Measuring nonequilibrium retarded spin-spin Green's functions in an ion-trap-based quantum simulator

Recently a variant on Ramsey interferometry for coupled spin-$1/2$ systems was proposed to directly measure the retarded spin-spin Green's function. In conventional experimental situations, the spin system is initially in a nonequilibrium state before the Ramsey interferometry is performed, so we examine the nonequilibrium retarded spin-spin Green's functions within the transverse-field Ising model. We derive the lowest four spectral moments to understand the short-time behavior and we employ a Lehmann-like representation to determine the spectral behavior. We simulate a Ramsey protocol for a nonequilibrium quantum spin system that consists of a coherent superposition of the ground state and diabatically excited higher-energy states via a temporally ramped transverse magnetic field. We then apply the Ramsey spectroscopy protocol to the final Hamiltonian, which has a constant transverse field. The short time allows us to extract the initial transport of many-body correlations, while the long-time behavior relates to the excitation spectra of the Hamiltonian. Compressive sensing is employed in the data analysis to efficiently extract that spectra.

Figure 1

Schematic of the Ramsey protocol. (a) Rotate a single spin $j$ by $\pi /2$. (b) Allow the resulting quantum spin state to freely evolve forward in time. (c) Apply a global rotation and immediately measure the $z$ component of the $i\mathrm{th}$ spin, ${\sigma }_i^{\left(z\right)}$.

Figure 2

Energy spectrum of the transverse-field Ising model with 10 spins and $\alpha =1.00$. Near $B^{\left(y\right)}/J_0=0.75$ there is a minimum energy gap between the ground state and the (red line) first coupled state.

Figure 3

Schematic of the complete protocol we use in implementing the Ramsey spectroscopy. (a) We initialize the state in the ground state of the Hamiltonian when $B^{\left(y\right)}\gg \left|J_{ij}\right|$. Then we decrease the transverse magnetic field via an exponential ramp in time. We also show the average magnitude of the nearest neighbor interaction, $J_{nn}$, and the next-nearest neighbor, $J_{nnn}$, in red (dashed) lines to give a relative idea of the strength of the transverse magnetic field to the spin-spin interaction. (b) For the time interval $[t_0,t]$, we apply the Ramsey spectroscopy protocol and perform signal processing on the resulting measurements as a function of the final time $t$.

Figure 4

Examples of the Ramsey spectroscopy as a function of time for $i=0$ and $j=0$ (black), 1 (red), 4 (blue), and 9 (green) with 4 different $B^{\left(y\right)}\left(t_0\right)/J_0$ values [$B^{\left(y\right)}\left(t_0\right)/J_0$ is equal to the following: (a) $0.94$, (b) $0.74$, (c) $0.49$, and (d) $0.35$].

Figure 5

First local minimum (blue circles), local maximum (red circles), and zero (black circles) of the pure state Green's function for $i=0$ and all other $j$ lattice sites. Here the $x$ axis is labeled at the equilibrium position from the left edge $i=0$, $|R_0-R_j|$.

Figure 6

Extraction of the first intercepts of 5 different values of $c$ for 4 values of $B^{\left(y\right)}\left(t_0\right)/J_0$: (a) $0.94$, (b) $0.74$, (c) $0.49$, and (d) $0.35$. The values of the intercepts $c$ are $-0.0002$ (black circles), $-0.0005$ (red squares), $-0.001$ (blue triangles), $-0.0015$ (purple plus signs), and $-0.002$ (green diamonds). In addition, the dashed lines show a power law fit, $|R_0-R_j|\propto t^{\gamma }|$, for each of these values. The $-0.0002$ power law fits are the most consistent among the other intercepts. Interestingly, the $-0.0005$ intercept is able to recover the power law behavior in (d). Here the $x$ axis is labeled at the spin's relative distance from $i=0$, $|R_0-R_j|$.

Figure 7

Fourier transform of the $i=j=0$ pure state retarded Green's function of the data taken in the time interval of $[0,6]$ ms. The results of the compressive sensing (black line) are compared to the scaled coefficients of the Lehmann representation, $1024G_{xx,00}^{\mathrm{ret}}(t,0)$ (blue and red circles), and to the partial Fourier transform (green line). The blue circles are energy differences that are associated with a $\delta$ function peak and the red circles are those that cannot be easily associated with a $\delta$ function peak. There are also spurious $\delta$ functions peaks that are due to the addition of two energy differences.