Quantum Simulation of a System with Competing Two- and Three-Body Interactions

Quantum phase transitions occur at zero temperature, when the ground state of a Hamiltonian undergoes a qualitative change as a function of a control parameter. We consider a particularly interesting system with competing one-, two-, and three-body interactions. Depending on the relative strength of these interactions, the ground state of the system can be a product state, or it can exhibit genuine tripartite entanglement. We experimentally simulate such a system in a NMR quantum simulator and observe the different ground states. By adiabatically changing the strength of one coupling constant, we push the system from one ground state to a qualitatively different ground state. We show that these ground states can be distinguished and the transitions between them observed by measuring correlations between the spins or the expectation values of suitable entanglement witnesses.

Figure 1

Schematic phase diagram of the ground state of the Hamiltonian $\mathcal{H}$ for $N=3$. Panels (a)–(c) represent two-dimensional sections through parameter space perpendicular to the ${\omega }_z$ axis, and the table (d) identifies the ground states in the different areas. Here $|{\mathrm{GHZ}}_{\pm }⟩=(|\uparrow \uparrow \uparrow ⟩\pm |\downarrow \downarrow \downarrow ⟩)/\sqrt{2}$ and $H$ is the Hardamard gate. The arrows in (a) and (b) represent the adiabatic evolutions discussed in the experimental section. For (a) and (c), ${\omega }_x$ is assumed to be small, i.e., $|{\omega }_x|\ll |{\omega }_z|$.

Figure 2

(a) Molecular structure and properties of the quantum register: diethyl-fluoromalonate. The oval marks the three spins used in the experiment. The table on the right summarizes the relevant NMR parameters measured at room temperature on a Bruker Avance II 500 MHz (11.7 T) spectrometer, i.e., the Larmor frequencies ${\omega }_i/2\pi$ (on the diagonal), the $J$ coupling constants $J_{ij}$ (below the diagonal), and the relaxation times $T_1$ and $T_2$ in the last two columns. (b) Pulse sequence for simulating the Hamiltonian of Eq. (1). The narrow black rectangles represent small-angle rotations, the narrow empty rectangles denote 90° rotations, and the wide ones denote the refocusing 180° pulses. The delays are ${\tau }_i=J_2\tau /[1/(\pi J_{ij})+1/(\pi J_{jk})]$ with ($i,j,k$) an even permutation of ($1,2,3$) and $d_1=2J_3\tau /(\pi J_{12})$. The offsets between the irradiation frequencies and the corresponding Larmor frequencies are $\mathrm{FQ}1=2{\omega }_z\tau /({\tau }_1-{\tau }_2+3{\tau }_3)$, $\mathrm{FQ}2=2{\omega }_z\tau /({\tau }_1+{\tau }_2-{\tau }_3)$, and $\mathrm{FQ}3=2{\omega }_z\tau /({\tau }_1+{\tau }_2+{\tau }_3)$.

Figure 3

Numerical simulation of the minimum fidelities during the adiabatic passage versus the number of steps for the ideal (piecewise constant) Hamiltonian (○) and for the more realistic case that includes the actual pulse sequence generated by average Hamiltonian approximation (AHA) and the effect of decoherence (□) for both cases A and B.

Figure 4

Experimental detection of quantum transition points by two-spin correlations $C_{xx}^{\exp }$ ($*$ and $+$), together with the simulated correlations $C_{xx}^{\mathrm{sim}}$ (○ and □), which take the effect of decoherence into account. The effective decoherence time $T_2^{\mathrm{eff}}$ was estimated as 150 ms for case A and 600 ms for case B.

Figure 5

Experimental detection of quantum transition points by entanglement witnesses for case A (left-hand side) and case B (right-hand side). The solid lines represent the theoretical expectations.