Perturbation Independent Decay of the Loschmidt Echo in a Many-Body System

When a qubit or spin interacts with others under a many-body Hamiltonian, the information it contains progressively scrambles. Here, nuclear spins of an adamantane crystal are used as a quantum simulator to monitor such dynamics through out-of-time-order correlators, while a Loschmidt echo (LE) asses how weak perturbations degrade the information encoded in these increasingly complex states. Both observables involve the implementation of a time-reversal procedure which, in practice, involves inverting the sign of the effective Hamiltonian. Our protocols use periodic radio frequency pulses to modulate the natural dipolar interaction implementing a Hamiltonian that can be scaled down at will. Meanwhile, experimental errors and strength of perturbative terms remain constant and can be quantified through the LE. For each scaling factor, information spreading occurs with a timescale, $T_2$, inversely proportional to the local second moment of the Hamiltonian. We find that, when the reversible interactions dominate over the perturbations, the information scrambled among up to ${10}^2$ spins can still be recovered. However, we find that the LE decay rate cannot become smaller than a critical value $1/T_3\approx (0.15\pm 0.02)/T_2$, which only depends on the interactions themselves, and not on the perturbations. This result shows the emergence of a regime of intrinsic irreversibility in accordance to a central hypothesis of irreversibility, hinted from previous experiments.

Figure 1

Scaled $XYX$ dynamics $\delta {\mathcal{H}}_d^{yy}$. (a) Forward and (b) backward eight-pulse sequences, ($\mathbf{8}{\mathbf{P}}_{\delta }\text{−}\mathbf{F}$ and $\mathbf{8}{\mathrm{P}}_{\delta }\text{−}\mathbf{B}$), consist of $\pi /2$ rf pulses along $X$ or $Y$. The interpulse delays ${\mathrm{\Delta }}_{1,F}=\tau (1-\delta )$, ${\mathrm{\Delta }}_{2,F}=\tau (1+2\delta )$, ${\mathrm{\Delta }}_{1,B}=\tau (1+\delta )$, ${\mathrm{\Delta }}_{2,B}=\tau (1-2\delta )$, resulting in a cycle time $t_c=12\tau$. An evolution time $t$, requires $n=t/t_c$ cycles. In $\mathbf{8}{\mathbf{P}}_{\delta }\text{−}\mathbf{B}$, pulses are rotated by $\phi$. (c) Polarization dynamics $⟨I^z(t)I^z(0){⟩}_{\beta }$. The longitudinal total magnetization is probed after a last $\pi /2$ pulse [41]. (d) Loschmidt echo (LE), $M^{\delta }(t)$, concatenates the forward, $\mathbf{8}{\mathbf{P}}_{\delta }\text{−}\mathbf{F}$, and backward, $\mathbf{8}{\mathbf{P}}_{\delta }\text{−}\mathbf{B}$, dynamics. OTOC protocol. It inserts a perturbation, $\mathrm{\Phi }=e^{\mathrm{i}{I}^z(t)\phi }$, before time reversal, achieved by shifting all pulses in the backward block by $\phi$ to get. $S_{\phi }(t)=⟨{\mathrm{\Phi }}^{†}(t)I_0^z(0)\mathrm{\Phi }(t)I_0^z(0){⟩}_{\beta }$ with $S_{\phi =0}(t)\equiv M^{\delta }(t)$.

Figure 2

Polarization dynamics: (a) forward and backward evolution, (b) backward evolution as a functions of self-time, $\delta t=\delta \times t$. Scaled dynamics are obtained with the $\mathbf{8}{\mathbf{P}}_{\delta }$ sequences for $\delta \in [0,0.4]$ and the ME forward, $\delta =1$ (spin-echo) and backward $\delta =0.5$ (on-resonance irradiation) portions.

Figure 3

Scrambling growth from OTOCs using the 16-pulse sequence for different $\delta$. The number of correlated spin $N(\delta t)$ (circles) fit a power law (black solid line), $N(\delta t)=A\delta t^b$, with $b=1.49\pm 0.04$.

Figure 4

Loschmidt Echo decay normalized by $M^{\delta =0}(t)$ (in the inset), as a function of self-time, using the $\mathbf{8}{\mathbf{P}}_{\delta }$ sequence, for different $\delta$. Lines are fits to a sigmoid.

Figure 5

Scaled LE decay rates $T_2^{\delta }/T_3^{\delta }$ versus perturbation rate $T_2^{\delta }/T_{\mathrm{\Sigma }}$, for $\mathbf{8}{\mathbf{P}}_{\delta }$ (blue) and $\mathbf{16}{\mathbf{P}}_{\delta }$ (green). Also shown is the ME result ($\delta =0.5$, red). Blue and green lines are fits to the function $(T_2^{\delta }/T_3^{\delta })=\sqrt{R^2+(T_2^{\delta }/T_{\mathrm{\Sigma }}{)}^2}$. The black line is a guide to the eye. The inset is an enlargement of the region of saturation of $(T_2^{\delta }/T_3^{\delta })$.