Microwave-to-optical conversion in a room-temperature ⁸⁷Rb vapor for frequency-division multiplexing control

Abstract

Coherent microwave-to-optical conversion is crucial for transferring quantum information generated in the microwave domain to optical frequencies, where propagation losses can be minimized. Coherent, atom-based transducers have shown rapid progress in recent years. This paper reports an experimental demonstration of coherent microwave-to-optical conversion that maps a microwave signal to a large, tunable 550(30) MHz range of optical frequencies using room-temperature ⁸⁷Rb atoms. The inhomogeneous Doppler broadening of the atomic vapor advantageously supports the tunability of an input microwave channel to any optical frequency channel within the Doppler width, along with the simultaneous conversion of a multi-channel input microwave field to corresponding optical channels. In addition, we demonstrate phase-correlated amplitude control of select channels, providing an analog to a frequency domain beam splitter across five orders of magnitude in frequency. With these capabilities, neutral atomic systems may also be effective quantum processors for quantum information encoded in frequency-bin qubits.

Introduction

Quantum information processing using solid-state qubits, such as superconducting qubits or spins associated with defects in solid state systems, has shown rapid progress in the field of quantum information technology^1,2. The typical operation frequency of these devices is ~10 GHz, in the microwave region of the electromagnetic spectrum. Due to the inherent room-temperature thermal-photon occupation of cables at GHz frequencies that leads to signal noise, practical implementation of scalable and distributed quantum networks at these frequencies poses a formidable challenge³. In recent years, progress has been made in addressing coherent conversion of signals from the microwave to optical regimes^4,5: at optical frequencies, not only are the thermal noise photons negligible at room temperature, but there exist efficient single-photon detectors and technologies for quantum state storage and reconstruction.

Several systems have been used to coherently convert signals at microwave frequencies to the optical regime. These include electro- and magneto-optical devices^6,7, opto-mechanical structures,⁸ and atoms with suitable internal energy level spacings^9,10,11,12. Impressive progress on improving the efficiency of the conversion process led to efficiency near 80%¹¹, and the largest reported conversion bandwidths of an input microwave signal are 15–16 MHz^12,13,14. Along with conversion bandwidth, a transducer that can accommodate several channels of input and output frequencies is advantageous, and tunability over input microwave frequencies across 3 GHz width was recently reported¹⁵. Yet, transducers that operate over multiple optical output frequencies have not been demonstrated to date.

In this article we show that a limited bandwidth input microwave signal can be converted to a large tunable range of output optical frequencies, across 550(30) MHz. This capacity enables information encoded in a narrow-bandwidth microwave field to be mapped to several frequencies in the optical domain, achieving coherent frequency-division multiplexing (FDM). The ability to tune the optical signal output over such a large frequency range arises because of the presence of a large inhomogeneous optical Doppler width for Rb atoms at room temperature. The Doppler width also enables an input multiple-channel microwave idler to be simultaneously and coherently converted to a multiple-channel optical-signal output. In addition, we demonstrate correlated amplitude control of the converted multi-channel optical field with the ability to selectively extinguish a desired output frequency channel. This action is very similar to the frequency domain beam-splitting operation for frequency bins, demonstrated using electro-optic modulators (EOMs) and pulse shapers¹⁶.

Results

Coherent frequency-division multiplexing enables qubits encoded in a microwave frequency to be placed in any desired output optical channel within the range of tunability. Multiple qubits in different input microwave channels can also be coherently mapped to the optical domain. The ability to control the amplitude of selected output optical channels is very useful in performing single-qubit-gate operations spanning involving both microwave and optical frequencies. With a channel linewidth of about 30 Hz, this results in a large channel capacity of about 10⁷ channels over the output tunable range. Thus neutral atom systems are a promising candidate as a quantum frequency processor platform¹⁷ for quantum information encoded in frequency bin qubits.

