Cite this Article
Zheng Ya-Mei, Hu Chang-Sheng, Yang Zhen-Biao, Wu Huai-Zhi. Photon bunching and anti-bunching with two dipole-coupled atoms in an optical cavity. Chinese Physics B, 2016, 25(10): 104202  
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Photon bunching and anti-bunching with two dipole-coupled atoms in an optical cavity
Zheng Ya-Mei, Hu Chang-Sheng, Yang Zhen-Biao, Wu Huai-Zhi†,
Department of Physics, Fuzhou University, Fuzhou 350116, China

 

† Corresponding author. E-mail: huaizhi.wu@fzu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11305037, 11347114, and 11374054) and the Natural Science Foundation of Fujian Province, China (Grant No. 2013J01012).

Abstract
Abstract

We investigate the effect of the dipole–dipole interaction (DDI) on the photon statistics with two atoms trapped in an optical cavity driven by a laser field and subjected to cooperative emission. By means of the quantum trajectory analysis and the second-order correlation functions, we show that the photon statistics of the cavity transmission can be flexibly modulated by the DDI while the incoming coherent laser selectively excites the atom–cavity system’s nonlinear Jaynes–Cummings ladder of excited states. Finally, we find that the effect of the cooperatively atomic emission can also be revealed by the numerical simulations and can be explained with a simplified picture. The DDI induced nonlinearity gives rise to highly nonclassical photon emission from the cavity that is significant for quantum information processing and quantum communication.

PACS: 42.50.Pq;32.80.Qk;32.80.Ee
Keyword:photon bunching and anti-bunching;dipole–dipole interaction;cooperative emission;cavity QED
1. Introduction

The standard Jaynes–Cummings (JC) model in cavity quantum electrodynamics describes the interaction of a two-level atom with a quantized cavity mode.[1] The atom–cavity strong coupling results in the anharmonicity of the JC ladder of eigenstates,[2] which gives rise to photon blockade[3] and thereby nonclassical output photon statistics. The JC model along with the quantum light sources appears as one of the key ingredients for applications in quantum information processing and secure quantum communication.[4]. Generalizing the standard JC model to the case with two atoms in a cavity, the photon emitted by an excited atom can be absorbed by the other in the ground state and the dipole–dipole interaction (DDI) between the two atoms can be induced through exchange of virtual photons.[5,6] The DDI with long-range and anisotropic nature can also be modulated by the electromagnetic environment[7] and significantly affects the dynamical properties[8–10] and the transmission spectrum in the driven atom–cavity systems.[11,12]

Cooperative emission is a fundamental effect in quantum optics arising in systems that are symmetric under the interchange of any pair of particles.[13,14] For few-particle systems where each particle’s state and position can be precisely controlled, the collective emission effects can be tailored.[15–17] In particular, by means of cavity quantum electrodynamics, when the emitters are equally coupled to the cavity mode, the spontaneous emission from the particles is indistinguishable even if they are spatially separated. The preferential emission into a single cavity mode has been recently observed by coupling two atoms[18] or two ions[19] to the normal mode of an optical cavity, where the cooperative emission from the ion–cavity or atom–cavity system can be flexibly tuned and can be used for preparation of the two-particle entangled state. The efficient atom/ion–cavity coupling is significant for the realization of photonic quantum memories[20] and long distance quantum networks.[21,22]

Recently, there has been increased interest in engineering nonclassical features of light since its importance for quantum information devices.[20,23,24] In a cavity QED system, the nonclassical photon statistics of light can be controlled via the photon blockade and electromagnetically induced transparency with emitters strongly coupled to an optical resonator.[25–33] Here, we will discuss the similar physics in a different way that involves the interatomic interaction. We consider two dipole-coupled atoms trapped in an optical cavity driven by an external laser field and take the cooperative emission into account. The effect of the dipole–dipole interaction between the two atoms on the photon statistics and correlations of the cavity transmission is numerically studied via the quantum jump method. If an incoming photon resonantly excites the atom–cavity system from its ground state to the n-excitation dressed state consisting of the two-atom collective state and the cavity field, then a second photon at the same frequency will be detuned from or resonant with the next steps up the ladders depending on the DDI strength. The photon statistics can therefore be modulated by the DDI, which is revealed by the sudden changes of cavity intensity in a random quantum jump trajectory, and by the second order correlation functions. The method can also reveal the effect of the cooperative emission on the photon statistics. The results will be of great interest for the two-atom cavity QED experiment considering the recent progress in observation of collective emission effects.[18,19]

