Two-way interconversion of millimeter-wave and optical fields in Rydberg gases

We show that cold Rydberg gases enable an efficient six-wave mixing process where terahertz or microwave fields are coherently converted into optical fields and vice versa. This process is made possible by the long lifetime of Rydberg states, the strong coupling of millimeter waves to Rydberg transitions and by a quantum interference effect related to electromagnetically induced transparency (EIT). Our frequency conversion scheme applies to a broad spectrum of millimeter waves due to the abundance of transitions within the Rydberg manifold, and we discuss two possible implementations based on focussed terahertz beams and millimeter wave fields confined by a waveguide, respectively. We analyse a realistic example for the interconversion of terahertz and optical fields in rubidium atoms and find that the conversion efficiency can in principle exceed 90\%.


I. INTRODUCTION
Two-way conversion between optical fields and terahertz/microwave radiation is a highly desirable capability with applications in classical and quantum technologies, including the metrological transfer of atomic frequency standards [1], novel astronomical surveys [2], long-distance transmission of electronic data via photonic carriers [3], and signal processing for applications in radar and avionics [4]. Efficient conversion of terahertz radiation into visible light would facilitate the generation, detection and imaging of terahertz fields [5,6] for stand-off detection, biomedical diagnostics and spectroscopy. In the quantum domain, coherent microwaveoptical conversion could enable quantum computing via optically-mediated entanglement swapping [7][8][9] in solid state systems such as spins in silicon [10] or superconducting qubits [11], which lack optical transitions but couple strongly to microwaves. Moreover, Josephson junctions can mediate microwave photonic non-linearities that cannot easily be replicated for optical photons [12] so that coherent microwave-optical conversion also provides a route to freely-scalable all-photonic quantum computing. Recent proposals for conversion between the optical and mm-wave domains have been based on optomechanical transduction [13][14][15], or frequency mixing in Λ-type atomic ensembles [16][17][18][19][20]. Both approaches require high quality frequency-selective cavities limiting the conversion bandwidth, as well as aggressive cooling or optical pumping to bring the conversion devices into their quantum ground states. In this paper, we propose instead to use frequency mixing in Rydberg gases [21][22][23] for the conversion of millimeter waves to optical fields (MMOC) (see Fig. 1). We use the terminology 'mm-wave' broadly to refer to fields with carrier frequencies between 10 and 10, 000 GHz, corresponding to resonant transitions between highly excited Rydberg states in an atomic vapour. Our scheme benefits from the strong coupling between Rydberg atoms and millimeter waves which has previously been used for detection and magnetometry [24][25][26], storage of mi-crowaves [27] and hybrid atom-photon gates [28]. Here we show how to achieve efficient and coherent MMOC without the need for cavities, microfabrication or cooling. Our MMOC scheme is made possible by an EIT-related quantum interference effect and the long lifetime of the Rydberg states. In contrast to previous frequency mixing schemes in EIT media [29][30][31], this quantum interference effect implements a coherent beam splitter interaction between the millimeter and optical fields which underpins the conversion effect. Our main result is a theoretical model establishing the principle of operation of the proposed device, which it is shown could be implemented in an ensemble of cold trapped Rb atoms. The paper is organised as follows. We introduce our theoretical model based on the standard framework of coupled Maxwell-Bloch equations in Sec. II, where we also describe how to include interactions between Rydberg atoms. In Sec. III we discuss the principle of operation of our scheme and show that both time-independent and pulsed input fields of arbitrary (band-limited) shape can be efficiently converted. We go on to consider the simultaneous spatial confinement of mm-wave and optical fields, and we show that high conversion efficiencies are predicted for a realistic implementation in trapped Rb vapour. A brief summary of our work is presented in Sec. IV.