Numerical model

Microwave-to-optical conversion in a room-temperature ⁸⁷Rb atomic vapor is possible through a sum-frequency generation (SFG) process analogous to a RF mixer [Fig. 1a] and depicted in Fig. 1c. The nonlinear process uses a hybrid, magneto-optical second-order [χ⁽²⁾] susceptibility induced in the vapor by the interaction of three energy levels with an optical pump (electric-dipole transition) and a microwave idler (magnetic-dipole transition) fields. (Note that isotropic atomic systems with only electric-dipole transitions have a χ⁽²⁾ of zero.) The idler field (ω_I) is supported by the TE₀₁₁ mode of a cylindrical copper microwave cavity^18,19 [Fig. 1b] and connects the $\vert b\rangle \equiv \vert 5{S}_{1/2},F=1\rangle$ and $\vert c\rangle \equiv \vert 5{S}_{1/2},F=2\rangle$ ground state levels. The pump frequency ω_P is, chosen anywhere within the Doppler width [Fig. 1c], and connects the $\left\vert c\right\rangle$ and $\vert a\rangle \equiv \vert 5{P}_{3/2},{F}^{{\prime} }=1,2\rangle$ levels. We define δ as the detuning of the pump field from the the mid-energy point between the $\left\vert {F}^{{\prime} }=1\right\rangle$ and $\left\vert {F}^{{\prime} }=2\right\rangle$ excited levels. In SFG, the pump combines coherently with the idler to generate a new optical signal field at frequency ω_S = ω_P + ω_I.

Fig. 1: Main components of the experiment.

a Schematic of the atomic frequency-division multiplexing scheme. The microwave cavity-vapor cell system acts analogously to an RF mixer, and information encoded in idler (yellow) channels (▲, ●, ■) is combined coherently with multiple pump (red) channels to produce many output signal (blue) channels over a range of optical frequencies. b A representation of our microwave cavity and the generation process. The optical pump beam passes transversely through the cavity and cell and is polarized along the z-axis, parallel to a weak magnetic field. The cavity’s TE₀₁₁ mode overlaps with the pump interaction region inside the vapor cell. The arrows define the beam propagation and polarization directions. c Three-level energy diagram showing the cyclical transition scheme. The idler (ω_I) and pump (ω_P) fields combine coherently in the vapor to produce the signal field (ω_S). The frequencies shown are for ⁸⁷Rb; in a room-temperature vapor, the excited state addressed by the pump field is Doppler-broadened by ~550 MHz.

We begin by establishing a numerical model that explains the large frequency tunability and multi-channel conversion capability observed in this experiment, similar to that discussed for a generic, three-level, room-temperature atomic system, with interacting idler, pump, and signal fields²⁰. The Rabi frequencies of the idler and pump fields at position z are Ω_I(z) = μ_cb ⋅ B_I(z) and Ω_P(z) = d_ac ⋅ E_P(z), where Δφ is the relative phase between the fields, and μ_cb and d_ac are the dipole matrix elements between $\left\vert c\right\rangle \,\& \,\left\vert b\right\rangle$ and $\left\vert a\right\rangle \,\& \,\left\vert c\right\rangle$ levels. The signal field is generated due to a sum-frequency interaction between the idler and pump fields. The Rabi frequency of the generated signal field Ω_S(ℓ) after a medium of length ℓ, under a rotating wave and three-photon transformation, is (see Supplementary Note 4)

$${\Omega }_{{{{{{{{\rm{S}}}}}}}}}(\ell )=\frac{i{\Omega }_{{{{{{{{\rm{P}}}}}}}}}(0){\Omega }_{{{{{{{{\rm{I}}}}}}}}}(0){e}^{-i\Delta \varphi }}{2{\Gamma }_{2}}({e}^{-\beta \ell }-1),$$

(1)

where the constant

$$\beta =\frac{2{\eta }_{{{{{{{{\rm{S}}}}}}}}}{\Gamma }_{2}{\Gamma }_{3}}{4{\Gamma }_{1}{\Gamma }_{2}{\Gamma }_{3}+{\Gamma }_{3}| {\Omega }_{{{{{{{{\rm{P}}}}}}}}}(0){| }^{2}+{\Gamma }_{2}| {\Omega }_{{{{{{{{\rm{I}}}}}}}}}(0){| }^{2}},$$

