Quantum statistics of light transmitted through an intracavity Rydberg medium

We theoretically investigate the quantum statistical properties of light transmitted through an atomic medium with strong optical non-linearity induced by Rydberg-Rydberg van der Waals interactions. In our setup, atoms are located in a cavity and non-resonantly driven on a two-photon transition from their ground state to a Rydberg level via an intermediate state by the combination of the weak signal field and a strong control beam. To characterize the transmitted light we compute the second-order correlation function $g^{\left(2\right)}\left(\tau\right)$. The simulations we obtained on the specific case of rubidium atoms suggest that the bunched or antibunched nature of the outgoing beam can be chosen at will by appropriately tuning the physical parameters.


Introduction
In an optically nonlinear atomic medium, the dispersion and absorption of a classical light beam depends on the powers of its amplitude [1]. At the quantum level, dispersive optical nonlinearities translate into effective interactions between photons. The ability to achieve such strong quantum optical nonlinearities is of prominent importance in quantum communication and computation, as it allows us to implement photonic conditional two-qubit gates. The standard Kerr dispersive non linearities obtained in noninteracting atomic ensembles, either in off-resonant two-level or resonant three-level configurations involving electromagnetically induced transparency (EIT), are too small to allow for quantum nonlinear optical manipulations. To further enhance such nonlinearities, EIT protocols were put forward in which the upper level of the ladder is a Rydberg level. In such schemes, the strong van der Waals interactions between Rydberg atoms result in a cooperative Rybderg blockade phenomenon [2][3][4], where each Rydberg atom prevents the excitation of its neighbours inside a 'blockade sphere'. This Rydberg blockade deeply changes the EIT profile and leads to magnified nonlinear susceptibilities [5][6][7][8]. In particular, giant dispersive nonlinear effects were experimentally obtained in an off-resonant Rydberg-EIT scheme using cold rubidium atoms placed in an optical cavity [9,10]. In this paper, we theoretically investigate the quantum statistical properties of the light generated in the latter protocol. Note that, contrary to other theoretical works, e.g. [11,12], here, we are interested in the dispersive region. Moreover, since we place the atoms in a cavity rather than in free-space, the theoretical framework and calculations we perform also differ from [11,12]. In particular, a technical benefit of our approach is that we are not restricted to considering only photon pairs but could, in principle, investigate higher-order correlations.
We first write the dynamical equations for the system of interacting three-level atoms coupled to the strong control field and the nonresonant cavity mode fed by the probe beam. We show that, under some assumptions, the system effectively behaves as a large spin coupled to the cavity mode [13]. We then compute the steady-state second-order correlation function to characterize the emission of photons out of the cavity. Our numerical simulations suggest that the bunched or antibunched nature of the outgoing light, as well as its coherence time, may be controlled through adjusting the detuning between the cavity mode and probe field frequencies.
The paper is structured as follows. In section 2, we present our setup and the assumptions we make to compute its dynamics. We also explain the analytical and numerical methods we employ to calculate the second-order g ( ) 2 correlation function of the outgoing light beam. In section 3, we present and interpret the results of the simulations we obtained for on the specific experimental case considered in [9]. Finally, we conclude in section 4 by evoking open questions and perspectives of our work. The appendices address the supplementary technical details that are omitted in the text for readability.

