Microwave–optical conversion is key to future networks of quantum devices, such as those based on superconducting technology. Conversion at the single quantum level requires strong nonlinearity, high bandwidth, and compatibility with a millikelvin environment. A large nonlinearity is observed in Rydberg atoms, but combining atomic gases with dilution refrigerators is technically challenging. Here, we demonstrate a strong microwave–optical nonlinearity in a cryogenic, solid-state system by exploiting Rydberg states of excitons in Cu2O. We measure a microwave–optical cross-Kerr coefficient of B0 = 0.022 ± 0.008 m V−2 at 4 K, which is several orders of magnitude larger than other solid-state systems. The results are in quantitative agreement with a nonlinear susceptibility model based on the giant microwave dipole moment between nearby excitonic states. Our results highlight the potential of Rydberg excitons for nonlinear optics and form the basis for a microwave–optical frequency converter based on Cu2O.

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 

In this Letter, we combine the advantages of atomic Rydberg states (giant nonlinearity) and the solid state (milliKelvin compatibility19,20) by using Rydberg states of excitons in the bulk semiconductor Cu2O21 (Fig. 1). Excitons are optically excited bound states of an electron and a hole, with an internal structure that resembles a hydrogen atom. In Cu2O, the resulting Rydberg series of excitonic states has been measured up to principal quantum number n = 30 at T ≈ 40 mK.20 Transitions between neighboring states of opposite parity are located in the microwave spectral region. The associated dipole moment scales as dn2, reaching 180e nm at n = 11, which is more than 30 times larger than in quantum dot molecules.11 Thus, Rydberg excitons in Cu2O provide a unique platform for MO coupling in the solid-state,22 with recent experiments demonstrating microwave control of the linear and nonlinear optical response.23 Here, we extend this work to a measurement of the optical phase shift induced by the presence of a microwave field, characterized by the microwave–optical Kerr coefficient B0,
B0=Δϕ2πLEMW2,
(1)
where Δϕ is the optical phase shift induced by the microwave field, L is the length of the material, and EMW is the amplitude of the applied microwave field.
FIG. 1.

(a) Excitons are created in state nP via laser excitation at λ ≈ 571 nm. A microwave field at frequency ωMW couples odd- and even-parity exciton states. The exciton wavefunction spans many lattice sites, giving rise to a large dipole moment. (b) Excitons are created in a thin slab of Cu2O located between the conductors of a microwave strip line. The transmitted light is filtered with a Fabry–Pérot etalon (FPE) and sent to a detector. (c) Example spectrum of the transmitted light for E = hc/λ = 2.170 927 eV and EMW=640 V/m, showing an array of sidebands at ± 2ωMW/(2π) and ±4ωMW/(2π) when microwaves are applied (ION).

FIG. 1.

(a) Excitons are created in state nP via laser excitation at λ ≈ 571 nm. A microwave field at frequency ωMW couples odd- and even-parity exciton states. The exciton wavefunction spans many lattice sites, giving rise to a large dipole moment. (b) Excitons are created in a thin slab of Cu2O located between the conductors of a microwave strip line. The transmitted light is filtered with a Fabry–Pérot etalon (FPE) and sent to a detector. (c) Example spectrum of the transmitted light for E = hc/λ = 2.170 927 eV and EMW=640 V/m, showing an array of sidebands at ± 2ωMW/(2π) and ±4ωMW/(2π) when microwaves are applied (ION).

Close modal

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 ωMW = 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 ωL = 2πc/λ and the appearance of sidebands at frequencies ωL ± 2ωMW 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., ωL ± 4ωMW) 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.

FIG. 2.

(a) Rydberg series of exciton absorption lines. (b) Measured (red points) and predicted (solid line) change in absorption, ΔαL at EMW=38 V/m. The dashed lines indicate the zero crossings used to extract the Kerr coefficient. The inset shows |ΔαL| vs microwave power PMW at the n = 9, 10, 12P resonances. The solid lines are fits to the linear region.

FIG. 2.

(a) Rydberg series of exciton absorption lines. (b) Measured (red points) and predicted (solid line) change in absorption, ΔαL at EMW=38 V/m. The dashed lines indicate the zero crossings used to extract the Kerr coefficient. The inset shows |ΔαL| vs microwave power PMW at the n = 9, 10, 12P resonances. The solid lines are fits to the linear region.

Close modal
The MO Kerr nonlinearity can be described in terms of the dielectric polarization at the laser frequency,
P(ωL)=ε0χ(1)(ωL)EL+χ(3)(ωL)ELEMWEMW+,
where χ(1) and χ(3) are the linear and Kerr nonlinear susceptibilities, respectively. Experimentally, we observe that the nonlinearity depends very weakly on the laser and microwave polarizations, and so a scalar treatment is used. We obtain an expression for χ(3) by treating the exciton states as hydrogen-like.23 Δα is related to the imaginary part of the susceptibility,
ΔαcωLEMW2=Imχ(3)(ωL)=Imn,n,l,n,±χnPnl,nP(3)(ωL).
Each term in the sum is given by
χnPnlnP(3)(ωL)=Nd8ε03DVBnPdnPnldnlnPDnPVB(δnPiΓnP)(δnl±ωMWiΓnl)(δnPiΓnP),
(2)
where Nd is the effective density of exciton states, D are the effective matrix elements for transitions from the valence band to the P states, d are the dipole matrix elements for transitions between exciton states, Γ are the exciton linewidths, and δ are detunings, given by δnP = ωnPωL and δnl±ωMW=ωnl(ωL±ωMW). Crucially, all these parameters are either known theoretically or can be derived from the analysis of data without a microwave field applied (see  Appendix B), enabling the use of Eq. (2) for quantitative predictions of the nonlinear response.