II. MODEL
We consider an ensemble of cold trapped atoms interacting with laser fields and mm-waves and model these interactions using the standard framework of coupled Maxwell-Bloch equations. A summary of the general approach is presented in Sec. II A, and a detailed derivation can be found in the Supplementary Information. The analytical solution of the Maxwell-Bloch equations is outlined in Sec. II B and complemented by Appendix A. In Sec. II C we include Rydberg-Rydberg interactions into our model. This allows us to identify parameter regimes in Sec. III where these interactions are negligible. Transition frequencies and detunings are not to scale. ΩM and ΩL are the Rabi frequencies associated with the mm-wave and the optical fields, respectively. ΩP, ΩR, ΩC and ΩA are Rabi frequencies of the auxiliary fields, and ∆ k is the detuning of the fields with state |k (k ∈ {4, 5, 6}). Levels |3 , |4 and |5 are Rydberg states with decay rate Γ γ, where γ is the decay rate of states |2 and |6 .

A. Maxwell-Bloch equations
In a first step we neglect atom-atom interactions and consider the Bloch equations for a single atom with level scheme as shown in Fig. 1. The millimeter wave Ω M of interest couples to the transition |3 ↔ |4 , where |3 and |4 are Rydberg states with principal quantum number n 20. The optical field Ω L of interest couples to the |1 ↔ |6 transition, and the conversion between Ω M and Ω L is facilitated by four auxiliary fields. The resonant fields Ω P and Ω R create a coherence on the |1 ↔ |3 transition through coherent population trapping [32]. The two other auxiliary fields Ω C and Ω A are in general off-resonant and establish a coherent connection between the |3 ↔ |4 and |1 ↔ |6 transitions. We model the time evolution of the atomic density operator by a Markovian master equation In the electric-dipole and rotating-wave approximations, the Hamiltonian H in Eq. (1) is given by and A ij = |i j| are atomic transition operators. The detuning ∆ k in Eq. (2) is defined as where ω k denotes the energy of state |k with respect to the energy of level |1 and ω X is the frequency of field X with Rabi frequency Ω X (X ∈ {P, R, C, A, M, L}). The term L γ in Eq. (1) accounts for spontaneous emission from the excited states. These processes are described by standard Lindblad decay terms. The full decay rate of the states |2 and |6 is γ, and the long-lived Rydberg states decay with Γ γ. The six fields drive a resonant loop, and we impose the phase matching condition In the following, we assume that Ω M and Ω L are copropagating, while the directions of the auxiliary fields are chosen such that Eq. (5) holds. Note that this phase matching condition is automatically fulfilled by virtue of Eq. (4) if all fields are co-propagating. The strong auxiliary fields are not significantly affected by their interaction with the mm-wave and optical signals. We therefore consider only these signal fields and the atomic coherences as dynamical variables. In the paraxial approximation we find where k M (k L ) is the wavenumber of the mm-wave (optical) field and ∆ ⊥ = ∂ 2 x + ∂ 2 y is the transverse Laplace operator. The coupling constants η M and η L are given by where d kl = k|d|l is the matrix element of the electric dipole moment operatord on the transition transition |k ↔ |l , c is the speed of light and N is the density of atoms. In the following the ratio of the coupling constants is denoted by Numerical solutions of Eqs. (1), (6) are presented in Sec. III C, but first it is instructive to derive analytic solutions in the limit that diffraction over the length of the atomic ensemble can be neglected.

B. Analytical solution
The first-order solution of Eq. (1) with respect to the Rabi frequencies Ω M , Ω L takes the form (see Appendix A) The response of the atomic system on the |3 ↔ |4 transition induced by the mm-wave field is described by χ M 43 , and χ L 61 accounts for the atomic response on the transition |1 ↔ |6 due to the optical field. In addition, the mm-wave field can induce a coherence proportional to χ M 61 on the optical transition |1 ↔ |6 , and the optical field can create a coherence proportional to χ L 43 on the transition |3 ↔ |4 . The cross-terms proportional to χ L 43 and χ M 61 in Eq. (9) originate from the closed-loop character of the atomic level scheme. Next we combine Eq. (9) with Eq. (6) and make the simplifying assumption that diffraction over the ensemble length can be neglected, so that the transverse Laplacians can be dropped. Making a coordinate transformation from the laboratory frame (t, z) to a frame (τ = t−z/c, z) co-moving with the signal fields, the evolution equation for the mm-wave and optical fields can then be written as where When the auxiliary fields are time-independent and spatially uniform, the solution to Eq. (10) is where Ω 0 is the initial condition Ω evaluated at z = 0. The matrix exponential in Eq. (12) can be expressed in terms of the 2×2 identity matrix 1 and the Pauli matrices σ k [33], where The solution presented here treats the signal fields Ω M , Ω L as c-numbers. However, the generalisation to quantum fields is straightforward since the coherences in Eq. (9) are linear in the signal fields. Apart from quantum noise operators, our calculations are thus equivalent to a Heisenberg-Langevin approach where the signal Rabi frequencies Ω M , Ω L are replaced by quantum fields [34][35][36]. Since the Langevin noise operators represent only vacuum noise, they do not contribute to normally ordered expectation values, which determine the conversion efficiency.

