Quantum phases in circuit QED with a superconducting qubit array

Circuit QED on a chip has become a powerful platform for simulating complex many-body physics. In this report, we realize a Dicke-Ising model with an antiferromagnetic nearest-neighbor spin-spin interaction in circuit QED with a superconducting qubit array. We show that this system exhibits a competition between the collective spin-photon interaction and the antiferromagnetic nearest-neighbor spin-spin interaction, and then predict four quantum phases, including: a paramagnetic normal phase, an antiferromagnetic normal phase, a paramagnetic superradiant phase, and an antiferromagnetic superradiant phase. The antiferromagnetic normal phase and the antiferromagnetic superradiant phase are new phases in many-body quantum optics. In the antiferromagnetic superradiant phase, both the antiferromagnetic and superradiant orders can coexist, and thus the system possesses symmetry. Moreover, we find an unconventional photon signature in this phase. In future experiments, these predicted quantum phases could be distinguished by detecting both the mean-photon number and the magnetization.

C ircuit quantum electrodynamics (QED) based on superconducting qubits is a fascinating topic in quantum optics and quantum information 1-3 . This artificial spin-1/2 particle can be controlled by tuning the external magnetic flux and gate voltage [4][5][6] . Moreover, a strong spin-photon coupling has been achieved, which allows to implement quantum operations for long coherence times 7,8 . Recently, many important quantum effects in atomic physics and quantum optics have been observed in this artificial spin-photon interaction 9,10 . Particularly, experiments realized multiple superconducting qubits interacting with a transmission-line resonator 11 . These experiments allow to explore many-body phenomena via circuit QED [12][13][14][15][16][17][18][19][20][21] . For example, the challenging Dicke quantum phase transition from a normal phase to a superradiant phase, which was predicted more than 30 years ago [22][23][24] , can be realized by controlling the gate voltage or external magnetic flux [25][26][27][28] and the no-go theorem arising from the Thomas-Reich-Ruhn sum rule may be overcome 29,30 .Moreover, the Jaynes-Cummings lattice model 31 can also be simulated by an array of transmission-line resonators, each coupled to a single artificial particle 32,33 . In addition, by measuring the microwave photon signature, the many-body nonequilibrium dynamics, as well as the known phase diagrams, could be derived [34][35][36] .
On the other experimental side, superconducting qubits can couple with each other, forming an array with an effective nearest-neighbor spin-spin interaction 37,38 . Thus, it is meaningful to explore the many-body physics when a superconducting qubit array couples with a transmission-line resonator because there exists a competition between the collective spin-photon interaction and the nearest-neighbor spin-spin interaction. Recently, sudden switchings, as well as a bistable regime between a ferromagnetic phase and a paramagnetic phase, have been predicted 39 , attributed to this competition.
In this report, we investigate the quantum phases in circuit QED with a superconducting qubit array, which is governed by a Dicke-Ising model with an antiferromagnetic nearest-neighbor spin-spin interaction. By considering the competition between the collective spin-photon interaction and the antiferromagnetic nearest-neighbor spin-spin interaction, we predict four quantum phases, including: a paramagnetic normal phase (PNP), an antiferromagnetic normal phase (ANP), a paramagnetic superradiant phase (PSP), and an antiferromagnetic superradiant phase (ASP). The ANP and the ASP are new phases in many-body quantum optics. In the ASP, both the antiferromagnetic and superradiant orders can coexist, and thus the system possesses Z z 2 6Z 2 symmetry, i.e., both U(1) and translation symmetries are broken simultaneously. Moreover, we find an unconventional photon signature in this phase which could increase from zero to a finite value and then decrease when increasing an effective magnetic field. In future experiments, these predicted quantum phases could be identified by detecting both the meanphoton number and the magnetization.

