Coherent microwave-to-optical conversion via six-wave mixing in Rydberg atoms

TL;DR

This paper demonstrates a coherent microwave-to-optical conversion using six-wave mixing in Rydberg atoms, achieving 0.3% efficiency and a broad bandwidth, with potential for near-unit efficiency in future work.

Contribution

The study experimentally realizes microwave-to-optical conversion via six-wave mixing in Rydberg atoms, showing coherence transfer and high potential efficiency.

Findings

Photon conversion efficiency of ~0.3% at low microwave intensities

Conversion bandwidth exceeds 4 MHz

Theoretical models predict near-unit efficiency achievable in future experiments

Abstract

We present an experimental demonstration of converting a microwave field to an optical field via frequency mixing in a cloud of cold $^{87}$Rb atoms, where the microwave field strongly couples to an electric dipole transition between Rydberg states. We show that the conversion allows the phase information of the microwave field to be coherently transferred to the optical field. With the current energy level scheme and experimental geometry, we achieve a photon conversion efficiency of ~0.3% at low microwave intensities and a broad conversion bandwidth of more than 4 MHz. Theoretical simulations agree well with the experimental data, and indicate that near-unit efficiency is possible in future experiments.

Equations

$${\partial }_t\varrho =-\frac{i}{\hslash }[H,\varrho ]+{\mathcal{L}}_{\gamma }\varrho +{\mathcal{L}}_{\text{deph}}\varrho ,$$

$$|D⟩\propto \left({\Omega }_{\text{M}}^{\ast }{\Omega }_{\text{C}}^{\ast }|1⟩-{\Omega }_{\text{M}}^{\ast }{\Omega }_{\text{P}}|3⟩+{\Omega }_{\text{A}}^{\ast }{\Omega }_{\text{P}}|5⟩\right)$$

$$\eta =\frac{P_{\text{L}}/\hslash {\omega }_{\text{L}}}{I_{\text{M}}{S}_{\text{M}}/\hslash {\omega }_{\text{M}}},$$

Full text

Coherent and efficient conversion from microwave and terahertz radiation into optical fields and vice versa has tremendous potential for developing next-generation classical and quantum technologies. For example, these methods would facilitate the detection and imaging of millimeter waves with various applications in medicine, security screening and avionics Adam (2011); Chan et al. (2007); Tonouchi (2007); Zhang et al. (2017). In the quantum domain, coherent microwave-optical conversion is essential for realizing quantum hybrid systems Xiang et al. (2013) where spin systems or superconducting qubits are coupled to optical photons that can be transported with low noise in optical fibers Kimble (2008). The challenge in microwave-optical conversion is to devise a suitable platform that couples strongly to both frequency bands, which are separated by several orders of magnitude in frequency, and provides an efficient link between them. Experimental work on microwave-optical conversion has been based on ferromagnetic magnons Hisatomi et al. (2016), frequency mixing in $\Lambda$-type atomic ensembles Williamson et al. (2014); O’Brien et al. (2014); Blum et al. (2015); Hafezi et al. (2012), whispering gallery resonators Strekalov et al. (2009); Rueda et al. (2016), or nanomechanical oscillators Bochmann et al. (2013); Andrews et al. (2014); Bagci et al. (2014). All of these schemes include cavities to enhance the coupling to microwaves. The realization of near-unit conversion efficiencies as e.g. required for transmitting quantum information remains an outstanding and important goal. Recently, highly excited Rydberg atoms have been identified as a promising alternative Kiffner et al. (2016); jacobs:17 as they feature strong electric dipole transitions in a wide frequency range from microwaves to terahertz Gallagher (1994).

In this letter, we demonstrate coherent microwave-to-optical conversion of classical fields via six-wave mixing in Rydberg atoms. Due to the strong coupling of millimeter waves to Rydberg transitions, the conversion is realized in free space. In contrast to millimeter-wave induced optical fluorescence Wade et al. (2016), frequency mixing is employed here to convert a microwave field into a unidirectional single frequency optical field. The long lifetime of Rydberg states allows us to make use of electromagnetically induced transparency (EIT) Mohapatra et al. (2007), which significantly enhances the conversion efficiency Fleischhauer et al. (2005). A free-space photon-conversion efficiency of 0.3% with a bandwidth of more than 4 MHz is achieved with our current experimental geometry. Optimized geometry and energy level configurations should enable the broadband inter-conversion of microwave and optical fields with near-unit efficiency Kiffner et al. (2016). Our results thus constitute a major step towards using Rydberg atoms for transferring quantum states between optical and microwave photons.