C. Interaction-induced imperfections
Next we consider the effects of dipole-mediated interactions between atoms excited into their Rydberg manifolds. In general, Rydberg interactions will prevent some fraction of atoms from participating in the conversion process and lead to absorption of the signal fields, and therefore will reduce the conversion efficiency. The atomic level scheme in Fig. 1(b) contains three Rydberg states, and the population in state |3 is continuously kept at ρ 33 ≈ |Ω P /Ω R | 2 via coherent population trapping. On the other hand, the population in the other Rydberg states |4 and |5 is negligibly small for weak fields Ω M and Ω L . The dominant perturbation to the conversion mechanism will thus stem from nearby Rydberg atoms in state |3 . In order to model this, we consider a system of two atoms where atom A is located at the coordinate origin. The conversion process in atom A is disturbed by Rydberg-Rydberg interactions with atom B, which is prepared in state |3 and positioned at R. Next we discuss the two dominant effects caused by the presence of atom B. First, atom B gives rise to a van der Waals shift of state |3 in atom A [37], where the coefficient C 6 depends on the quantum numbers of state |3 . If R is smaller than the blockade radius R b , atom A cannot be excited to the Rydberg state and thus does not participate in the conversion. The blockade radius is determined by the single-atom EIT linewidth γ EIT = |Ω R | 2 /γ and given by R b = [2|C 6 |/( γ EIT )] 1/6 [38]. Second, atom B gives rise to a frequency shift of state |4 in atom A via the resonant dipole-dipole interaction [39], where R = R/R. In contrast to the van der Waals shift in Eq. (15), ∆ DD depends on the relative orientation of the two atoms. In principle, state |5 in atom A can experience a similar shift ∆ DD if the dipole moment d 53 is different from zero. Here we assume that states |5 and |3 have the same parity so that d 53 = 0, consistent with the example implementation in Rb that we introduce below in Sec. III B.
The preceding discussion shows that Rydberg-Rydberg interactions change the energies of states |3 and |4 . In order to incorporate these frequency shifts into our model, we find the general first-order solution of the atomic coherences in Eq. (9) for arbitrary detunings and Rabi frequencies of the auxiliary fields. We then intro-duce the effective detuning parameters and replace ∆ 3 and ∆ 4 in the general expression for the matrix M in Eq. (11) by ∆ 3 and ∆ 4 . Since ∆ vdW and ∆ DD depend on the relative position R, we average M over R, where the distribution of nearest neighbours in a random sample of Rydberg atoms follows the probability density [40], with the parameter the Wigner-Seitz radius for a given density of Rydberg atoms N Ry . This account of Rydberg-Rydberg interactions is expected to work well for weak optical and mm-wave fields. If the intensities of Ω M and Ω L are increased such that the population in |4 and |5 is not negligible, other dipoledipole interactions can occur that are not captured by our model. Furthermore, our model neglects cooperative effects like superradiance [41] and frequency shifts due to a ground state atom within the electron orbit of a Rydberg state [42]. However, experimental results [43][44][45] for EIT involving a Rydberg state show that these effects can be negligible for low principal quantum numbers n 40, for weak probe fields and low atomic densities.

III. RESULTS
In a first step we analyse the simplified analytical model of Sec. II B in order to explain the principle of the conversion mechanism. This is presented in Sec. III A where we also investigate the maximally achievable conversion efficiencies. We then introduce one possible implementation of our scheme in rubidium vapour in Sec. III B and find a set of parameters for which Rydberg-Rydberg interactions are negligibly small. Finally, we present numerical results for MMOC in the physical systems shown in Figs. 1 (a) and (b) in Sec. III C.

