Giant microwave–optical Kerr nonlinearity via Rydberg excitons in cuprous oxide

Superconducting microwave devices play a key role in quantum computation.¹ To eliminate thermal noise, these devices must be cooled to T ≈ 10 mK, a requirement that makes direct quantum networking impractical over distances larger than $∼1$ m.² Microwave–optical (MO) conversion is, therefore, a critical enabling technology,^3–5 with current approaches, including electro-optics,^6,7 rare-earth ions,⁸ optomechanical systems,^9,10 and quantum dot molecules.¹¹ Very strong nonlinearity can be achieved by exploiting the large microwave dipole moment associated with highly excited atomic Rydberg states,^12,13 leading to the largest observed cross-Kerr effect of any material.¹⁴ However, interfacing Rydberg atoms with planar superconducting quantum devices in a millikelvin environment is an outstanding challenge, and coupling has so far been achieved only at much higher temperatures.^15–18 

A schematic of the experiment is shown in Fig. 1. A thin slab of Cu₂O (55 ± 10 μm) is mounted between the conductors of a microwave strip line and cooled to T = 4 K. Excitons in a quantum state nP (where P denotes the orbital angular momentum l = 1) were excited from the ground state (valence band) using a single-frequency laser at wavelength λ ≈ 571 nm. A microwave field at angular frequency ω_MW = 2π × 7 GHz couples nP states to the nearby states of opposite parity (e.g., n′S and n′D, where S and D indicate l = 0 and l = 2). The spectrum of the transmitted laser light was resolved using a scanning Fabry–Pérot etalon. Further experimental details are provided in  Appendix A. Applying the microwave field leads to a change in transmission at the laser frequency ω_L = 2πc/λ and the appearance of sidebands at frequencies ω_L ± 2ω_MW on the light transmitted through the sample [Fig. 1(c)] that are indicative of a cross-Kerr nonlinearity. As the microwave field amplitude is increased further, higher order sidebands (e.g., ω_L ± 4ω_MW) become visible.

The optical absorption spectrum close to the bandgap is shown in Fig. 2(a). Peaks corresponding to nP excitonic states are visible up to n = 16, limited by a combination of temperature and sample quality. As Cu₂O is centrosymmetric, the MO coupling is absent from the ground-state symmetry but is created by the Rydberg excitons, as shown in Fig. 2(b), which plots the spectrum of the microwave-induced change in absorption ΔαL. Each exciton resonance shows decreased absorption on-resonance and increased absorption on each side at the location of the nS and nD states. In contrast to Rydberg atoms, a response is observed over a wide range of n, even for a single microwave frequency. This is due to non-radiative broadening of the exciton lines, which means that the microwave field is near-resonant with many transitions simultaneously. The same effect results in a broad microwave frequency dependence (see  Appendix A and Ref. 23). |ΔαL| increases with n up to n = 13 before decreasing toward the band edge. As shown in the inset of Fig. 2(b), |ΔαL| increases linearly with the microwave power as expected for a Kerr nonlinearity, before saturating at a value that depends on n. Saturation occurs due to multi-photon processes and is accompanied by the appearance of the higher-order (fourth-order) sidebands in Fig. 1(c). For the remainder of this paper, we consider only the linear regime.

The predicted Δα is compared with the data in Fig. 2(b). The only adjustable parameter is the amplitude of the microwave electric field $EMW$⁠. The prediction is in excellent agreement with the data over the full range of n. Similar levels of agreement are observed up to field amplitudes of 60 V/m, above which saturation occurs.

We exploit the quantitative agreement Fig. 2(b) to calibrate the microwave electric field amplitude $EMW$⁠. The model was fitted to the change in absorption over a range of applied microwave powers. From these fits, we obtain a calibration between electric field amplitude and applied microwave power of 43 ± 3 (V/m)/mW^1/2, in reasonable agreement with a value of 70 (V/m)/mW^1/2 obtained from a finite-element simulation of the antenna that excludes connection losses.²³ More details are given in  Appendix C.

To independently measure B₀, we make use of the zero-crossings in Fig. 2. At these points Δα = 0, the imaginary part of the susceptibility is zero, and sidebands are generated via phase modulation only. By using the conventional expansion of a phase-modulated wave in terms of Bessel functions, it can be shown that the ratio of the intensity of the sidebands to the carrier is $ISB/IC=J1(Δϕ)2$⁠, where J₁ is the first-order Bessel function of the first kind. Thus, the microwave-induced phase shift Δϕ can be directly extracted from spectra like Fig. 1(c).

