Corrigendum: Two-way interconversion of millimeter-wave and optical fields in Rydberg gases (2016 New J. Phys. 18 093030)

We show that cold Rydberg gases enable an ef ﬁ cient six-wave mixing process where terahertz or microwave ﬁ elds are coherently converted into optical ﬁ elds 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. 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 ﬁ elds con ﬁ ned by a waveguide, respectively. We analyse a realistic example for the interconversion of terahertz and optical ﬁ elds in rubidium atoms and ﬁ nd that the conversion ef ﬁ ciency can in principle exceed 90%.

. Realisation of the level scheme in figure 1 based on transitions in 87 Rb. All quantum numbers of the employed states as well as intensities, polarisations and detunings are indicated. Note that energy spacings are not to scale. The intensities and detunings correspond to the parameters in figure 2. The decay rate ɣ = 2π × 6.1 MHz corresponds to the D 2 line. We set the decay rate Γof all Rydberg states equal to the decay rate of the | ⟩ S 23 1 2 / state at T = 300 Kwhich is faster than the decay rate of the | P 24 1 2

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 microwave-optical 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 nonlinearities 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 figure 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 microwaves [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 electromagnetically induced transparency (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 section 2, where we also describe how to include interactions between Rydberg atoms. In section 3 we discuss the principle of operation of our scheme and show that both timeindependent 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 section 4. ¶ = - Figure 1.  In the electric-dipole and rotating-wave approximations, the Hamiltonian H in equation (1) (2) is defined as where w  k denotes the energy of state ñ |k with respect to the energy of level ñ |1 and w X is the frequency of field X with Rabi frequency W X ( Î { } X P, R, C, A, M, L ). The term g   in equation (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 ñ |2 and ñ |6 is γ, and the long-lived Rydberg states decay with g G  . The six fields drive a resonant loop and we impose the phase matching condition In the following, we assume that W M and W L are co-propagating, while the directions of the auxiliary fields are chosen such that equation (5) holds. Note that this phase matching condition is automatically fulfilled by virtue of equation (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 k M L is the wavenumber of the mm-wave (optical) field and D = ¶ + ¶ is the matrix element of the electric dipole moment operatord on the transition transition ñ « ñ | | k l , c is the speed of light and  is the density of atoms. In the following the ratio of the coupling constants is denoted by . 3, but first it is instructive to derive analytic solutions in the limit that diffraction over the length of the atomic ensemble can be neglected.

Analytical solution
The first-order solution of equation (1) with respect to the Rabi frequencies W M , W L takes the form (see appendix) The response of the atomic system on the ñ « ñ | | 3 4 transition induced by the mm-wave field is described by c 43 M , and c 61 L accounts for the atomic response on the transition ñ « ñ | | 1 6 due to the optical field. In addition, the mm-wave field can induce a coherence proportional to c 61 M on the optical transition ñ « ñ | | 1 6 , and the optical field can create a coherence proportional to c 43 L on the transition ñ « ñ | | 3 4 . The cross-terms proportional to c 43 L and c 61 M in equation (9) originate from the closed-loop character of the atomic level scheme. Next we combine equation (9) with (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 t = -( ) 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 When the auxiliary fields are time-independent and spatially uniform, the solution to equation (10) is where W 0 is the initial condition W evaluated at z=0. The matrix exponential in equation (12) can be expressed in terms of the 2×2 identity matrix  and the Pauli matrices s k [33] The solution presented here treats the signal fields W M , W L as c-numbers. However, the generalisation to quantum fields is straightforward since the coherences in equation (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 W M , W 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.

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 figure 1(b) contains three Rydberg states, and the population in state ñ |3 is continuously kept at r » W W | | 33 P R 2 via coherent population trapping. On the other hand, the population in the other Rydberg states ñ |4 and ñ |5 is negligibly small for weak fields W M and W L . The dominant perturbation to the conversion mechanism will thus stem from nearby Rydberg atoms in state ñ |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 ñ |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 ñ where the coefficient C 6 depends on the quantum numbers of state ñ |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 g g = W | | EIT R 2 and given by g 6 EIT 1 6 [38]. Second, atom B gives rise to a frequency shift of state ñ |4 in atom A via the resonant dipole-dipole interaction [39] pe In contrast to the van der Waals shift in equation (15), D DD depends on the relative orientation of the two atoms. In principle, state ñ |5 in atom A can experience a similar shift D DD if the dipole moment d 53 is different from zero. Here we assume that states ñ |5 and ñ |3 have the same parity so that = d 0 53 , consistent with the example implementation in Rb that we introduce below in section 3.2.
The preceding discussion shows that Rydberg-Rydberg interactions change the energies of states ñ |3 and ñ |4 . In order to incorporate these frequency shifts into our model, we find the general first-order solution of the atomic coherences in equation (9) for arbitrary detunings and Rabi frequencies of the auxiliary fields. We then introduce the effective detuning parameters and replace D 3 and D 4 in the general expression for the matrix  in equation (11) by D 3 and D 4 . Since D vdW and D DD depend on the relative position R, we average  over R where the distribution of nearest neighbours in a random sample of Rydberg atoms follows the probability density [40]  the Wigner-Seitz radius for a given density of Rydberg atoms  Ry . This account of Rydberg-Rydberg interactions is expected to work well for weak optical and mm-wave fields. If the intensities of W M and W L are increased such that the population in ñ |4 and ñ |5 is not negligible, other dipole-dipole 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.

