Microwave-optical coupling via Rydberg excitons in cuprous oxide

We report exciton-mediated coupling between microwave and optical fields in cuprous oxide (${\mathrm{Cu}}_2\mathrm{O}$) at low temperatures. Rydberg excitonic states with principal quantum number up to $n=12$ were observed at 4 $\mathrm{K}$ using both one-photon (absorption) and two-photon (second harmonic generation) spectroscopy. Near resonance with an excitonic state, the addition of a microwave field significantly changed the absorption line shape, and added sidebands at the microwave frequency to the coherent second harmonic. Both effects showed a complex dependence on $n$ and angular momentum $l$. All of these features are in semiquantitative agreement with a model based on intraband electric dipole transitions between Rydberg exciton states. With a simple microwave antenna we already reach a regime where the microwave coupling (Rabi frequency) is comparable to the nonradiatively broadened linewidth of the Rydberg excitons. The results provide an additional way to manipulate excitonic states, and open up the possibility of a cryogenic microwave to optical transducer based on Rydberg excitons.

Figure 1

Effect of a microwave field on the one-photon absorption spectrum of ${\mathrm{Cu}}_2\mathrm{O}$. (a) Antenna A1 and (b) antenna A2 used to deliver microwave fields to the sample. (c) Energy level diagram for the one-photon experiment. LED light probes odd-parity (P) states. A microwave field of frequency $f_{\mathrm{MW}}$ introduces coupling between the even- and odd-parity states. (d) Broadband transmission spectrum with (blue) and without (purple) a microwave field at frequency $f_{\mathrm{MW}}=15.0\mathrm{G}\mathrm{Hz}$ using antenna A2. Exciton states from 5P to 11P are visible. The effect of the microwave field is more prominent at higher $n$. (e) Relative change in transmission $\mathrm{\Delta }T/T_{\mathrm{off}}$, due to the microwave field at $f_{\mathrm{MW}}=15.0\mathrm{G}\mathrm{Hz}$ using antenna A2. The intensity is increased at P resonances and decreased at the energies of the even-parity states, indicating a mixing between the even- and odd-parity states. (f) Heat map of $\mathrm{\Delta }T/T_{\mathrm{off}}$ as a function of excitation energy $E$ and microwave frequency $f_{\mathrm{MW}}$. Note that (e) is a cross section of this heat map at $f_{\mathrm{MW}}=15\mathrm{G}\mathrm{Hz}$. Fine structure in $f_{\mathrm{MW}}$ dimension is attributed to the frequency response of the antenna (see Appendix pp1).

Figure 2

Second harmonic generation spectroscopy of ${\mathrm{Cu}}_2\mathrm{O}$ with a microwave field. (a) Energy level diagram of the SHG experiment. Labels $D$ and $Q$ indicate whether the step occurs through a dipole or quadrupole process. Two-photon excitation at frequency $f_{\mathrm{IN}}$ excites an even-parity (S or D) exciton. Emission can occur from this state through a quadrupole process at frequency $2f_{\mathrm{IN}}$. The addition of a microwave field of frequency $f_{\mathrm{MW}}$ couples the even- and odd-parity exciton states through electric dipole transitions leading to a four-wave mixing type process and new emission pathways at $2f_{\mathrm{IN}}\pm f_{\mathrm{MW}}$. (b) Experiment block diagram. The seed laser light is sliced into pulses by an electro-optic modulator (EOM) and amplified by a Raman fiber amplifer (RFA), before being focused onto the sample. Backscattered light is collected and detected using a photon counter. A scanning etalon may be inserted to provide additional filtering. (c)(i) Emitted second harmonic intensity $I$ as a function of two-photon excitation energy $E$, with ($I_{\mathrm{on}}$; blue) and without ($I_{\mathrm{off}}$; purple) a microwave field at 19.5 $\mathrm{G}\mathrm{Hz}$. Resonance from $n=7$ to 12 are visible. Solid purple line shows fit to $I_{\mathrm{off}}$. (ii) Fractional change in intensity, $\mathrm{\Delta }I/I_{\mathrm{off}}$, of the excitation spectrum with $f_{\mathrm{MW}}=19.5\mathrm{G}\mathrm{Hz}$. The microwave field alters the SHG spectrum throughout the range of two-photon excitation spectrum. (d) Spectrally resolved emitted second harmonic intensity $I$ at $E=E_{8\mathrm{S}}$, with (blue) and without (purple) a microwave field at $f_{\mathrm{MW}}=19.5 \mathrm{G}\mathrm{Hz}$ as function of etalon detuning $f_{\mathrm{E}}$. Red shaded area shows light doubled through a ppLN crystal for comparison. The microwave field causes the appearance of sidebands on the second harmonic.