(2)

and

$${\Gamma }_{1}= \frac{{\gamma }_{{{{{{{{\rm{P}}}}}}}}}}{2}+\frac{{\gamma }_{{{{{{{{\rm{S}}}}}}}}}}{2}-i\,{\Delta }_{{{{{{{{\rm{S}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}}),\\ {\Gamma }_{2}= \frac{1}{2}{\gamma }_{{{{{{{{\rm{I}}}}}}}}}-i\left[{\Delta }_{{{{{{{{\rm{S}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}})-{\Delta }_{{{{{{{{\rm{P}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}})\right],\\ {\Gamma }_{3}= \frac{{\gamma }_{{{{{{{{\rm{P}}}}}}}}}}{2}+\frac{{\gamma }_{{{{{{{{\rm{S}}}}}}}}}}{2}+\frac{{\gamma }_{{{{{{{{\rm{I}}}}}}}}}}{2}-i\,{\Delta }_{{{{{{{{\rm{P}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}}),\\ {\eta }_{{{{{{{{\rm{S}}}}}}}}}(v)= {{{{{{{{\boldsymbol{d}}}}}}}}}_{{{{{{{{\rm{ac}}}}}}}}}^{2}{\omega }_{s}N(v)/2{\epsilon }_{0}c,$$

(3)

with N(v) is the number density of atoms in the velocity class with v = ∣v∣. The value N(v) is the Maxwell-Boltzmann (MB) velocity distribution of atoms at room temperature. The Doppler-shifted detunings of the pump and the generated signal fields, as seen from the atom frame [Fig. (2a, iii–v)], are denoted as Δ_P = δ_P − k_P ⋅ v and Δ_S = δ_S − k_S ⋅ v, while the corresponding detunings in the lab frame are δ_P & δ_S [Figs. 1c and 2a, i]. The microwave idler field has negligible Doppler shift for all velocity values. The natural decay rates from the $\left\vert a\right\rangle$ level to both the $\left\vert c\right\rangle$ and $\left\vert b\right\rangle$ levels are denoted by γ_P and γ_S, respectively [Fig. 1c].

Fig. 2: Numerical simulations.

a i Lab-frame cartoon depicting multiple pump frequency channels (central-green, upper-purple, lower-organge) combining with a single idler (yellow) to produce multiple signal channels. a ii These pump frequencies access independent atomic velocity classes from a thermal MB distribution. (a-iii to a-v) In their reference frame, atoms in these velocity classes see the respective pump frequency Doppler-shifted to resonance. Generation occurs in the velocity class for which the three-photon resonance condition (i.e. energy conservation) is satisfied. b Calculated spectra of an amplitude-modulated idler field showing sidebands (SBs); the carrier and SBs combine with the pump to produce a multichannel signal spectrum. c When the idler and pump are both modulated at the same frequency, multiple generation pathways interfere producing an upper and lower generated signal SBs. The input fields each have lower (▲), central (●), and upper (■) spectral peaks. d Calculated curves showing correlated signal SB amplitudes, which vary coherently with the relative phase of the two modulations depicted in (c). Both AM and PM modulation indices are 0.2.

To maximize the generated signal field, sufficiently high input pump and idler field intensities and a small value for Γ₂ [Eq. (1)] are required. From Eq. (3) this is achieved when

$${\Delta }_{{{{{{{{\rm{S}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}})-{\Delta }_{{{{{{{{\rm{P}}}}}}}}}({{{{{{{\boldsymbol{v}}}}}}}})=0.$$

(4)

Typically, the wavelengths of optical fields are such that k ≡ ∣k_S∣ ≈ ∣k_P∣ ≫ ∣k_I∣, in which case Eq. (4) reduces to (δ_S − δ_P) = 0. Since the signal and pump fields travel in the same direction, the phase-matching condition k_I + k_P = k_S is automatically satisfied for all atomic velocity classes.