Results
System and Hamiltonian. Figure 1 shows our proposed quantum network. Many superconducting qubits connected in a chain couple capacitively to their neighboring qubits and also interact identically with a one-dimensional transmission-line resonator. The corresponding Hamiltonian is given by [4][5][6] H 1~e where s i z~:i j i : i h j{ ; i j i ; i h j and s i x~:i j i ; i h j{ ; i j i : i h j are the Pauli spin operators, is an effective magnetic field, and describes the capacitance-induced nearest-neighbor spin-spin interaction. Now a one-dimensional transmission-line resonator is placed in parallel to the superconducting qubits. All superconducting qubits are situated at the antinode of the magnetic field induced by the oscillating supercurrent in the transmission-line resonator 2,3 . Due to the boundary condition at the end of the transmission-line resonator, these superconducting qubits are controlled only by the magnetic component, which shifts the original magnetic flux W x bỹ where l is the inductance per unit length, and S 0 is the enclosed area of the superconducting qubit. In the Lamb-Dicke limit (g 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi a { a h iz1 q =1), together with the condition W x 5 W 0 /2, an effective Hamiltonian for Fig. 1 is obtained by where is the collective spin-photon coupling strength. In this Hamiltonian, all parameters can be controlled independently. For example, the effective magnetic field e can be tuned via the gate voltage V g from the negative to the positive. For simplicity, we address mainly the case e $ 0 in the following discussions. The Hamiltonian (10) is a Dicke-Ising model with an antiferromagnetic nearest-neighbor spin-spin interaction. This Hamiltonian shows clearly that when both g and J coexist, the collective spinphoton interaction has a competition with the nearest-neighbor spin-spin interaction. As a result, it exhibits exotic phase transitions  beyond the previous predictions of the standard Dicke (Ising) model. For example, a first-order superradiant phase transition has been predicted 40,41 , when J , 0. In this report, we will find rich quantum phases including the PNP, the ANP, the PSP, and the ASP for J . 0.
Quantum phases. For the Hamiltonian (10), the quantum phases can be revealed by calculating the ground-state energy and the order parameters via a mean-field approach 42 . In the classical picture, the spin in the Hamiltonian (10) can be represented as a vector line in the xz plane with the unit vectorñ~cos Q,0, sin Q ð Þ . Thus, we can introduce a variational ground-state wave function where and are the spin and boson coherent states, to describe both the antiferromagnetic and superradiant properties.
Since the antiferromagnetic exchange interaction (J . 0) leads to a staggered arrangement of all spins in the z direction, we should consider two sublattices with Q 1 and 2Q 2 , which corresponds to the odd and even sites of spins, respectively, in the ground-state wave function. After a straightforward calculation, the scaled ground-state energy is given by where l~l 0 . ffiffiffiffi N p and the parameters (l, Q 1 and Q 2 ) are to be determined. As shown in the Methods section, by minimizing the ground-state energy E with respect to the variational parameters (l, Q 1 , Q 2 ), we obtain three equilibrium equations: where b~1 These equilibrium equations, together with the stable conditions (see the Methods section), determine the ground-state energy in Eq. (16) and the order parameters, such as the mean-photon number AEa { aae, the magnetization AES z ae and the staggered magnetization AEM s ae, which are given respectively by The introduction of the order parameter, the staggered magnetization AEM s ae, is to conveniently discuss the antiferromagnetic properties of the Hamiltonian (10). After the ground-state energy, and especially, the order parameters, are obtained, several rich phase diagrams can be obtained. We first address two known limits. The first is the case when J 5 0, in which the Hamiltonian (10) reduces to the standard Dicke model 43 By means of the equilibrium equations (17)- (19) and the stable conditions, we find for g , g c and for g , g c and for g . g c . This means that a second-order quantum phase transition from the normal phase (g , g c ) to the superradiant phase (g . g c ) occurs 44,45 , as shown in Fig. 2(a). Moreover, the Dicke model has U(1) symmetry in the normal phase. Whereas, in the superradiant phase the system acquires macroscopic collective excitations governed mainly by the collective spin-photon interaction term g X i s i x aza { À Á , and thus it has Z z 2 6Z 2 symmetry, where Z z 2 is the global rotation of p around the z axis 46 and Z 2 is the change of sign of the boson coherent state (jl 0 ae R 2jl 0 ae). In experiments, this quantum phase transition has been observed [47][48][49] in an optical cavity with a Bose-Einstein condensate by measuring the mean-photon number AEa { aae and the magnetization AES z ae/N. Recently, it has been well investigated in many-body circuit QED [25][26][27][28][29][30] and spin-orbit-driven Bose-Einstein condensate 50 .
For g 5 0, the Hamiltonian (10) turns into the Ising model 51 in which a first-order phase transition from the paramagnetic phase to the antiferromagnetic phase at the critical point J c 5 e/2 can be recovered. In the paramagnetic phase (J , J c ), the ground-state wave function is j… ######## …ae, which implies that the system has translation symmetry and AES z ae/N 5 21 and AEM s ae 5 0. In the antiferromagnetic phase (J . J c ), the ground-state wave function becomes j… "#"#"#"# …ae, in which translation symmetry is broken and AES z ae/N 5 0 and AEM s ae 5 1.
