Instabilities of interacting matter waves in optical lattices with Floquet driving

We experimentally investigate the stability of a quantum gas with repulsive interactions in an optical one-dimensional lattice subjected to periodic driving. Excitations of the gas in the lowest lattice band are analyzed across the complete stability diagram, from slow to fast driving frequencies and from weak to strong driving strengths. To interpret our results, we expand the established analysis based on parametric instabilities to include modulational instabilities. Extending the concept of modulational instabilities from static to periodically driven systems provides a convenient mapping of the stability in a static system to the cases of slow and fast driving. At intermediate driving frequencies, we observe an interesting competition between modulational and parametric instabilities. We experimentally confirm the existence of both types of instabilities in driven systems and probe their properties. Our results allow us to predict stable and unstable parameter regions for the minimization of heating in future applications of Floquet driving.

Figure 1

Experimental setup and absorption images. (a) Sketch of the experimental setup. (b) Absorption images showing the real momentum distribution after release from the lattice (average of eight repetitions, images are compressed horizontally by factor 0.55). Red and blue patches indicate atoms in the carrier and in excited modes of the wave packet, respectively. (c) Integrated density profile to count the atoms in regions A and B.

Figure 2

Measured stability diagram for a driven BEC in a 1D lattice potential. The number of atoms, $N_A$, close to the initial quasimomentum of the quantum gas is measured after driving for approximately 30 ms with strength $K$ and driving period $T_D$. Colors from yellow to blue indicate stable to unstable regions. Left and right panels indicate different initial states with $k_0=0$ and $k_0=k_L$ for positive and negative values of $J_{\text{eff}}$.

Figure 3

Excitation energy of phonon modes. (a) Real part of the energy for the phonon (solid lines) and antiphonon (dashed lines) modes for $k=0$ (dark blue) and $k=0.3k_L$ (light blue) in Eq. (3). All imaginary components are zero (red line) for parameters $V=9E_r, U/J=1$. (b) Phonon and antiphonon modes for $k=0.9k_L$. The energy's imaginary component is nonzero when real energies of phonon and antiphonon modes match.

Figure 4

Parametric resonances. (a) Calculated excitation energy $\hslash {\omega }_q^f$ of modes with momentum $q$ ($V=8E_r, U=4J, K=1.4$). Resonances occur when integer multiples of the driving frequency ${\omega }_D$ match ${\omega }_q^f$. (b) Calculated growth rate $\mathrm{\Gamma }$ for excitation modes with momentum $q$ and driving frequency ${\omega }_D$. Red lines indicate the frequency-matching condition ${\omega }_q^f=n{\omega }_D$.

Figure 5

Calculated time evolution of an excitation mode $(u=1,v=0)$. (a) Components of $u\left(t\right)$ oscillate with the mode's intrinsic frequency ${\omega }_q^f$ in the fast-driving limit. Dotted, dashed, and red lines show $\text{Im}\left[u\right(t\left)\right], \text{Re}\left[u\right(t\left)\right]$, and ${\left|u\left(t\right)\right|}^2$, for parameters ${\omega }_D=4{\omega }_q^f, K=1.4, q=k_L$, and $U=10J$. (b) A parametric instability appears when ${\omega }_D$ matches this intrinsic frequency and the excitation mode grows exponentially. (c) Modulational instabilities require the micromotion (black line, top panel) to cross into regions with negative effective mass (red patches). The system is modulationally unstable for $T_D=11.3$ ms (solid red line), but stable for $T_D=8.3$ ms (dashed red line) with parameters $K=2.2, k_0=0$, and $U=10J$. (d) Sketch of the oscillation of ${\left|u\right|}^2$ (red line) and its derivative (blue line). The system is stable when the effective mass turns negative (arrows) in intervals with alternating slope of ${\left|u\right|}^2$.

Figure 6

Calculated growth rate for strong driving strength. (a) Imaginary component of time-averaged phonon energy $\hslash {\omega }_q^s$ (top) and the growth rate (bottom) for strong interactions $U=10J$ ($K=2.2, V=8E_r$). Black and white lines indicate integer and half-integer multiples of $T_q^s$. (b) Same panels for weak interactions with $U=0.5J$. The red lines and black lines indicate multiples of $2\pi /{\omega }_q^f$ and $T_q^s$.

Figure 7

Momentum distribution of MIs without driving. (a) Example images of MIs for different lattice depths $V$ and $k=k_L$ after approximately 10 ms ($a_s=6a_0, N=3\times {10}^4$). We observe images with matching pairs of excitation modes ($V=3.8E_r$–$12.5E_r$), but also images with multiple modes ($V=12.5E_r$, right). (b) Histogram of peak positions for increasing lattice depths and $U/J=0.5$, 1.2, 2.5, and 6.5 (top to bottom). Red lines indicate $\mathrm{\Gamma }=\text{Im}\left({\omega }_q\right)$. (c) The measured peak positions of the histograms match well to the predicted positions in Eq. (9) (red line).

