Quantum phase transition of nonlocal Ising chain with transverse field in a resonator

We study the quantum phase transition in a spin chain with variable Ising interactions and position-dependent coupling to a resonator field. Such a complicated model, usually not present in natural physical systems, can be simulated by an array of qubits based on man-made devices and exhibits interesting behavior. We show that, when the coupling between the qubit and field is strong enough, a superradiant phase transition occurs, and it is possible to pick a particular field mode to undergo this phase transition by properly modulating the strength of the Ising interaction. We also study the impact of the resonator field on the magnetic properties of the spin chain and find a rich set of phases characterized by distinctive qubit correlation functions.

Figure 1

(a) An array of charge qubits capacitively coupled to the TLR. The TLR consists of a center conductor and two ground planes. The voltage between the center conductor and the ground planes is position dependent, as indicated by the cosine curve in the figure. The charge qubits located between the center conductor and ground plane are capacitively coupled to the center conductor. The nearest-neighbor interaction between charge qubits is realized by an rf superconducting quantum interference device (in the dashed box). (b) The equivalent distributed circuit of panel (a).

Figure 2

${\varphi }_l^g$ versus ${\lambda }_0$ for different resonator modes $l$. In panels (a) and (b), only one mode is considered in the calculation, although a different mode is used for each curve. The Ising interaction is homogeneous, $J\left(j\right)=J$, and $J=0.05$ and $0.35$, respectively. In panel (c), the ground-state values of three modes $\left(l=1,2,3\right)$ are plotted for a homogeneous Ising interaction $J=0.05$ by considering all modes simultaneously in the calculation. In panels (d) and (e), $J\left(j\right)$ has a rectangular waveform as in Eqs. (11) and (12), and $J_{\max }=0.35,J_{\min }=0.05$ are used. The modes $l=2$ and $l=3$ are singled out since the critical value ${\lambda }_0^c$ for them to undergo the superradiant transition is the lowest. Only one mode is considered in each calculation. In panel (f), three modes are considered simultaneously in the calculation. $J\left(j\right)$ has a rectangular waveform as in Eq. (11), and $J_{\max }=0.35,J_{\min }=0.05$. In panel (a)–(f), the parameters, $\frac{e^2}{2C_{\Sigma }}=8$ and $E_z=0.8$, are chosen to be accessible values in typical experiments. The size of the system is $N=200$.

Figure 3

Phase diagram of the superradiant quantum phase transition (QPT) in the $J_{\min }\text{−}{\lambda }_0$ plane. When the phase transition occurs, the mean-field value ${\varphi }_2^g$ changes from 0 (the normal phase) to nonzero (the superradiant phase). When $J_{\min }$ is small, the change of ${\varphi }_2^g$ from 0 to nonzero is continuous. When $J_{\min }$ is greater than $0.35$ (roughly), this change becomes discontinuous. Correspondingly, the phase transition changes from second order to first order. In the upper-left and lower-right corner, the Ising interaction strength $J\left(j\right)$ and effective transverse field $\Omega \left(j\right)$ at points A and B in the phase diagram are plotted. $J\left(j\right)$ is modulated as in Eq. (11) to single out the mode $l=2$ for the superradiant phase transition. The effective transverse field $\Omega \left(j\right)={\{{\left(\frac{E_z}{2}\right)}^2+{\left[2{\lambda }_2\left(j\right){\varphi }_2^g\right]}^2\}}^{1/2}$ [see Eq. (B4)] is dependent on the order parameter ${\varphi }_2^g$. At point A in the normal phase, ${\varphi }_2^g=0$ and $\Omega \left(j\right)$ is constant. At point B in the superradiant phase, ${\varphi }_2^g$ is nonzero and $\Omega \left(j\right)$ is oscillatory because of the position dependence of ${\lambda }_2$, as in Eq. (5). Parameters used in the simulation are $\frac{e^2}{2C_{\Sigma }}=8,E_z=0.8,l=2,N=200$, and $\Delta J=0.3$.

Figure 4

(a)–(f) The single-particle energy $e_g$, ground-state energy $e_{gg}$, and the first and second derivative of $e_{gg}$. In panels (a)–(c), $J_{\min }=0.3$, and the phase transition is second order. In panels (d)–(f), $J_{\min }=0.5$, and the phase transition is first order. Other parameters are the same as in Fig. 3. (g) $|{\Sigma }_2^x|$ as a function of ${\lambda }_0$ for different values of $J_{\min }$. The red, blue, and black solid lines refer to the values of $|{\Sigma }_2^x|$ for the stable points of the minimum energy in the $e_g\text{−}{\varphi }_2$ plane. The dashed lines refer to the values of $|{\Sigma }_2^x|$ for the unstable points of the maximum energy in the $e_2\text{−}{\varphi }_2$ plane. The vertical dotted line marks the critical point of the first order QPT for $J_{\min }=0.5$. To the left of the dotted line, $|{\Sigma }_2^x|=0$ for the ground state of the system. To the right of this line, the value of $|{\Sigma }_2^x|$ for the ground state becomes nonzero.

Figure 5

The full phase diagram in the parameter space $(E_z,{\lambda }_0,J_{\min })$. The ${\lambda }_0-J_{\min }$ plane at $E_z=0.8$ (which corresponds to Fig. 3) is shown, and the black dashed line marks the boundary between the first and second order phase transition. The effective transverse field is constant in the normal phase and oscillatory in the superradiant phase as in Fig. 3. Here, we set $\Delta J=0.375E_z$. Other parameters are the same as Fig. 3.

Figure 6

(a) Phase diagram of the system. All parameters are the same as in Fig. 3. (b)–(f) Values of $\langle {\overline{\sigma }}_j^z\rangle$ and the correlation length ${\xi }_{LR}\left(j\right)$ at points B–F in panel (a).

Figure 7

(a) $|{\varphi }_l^g|$ versus ${\lambda }_0$ for different values of $l$ with $E_z=0.1$ (in units of ${\omega }_1)$, and $J_{\max }=0.26E_z,J_{\min }=0.01E_z$. In the normal phase, the order parameter ${\varphi }_l^g=0$. As ${\lambda }_0$ crosses the critical value, ${\varphi }_l^g$ becomes nonzero, and the system enters superradiant phase. Panels (b) and (c) show $\langle {\overline{\sigma }}^z\left(j\right)\rangle$ and ${\xi }_{RL}\left(j\right)$ for the normal phase (point A) and superradiant phase (point B). Both are in the paramagnetic regime.