1. Introduction

Dicke model was first proposed by Dicke in the 1950s to describe the coherent interaction between an ensemble of two-level atoms and a single-mode bosonic field [1]. It has been shown that under collective spontaneous emission of atoms, the atomic system in Dicke model undergoes a quantum phase transition from the normal to superradiant phase in the thermodynamic limit [2–5]. In recent years, Dicke model has been realized in various systems [6–11], such as atom-cavity setup [12], waveguide-emitter [13] superconducting circuits [14] setups. Besides quantum phase transition, new interesting features, like quantum critical behavior [15,16], quantum chaos [17,18] and quantum entanglement and correlations [19–23], have also been discovered in the Dicke model. In paricular, the entanglement in the model has a critical point similar to the phase transition, showing interesting characteristics [8,12,24–31]. However, in much work on the open Dicke model only the cavity dissipation has been considered to perform the Holstein-Primakoff approximation [32]. Recent studies has shown that the decay of atomic internal states due to spontaneous emission has nontrivial effects on the mean-field nonequilibrium steady states for spin and photon observables [33–35] and quantum-fluctuation correlations of atoms and field [29].

In this paper, we intend to explore a new type of quantum correlations — quantum steering — in an open Dicke model in which not only the cavity dissipation but also the spontaneous emission and dehasing of individual atoms are taken into account. We consider the situation that each atom is immersed in an independent squeezed bath and undergoes dephasing process. Quantum steering originates from the well-known Einstein-Podolsky-Rosen (EPR) paradox, which describes the situation where two remote observers, say Alice and Bob, share a pair of perfect entangled particles and Alice can nonlocally manipulate the particle’s states in Bob via different types of local measurements on his own particle [36]. Such ability to nonlocally steer quantum states of remote particles was originally termed by Schrödinger as steering [37]. It has recently been revealed by Wiseman et al. [38] that EPR steering is a type of quantum correlation intermediate between entanglement (nonseparability) [39] and Bell nonlocality (the violation of Bell inequality) [40]. This unique feature makes steering a physical resource for one-sided device-independent quantum cryptography [41], subchannel discrimination [42], and secure quantum teleportation [43].

Specifically, here we found that in both normal and superradiant phases, the cavity dissipation and individual atomic decohernece can improve the entanglement and steering between the cavity field and atomic ensemble. The individual atomic spontaneous emission leads to the appearance of the steering between the cavity field and atomic ensemble but the steerings in two directions cannot be simultaneously generated. The maximal achievable steering in normal phase is stronger than that in superradiant phase. Furthermore, the entanglement and steering between the cavity output field and atomic ensemble is much stronger than that with the intracavity and the steerings in two directions can be achieved even with the same parameters.

This paper is organized as follows: In Sec. II, we introduce the model. In Sec. III and IV, the steady-state solution of the mean field and quantum fluctuations of the system are presented. In Sec. V, the quantum steering and entanglement with the intracavity field and output field are investigated. In the last section, some discussion and the conclusion are presented.

2. Model

As shown in Fig. 1, we consider a Dicke system in which an ensemble of two-level atoms is strongly coupled to a single-mode cavity field with the resonance frequency $\omega _0$ and denoted by the annihilation (creation) operator $\hat {a}_f$ ($\hat {a}^{\dagger}_f$). Each atom is described by the Pauli operator $\hat {\sigma }_j$ ($j=x,y,z$), where the operators $\hat {\sigma }_z=\frac {1}{2}(|\uparrow \rangle |\uparrow \rangle -|\downarrow \rangle |\downarrow \rangle )$ and $\hat {\sigma }_{x, y}=\frac {1}{2}(|\uparrow \rangle |\downarrow \rangle \pm |\downarrow \rangle |\uparrow \rangle )$, with $|\uparrow \rangle$ and $|\downarrow \rangle$ respectively denoting the spin-up and spin-down states of the frequency difference $\omega _z$. The interaction between the cavity field and the collective atoms can be described by the Hamiltonian ($\hbar =1$)

 

Fig. 1. Schematic plot of the system. An ensemble of identical two-level systems is collectively coupled to a cavity with loss rate $\kappa$. Each atom undergoes independent spontaneous and dephasing processes with rates $\gamma _k$ and $\gamma _c$ respectively. The output field is filtered to define temporal modes to study the quantum correlations between the ensemble and output field.

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When considering that each atom is immersed in individual decoherencing environments, we can write the master equation for the density matrix $\hat \rho$ of the system as

3. Equations of mean fields

We at first study quantum phase transition of the open Dicke model in the thermodynamic limit ($N_a\rightarrow \infty$). By defining the following scaled operators

The stationary solutions of the above equations can be found. When the coupling $\lambda <\lambda _c$, where the critical coupling strength

4. Quantum-fluctuation matrix correlations

We next consider quantum fluctuations of the system. Since the total spin number is no longer conservative due to individual atomic decoherence [44,45], the conventional approach of the Holstein-Primakoff approximation [28,32,46] to bosonify atomic variables is invalid. Instead, we consider the rotation of the reference frame for the atomic variables by rotating $\hat m_x$ and $\hat m_y$ an angle $-\varphi$ around $z$-axis and then rotating $\hat m_z$ and $\hat m_x$ an angle $-\theta$ around $y$-axis. In the new frame, the rotated spin operator $\hat {\tilde {m}}_z$ is aligned to the spin vector $\hat {\vec {m}}$, and therefore the rotated spin variables $\hat {\tilde {m}}_{x,y}$ are pure fluctuations and for $N_a\rightarrow \infty$ we have $\hat {\tilde {{m}}}_z\simeq \lvert \hat {\vec {m}}\rvert$. From the commutation relation $[\hat {\tilde {{m}}}_x, \hat {\tilde {{m}}}_y]=i\lvert \hat {\vec {m}}\rvert$ and by defining

This means that in the rotated frame the atomic ensemble can be treated as a harmonic oscillator and we thus can study continuous variable quantum correlations between the cavity field and the atomic ensemble [22,29].

The above rotation corresponds to a local unitary operation on the spin, which can be described by the matrix

It shows that if $m_z\rightarrow |\overrightarrow {m}|$, i.e., the spins are almost aligned to the $z$ component, the rotation matrix is unnecessary since $R_{\theta \varphi }\simeq 1$.

We proceed to determine the correlation matrix between the atomic and field fluctuations characterized by $\delta \hat o=\sqrt {N_a}(\hat o-\langle \hat o\rangle _s)$ ($\hat o$ being $\hat {m}_j$, $\hat {X}_f$ or $\hat {Y}_f$), which is rescaled by a factor of $\sqrt {N_a}$. For the fluctuation operators $u(t)=(\delta \hat {m}_x, \delta \hat {m}_y, \delta \hat {m}_z, \delta \hat {Y}_f, \delta \hat {X}_f)^{T}$, from Eq. (2), the equation of motion for $u(t)$ can be derived as

For the fluctuation operators $\tilde {u}(t)=(\delta \hat {q}_a, \delta \hat {p}_a, \delta \hat {\tilde {m}}_z, \delta \hat {Y}_f, \delta \hat {X}_f)^{T}$, with Eq. (8) and Eq. (10) we have $\tilde {u}(t)=WR_{\theta \varphi }u(t)$ and thus the corresponding correlation matrix $\tilde {V}_{ii'}\equiv \frac {1}{2}\langle \tilde {u}_i \tilde {u}_{i'}+\tilde {u}_{i'}\tilde {u}_i\rangle$ can be given by

We consider the stationary solution $\tilde {V}(t\rightarrow \infty )$ of the Eq. (16), omit the third row and the third column and a $4\times 4$ reduced correlation matrix which carries the information we need for the quantum correlations can be finally obtained as

With the steady-state correlation matrix we can investigate quantum correlations between the cavity field and the atomic ensemble. The atom-field bipartite entanglement can be quantified by the logarithmic negativity [48–50]

5. Results

5.1 Intracavity case

We first investigate the entanglement and steering between the intracavity field and atomic ensemble. Fig. 2 plots the dependencies of the steering $S_{f|a}$, $S_{a|f}$ and entanglement $E_n$ on the cavity dissipation $\kappa$ and individual atomic spontaneous emission rate $\gamma$ and dephasing rate $\gamma _c$. Since the critical point increases with the increase of $\gamma$ and $\kappa$, we consider temporally the regime in normal phase for which the coupling $\lambda$ is fixed at $0.1$. It is shown that in the absence of individual atomic spontaneous emission and dephasing, the steering in either direction is unachievable. However, when the spontaneous emission is present ($\gamma \neq 0$ and $\gamma _c=0$), we have $S_{f|a}\neq 0$ and $S_{a|f}\neq 0$, meaning that the steering in two directions can occur, as shown in Fig. 2 ($a_1$) and ($a_2$). This is because that the individual atomic decoherence processes decrease the fluctuations of the atomic ensemble. Further, we see from Fig. 2 ($a_{1,2}$) and ($b_{1,2}$) that the steering in two directions cannot be simultaneously achieved for the parameters $\kappa$, $\gamma$ and $\gamma _c$. This means that one can only achieve one-way steering between the intracavity field and atomic ensemble. The atom-to-field steering $S_{f|a}$ only exists for $\{\gamma,~\gamma _c\}\lesssim 1$, whereas the inverse steering $S_{a|f}$ is just absent for $\{\gamma, \gamma _c\}\lesssim 0.3$. Likewise, the steering $S_{f|a}$ occurs for $\kappa \gtrsim 1.5\omega _z$ and the steering $S_{a|f}$ can exist for much less values of $\kappa$. Note that for the same rates $\gamma =\kappa$, only the steering $S_{a|f}$ is present, which means that the local states of the atomic ensemble and intracavity field is asymmetric even for the balanced dissipation rates. Different from the steering, the entanglement exists even when $\gamma =0$ and $\gamma _c=0$, as already studied in Ref. [28]. More importantly, we can see that the steering and entanglement can be increased by the individual atomic decoherence and cavity decoherence. It is clearly shown that the steering and entanglement increases at first and then decreases as $\gamma$, $\gamma _c$ and $\kappa$ arise continuously. Thus, for the open Dicke model, individual atomic decoherence and cavity dissipation may be constructive to the quantum correlations between the field and atomic ensemble. In addition, the maximal achievable entanglement and steering are achieved for different parameters.

 

Fig. 2. Density plot of the steering $S_{f|a}$, $S_{a|f}$ and entanglement $E_n$ between intracavity field and spin field versus $\gamma$ and $\kappa /\omega _z$ in ($a_1$)-($a_3$), $\gamma _c$ and $\kappa /\omega _z$ in ($b_1$)-($b_3$), $r$ and $\kappa /\omega _z$ in ($c_1$)-($c_3$). Related parameters in plots: $\omega _0=\omega _z=0.2$, $\lambda =0.1$. In (a1)-(a3) $\gamma _c=r=0$; in ($b_1$)-($b_3$) $\gamma =0.1$, $r=0$; in ($c_1$) and ($c_3$) $\gamma =\gamma _c=0.1$; in ($c_2$) $\gamma =0.1$, $\gamma _c=1.5$.

