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Abstract
We theoretically propose a scheme to generate a strong continuous-variable quantum entangled light source in four-wave mixing (FWM) process by increasing the optical density of atomic medium. By properly choosing the input coupling field Rabi frequency and detuning, the optimized entanglement can be achieved to be better than −17 dB at an optical density of approximately 1, 000, which has been realized in atomic media. Besides, with the optimized one-photon detuning and coupling Rabi frequency, the optimum entanglement degree can be greatly enhanced with the increment of optical density. We also examine the effects of atomic decoherence rate and two-photon detuning on entanglement in a realistic setting, and evaluate the experimental feasibility. We find that the entanglement can be further improved by considering two-photon detuning. In addition, with optimum parameters the entanglement is robust against the decoherence. The strong entanglement provides a promising applications in continuous-variable quantum communications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum entanglement is one of the most important quantum properties and resources in the development of modern quantum optics and quantum information sciences. Some relevant technologies using quantum entanglement such as quantum teleportation [1], quantum key distribution [2], quantum communication [3,4], quantum computing [5] and quantum information processing [6] have been paid great attentions and studied intensively in recent decades. Apart from some entangled states using individual qubits, continuous-variable (CV) entanglement based on the two quadratures in amplitude and phase of electromagnetic fields offers great advantages in the progress of quantum sciences and technologies [7,8].
Regarding the quantification of quantum entanglement, it should be noted that the entanglement degree depends on the entanglement witness of the used criterion. There are two main popular criteria in CV system [11,12]. Here we use Duan-Giedke-Cirac-Zoller (DGCZ) entanglement criterion [11] to evaluate the entanglement degree. The DGCZ entanglement criterion between two fields, probe and conjugate, is required to satisfy the condition as shown as
For generating entangled light source, the mature technique using optical parametric oscillation (OPO) or optical parametric amplification (OPA) with nonlinear crystals (NLCs) is popular in the laboratory [9,10]. The first experiment to realize Einstein-Podosky-Rosen (EPR) entangled light source was using nondegenerate OPA [13]. It shows that the entanglement degree is only about −0.78 dB. Later, T. C. Ralph et al. generated entangled light by using a beam splitter to mix two squeezed lights, which are generated from two OPA processes [14]. The corresponding entanglement degree can be achieved to −1.16 dB. The entangled light with $-2.11$ dB generated by OPO process above threshold was realized in 2007 [15]. Despite the simple implementations, it has no flexibility to manipulate the entanglement degree and the carried frequency due to the fixed physical parameters of NLCs. In addition, it also needs to use a cavity to increase the interaction times for reaching higher entanglement. In recent decades, thanks to the great progress of experimental techniques in the cold atoms, coherent control in atomic ensembles becomes an important topic in quantum optics. One of the important phenomena is EIT [16], which can reduce the field absorption on resonant frequency [17] and greatly slows down the group velocity of an optical pulse propagating through the atomic medium [18] simultaneously. With the EIT scheme as well as some other coherent processes using atomic ensembles, the field correlations established in EIT have been studied [19,20]. Following these studies, it was suggested that some non-classical states of light such as quantum squeezed light [21] and entangled light [22–24] can be generated from EIT-based media. On the other hand, controlling one light by the other light using all-optical switching or cross-phase modulation based on EIT brings nonlinear optics and coherent control together [25], and naturally four-wave mixing (FWM) processes in the atomic ensemble have been paid great attention in theoretical [26–29] and experimental [30–32] researches. Based on the FWM scheme, quantum correlation between two fields in atomic media has been studied theoretically [33,34], and also realized in experiments [35–39]. It shows that FWM processes in atomic media provide a good way to generate CV entangled light sources.
