Due to its fundamental importance and potential applications in quantum technologies, quantum phase transition (QPT) in, e.g., the Dicke model has attracted much attention.^[1–19] In quantum optics, the Dicke model, which describes the collective behavior of two-level systems (TLSs) interacting with a cavity mode, was first used to study the superradiance.^[20] This model includes both rotating and counter-rotating interactions between the TLS ensemble and the cavity mode. In the thermodynamical limit, when continuously varying the strength of the collective interactions, the system undergoes a QPT at a critical coupling strength (equal to the half of the geometric mean of the resonance frequencies of the TLS ensemble and the cavity mode).^[21–27] With maller (larger) than the critical coupling strength, the TLS ensemble is in the normal (superradiant) phase. When neglecting the counter-rotating coupling terms via the rotating-wave approximation (RWA),^[28] the Dicke model can be reduced to the Tavis–Cummings (TC) model.^[29] Similar to the Dicke model, the TC model can also exhibit a QPT, but the corresponding critical coupling strength is twice the critical coupling strength of the Dicke model.^[30] However, the very large critical coupling strength (in either the Dicke or TC model) is not accessible in a realistic physical system.

To circumvent this difficulty, the nonequilibrium QPT was theoretically proposed^[31–34] and experimentally implemented^[35–38] by simulating the superradiant QPT with drive physical systems. For instance, an effective Dicke Hamiltonian was designed by pumping the ensemble of four-level atoms coupled to an optical cavity mode with two drive fields.^[31] In this engineered Dicke model, the QPT is controlled by tuning the coupling strength between the TLS ensemble and the cavity mode via varying the drive-field frequencies and intensities. Then, this proposed scheme was implemented using a hybrid system consisting of a Bose–Einstein condensate in a dilute atomic gas and an optical cavity.^[35] In addition, by off-resonantly pumping the standard TC model with a drive field, an effective TC Hamiltonian can also be engineered in a rotating frame with respect to the drive field.^[34] In the ideal case without decoherence for the system, the driven TC model can have a QPT in the weak drive-field limit, where the experimentally achievable critical coupling strength is equal to the geometric mean of the frequency detunings of both the cavity mode and the TLS ensemble from the drive field. Usually, both cavity mode and TLS ensemble are dissipative. The losses of the cavity mode and the TLS ensemble then greatly obscure the QPT in the TC model,^[39–41] as observed in the experiment.^[38]

In this paper, we present a scheme to demonstrate the QPT in a biased TC model by including the decoherence of the system. In our proposal, the biased TC model describes a dissipative TLS ensemble pumped by a drive field and coupled to an active cavity mode (instead of a lossy cavity mode^[38]). In the rotating frame with respect to the drive field, the effective frequencies of the TLS ensemble and the cavity mode become their frequency detunings from the drive field, respectively. In the limit of a weak drive field, we show that the biased TC model tends to exhibit the QPT owing to the combined actions of the loss and gain of the system. Furthermore, we find that the active cavity can greatly improve the QPT behavior even at a finite strength of the drive field. We also propose to implement the scheme by using an ensemble of nitrogen-vacancy (NV) centers in diamond coupled to an active optical cavity and show that the QPT can be probed by measuring the transmission spectrum of the cavity embedding the TLS ensemble. Our work provides an approach to reduce the adverse effect of the decoherence on the QPT in a biased TC model and can make it promising to experimentally observe the QPT in a realistic system.

For an ensemble of N TLSs in a cavity, when the coupling between each TLS and the cavity photon is identical, the coupled system can be described in the RWA by a TC model (we set ħ = 1)

The TC model has a conserved parity,^[25] [H_TC,Π] = 0, where Π = exp{iπ(a^† a + J_z + N/2)}. In the thermodynamic limit N → + ∞ and without the decoherence of the system, by varying the collective coupling g from the regime to , where , the system undergoes a QPT from the normal to the super-radiant phase at zero temperature.^[21,22] However, it is extremely difficult to experimentally demonstrate this QPT due to the decoherence of the system^[39–41] as well as the difficulty in accessing the very large critical coupling strength .