The maximum signal generation for each velocity class is obtained when δ_S of the signal-field is the same as the pump-field detuning δ_P, such that different velocity classes participate in the maximum signal-field generation at different pump-field detunings. Therefore, changing the laboratory pump frequency within the Doppler width to ${\omega }_{{{{{{{{\rm{P}}}}}}}}}^{{\prime} }={\omega }_{{{{{{{{\rm{P}}}}}}}}}+\delta$ gives rise to a corresponding signal field at ${\omega }_{{{{{{{{\rm{S}}}}}}}}}^{{\prime} }={\omega }_{{{{{{{{\rm{S}}}}}}}}}+\delta$, thus enabling a large tunable generation bandwidth. This is depicted numerically in Fig. 2a, ii for atoms at T = 300 °C. The shape of the generated signal field is primarily determined by the MB velocity distribution.

The broad Doppler width at room temperature and guaranteed phase-matching for sum-frequency generation [Eq. (4)] make possible the simultaneous conversion of multi-channel microwave field inputs to a multi-channel optical signal outputs [Fig. 2b]. Multiple frequency channels in the idler and pump fields are obtained by either amplitude or phase modulation. Specifically, the idler field amplitude is modulated as ${\Omega }_{{{{{{{{\rm{I}}}}}}}}}(t)={\Omega }_{{{{{{{{\rm{I0}}}}}}}}}(1+{a}_{m}\sin ({\omega }_{{{{{{{{\rm{AM}}}}}}}}})t)$ and phase modulation of the pump field takes the form ${\Omega }_{{{{{{{{\rm{P}}}}}}}}}(t)={\Omega }_{{{{{{{{\rm{P0}}}}}}}}}\sin ({\omega }_{{{{{{{{\rm{P}}}}}}}}}t+{p}_{m}\sin ({\omega }_{{{{{{{{\rm{PM}}}}}}}}})t+\Delta \phi )$. The symbols a_m and p_m denote amplitude and phase modulation indices. The modulation frequencies are ω_AM and ω_PM and Δϕ denotes the relative phase between the two modulation fields. By multiplying the modulated idler and pump fields to obtain the signal field, we derive the amplitudes for the generated signal field and its first-order side-bands for the case of equal modulation frequencies ω_AM = ω_PM = δ and for equal modulation indices a_p = a_m = 0.2. At equal modulation frequencies, the pump and idler fields associated with the sideband channels interfere with each other [Fig. 2c] enabling control of their amplitudes through Δϕ²¹. The powers of the positive and negative sideband channels at ω_S ± δ are proportional to ${a}_{m}^{2}+{a}_{p}^{2}+2{a}_{m}{a}_{p}\sin (\Delta \phi )$ and ${a}_{m}^{2}+{a}_{p}^{2}-2{a}_{m}{a}_{p}\sin (\Delta \phi )$ (see Supplementary Note 7). The power variations at ±δ sidebands, as a function of Δϕ, are shown in Fig. 2d.

Experimental results

First, we show that light is generated on the $\left\vert a\right\rangle \to \left\vert b\right\rangle$ transition. The χ⁽²⁾ interaction mediated by these atoms between the pump and microwave fields results in a sum-frequency process that generates a corresponding signal optical field. By combining the generated optical light with an optical local oscillator, we detect the generated light at the heterodyne beat frequency f_het,S (see Methods and Fig. 3a).

