In contrast to classical phase transition, quantum phase transition (QPT),^[1] which occurs at absolute zero temperature, can only be accessed by changing the physical parameters, for instance, the magnetic field strength. QPT takes place in many-body system and such QPT is a second-order phase transition.

Dicke model (DM)^[2] is mainly used to describe the interaction between an ensemble of N identical two-level atoms and a quantized single-mode bosonic field. Although the form of DM is simple, it contains a rich physical meaning. Moreover, the Dicke Hamiltonian, which contains a phase transition from the normal phase to the super-radiant phase,^[3–6] is extensively used in theoretical research. In the atomic physics and solid state physics, the Dicke model has a wide range of research. In the processing of quantum information and quantum computing, QPT has a certain relationship between quantum entanglement and quantum chaos.^{[3–4,7–9]}

The Dicke-type QPT,^[10–11] which is useful to understand quantum many body systems, has been observed in recent experimental setup in 2010.^[12] At the same time, a series of theoretical analysis are presented,^[10,13–20] especially controlling the atom-atom interactions by the Feshbach resonances.^[21] It is highlighted that the effective atomic interaction strength can be manipulated by Feshbach resonance, and in Feshbach resonance point, that is near its size and even up to the magnitude of MHz. Noting that the effective dipole-dipole interactions between the atoms are ignored in the above mentioned DM. In fact, the dipole-dipole interactions, which are spatially dependent on the leading term varying as for small samples, where r_ij is the distance between i-th and j-th atoms, have a significant contribution to dephasing.^[22] Meanwhile, the fluorescent spectrum and the second-order intensity correlation function are strongly dependent on the inter-atom interactions.^[22–24] The hamiltonian of DM shows the great importance of the spin-spin interactions.^[24]

In this paper, we discuss the extended Dicke model including the atom-atom interaction. Based on the spin coherent-state (SCS) variational method,^[25–26] we reveal the exact solutions of the ground state about the extended Dicke Hamiltonian. According to the analytic solutions, the mean photon numbers, the scaled atomic population and the average energy have been plotted. We also plot the critical curves and ground state phase figure. For the item, the SCS variational method can not deal with the square item of the atom-atom interaction. In order to solve the item, the self-consistent-field method^[27–36] are introduced. Finally, a first-order phase transition^[37] for the extended DM, which is different from the previous conclusion,^[24] are revealed. It is demonstrated that the critical phase transition point from the normal phase to the superradiant phase is shifted by the dipole-dipole interaction and the self-consistent method applies to the case of small change.

The proposed DM Hamiltonian describing the interaction of an ensemble of N identical two-level atoms coupled to a single mode high-finesse optical cavity is given. For the atom-atom interaction, we only consider a simple two-body interactional potential and obtain the following Hamiltonian, which is expressed as the following three parts

In order to properly represent the elastic collision between the atoms, we only consider the two-body interaction with the δ potential and introduce the wave functions of the macroscopic condensate states . According to the second quantization of Gross–Pitaevskii equation, the Hamiltonian of the atomic part in Eq. (3) could be obtained that

Based on the two-mode approximation and , where two boson annihilation operators b₁ and b₂ respectively represent two atomic modes and they satisfy the commutation , Eqs. (4)–(7) of the atom-photon interaction and the atomic part reduce to the following

Substituting Eqs. (2)–(8) into the total Hamiltonian (1), finally the system’s Hamiltonian takes the form

For simplicity, we introduce the Schwinger angular momentum operators

From Eq. (10), we can obtain that , . On the basis of and , we get

Substituting Eq. (11) into Eq. (9) and using the boson commutation relation, the Hamiltonian (9) is rewritten as

The Dicke Hamiltonian for N identical two-level atoms interacting with a single mode of the electromagnetic field is written by^[12,38]

Under the optical coherent state , we make the average of the DM’s Hamiltonian (13)

The average Hamiltonian possesses two macroscopic eigenstate, i.e., the SCS of south and north pole gauge. The SCS, which correspond to the normal and inverted pseudospin states in the dynamics of the DM, can be generated from the extreme Dicke states () with the SCS transformation .^[25–26] And the unitary operator is explicitly given by

In fact, the SCS of the north and south pole gauges are the eigenstates of the spin projection operator , where is the unit vector with the directional angles θ and φ. And the spin operators satisfy the minimum uncertainty relation () and therefore the SCSs are called the macroscopic quantum states, which are orthogonal. Using the unitary transformations for the spin operators J_z, ,

Through a series of complicated and tedious calculation, we finally get the following spin Hamiltonian

In order to obtain the diagonalized spin Hamiltonian, it needs to satisfy the conditions , i.e.,

In principle, the angle parameters θ, φ can be determined based on the above conditions Eq. (17). Meanwhile, we get the following relations of the parameters θ and A