To solve these problems, we first reduce the critical coupling strength by applying a field with frequency ω_d to drive the TLS ensemble. This corresponds to adding a drive term to the Hamiltonian (1), where denotes the collective coupling strength between the drive field and the TLS ensemble, with Ω_s being the TLS–field coupling strength. In the rotating frame with respect to this drive field, the Hamiltonian of the system can be converted, in the RWA, to a biased TC model

In addition, the QPT can be greatly affected by the decoherence of the system. In fact, for the TC model in Eq. (1), the QPT disappears due to the decoherence of the system.^[39–41] In previous studies, the QPT is investigated by diagonalizing the Hamiltonian of the system,^[25,34] but it is difficult to diagonalize the Hamiltonian of the system when the decoherence of the system is included. Therefore, we use a quantum Langevin approach^[28] to study the QPT in the biased TC model. For the Hamiltonian in Eq. (2), the dynamics of the system is governed by the following quantum Langevin equations:

To study the steady behavior of the system, we write each of the operators a, J₋, and J_z as a sum of its mean value and fluctuation, a = ⟨a⟩ + δ a, J₋ = ⟨J₋⟩ + δ J₋, and J_z = ⟨J_z⟩ + δ J_z. From Eq. (3), it follows that the mean values ⟨a⟩ and ⟨J₋⟩ satisfy

For the ideal case without decoherence, i.e., γ = κ_c = 0, the above equation is reduced to . Solving it in the weak drive-field limit , we obtain the steady solutions χ = 0 and , which correspond to the normal and super-radiant phases for g < g_c and g ≥ g_c, respectively. Similar to the driven TC model in Ref. [34], in the weak drive-field limit and in the absence of decoherence, the biased TC model also tends to exhibit a QPT from the normal to the super-radiant phase at the critical coupling strength g ≡ g_c by tuning g_c (via the frequency ω_d of the drive field) from g < g_c to g > g_c, with ⟨J_z⟩/(N/2) = 2χ −1 given by

Fig. 1.

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Fig. 1. Numerical results of ⟨Jz⟩/(N/2) versus the coupling strength g/gc for different amplitudes of the drive field. (a) ⟨Jz⟩/(N/2) versus g/gc without the gain, calculated using Eq. (8). Here κc = γ = 0 and for the (black) solid curve; κc/γ = 0.5 and for the (red) dotted curve; and κc/γ = 0.5 and for the (blue) dashed curve. (b) ⟨Jz⟩/(N/2) versus g/gc with the gain, calculated using Eq. (11). There is no ⟨Jz⟩/(N/2) when g/gc < 0.2 (grey region) because in this region [cf. Eq. (13)]. The gain rate κg in Eq. (13) is determined with and the same value of κg is used for the two curves. Here for the (red) dotted curve and for the (blue) dashed curve. In both (a) and (b), we choose Δc = Δs and g/γ = 10.

To circumvent this situation, we can harness an active cavity mode with the effective gain rate coupled to a dissipative TLS ensemble with damping rate γ, where a gain rate is introduced by using a gain medium in the cavity (cf. Section 3). This corresponds to the case with κ_c in the first equation of Eq. (3) replaced by −κ_g. In this active-cavity case, equation (3) becomes

With an appropriate gain rate satisfying γ − ηκ_g = 0, the steady-state equation (11) of the system is also reduced to , as in the case without decoherence for the system. In the weak drive-field limit , the steady-state solution in Eq. (9) can also be obtained in the presence of decoherence and ⟨J_z⟩/(N/2) has the same critical behavior, i.e., , with ν_z = 1, but the critical coupling strength becomes

In Fig. 1, we numerically show the QPT in a biased TC model using Eqs. (8) and (11), respectively. Different from Fig. 1(a), no ⟨J_z⟩/(N/2) exists in Fig. 1(b) when g/g_c < 0.2 (see the grey region) because in this region [cf. Eq. (13)]. In the region of 1.2 < g/g_c < 3, ⟨J_z⟩/(N/2) gives rise to bistability at a finite drive-field strength [cf. the blue dashed curve in Fig. 1(b)], where the higher and lower branches are stable and the intermediate branch is unstable. When sweeping the coupling strength g/g_c rightwards (leftwards), the higher (lower) branch is approachable. Hereafter, we only focus on the higher stable branch of the blue dashed curve in the bistable region, which corresponds to the case of sweeping g/g_c rightwards. As expected, for the ideal case of κ_c = γ = 0, there is the QPT in the weak drive-field limit [see the black solid curve in Fig. 1(a)]. When including the losses of both cavity mode and TLS ensemble, the QPT disappears either in the weak drive-field limit or at a finite drive-field strength [see the red dotted curve and the blue dashed curve in Fig. 1(a)]. Instead of a lossy cavity, if an active cavity is coupled to the lossy TLS ensemble and their gain and loss rates satisfy Eq. (13), the system tends to approach the QPT behavior in the weak drive-field limit, even for the finite drive-field strength in Fig. 1(a) [comparing the red dotted curve and the higher stable branch of the blue dashed curve in Fig. 1(b) with the red dotted line and the blue dashed curve in Fig. 1(a)]. Therefore, an active cavity can much improve the QPT behavior in a biased TC model.