Compared with the schemes with OPO, the FWM scheme using atomic media has to face some problems in practice. The optical nonlinearity of an atomic medium can be greatly enhanced by atom-field interactions near the resonant frequency, while the strong gain or loss would introduce extra quantum noises, which greatly restrict the increment of quantum entanglement. In order to enhance entanglement degree, people used some schemes to improve the entanglement degree. One is using OPO scheme based on the FWM process [40], and the entanglement degree can be achieved to $-3.5$ dB in hot atomic ensembles. The other scheme is using cascaded FWM processes [41–43], and it can successfully increase the entanglement up to −6.6 dB, but the essential problem mentioned above still exists and the experimental layout becomes more and more complicated. Therefore, in order to have strong CV entangled light generated from atomic medium, it is necessary to propose an ideal physical process with strong correlated interactions between fields, and also can avoid the severe absorption at the same time.
Here we propose a FWM scheme using a four-level atomic system in presence of four interacting fields. The four-level system can be consist of two $\Lambda$-type configurations. For each $\Lambda$, a quantized field ($\hat {E}_p~\text {or}~\hat {E}_s$) and a strong coupling field ($\Omega _2~\text {or}~\Omega _1$) couple with it, see Fig. 1. We consider the condition of two-photon on resonance is satisfied in both two $\Lambda 's$, but their one-photon detunings can be different. Then we set one-photon on resonance condition in one of $\Lambda 's$, and large one-photon (far detuning) in the other $\Lambda 's$. It turns out that the whole system is consist of an EIT and a Raman process. By scanning the parameters space spanned by input Rabi frequency of strong fields and detuning, the entanglement degree can be optimized. By increasing the optical density, we find that the optimum entanglement and the corresponding optical gain can be greatly enhanced simultaneously. The entanglement degree can be achieved up to −14 dB at an optical density (OD) of approximately 1,000, which has been realized in atomic media with a decoherence rate of about $10^{-3}\Gamma$ [44,45]. As compared with the Rabi frequencies in this study, the decoherence rate is negligible. However, in order to make this scheme close to realistic case, we also included the effects of atomic ground decoherence and two-photon off-resonance in our discussions.
Fig. 1. The atomic configuration. The two strong fields, $\Omega _1$ and $\Omega _2$, couple to the two atomic transitions from $\vert 1 \rangle$ to $\vert 4 \rangle$ and $\vert 2 \rangle$ to $\vert 3 \rangle$. The two quantized fields, probe $\hat {E}_p$ and conjugate $\hat {E}_c$, making the corresponding transitions. $\Gamma _{\mu \nu }$ is the spontaneous emission rate from $\vert \mu \rangle$ to $\vert \nu \rangle$. $\Delta _1$ and $\Delta _2$ are the one-photon detunings of two strong fields, and $\Delta _p$ and $\Delta _c$ are the one-photon detuning of probe and conjugate fields, respectively. In this work, we focus on the case of $\Delta _p = \Delta _2 \equiv \Delta _a$ and $\Delta _c = \Delta _1 \equiv \Delta _b$, i.e., the two-photon resonance condition, as well as $\Omega _1 = \Omega _2 \equiv \Omega$, $\Omega _p = \Omega _c \equiv \Omega _w$, and $\Omega \gg \Omega _w$.
The paper is organized as follows. In Sec. II, we introduce our system model from a microscopic point of view. The complete mathematical descriptions of the atom-field interactions are given by Heisenberg-Langevin equations and Maxwell-Schrödinger equations. In Sec. III, we show our main results and some discussions. In Sec. IV, we discuss the experimental feasibility with this scheme. Finally, we present our conclusions in Sec. V.
