Parametric Instabilities of Interacting Bosons in Periodically Driven 1D Optical Lattices

Periodically driven quantum systems are currently explored in view of realizing novel many-body phases of matter. This approach is particularly promising in gases of ultracold atoms, where sophisticated shaking protocols can be realized and interparticle interactions are well controlled. The combination of interactions and time-periodic driving, however, often leads to uncontrollable heating and instabilities, potentially preventing practical applications of Floquet engineering in large many-body quantum systems. In this work, we experimentally identify the existence of parametric instabilities in weakly interacting Bose-Einstein condensates in strongly driven optical lattices through momentum-resolved measurements, in line with theoretical predictions. Parametric instabilities can trigger the destruction of weakly interacting Bose-Einstein condensates through the rapid growth of collective excitations, in particular in systems with weak harmonic confinement transverse to the lattice axis. Understanding the onset of parametric instabilities in driven quantum matter is crucial for determining optimal conditions for the engineering of modulation-induced many-body systems.

Popular Summary

New states of matter can arise when a system is exposed to a periodic drive, such as by irradiating a solid with light or by mechanically shaking a gas of ultracold atoms. While experiments have validated this method, the target often exhibits a drastically reduced lifetime because it continuously absorbs energy from the drive. Moreover, in systems of bosonic particles the periodic drive can trigger the rapid growth of collective excitations, known as parametric instabilities, which typically decay and result in additional heating. Here, we report on observations of collective modes in the early-stage dynamics of ultracold bosons in shaken optical lattices.

In our experiment, we monitor the momentum distribution of a Bose-Einstein condensate after releasing it from the lattice. The appearance of additional momentum components upon shaking directly signals the presence of collective excitations. The dominant mode is known to appear at very specific momenta, which depend on the parameters of the drive. Our observations confirm recent theoretical predictions and directly show the important role played by parametric instabilities during early evolution times for a wide class of shaken quantum systems with bosonic elementary excitations.

Understanding the onset of parametric instabilities in driven quantum matter is crucial for determining stable parameter regimes, thereby opening the door to a panoply of exciting new phenomena including the engineering of genuine out-of-equilibrium many-body systems with exotic properties that have no static counterpart.

Figure 1

Illustrations of the experimental setup and the properties of parametric instabilities. (a) Schematic of the driven 1D optical lattice with tunneling $J$, generated by two laser beams with frequencies ${\omega }_{1,2}$, and weak harmonic transverse confinement. Modulating ${\omega }_2(t)$ periodically with frequency $\omega$ generates a force $F(t)$ with period $T=2\pi /\omega$. (b) Illustration of the effective 1D Bogoliubov dispersion $E_{\mathrm{eff}}^B(q^x,J_{\mathrm{eff}})$, with width $W_{\mathrm{eff}}$ reduced according to the Floquet-renormalized tunnel coupling $J_{\mathrm{eff}}$ (blue) compared to the static lattice case (gray). Parametric resonances at $E_{\mathrm{eff}}^B=\hslash \omega$ induced by the modulation lead to instabilities centered around the most unstable mode $q_{\mathrm{mum}}^x$; $\hslash =h/(2\pi )$ is the reduced Planck’s constant. (c) Momentum of the most unstable mode, ${\mathbf{q}}_{\mathrm{mum}}=(q_{\mathrm{mum}}^x,{\mathbf{q}}_{\mathrm{mum}}^{\perp })$, as predicted by Bogoliubov theory, which shows a clear separation between lattice (I) and transverse (II) degrees of freedom (d.o.f.), that occurs at the saturation frequency $\hslash {\omega }_{\mathrm{sat}}\approx W_{\mathrm{eff}}$; $d$ is the lattice constant.

Figure 2

Momentum-resolved images of the most unstable mode for $\alpha =1.78$, $\omega /(2\pi )=720\mathrm{Hz}$, and $\phi =0$. (a) Time series of absorption images after 6 ms TOF. The edges and the center of the first Brillouin zone (BZ) are marked by dashed lines. (b) Difference image at $t=14T$, obtained by subtracting the condensate profile at $t=0$. The lower panels show the 1D profiles along $q^x$, resulting from summation of the difference image perpendicular to the lattice axis for the left and right excitation peak together with asymmetric Lorentzian fits (black solid lines), used to determine the amplitude $A^x$ and position $q_{\max }^x$ (dashed lines) of the excitation peaks [52]. The left- and right-hand plots show the profiles resulting from summation along the lattice axis with Lorentzian fits to determine the transverse full width at half maximum $\mathrm{\Delta }{q}^y$ (dashed lines) of the peaks. Here, we integrate the density profile in a region of interest containing the left or right peak only.