If both g and J are non-zeros, we find four different regions: (i) a 5 0, b 5 2p/2, (ii) a 5 1p/2, b 5 0, (iii) a 5 0, b ? 0, and (iv) a ? 0, b  ? 0. Specially, the order parameters in these four regions are given respectively by where cos a~{ e sin b 2 J{g 2 cos 2 b=v ð Þ , ð33Þ In terms of the different properties of the order parameters, the cases (i)-(iv) are denoted by PNP, ANP, PSP, and ASP, respectively. The ANP (AEa { aae/N 5 0, AES z ae/N 5 0 and AEM s ae 5 1) and the ASP (AEa { aae/N ? 0, AES z ae/N ? 21 and AEM s ae ? 1) are new phases in many-body quantum optics. In Fig. 2, we plot phase diagrams for different antiferromagnetic spin-spin nearest-neighbor interactions. This figure shows clearly that these predicted quantum phases can be driven by the collective spinphoton coupling strength g, the antiferromagnetic nearest-neighbor spin-spin interaction J, and the effective magnetic field e. Especially, the region of the ASP becomes larger when increasing J.
In Fig. 3, we plot phase diagrams as functions of the antiferromagnetic nearest-neighbor spin-spin interaction J and the collective spinphoton coupling strength g for different effective magnetic fields (a) e 5 0 and (b) e 5 v/4. In the absence of e, Eq. (10) reduces to the form In such a case, only the ANP and the PSP can be found, as shown in Fig. 3(a). When increasing e, four quantum phases are predicted again, as shown in Fig. 3(b). In addition, by means of the groundstate energy, we find that all transitions between these different quantum phases in Figs. 2 and 3 are of second order.
Symmetry. In order to better understand these predicted quantum phases, it is necessary to discuss the corresponding symmetries. For the PNP and the PSP, the system properties are similar to those of the normal phase and the superradiant phase in the standard Dicke model, i.e., the system displays both U(1) and translation symmetries in the PNP, and becomes Z z 2 6Z 2 and translation symmetries in the PSP. However, in the ANP, though no photon is excited, the antiferromagnetic order emerges. This implies that in such a case only U(1) symmetry can be found. Interestingly, in the ASP the Hamiltonian (10) is governed mainly by the term in which there exists a competition between the antiferromagnetic nearest-neighbor spin-spin interaction and the collective spinphoton interaction. As a result, both the antiferromagnetic and superradiant orders coexist and the system possesses Z z 2 6Z 2 symmetry, i.e., both U(1) and translation symmetries are broken simultaneously.
Possible experimental observation. We first estimate the parameters for experiments. When we choose C S , 600 aF, C , 20 aF, E 0 J *2p|3:5 GHz, S 0 , 1 mm 2 , d , 10 mm, L 0 , 19 mm,  v , 2p 3 6.729 GHz, and N 5 100 11 , the antiferromagnetic nearestneighbor spin-spin interaction parameter and the collective spinphoton coupling strength are given respectively by J , 2p 3 2.2 GHz and g , 2p 3 1.5 GHz (g 0 5 0.01 is responsible for the Lamb-Dicke approximation). In addition, the effective magnetic field e can range from 0 to 2p 3 6.8 GHz by controlling the gate voltage V g . These parameters ensure that the system should probe the predicted phase transitions. To observe these phase transitions, the relaxation time T 1 and the coherence time T 2 should be much smaller than the lifetime 1/k of the photon, i.e., T 1 . T 2 . 1/k 5 23.4 ns, where k is the decay rate of the photon. This restriction can be easy to satisfy in current experimental setups (for example, T 1 5 7.3 ms and T 2 5 500 ns in Ref. 52). We now illustrate how to identify these different quantum phases. Here we propose to detect four phases by measuring both the meanphoton number AEa { aae and the magnetization AES z ae. For the PNP and the ANP, we can separate these by directly observing the magnetization because AES z ae/N 5 21 in the PNP and AES z ae/N 5 0 in the ANP, as shown in Fig. 4(a). For the PSP and the ASP, both the photon and the spin are collectively excited. Moreover, when increasing the effective magnetic field e, AES z ae/N always decreases. This means that it is difficult to distinguish the PSP and the ASP by measuring AES z ae/N. Fortunately, we find that in the ASP the mean-photon number has an unconventional behavior that could increase it from zero to a finite value and then decrease, as shown in Fig. 4(b). The relevant physics can be understood as follows. When the effective magnetic field e is applied, it can initially promote the arrangement of all spins from the antiferromagnetic to the paramagnetic terms 53 . For example, in the case of a weak e, the spin arrangement becomes j… "#" {###} "#" …ae from the antiferromagnetic case j… "#"#"#"# …ae. This process is helpful for achieving photon-induced collective excitations. Thus, the mean-photon number can increase. However, the rearrangement of spins gives rise to an opposite result of the magnetization, i.e., it decreases when increasing e. For strong e, this effective magnetic field in the z axis leads to a large spin imbalance j… ""###### …ae and thus suppresses the spin-photon collective excitations, i.e., both AEa { aae/N and AES z ae/N decrease when increasing e. In terms of the different behaviors of both AEa { aae/N and AES z ae/N, we argue that our predicted quantum phases can be identified.