2. Theoretical model
We consider an atomic ensemble with the four-level configuration, as shown in Fig. 1. The labels $\vert 1\rangle$, $\vert 2\rangle$, $\vert 3\rangle$ and $\vert 4\rangle$ can be in general for the chosen energy levels in practical experiment. There are four electromagnetic fields, two strong fields and two weak fields, interacting with the atomic ensemble by using the proper frequencies. The strong coupling fields with the Rabi frequencies $\Omega _1$ and $\Omega _2$ make the dipole transitions from $\vert 1\rangle$ to $\vert 4\rangle$ and $\vert 2\rangle$ to $\vert 3\rangle$, respectively. The detunings of the two fields are given by $\Delta _1$ and $\Delta _2$, which are defined by the frequency difference between optical field frequency $\omega _i,~i\in {1,2}$ and atomic transition frequency $\omega _{\mu \nu } = \left ( E_\mu - E_\nu \right )/\hbar$, where $E_{\mu (\nu )}$ is the energy of the atomic level $\vert \mu \rangle ~(\vert \nu \rangle )$. On the other hand, the two weak fields, probe $(\hat {E}_p)$ and conjugate $(\hat {E}_c)$ individually couple to the two transitions of $\vert 1\rangle \rightarrow \vert 3\rangle$ and $\vert 2\rangle \rightarrow \vert 4\rangle$ with corresponding frequencies of $\omega _p$ and $\omega _c$. The two detunings are $\Delta _p = \omega _p - \omega _{31}$ and $\Delta _c = \omega _c - \omega _{42}$. The spontaneous emission rates from the two excited states $\vert 3 \rangle$ and $\vert 4 \rangle$ to the two lower states $\vert 1 \rangle$ and $\vert 2 \rangle$ are given in $\Gamma _{31}$, $\Gamma _{32}$, $\Gamma _{41}$ and $\Gamma _{42}$. In simply, we assume that $\Gamma _{31} = \Gamma _{32} = \Gamma _{41} = \Gamma _{42} = \Gamma /2$, where $\Gamma$ is the total spontaneous emission rate from the excited state. In realistic case, the atomic ground state decoherence rate $\gamma$ is non-zero even though it is small enough in ultracold atomic system. We set $\gamma = 0$ in our theoretical simulation at first, and then discuss how its influence is later. In our case, we can treat the probe and the conjugate fields as quantized fields because of their weakness, so that we can study some quantum properties of the two fields in this system.
According to the atom-fields interactions in this system, we can write down each part of Hamiltonian as follows
It should be noted that the system Hamiltonian is derived in the rotating optical frequency frame, thus our detunings for the four interacting fields have a constraint given by $(\Delta _p - \Delta _2) + (\Delta _c-\Delta _1) = 0$.
According to Eq. (6) with the help of Eq. (3–5), we can write down the Heisenberg-Langevin equations for atomic operators as
On the other hand, we also can write down the Maxwell-Schrödinger equations of motion for the two interacting quantum fields as following.
As we can see, Eq. (8), (9) are coupled with Eq. (7) so that we have to solve the field propagation equations and the atomic operator equations of motion simultaneously.
For simplicity, we consider the steady state case, so that we can ignore the time dependent parts. Thus Eq. (8) and (9) can be re-written as
where $\xi \equiv z/L$ is the dimensionless propagation distance. $\alpha$ is the optical density. $\Gamma$ is the total spontaneous emission rate of the excited state. In our case $\Gamma _{41}+\Gamma _{42} = \Gamma = \Gamma _{31}+\Gamma _{32}$. The optical density is $\alpha _i \equiv n\sigma _i L$, where $n$ is the atomic number density, $L$ is the total length of the atomic medium, and $\sigma _i = 3\lambda _i^2/2\pi$ is the absorption cross section of field with the wavelength $\lambda _i$.In Eq. (10) and (11), $\hat {\sigma }_{13}$ and $\hat {\sigma }_{24}$ are the source terms from two atomic dipole transitions, which provide the connections between the two fields. At the same time, the quantum noises of the atomic system would be migrated into the two fields system, causing the extra fluctuations.
3. Results and discussions
In this section, we analyze the quantum entanglement between probe and conjugate fields with respect to the system parameters numerically. As we have mentioned in Sec. II, the entanglement is quantified by the quantity $V$, which depends on the fields’ and atomic parameters in our system. In general, we have $V = V(\Omega _\mu, \Delta _\mu, \alpha )$, $\mu \in {1,2,p,c}$, where $\Omega _\mu$ is the Rabi frequency of the interacting field $E_\mu$, $\Delta _\mu$ corresponds to the one-photon detuning of field, and $\alpha$ is the optical density of the atomic medium. The parameter space is quite large. For simply, we restrict our discussions on the some conditions. For the two weak fields, probe and conjugate, we set their input Rabi frequency $\Omega _p(0) = \Omega _c(0) \equiv \Omega _w$, and the input Rabi frequencies of the two strong coupling fields $\Omega _1(0) = \Omega _2(0) \equiv \Omega$. These Rabi frequencies satisfy $\Omega \gg \Omega _w$. Physically, the Rabi frequencies of interacting fields will change after propagating through the atomic medium. Thus it is more relevant to use the input Rabi frequency for the free parameters.
