Freezing a quantum magnet by repeated quantum interference: An experimental realization

We experimentally demonstrate the phenomenon of dynamical many-body freezing in a periodically driven Ising chain. Theoretically [A. Das, Phys. Rev. B 82, 172402 (2010)], for certain values of the drive parameters all fundamental degrees of freedom contributing to the response dynamics freeze for all time and for arbitrary initial state. Also, since the condition of freezing involves only the drive parameters and not the quantization of the momentum (i.e., the system size), our simulation with a small (3-spin) chain captures all salient features of the freezing phenomenon predicted for the infinite chain. Using optimal control techniques, we realize high-fidelity cosine-modulated drive, and observe nonmonotonic freezing of magnetization at specific frequencies of modulation. Time evolution of the excitations in momentum space has been tracked directly through magnetization measurements.

Figure 1

Comparison of analytical results for $Q$ obtained for the $\omega \gg \mathcal{J}$ limit for $L\to \infty$ [Eq. (2)] and $L=3$ [Eq. (12)] for $h_0=5\pi \mathrm{rad}/\mathrm{s},\mathcal{J}=h_0/20$. Strong nonmonotonic freezing behavior with respect to the drive frequency $\omega$ is visible, in contrast to the intuition of faster drive $\Rightarrow$ lesser time to respond $\Rightarrow$ more frozen response. Almost absolute freezing $Q\approx 1$ is visible for certain special values of $\omega ,h_0$ satisfying the freezing condition [Eq. (3)]. Our experiment captures all these features remarkably well (Fig. 4).

Figure 2

(a) Molecular structure and (b) chemical shifts (diagonal terms) and $J$ coupling (off-diagonal terms) in Hz of trifluoroiodoethylene.

Figure 3

Evolution of $m^x$ with time for $m^x\left(0\right)=1$. The period $\left(\tau \right)$ of oscillation of $m^x$ visible from the experimental data (marked for $\omega =8.40,24.54$ rad/s) corresponds to the frequency $2{\varphi }_k$ (period $T_k=2\pi /{\varphi }_k$) of population oscillation for the fundamental excitation in momentum space for $k=\pi /3$ given in Eq. (9). The analytical curves represent Eq. (11).

Figure 4

$Q$ vs $\omega$ for different initial conditions: Left panels for the initial state corresponding to $m^x\left(0\right)=1$, right panels for the initial state corresponding to $m^x\left(0\right)=0.5$. Top frames compare the final experimental data (with inverse decay correction) for 3-spins with the theoretical results for $L=3$ and $L\to \infty$. Simulation for $L=3$ is done by solving the exact time-dependent Schrödinger equation numerically. $Q$ is obtained by measuring $m^x$ after each cycle and carrying out the stroboscopic average over 30 cycles (as done in the experiment). The vertical lines indicate the position of the freezing peaks. The bottom frames compare raw experimental data with the decay-corrected data. Both the data capture all the essential features of those predicted theoretically for $L\to \infty .$

Figure 5

${}^{19}\mathrm{F}$ spectra corresponding to a nonfreezing condition (left column; $\omega =24.54$ rad/s) and a freezing condition (right column; $\omega =5.61$ rad/s) for $m^x\left(0\right)=1$. Only spectra for the indicated time instants $\left(n=t/\tau \right)$ are shown. In contrast to the classical intuition, the response is clearly seen to be much strongly frozen for the much lower driving frequency ($\omega =5.61$ rad/s) due to quantum interference, as predicted by the theory.

Figure 6

Plot of energies of the instantaneous ground state (denoted by $|G\rangle$) and excited state $\left(|Ex\rangle \right)$ of the $2\times 2$ Hamiltonian [Eq. (8)] as a function of $\omega t$. Our first sweep $(m=0)$ starts from the left end with the initial state ${|\psi \left(0\right)\rangle }_{\pi /3}=\{u_{\pi /3}\left(0\right)=0;v_{\pi /3}\left(0\right)=1\}\approx |G\left(0\right)\rangle$. $P_{ex}$ denotes the probability that starting from this initial state the system ends up with the same state (i.e., $\{u_{\pi /3}=0;v_{\pi /3}=1\}$) at the end of half a cycle, instead of following the drive adiabatically and ending up with the ground state $\left(\approx |0_{\pi /3}\rangle \right)$ of the final Hamiltonian.

Figure 7

Qualitative feature of the theory that uses only transition probabilities between the states $|u_{\pi /3}\rangle ,|v_{\pi /3}\rangle$ after each half cycle, neglecting the phase between them. This illustrates the importance of quantum interference—the role those phases play in shaping the qualitative characteristics of the freezing phenomenon. The full quantum treatment provides excellent agreement with the experimental data (Fig. 4). The results correspond to experimental parameter values $h_0=5\pi ,\mathcal{J}=h_0/20$ (in rad/s).