Figure 3

Comparison of the predicted and measured change in one-photon absorption due to the microwave field. (a) Change in absorption due to the microwave field, $\mathrm{\Delta }\overline{\alpha }L$, as a function of excitation energy $E$ at microwave frequency $f_{\mathrm{MW}}=15.0\mathrm{G}\mathrm{Hz}$. The range of $E$ spans from the $n=5$ state up to the band edge. Experimental data are shown as solid red line, and theoretical predictions as dashed blue line for ${\mathcal{E}}_{\mathrm{MW}}=400 \mathrm{V}{\mathrm{m}}^{-1}$. (b) Predicted change in absorption as a function of microwave frequency $f_{\mathrm{MW}}$ (shaded background, right axis) at excitation energy $E=E_{7\mathrm{D}}$. Also shown is the measured (points) and predicted (lines) ratio of the change in absorption at $E=E_{nl}$ to $E_{7\mathrm{D}}$ as a function of microwave frequency (left axis). Three different values of $E_{nl}$ are shown: $E_{nl}=E_{8\mathrm{D}}$ (blue circles, solid line), $E_{nl}=E_{9\mathrm{D}}$ (orange squares, dotted line), and $E_{nl}=E_{10\mathrm{D}}$ (gold diamonds, dashed line) states.

Figure 4

Laser and microwave power dependencies of sideband and carrier amplitudes in the SHG experiment. Here the two-photon excitation energy is $E=E_{8\mathrm{S}}$ and the microwave frequency is $f_{\mathrm{MW}}=19.5\mathrm{G}\mathrm{Hz}$. (a) Laser power, $P_{\mathrm{IN}}$, dependency of SHG (triangles), magnitude of carrier depletion (circles), blue sideband (squares) and red sideband (diamonds) at microwave power $P_{\mathrm{MW}}=25\mathrm{m}\mathrm{W}$. Diagonal lines show the predicted quadratic dependencies on laser power. (b) Microwave power dependencies of the magnitude of carrier depletion (circles), blue sideband (squares) and red sideband (diamonds) at $P_{\mathrm{IN}}=50\mathrm{m}\mathrm{W}$. Solid lines show predicted power dependencies from equations (6) and (7).

Figure 5

Dependence of sideband and carrier intensities on two-photon excitation energy $E$. (a)–(c) Show the spectrally resolved change in second harmonic intensity due to a microwave field at $f_{\mathrm{MW}}=19.0\mathrm{G}\mathrm{Hz}$ for three different two-photon excitation energies $E_{8\mathrm{S}}, E_{8\mathrm{P}}$, and $E_{8\mathrm{D}}$, respectively. An asymmetry is observed, with the blue sideband larger when two-photon resonant with a S state and the red sideband larger when two-photon resonant with a D state. (d) Theory curve (solid) from Eq. (6) and experimental data (points) showing the intensity of the blue and red sidebands at microwave frequency $f_{\mathrm{MW}}=19\mathrm{G}\mathrm{Hz}$ and microwave power $P_{\mathrm{MW}}=3\mathrm{m}\mathrm{W}$ as a function of two-photon excitation energy. (e) Theory curve (solid) from Eq. (8) and experimental data (points) showing change in carrier intensity at $f_{\mathrm{MW}}=19\mathrm{G}\mathrm{Hz}$ and $P_{\mathrm{MW}}=3\mathrm{m}\mathrm{W}$ as a function of two-photon excitation energy. The shaded background in (d) and (e) is the fit to the SHG excitation spectrum in Fig. 2 to show the range of exciton states explored. Dashed vertical lines indicate the three two-photon excitation energies of the etalon scans in parts (a)–(c).

Figure 6

Time dependence of the second harmonic microwave response compared with turn on and off times of microwave pulse. Normalized SHG intensity (not spectrally resolved) as a function of time relative to the microwave pulse trigger. Data taken with $E=E_{8\mathrm{S}}, f_{\mathrm{MW}}=19.5\mathrm{G}\mathrm{Hz}$, and $P_{\mathrm{MW}}=25\mathrm{m}\mathrm{W}$. The microwave field causes a decrease in the intensity of the light detected (blue data). The red curve shows the microwave pulse shape as measured on an oscilloscope. Insets show zoom of the turn on and turn off times.

Figure 7

Microwave frequency dependence of antennas. (a) Renderings of antenna A1 (i) and A2 (ii). (b) Rendering of components included in microwave field simulations of antenna A1 (i) and A2 (ii). The antenna is positioned in the center between two stainless steel lens tubes (translucent gray) and support by a copper mounting plate (yellow). (c) Measured microwave frequency dependence of the change in absorption at excitation energy $E=E_{9\mathrm{D}}$ (dashed red, left axis), and the corresponding variation of the modulus squared of the simulated microwave field strength inside the sample (solid orange, right axis) for antenna A1 (i) and A2 (ii).

Figure 8

Microwave frequency dependence of sideband intensities. (a) Predicted microwave frequency dependence of blue sideband intensity. Excitation energies resonant with three different states are plotted: $E=E_{6\mathrm{S}}$ (solid), $E=E_{8\mathrm{S}}$ (dashed), $E=E_{10\mathrm{S}}$ (dotted). (b) Ratio of red sideband intensity at two excitation energies ($E=E_{8\mathrm{D}}$ and $E=E_{9\mathrm{D}}$) as function of microwave frequency. Solid line shows predictions from Eq. (6) with ${\mathcal{E}}_{\mathrm{MW}}=200\mathrm{V}{\mathrm{m}}^{-1}$. Points are experimentally measured.