In addition, the conditions of field detunings are required as $\Delta _p=\Delta _2=\Delta _a$ and $\Delta _s=\Delta _1=\Delta _b$, which implies the two-photon on resonance conditions are satisfied. $\Delta _a$ and $\Delta _b$ are two free parameters, which can be positive or negative, and can be controlled independently. We study the entanglement in one-photon detuning space, i.e. $\Delta _a$ and $\Delta _b$ first. On the other hand, two-photon detuning plays an important role in our system. The two-photon detuning can be introduced with $\Delta _p \neq \Delta _2$ or $\Delta _c \neq \Delta _1$. For simply, we consider the case of two-photon on resonance in this section.
In Fig. 2(a), we’ve depicted the 3D mesh plot of the entanglement quantity $V$ versus $\Delta _a$ and $\Delta _b$ ranging from $-40\Gamma$ to $40\Gamma$, respectively. All the points on the surface satisfy the two-photon on resonance condition. In 2(b), the blue curve shows the relation between $V$ and $\Delta _a$ under $\Delta _b = 0$, while the red-dashed curve is plotted with $\Delta _b$ under $\Delta _a = 0$. In addition, if we follow the two diagonal directions in Fig. 2(a), we can obtain the results in Fig. 2(c). It is obvious that no entanglement can be found under the conditions of $\Delta _a = \Delta _b$ (blue curve) or $\Delta _a = -\Delta _b$(red dashed curve). The used parameters are given by $\Omega _w = 0.05\Gamma$, $\Omega = 10\Gamma$, and $\alpha = 100$. From the results in Fig. 2, it shows that the strong entanglement occurs in the region where one of detuings is on one-photon resonance and the other one is far off resonance. In the atomic configuration shown in Fig. 1, it can treat that the four-level system is consist of two-$\Lambda$ systems: one is $\vert 1\rangle$ - $\vert 2\rangle$ - $\vert 3\rangle$, and the other is $\vert 1\rangle$ - $\vert 2\rangle$ - $\vert 4\rangle$. According to the detuning condition for strong entanglement, it implies that one-photon on resonance in one of $\Lambda$’s, i.e. EIT system, while large one-photon detuning in the other $\Lambda$’s, which corresponds to a Raman process.
Fig. 2. (a) The contour plot of entanglement quantity $V$ versus $\Delta _a$ and $\Delta _b$. (b) The entanglement is plotted with $\Delta _a$($\Delta _b$) under the condition of $\Delta _b = 0$($\Delta _a = 0$). (c) The entanglement quantities are plotted along diagonal lines of $\Delta _a = \Delta _b$ and $\Delta _a = -\Delta _b$. The valus of initial field Rabi frequencies are $\Omega = 10\Gamma$ and $\Omega _w = 0.05\Gamma$. The optical density $\alpha = 100$.
Next, we consider the influence of input Rabi frequency of two weak fields $\Omega _w$ under the condition of detuning of $\Delta _a = 0$ and $\Delta _b \gg 1$. The relation between the output entanglement and $\Omega _w$ is depicted in Fig. 3. The blue solid, red dashed, green dotted, purple dash-dotted, and brown solid curves represent $\Omega _w/\Gamma = 0.05,~0.1,~0.5,~1.0$ and 2.0, respectively. $\Omega = 10\Gamma$, and $\alpha =100$. As we can see, the largest entanglement between two weak fields decreases when the strength of $\Omega _w$ increases, and the corresponding detuning $\Delta _b$ becomes large. The results of $\Omega _w = 0.1\Gamma$ and $\Omega _w = 0.05\Gamma$ are very close, which reflects the asymptotic limit $\Omega _w$. It implies that the entanglement would approach to a limit as long as the $\Omega _w$ is far smaller than $\Omega$. Since the gain is large in the system, the interference effect due to the relative phase of the two weak fields is negligible at the output. As a result, the entanglement is phase insensitive under low-light level, and we can avoid the influence of phase difference between two fields.