The paper is organized as follows. In Section 2, we derive the effective model of the system with two identical dipole-coupled two-level atoms trapped in a single-mode optical cavity. In Section 3, we briefly review the theory of the quantum-trajectory approach and introduce the method for calculating the second order correlation function. In Section 4, we study the influence of the cavity decay on the cavity intensity and thus the photon statistics involving the interatomic DDI, via the method presented in Section 3. In Section 5, we investigate the effect of the cooperative emission on the cavity intensity. Finally, we summarize our results in Section 6.

2. Theoretical model

Consider a system composed of two identical two-level atoms that are trapped in a single-mode high-finesse optical cavity, and are localized at positions r1 and r2 along the cavity axis, as shown in Fig. 1.

Fig. 1. The schematic diagram of the scheme. Two identical dipole-coupled two-level atoms trapped in a single-mode high-finesse optical cavity, which is pumped by a coherent laser field with strength η. The two atoms experience the cooperative emission with the rate γ. The dipole–dipole interaction strength, individually spontaneous emission rate of the atoms, and the cavity decay rate are J, Γ, and κ, respectively. The transmitted photons are detected by a photon detector.

The cavity field is driven by a coherent laser field with the strength η and the detuning from cavity resonance Δcl. The two atoms interact with each other via a dipole–dipole potential with the interaction strength defined by[8]

where Γ0 is the spontaneous emission rate in free space, r = |r1 − r2|, ωa is the transition frequency, θ is the atomic dipole moments with respect to the interatomic axis, and c is the velocity of light. The Hamiltonian for the system in the rotating wave approximation and in the interaction picture is given by[5,8,34]

where a and a† are creation and annihilation operators of the cavity field, σi = |g〉i〈e| (i = 1,2) are the lowering operators of the ith atom, g(ri) is the ri-dependent atom–cavity coupling strength, and Δal is the laser detuning from atomic resonance.

For the atom–cavity interaction part, it is convenient to rewrite the Hamiltonian Eq. (2) in terms of the two-atom collective operators

which applied to the collective ground state |gg〉 will create the symmetric and antisymmetric Dicke states

Thus, the Hamiltonian after rotating with the frame becomes:[18,35,36]

with and Δac = Δal − Δcl the atom–cavity detuning. Here Hd+ and Hd− indicate the interatomic distance-dependent coupling of the cavity mode with the symmetric and antisymmetric Dicke state, respectively. Without loss of generality, we assume that the interatomic separation r is on the order of the cavity wavelength giving rise to cos(kr1) = cos(kr2), and g− = 0. Therefore, the coherent dynamics of the system excludes the antisymmetric Dicke state and the Hamiltonian Eq. (3) reduces to

without the laser driving HL = ηae−iΔclt + h.c.. In this case, the dipole–dipole interaction plays a role in equivalence to an additional atom–cavity detuning. We now introduce the new notations for the collective atomic states defined as |G〉 ≡ |gg〉, |E〉 ≡ |ee〉, and . For a closed system with the excitation number n, the coherent dynamics of the system will remain in the subspace spanned by {|G,n〉,|S,n − 1〉,|E,n − 2〉}, which constructs an energy structure like a Jaynes–Cummings ladder. For n = 1, there exists only the doublet dressed states |1,±〉 with the eigenenergies given by

The energy gap ΔE = |E1,+ − E1,−| between these two dressed levels reaches the minimum for Δac + J = 0, which is shown in Fig. 2(a). While for n ≥ 2, the dressed states are indeed triplets, and are denoted as |n,0〉 and |n,±〉, respectively. The dressed energy level for n = 0,1,2 as a function of J and as a function of Δac are shown in Figs. 2(b)–2(d).

