantum version of a magnetic Newton’s cradle. In a traditional cradle, metal balls collide with their neighbors to produce a periodic motion. In our version, we trap an ultracold gas of highly magnetic dysprosium atoms in a one-dimensional tube made of laser light. The atoms oscillate periodically when kicked, but unlike the toy, the atoms can collide and pass through each other simultaneously. When we increase the magnetic interaction strength between atoms by changing their orientation, we observe that the gas thermalizes more rapidly. Magnetic balls behave similarly in a classical Newton’s cradle, but we find that the way chaotic motion emerges differs in our quantum cradle. A complex numerical simulation points to the generality of our results.

This work provides us with new ideas about how quantum systems thermalize. These ideas may one day allow us to design useful devices for quantum memory storage and information processing.

Figure 1

Dipolar Newton’s cradle setup. (a) Two of the 1D tubes of atoms in a dipolar quantum Newton’s cradle. Application of an optical phase grating (not shown) kicks atoms along the tubes, and the weak harmonic confinement induces periodic collisions. The highly magnetic dysprosium atoms (silver spheres) are trapped in a 2D optical lattice (red) defining the tubes. (b) DDI strength between the central atom and atoms in neighboring tubes in one quadrant of the square lattice when $\theta =90°$, where $\theta$ is the angle between the B-field in the $xz$-plane and the axis of the 1D tubes along $\widehat{x}$. The patterns of strengths in the other three quadrants are the same by symmetry. The DDI strength is tunable via changing $\theta$. Numbers labeled on the atoms and the tubes are the pairwise DDI strength and integrated DDI along the tubes, respectively, in Hz. Blue indicates large positive strength, and red large negative strength. (c) Dependence of integrability-breaking dipolar interaction strength and $\gamma$ on $\theta$. Solid curves: Integrability-breaking contributions of the DDI energy $U_{\mathrm{DDI}}$ (added in quadrature and defined in Appendices pp1, pp2, pp3) versus $\theta$. Shown are the total, total intertube, and total integrability-breaking intratube DDI energies. While the intratube DDI is maximally repulsive (attractive) for $\theta =90°$ (0°), it vanishes among intratube atoms for $\theta =55°$. Integrability-breaking DDI contributions come not just from the intratube 1D DDI along $\widehat{x}$, but also from the 3D DDI between atoms in all neighboring tubes along $\widehat{y}$ and $\widehat{z}$. This dilutes the tunablility of the DDI, reducing the contrast to a factor of approximately 1.5 between $\theta =0°$ and 90°. Dashed curve: Lieb-Liniger parameter $\gamma (\theta )$; see Appendix pp4 for calculation.

Figure 2

Evolution of postkick momentum distributions at multiples of $T$. (a) 3D gas at $\theta =35°$. (b) Regime I, fast dephasing of the 1D gas at $\theta =35°$. While the momentum distribution of the 3D gas thermalizes after about $5T$, the 1D gases exhibit nonthermal (i.e., non-Gaussian) distributions far longer. The remaining panels show 1D gases in regime II for $\theta$’s of (c) 0°; (d) 55°; and (e) 90°. We diffract and evolve with $\theta =35°$ until rotation at $5T$ to the target $\theta$. As can be seen in panels (c)–(e), this procedure produces nearly identical momentum distributions after field rotation regardless of $\theta$. The color scale is proportional to the distance to thermalization. The best-fit Gaussian curve and the corresponding log(DT) value are shown for the 90° data at the earliest and latest times.

Figure 3

The full $\theta =35°$ evolution showing the boundary between regime I and regime II. The solid blue line is fit to the data between the beginning of regime II [$\log (\mathrm{DT})\approx 1.5$] and the noise floor [$\log (\mathrm{DT})=1$]. Vertical bars indicate standard error. The light blue horizontal band is the standard uncertainty of the noise floor. The inset contains regime II decay results for the same angles as in Figs. 2.

Figure 4

Thermalization rate data in red versus $\theta$. See Eq. (1) for the definition of the thermalization rate $1/{\tau }_{\mathrm{th}}$. The gray curve is the scaling estimate ${\gamma }^{\prime 2}(\theta )U_{\text{total}}^2(\theta )/(E_{\mathrm{c}})$ with no free parameters or offset. Vertical bars and the light blue band indicate standard error; atom number noise and $a_{3\mathrm{D}}$ uncertainty dominate the latter. Evidently, the thermalization rate in the dipolar quantum Newton’s cradle is well described by terms dependent on both the long-range and short-range parts of the DDI, the former through total (intertube plus intratube) $U_{\text{total}}^2(\theta )$ and the latter through the intratube DDI dependence of $a_{1\mathrm{D}}^{\text{intra}}(\theta )$ in ${\gamma }^{\prime 2}(\theta )$.

Figure 5

Two-rung lattice made of two identical chains with nearest neighbor hopping ($t$), interaction ($V$), and next-nearest neighbor hopping ($t^{\prime }$). The two chains interact along the rungs ($V_r$).