Fig. 3. The entanglement degree $V$ versus $\Delta _b$ with different input Rabi frequencies $\Omega _w$’s. The blue solid, red dashed, green dotted, purple dash-dotted, and brown curves correspond to $\Omega _w/\Gamma = 0.05~0.1,~0.5,~1.0$ and 2.0, respectively. The other parameters are $\Delta _a = 0$, $\Omega = 10\Gamma$ and $\alpha =100$.
In Fig. 4, we plot the entanglement degree $V$ versus the input coupling Rabi frequency $\Omega$ with $\Delta _b/\Gamma$= 10, 13 and 15, which correspond to red dashed, blue and green curves, respectively. The other parameters are $\alpha = 100$ and $\Omega _w = 0.05\Gamma$. As we can see, the entanglement degree has a minimum value for a given $\Delta _b$. Among these $\Delta _b$’s, we can find an optimum $\Delta _b$ such that the minimum of $V$ is smallest. We call $V_{\rm opt}$ as the optimized entanglement, which is shown as the red point in Fig. 4, and the corresponding $\Omega$ and $\Delta _b$ are denoted by $\Omega _{\rm opt}$ and $\Delta _{\rm b,opt}$.
Fig. 4. The entanglement degree $V$ versus input coupling Rabi frequency $\Omega$ with different $\Delta _b$’s at $\alpha = 100$ and $\Omega _w = 0.05\Gamma$. The red dashed, blue and green curves correspond to $\Delta _b/\Gamma =$10, 13 and 15, respectively. The red point shows the optimum entanglement $V_{\rm opt}$ and the corresponding optimum Rabi frequency $\Omega _{\rm opt}$.
It is interesting to find the optimized entanglement by choosing proper parameters of the system. According to Fig. 3, the entanglement degree becomes better when the Rabi frequency $\Omega _w$ is smaller. In the asymptotic limit, we set $\Omega _w = 0.05\Gamma$ and $\Delta _a = 0$, and find the minimum value of $V$ by scanning $\Omega$ and $\Delta _b$ at a given optical density $\alpha$.
In Fig. 5(a), we’ve shown the relation of the optimized entanglement $V_{\text {opt}}$ with respect to the given optical density is ranging from 10 to 1000. The blue line is the numerical simulation result, and the red dashed line is the fitting line for large optical density. It can see that $V_{\rm opt} \propto 1/\sqrt {\alpha }$ when $\alpha > 100$. When $\alpha \leq 100$, the numerical simulation gradually deviates from the fitting line. The corresponding optical gain of probe field under the optimized entanglement is shown in Fig. 5(b). It shows that probe gain $G \propto \sqrt {\alpha }$ when $\alpha > 100$, and the simulation line gradually deviates from the fitting line when $\alpha \leq 100$. From the results of Fig. 5(a, b), one can find that the quantity of $V_{\rm opt}\times G$ is almost independent of optical density as $\alpha > 100$. It shows that $V_{\rm opt} \simeq 8/G$. Different than the result obtained from ideal two-mode squeezing [35], we include the quantum noises from optical gains in our theoretical model, and the physical picture is given as follows. The probe and conjugate fields become entangled through FWM process, in which the two fields are correlated by the energy transfer from two strong coupling fields. At the same time, the optical gain will introduce extra quantum noises into probe and conjugate fields, and degrade the entanglement. The two processes compete with each other, and finally the output entanglement is limited.
Fig. 5. (a) The optimized entanglement degree V versus $\alpha ^{-1/2}$. (b) The corresponding optical gain of probe and conjugate fields versus $\alpha ^{1/2}$. Here we use $\Omega _w = 0.05\Gamma$. The results in (a) and (b) are nearly the same as long as $\Omega _w \ll \Omega$.