The energy levels for the six-wave mixing are shown in Fig. 1(a), and the experimental setup is illustrated in Fig. 1(b). The conversion of the input microwave field M into the optical field L is achieved via frequency mixing with four input auxiliary fields P, C, A, and R in a cold atomic cloud. Starting from the spin polarized ground state $|1\rangle$, the auxiliary fields and the microwave field M, all of which are nearly resonant with the corresponding atomic transitions, create a coherence between the states $|1\rangle$ and $|6\rangle$. This induces the emission of the light field L with frequency $\omega_{\text{L}}=\omega_{\text{P}}+\omega_{\text{C}}-\omega_{\text{A}}+\omega_{\text{M}}-\omega_{\text{R}}$ such that the resonant six-wave mixing loop is completed, where $\omega_{\text{X}}$ is the frequency of field X ($\text{X}\in\{\text{P},\text{R},\text{M},\text{C},\text{L},\text{A}\}$). The emission direction of field L is determined by the phase matching condition $\mathbf{k}_{\text{L}}=\mathbf{k}_{\text{P}}+\mathbf{k}_{\text{C}}-\mathbf{k}_{\text{A}}+\mathbf{k}_{\text{M}}-\mathbf{k}_{\text{R}}$, where $\mathbf{k}_{\text{X}}$ is the wave vector of the corresponding field. The wave vectors of the microwave fields $\mathbf{k}_{\text{A}}$ and $\mathbf{k}_{\text{M}}$ are negligible since they are much smaller than those of the optical fields and to an excellent approximation, they cancel each other. Moreover, we have $\mathbf{k}_{\text{C}}\approx\mathbf{k}_{\text{R}}$, thus the converted light field L propagates in the same direction as the input field P. The transverse profile of the converted light field L resembles that of the auxiliary field P due to pulse matching Harris (1993, 1994) as illustrated in Fig. 1(b).

An experimental measurement begins with the preparation of a cold cloud of 87Rb atoms in the $|5S_{1/2},F=2,m_{F}=2\rangle$ state in a magnetic field of 6.1 G, as described previously in Han et al. (2015). At this stage, the atomic cloud has a temperature of about $70\ \mu\text{K}$, a $1/e^{2}$ radius of $w_{z}=1.85(10)\,\textrm{mm}$ along the $z$ direction, and a peak atomic density $n_{0}=2.1(2)\times 10^{10}\,\textrm{cm}^{-3}$. We then switch on all the input laser and microwave fields simultaneously for frequency mixing. The beams for both C and R fields are derived from a single 482 nm laser, while that of the P field comes from a 780 nm laser, and the two lasers are frequency locked to a single high-finesse temperature stabilized Fabry-Perot cavity Han et al. (2015). The $1/e^{2}$ beam radii of these Gaussian fields at the center of the atomic cloud are $w_{\text{P}}=25(1)\ \mu\text{m}$, $w_{\text{C}}=54(2)\ \mu\text{m}$, and $w_{\text{R}}=45(1)\ \mu\text{m}$, respectively; and their corresponding peak Rabi frequencies are $\Omega^{(0)}_{\text{P}}=2\pi\times 1.14(7)\ \text{MHz}$, $\Omega^{(0)}_{\text{C}}=2\pi\times 9.0(5)\ \text{MHz}$, and $\Omega^{(0)}_{\text{R}}=2\pi\times 6.2(3)\ \text{MHz}$. The two microwave fields M and A, with a frequency separation of around 450 MHz, are generated by two different microwave sources via frequency multiplication. They are emitted from two separate horn antennas, and propagate in the horizontal plane through the center of the atomic cloud, as shown in Fig. 1(b). The Rabi frequencies $\Omega_{\text{M}}$ and $\Omega_{\text{A}}$ are approximately uniform across the atomic cloud volume that intersects the laser beams. The Rabi frequency of the A field is $\Omega_{\text{A}}=2\pi\times 1.0(1)\ \text{MHz}$, while the Rabi frequency of the M field $\Omega_{\text{M}}$ is varied in different measurements. The details of the microwave Rabi frequency calibrations are presented in Note (1). The P and L fields that emerge from the atomic cloud are collected by a diffraction-limited optical system Han et al. (2015), and separated using a quarter-wave plate and a polarization beam splitter (PBS). Their respective powers are measured with two different avalanche photodiode detectors. Each optical power measurement is an average of the recorded time-dependent signal in the range from 6 to $16\ \mu$s after switching on all the fields simultaneously, where the delay ensures the steady state is fully reached.