Fig. 2. (a) The energy separation between the doublet dressed state for n = 1 as a function of dipolar coupling strength J and the atom–cavity detuning Δac. The separation reaches the minimum for Δac + J = 0, as indicated by the dashed line. The dressed energy level for n = 1,2 versus the dipolar coupling strength J for (b) Δac = 0, and versus the atom–cavity detuning Δac for (c) J = 1 and (d) J = 5. Here, the dressed energy for n = 2 has been divided by a factor of 2. Further parameters are (κ,Γ,γ,η) = (0.2,0.1,0.1,0.1) and we have fixed units of g = 1.
3. The numerical method

The dissipative dynamics of the system including the atomic spontaneous emission and cavity photonic leakage can be described by the master equation of the Lindblad form in the Born–Markov approximation[37]

with the relaxation operator induced by system-reservoir coupling

where κ,Γ,γ are decoherence rates for the cavity decay, atomic spontaneous emission, and cooperative atomic decay, respectively.

The dissipation in the system can be alternatively treated by the quantum-trajectory method,[38–41] which presents stochastic traces of photon detection events followed by a transient evolution until the next quantum jump is detected. Averaging over many independent realizations of these random trajectories, one recovers the predictions of the corresponding master equation. In addition, the individual random simulations are also available, offering representative records of the randomly selected detection events that directly correspond to what would be observed in a veritable transmission experiment. Thus, by measuring the times for photons leaving the cavity through the output mirror,[38] we are able to analyze the photon statistics and photon correlations.

In this paper, we will focus on the statistical properties of photons under the effect of the interatomic dipolar interaction, where the photon correlation for the cavity transmission in the steady state is described by the second-order correlation function

where τ is the time delay for two different photons arriving at the detector, and it should be noted that g(2) (0) cannot be less than unity classically. Then, if the transmitted photons tend to arrive in groups, we have g(2) (0) > 1 corresponding to the phenomenon named photon bunching. On the contrary, the anti-bunching effect is reflected as the case that it is less probable to detect a further photon right after the detection of the first photon, and the correlation function will be g(2) (0) < 1.[42] If the correlation function is zero at τ = 0, then it implies that two photons are never detected immediately after one another. This photon anti-bunching effect has been regarded as the basic characteristic of a non-classical field state.

Numerically, we first calculate the density matrix that conditioned on the act of detecting and annihilating a photon from the cavity field in the steady state ρss,

Suppose that the mean photon number right before the photon jump is 〈a†a〉 (τ = 0−) = Tr[ρssa†a] and the intracavity photon number conditioned on a photon jump at τ = 0 is indicated as

then the equal-time second order correlation function g(2) (0) for a weak driving regime can be connected to the variance of the intracavity photon number and is given by[43]

The two-photon correlation function g(2)(τ) can then be reconstructed as an ensemble average over many quantum jump trajectories conditioned on a photon register at τ = 0. In the following, we will analyze the effect of the dipole-dipole interaction induced nonlinearity and the collective atomic emission on the photon statistics and the cavity intensity under the condition of weak driving, via the quantum trajectory approach and the method above.[40]

4. The photon statistics under interatomic DDI
4.1. Atom–cavity resonant interaction

We first consider the resonant interaction of the atoms with the cavity field, i.e., Δac = 0, and simply modulate the interatomic interaction J. By sweeping the driving laser frequency, we can find a sudden increase of the cavity intensity resulting from the single-photon or two-photon resonant transition from the ground state to the corresponding dressed energy levels. The effect of the nonlinearity induced by J on the photon statistics can be revealed by the equal-time correlation function g(2) (0), as shown in Fig. 3. Note that we mainly focus on the lowest two dressed levels |1,−〉 and |2,−〉 since the dipolar induced nonlinearity will make the transition of the system to both states under laser driving and therefore significantly influence the photon statistics.

Fig. 3. The equal-time correlation function g(2) (0) as a function of the dipolar coupling strength J for Δac = 0. Further parameters in units of g are (κ,Γ,γ,η) = (0.2,0.1,0.1,0.1).

