Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins

We perform a quantum simulation of the Ising model with a transverse field using a collection of three trapped atomic ion spins. By adiabatically manipulating the Hamiltonian, we directly probe the ground state for a wide range of fields and form of the Ising couplings, leading to a phase diagram of magnetic order in this microscopic system. The technique is scalable to much larger numbers of trapped ion spins, where phase transitions approaching the thermodynamic limit can be studied in cases where theory becomes intractable.

Figure 1

Energy level diagrams for Eq. (1) with two different types of spin-spin interactions. For both panels, the nearest-neighbor interactions are FM $\left(J_1<0\right)$. (a) The next-nearest-neighbor interaction is FM with $J_2/\left|J_1\right|\sim -2$ and (b) AFM with $J_2/\left|J_1\right|\sim 0.9$. The arrow in both diagrams indicates the trajectory in the simulation, initialized at $B_y/\left|J_1\right|\sim 10$. Under this condition, the initial ground state is an eigenstate of the second term in Eq. (1), a polarized state along $B_y$. In both examples, at $B\approx \left|J_1\right|$ some high-energy states cross, but the ground state (black solid line) has no level crossings with any excited state. Likewise, the highest energy state does not cross any other levels, allowing one to also adiabatically follow this state. The dotted lines represent excited states which are coupled to the ground state along the path. In the large field limit, the energy difference between ground and excited states ${\Delta }_{ge}$ (here, scaled by $\sqrt{B_y^2+J_1^2}$) is proportional to $B_y$ but as $B_y/\left|J_1\right|$ decreases the spin-spin couplings determine the energy difference and the form of the ground state. In both (a) and (b), the final ground state is FM (defined along the $x$ axis of the Bloch sphere), however in the case of (a), the gap to the nearest allowed excited state at the crossover is $\sim 15$ times larger.

Figure 2

(Color online) Theoretical phase diagram for Eq. (1). The color scale indicates the amplitude of the FM order parameter, $\text{P}\left(\text{FM}\right)=P_{|\uparrow \uparrow \uparrow ⟩}+P_{|\downarrow \downarrow \downarrow ⟩}$. Here, $J_1$ is always negative, yielding FM order in that coupling. In the region where $J_2/\left|J_1\right|<0$, there is a crossover to FM order as $B_y/\left|J_1\right|$ is lowered [corresponds to Fig. 1a]. When $J_2/\left|J_1\right|>0$, the AFM and FM interactions compete [also shown in Fig. 1b]. This gives rise to a special sharpened point at $J_2/\left|J_1\right|=1$ and $B_y=0$. Here, the ground state is comprised of six states: four asymmetric AFM and two FM states.

Figure 3

(Color online) Experimental measurements of the phase diagram for Eq. (1) (solid bars) compared to the theoretical prediction from Fig. 2 (surface). The vertical amplitude is the FM order parameter $\text{P}\left(\text{FM}\right)=P_{|\uparrow \uparrow \uparrow ⟩}+P_{|\downarrow \downarrow \downarrow ⟩}$. The ratio of $B_y/\left|J_1\right|$ was varied from $\sim 10$ to $\sim 0.1$ for $J_2/\left|J_1\right|$ values of $-1.3$, $-2.0$, $-3.6$, 4.2, 2.0, 1.3, 0.92, 0.74, and 0.62. $J_1<0$ for all traces. (b) As $B_y/\left|J_1\right|\to 0$ in the region where $J_2/\left|J_1\right|<-1$, we observe a smooth crossover to FM order. The filled circles and solid line are the data and theory for $J_2/\left|J_1\right|=-1.3$, respectively. (c) When changing $J_2$ for a fixed and small value of $B_y/\left|J_1\right|$ the system undergoes a sharp transition. The data (filled circles) shown is for a scan of $B_y/J_y=0.57$. The average deviation per scan of $B_y/\left|J_1\right|$ from the exact ground state is $\sim 0.09$.

Figure 4

Adiabaticity for two cuts in the phase diagram, where (a) $J_2/\left|J_1\right|=-2$ and (b) $J_2/\left|J_1\right|=0.9$. The upper plots show the theoretical ground-state adiabaticity parameter ${\dot{B}}_y\left(t\right)ϵ/{\Delta }_{ge}^2$ of Eq. (3) (dashed lines) for each of the two coupled excited states (see Fig. 1). We use a pure exponential decay ramp for the field $B_y$ with time constant $100\mu \text{s}$ in order to match the experiment for large values of $B_y/J_1$ while also extending the theory curves below the minimum value of $B_y$ reached in the experiment. At every point in the trajectory of (a), we find that the adiabatic condition [Eq. (3)] is satisfied for typical experimental times $\left(300\mu \text{s}\right)$. On the other hand, in (b), there is a significant probability of diabatic transitions to excited states for $B_y/J_1⪡1$. In the lower plots, we compare the observed FM order parameter (points) with theory. In (a), the theoretical order from the exact experimental ramp with a $35\mu \text{s}$ time constant and final offset value given in the text (gray solid line) is in reasonable agreement with the order in the true ground state (black solid line) for $B_y/J_1>0.5$. The dotted line is the expected state evolution for a pure exponential decay ramp with a $100\mu \text{s}$ time constant, allowing $B_y\to 0$. In (b), the data also matches well to theory, as we avoided the regions where diabatic transitions are expected for $B_y/J_1⪡1$. According to the calculations, the duration of three-spin experiments near the special point should be on the order of milliseconds.