Model and methods
The system we consider comprises N atoms that present a three-level ladder structure with a ground g , intermediate e , and Rydberg states r (see figure 1). The energy of the atomic level = k g e r , , is denoted by ℏω k (by convention ω = 0 g ), and the dipole decay rates from the intermediate and Rydberg states are denoted by γ e and γ r , respectively. The transitions ↔ g e and ↔ e r are, respectively, driven by a weak probe field of frequency ω p and a strong control field of frequency ω cf . e is supposed to be in the neighbourhood of a cavity resonance. The frequency and annihilation operator of the corresponding mode are denoted by ω c and a, respectively; the detuning of this mode with the probe laser is defined by Δ ω ω ≡ − ( ) c p c , and α denotes the feeding rate of the cavity mode with the probe field, which is assumed real for simplicity. Finally, we introduce g and Ω cf , which are the single-atom coupling constant of the transition ↔ g e with the cavity mode and the Rabi frequency of the control field on the transition ↔ e r , respectively. In the following paragraphs, we study the dynamics of the system, which, under some assumptions, is equivalent to a damped harmonic oscillator, i.e. the cavity mode, coupled to an assembly of spins 1 2 , and the Rydberg bubbles corresponding to the 'super-atoms' delimited by the Rydberg blockade spheres.
Starting from the full Hamiltonian, we perform the rotating wave approximation and adiabatically eliminate the intermediate state e as described in appendix A. Note that the result we obtain coincides with the lowest order of the EIT model-the nonlinearity of the three-level atoms is neglected, and the leading nonlinear effect comes from the Rydberg-Rydberg collisional effects. The system therefore consists of N effective two-level atoms { }   In this expression, we have introduced the atomic operators , , as well as the effective detunings is the effective coupling strength of the two-photon transition → g r driven by the cavity mode and the control laser.
At this point, following [13], we introduce the Rydberg bubble approximation. In this approach, the strong Rydberg interactions are assumed to effectively split the sample into  b atoms but can only accomodate a single Rydberg excitation, delocalized over the bubble. Note that, within this approximation, all New J. Phys. 16 (2014) 043020 A Grankin et al bubbles have the same radius, which is fixed by the coefficient C 6 and the detunings Δ r and Δ e , as well as the control field Rabi frequency Ω cf ; assuming that the atomic ensemble is homogeneous, the number of atoms per bubble n b is approximately given by [9] π ρ where ρ at is the atomic density. Each bubble can, therefore, be viewed as an effective spin 1 2 whose Hilbert space is spanned by the ground state of the bubble α  and its symmetric singly Rydberg excited state, respectively.
Introducing the bubble spin- , where the operator α − s ( ) corresponds to the lowering operator of the spin and the annihilation of a Rydberg excitation, one can write the Hamiltonian under the approximate form (see appendix A) The system is, therefore, equivalent to a large spin, i.e. the assembly of spin-1 2 Rydberg bubbles, coupled to a harmonic oscillator. Its density matrix satisfies the master equation One can also write the Heisenberg-Langevin equations for the time-dependent operators are the Langevin forces associated with a and − J , respectively. Note that we neglected the effect of extra dephasing due to, for example, collisions or laser fluctuations. To study the quantum properties of the light transmitted through the cavity, we shall compute the function g ( ) out 2 , which characterizes the two-photon correlations. In the input-output formalism [14], one shows that this function simply equals the function g ( ) 2 for the intracavity field (see appendix B for details) given by where ρ ss denotes the steady state of the system defined by ρ =  0 ss ; see equation (1). In the region of the small feeding parameter α, one can compute ρ ss numerically by propagating in time the initial state ρ ≡ ). The steady state is reached in the limit of large times, ideally when → ∞ t . The denominator of the ratio in equation (2) is directly obtained from ρ ss . To compute its numerator, one first propagates in time ρ † a a ss from t = 0 to τ, using the same procedure as earlier, then applies the operator † a a and takes the trace. In the region of weak feeding, it is also possible to get a perturbative expression for that is too cumbersome to be reproduced here, but allows for faster calculations than the numerical approach. Such a fully analytical treatment, however, cannot, to our knowledge, be extended to τ > Finally, we get the following approximate expression for the effective Hamiltonian where κ Δ ≡˜ 2 r b . In this region, the system thus behaves as two coupled oscillators: one is harmonic, the cavity field; the other is anharmonic, the Rydberg bubble field. The cavity resonance is therefore shifted depending on the number of excitations in the system.
In the following section, we present and discuss the results we obtained with the specific system used in [9]. It appears that one can choose the bunched or antibunched behaviour of the light transmitted through the cavity by adjusting the detuning Δ c . We also show that the time behaviour of the function τ ( ) depends on the region considered, and can be roughly understood as resulting from the damped exchange of a single excitation between the atoms and field.