We experimentally demonstrate the coherent microwave-to-optical conversion via the six-wave mixing process by two measurements. First, we scan the detuning $\Delta_{\text{P}}$ of the P field across the atomic resonance and measure the power of the transmitted field P ($P_{\text{P}}$), and the power of the converted optical field L ($P_{\text{L}}$) simultaneously. All other input fields are held on resonance. The results of this measurement are shown in Fig. 2(a), where the spectrum of the transmitted field P (red squares) exhibits a double peak structure. The signature of the six-wave mixing process is the converted field L (purple circles), and its spectrum features a pronounced peak around $\Delta_{\text{P}}=0$.

Second, to verify the coherence of the conversion, we perform optical heterodyne measurements between the L field and a reference field that is derived from the same laser as the P field. Fig. 2(b) shows that the Fourier spectrum of a 500$\ \mu$s long beat note signal has a transform limited sinc function dependence. The central frequency of the spectrum confirms that the frequency of the converted field L is determined by the resonance condition for the six-wave mixing process. Furthermore, we phase modulate the M field with a triangular modulation function and observe the recovery of the phase modulation in the optical heterodyne measurements, as shown in Fig. 2(c). This demonstrates that the phase information is coherently transferred in the conversion, as expected for a nonlinear frequency mixing process.

We simulate the experimental spectra by modelling the interaction of the laser and microwave fields with the atomic ensemble within the framework of coupled Maxwell-Bloch equations Note (1). The time evolution of the atomic density operator $\varrho$ is given by a Markovian master equation ($\hbar$ is the reduced Planck constant),

[TABLE]

where $H$ is the Hamiltonian describing the interaction of an independent atom with the six fields, and the term $\mathcal{L}_{\gamma}\varrho$ describes spontaneous decay of the excited states. The last term $\mathcal{L}_{\text{deph}}\varrho$ in Eq. (1) accounts for dephasing of atomic coherences involving the Rydberg states $|3\rangle$, $|4\rangle$, and $|5\rangle$ with the dephasing rates $\gamma_{d}$, $\gamma_{DD}$, and $\gamma_{d^{\prime}}$, respectively Note (1). The sources of decoherence are the finite laser linewidths, atomic collisions, and dipole-dipole interactions between Rydberg atoms. The dephasing rates affect the P and L spectra and are found by fitting the steady state solution of coupled Maxwell-Bloch equations to the experimental spectra in Fig. 2(a). All other parameters are taken from independent experimental measurements and calibrations. We obtain $\gamma_{d}=2\pi\times$150 kHz, $\gamma_{DD}=2\pi\times$ 150 kHz and $\gamma_{d^{\prime}}=2\pi\times$ 560 kHz and keep these values fixed in all simulations.

The system in Eq. (1) exhibits an approximate dark state Note (1)

[TABLE]

for $\Omega_{\text{L}}/\Omega_{\text{P}}=-\Omega_{\text{A}}^{*}\Omega_{\text{R}}^{*}/(\Omega_{\text{M}}^{*}\Omega_{\text{C}}^{*})$, where $\Omega_{\text{L}}$ is the Rabi frequency of field L. This state has non-zero population only in metastable states $|1\rangle$, $|3\rangle$, and $|5\rangle$, and is decoupled from all the fields. The population in $|D\rangle$ increases with the build-up of the converted light field along the $z$ direction, and thus $P_{\text{L}}$ saturates when all atoms are trapped in this state. Fig. 3 shows the dependence of the output power $P_{\text{L}}$ on the optical depth $D_{\text{P}}\propto n_{0}w_{z}$ of the atomic cloud, and the theory curve agrees well with the experimental data. The predicted saturation at $D_{\text{P}}\approx 20$ is consistent with the population in $|D\rangle$ exceeding 99.8% at this optical depth.

Next we analyze the dependence of the conversion process on detuning and intensity of the microwave field M. All auxiliary fields are kept on resonance and at constant intensity. Fig. 4(a) shows $P_{\text{L}}$ as a function of the microwave detuning $\Delta_{\text{M}}$. We find that the spectrum of the L field can be approximated by a squared Lorentzian function centered at $\Delta_{\text{M}}=0$, and its full width at half maximum (FWHM) is $\approx 6\ \text{MHz}$. The FWHM extracted from microwave spectra at different intensities $I_{\text{M}}$ is plotted in Fig. 4(b). The FWHM has a finite value $>4$ MHz in the low intensity limit, and increases slowly with $I_{\text{M}}$ due to power broadening. This large bandwidth is one of the distinguishing features of our scheme and is essential for extending the conversion scheme to the single photon level Rueda et al. (2016). In Fig. 4(c), we show measurements of $P_{\text{L}}$ vs. the intensity of the microwave field $I_{\text{M}}$ at $\Delta_{\text{M}}=0$. We find that the converted power $P_{\text{L}}$ increases approximately linearly at low microwave intensities, and thus our conversion scheme is expected to work in the limit of very weak input fields. The decrease of $P_{\text{L}}$ at large intensities arises because the six-wave mixing process becomes inefficient if the Rabi frequency $\Omega_{\text{M}}$ is much larger than the Rabi frequency $\Omega_{\text{A}}$ of the auxiliary microwave. All the theoretical curves in Fig. 4 agree well with the experimental data.

