^{†}Corresponding author. Email: guoyujie@csrc.ac.cn
^{*}Project supported by the National Natural Science Foundation of China (Grant No. 11304010).
We study the optical properties of a twolevel atomic ensemble controlled by a highfinesse cavity. Even though the cavity is initially in the vacuum state in the absence of external driving, the probe response of the atomic ensemble can be dramatically modified. When the collectively enhanced atom–cavity coupling is strong enough and the cavity decay rate is much smaller than the atomic damping rate, an electromagnetically induced transparencylike coherent phenomenon emerges with a dip absorption for the response of the twolevel atoms in the cavity without driving, and thus is called vacuum induced transparency. We also show the slow light with very low group velocity in such an atomic ensemble.
The quantum coherent effect in a light– atom interaction system plays an important role in controlling the optical property of the medium. An outstanding instance is electromagnetically induced transparency (EIT), ^{[1– 4]} which occurs in the Λ type threelevel atomic system. The transition between the excited and the ground states of the threelevel atom is coupled by a weak probe laser, while the transition between the excited and the metastable states is coupled by a strong coherent controlling laser. Under appropriate conditions, the atomic medium becomes transparent for the weak probe field in a very narrow frequency range owing to the destructive interference between two absorption pathways. Recently, the concept of EIT has been extended to the fewphoton limit within a highfinesse cavity, ^{[2]} where the controlling laser is replaced by a single quantized cavity mode with only a few photons. Furthermore, the extreme situation is called the vacuum induced transparency (VIT), ^{[5, 6]} where the controlling laser is replaced by a vacuum cavity field with strong atom– cavity coupling.
Besides, there is a quantum coherent effect in the twolevel atomic system, which is similar to the EIT phenomenon and is called the coherent population oscillation (CPO).^{[7, 8]} The CPO occurs in the twolevel system coupled to a weak probe laser and a strong coherent driving laser, where the couplings lead to the oscillation of the ground state population at the beat frequency of the two lasers. If the driving field is strong enough and the damping rate of the atomic coherence is much larger than the decay rate of the excited state population, then the probe response of the atomic medium will be significantly modified, i.e., a narrow dip will be created in the absorptive spectrum.^{[9]} The result of the CPO is similar to that of the EIT, except that the absorption dip of the CPO is not close to zero.
In this paper, we find that VIT, an outstanding instance of an EITlike phenomenon, can be realized not only in the Λ type threelevel atomic system but also in a relatively simpler twolevel atomic system. As mentioned above, the VIT in the threelevel system can be regarded as the fewphoton case of the EIT. The VIT realized in our twolevel system in fact extends the CPO to the fewphoton limit. Moreover, the absorption dip of the VIT in our twolevel system can be close to zero, which is superior to that of the CPO. The VIT realized in our scheme originates from the coupling between the atomic ensemble and the cavity field in the absence of external driving. When the collectively enhanced atom– cavity coupling is strong enough and the cavity decay rate is much less than the atomic damping rate, the VIT emerges clearly in the twolevel system. The EITlike phenomenon, which shows greatly reduced absorption and enhanced normal dispersion, plays a central role in the generation of slow light. There have been a series of researches on the slow light based on the EIT, ^{[10– 13]} VIT, ^{[5]} and CPO.^{[9, 14– 17]} Our scheme also provides another candidate. We demonstrate that the VIT in the twolevel system is particularly suitable for the generation of slow light, and a group velocity close to 10 m/s is obtained with realistic parameters.
This paper is organized as follows. In Section 2, we present our model and its corresponding Hamiltonian. In Section 3, the dynamics and the optical response of the atomic ensemble are studied, including the EITlike phenomenon, the analysis of the vacuum Rabi splitting, and the slow light. Finally, a discussion and a brief conclusion are given in Section 4.
The studied system is comprised of a twolevel atomic ensemble and a singlemode cavity in the absence of an external driving field. The atoms coupled to the cavity mode are probed by a weak laser field. The model Hamiltonian reads^{[18]}
Here
To simplify Hamiltonian (1), we define the collective operators of the atomic ensemble
where N is the number of the atoms. In the limit of large N and under the lowexcitation condition, the collective operators approximatively satisfy the bosonic commutation relation [A, A^{† }] ≈ 1, and further
Here the existence of the factor
The dynamics of the system is determined by the quantum Langevin equations^{[23]}
where γ and κ are the atomic damping rate and the cavity decay rate, respectively; A_{in} (c_{in}), whose mean value is equal to zero, is the noise operator indicating the action of the reservoir on the atomic system (cavity). Here we have assumed that the atomic collisional dephasing rate is negligible.