Numerical results and discussion
We now consider the physical setup presented in [9], i.e. an ensemble of Rb  The dynamical features observed can be understood and satisfactorily accounted for by a simple three-level model. Indeed, due to the weakness of α, the system in its steady state is expected to contain, at most, two excitations (either photonic or atomic). After a photon detection at t = 0, it contains, at most, one excitation, which can be exchanged between the cavity field and atoms, as it has been known for a long time [15,16]. In other words, the operator ρ † a a ss can be expanded in the space restricted to the three states 0, 0 , 01 0, 1 , 10 1, 0 r c r c r c and the effective non-Hermitian Hamiltonian for the system, in this subspace, takes the following form: The order of magnitude of the frequencies and decay rates of the oscillations observed for g t ( ) ( ) 2 in the specific cases (A,B) are satisfactorily recovered by this Hamiltonian, which validates the schematic model we used and suggests it comprises the main physical processes at work. To conclude this section, it is worth mentioning that the two-boson approximation, though strictly speaking not applicable here-the parameters considered in this section indeed correspond to a number of bubbles ≃  2 b -yields, however, the qualitative behaviour for The minimum is correctly located, though slightly higher than in the spin model; the antibunching peak is slightly shifted towards positive detunings and is weaker than in the previous treatment. These discrepancies result from too low a value of the nonlinearity parameter κ; they can be corrected by replacing κ in the twoboson Hamiltonian. We first note that κ and κ′ coincide in the region of a large number of bubbles. Moreover, κ′ makes sense in the region of a low number of bubbles: in particular, when →  1 b , i.e. when only one bubble is available, the nonlinearity proportional to κ′ diverges accordingly, thus forbidding the boson field to contain more than one excitation. Finally, let us mention that κ′ can also be recovered via a perturbative treatment of the full model, which will be presented in a future paper.

Conclusion
In this work, we studied how the strong Rydberg-Rydberg van der Waals interactions in an atomic medium may affect the quantum statistical properties of an incoming light beam. In our model, atoms are located in a low-finesse cavity and subject to a weak signal beam and a strong control field. These two fields nonresonantly drive the transition from the ground to a Rydberg level. The system was shown to effectively behave as a large spin coupled to a damped harmonic oscillator, i.e. the assembly of Rydberg bubbles and the cavity mode, respectively. The strong anharmonicity of the atomic spin affects the quantum statistics of the outgoing light beam. To demonstrate this effect, we performed analytical and numerical calculations of the second-order correlation function The results we obtained on a specific physical example with rubidium atoms indeed show that the transmitted light presents either bunched or antibunched characters, depending on the detuning between the cavity mode and the probe field. This suggests that in such a setup, one could design light of arbitrary quantum statistics through appropriately adjusting the physical parameters.
In this work, we performed the Rydberg bubble approximation, which allowed us to derive a tractable effective Hamiltonian. This scheme is, however, questionable: interactions between bubbles are indeed neglected, and the different spatial arrangements of the bubbles in the sample are not considered. Though challenging, it would be interesting to run full simulations of the system, rejecting those states that are too far off-resonant due to Rydberg-Rydberg interactions. Besides validating the assumption of the present work, this would indeed enable us to consider other regions, such as, for instance, the case of resonant transition towards the Rydberg level. We also implicitly made the assumption that the cavity mode and control beam were homogeneous. Spatial variations should be included in the model, and their potential influence studied in future work. Finally, due to the very weak probe field region considered in this paper, we only presented results on the function τ ( ) : the production of = … n 3, 4, correlated photons is indeed very unlikely. In principle, we can, however, numerically compute τ ( ) for any > n 2, which might be relevant in a future work, if addressing stronger probe fields. where    approximately given by [9] π ρ Δ Ω Δ = − n C 2

(4 )
b r c f e 2 at 6 2 where ρ at is the atomic density. Each bubble can, therefore, be viewed as an effective spin 1 2 whose Hilbert space is spanned by where we used + + ≡ +