Direct observation of quantum criticality in Ising spin chains

We use NMR quantum simulators to study antiferromagnetic Ising spin chains undergoing quantum phase transitions. Taking advantage of the sensitivity of the systems near criticality, we detect the critical points of the transitions using a direct measurement of the Loschmidt echo. We test our simulators for spin chains of even and odd numbers of spins, and compare the experimental results to theoretical predictions.

Figure 1

Phase diagram of the antiferromagnetic Ising chain with transverse and longitudinal fields, $B_x$ and $B_z$, respectively, in the thermodynamic limit of infinite chain size [22, 24]. The coupling strength is chosen as the unit for $B_x$ and $B_z$ [see Eq. (1) in the text]. In the shadowed region inside the circle the ground state is (doubly degenerate) antiferromagnetic (AF), and in the clear region outside it the ground state is paramagnetic (PM). The transition line between both phases is a second-order critical line, while the points at $B_x=0$ are first-order transitions. The phase diagram corresponds to that of a two-dimensional classic Ising model with field equal to $B_z$ and effective temperature proportional to $B_x$. The dashed line shows qualitatively the region we explore experimentally, where the critical points in the thermodynamical limit are close to $B_z=\pm 2$. For the finite systems used in our experiments, we need to consider boundary effects, which show up like extra subphases inside the AF phase. For odd $N$, a new critical point appears at $B_z=0$, while for even $N$ two extra critical points appear at $B_z=\pm 1$.

Figure 2

Phase diagrams without transverse field [(a) and (b)] and Loschmidt echo (dimensionless) with small transverse field [(c) and (d)] for the Ising chains with odd and even spins, shown in the left and right columns, respectively. The dark and light curves in (a) and (b) represent the two lowest energy levels, by setting the coupling strength and $\hslash$ to unity. The phases and energy levels are listed in Eqs. (2, 3, 4, 5). The crossover points are $B_c=\pm 2,0$ in the odd spin system, and $B_c=\pm 2,\pm 1$ in the even spin system. The minima of the Loschmidt echo in (c) and (d) indicated the critical points. Without loss of generality, we choose $N=7$ and 8 to illustrate the odd and even cases, where $\epsilon =0.1$, $\tau =\pi$, and $B_x=0.1$, for calculating $L$. In (c) and (d) the light thick curves show the numerical results from Eq. (6), while the dark thin curves show the approximate analytical results from Eq. (12).

Figure 3

Gate sequence to prepare the effective pure state ∣000⟩ by spatial averaging from the thermal equilibrium state of TCE, where $\cos \alpha =2{\gamma }_C∕{\gamma }_H$. Here ${\gamma }_H$ and ${\gamma }_C$ denote the gyromagnetic ratios of proton and carbon, respectively. The single-qubit gates are implemented through radio frequency pulses denoted by the rectangles. The rotation angles and directions are shown inside and above the rectangles. The bold vertical lines denote the gradient pulses along the $z$-axis. The two filled circles connected by a line denote the $J$-coupling evolution $e^{-i\varphi {\sigma }_z^l{\sigma }_z^k}$ between qubits $l$ and $k$, where $\varphi$ is shown next to the line.

Figure 4

Quantum networks for measuring critical points in intervals $B_z∊[-3,-1]$, $(-1,1)$, and [1, 3] in the three-qubit system, shown in (a)–(c), respectively. $R=e^{i\phi {\sigma }_y}$, where $\phi$ is given by Eq. (22), and $R_0=\mathbf{1}$ (unit operator), $e^{i\pi {\sigma }_y∕4}$ or $e^{i\pi {\sigma }_y∕2}$ for $B_z=-0.5$, 0, or 0.5, respectively. $U_0$ and $U_0^{†}$ are indicated by the dashed rectangles, and $U_p^{†}U\approx e^{-i\tau \epsilon ({\sigma }_z^1+{\sigma }_z^2+{\sigma }_z^3)}$. ⊕ and the black dot connected by a line denote a controlled-NOT gate, and $N$ denotes a NOT gate. $D$ denotes the operation to eliminate the nondiagonal elements of the density matrix. The last operation in each figure denotes the measurement, which can be applied to an arbitrary qubit of the system.

Figure 5

Gate sequences (a)–(c) to implement Figs. 4a, 4b, 4c, respectively. A 16-step average over a random delay, denoted by $d$, between 0 and $10\mathrm{ms}$, dephases the residual zero-quantum coherence. The last $\pi ∕2$ pulse is the readout pulse, which can be applied to an arbitrary qubit of the system.

Figure 6

Experimental results in the three-qubit QPT system, where $\tau =\pi$. The four phases $\mid {\psi }_k^o⟩$ with $k=1,\dots ,4$ are represented as $\mid {\psi }_k^o⟩=\mid 000⟩$, ∣010⟩, ∣101⟩, and ∣111⟩, respectively. The experimentally measured amplitudes of the signals are marked by × and + for $\epsilon =0.2$ and 0.125, respectively. The minima of the amplitudes indicate the critical points. The theoretical results are shown as the light and dark curves. The experimental results show a good agreement with theory.

Figure 7

Gate sequence to prepare the effective pure state ∣0000⟩ by spatial averaging from the thermal equilibrium state of the four carbons in crotonic acid, where $\cos \alpha =1∕8$ and $\cos \beta =1∕4$. The filled rectangles in pairs connected by a line denote a state specific swap gate between qubits $l$ and $k$, i.e., it transforms ${\sigma }_z^l$ to ${\sigma }_z^k$, or ${\sigma }_z^k$ to ${\sigma }_z^l$.

Figure 8

Quantum network for measuring critical points in intervals $B_z∊[-3,-1.44]$ and $(-1.44,0]$ in a four-qubit system, shown in (a) and (b), respectively. $H$ denotes the Hadamard transform gate, and $U_p^{†}U\approx e^{-i\tau \epsilon ({\sigma }_z^1+{\sigma }_z^2+{\sigma }_z^3+{\sigma }_z^4)}$. $R_2=e^{i{\phi }_2{\sigma }_y}$ and $R_1=e^{i{\phi }_1{\sigma }_y}$, where ${\phi }_2$ and ${\phi }_1$ are chosen as Eqs. (24, 25). The rectangle and the dot connected by a line denote a controlled operation that is shown inside the rectangle. The filled rectangles in pairs connected by a line denote a SWAP gate. The networks for intervals $B_z∊[1.44,3]$ and (0, 1.44) can be obtained by adding NOT gates to all qubits at the end of the networks for implementing $U_0$ in (a) and (b), respectively.

Figure 9

Quantum gate sequences (a) and (b) to implement Figs. 8a, 8b respectively. Through replacing $\epsilon$ by $-\epsilon$ in (a) and (b) one can obtain the gate sequences for the intervals $B_z∊[1.44,3]$ and (0, 1.44), respectively.

Figure 10

Experimental results in the four-qubit QPT system, where $\tau =\pi ∕2$. The five phases $\mid {\psi }_k^e⟩$ with $k=1,\dots ,5$ are represented as $\mid {\psi }_k^e⟩=\mid 0000⟩$, $(\mid 0100⟩+\mid 0010⟩)∕\sqrt{2}$, $(\mid 0101⟩+\mid 1010⟩)∕\sqrt{2}$, $(\mid 1101⟩+\mid 1011⟩)∕\sqrt{2}$, and ∣1111⟩, respectively. The experimentally measured amplitudes are marked by × and + for $\epsilon =0.5$ and 0.4, respectively. The minima of the amplitudes indicate the critical points. The theoretical results are shown as the light and dark curves, in good agreement with the experimental results.