In Fig. 6(a), we plot the optimized entanglement degree in dB with the logarithmic scale of optical density. It is clear to see that one can achieve $\sim 10~\text {dB}$ when $\alpha = 200$, and $\sim 14~\text {dB}$ when $\alpha = 1,000$. This is the main result of this work. It is worth noted that the result of $V_{\rm opt}$ versus $\alpha$ shown in Fig. 5(a) (or Fig. 6(a)) is nearly the same as long as $\Omega _w \ll \Omega$. The phases of probe and conjugate field can’t affect the optimized entanglement and the optimum quadrature angle at output. Nevertheless, the values of $\Omega _{\rm opt}$ and $\Delta _{\rm b,opt}$ still depend on $\Omega _w$ but not on their phases. Therefore, OD-enhanced entanglement is robust against the probe and conjugate fields at input, and one only need to tune $\Omega$ and $\Delta _{\rm b}$ to optimize entanglement. We also provide a recipe to get the optimized entanglement for FWM experiments in Fig. 6(b). It shows that the optimized Rabi frequency $\Omega _{\rm opt}$ and one-photon detuning $\Delta _{\rm b,opt}$ for optical densities ranging from 10 to 1,000. It is clear to see that the optimum condition at each optical density satisfies the condition of $\Omega ^2/\Gamma > \sqrt {\alpha }/2$, which refers to EIT, not Aulter-Townes splitting.
Fig. 6. (a) The result of Fig. 5(a) expressed in decibel by using Eq. (2) The x-axis is expressed as $\text {log}_{10} \alpha$ under optical density $\alpha$ are 10, 30, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1,000. It is clearly to display the values of $V_{\rm opt}$ at high ODs. (b) The corresponding optimum parameters of $\Omega _{\rm opt}^2$ versus $\Delta _{\text {b,opt}}$ at each OD. In (a) and (b), we use $\Omega _w = 0.05\Gamma$, and we can find that $\Omega _w \ll \Omega _{\rm opt}$.
4. Discussion of experimental feasibility
We now discuss the experimental feasibility of the OD-enhanced entanglement via the FWM process proposed in this work, and evaluate practical issues in realistic experiments. In Fig. 1, the transitions driven by $\Omega _2$ and $\hat {E}_p$ form the EIT effect due to $\Delta _a = 0$. The EIT effect plays a dominate role in the FWM process in the case of a large OD as demonstrated by Refs. [46,47]. A large OD enhances the interaction time between light and matter due to the EIT effect. The propagation time of slow light in EIT effect is given by
Such an enhancement is similar to the effect of an optical cavity [21]. As mentioned in Sec. 1, an OD of about 1000 as well as a decoherence rate of around $10^{-3}\Gamma$ or 10 kHz has been achieved in experiments and utilized to realize a high storage efficiency of the EIT-based coherent optical memory [44,45]. Nevertheless, the transparency bandwidth, $\Delta \omega _{\rm EIT}$, and the attenuation coefficient, $\beta$, due to the decoherence rate are the two important issues in the EIT experiments, and they are given by [48]
where $\gamma$ is the decoherence rate in the experimental system. In an EIT spectrum, i.e., the probe transmission as a function of the two-photon detuning under the presence of $\Omega _2$, the peak transmission is given by $\exp (-\beta )$ and the $e^{-1}$ full width determines $\Delta \omega _{\rm EIT}$. We will discuss the influences of the two-photon detuning and the decoherence rate on the degree of entanglement, i.e., $V$, in the next two paragraphs.Referring to Fig. 1, the two-photon detuning, $\delta$, is defined as $\Delta _p - \Delta _a$ or $\Delta _b - \Delta _s$, where $\Delta _p$ and $\Delta _s$ are the one-photon detunings of the probe and conjugate