Fig. 3: Experimental setup and main results.

a Schematic of the experimental setup: Central to the experiment is a microwave cavity supporting a TE₀₁₁ mode, polarized along the y-axis. The light incident on the cavity is polarized along the z-axis and propagates along the x-axis. The vapor cell inside the cavity has enriched ⁸⁷Rb. Unshifted light from the external cavity diode laser (ECDL) is used as a local oscillator for heterodyne detection; it is combined with the other beam in a fiber 50:50 beam splitter (BS), producing two beat signals at f_het,P and f_het,S. AOM - Acousto-optic modulator; EOM - Electro-optic modulator; SG - signal generator; SA - spectrum analyzer; PD - photodiode. b The generated signal measured with the spectrum analyzer in zero-span mode, showing a FWHM of 550(25) MHz. The dashed curve shows the calculated Doppler-broadened absorption coefficient of the pump light; the individual components due to the ${F}^{{\prime} }=1$ and ${F}^{{\prime} }=2$ excited hyperfine levels are shown as dotted curves. Without any fitting or free parameters, the generated signal data is proportional to the MB absorption distribution of the pump transitions. c The generated signal amplitude responds roughly linearly to increased pump optical power. d The response of the generated signal amplitude to microwave powers from SG1. The largest pump and idler powers measured correspond to intensities of about 3 & 7 mW/cm². Error bars represent the standard deviation.

To probe the frequency-tunable nature of the conversion process, we scan the pump laser frequency across the Doppler width. At any instant during the scan, the pump field interacts with one velocity class of atoms, which see this frequency on resonance with the pump transition. The generated signal reaches a maximum amplitude between the $F=2\to {F}^{{\prime} }=1$ and ${F}^{{\prime} }=2$ transitions [Fig. 3b], with a full width at half maximum (FWHM) of 550(25) MHz. This generated spectrum corresponds to the combined Doppler-dependent absorption coefficients for the $F=1\to {F}^{{\prime} }=1,\,2$ transitions, which has a FWHM of 540 MHz. The ${F}^{{\prime} }=3$ state does not participate in this nonlinear process, since selection rules prohibit an electric-dipole transition to the F = 1 ground state. Similarly, the pump transition from $F=2\to {F}^{{\prime} }=0$ is electric-dipole forbidden. We also characterize the effects of various pump and idler powers as shown in Fig. 3c, d. For a good signal-to-noise ratio, we typically operated with a pump power near 76 μW and idler power of 3.2 mW.

Next, we explore the multi-channel character of the conversion process. The pump laser is tuned halfway between the $F=2\to {F}^{{\prime} }=2\,\& \,3$ transitions. By modulating either the idler or the pump, multiple frequency components are generated in the signal: First, we amplitude-modulate (AM) the microwave idler signal at ω_AM and observe sidebands in the signal at ω_S ± ω_AM. The conversion bandwidth from idler modulation is limited to a maximum of 1 MHz by the width of the cavity resonance. Next, we additionally phase-modulate (PM) the pump using a fiber EOM in the optical path and observe that the idler and pump combine together as dictated by the sum-frequency process [Eq. (1)]. The MB width and density determine the bandwidth and amplitudes of the pump-modulated generated sidebands. Both types of sideband generation are illustrated in Fig. 4a, including simultaneous idler AM and pump PM. In the three cases depicted, ω_AM/2π = 50 kHz is fixed and ω_PM/2π is varied from 10 MHz to 100 MHz.

Fig. 4: Sideband generation and interference.

a–i Multichannel generation using a fixed microwave amplitude modulation frequency of ω_AM/2π = 50 kHz and a variable pump phase modulation frequency ω_PM/2π of 10 (blue), 50 (red) and 100 (black) MHz. The relative sideband amplitude decreases for increased ω_PM because the pump light accesses velocity classes with decreased occupation. a–ii Zoomed-in data revealing AM sidebands flanking each of the central and SB PM peaks. b When the two modulation frequencies are equal, i.e. ω_PM = ω_AM, the PM and AM sidebands overlap and interfere. The relative phase difference Δϕ between the modulating waves controls the height of the correlated upper (■, blue) and lower (▲, red) generated sidebands. (b-i) The generated spectrum at zero relative phase. b–ii The relative power amplitude response of the peaks have a visibility of 97%, while the central peak (●, purple) is independent of the relative phase. The two SB amplitudes differ from each other by (π − 0.38) radians of phase. The dashed lines show independent fits to a cosine.