A. Conversion mechanism
The conversion efficiency between mm-wave and optical fields according to Eq. (12) will be small for a generic matrix M, but complete conversion can be achieved if the atomic ensemble realises a beam splitter interaction where the 'hat' notation emphasises the operator nature of the fields. Formally, such an interaction corresponds to the case where the diagonal elements of M vanish. We find that this condition, such that χ M 43 ≈ χ L 61 ≈ 0, can indeed be met if the intensities and detunings of the auxiliary fields satisfy To first order in Γ/γ the susceptibilities in Eq. (9) are then given by where ε and ε Γ are dimensionless parameters that are generally smaller than unity. Since ε Γ ∝ Γ, ε Γ is typically of the order of ε 2 . On the other hand, |α| ∝ ε and hence the off-diagonal elements of the matrix M are indeed much larger than the diagonal elements. This result can be understood as follows. The level scheme in Fig. 1 can be regarded as three consecutive EIT systems where the weak probe fields are represented where κ = (ε 2 +ε Γ )/l abs and k = ε/l abs determine the loss and the spatial oscillation period of the interconversion, respectively, l abs = γ/(4η L ) is the resonant absorption length on the |6 ↔ |1 transition and we assumed ε 1 and ε Γ /ε 1. The spatial oscillations of optical and mm-wave intensities according to Eq. (25) are shown in Fig. 2. Our model is in excellent agreement with a full numerical solution of the Maxwell-Bloch equations. Note that the small deviations for large z vanish if the approximations leading to Eq. (25) are omitted. Complete MMOC occurs after a length L c = π/(2k), and thus requires an optical depth D c = L c /l abs that is inversely proportional to ε in Eq. (24b), D c = π/(2ε). Since the value of ε can be adjusted through the intensities and frequencies of the auxiliary fields, the condition for complete MMOC can be met for various densities and sizes of atomic gases. In the example in Fig. 2, we find L c = 100l abs . The efficiency F = e −2κLc for complete conversion can be ex- pressed in terms of the optical depth D c , and F (D c ) is shown in Fig. 3 for three different values of ε Γ . The maximum efficiency is attained at an optical depth D max c = π/(2 √ ε Γ ) and tends to unity for ε Γ → 0. Since ε Γ ∝ Γ, efficiencies close to unity are only possible because of the slow radiative decay rate Γ of the Rydberg levels |3 , |4 and |5 . Γ decreases with increasing n as Γ ∝ n −3 [46] and is thus typically several orders of magnitude smaller than the decay rate γ of the low-lying states |2 and |6 . The efficiency for complete MMOC for the parameters in Fig. 2 is F ≈ 92.1%. Note that our definition of the efficiency is based on photon fluxes and not intensities as required for a coherent conversion scheme that conserves the total photon flux. In order to see this, we consider the perfectly coherent conversion of an optical field to a millimeter wave with F (D c ) = 1. According to Eq. (25), we obtain |Ω M (L c )| 2 = b 2 |Ω L (0)| 2 . With the definition of b in Eq. (8) and the definition of the Rabi frequencies we get where E M and E M are the electric field amplitudes of the mm-wave and optical fields, respectively. The ratio of the intensities (I ∝ |E| 2 ) is thus I out M /I in L = ω M /ω L , as it should be. Similarly, we obtain I out L /I in M = ω L /ω M for the conversion of mm-waves into optical fields. Next we consider the conversion of pulsed fields. The derivation of Eq. (25) shows that our scheme is not modeselective and works for broadband pulses. The only requirement is that the atomic dynamics remains in the adiabatic regime, which holds if the bandwidth δν of the input pulse is smaller than all detunings ∆ k and the Rabi frequencies Ω R , Ω C and Ω A (see Sec. A). In order to demonstrate this, we present numerical solutions of the Maxwell-Bloch equations for a mm-wave input pulse as shown in Fig. 4. The intensity of a mm-wave input pulse with Gaussian envelope is shown in Fig. 4(a), and the corresponding optical output field is shown in Fig. 4(b). The input pulse has a bandwidth on the order of ∆ν ≈ 2π × 80 kHz and is converted without distortion of its shape. We thus find that the bandwidth of our conversion scheme is at least ∼ 80 kHz for the chosen parameters. This bandwidth can be significantly increased by increasing the detunings and Rabi frequencies of the auxiliary fields. Finally, we note that the conversion of optical pulses to mm-waves works equally well.