It should be pointed out that the microwave photon in superconducting circuits is not easy to measure directly 54 via photon-number detectors, because its energy ( hv) is very small. However, in the dispersive region D ? g, where D 5 e 2 v, the photon number can be detected by the photon-number-dependent light shift (the Stark plus Lamb shifts) of the atom transition frequency 55 . Unfortunately, to achieve the predicted phase diagrams, the system should work at the quasi-dispersive-strong region D/g , 4. In such a region, the above approach to detect photons does not work. Recently, it has been proposed 56,57 to detect the photon by irreversible absorption of photons. In these proposals 56,57 , the absorbers along the waveguide are built with bistable quantum circuits, and can produce a large voltage pulse when the photon decays into a stable state. This suggests that in future experiments the mean-photon number could be detected, and then our predicted phase diagrams could also be observed.

Discussion
Let us here address the no-go theorem in quantum optics. This no-go theorem, demonstrated 58 in 1975, shows that in a typical optical cavity with an ensemble of natural two-level atoms, the phase transition from the normal phase to the superradiant phase is forbidden by the A 2 term, where A is the vector potential. Recently, the no-go theorem has been addressed 29,30 in circuit QED, with many superconducting qubits interacting with a quantized voltage (microwave photon). However, in this report, the required microwave photon is generated from the quantization of the magnetic flux. In such a case, no A 2 term can occur, i.e., the no-go theorem is not valid.
We now consider how the decay of the photon and the disorder in fabrication affect the predicted phase diagrams. When considering the decay of the photon, the stationary mean-photon number, which can be derived from becomes whereas the stationary value of AES x ae remains unchanged (i.e., it is identical to the case without photon decay). This means that we can use an effective Hamiltonian with to discuss the phase diagrams 49 induced by the decay of the photon. Since the decay rate k of the photon (,MHz) is far smaller than the other parameters (,GHz), the decay of the photon has almost no effect on the predicted phase diagrams.
In addition, the imperfections in fabrication result in a weak randomness in the antiferromagnetic nearest-neighbor spin-spin interaction 14 . Moreover, the disorder antiferromagnetic interaction generates a disordered phase, such as a random singlet phase 59 . In this phase, most spins form a singlet pair with nearby spins, and the residual induce weak long-distance pairs. Unfortunately, the disordered phase is only a local correlation, and is thus unstable in the presence of a strong external magnetic field (i.e., for a weak magnetic field, the disordered phase can occur). The phase diagrams predicted here should be observable under a strong magnetic field (See Figs. 2(c) and 2(d)). This means that these predicted quantum phases will not be qualitatively affected by weak disorder.
Mean-field predictions become more accurate for larger number of spins. However, mean-field is often a good starting point, and provides some basic insight in the system. Moreover, for current experimental techniques, the spin number is not sufficiently large, but this should change in the future. For smaller number of spins, we can perform direct numerical diagonalization to discuss the groundstate properties. In Fig. 5, we plot the order parameters AEa { aae/N, AES z ae/ N, and AEM s ae as functions of the collective spin-photon coupling strength g and the effective magnetic field e. This result shows clearly that for a small number of spins, the predicted quantum phases still exist, but the phase boundaries are affected significantly.
In summary, we have investigated the Dicke-Ising model with an antiferromagnetic nearest-neighbor spin-spin interaction in circuit QED for a superconducting qubit array and predicted four quantum phases, including the PNP, the ANP, the PSP and the ASP, with different symmetries. Moreover, all transitions between these different quantum phases are of second order. We have also found an unconventional photon signature in the ASP, where both the antiferromagnetic and superradiant orders coexist. We believe that this system allows to explore exotic many-body physics in quantum optics and condensed-matter physics because it has an interesting competition between the collective spin-photon interaction and the nearest-neighbor spin-spin interaction.

Methods
Three equilibrium equations and the corresponding stable conditions. Here we present detailed calculations on how to derive the three equilibrium equations (17)- (19) and the corresponding stable conditions. After minimizing the ground-state energy E in Eq. (16) with respect to the variational parameters (l, Q 1 and Q 2 ), we obtain three equations: