1. Introduction

Continuous-variable (CV) quantum entanglement is an important resource which has been paid great attention in modern quantum optics and quantum information sciences, possessing many potential applications in quantum teleportation [1], quantum key distribution [2], quantum communication [3,4], quantum information processing [5], etc. Carrying quantum information onto the quadratures of optical fields, such as amplitude and phase, has higher tolerance to dissipation during light propagation processes. In addition, CV quantum entanglement can be realized in other degree of freedom of optical fields, for instance, the polarization state of light has also been extensively studied in CV regime by transforming the quadrature entanglement onto polarization basis [6–9], and the quadrature entanglement using quantum orbital angular momentum with spatial Laguerre-Gauss mode has been discussed in experiment [10]. Furthermore, CV entanglement light source is essential in quantum imaging [11–13], which is an extension of quantum nature to transverse spatial degree of freedom. According to these previous researches, it is believed that optical field in CV entanglement plays an ideal information carrier, which is robust in quantum information sciences.

In order to generate entangled light, it is well known that using of optical nonlinear crystal is a typical scheme to generate light sources in CV regime. In theory, quantum correlation based on nondegenerate parametric oscillation was proposed [14]. Later, the generation of CV entanglement with nondegenerate parametric amplification was first observed in experiment by Ou et al. in 1992 [15]. Recently, the CV quantum entanglement at a telecommunication wavelength of 1550 nm had been realized using nondegenerate optical parametric amplifer [16]. On the other hand, mixing two independent squeezed lights which are generated from optical parametric amplifiers individually provides a practical method to generate quadrature entanglement [17]. These studies above clearly indicate that there is a connection between nonlinear optical processes and CV entanglement generation, so that the integration of all optical elements on chip has been proposed in order to further approach the goal of implementation of quantum computer in future [18].

Although the generation of quantum light sources from optical parametric processes, especially using $\chi ^{(2)}$ optical susceptibility of nonlinear crystal is popular, the light-matter interaction strength is difficult to control. In contrast, the nonlinear optical processes based on the interactions between fields and atomic systems can produce not only large amounts of quantum correlations between intense fields, but also controllability with accessible physical parameters. Recent decades, research reported on entangled light generation by four-wave mixing (FWM) has been intensely studied in hot atomic vapors [19–25]. Moreover, it has some potential applications, including the production of multiple quantum correlated beams [26], enhancement of the degree of entanglement [27], quantum metrology [28], etc. Meanwhile, electromagnetically induced transparency (EIT) [29] which is a coherent medium also plays an important role in atom-field interactions. Some peculiar features such as low-absorption, slow-light [30] and quantum memory [31–33] make EIT to be a promising ingredient in the development of quantum technologies. In the scenario, quantum optical pulse propagation in EIT [34], quantum squeezing generation in coherent population trapping media [35], large cross-phase modulation at few-photon level [36], and quantum correlated light generation as well as multiple fields correlation have been actively studied [37–46].

CV quantum entanglement arising from atom-field interactions is a good platform to investigate the connections between quantum coherence and correlations. Despite the fact that many papers have discussed the quantum entanglement generation in EIT systems, a systematic understanding of the physics behind the entanglement generation is still lacking. In this paper, we will discuss about the following questions: how do the tunable physical parameters in EIT system, which are photon-detunings, field Rabi frequencies, and atomic optical density, influence the entanglement degree between interacting fields? And how is the entanglement affected by the “EIT degree", which is related to ratio between two interacting field strengths. By solving the coupled equations of atomic and field operators numerically, we are able to study these questions.

The paper has been organized in the following way. In Sec. II, we start from a standard analysis of typical EIT interaction Hamiltonian, and derive the equations of motion for atomic operators, as well as the propagation equations for two quantized fields. The results from numerical calculation are given in Sec. III. Then, to reveal the underline physics, Sec. IV is concerned with the analytical approach for the output entanglement. Finally, a conclusion is given in Sec. V.

2. Theoretical model

We consider a collection of atoms having the three-level $\Lambda$-type configuration as shown in Fig. 1(a). Two ground states $\vert 1 \rangle$ and $\vert 2 \rangle$ are coupled to the common excited state $\vert 3 \rangle$ by probe and coupling fields, respectively. The Rabi frequency of probe field $\Omega _p$ is much weaker than that of the coupling $\Omega _c$, and the whole system forms a standard EIT. The one-photon detuning of probe and coupling fields are defined by $\Delta _p = \omega _p - \omega _{31}$ and $\Delta _c = \omega _c - \omega _{32}$, where $\omega _p$ and $\omega _c$ denote the light frequencies of probe and coupling, and $\omega _{\mu \nu } \equiv (E_\mu - E_\nu )/\hbar$ is the energy difference between any two states $\vert \mu \rangle$ and $\vert \nu \rangle$. The two-photon detuning is an important parameter in EIT system, defined by $\delta = \Delta _p - \Delta _c$. The decay rates from $\vert 3\rangle$ to the two ground states are $\Gamma _{1}$ and $\Gamma _{2}$, which are assumed to be the same in this work, so that we have $\Gamma _{1} = \Gamma _{2} = \Gamma /2$, where $\Gamma$ is the total decay rate of $\vert 3\rangle$. The damping rate from $\vert 2 \rangle$ to $\vert 1 \rangle$ is $\Gamma _{12}$. In addition, due to some practical effects such as atom collisions, we have to introduce the dephasing rate between two ground states, which is given by $\gamma _p$.

 

Fig. 1. (a) Atomic system configuration. (b) Probe and coupling are interacting with EIT atomic ensemble, and the entanglement measurement is performed at output.

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The system is arranged as shown in Fig. 1(b). Probe and coupling fields are propagating along the same direction, illuminating the EIT atomic ensemble, which is cooled to several hundred micro-Kelvin. Both probe and coupling fields are coherent states at input, and the entanglement measurement is performed at output for the two fields after propagating though EIT ensemble. In experiments, a closed three-level system in the $\Lambda$-type configuration is feasible. For example, in the energy levels of rubidium-87 atoms, the two hyperfine ground states of $5S_{1/2},F=1$, $5S_{1/2},F=2$, and the excited state of $5P_{3/2},F=2$ can form the closed three-level system.

Next, we start to study the system theoretically. We write the atom-field interaction Hamiltonian $\hat {H}$ in the rotating wave approximation [34,35]

Together with atomic equations in Eqs. (2)–(10) and field equations in Eqs. (11) and (12), we have a set of coupled equations between atomic and field operators. To calculate entanglement properties between two fields, we apply the mean-field approximation, dividing each operator $\hat {A}$ into two parts, i.e., $\hat {A} = A + \hat {a}$, where $A$ represents the mean-field value and $\hat {a}$ corresponds to the quantum fluctuation operator. Thus, we can decompose atomic and field operators as $\hat {\sigma }_{\mu \nu } = \sigma _{\mu \nu } + \hat {s}_{\mu \nu },~\mu ,\nu \in 1,2,3$, and $\hat {\mathcal {E}}_{\mu } = \mathcal {E}_\mu + \hat {a}_\mu ,~\mu \in p,c$, where $\hat {s}_{\mu \nu }$ and $\hat {a}_{\mu }$ are dimensionless atomic and field fluctuation operators, respectively. The detail derivations are given in Appendix.

In order to quantify the entanglement between two fields, we use Duan’s inseparability [47–49], which is a sufficient condition for continuous-variable entanglement demonstrated by many experiments, i.e.,

By scanning all quadrature angles, one can find an optimum quadrature angle $\theta _{\textrm {opt}}$, which minimizes the entanglement quantity $V$. The entanglement quantity $V(\theta )$ at $\theta _{\textrm {opt}}$ is given by

The entanglement degree depends on some tunable parameters. In Sec. III, we will show the results of entanglement under various physical quantities, and compare the corresponding entanglement degree.

3. Numerical results

According to the theoretical model in Sec. II, we know that the entanglement is the function of optical density ($\alpha$), two-photon detuning ($\delta$), input Rabi-frequency of two fields ($\Omega _{p,c}$), and ground-state relaxation rate ($\Gamma _{12}$), which are measurable physical quantities in experiments. For simplicity, we consider asymmetric one-photon detuning, which is arranged as $\Delta _p = -\Delta _c = \delta /2$. In this section, we will compare the two entanglement generation processes: one is by ground-state relaxation rate, and the other one is by two-photon detuning. All the results in this section are obtained numerically.

In Fig. 2(a), we have shown the relation between entanglement quantity $V$ and a ground-state relaxation rate $\Gamma _{12}$. For given optical density $\alpha$ and input Rabi frequencies of probe and coupling fields $\Omega _{p,c}$, we can find an optimum ground-state relaxation rate $\Gamma _{12, \textrm {opt}}$ to maximize the output entanglement (the minimum value of $V$, i.e., $V_{\textrm {opt}, \Gamma _{12}}$). If we give an optical density and a ground-state relaxation rate, there exists the optimum input Rabi frequency of coupling field $\Omega _{c, \textrm {opt}}$, such that the output entanglement is maximum ($V_{\textrm {opt}, \Omega _c}$), as shown in Fig. 2(b). EIT condition in Fig. 2 has been used by setting $\Omega _p = 0.1 \Omega _c$, and we consider on-resonance case, i.e., two-photon detuning $\delta = 0$.

 

Fig. 2. (a) Entanglement versus ground-state relaxation rate under the parameters given by $\alpha = 1000$ and $\Omega _c = 1\Gamma$. (b) Entanglement versus input coupling Rabi frequency with parameters given by $\alpha = 1000$ and $\Gamma _{12} = 0.01\Gamma$. The input Rabi frequency of probe field is used by setting $\Omega _p = 0.1\Omega _c$, and $\delta = 0$ for the two figures.

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Similarly, we consider how the two-photon detuning influences output entanglement. As shown in Fig. 3(a), we can obtain the maximum entanglement by scanning two-photon detuning $\delta$ for given optical density and input Rabi frequencies of probe and coupling fields. It is shown that one can find the optimum two-photon detuning $\delta _{\textrm {opt}}$ and the corresponding entanglement quantity $V_{\textrm {opt}, \delta }$. With the same process, there exists an optimum $\Omega _{c,\textrm {opt}}$ to maximize output entanglement, which is $V_{\textrm {opt}, \Omega _c}$ for given $\alpha$ and $\delta$, as shown in Fig. 3(b). As Fig. 2, we have set $\Omega _p = 0.1 \Omega _c$ in order to satisfy EIT condition, and we let $\Gamma _{12} = 0$ to ensure that the entanglement is coming from the fact of $\delta$.

 

Fig. 3. (a) Entanglement versus two-photon detuning under the parameters given by $\alpha = 1000$ and $\Omega _c = 1\Gamma$. (b) Entanglement versus input coupling Rabi frequency with parameters given by $\alpha = 1000$ and $\delta =0.01\Gamma$. The input Rabi frequency of probe field is used by setting $\Omega _p = 0.1\Omega _c$, and $\Gamma _{12} = 0$ for the two figures.

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From Fig. 2(a) and Fig. 3(a), we can find that no entanglement is generated at output when $\Gamma _{12} = 0 = \delta$, and entanglement between two fields is generated in the presence of $\Gamma _{12}$ or $\delta$. Compared with the entanglement generated by the two processes, it is clear to see that the entanglement degree is larger and more efficient in two-photon detuning scheme. In addition to the factors of $\Gamma _{12}$, $\delta$, and $\Omega _c$, entanglement also depends on the optical density, which is tunable and available in experiments. We are interested in maximum entanglement at different values of $\alpha$. Figures 4(a) and 4(b) show the results based on the scheme by using $\Gamma _{12}$, and the results of $\delta$ scheme are depicted in Figs. 4(c) and (d) under various $\alpha$’s. Figure 4(a) illustrates $V_{\textrm {opt},\Omega _c}$ as a function of $\Gamma _{12}$, and the values of $V_{\textrm {opt},\Omega _c}$ are insensitive to $\Gamma _{12}$. For different values of optical densities, it shows that the entanglement values do not have significant changes . Similar tendency has been found in $V_{\textrm {opt},\Gamma _{12}}$, which is insensitive to $\Omega _c$ and $\alpha$, see Fig. 4(b). The entanglement values are around 3.7, just a little smaller than the upper bound value 4. In contrast, Figs. 4(c) and 4(d) illustrate $V_{\textrm {opt}, \Omega _c}$ and $V_{\textrm {opt}, \delta }$ as the functions of $\delta$ and $\Omega _c$, respectively. The values of $V$’s are also insensitive to the corresponding variables, but change significantly with $\alpha$’s. The black dotted, green dashed-dotted, blue solid, and red dashed lines represent the values of $\alpha$’s given by 100, 300, 1000, and 3000, respectively. From Figs. 4(c) and (d) , it manifests that the entanglement degree increases when optical density is increasing. The optical density can enhance the output entanglement in the two-photon detuning scheme.

 

Fig. 4. Optimized entanglement at different $\alpha$’s for the two different processes: by ground state population damping (a) and (b), and by two-photon detuning (c) and (d). Black dotted, green dashed-dotted, blue solid, and red dashed lines represnt $\alpha$’s values of 100, 300, 1000, and 3000, respectively. (a) The minimum value of entanglement quantity obtained by scanning all input Rabi frequencies, $V_{\textrm {opt},\Omega _c}$, as a function of ground state population damping $\Gamma _{12}$. (b) The minimum value of entanglement quantity obtained by scanning all ground state damping, $V_{\textrm {opt},\Gamma _{12}}$, as a function of input Rabi frequency $\Omega _c$. (c) The minimum entanglement quantity obtained by scanning all input Rabi frequencies, $V_{\textrm {opt},\Omega _c}$, as a function of two-photon detuning $\delta$. (d) The minimum entanglement quantity obtained by scanning all two-photon detunings, $V_{\textrm {opt},\delta }$, as a function of input Rabi frequency $\Omega _c$. $\Omega _p = 0.1\Omega _c$ is used in all the figures.

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Since the entanglement degree is much larger by using two-photon detuning scheme, we focus on the results of Figs. 4(c) and (d). We can see that the value of optimum entanglement, $V_{\textrm {opt},\Omega _c}$ and $V_{\textrm {opt},\delta }$, under a given optical density $\alpha$ is almost a constant.

In Fig. 5, we have numerically plotted the contour plot of entanglement quantity $V$ with respect to $\Omega _p$ and $\delta$ under three different $\Omega _c$’s, which are $0.85\Gamma$, $1.2\Gamma$, and $1.7\Gamma$, respectively. As shown in the three plots, we can see that the values of $V$ are the same, but with different ranges of $\Omega _p$ and $\delta$. The positions of $\Omega _{p}$ corresponding to the same values of $V$ is clearly proportional to $\Omega _c$. Similarly, the positions of $\delta$ is proportional to $\Omega _c^{2}$. It implies that there exists a relationship among entanglement quantity $V$ and the ratios of $\Omega _p/\Omega _c$ and $\delta /\Omega _c^{2}$. The deeper understanding to the results in Fig. 5 will be discussed in Sec. IV.

 

Fig. 5. Contour plot of entanglement quantity $V$ versus $\Omega _p$ and $\delta$ under different $\Omega _c$’s: (a) $\Omega _c = 0.85\Gamma$, (b) $\Omega _c = 1.2\Gamma$, and (c) $\Omega _c = 1.7\Gamma$. For the three plots, optical density is set to be $1,000$. Please note that the ranges of $\Omega _p$ and $\delta$ are different in the three plots.

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4. Discussions

Generation of CV quantum entanglement between probe and coupling fields using atomic EIT system is the key point of this paper. We have proposed a theoretical model in Sec. II, deriving equations of motion for atomic and field operators, and showing the numerical simulation results in Sec.III. In order to understand the physics behind these results thoroughly, in this section we study the system analytically from the framework given in Sec. II. Using Eqs. (2)–(12), one can obtain

Now we turn to consider the entanglement between probe and coupling fields. From Eq. (18), it is clearly to see that the terms of $R_1$ and $S_1$ link coupling field operators ($\hat {a}_c$, $\hat {a}_c^{\dagger }$) and probe field operators ($\hat {a}_p$, $\hat {a}_p^{\dagger }$); while $P_2$ and $Q_2$ make the correlation between probe field operators and coupling field operators. On the other hand, the coefficients of $P_1,~Q_1,~ R_2,~\textrm {and}~S_2$ correspond to the self-interaction processes for each field. Let’s discuss the physical meaning of these coefficients. First, if there only exists the coefficients of $S_1$ and $Q_2$, we can obtain an ideal entangled state, which is coming from two-mode squeezed state formed by $\hat {a}_p$ and $\hat {a}_c$, and the output entanglement is given as $V = 4 e^{-2\vert S\vert \zeta }$, where $\vert S_1\vert = \vert Q_2\vert \equiv S$. Second, the physics of the terms of $R_1$ and $P_2$ is the cross-phase coupling, which can’t produce entanglement at output. Then, when we only consider the coefficients of $Q_1$ and $S_2$, which correspond to the single-mode squeezing processes, we can obtain the two independent squeezed lights, for which the entanglement is given by $V = 4\cosh (2\vert Q_1\vert \zeta )$. Finally, the coefficients of $P_1$ and $R_2$ are related to self-phase or damping/amplification processes, which also do not have the abilities to produce entanglement.

In order to obtain the analytical expression for entanglement, we consider the coefficients coming from the $0^{\textrm {th}}$ and $1^{\textrm {st}}$ order terms of $r$, i.e., $P_1,~R_1,~S_1, P_2$, and $Q_2$, as well as the Langvin noise contributions from $\hat {n}_p$ and $\hat {n}_c$. We yield a form as follows.

According to Fig. 5, we can plot the entanglement quantity $V$ with respect to $r$ and $\epsilon$. After plotting with the new variables, we can find that the three plots of Figs. 5(a)-(c) correspond to the same result shown in Fig. 6. It implies that $V$ depends only on two independent parameters of $\epsilon$ and $r$, i.e., as long as $\epsilon$ and $r$ are given. Any combination of $\delta$, $\Omega _c$, and $\Omega _p$ results in the same value of $V$. One can clearly see that in the plot the condition of $\mu = 1/2$, represented by the dashed line of a hyperbolic function, crosses the minimum or optimum value of $V$.

 

Fig. 6. Contour plot of entanglement quantity $V$ versus $r$ and $\epsilon$. The optical density is given by $\alpha = 1,000$. The black dashed curve is plotted with the condition of $\mu = \alpha \epsilon r = 1/2$.

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However, $\mu = 1/2$ is not a sufficient condition to find the optimum entanglement, and Eq. (19) can’t explain our results completely. The main reason for this problem is that the higher order terms of $r$ become important when $r$ is getting large. Thus, we have to consider the terms of $Q_1$ and $S_2$, which are related to the single-mode squeezing coefficient. According to Eq. (15), one can see that the entanglement degree depends on the average photon numbers of probe and coupling fields. For single-mode squeezed state of probe and coupling, the average photon numbers are $\sinh ^{2}(\vert Q_1\vert \zeta )$ and $\sinh ^{2}(\vert S_2\vert \zeta )$ rather than 0. As a reason, we can naively modify output entanglement by considering the external photon numbers coming from the single-mode squeezing terms, but without introducing the corresponding extra noises. Thus it reads as

For the case $\lambda \ll 1$, we can expand $V_1$ to $\mathcal {O(\lambda )}$. With these approximations, we can obtain a closed form of output entanglement shown below:

From Eq. (21), we can obtain the optimum entanglement by substituting $\mu = 1/2$, and express $V$ as the function of $\alpha$, $\epsilon$, and $r$ by using $\lambda = 2\alpha \epsilon ^{2}$. It will be

According to Eq. (22), we can find the best entanglement by using Lagrangian multiply with the constraint condition given by $\mu = 1/2$. It shows that

From Eqs. (23) and (24), we can see that $\epsilon _{\textrm {opt}}$ and $r_{\textrm {best}}$ are constants when $\alpha$ is given. Under these conditions, the best entanglement value is

In comparison with the entanglement generation from ground-state dephasing or population decay, we have seen that the scheme of two-photon detuning is more efficient from Fig. 4. The physics behind the result can be understood as follows. For the scheme of ground-state dephasing or population decay, the entanglement coefficients are $\vert S_1\vert = \vert Q_2\vert = \alpha \varepsilon r e^{-\alpha \varepsilon \zeta }$, where $\varepsilon \equiv \Gamma \Gamma _{12}/(2\vert \Omega _c\vert ^{2})$. Since the probe field suffers strong dissipation from $\Gamma _{12}$, the entanglement coefficient decays exponentially. However, the depletion rate of probe field due to two-photon detuning is $\exp \left ( -2\alpha \epsilon ^{2}\zeta \right )$, which is much smaller than that of probe field due to ground-state dephasing or poluation decay, and the entanglement degree $\vert S_1\vert = \vert Q_2\vert = \alpha \epsilon re^{ -2\alpha \epsilon ^{2}\zeta } \simeq \alpha \epsilon r$. The result implies that the entanglement degree can be enhanced by optical density in two-photon detuning scheme. In contrast, the entanglement degree in ground-state dephasing or population decay scheme is insensitive to optical density, as shown in Figs. 4(a) and (b).

The existence of the optimum ratio $r$ is coming from the competition between dissipation and single-mode squeezing. From Eq. (22), we can rewrite the optimum entanglement as $V_{\textrm {opt}} = 2 + (3\alpha )^{-1}r^{-2} + 8r^{2}$, from which it is clearly to see that the second and the third terms are attributed to fluctuation noise and single-mode squeezing term, respectively. When $r$ is small, the extra noise dominates the output entanglement value, while the effect of single-mode squeezing term becomes important when $r$ is getting larger. As a result, there exists an optimum value of $r$ to minimize entanglement value $V$. On the other hand, for given optical density $\alpha$, the optimum entanglement is given by $V_{\textrm {opt}} = 2 + (4\alpha /3)\epsilon ^{2} + (2/\alpha ^{2})\epsilon ^{-2}$. It implies that there exists a best entanglement when the sum of extra noise from dissipation and single-mode squeezing is minimized. Generally, two-mode squeezing ($S_1,~Q_2$) and the cross-coupling terms ($R_1,~P_2$) limit the best entanglement to be 2, which is $50\%$ of ideal entangled state. The presence of the probe dissipation $P_1$ will introduce extra noise fluctuation which is proportional to $r^{-2}$, and degrade the output entanglement in the region of $r\ll 1$. In contrast, the single-mode squeezing, $Q_2$ and $S_2$, will degrade the entanglement degree with extra term being proportional to $r^{2}$, which destroys entanglement when $r$ is getting larger. Similarly, we also have optimum $\epsilon$ from the condition of $\mu = 1/2$. As a result, we can find the best strength ratio $r$ and detuning-coupling field ratio $\epsilon$ to minimize the output entanglement.

5. Conclusion

In the present work, we have discussed the generation of quantum entanglement between probe and coupling fields under EIT condition. We compare the entanglement degree arising from two different mechanisms, which are ground-state relaxation rate and two-photon detuning. Our study has identified that it is more efficient to obtain higher output entanglement degree by introducing two-photon detuning. Furthermore, we have numerically studied the influence of the EIT parameters, which are two-photon detuning, field Rabi frequencies, and optical density to entanglement degree. Also, the conditions of the corresponding parameters for obtaining the optimal entanglement have been found from theoretical analysis. It shows that the two-mode squeezing and cross-coupling terms give us a constraint for the parameters to obtain the best entanglement, i.e. $\mu = 1/2$. The noise fluctuation from probe field dissipation and the single-mode squeezing from probe and coupling fields will reduce the entanglement degree. The optimum condition of $r$ and $\epsilon$ for the best entanglement have been found. The study contributes to our understanding of the origin of entanglement induced by atom-field interaction in EIT system, as well as a deeper connection between quantum coherence and entanglement. The work can be further extended to more complicated atomic systems, which have possibilities to produce higher entanglement degree, conducing the progresses in the development of CV quantum information sciences.