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In order to observe the characteristics of quantum steerable correlations in the superradiant phase, we plot the dependence of the steering and entanglement on the coupling $\lambda$ for different values of $\gamma$ and $\gamma _c$ in Fig. 3, where the critical coupling point $\lambda _c$ is marked with asterisks. The value of $\kappa$ is fixed at $6\omega _z$. Note that the values of $\gamma$ are different in Fig. 3 ($a_1$) and ($a_2$), since the steerings in two directions cannot be achieved simultaneously for the same value. It is found that the steering $S_{a|f}$ only occurs when $\gamma >6\omega _z$ and always be zero in the superradiant phase, while $S_{a|f}$ only occurs when $\gamma <6\omega _z$ and has nonzero values in the superradiant phase. The maximal achievable steering is much weaker than that in the normal phase. It is also shown that the steering is almost absent near the critical coupling points. This is because that near the critical coupling point, the quantum fluctuations of quadratures of the cavity field and atomic ensemble (described by their variances) become much larger, which makes the Gaussian quantum states of the two subsystems are difficult to be locally steered (by Gaussian measurement) via the finite quantum correlations. Further, when the system is in the superradiant phase, the spontaneous emission process can also improve the steering, as shown in Fig. 3 ($a_1$). For the same value of $\gamma$, the optimal entanglement in the superradiant phase can be larger than that in the normal phase, as shown in Fig. 3 ($a_3$), different from the behavior of the steering. When $\gamma =0$, the entanglement peaks around the critical coupling point $\lambda _c$, but it may become local minimal around $\lambda _c$ for nonzero $\gamma$. In both normal and superradiant phases, the steerings in two directions and entanglement do not increase monotonically with the coupling $\lambda$ and they exhibit local peaks with respect to $\lambda$. But for the entanglement, it has local dips or peaks near the critical coupling $\lambda _c$. Figures 3 ($b_1$)-($b_3$) depicts the effect of the dephasing processes on the steering and entanglement. Since quantum phase transition disappears in the open Dicke model only with dephasing processes, we set $\gamma =0.5\omega _z=0.1$ in Figs. 3 ($b_1$) and ($b_3$) and $\gamma =7.5\omega _z=1.5$ in Fig. 3 ($b_2$). We see that when $\gamma _c$ increases, the steering $S_{f|a}$ increases at first and then decreases and $S_{a|f}=0$. As shown in Figs. 2 and 3 ($c_1$)-($c_3$), the local squeezed environment with small squeezing can also lead to the enhancement of the entanglement and steering. This is due to that stronger squeezing causes the amplification of the noise and the decrease of the entanglement and steering.

 

Fig. 3. The dependencies of the steering $S_{f|a}$ and $S_{a|f}$, entanglement $E_n$ between intracavity field and spin field on $\lambda$ for different $\gamma$ in ($a_1$)-($a_3$), $\gamma _c$ in ($b_1$)-($b_3$), $r$ in ($c_1$)-($c_3$). Related parameters in plots: $\kappa =1.2$. In ($a_1$)-($a_3$) $\gamma _c=0$, $r=0$; in ($b_1$) and ($b_3$) $\gamma =0.1$, $r=0$; in ($b_2$) $\gamma =1.5$, $r=0$; in ($c_1$) and ($c_3$) $\gamma =\gamma _c=0.1$; in ($c_2$) $\gamma =\gamma _c=1.5$; the other parameters are the same as Fig. 2.

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5.2 Output case

Since the intracavity field in the open system cannot be observed directly, we then study the quantum steering and entanglement between the output field emitted from the cavity and the atomic ensemble. The output field is related to the intracavity field via the input-output relation [52]

Since the output field is spectral continuous, we define temporal mode by

Here $\tilde {M}(\omega )=(WR_{\theta \varphi }AR^{T}_{\theta \varphi }W^{-1}+i\omega )^{-1}$, $P^{out}=Diag[0,0,0,1,1]$,

In Fig. 4, we plot the dependence of the steering and entanglement between the filtered output field and the atomic ensemble on the spectral frequency $\Omega _f$ and bandwidth $\tau ^{-1}$, by considering the coupling strengthes respectively in the normal phase and superradiant phase. Different from the intracavity case, one can see that the steerings in both directions can be achieved when $\gamma$ and $\gamma _c$ deviate from zero. By comparing Fig. 3 and Fig. 4, we can see that the steering and the entanglement with the output field are much stronger than that with the intracavity field for the same parameters. This is because that due to the fact that the latter can be considered as the sum of the steering and entanglement over all the spectral components. Therefore, robust atom-field steering and entanglement are built up by filtering the cavity output, which also means an effective entanglement and steering distillation. It is also shown that the steering and entanglement become optimal around $\Omega _f=-\omega _z$ and $\tau ^{-1}=\omega _z$ in the normal phase, while in the superradiant phase the optimal point of $\Omega _f$ shifts to the left on $\Omega _f=-2.5\omega _z$ and $\tau ^{-1}$ still keeps the value of $\omega _z$.