From the quantum Langevin equations (4), we can obtain the susceptibility, which actually reflects the optical response of the atomic ensemble. Taking the average
and supposing 〈 A〉 and 〈 c〉 ∝ e^{− iω pt}, we can immediately obtain the mean value of the collective operator A at the steady state
where δ = ω _{p} – ω _{a} is the detuning between the probe field and the atomic transition, and Δ = ω _{c} – ω _{a} is the detuning between the cavity field and the atomic transition. The polarization of the atomic ensemble is
where C = (d^{2}/(ħ ε _{0}))(N/V). The term proportional to g^{2} in the denominator of Eq. (7) reflects the role of the atom– cavity coupling, and N indicates that the coupling can be collectively enhanced. Equation (7) can be reduced to the result of bare atoms when g = 0, which means the existence of the cavity effectively modifies the optical response of the atoms, although the cavity is initially in the vacuum state. In this sense, we call it the vacuum induced transparency phenomenon.
The real and imaginary parts of the susceptibility are related to the dispersion and absorption, respectively. For the case of bare atoms in the absence of the cavity, i.e., g = 0, Imχ has a Lorentzian shape with an absorption peak located at the atomic resonance, while Reχ displays an anomalous dispersion in the vicinity of the line center, as shown in Fig. 2(a). Under the control of the cavity vacuum field, the EITlike coherent phenomenon emerges when κ ≪ γ and the strong collectively enhanced coupling Ng^{2} ⩾ κ γ . For simplicity, we first consider the case of Δ = 0. From Fig. 2(b), we can see that Imχ has a narrow dip, while Reχ displays the strong normal dispersion in the vicinity of the line center. To show how Ng^{2} and κ influence the susceptibility, we first fix the cavity decay rate at κ = 0.01γ ; we can see in Fig. 3 that the absorption at the line center becomes markedly reduced with increasing strength of the atoms– cavity coupling, while the slope of the normal dispersion at δ = 0 reaches its maximum when Ng^{2} = κ γ + 2κ ^{2} ≈ κ γ . Then we turn to confine the strong coupling condition, e.g., Ng^{2} = 10κ γ ; in Fig. 4, it is clear that with decreasing decay rate of the cavity, the width of the dip in the absorption spectrum becomes narrower and the slope of the normal dispersion at δ = 0 becomes larger.
Note that in Figs. 2– 4, we have considered the simplest resonant case of Δ = 0. In this case, the imaginary (real) part of the susceptibility is a symmetric (antisymmetric) function of the probe detuning δ = 0. In fact, in the nonresonant case of Δ ≠ 0, the vacuum induced transparency phenomenon still appears even though the corresponding imaginary (real) part of the susceptibility is not symmetric (antisymmetric), as seen in Fig. 5. The corresponding transparency window appears near δ = Δ , wherein the socalled twophoton resonant condition (ω _{p} = ω _{c}) is satisfied.
We note that the EITlike phenomenon in our system essentially has the same physical origin as the vacuum Rabi splitting (VRS), ^{[22, 24]} i.e., the sharp exchange of energy between two oscillators results in a normalmode splitting in the eigenvalues, though the original VRS is shown by observing the transmission spectrum of a cavity^{[25, 26]} and our scheme focuses on the probing of the atomic susceptibility. A transparency phenomenon in a system consisting of a twolevel atom and a cavity has been studied in Ref. [27], however the transparency window is not due to the VRS as the atom– cavity coupling is not strong enough. In our scheme, the atom– cavity coupling is collectively enhanced by the number of the atoms, so we can reach the VRS regime relatively easily. To clearly demonstrate this normalmode splitting, as shown in the profile of Imχ in Figs. 2(b) and 5, we define two new independent bogoliubov modes as follows:
where θ satisfies
which means that the two coupled modes A (atoms) and c (cavity) split into two independent modes a_{1} and a_{2} with frequencies
As is known, the strong normal dispersion combining with low absorption is particularly suitable for the realization of slow light, which has an extremely small group velocity compared with the velocity of light in a vacuum. In order to calculate the group velocity v_{g} = c/n_{g}, we need to compute the group index n_{g} which is defined as
where
It has been emphasized that the strong coupling Ng^{2} ⩾ κ γ and the small cavity decay rate κ ≪ γ are the key conditions to trigger the EITlike phenomenon. Realizing the strong coupling condition is relatively easy, since it can be collectively enhanced by the atom number. The main difficulty lies in the small cavity decay rate, which requires high finesse. Moreover, as shown in Fig. 4(b), a too small κ results in a too narrow transparency window, which may broaden the short probe laser pulse in passing through the atomic ensemble. Alternatively, we can choose other atoms with a relatively large atomic damping rate to realize γ ≫ κ . In addition, as mentioned in Section 3, a large number density can result in a low group velocity. However, we must emphasize that we cannot immoderately enlarge N/V, because it can lead to a remarkable atomic collisional dephasing rate, ^{[29]} then our method based on the quantum Langevin equations (4) will fail.