fields, respectively. For simplicity, we set $\delta = 0$ in the earlier sections. This does not affect the conclusion of the OD-enhanced entanglement via the FWM process. We now vary $\delta$ to optimize $V$ at $\alpha =$ 1000, which is the largest OD considered in this work. Figure 7(a) shows $V$ as a function $\delta$ at $\Omega _1 = \Omega _2 =$ 4.86$\Gamma$ and $\Delta _b$ = 70.4$\Gamma$, i.e., at the optimum condition for $\alpha = 1000$ shown in Fig. 6(b). Please note that the optimum two-photon detuning, $\delta _{opt}$, is not zero, but is a negative value. This is due to the AC Stark shift of the ground state $|1\rangle$ induced by $\Omega _1$. When we make $\Omega _1$ and $\Omega _2$ constant in the medium, i.e., do not consider their propagation equations, and vary $\Omega _1$ and $\Delta _b$, the optimium two-photon detuning is always given by
where $-\Omega _1^2/(4\Delta _b)$ is the amount of the AC Stark shift [49]. As $\Omega _1$ and $\Omega _2$ can attenuate during the propagation in the medium, $|\delta _{\rm opt}|$ does not follow the above equation, but still increases with $\Omega _1^2$ and decreases with $|\Delta _b|$. Regarding the prediction in Fig. 7(a), the propagation equations of $\Omega _1$ and $\Omega _2$ are considered in the calculation. The minimum or optimum $V$ is improved to $-17.3$ dB, and the 3-dB bandwidth of $V$ is $0.18\Gamma$. According to Eq. (13), the EIT bandwidth is 0.75$\Gamma$. Therefore, the EIT bandwidth is not a limitation factor on the FWM-based entanglement generation, and one needs to tune the two-photon detuning to obtain maximum entanglement in experiments.Fig. 7. (a) The entanglement versus two-photon detuning. (b) The optimized entanglement versus ground state decoherence rate.
There always exists a nonzero decoherence rate in an experimental system. The decoherence rate $\gamma$ causes decay or attenuation, and consequently introduces random noise to the system, degrading the entanglement. To see the effect of $\gamma$ on $V$ at the OD of 1000, we optimize $\Omega _1$ ($= \Omega _2 \equiv \Omega$), $\Delta _b$, and $\delta$ at each value of $\gamma$. Figure 7(b) shows the optimum $V$ as a function of $\gamma$. To achieve the optimum $V$, a larger $\gamma$ needs a larger $\Omega _{\rm opt}$ and a smaller $\Delta _{b,{\rm opt}}$. Since the increasing rate of $\Omega _{\rm opt}^2$ is larger than the decreasing rate of $\Delta _{b,{\rm opt}}$, $\delta _{\rm opt}$ increases with $\gamma$. The increasing of $\Omega _{\rm opt}$ (or $\Omega _{\rm 2,opt}$) and the decreasing of $\Delta _{b,{\rm opt}}$ can be understood. According to Eq. (14), a large value of $\Omega _2$ reduces the attenuation caused by $\gamma$. A larger $\gamma$ makes $\Omega _{2,{\rm opt}}$ (or $\Omega _{\rm opt}$) shift toward a larger value. Furthermore, as $\Delta _b$ decreases, the interaction between light and matter becomes stronger, enhancing $V$, but the incoherent absorptions or attenuations of $\Omega _1$ and $\hat {E}_c$ becomes larger, degrading $V$. Consequently, there must be an optimum $\Delta _b$ as depicted by Fig. 3. Since a nonzero $\gamma$ causes an extra attenuation, it makes $\Delta _{b,{\rm opt}}$ shift toward a smaller value to increase the interaction. Figure 7(b) also reveals that the degradation of the entanglement is less than 3 dB as long as $\gamma <$ 0.07$\Gamma$. The condition of $\gamma$ less than $0.07\Gamma$ together with the OD of 1000 has been realized in the experimental system of laser-cooled atoms [44,45]. Hence, an entanglement better than $-17$ dB via the FWM process is experimentally achievable.