The gain has been widely studied in optical cavities.^[43,44] Below we propose to implement our scheme using an ensemble of NV centers in diamond coupled to an active optical whispering-gallery-mode (WGM) cavity,^[45–47] where each NV center acts as a TLS with the ground and excited states being ³A₂ and ³E of the electronic spin.^[48]

In the system proposed, the ensemble of NV centers is driven by an optical field with frequency ω_d, and the WGM cavity is made from the silica doped with a gain medium (e.g., rare-earth-metal ions).^[43,44] This gain medium can be modeled as an auxiliary spin ensemble with population inversion,^[49] as schematically shown in Fig. 2(a). Without the auxiliary spin ensemble, the coupled system is also described by the Hamiltonian H_s in Eq. (2).^[50,51] In the rotating frame with respect to the drive field on the NV centers, the Hamiltonian of the auxiliary spin ensemble is

Fig. 2.

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Fig. 2. (a) Schematic illustration of our proposal for improving the QPT in a hybrid system consisting of a driven ensemble of dissipative NV centers (blue oval) coupled to an active WGM cavity with gain modeled as an auxiliary spin ensemble with population inversion (yellow oval). The input and output fields of the cavity are denoted as and , respectively. (b) Energy-level diagram for the Hamiltonian in Eq. (16).

With the auxiliary spin ensemble included, the quantum Langevin equations of the total system are

In the experiment, the dissipative QPT can be measured via the transmission spectrum of the active WGM cavity embedding an ensemble of NV centers. When considering the probe field , it corresponds to adding a probe term to the first equation in Eq. (10), where κ_i is the decay rate of the cavity mode induced by the input port. Then, from Eq. (10), it follows that

According to the input–output theory,^[28] when no input field is applied to the output port, the output field can be written as

Using Eqs. (11) and (26), we can simulate the transmission spectrum |S₂₁(ω_p)|² of the active WGM cavity embedding the dissipative ensemble of NV centers. Figures 3(a) and 3(b) show the simulated transmission spectra versus ω_c − ω_p and g/g_c in the weak drive-field limit and at a finite drive-field strength, respectively, where the corresponding ⟨J_z⟩/(N/2) can be found in Fig. 1(b). The maximal values in each transmission spectrum (see the yellow pattern) represent the two Rabi-splitting peaks. At a fixed coupling strength g/g_c in the transmission spectrum, ⟨J_z⟩/(N/2) can be obtained from the separation between the two Rabi-splitting peaks [cf. Eq. (27)].

Fig. 3.

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	Fig. 3. Simulated transmission spectrum |S21(ωp)|2 of the system versus (ωc − ωp)/γ and g/gc when (a) and (b) , calculated using Eqs. (11) and (26). Here we choose κi/γ = κo/γ = 0.1, and other parameters are the same as those in Fig. 1(b).

For a TC model, one can also use a pump field to directly drive the cavity instead of the TLS ensemble.^[34] Actually, both cases of driving the cavity and the TLS ensemble are essentially equivalent. Using a unitary transform D(α) = e^{α a† − α*a} (with α = − Ω_d/g) on the biased TC model in Eq. (2), we obtain

In summary, we have theoretically studied the dissipative QPT in a biased TC model. We propose to couple a dissipative ensemble of TLSs with an active cavity field, where the TLS ensemble is pumped by a drive field. Using a quantum Langevin approach, we demonstrate the existence of QPT in the weak drive-field limit when the loss and gain rates of the system satisfy a certain constraint. Also, we propose to experimentally implement our scheme with a lossy ensemble of NV centers coupled to an active optical cavity and also show that the QPT can be measured using the transmission spectrum of the cavity embedding the TLS ensemble.