B. Rubidium parameters
Here we discuss one possible realisation of our scheme based on an ensemble of 87 Rb atoms. The atomic level scheme is shown in Fig. 5, where the optical field L couples to the D 2 line, and the auxiliary P field couples to the D 1 line. The transition dipole matrix elements for the optical transitions can be found in [47], and for transitions between Rydberg states we follow the approach described in [48]. The intensities of the auxiliary fields are chosen such that they correspond to the Rabi frequencies in Fig. 2, and the values of the detuning parameters in Figs. 5 and 2 are also equivalent. Next we show that Rydberg-Rydberg interactions are negligible for the level scheme in Fig. 5 and for an atomic density of N = 2 × 10 17 m −3 . First we note that the Rydberg blockade radius is R b ≈ 0.63µm for the parameters of Fig. 5. This is significantly smaller than the mean distance between atoms, and hence the density of Rydberg atoms is simply given by N Ry ≈ ρ 33 N [49,50]. By carrying out the average in Eq. (18), we find that the matrix M leads to the same conversion efficiency as M, i.e., there is no notable difference between the curves in  Fig. 2. The decay rate γ = 2π × 6.1MHz corresponds to the D2 line. We set the decay rate Γ of all Rydberg states equal to the decay rate of the |23S 1/2 state at T = 300K, which is faster than the decay rate of the |24P 1/2 state. We find [46] Γ/γ = 1/285 and the ratio of the coupling constants is b = √ 0.72.
duced with M. On the other hand, if we choose |3 = |24S 1/2 , m J = 1/2 instead of |3 = |23S 1/2 , m J = 1/2 , the conversion efficiency drops to 61%. In order to obtain more insight into these results, we consider the distance R 90 where 90% of all Rydberg atom pairs will have a larger separation than R 90 , This is much smaller than all detuning parameters and Rabi frequencies entering the matrix M. Since the frequency shifts for 90% of all atoms are even smaller, averaging over all nearest neighbour distances does not change the matrix M. Similarly, the dipole-dipole shifts in Eq. (16) with |3 = |24S 1/2 , m J = 1/2 are on the order of which is also small compared to the detuning parameters and Rabi frequencies of the auxiliary fields. On the other hand, ∆ DD increases by a factor of 100 by choosing |3 = |24S 1/2 , m J = 1/2 instead of |3 = |23S 1/2 , m J = 1/2 . This explains why the conversion efficiency drops significantly by using the strong |ns ↔ |np transition instead of |(n − 1)s ↔ |np as in Fig. 5. The absorption length for the field L is l abs = 5.1 × 10 −2 mm for our parameters. Since full conversion requires an optical depth of ∼100, the length of the medium needs to be L c ≈ 5.1mm. These parameters are experimentally achievable. For example, much higher optical depths ∼1000 have been reported [51,52], and the atomic cloud size considered here is similar to the dimensions of the experiment in [52], where cold Rb atoms were trapped in a cylindrical geometry of length 4.6 mm and width 0.45 mm.