It is interesting to estimate what degree of entanglement in the presence scheme can be achieved with a Doppler-broadened medium, i.e., a room-temperature or hot atomic vapor. In a hot atomic vapor, the crucial problem is the Doppler shift. Fortunately, the problem can be resolved by the all-copropagation scheme, i.e., all of $\Omega _1$, $\hat {E}_c$, $\Omega _2$, and $\hat {E}_p$ propagate in the same direction [50,51]. For example, consider the ground states $|1\rangle$ and $|2\rangle$ are the two hyperfine levels of $^{87}$Rb atoms, and the excited states $|3\rangle$ and $|4\rangle$ are those in the atoms’ $|5P_{1/2}\rangle$ or $|5P_{3/2}\rangle$ state. Then, the angular frequency difference, $\omega _{21}$, between $|1\rangle$ and $|2\rangle$ is 2$\pi$ $\times$6.8 GHz, and the spontaneous decay rate, $\Gamma$, of $|3\rangle$ and $|4\rangle$ is about 2$\pi$ $\times$6 MHz. The two-photon detuning, $\delta _D$, due to the Doppler shift of the root-mean-square atom velocity in the propagation direction of light, $\bar {v}$, is given by
where $k_1$, $k_c$, $k_2$, and $k_p$ are the wave vectors of $\Omega _1$, $\hat {E}_c$, $\Omega _2$, and $\hat {E}_p$, respectively, and $c$ is the speed of light. At an atom temperature of 100 $^\circ$C, $\delta _D {\approx } 2{\pi }{\times }6\;\textrm {kHz} (= 0.001{\Gamma })$ which is far less than the bandwidth of the entanglement mentioned earlier. Thus, the all-copropagation scheme not only resolves the problem of the Doppler shift induced by the atoms’ thermal motion, but also maintains an excellent phase-match condition in the FWM process [50,51].In Ref. [51], $\alpha$ of about 370 has been achieved at the atom temperature of 65 $^\circ$C, and $\gamma$ of the experimental system was 0.02$\sim$0.03$\Gamma$. According to the study on the EIT spectrum of a hot atomic vapor in Ref. [52], the $\alpha$ of Doppler-broadened atoms can be equivalent to the $\alpha$ of stationary atoms under the criterion of $\Omega _2^2/(2\gamma ) \gg \Gamma _D$, where $\Gamma _D (= k_p \bar {v})$ is the Doppler width. Using $\Omega _{\rm c,opt}$ (or $\Omega _{\rm 2,opt}$) of about $3.2\Gamma$ at $\alpha =$ 400 in Fig. 6(b) and taking $\gamma$ as 0.025$\Gamma$, we find the criterion is approximately satisfied. After $\delta$ is optimized, $V$ is −12.6 dB at $\alpha =$ 370 and $\gamma =0.025\Gamma$. Such a degree of entanglement has surpassed the best result to date, and can be realized in a Doppler-broadened medium like the one in Ref. [51].
5. Conclusion
In this work, we study the quantum entanglement under FWM process, which is consist of two $\Lambda$ systems. At first, we have found that the strong entanglement occurs in the region of one-photon on resonance in EIT system, and the other $\Lambda 's$ is under far-off resonance, corresponding to a Raman process. For a given optical density, one can find an optimized input Rabi frequency of the strong fields and the corresponding one-photon detuning. It shows that optimum entanglement degree can be greatly enhanced by increasing optical density. The entanglement degree can achieve to be better than $-17\text {dB}$ when $\alpha = 1,000$. It provides a strong entangled light source with high optical density of atomic system in FWM process. Finally, in order to make our scheme more realistic, we include the effects of ground state decoherence rate and two-photon off-resonance in our system, and we find that we can obtain stronger entanglement with non-zero two-photon detuning. Besides, the ground state decoherence limits the optimum entanglement, but the entanglement degree can still sustain about 15 dB when $\gamma < 0.07\Gamma$. It reflects that the entangled source generated from the FWM scheme is robust against the atomic decoherence.
In addition to the realistic effects in the FWM system, there are some issues in practical measurements. Since the fields get large gains in the FWM process, their one-mode squeezing become exceedingly large. It implies that such noises of anti-squeezing will leak into the measurements due to some imperfect conditions in homodyne detections, resulting in the great degradation of entanglement degree. It has great challenges in experiments to observe as high as the entanglement degree based on our theoretical scheme.
Acknowledgements
This work was supported by the Ministry of Science and Technology, Taiwan under Grant No. 110-2639-M-007-001-ASP and No. 111-2639-M-007-001-ASP. We appreciate National Tsing Hua University for offering the license of simulation software, and the hardware facilities in the National Center for Theoretical Sciences (NCTS), Taiwan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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