Figure 3

Time-resolved growth of the most unstable mode for $\alpha =1.78$ and $\omega /(2\pi )=720\mathrm{Hz}$. (a) Logarithmic plot of the peak amplitude $A^x$ as a function of modulation time. The solid lines are fits to the data to extract the end of the short-time regime $t_s$ (black dashed line). (b) Position of the peak maximum $q_{\max }^x$ along $q^x$ as a function of modulation time. The most unstable mode $q_{\mathrm{mum}}^x$ is defined as the weighted mean of all $q_{\max }^x$ for $t\le t_s$. Its value is shown as the solid black line; the shaded blue bars denote its standard error of the weighted mean. The shaded gray area defines the range of hold times over which the transverse width $\mathrm{\Delta }{q}^y$ is averaged. Each data point is an average over $\sim 10$ individual experimental realizations; error bars indicate the standard deviation. To minimize systematic deviations in the band-mapping images, we average $\lesssim 5$ realizations for a modulation phase $\phi =0$ and $\lesssim 5$ with $\phi =\pi$ (Fig. S1 of Supplemental Material [52]).

Figure 4

Position of the most unstable mode and influence of transverse modes. Upper panel: Position of the most unstable mode $q_{\mathrm{mum}}^x$ as a function of modulation frequency for $\alpha =1.44$ and $\alpha =1.78$. The error bars denote the standard error of the mean. The solid lines show Eq. (3) for $g=11.5J$. The vertical dashed lines mark the corresponding saturation frequency. Lower panel: Transverse width $\mathrm{\Delta }{q}_{\mathrm{lin}}^y(\omega )$. Error bars denote the standard error of the weighted mean. The thin lines are guides to the eye.

Figure 5

Comparison of the perturbative analytic treatment calculated to zeroth [solid blue line, Eqs. (3) and (4)] and second order (dashed blue line) with numerical simulations of the exact BdG equations (green dots) for $g/J=11.5$ and two modulation amplitudes, $\alpha =1.44$ (left) and $\alpha =1.78$ (right). The steps in the numerical results are due to the discretization in momentum space.

Figure 6

Energy spectrum of the noninteracting time-averaged Hamiltonian (orange dots) and corresponding effective Bogoliubov dispersion (blue dots) for $\alpha =1.4$ and $g_{\max }=10.2J$ in a harmonic trap with ${\omega }_x=0.26J/\hslash$. The dashed lines indicate the respective bandwidths in the homogenous case, i.e., $\hslash {\omega }_{\mathrm{sat}}=2.2J$ (orange) and $\hslash {\omega }_{\mathrm{sat}}=7.17J$ (blue) according to Eq. (b1). Inset: As the energy reaches the homogenous bandwidth, the spectrum becomes quasidegenerate.

Figure 7

Numerical simulation of the saturation frequency ${\omega }_{\mathrm{sat}}$ versus $\alpha$ (dots) compared to the BdG prediction (solid line) from Eq. (b1). The presence of a harmonic trap does not affect the saturation frequency significantly. A slight deviation from the BdG prediction is observed compared to the numerical simulation for large values of $\alpha \sim 2$ and sufficiently strong effective interaction $g/J$. The trapping frequency is ${\omega }_x=0.26J/\hslash$, the system size is $L_x=201d$, and the atom number is $N_0=1000$.

Figure 8

Numerical simulations of the instability rate in homogeneous 1D lattices (light red and light blue, $g=10.2J$) in comparison with the rates in 1D lattices with harmonic confinement (dark red and dark blue, ${\omega }_x=0.26J/\hslash$ and $g_{\max }=10.2J$) for two different driving parameters, $\alpha =1.4$ (blue) and $\alpha =1.8$ (red). The system parameters are the same as for the simulations in Fig. 7; i.e., $L_x=201d$ and $N_0=1000$. The instability rate was extracted from an exponential fit to the numerical data from the last five out of 24 driving cycles of evolution, to make sure the maximally unstable mode dominates. The dashed vertical lines show the BdG predictions for ${\omega }_{\mathrm{sat}}$ at $g=10.2J$.

Figure 9

Numerical simulations of the occupation of the most unstable mode $n_{q_{\mathrm{mum}}}$ for a homogeneous 2D system (1D lattice and one continuous direction) for $g=9.52J$, $\hslash \omega =9.25J$, and $\alpha =1.44$. The solid line displays the rate obtained from the analytic formulas in Eqs. (a5) and (a6), which is in agreement with the BdG simulations (dark blue) for $t\gtrsim 5T$. The TWA (blue) and WCCA (light blue) partially capture additional nonlinear effects, which result in a time-dependent instability rate.

Figure 10

Instability rates extracted from linear fits to the logarithmic peak amplitudes as a function of modulation time for $\alpha =1.44$ and $\alpha =1.78$. The dark solid lines are the theoretical rates calculated from Eqs. (a5) and (a6) for $g=11.5J$, and the bright solid lines are a FFGR calculation for the same modulation parameters according to Eq. (c1). The inset shows the same data on a linear scale to reveal the frequency dependence of the measured rates. Error bars indicate the standard deviation.