The variation in Δϕ with $|EMW|2$ is shown for two zero-crossings in Fig. 3(b). As expected, the phase shift increases linearly in the Kerr regime, before saturating. The Kerr coefficient is obtained from the gradient in the linear regime using Eq. (1) and the electric field calibration described previously.

As shown in Fig. 3(a), the measured Kerr coefficient is in agreement with the prediction. We emphasize that the phase shift measurement does not depend on the model for χ⁽³⁾ [Eq. (2)] and its input parameters; these affect the measured value of B₀ only through the field calibration. The dominant experimental uncertainties come from the electric field calibration and the thickness of the sample. The analysis also assumes that χ⁽³⁾(ω_L ± 2ω_MW) = χ⁽³⁾(ω_L), which is not strictly true due to the strong energy dependence near resonance. An indication that this assumption does not fully hold is the asymmetry in the red and blue sideband amplitudes in Fig. 3. We, therefore, include the difference between the red and blue values as part of the uncertainty in B₀ and present the average value. The analytic expressions for χ⁽³⁾(ω_L ± 2ω_MW) are provided in  Appendix D.

Our highest measured value of B₀ = 0.022 ± 0.008 V m⁻² at n = 12 is compared to other low-frequency (DC) Kerr coefficients in Table I. Our measured Kerr coefficient is extremely large due to a combination of the underlying resonant nature of the nonlinearity and the large dipole moment. The other values in Table I were measured far from resonance—a key feature of Cu₂O is that a much larger but still broadband response can be obtained via multiple resonances. Cu₂O also has the highest absorption coefficient among the materials in Table I. An alternative figure of merit is the phase shift per unit absorption length and microwave intensity, described by the ratio $F=B0/α$⁠, where α is the linear absorption coefficient. Here, Cu₂O has the largest $F$⁠, with only Rydberg atoms in Rb vapor being comparable, highlighting the importance of Rydberg physics in the solid state.

Despite the large values of B₀ and $F$⁠, the maximum phase shift remained limited to around 0.1 rad as shown in Fig. 3(b). This is due to a combination of the saturation of the phase shift at high electric fields and the high background absorption, which together limit the maximum phase shift that can be measured. The saturation is a consequence of the extremely strong nonlinearity and corresponds to the emergence of higher order terms (e.g., χ⁽⁵⁾) in the susceptibility. In fact, the experiment enters the ultra-strong driving regime, where the coupling strength $ΩMW=dEMW/ℏ≫ωMW,Γ$⁠. Microwave–optical conversion still occurs in this regime, but it is no longer given by a simple Kerr effect. A reduction in absorption would enable the use of thicker samples, resulting in larger phase shifts. The absorption is dominated by phonon-assisted processes that do not involve Rydberg states, and which are unaffected by the microwave field.³⁰ At n = 11, 80% of the absorption coefficient is due to the background. As in atomic Rydberg gases, the background may be reduced by nonlinear spectroscopy techniques, such as second harmonic generation^31,32 or electromagnetically induced transparency,^29,33 or by exploiting Rydberg exciton–polaritons.³⁴ 

Finally, we discuss our results in the context of MO conversion, where the standard figures of merit are bandwidth and conversion efficiency.³ Large bandwidth is the major advantage of Cu₂O; our model predicts that the Kerr coefficient varies by less than a factor of two over the entire range ω_MW/2π = 1 − 20 GHz (see  Appendix A). For comparison, most platforms have a bandwidth of less than 10 MHz³ with quantum dot molecules achieving 100s of MHz.¹¹ Conversion efficiency is critically dependent on the optimization of the device parameters (e.g., mode volume), which we did not study here. Instead, we compare the intrinsic strength of the nonlinearity with lithium niobate (LN), which can also provide a broadband response.^35–37 The refractive index change in a Kerr medium becomes larger than that in a linear electro-optic medium, such as LN above a critical field given by $Ecrit=cηEO3r/(2ωLB0)$⁠, where r_EO and η_EO are the electro-optic coefficient and refractive index of the linear medium, respectively.³⁵ Comparing Cu₂O with LN, we find E_crit = 0.01 V m⁻¹, which is 20 times lower than the rms vacuum field in typical superconducting quantum circuits.³⁸ Given the nonlinearity in Cu₂O should improve as the temperature is reduced (due to increased exciton oscillator strength¹⁹ and the accessibility of higher n²⁰), Cu₂O appears to be a promising candidate to achieve a high-bandwidth, efficient microwave–optical conversion.