C. Physical implementation
We first consider the setup in Fig. 1(a) where the mmwave field is focussed into the atomic ensemble by lenses. We assume that the focal spot is at z = 0 such that the mm-wave beam profile is where r = x 2 + y 2 is the radial coordinate in the x-y plane, σ M is the beam waist and Ω frequency at the center of the beam. We model the transverse density profile of the atom cloud by a Gaussian with peak density N (0) and width σ c , In order to calculate the conversion efficiency, we find the stationary solution of Eq. (6) with the boundary condition in Eq. (31), the density profile in Eq. (32) and with the analytical expression for the atomic coherences in Eq. (23). The result for the parameters specified in Sec. III B is shown in Fig. 6, where we consider a beam waist of σ M = 1.9λ M and an atomic cloud with transverse size σ c ≈ 413 µm. The intensity of the millimeter wave is shown in Fig. 6(a) and decreases due to the conversion mechanism. In addition, it broadens slightly with increasing z which can be understood as follows. For the given parameters the Rayleigh length z M = πσ 2 M /λ M of the mm-wave is z M ≈ 11.3λ M , which is about half the length of the medium. The broadening is thus caused by the strong focussing of the beam before it enters the atomic ensemble. Note that the Rayleigh length is much larger than the wavelength λ M , and hence the paraxial approximation is justified. The intensity of the optical wave is shown in Fig. 6(a) and increases with increasing z. In order to quantify the conversion efficiency, we consider the total power of the incoming mm-wave and of the outgoing optical field, where ε 0 is the dielectric constant. We then define the conversion efficiency by which is consistent with our definition of the conversion efficiency in Sec. III A. We find F ≈ 26% for the parameters in Fig. 6, and this value can be further increased by increasing the transverse size of the atomic cloud. For example, for an atomic ensemble with transverse size σ c ≈ 1mm we obtain F ≈ 61%. In addition, the conversion of optical fields to mm-waves works equally well. For a Gaussian optical beam of width σ L ≈ 509 µm and all other parameters as in Fig. 6, we find F ≈ 24%. This value increases to F ≈ 72% if the atomic cloud size is increased to σ c ≈ 1mm. However, increasing the transverse size of the atomic ensemble requires auxiliary fields with higher power in order to maintain the intensities shown in Fig. 5. Next we discuss the implementation shown in Fig. 1(b), where the mm-waves are confined by a waveguide and an elongated atomic cloud is trapped inside the waveguide core. This setting can be approximately described by the one-dimensional model in Eq. (10) if the ratio of the coupling constants in Eq. (8) is replaced by where A M is the effective area of the mm-wave guided mode, and A L is the transverse size of the optical beam which is assumed to match the transverse density profile of the atoms [35,53]. In principle, the setup in Fig. 1(b) can thus be employed to interconvert mm-waves with longer wavelengths that cannot be focussed down to realistic dimensions of cold atom clouds. However, since A L /A M 1, this results in smaller values of the parameter ε ∝ b wg defined in Eq. (24) and thus in larger values of the optical depth required for complete conversion, D c = π/(2ε). In order to achieve the required optical depths, the atoms could be confined inside hollow core fibres where extremely large optical depths have been observed [54,55]. In addition, mm-waves can similarly be guided by photonic crystal fibres [56]. The strong coupling of atoms with mm-wave and optical fields required for efficient conversion might then be achievable by em-bedding a small hollow-core photonic crystal fibre into a larger mm-wave photonic crystal fibre.

IV. SUMMARY
We have shown that frequency mixing in Rydberg gases enables the coherent conversion between mm-wave and optical fields. Due to the numerous possibilities for choosing the |3 ↔ |4 transition within the Rydberg manifold, our proposed MMOC scheme enables the conversion of various frequencies ranging from terahertz radiation to the microwave spectrum, that is for frequencies in the range 10 − 10, 000 GHz. The degree of conversion can be adjusted through the atomic density and the ancillary drive field intensities and frequencies. Conversion efficiencies are limited by the lifetime of the Rydberg levels and dipole-dipole interactions between Rydberg atoms. Imperfections due to Rydberg interactions can be minimised in ensembles with low atomic densities and by the choice of the atomic states and parameters of the auxiliary fields. We have analysed a realistic implementation for the interconversion of terahertz and optical fields with an ensemble of trapped rubidium atoms, and find that the conversion efficiency can exceed 90%. Efficient conversion requires a large spatial overlap between the mm-wave and optical fields, and we have discussed two possible scenarios how to achieve this. First, we have considered focussed terahertz beams and found that high conversion efficiencies are possible if the Rayleigh length of the beams is comparable to the length of the atomic cloud. Second, we investigated a setup where the mm-wave fields are transversally confined by a waveguide and the atoms are trapped inside the waveguide core. The optical depth required for complete conversion increases by A M /A L compared to the free-space implementation, where A M is the effective area of the mm-wave guided mode and A L is the transverse size of the atomic cloud. This waveguide setting enables high conversion efficiencies close to the theoretical limit set by the lifetime of the Rydberg states and Rydberg interactions. Here we derive the adiabatic solutions for the atomic coherences 43 and 61 in Eq. (9). To this end, we assume that the fields Ω M and Ω L are sufficiently weak and expand the atomic density operator as follows [57,58],