Figure 8

Growth rate of MIs without driving. (a) Time evolution of excitations with $N_B$ atoms at the center of the Brillouin zone (red circles, red patches) and total atom number, $N_{\text{tot}}$ (blue squares), $k=0.95k_L, V=10E_r, a_s=90a_0$. Lines are added to guide the eye. (b) Absorption images of the time evolution after seeding excitations at $q=\pm 0.46$ ($k=k_L, V=10E_r, a_s=14a_0, N_{\text{tot}}\approx 41000, U/J=8.9$, five repetitions). (c) Relative atom number in seeded excitation modes for $V=10E_r$ (blue squares) and $V=6.3E_r$ (red circles). The lines are exponential fits to the data points with $t<1.5$ ms. (d) Measured growth rate ${\mathrm{\Gamma }}_q$ (red circles) and range of expected rates $\text{Im}\left({\omega }_q\right)$ assuming lattice site occupations between 60% and 100% of the peak value (red patch).

Figure 9

Decay of excitations in stable regions without driving. (a) Absorption images of the time evolution of excitation modes $k=0$ ($a_s=31a_0, V=10E_r, q=0.47k_L$). The excitation modes are created by a Bragg pulse which excites approximately 40% of the atoms. Images are centered to the $q=0$ peak and averaged over eight repetitions. (b) Fraction of atomss $N_B/N_{\text{tot}}\mathrm{s}$ in excited mode for data in (a). We determine the time at the minimum $T_h$ by locally fitting a second-order polynomial. (c) $T_h$ for various scattering lengths (red circles) and period $2\pi /{\omega }_q$ in Eq. (3) (blue squares). The inset shows a sketch of two density profiles (red and blue) for a pair of excitation modes with $\pm q$.

Figure 10

Excitation growth due to MIs with driving. (a) Time evolution of a seeded excitation mode without driving (blue squares) and with driving (red circles), $q=0.48k_L, V=6.3E_r, a_s=105a_0, T_D=3.3$ ms. Top: Micromotion for $K=1.8, k_0=0$. Critical regions of the Brillouin zone are indicated by red patches and time intervals of $k\left(t\right)$ in those critical regions by gray patches. The lines next to data sets are smoothing splines to guide the eye. (b) Averaged absorption images for the driven system. We observe no significant heating during the initial evolution time.

Figure 11

Measurement of parametric instabilities. (a) Absorption images with momentum distribution after driving for approximately 30 ms ($a_s=73a_0, K=1.3, V=9E_r, N_{\text{tot}}=3.4\times {10}^4$). An initial Bragg pulse seeds excitation modes at $q=0.46\hslash k_L$. (b) Number of atoms that are not in the ground state, $N_{\overline{A}}$, for the data in (a). (c) Measured driving periods of the first (blue squares) and second (red circles) peaks of $N_{\overline{A}}$ for varying lattice depths. Solid lines show calculated peak positions. (d) Numerically calculated growth rates using Eq. (5) and parameters in (a). The dashed line indicates the quasimomentum of the exited excitation mode.

Figure 12

Stability diagram. Numerically calculated stability using $\overline{\mathrm{\Gamma }}(K,T_D)$ ($V=8.8E_r, U/J=22$). Solid red and black lines show $2\pi /{\omega }_q^f$ and $2\pi /{\omega }_q^s$ for $q=\hslash k_L$; dashed lines show half of the corresponding values. The gray vertical line at $K=\pi /2$ separates weak- from strong-driving regimes. Solid and dashed horizontal blue lines provide $2\pi /{\omega }_{\text{max}}$ and $h/8J$, respectively. Regions indicated by roman numerals are discussed in the text.

Figure 13

Comparison of MIs and PIs. (a) Calculated growth rate for mode $q=\hslash k_L$ with the same parameters as in Fig. 12. (b) Calculated growth rate with interactions switched off whenever the micromotion passes through a region with negative effective mass. Circles indicate values of $(K,T_D)$ for lines with matching colors in (c) and (d). (c) Time evolution $\left|u\right(t\left)\right|$ for an initial state $(u=1,v=0)$ and $K=1.572$ with switched interactions (dashed red line) and without (blue line). (d) Time evolution for $K=2.1$ with (red line) and without (blue line) switched interactions, and in the slow-driving regime with $T_D=30.6$ ms (green line).