In conclusion, we have demonstrated that the strong microwave–optical nonlinearity observed in atomic gases can be realized in a cryogenic, solid-state setting using Rydberg excitons in Cu₂O. The experimentally measured Kerr coefficient B₀ = 0.022 ± 0.008 m V⁻² is four orders of magnitude larger than other solid-state platforms, with only Rydberg atoms being comparable. The observations are in agreement with a model based on the conventional hydrogen-like theory of Wannier–Mott excitons, enabling the quantitative design of future devices that exploit the giant Kerr effect; in addition, we highlight that the effect is visible at relatively low n, meaning that synthetic materials could be utilized.^39–41 Our work complements efforts to prepare quantum states of light using Rydberg-mediated interactions in Cu₂O^42–45 and opens a potential route to microwave–optical conversion in the quantum regime.

We thank I. Chaplin and S. Edwards for preparing samples and V. Walther for calculating the dipole matrix elements. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom, through research Grant Nos. EP/P011470/1, EP/P012000/1, EP/X038556/1, and EP/X03853X/1.

The authors have no conflicts to disclose.

J.D. and L.A.P.G. contributed equally to this work.

Jon D. Pritchett: Formal analysis (equal); Investigation (lead); Methodology (equal); Writing – review & editing (equal). Liam A. P. Gallagher: Formal analysis (equal); Investigation (supporting); Methodology (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Alistair Brewin: Formal analysis (equal); Writing – review & editing (equal). Horatio Q. X. Wong: Formal analysis (supporting); Investigation (supporting); Writing – review & editing (supporting). Wolfgang Langbein: Methodology (equal); Writing – review & editing (equal). Stephen A. Lynch: Methodology (supporting); Writing – review & editing (equal). C. Stuart Adams: Conceptualization (supporting); Methodology (supporting); Supervision (supporting); Writing – review & editing (supporting). Matthew P. A. Jones: Conceptualization (lead); Methodology (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are openly available at https://doi.org/10.15128/r27d278t10x.

The Cu₂O sample was prepared from a natural gemstone (Tsumeb mine) using the procedure detailed in Ref. 39 and glued to a CaF₂ window. The light had an incident intensity of 20 μW/mm², propagated along the [111] crystallographic axis, and was linearly polarized. Finite element analysis showed that the microwave electric field was aligned in the plane of the sample and orthogonal to strip line. Δα was observed to depend very weakly on the angle between microwave and optical electric fields with a less than 10% difference observed when varying the angle by π/2.⁴⁶ No microwave-induced change in the polarization of the transmitted light was observed within the experimental uncertainty. The etalon had a free spectral range of 60.1 ± 0.2 GHz and a finesse of 44.5 ± 0.7 and was temperature-tuned. The carrier and sideband amplitudes were extracted by fitting the Lorentzian response function of the etalon to spectra like those in Fig. 1(c).

As stated in the text, the nonlinear response is very broadband. The predicted Kerr coefficient as a function of microwave frequency is shown in Fig. 4 over the range 1–20 GHz. Measurements at multiple frequencies²³ are challenging in our current setup due to parasitic resonances in the cryostat. We, therefore, fixed ω_MW = 2π × 7 GHz to give good resolution of the sidebands while remaining in the relevant range for superconducting quantum circuits.

Here, we give details of the parameters in the susceptibility model [Eq. (2)]. States from n = 5–17 and l = 0–2 are included in the model.

Matrix elements for transitions between exciton states, d, were calculated using a spinless hydrogen-like model³³ for the exciton states, with quantum defects obtained from dispersion relations for the electron and the hole.⁴⁷ 

The widths and energies were extracted from experimental absorption spectra without microwaves. The product $NdD2$ was determined from the area of the exciton peaks in the absorption spectra without microwaves.

Here, we provide further details of the electric field calibration. By fitting the model to the change in absorption [see Fig. 2(b) for an example] at different applied microwave powers P_MW, it is possible to extract the calibration between the externally applied power and $EMW$⁠. The result is shown in Fig. 5. As expected, the relation between $|EMW|2$ and power is linear at low electric fields, before a deviation occurs as the susceptibility model breaks down due to the higher-order processes discussed in the text. The quantitative agreement between the model and the data illustrated in Fig. 2(b) holds for all powers within the linear range.

Fitting the linear region in Fig. 5 (purple points) gives a value of 43 ± 3 (V/m)/mW^1/2. A simulation of the strip line antenna and the surrounding environment using a commercial finite element analysis software package (see Ref. 23) yields a value of 70 (V/m)/mW^1/2. We consider that these values are in reasonable agreement, especially since no connection losses were included in the simulation. Using the simulated value for the calibration would reduce the measured Kerr coefficient by a factor of $∼3$⁠, which remains the orders of magnitude larger than in other solid-state systems.