Finally, the ground state energy functional of one variable is obtained:

By the usual extremum condition of the energy function for the first-order derivative equal to zero, the macroscopic many-particle quantum state solutions are found:

From Eq. (18), we find it possesses two solutions. One solution is the parameter , corresponding to a zero-photon-number solution and giving rise to the normal phase (NP) only if the second-order derivative is positive, namely, . The other solution satisfies the condition

If the second-order derivative in the solution is positive, we define the system into the superradiant phase (SP). In the critical phase transition point , the Dicke system undergoes a QPT from the NP to the SP, which has a wide range of research.^{[2–4,6,11–14,18–19,24]}

The ground-state energy of the DM system is

The atomic population difference is proportional to the atom-field interaction strength in the SP region and satisfies the following relation

The mean photon number distribution meets the following nonlinear relationship when

It is worth while to emphasize that the SCS method is completely consistent to the Holstein–Primakoff transformation^[3] in deal with the ground state properties of the standard DM and is valid for arbitrary atom number N, which is superior to the Holstein–Primakoff transformation under the condition of the thermodynamic limit.

In some physical theories, mean field theory also can be seen as the self-consistent field theory,^[27] which can be used to research the behavior of the large and complex stochastic models in a simple manner. Both the strong-interaction^[34] and the weak-interaction^[29] are compatible with the self-consistent field theory. The self-consistent field theory has been applied in the atoms,^[30] the condensed matter,^[31] the Lorentz gas^[32] and so on. A series of literature have studied the exact-energy self-consistent field,^[28] the self-consistent-field perturbation theory for the Schrödinger equation,^[39] the self-consistent mean-field theory,^[35] the self-consistent mean-field theory of asymmetric first-order structural phase transition,^[33] the quantum quench in interacting field theory,^[40] polariton condensation with nonlinear photons described by the generalized Dicke Hamiltonian on the mean-field level.^[41]

In this article, we begin the following discussion based on the mean field approximation and the self-consistent field theory to solve the extended DM containing the atom-atom interaction.

For the atom-atom interaction , by the means of the self-consistent mean field theory,^[27] we obtain

Substituting Eq. (22) for into Hamiltonian (12), it turns into

It is to be highlighted that Hamiltonian (23) only satisfies the case of the normal phase. And the corresponding scaled ground state energy

Substituting Eq. (22) for again into Hamiltonian (12), it turns into

For Hamiltonian (25), the atomic population difference has such form

Once again, we substitute Eq. (27) into Hamiltonian (12), it turns into

Hamiltonian (12) repeats the above steps (26), (27), and (29), (30) we obtain the third order case

Repeating the above iteration until the k-th grade and -th grade approximation have almost no influence on to the system’s total energy and meet the permissible error of the self-consistent field theory. Finally we obtain the n-th atomic effective frequency and the coresponding atom-atom interaction are respectively

And the coresponding total Hamiltonian

When , Hamiltonian (31) returns to

When the atom-atom interaction is not considered, i.e., q = 0, Hamiltonian (32) returns to the standard Dicke hamiltonian (13). If n changes into infinity, the self-consistent method has quite high accuracy and there is almost no error. In fact, there is so little difference between ω₂ and ω₁ or between ω₃ and ω₂. From Fig. 1, we may more clearly see this small differences. In the whole paper, the unite vector is ω_a and we choose .

Fig. 1

	Figure Option ViewDownloadNew Window
	Fig. 1 (Color online) The ratio of (a) and (b) as the function of the atom-atom interaction q. The plotted collective coupling strength are respectively 1.2 (the red solid line), 1.5 (the green solid line), 1.8 (the violet solid line). The frequency of the light field is .

Based on the above SCS method and Hamiltonian (32), the ground state energy functional

We also obtain that the extended DM goes through a quantum phase transition with the phase transition point

We also calculate the three corresponding physical quantity: the mean photon number, the atomic population imbalance, and the quasi-ground-state energy.

The mean photon number

The atomic population imbalance

The quasi-ground-state energy

Figure 2 shows the influence of the atom-atom interaction parameter on the mean photon number (a), the atomic population imbalance (b), and the average ground-state energy (c). The average ground-state energy are continuous and slippy at the critical phase transition point g_cq, which shifts toward the lower g direction with the increasing of the atom-atom interaction. In NP, the mean photon number and the atomic population imbalance have not been excited and the average energy is stable. On the opposite of NP, in the SP area, the excited photon or atom number are non-zero and increases as the collective coupling strength increases. At the same time, the average energy become low, but is continuous with the case of the NP zero. The three physical quantities are more and more close to the each other with the increasing of the collective coupling strength in the case of the atom-atom interaction , 0.05, 0.10. The existence of the atom-atom interaction makes the phase transition point move left. Especially when , there is a big movement.