ACKNOWLEDGMENTS
where (k) denotes the contribution to in k th order in the Hamiltonian The solutions (k) can be obtained by re-writing the master equation (1) as where the linear super-operator L 0 is independent of Ω M and Ω L . Inserting the expansion (A1) into Eq. (A3) leads to the following set of coupled differential equationṡ Equation (A4) describes the interaction of the atom with the fields Ω P , Ω R , Ω C and Ω A to all orders and in the absence of H 1 . Higher-order contributions to can be obtained if Eq. (A5) is solved iteratively. Equations (A4) and (A5) must be solved under the constraints Tr( (0) ) = 1 and Tr( (k) ) = 0 (k > 0). The zeroth-order solution (0) is the EIT dark state of the three-level ladder system |1 , |2 and |3 . For the special case ∆ 3 = 0 and if the small decay rate Γ of state |3 is neglected, we find For |Ω P | |Ω R | the steady state is reached within several inverse decay times 1/γ. In general, we obtain the zeroth-order solution (0) for ∆ 3 = 0 and substitute it in the first-order equation (A5) with k = 1. The formal solution of this differential equation is given by (1) where we assumed H 1 (0) = 0. If H 1 (t) varies sufficiently slowly with time, the second term on the right-hand side in Eq. (A7) involving the time derivative of H 1 can be neglected. More precisely, this approximation is justified if the bandwidth δ ν of the pulses Ω M and Ω L is small as compared to the relevant differences between eigenfrequencies of H 0 . Through a numerical study we find that this condition is satisfied if all detunings ∆ k (k ∈ {4, 5, 6}) and the Rabi frequencies Ω R , Ω C and Ω A are large as compared to the bandwidth δ ν . In general, the analytical expression for the first-order density operator is too bulky to display here. A special solution if the conditions in Eq. (22) are met is given in Eq. (23).

Supplemental Material for:
Two-way interconversion of millimeter-wave and optical fields in Rydberg gases

Detailed model
Here we derive the Maxwell-Bloch equations for our system from first principles. The electric field amplitude of the millimeter wave is E M , and the optical field is denoted by E L . The other fields E P , E R and E C are auxiliary fields facilitating the frequency conversion. We decompose all electric fields as (X ∈ {P, R, M, C, L}) where e X , E X and ω X is the unit polarisation vector, envelope function and central frequency of field E X , respectively (X ∈ {P, R, C, A}). In order to simplify the notation, we introduce atomic transition operators In electric-dipole and rotating-wave approximation, the Hamiltonian of each atom interacting with the six laser fields is where ω k denotes the energy of state |k with respect to the energy of level |1 . The matrix element of the electric dipole moment operatord on the transition transition |k ↔ |l is defined as We model the time evolution of the atomic system by a master equation for the reduced density operator R, The last term in Eq. (A14) describes spontaneous emission and is given by While the ground states |1 is assumed to be (meta-) stable, the states |2 , |3 , |4 and |5 decay through spontaneous emission. The decay rate γ is the full decay rate of states |2 and |5 , and Γ is the decay rate on the Rydberg transitions. In our scheme, Γ is much smaller than the decay rate γ of the low-lying electronic states. In order to remove the fast oscillating terms in Eq. (A14), we transform the latter equation into a rotating frame W = exp {i[ ω P A 22 + (ω P + ω R )A 33 + (ω P + ω R + ω M )A 44 + (ω P + ω R + ω M − ω C )A 55 +(ω P + ω R + ω M − ω C − ω A )A 66 ]t} exp {−i[ k P · rA 22 + (k P + k R ) · rA 33 + (k P + k R + k M ) · rA 44 We assume that the central frequencies of all fields are resonant with the loop |1 ↔ |2 ↔ |3 ↔ |4 ↔ |5 ↔ |6 ↔ |1 , In addition, we impose the phase matching condition