Figure 6

Numerical results for the approach to equilibrium [see Eq. (9)] in the two-rung hard-core boson model calculations with 22 lattice sites and nearest neighbor coupling $V=1.6$. The symbols show results for a quench in which the system is initialized in a state with a two-peaked momentum distribution (created through an initial Hamiltonian with strong next-nearest neighbor coupling $t^{\prime }=50$), and the integrability-breaking interaction is turned on postquench. Dashed lines show results for evolution under the same final Hamiltonian, but from an initial state that has already dephased under the fast integrable dynamics. Specifically, the initial state is a diagonal-ensemble state generated by a quench in which $t^{\prime }=50\to t^{\prime }=0$ is changed but dipolar interactions are absent. The fast dephasing at short times in the simulation depends weakly on the strength of the integrability-breaking perturbation.

Figure 7

Atom number during the momentum evolution at ${\gamma }_d^{\mathrm{VdW},\mathrm{avg}}=5.7(7)$ for $\theta =0°$ (triangle), 55° (square), and 90° (diamond).

Figure 8

Thermalization distance at $\theta =90°$ measured as a function of lattice depth for different observation times: $10T$ (triangle), $15T$ (square), and $20T$ (diamond). The blue band represents the noise floor and its $1\sigma$ uncertainty. The ${\gamma }_d^{\mathrm{VdW},\mathrm{avg}}$’s associated with the lattice depths are shown on top.

Figure 9

Measured heating rate of the undiffracted BEC at equilibrium in a $V_0=18E_R$ 2D lattice at various angles.

Figure 10

Distance-to-thermalization (DT) of the field-rotated state near the end of regime I for each $\theta$ value investigated. The blue horizontal line is the mean noise floor, and the light blue band represents its $1\sigma$ uncertainty.

Figure 11

Time evolution of DT at $\theta =0°$ for rotating the field after a waiting time of $10T$ (triangle) and $20T$ (square).

Figure 12

Time evolution of distance to dephasing (DD) predicted by the collisionless classical dynamics simulation for two cases: the first for when only process (ii) is present (circle), and second when both (i) and (ii) are present (triangle). The inset shows the dephased momentum distribution.

Figure 13

Calculated time evolution of the fractional populations of the $|\hslash (k_s\pm 2k_D)⟩$ states and the undiffracted state $|\hslash k_s⟩$ during the single-sided kick pulse sequence. See text for detailed settings for each parameter.

Figure 14

Time evolution of an initially single-sided momentum distribution, whose dephasing timescale accounts for the technical dephasing mechanisms while minimizing those from high-energy collisions.

Figure 15

Thermalization distance of synthesized symmetric distributions generated by adding to the single-sided distributions shown in Fig. 14 their own mirror images. The blue horizontal line is the mean noise floor, and the light blue band represents its $1\sigma$ uncertainty.

Figure 16

Examples of momentum distributions ($m_k$) obtained in the exact diagonalization calculations of the two-rung hard-core boson model with $L=22$. We show the momentum distributions for an initial state (labeled as “initial state”), the diagonal ensemble after the quench to the integrable part of ${\widehat{H}}_F$ (labeled as “initial DE”), the diagonal ensemble after the quench to ${\widehat{H}}_F$ (labeled as “DE”), and the grand canonical ensemble prediction for the thermal momentum distribution after the quench (labeled as “GE”). For the quenches, we set $V=1.6$, with (a) $V_r=0.2$ and (b) $V_r=0.8$. The momentum distribution of the initial state and of the diagonal ensemble after the quench to the integrable part of ${\widehat{H}}_F$ are the same in (a) and (b); only the diagonal and grand canonical ensemble predictions change due to the change of $V_r$.

Figure 17

(a) RMS distance ${\delta }_{\mathrm{GE}}(\tau )$ [see Eq. (l4)] versus $\tau$ for quenches in which $V=1.6$ and the integrability breaking $V_r$ takes different values. The results shown are for a system with $L=22$ sites. (b) RMS distance ${\delta }_{\mathrm{GE}}(\tau )$ versus $\tau$ for quenches in which $V=1.6$, $V_r=0.4$, and for different lattice sizes ($L=16$, 18, 20, and 22). Inset in (b): Distance between the diagonal and the grand canonical ensembles $\delta (\mathrm{DE}\text{−}\mathrm{GE})$ [see Eq. (l5)] versus $L$ for $V=1.6$ and $V_r=0.4$ (as in the main panel). The black dashed line depicts exponential behavior.

Figure 18

(a) RMS distance ${\delta }_{\mathrm{DE}}(\tau )$ [defined in Eq. (9)] versus $\tau$ for two-step quenches in which $V=1.6$, $V_r=0.4$, and $L=22$. The distance is computed, in all cases, from the DE result for the single quench. Results are shown when the second quench is carried out at $\tau =0.8$ and 9.0 following the first quench, and at $\tau =0$ when starting with the DE of the integrable system after the first quench (labeled as “initial DE”). The solid lines at long times depict exponential behavior indicating the same relaxation rate in all cases. (b) Distance to equilibration in single quenches for different systems sizes $L$. These results show the finite-size effect on the relaxation rates.