Probing eigenstate thermalization in quantum simulators via fluctuation-dissipation relations

The eigenstate thermalization hypothesis (ETH) offers a universal mechanism for the approach to equilibrium of closed quantum many-body systems. So far, however, experimental studies have focused on the relaxation dynamics of observables as described by the diagonal part of ETH, whose verification requires substantial numerical input. This leaves many of the general assumptions of ETH untested. Here, we propose a theory-independent route to probe the full ETH in quantum simulators by observing the emergence of fluctuation-dissipation relations, which directly probe the off-diagonal part of ETH. We discuss and propose protocols to independently measure fluctuations and dissipations as well as higher order time-ordered correlation functions. We first show how the emergence of fluctuation-dissipation relations from a nonequilibrium initial state can be observed for the two-dimensional (2D) Bose-Hubbard model in superconducting qubits or quantum gas microscopes. Then we focus on the long-range transverse field Ising model (LTFI), which can be realized with trapped ions. The LTFI exhibits rich thermalization phenomena: For strong transverse fields, we observe prethermalization to an effective magnetization-conserving Hamiltonian in the fluctuation-dissipation relations. For weak transverse fields, confined excitations lead to nonthermal features, resulting in a violation of the fluctuation-dissipation relations up to long times. Moreover, in an integrable region of the LTFI, thermalization to a generalized Gibbs ensemble occurs and the fluctuation-dissipation relations enable an experimental diagonalization of the Hamiltonian. Our work presents a theory-independent way to characterize thermalization in quantum simulators and paves the way to quantum simulate condensed matter pump-probe experiments.

Figure 1

Measuring two-time correlation functions out of equilibrium. The statistical function $F$ can be measured by employing a nondestructive measurement on site $i$ at time $t_1$ before measuring site $j$ at time $t_2$. The measurement at $t_1$ can be deferred to $t_2$ by shelving. Alternatively, measurements of two independent experimental realizations can be combined to yield $F$ by averaging over global random unitaries $U$ acted on the initial state. The spectral function $\rho$ can be measured by nonequilibrium linear response (e.g., Bragg or tweezer spectroscopy), employing light pulses on lattice site $i$ at time $t_1$ before measuring at time $t_2$. Alternatively, a Ramsey-type sequence works similarly. The protocols for $F$ and $\rho$ can be realized in quantum simulators of spin models such as trapped ion experiments as well as simulators of Bose- and Fermi-Hubbard models such as quantum gas microscopes and superconducting qubits. The nondestructive measurement and Ramsey protocols can be combined to measure higher order time-ordered correlation functions.

Figure 2

Emergence of FDRs in the 2D Bose Hubbard model. (a) Central-time averaged equal-site density-density spectral function $\rho$ as a function of central time $JT$ and frequency. (b) Late time spectral and statistical functions ($JT=40$, dark) compared to early times ($JT=2$, bright). (c) Fluctuation dissipation relation function defined in Eq. (23) at time $JT=40$ compared to the equilibrium expectation (dashed black line), with the inverse temperature $\beta$ set by energy of the initial state according to Eq. (24). While the dark red line shows the ideal result, the bright dashed line is the result measured by the linear response and nondestructive projective measurement schemes. The inset shows the location of the initially occupied sites (black) and the probed lattice site (red) on the $4\times 4$ lattice. The on-site repulsion is given by $U/J=6$. We used a Gaussian frequency broadening with standard deviation ${\sigma }_{\omega }=0.05J$ for the Fourier transform.

Figure 3

Prethermal FDRs in the long-range transverse-field Ising model. (a) The equal-site spectral and statistical functions $\rho ,F$ of the local spin raising operators ${\widehat{\sigma }}_i^{+}$ with initial state $|{\psi }_0{\rangle =|\uparrow \downarrow \cdots \downarrow \uparrow \rangle }_x$ and $i=2$. Central times $JT$ increase from bright to dark. (b) Fluctuation-dissipation relation for different lattice sites $i$ (increasing from bright to dark) along with the expected inverse temperatures $\beta$ in thermal equilibrium of the LTFI and the prethermal Hamiltonian ($XY$ model). We used a Gaussian broadening with standard deviation ${\sigma }_{\omega }=0.2J$ for the Fourier transform. Parameters used are $L=13$ (open boundary conditions), long-range exponent $\alpha =1.5$, transverse field $g=12J$.

Figure 4

Violation of the FDR due to confined excitations in the LTFI. (a) Two-domain wall spectral function averaged over central time $T$ starting from a completely $x$-polarized state. Black dots and crosses indicate the difference $E_n-E_0$ of the eigenenergies $E_n$ with the ground-state energy $E_0$; gray dots are $E_n-E_1$. (b) Time- and frequency-resolved spectral function. Black arrow and star indicate oscillatory features, which are nonthermal as they depend on central time. Inset: The oscillation frequency of the peak at $\omega =0.9J$ (black) matches the energy difference between the first and second excited states (sinusoidal fit with fixed frequency in dashed gray). (c) Test of the FDR by comparing $F_{q=0}(T,\omega )$ with the corresponding right-hand side of the FDR in Eq. (4) for a fixed time $JT=61$. The black arrow and star indicate the location of the two nonthermal features in panel (b). For all subplots, we used a Gaussian broadening with standard deviation $40/J$ in time and do not plot very small frequencies in panel (c) due to artifacts from the Fourier transform. Here, the transverse field is $g=0.53J$ and the long-range exponent $\alpha =2.3$.

Figure 5

FDR for a single eigenstate. (a) Eigenstate statistical $F_{nn}$ and spectral functions ${\rho }_{nn}$ as well as the right-hand side of the FDR $n_{\beta }{\rho }_{nn}$ for an eigenstate $n$ with eigenenergy $E_n\approx -9.06J$ in the 2D Bose-Hubbard model at $U/J=4$ with the same initial state and observable as in Fig. 2 in the main text. The corresponding inverse temperature $\beta \left(E_n\right)\approx 0.95J$ expected in thermal equilibrium is set by the eigenenergy of the state via Eq. (24). To evaluate Eq. (B4), we used a Lorentzian broadening with FWHM of $0.2J$. (b) FDR function as defined in Eq. (23) for low frequencies, showing the expected linear behavior with a slope matching the inverse temperature.

Figure 6

Closed time contour depiction of the subclass of $(2n+1)$ point correlation functions accessible via protocols with $n$ $\pi$ pulses. Starting from the initial time $t_0$, operators are inserted along the contour at times $t_1,t_2,\cdots ,t_n$ by local pulses. At $t_{n+1}$, the operator $\widehat{C}$ gets measured and the evolution is stopped. The operator measured by the protocol can then be obtained by starting from $t_0$ on the upper branch up to $t_{n+1}$ and then backward on the lower branch back to $t_0$, i.e., $\langle \widehat{A}\left(t_1\right)\widehat{B}\left(t_2\right)\cdots \widehat{C}\left(t_{n+1}\right)\cdots \widehat{B}\left(t_2\right)\widehat{A}(t_1\rangle$. Note that this is only a subclass of the correlation functions obtainable by the protocols presented in this section and all operators are given by Pauli matrices.