 

Fig. 4. The dependencies of the steering $S_{f|a}$ and $S_{a|f}$, entanglement $E_n$ between output field and spin field on $\Omega _f/\omega _z$ in ($a_1$), ($b_1$) and $\omega _z\tau$ in ($a_2$), ($b_2$). In ($a_1$)-($a_2$) the system is in normal phase and $\lambda =\omega _z=0.2$, in ($b_1$)-($b_2$) the system is in superradiant phase and $\lambda =2\omega _z=0.4$. Related parameters in plots: $\gamma =0$, $\gamma _c=0$, $r=0$, in ($a_1$)($b_1$) $\tau ^{-1}=\omega _z=0.2$; in ($a_2$) $\Omega _f=-\omega _z=-0.2$; in ($b_2$) $\Omega _f=-2.5\omega _z=-0.5$; the other parameters are the same as Fig. 3.

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In Fig. 5, we plot the dependence of the output steering and entanglement with the filtered output of frequency $\Omega _f=-\omega _z$ on the coupling $\lambda$. Similar features are shown to the intracavity case. For the same rates $\gamma$ and $\gamma _c$ and coupling $\lambda$, the steering from the atomic ensemble to the output field $S_{f|a}$ is larger than the steering reverse $S_{a|f}$. The steering $S_{f|a}$ is more sensitive to $\gamma$ and $\gamma _c$ than the steering $S_{a|f}$. When $\gamma$ and $\gamma _c$ increase from $0$ to $\omega _z$, $S_{f|a}$ decreases considerably and approaches to zero, but the steering $S_{a|f}$ drops lightly. This result implies that the direction of the steering can be controlled by the dissipation of atoms.

 

Fig. 5. The dependencies of the steering $S_{f|a}$ and $S_{a|f}$, entanglement $E_n$ between output field and spin field on $\lambda$ for different $\gamma$ in ($a_1$)-($a_3$), $\gamma _c$ in ($b_1$)-($b_3$), $r$ in ($c_1$)-($c_3$). Related parameters in plots: $\kappa =1.2$, $\Omega _f=-\omega _z$, $\tau ^{-1}=2\omega _z=0.4$. In ($a_1$)-($a_3$) $\gamma _c=0$, $r=0$; in ($b_1$)-($b_3$) $\gamma =0.1$, $r=0$; in ($c_1$)-($c_3$) $\gamma =\gamma _c=0.1$; the other parameters are the same as Fig. 2.

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6. Conclusion

In conclusion, we study steady-state quantum entanglement and steering in an open Dicke model where cavity dissipation and individual atomic decoherence are taken into account. Specifically, we consider that each atom is coupled to independent dephasing and squeezed environments, which makes the conventional Holstein-Primakoff approximation invalid. We find that in both of normal and superradiant phases, the cavity dissipation and individual atomic decoherence can improve the entanglement and steering between the cavity field and atomic ensemble. The individual atomic spontaneous emission leads to the appearance of the steering between the cavity field and atomic ensemble but the steerings in two directions cannot be simultaneously generated. In addition, the maximal achievable steering in normal phase is stronger than that in superradiant phase. The entanglement and steering between the cavity output field and atomic ensemble is much stronger than that with the intracavity. Our findings reveal unique features of quantum correlations in the open Dicke model in the presence of individual atomic decoherence processes. The nonclassical atom-field states can be used for quantum information processing, e.g., quantum teleportation channel for quantum-state transfer, and quantum metrology, such as the sensing of weak magnetic field.