For J = 1, the driving laser with detuning Δcl = 1 meets both the one-photon resonance and two-photon resonance conditions, as can be seen in Figs. 2(b) and 2(c), where the dressed energy level (DEL) |1,−〉 and the rescaled DEL |2,−〉 intersect. The equal-time correlation function g(2) (0) is less than 1 indicating photon anti-bunching while the system moves towards the steady state. As J increases, the energy difference between the DELs for |1,−〉 and |2,−〉 diverges, but the laser driving frequency becomes closer to the two-photon resonance |2,−〉. This leads to the increase of g(2) (0) that might be greater than 1 and results in photon bunching.

The laser driving with Δcl = 0.37 and Δcl = 0.5 correspond to the one-photon resonance and two-photon resonance for J = 5, respectively. Due to the dipolar interaction induced nonlinearity, the former exhibits photon anti-bunching with g(2) (0) < 1 and the latter behaves the other way around. As J decreases from 5 to 1, both the DELs |1,−〉 and |2,−〉 decline, but finally the single-photon resonance condition dominates for J becoming less than 4, since the DEL |1,−〉 is closer to the laser driving frequency. The situation changes while J < 1, where the two-photon resonance in reverse plays the main role since the DEL |2,−〉 appears above the DEL |1,−〉, and g(2) (0) becomes larger than 1.

The photon counting statistics can be directly modelled by the quantum-trajectory approach, as shown in Fig. 4, where the mean photon number in the cavity 〈n〉 varies suddenly along with the quantum jumps corresponding to photon clicks at the detector. The clicks induced by photon decay are indicated by the green cross. If the cavity intensity increases after detection of a photon, then it implies photon bunching, and photon anti-bunching occurs for the other way. It is interesting to see that for J = 1, both increase and decrease of the cavity intensity after a photon jump are possible since the one-photon and two-photon resonance conditions are simultaneously satisfied with Δcl = 1 [see Fig. 4(a)]. This behavior is not revealed in Fig. 3 because the system dynamics in the stationary limit is mainly contributed by the single-photon transition. For laser detunings Δcl = 0.37 and Δcl = 0.5, the photon counting statistics shown in Figs. 4(b) and 4(c) behave as expected from the above analyses.

Fig. 4. The mean photon number of the cavity field versus time t for atom–cavity resonant interaction with (a) Δcl = 1, (b) Δcl = 0.37, and (c) Δcl = 0.5. In the right panels, the green crosses indicate the photon jump events resulted from cavity decay. The atom jumps are not indicated. Further parameters are (κ,Γ,γ,η) = (0.2,0.1,0.1,0.1). All parameters in units of g.
4.2. Atomic–cavity non-resonant interaction

As mentioned before, the dipolar interaction plays a role similar to atom-cavity detuning. So it is worthwhile investigating the system dynamics with non-resonant interaction between atom and cavity field, and we will focus on the condition Δac + J = 0 in particular. In Fig. 5, we show the dressed energies as a function of dipolar coupling strength J for Δac = −1 and Δac = −5, respectively. Comparing with the resonant case, we find that the intersection point for the DELs |1,−〉 and |2,−〉 shifts to J < 1 although the minimal energy separation for |1,+〉 and |1,−〉 remains at J = Δac. On the other hand, for increasing atom–cavity detuning Δac, the separation between the DEL |1,−〉 and the rescaled DEL |2,−〉 enlarges for J > 1, and meanwhile the DEL |1,+〉 and the rescaled DEL |2,+〉 become closer to overlap.

Fig. 5. The dressed energy level for n = 1,2 as a function of the dipolar coupling strength J for (a) Δac = −1 and (b) Δac = −5. The dressed energy for n = 2 has been divided by a factor of 2. Parameters in units of g.