The mechanism of VIT in the twolevel system can be explained by quantum coherence, which leads to the destructive interference. Initially, the cavity is in the vacuum state  0_{c}〉 , and the twolevel atomic ensemble is in the ground state  0_{A}〉 . When the system is probed by a very weak laser, only three states  0_{A}0_{c}〉 ,  0_{A}1_{c}〉 , and  1_{A}0_{c}〉 will be involved. Here  n_{A}m_{c}〉 is the Fock state of the two coupled modes, and n_{A} and m_{c} are the eigenvalues of A^{† }A and c^{† }c, respectively. The combined action of the probing and the controlling fields on the atoms generates quantum coherence between  0_{A}0_{c}〉 and  0_{A}1_{c}〉 (dark state). Then this quantum coherence leads to interference between two absorption pathways, which are  0_{A}0_{c}〉 →  1_{A}0_{c}〉 and  0_{A}1_{c}〉 →  1_{A}0_{c}〉 . Here  0_{A}0_{c}〉 →  1_{A}0_{c}〉 means that the atoms absorb a probing photon with frequency ω _{p}, and  0_{A}1_{c}〉 →  1_{A}0_{c}〉 means that the atoms absorb a controlling cavity photon with frequency ω _{c}. When ω _{p} = ω _{c} (twophoton resonant condition), the destructive interference between the two absorption pathways results in the transparency. As mentioned above, the high finesse (κ ≪ γ ) and the strong coupling (Ng^{2} ⩾ κ γ ) are the essential conditions to trigger the VIT. On one hand, the cavity decay leads to a dissipative process from  0_{A}1_{c}〉 to  0_{A}0_{c}〉 , which breaks the quantum coherence between  0_{A}0_{c}〉 and  0_{A}1_{c}〉 . On the other hand, the strong coupling gives rise to the quantum coherence. So the decay should be suppressed, whereas the coupling should be enhanced. Here we can see that  0_{A}0_{c}〉 ,  0_{A}1_{c}〉 , and  1_{A}0_{c}〉 are analogous to the ground, metastable, and exited states in the EIT, respectively.
The similarity between VIT in the twolevel system and the original EIT can also be seen from the corresponding Hamiltonians, which have the similar form of two coupled bosonic modes after defining the atomic collectiveexcitation modes in these two models. In our model, the quantized cavity field plays the same role as the collectiveexcitation mode from the ground state to the metastable state in the threelevel atomic ensemble. Physically, the additional cavity field (or the additional metastable state) with very small decay rate changes the response spectrum to the probe field which couples with the collective excitation mode from the ground to the excited states.
In conclusion, we have studied the optical response of a twolevel atomic ensemble controlled by a highfinesse cavity. We find that the EITlike coherent phenomenon emerges even when the cavity is in a vacuum. The VIT in our scheme can be regarded as the fewphoton case of the CPO, but their mechanisms are in fact different, though the twolevel atoms are involved in both models. The former originates from the VRS, while the latter arises from the interference of the controlling and the probing lasers. Note that in the case of CPO, the dip in the imaginary part of the susceptibility does not approach to zero, which means that the transparency is not perfect. Instead, in the present cavityassisted model, the vacuum induced transparency is near 100% at the twophoton resonant point in the case of a small cavity decay rate. Our scheme also provides another candidate for realizing slow light, which may help promote the quantum computing and optical communications.
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