When the pump and idler are modulated at the same frequency, ω_PM = ω_AM, the sidebands from each input overlap and interfere [Fig. 4b-i]. By tuning the relative phase of the modulating waves, one sideband or the other can be selectively enhanced or eliminated, with a relative visibility of 97% [Fig. 4b-ii], in qualitative agreement with the calculated variation shown in Fig. 2d.

The efficiency of conversion from microwave to signal photons is defined as the rate of signal photon production to the rate of microwave photons residing in the cavity mode^9,10

$$\eta \equiv \frac{{P}_{{{{{{{{\rm{signal}}}}}}}}}/\hslash {\omega }_{{{{{{{{\rm{signal}}}}}}}}}}{{P}_{{{{{{{{\rm{idler}}}}}}}}}/\hslash {\omega }_{{{{{{{{\rm{idler}}}}}}}}}}.$$

(5)

Further details on this calculation are included in Supplementary Notes 2, 3 and 6. We determine a maximum conversion efficiency of η₀ = 1.5(1) × 10⁻⁹. Efficiency in our system is primarily limited by a very low optical depth of 0.05(1). In comparison, cold atom conversion experiments have employed optical depths of up to 15⁹ and 120¹¹. Our optical depth corresponds to a number density of about 4 × 10¹⁶ cm⁻³ with 8.7(1) × 10⁸ atoms in the interaction region. This atom number is ~10⁹ times lower than the 3.2(1) × 10¹⁸ available microwave photons in the same volume. At room temperature, the thermal noise is at the level of 900 photons, which is indiscernible compared to the signals used here. We note that it is this noise, determined by the temperature of the microwave electronics and transmission components (rather than of the conversion medium), that sets the limit of low-photon-number operation. Moving toward true quantum transduction requires reducing the noise in these components by operating at cryogenic temperatures²².

Discussion

The frequency tunability of the generated signal field results from the existence of a large Doppler width at room temperature. While this work uses a warm vapor cell of enriched ⁸⁷Rb without an anti-relaxation coating or a buffer gas, a buffer gas would provide additional tunability due to collisional broadening^23,24,25. This tunability is ultimately limited to the order of the ground-state hyperfine splitting of the atoms used.

While other conversion schemes display degrees of tunability of the output frequency^9,11,14,15, these are often restricted by natural atomic or optical cavity resonance frequencies. In the warm-atom system, a continuous tunability over the entire Doppler width is possible. This frequency multiplexing capability enables multiple microwave sources to be interfaced with a single atomic transducer, and the conversion can be used to place differing sources in distinct optical frequency domains. Switching between different optical frequencies for a single source is also possible. Our scheme features a conversion bandwidth of 910(20) kHz [see Fig. 6b], which is primarily limited by the cavity linewidth. As previously demonstrated in a different context²⁶, the reverse optical-to-microwave conversion process is achievable in an atomic three-wave mixing system.

The high-contrast amplitude control of signal sidebands enabled by simultaneous pump and idler modulation [Fig. 4b] is equivalent to a tunable beam-splitter operating in the frequency domain¹⁶. The modulation indices and the relative phase act together as reflection and transmission coeffiecients. For particular values of these coefficients we can selectively generate a chosen sideband channel in the optical domain while completely suppressing the other, resulting in single-sideband conversion.

Thus, our neutral atom system is capable of performing as a “quantum frequency processor”^17,27,28 with particularly low input-field intensities. This allows us not only to encode and manipulate information in multiplexed frequency bins but also to convert it from microwave to optical frequency domains.

The measured conversion efficiency is comparable with magnon-based microwave-to-optical conversion schemes^14,15. The warm-atom efficiency is inherently limited by the intrinsic weakness of the magnetic-dipole microwave transition, as well as the ground-state spin decoherence γ_I which includes wall-collisions, transit-time decoherence, and Rb-Rb spin-exchange collisions (see Supplementary Note 5). A larger optical beam combined with a buffer gas^29,30 and anti-relaxation coating on the cell walls^31,32,33,34 would help reduce spin decoherence from the dark state and boost conversion efficiency. For example, we expect that a high-quality anti-relaxation coating and a modest 5 torr of neon would significantly reduce transit-time and wall effects, and reduce γ_I/2π by a factor of 10³ to 100 Hz, limited by Rb-Rb spin-exchange. Everything else being equal, this would provide a 10⁶-fold improvement to the generation efficiency. Furthermore, based on¹⁰, we expect that increased pump powers will produce a moderate 4-fold efficiency gain before the efficiency saturates due to absorption and incoherent scattering of the pump.