We evaluate the photon conversion efficiency of our setup by considering the cylindrical volume ${\mathcal{V}}$ where the atomic cloud and all six fields overlap. This volume has a diameter $\sim 2w_{\text{P}}$ and a length $\sim 2w_{z}$ [see Fig. 1(b)]. We define the conversion efficiency as

[TABLE]

where $S_{\text{M}}=4w_{\text{P}}w_{z}$ is the cross-section of the volume ${\mathcal{V}}$ perpendicular to $\mathbf{k}_{\text{M}}$. The efficiency $\eta$ gives the ratio of the photon flux in L leaving volume ${\mathcal{V}}$ over the photon flux in M entering ${\mathcal{V}}$. As shown in Fig. 4(d), the conversion efficiency is approximately $\eta\approx 0.3\%$ over a range of low intensities and then decreases with increasing $I_{\text{M}}$. Note that $\eta$ in Eq. (3) is a measure of the efficiency of the physical conversion process in the Rydberg medium based on the microwave power $I_{\text{M}}S_{\text{M}}$ impinging on $S_{\text{M}}$. This power is smaller than the total power emitted by the horn antenna since the M field has not been focused on $\mathcal{V}$ in our setup.

The good agreement between our model and the experimental data allows us to theoretically explore other geometries. To this end we consider that the microwave fields M and A are co-propagating with the P field, and assume that all other parameters are the same Note (1). We numerically evaluate the generated light power $P_{\text{L}}^{\parallel}$ for this setup and calculate the efficiency $\eta^{\parallel}$ by replacing $P_{\text{L}}$ with $P_{\text{L}}^{\parallel}$ and $S_{\text{M}}$ with $S_{\text{M}}^{\parallel}=\pi w_{\text{P}}^{2}$ in Eq. (3). We find $\eta^{\parallel}\approx 26\%$, which is approximately two orders of magnitude larger than $\eta$. This increase is mostly due to the geometrical factor $S_{\text{M}}/S_{\text{M}}^{\parallel}\approx 91$, since $P_{\text{L}}^{\parallel}\sim P_{\text{L}}$. Note that such a value for $\eta^{\parallel}$ is consistent with the efficiency achieved by a similar near-resonance frequency mixing scheme in the optical domain Merriam et al. (2000a).

In conclusion, we have demonstrated coherent microwave-to-optical conversion via a six-wave mixing process utilizing the strong coupling of electromagnetic fields to Rydberg atoms. We have established the coherence of the conversion by a heterodyne measurement and demonstrated a large bandwidth by measuring the generated light as a function of the input microwave frequency. Coherence and large bandwidth are essential for taking our scheme to the single photon level and using it in quantum technology applications. Our results are in good agreement with theoretical simulations based on an independent atom model thus showing a limited impact of atom-atom interaction on our conversion scheme.

This work has focussed on the physical conversion mechanism in Rydberg systems and provides several possibilities for future studies and applications. Alkali atom transitions offer a wide range of frequencies in the optical and microwave domain with properties similar to those exploited in this work. For example, the conversion of a microwave field to telecommunication wavelengths is possible by switching to different optical transitions and/or using different atomic species Gilbert (1993); Bouchiat et al. (1989); jacobs:17 , which makes our approach promising for classical and quantum communication applications. Moreover, it has been theoretically shown that bidirectional conversion with near-unit efficiency is possible by using a different Rydberg excitation scheme and well-chosen detunings of the auxiliary fields Kiffner et al. (2016). Such non-linear conversion with near-unit efficiency has only been experimentally realized in the optical domain Merriam et al. (1999). Reaching this level of efficiency requires good mode-matching between the millimeter waves and the auxiliary optical fields Kiffner et al. (2016), which can be achieved either by tightly focusing the millimeter wave, or by confining it to a waveguide directly coupled to the conversion medium Hafezi et al. (2012); Hogan et al. (2012). Eventually, extending our conversion scheme to millimeter waves in a cryogenic environment Hermann-Avigliano et al. (2014); cano:11 would pave the way towards quantum applications.