Fig. 2

	Figure Option ViewDownloadNew Window
	Fig. 2 (Color online) Variations of the average photon number npq (a) (the solid line), the atomic population imbalance (b) (shte dashed line), and the average energy (c) (the dotted line) with respect to the atom-field coupling constant for the given parameter (the black line), 0.05 (the red line), 0.10 (the green line).

Figure 3 displays variations of the average photon number n_pq (a), the atomic population imbalance (b), and the average energy ε_q (c) with respect to for the given , 1.2, 1.5. The average photon number and the atomic population imbalance have not been excited, which are respectively a horizontal line, meanwhile the average energy increases obviously as the increase of the atom-atom interaction in the case of . With the increase of , there exists the excited photon or atom number and the average energy decreases. But the excited particle number almost has no change and the average energy is the lowest and almost keeps a straight line when .

Fig. 3

	Figure Option ViewDownloadNew Window
	Fig. 3 (Color online) Variations of the average photon number npq (a) (the solid line), the atomic population imbalance (b) (the dashed line) and the average energy (c) (the dotted line) with respect to the atom-atom interaction for the given atom-filed coupling strength (the black line), 1.2 (the red line), 1.5 (the green line).

In Fig. 4 we plot the mean photon number γ ² as a function of the collective atom-photon coupling strength g and the atom-atom interaction strength q. The green region () belongs to the NP and the color region () is the SP. The critical line of the two regions are g_cq, which shifts toward the lower g direction when and the more moving to the left is seen when the q increases from 0 to 0.10.

Fig. 4

	Figure Option ViewDownloadNew Window
	Fig. 4 (Color online) Phase diagram in g-q space. The NP with belongs to the normal phase. The SP with is the right region of the boundary line gcq. The color index indicates the photon number .

In order to compared to the effect of the atom-atom interaction strength q and the collective atom-photon coupling strength g on the average ground state energy, in the following we introduce the difference in the SP between the average ground state energy and the case of q = 0, i.e., . At the same time, the relative difference has been introduced. The conclusion has been plotted in Fig. 5. For the given collective atom-photon coupling strength g, the difference of or increases or decreases with the increased atom-atom interaction. While the difference is more obvious when than the case of or .

Fig. 5

	Figure Option ViewDownloadNew Window
	Fig. 5 (Color online) (a) and (b) as the function of the atom-atom interaction strength for the given collective atom-filed couping strength (the red line), 1.5 (the green line), 1.8 (the violet line).

We give the results of and .

Based on the average ground-state energy Eq. (34), which is the function of the atom-field coupling constant g and the atom-atom interaction strength q, the first-order and second-order partial derivatives of the average ground-state energy are given

Based on Eqs. (35), (36) Fig. 6 and Fig. 7 give the first-order and second-order derivatives in different system’s parameters. In the absence of the atom-atom interaction, i.e., q = 0, the first-order derivative is continuous in the phase boundary of the NP and the SP, but the second-order derivative is discontinuous. So we come to the conclusion that the standard DM exists a second-order QPT from the NP to the SP in the critical phase transition point . While the atom-atom interaction is existent, the extented DM belongs to the first-order quantum phase transition, for the first-order and second-order derivative are both discontinuous in the critical phase transition point g_cq, which shifts to the left.

Fig. 6

	Figure Option ViewDownloadNew Window
	Fig. 6 The first-order derivative of the average ground-state energy (a) and the second-order derivative (b) as the function of the collective coupling strength for the given atom-atom interaction (the black line), 0.05 (the red line), 0.10 (the green line) with .

Fig. 7

	Figure Option ViewDownloadNew Window
	Fig. 7 The first-order derivative of the average ground-state energy (a) and the second-order derivative (b) as the function of the collective coupling strength and the atom-atom interaction () with .

Figure 7 displays the three-dimensional curves about the first-order derivative and the second-order derivative for the parameters and . They are discontinuous at the every critical phase transition point, which shifts to the left with the increasing q. They imply that the system undergoes a first-order QPT from the NP to the SP and the exclusive atom-atom interaction makes the phase transition point shift toward the lower atom-photon collective coupling strength.

In summary, an interesting prediction of the extended DM is that it undergoes a first-order QPT between the NP and the SP. In order to detect this first-order transition, we plotted the mean photon number, the scaled mean atom population and the average ground-state energy for different atom-atom interaction and discovered the QPT point shifts to left for the increasing atom-atom interaction. By the discontinuous first-order and second-order derivative, we come to the conclusion that the atom-atom interaction induces a first-order phase transition, which is different from the second-order QPT of the standard DM. We believe the atom-atom interaction can also have a deep influence on the physical properties such as the entanglement and the geometric phase in quantum information process.