In this proof-of-concept demonstration, we have shown that inhomogeneous Doppler broadening in a warm vapor cell provides significant tunability of the output optical frequency in a microwave-optical transduction, resulting in frequency division multiplexing. The inhomogeneous width also enables simultaneous frequency up-conversion of multi-channel microwave inputs. Amplitude control of up-converted optical channels demonstrates analogous frequency domain beam splitting action across five orders in frequency. The frequency division multiplexing capability combined with amplitude control can make neutral atoms as quantum processors for information encoded in frequency bin qubits.

Methods

Experimental setup

The experimental setup can be seen in Fig. 3a. The pump optical field is derived from an continuous-wave external cavity diode laser (MOGLabs CEL) operating at 780.24 nm. This laser is frequency-stabilized to the saturated absorption spectrum of a separate Rb reference vapor cell. We have the option to lock the laser frequency 80 MHz below the ${F}^{{\prime} }$ = 2–3 crossover transition with a locked linewidth of ~100 kHz. Alternatively, we can unlock and move the laser frequency off-resonance. The pump beam is then shifted upwards in frequency by f_AOM = 76.60 MHz with an acousto-optical modulator and transmitted to the cavity-vapor cell apparatus via optical fiber. After the optical fiber, the polarization is filtered with a polarizing beam splitter (PBS) and the power of the pump light is stabilized with a servo controller (NuFocus LB1005) to <1% fluctuations. The Rabi frequency of the pump laser Ω_P/2π has a value of 3.5 MHz³⁵ and it is linearly polarized along the z-axis.

Inside the cavity^18,19, the idler microwave field is polarized along the y-direction. Orthogonal Helmholtz coils null the external magnetic field along the x and y-axes and apply a weak DC magnetic field ${\overrightarrow{B}}_{z}$ along the z-axis. The magnetic field strength makes a striking impact on the conversion efficiency. As shown in Fig. 5a, conversion is strongly suppressed at zero magnetic field and reaches maxima at ~±200 mG. This behavior is well-described by a symmetric, generalized Fano lineshape³⁶ that has the simplified form

$$\eta ({B}_{z})={\eta }_{0}^{{\prime} }{\left[\frac{2{\gamma }_{B}({B}_{z}-{B}_{0})}{{\gamma }_{B}^{2}+{({B}_{z}-{B}_{0})}^{2}}\right]}^{2}.$$

(6)

This points to the participation and interaction of multiple Zeeman levels in the generation process. At Zeeman degeneracy, generation pathways may destructively interfere. The value γ_B × γ₀ = 140 kHz ~ γ_I, the ground state decoherence rate, where γ₀ is the ground state gyromagnetic ratio of ⁸⁷Rb³⁷. Further discussion goes beyond the scope of this work and will be the topic of future investigation.

Fig. 5: Magnetic field effect and generated linewidth.

a The generated signal’s response to a variable magnetic field strength ${\overrightarrow{B}}_{z}$ along the z-axis. Wte operate at 209 mG for maximum generation. We fit a symmetric Fano profile [Eq. (6)] to these data, with fit parameters ${\eta }_{0}^{{\prime} }=0.95(2)$, γ_B = 200(4)mG and B₀ = − 5(3)mG. This behavior might indicate the interaction and interference of multiple Zeeman level resonances within the decoherence width of our atoms. Error bars represent the standard deviation. b A close up of the unmodulated generated line revealing a FWHM of 26(2) Hz.