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)/mW1/2, in reasonable agreement with a value of 70 (V/m)/mW1/2 obtained from a finite-element simulation of the antenna that excludes connection losses.23 More details are given in  Appendix C.

Once the electric field is known, the real part of the susceptibility can be used to make a quantitative prediction of the microwave–optical Kerr coefficient,
B0=ωL4ηπcReχ(3)(ωL),
where η = 2.8 is the optical refractive index. The predicted values of B0 are shown in Fig. 3(a).
FIG. 3.

(a) Predicted (solid line) and measured (points) Kerr coefficient vs excitation energy E. The shaded area shows the exciton spectrum from n = 8 upward for reference. (b) ISB/IC vs EMW for positive frequency (blue) and negative frequency (red) sidebands. The right-hand side axis gives the corresponding phase shift.

FIG. 3.

(a) Predicted (solid line) and measured (points) Kerr coefficient vs excitation energy E. The shaded area shows the exciton spectrum from n = 8 upward for reference. (b) ISB/IC vs EMW for positive frequency (blue) and negative frequency (red) sidebands. The right-hand side axis gives the corresponding phase shift.

Close modal

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)(ωL ± 2ωMW) = χ(3)(ωL), 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)(ωL ± 2ωMW) 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.

TABLE I.

Comparison of Kerr (B0) and absorption (α) coefficients, and their ratio F=B0/α for various materials. For nitrobenzene, glass and Cu2λ = 570 nm; for PMN-PT (a transparent ceramic) and GO-LC (graphene oxide liquid crystals) λ = 633 nm; Rb vapor λ = 780 nm.

MaterialB0 (m V−2)α (m−1)F (m2 V−2)
Glass24  1014 10−1 1013 
Nitrobenzene24,25 1012 10−1 1011 
PMN-PT26  107 102 109 
GO-LC27,28 106 103 109 
Rb vapor14,29 106 101 107 
Cu2102 104 106 
MaterialB0 (m V−2)α (m−1)F (m2 V−2)
Glass24  1014 10−1 1013 
Nitrobenzene24,25 1012 10−1 1011 
PMN-PT26  107 102 109 
GO-LC27,28 106 103 109 
Rb vapor14,29 106 101 107 
Cu2102 104 106 

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/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 generation31,32 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 ωMW/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 rEO and ηEO are the electro-optic coefficient and refractive index of the linear medium, respectively.35 Comparing Cu2O with LN, we find Ecrit = 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 Cu2O42–45 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 ωMW = 2π × 7 GHz to give good resolution of the sidebands while remaining in the relevant range for superconducting quantum circuits.

FIG. 4.

Predicted dependence of the Kerr coefficient B0 on microwave frequency ωMW at excitation energy E = 2.171 411 eV.

FIG. 4.

Predicted dependence of the Kerr coefficient B0 on microwave frequency ωMW at excitation energy E = 2.171 411 eV.

Close modal

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.

The values for the energy and width of the S and D states were extracted from Ref. 32. In the two-photon experiments only, states up to n = 12 were observed, and so extrapolation was used to extend the range of n beyond that available from these experiments. The energies were extrapolated using a quantum defect model,
Enl=EgRX(nδl)2,
(B1)
where RX is the excitonic Rydberg energy and δl is the quantum defect.
The widths of the P states were fitted using the following equation:
ΓnP=Γn3+Γ0,
(B2)
where Γ0 is a constant offset attributed to an inhomogenous broadening due to charges and defects in the material.48 For the S and D widths, the Γ0 was fixed to the value fitted from the P state trend and the high n states were extrapolated.

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 PMW, 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.

FIG. 5.

Square of the fitted microwave electric field strength EMW vs microwave power applied at the input port of the strip line (output terminated at 50 Ω) PMW. By fitting the region where the model is valid (purple points) with a linear trend (purple line), the electric field calibration is extracted.

FIG. 5.

Square of the fitted microwave electric field strength EMW vs microwave power applied at the input port of the strip line (output terminated at 50 Ω) PMW. By fitting the region where the model is valid (purple points) with a linear trend (purple line), the electric field calibration is extracted.

Close modal

Fitting the linear region in Fig. 5 (purple points) gives a value of 43 ± 3 (V/m)/mW1/2. 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)/mW1/2. 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.

The nonlinear susceptibility at the frequency of the red and blue sidebands is
χnPnlnP(3)(ωL±2ωMW)=Nd2ε03DVBnPdnPnldnlnPDnPVB(δnPiΓnP)(δnl±ωMWiΓnl)(δnP±2ωMWiΓnP).
(D1)
The only difference between this and Eq. (2) is the detuning in the third term on the denominator, which is given by δnP±2ωMW=ωnP(ωL±2ωMW). In principle, it is possible to derive the sideband amplitude by using these susceptibilities as source terms. However, this calculation is strongly dependent on the unknown relative phase of each term in the summation.32 
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