Results
In a first step we analyse the simplified analytical model of section 2.2 in order to explain the principle of the conversion mechanism. This is presented in section 3.1 where we also investigate the maximally achievable conversion efficiencies. We then introduce one possible implementation of our scheme in rubidium vapour in section 3.2 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 figures 1 (a) and (b) in section 3.3.

Conversion mechanism
The conversion efficiency between mm-wave and optical fields according to equation (12) will be small for a generic matrix , 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  vanish. We find that this condition, such that c c » » 0 43 M 61 L , can indeed be met if the intensities and detunings of the auxiliary fields satisfy


To first order in g G the susceptibilities in equation (9) are then given by ε and e G are dimensionless parameters that are generally smaller than unity. Since e µ G G , e G is typically of the order of e 2 . On the other hand, a e µ | | and hence the off-diagonal elements of the matrix  are indeed much larger than the diagonal elements.
This result can be understood as follows. The level scheme in figure 1 can be regarded as three consecutive EIT systems where the weak probe fields are represented by W P , W M and W L , respectively. However, these three systems are coupled and hence the normal two-photon resonance condition for transparency of the W M and W L fields is changed. The conditions in equation (22)   , efficiencies close to unity are only possible because of the slow radiative decay rate Γ of the Rydberg levels ñ |3 , ñ |4 and ñ |5 . Γ decreases with increasing n as G µn 3 [46] and is thus typically several orders of magnitude smaller than the decay rate γ of the low-lying states ñ |2 and ñ |6 . The efficiency for complete MMOC for the parameters in figure 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 Next we consider the conversion of pulsed fields. The derivation of equation (25) shows that our scheme is not mode-selective and works for broadband pulses. The only requirement is that the atomic dynamics remains in the adiabatic regime, which holds if the bandwidth dn of the input pulse is smaller than all detunings D k and the Rabi frequencies W R , W C and W A (see appendix). In order to demonstrate this, we present numerical solutions of the Maxwell-Bloch equations for a mm-wave input pulse as shown in figure 4. The intensity of a mm-wave input pulse with Gaussian envelope is shown in figure 4(a), and the corresponding optical output field is shown in figure 4(b). The input pulse has a bandwidth on the order of n p D »2 80kHz 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.

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 figure 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 figure 2, and the values of the detuning parameters in figures 5 and 2 are also equivalent.
Next we show that Rydberg-Rydberg interactions are negligible for the level scheme in figure 5 and for an atomic density of =´- 2 10 m 17 3 . First we note that the Rydberg blockade radius is m for the parameters of figure 5. This is significantly smaller than the mean distance between atoms, and hence the density of Rydberg atoms is simply given by r »   Ry 33 [49,50]. By carrying out the average in equation (18), we find that the matrix leads to the same conversion efficiency as , i.e., there is no notable difference between the curves in figure 2 generated by  and the corresponding curves produced with. On the other hand, if we , 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 Our parameters give »´- 6.1MHz corresponds to the D 2 line. We set the decay rate Γ of all Rydberg states equal to the decay rate of the ñ | S 23 1 2 state at T=300K, which is faster than the decay rate of the ñ | P 24 1 2 state. We find [46] g G = 1 285 and the ratio of the coupling constants is which is consistent with our definition of the conversion efficiency in section 3.1. We find » F 26% for the parameters in figure 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 s » 1 mm c we obtain » F 61%. In addition, the conversion of optical fields to mm-waves works equally well. For a Gaussian optical beam of width s m » 509 m L and all other parameters as in figure 6, we find » F 24%. This value increases to » F 72% if the atomic cloud size is increased to s » 1mm c . However, increasing the transverse size of the atomic ensemble requires auxiliary fields with higher power in order to maintain the intensities shown in figure 5.
Next we discuss the implementation shown in figure 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 equation (10) if the ratio of the coupling constants in equation (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 figure 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 A 1 L M  , this results in smaller values of the parameter e µ b wg defined in equation (24) and thus in larger values of the optical depth required for complete conversion, 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, mmwaves 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 embedding a small hollow-core photonic crystal fibre into a larger mm-wave photonic crystal fibre.

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 ñ « ñ | | 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 A M 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. [57,58]  The solutions ( )  k can be obtained by re-writing the master equation (1) as where the linear super-operator  0 is independent of W M and W L . Inserting the expansion(A1) into equation (A3) leads to the following set of coupled differential equations  varies sufficiently slowly with time, the second term on the right-hand side in equation (A7) involving the time derivative of H 1 can be neglected. More precisely, this approximation is justified if the bandwidth d n of the pulses W M and W 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 D k ( Î { }) k 4, 5, 6 and the Rabi frequencies W R , W C and W A are large as compared to the bandwidth d n . In general, the analytical expression for the first-order density operator  is too bulky to display here. A special solution if the conditions in equation (22) are met is given in equation (23a).