The generated signal light emerges from the cavity polarized along the y-axis, orthogonal to the pump³⁸. Before recoupling to an optical fiber, a half-waveplate is manually adjusted to optimize the signals at f_het,P and f_het,S for each respective measurement. The light is then combined in a 50:50 fiber BS with local oscillator laser light that was not shifted by the acousto-optic modulator (AOM). The interference produces heterodyne beat notes at f_het,P = f_AOM and f_het,S = ω_μ/2π + f_AOM = 6.911282 GHz. A high bandwidth amplified photodiode (PD1, EOT ET-4000AF) detects the beating signals. This electronic signal is amplified (AMP, Mini-Circuits ZJL-7G+) and its amplitude is measured by a spectrum analyzer (SA, R&S FSV3013). This heterodyne peak at f_het,S has a FWHM of 26(2) Hz [see Fig. 5b], larger than the <2 Hz linewidth of our microwave source, SG1. We presume the presence of additional perturbations (e.g. 60 Hz line noise) broadens and modulates our generated signal.

Broad-tunability

To access all the frequencies within the Doppler profile, the pump frequency is scanned across the Doppler profile with a laser frequency sweep. We trigger the SA off the sweep ramp and measure the heterodyne signal in zero-span mode with a resolution bandwidth of 2 kHz. Simultaneous to the sweep, we also trigger and record the transmission of pump light through a reference vapor cell of rubidium (natural abundance) placed near the microwave cavity and at ambient 21.6(3) °C. We find good agreement between the measured spectrum and a theoretical transmission model³⁹. Using the ⁸⁷Rb and ⁸⁵Rb transmission dip extrema, we calibrate our sweep rate to 17.2 MHz/ms and convert the time axis of the series to frequency [Fig. 6a].

Fig. 6: Frequency sweep and conversion bandwidth.

a The Doppler-broadened transmission spectrum of a reference rubidium vapor cell (outside the microwave cavity) at room temperature, measured while sweeping the pump laser frequency. The dotted line shows a transmission model curve with good agreement to the measured trace. This trace was taken simultaneous to the data shown in Fig. 3b; the positions of the two Doppler peaks are used to calibrate the horizontal frequency scale. b The raw generated signal during a scan of δ_I, as measured with a vector network analyzer (VNA) in S21 mode. We measure a conversion bandwidth (FWHM) of 910(20) kHz when the laser is locked on-resonance and with a z-magnetic field of 209 mG. The dashed line is a smoothing spline used to determine the bandwidth. No generation is visible when the laser is off-resonance. The cavity resonance has a FWHM of 417 kHz, as measured with the VNA in S11 mode.

Multichannel conversion

We generate a multichannel optical signal field in two different ways: first, the signal generator (SG1) supplying the idler field is amplitude-modulated at a variable ω_AM. We note that due the finite microwave cavity resonance (see Supplementary Note 1), ω_AM/2π can be tuned up to 455(10) kHz before the sidebands become significantly suppressed. Second, we modulate the pump light by using a fiber phase-modulator EOM (EOSPACE PM-0S5-20-PFA-PFA-780) inserted into the pump light optical path. We drive the EOM with a second signal generator (SG2, SRS SG384) at a variable ω_PM. This produces sidebands in the pump optical spectrum, and corresponding sidebands are visible in the generated spectrum around the heterodyne frequency f_het,S. The SA and SG2 are both phase-locked to the internal 10 MHz clock reference of SG1. In this second case, ω_PM can be tuned over the Doppler FWHM before the sidebands become significantly suppressed.

Conversion bandwidth

We characterize the conversion bandwidth of our microwave-to-optical conversion process. To do this, we replace SG1 and SA with Ports 1 and 2, respectively of a VNA (Keysight E5063A). The generated signal is measured on the VNA in S21 mode and is shown in Fig. 6b. When the laser is locked on resonance and at the optimal magnetic field of 209 mG, a clear peak in the S21 signal is visible with FWHM = 910(20) kHz. This width is compared to the 417 kHz width of the cavity resonance. When the laser is off-resonance, no generation is apparent.