The photon statistics in the non-resonant case is shown in Fig. 6. For Δac = −1, g(2) (0) reaches the minimum at J = 1 and is less than 1 with . Compared to the resonant case, here the laser driving fulfills the single-photon resonance condition, and meanwhile, is detuned from the two-photon resonance. Therefore, the photon counting statistics show a different behavior that the detection of a photon leaked from the mirror uniquely corresponds to a reduction of the cavity intensity [see Fig. 7(a)]. As J increases from 1, because the DEL |2,−〉 approaches the two-photon resonance with respect to the driving laser frequency, the counting statistics is expected to be photon bunching evidenced by the peak (g(2) (0) > 1) near J = 2.5. Moreover, the dressed level |2,−〉 gets into the two-photon resonance for Δcl = 1.7 and J = 1 leading to g(2) (0) > 1 (photon bunching) [see Fig. 7(b)]. The photon counting statistics are expected to be anti-bunching as J decreases, and the DEL |2,−〉 and the rescaled DEL |2,−〉 approaches the intersection. g(2) (0) approaches to 1 for J increasing to be larger than 6.

Fig. 6. The equal-time correlation function g(2) (0) versus the dipolar coupling strength J for atom-cavity non-resonant interaction. Further parameters in units of g are (κ,Γ,γ,η) = (0.2,0.1,0.1,0.1).
Fig. 7. The mean photon number in the cavity versus time t for atom-cavity detuning Δac = −1 with (a) Δcl = 1.4, (b) Δcl = 1.7. In the right panels, the red crosses in the right panels indicate the quantum jump events resulted from the cooperative emission of the atoms. Further parameters are (J,κ,Γ,γ,η) = (1,0.2,0.1,0.1,0.1). All parameters in units of g.

For Δac = −5, the atoms interact with the cavity field dispersively. The eigenenergy for |2,−〉 is only weakly perturbed by the dipolar interaction J. The laser frequency tuned to the red-detuning Δcl = 5.1 from the cavity resonance is however far off-resonant with the intermediate DELs |1,±〉, therefore, the photons can hardly enter the cavity. While J decreases towards the intersection of |1,−〉 and the rescaled |2,−〉, the photon statistics become obviously anti-bunching.

5. The cavity intensity influenced by the cooperative atom jumps

In addition to the photon jumps, the cavity intensity is significantly influenced by the collective atom jumps. The events corresponding to the atomic cooperative emission are marked by the red cross in Fig. 7, which shows that the mean photon number drops when the collective jumps occur under the condition of one-photon resonance, nevertheless, the mean photon number increases under the condition of two-photon resonance. The interesting phenomenon can be simply explained as follows: Starting with the initial density operator ρ(0) of the system, for a short-time Δt without atomic emission, the non-unitary time evolution operator is given by

with the non-Hermitian Hamiltonian

Then, the system after the cooperative emission can be renormalized as

The cavity intensity can therefore be calculated by following the quantum jumps induced by the atomic emissions. For laser-driven single-photon resonant transition, the coherent dynamics mainly remains in the state space spanned by {|G,0〉,|G,1〉,|S,0〉}. When no atoms have emitted for a while, the wave function of the system approaches an entangled steady state due to the balance of laser driving and non-unitary decay for non-detection of photons, and can be approximately given by

with the mean photon number being n̄c > 0. The quantum jump due to the cooperative emission projects the system onto the state |ψ〉1,co = |G,0〉 with n̄c = 0, resulting in reduction of the cavity intensity. Similarly, under the condition of two-photon resonance, the coherent dynamics mainly remains in the state space spanned by {|G,0〉,|G,2〉,|S,1〉,|E,0〉} and the steady state of the system under non-unitary evolution can be approximately given by

with the mean photon number being 0 < n̄c < 1 for p2 ≪ p0. The quantum jump due to the cooperative emission projects the system onto the state |ψ〉2,co = |G,1〉 with n̄c = 1, resulting in enhancement of the cavity intensity.

6. Summary

In summary, we have studied the effect of the dipole-dipole interaction on the photon statistics in the laser-driven cavity QED system by including the cooperatively atomic emission. We have found that by modulating the interatomic dipolar coupling, the photon statistics described by the equal-time second order correlation function can vary from bunching to anti-bunching for both the resonant and non-resonant atom–field interaction. The behavior is revealed by the random quantum trajectory as well, which shows sudden increase or decrease of the cavity intensity soon after quantum jumps. This can be further used to explain the effect of the cooperatively atomic emission on the photon statistics and will be of great interest for the two-atom cavity QED experiment.

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