Spectrally reconfigurable quantum emitters enabled by optimized fast modulation

The ability to shape the spectrum of a single photon enables communication between disparate physical systems in a quantum network, as well as the creation of high-dimensional frequency-encoded quantum states of light. Spectral control in solid state platforms such as color centers, rare earth ions, and quantum dots is particularly attractive for realizing such protocols on-chip. Here we propose the use of frequency-modulated optical transitions for spectral engineering of single photon emission. Using a scattering-matrix formalism, we find that a two-level system, when modulated faster than its optical lifetime, can be treated as a single-photon source with a widely reconfigurable photon spectrum that is amenable to standard numerical optimization techniques. To enable the experimental demonstration of this spectral control scheme, we investigate the Stark tuning properties of the silicon vacancy in silicon carbide, a color center with promise for optical quantum information processing technologies. We find that the silicon vacancy possesses excellent spectral stability and tuning characteristics, allowing us to probe its fast modulation regime and demonstrate spectral engineering. Our results suggest that frequency modulation is a powerful technique for the generation of new light states with unprecedented control over the spectral and temporal properties of single photons.

tem (TLS). 9 However, since unperturbed spontaneous emission only produces photons with a Lorentzian spectrum, significant effort has been devoted to exploring more complex systems for spectral control of single photon emission. For instance, a TLS with a time-dependent coupling to a cavity has been studied for symmetrization of single-photon wavepackets, 10,11 and cascaded three-level systems and lambda systems have been used for partly-stimulated two-photon emission 12 and Raman emission, 13, 14 respectively. A system-agnostic approach to spectral shaping is post-processing emitted photons using electro-optic phase modulators 15 and nonlinear frequency conversion, 16,17 at the expense of additional system complexity and loss.
Here, we propose a comprehensive alternative to these techniques that requires only a TLS whose transition energy can be rapidly modulated. The time-modulated TLS has been studied in other contexts since the early days of quantum mechanics; 18 its application in atomic systems, 19 superconducting qubits, 20 and solid state defects [21][22][23] has allowed for the demonstration of fundamental phenomena such as spectral sideband formation, Landau-Zener-Stückelberg interference, and motional averaging. Here, we examine the fast time-modulated TLS as a single-photon source. To this end, we study the fewphoton scattering properties of the modulated TLS, as well as its single-photon-emission fidelity under pulsed resonant drive. We find that, remarkably, in the fast modulation regime (i.e., modulation faster than the optical lifetime) the modulated TLS can be treated like a conventional two-level system but with an exotic, reconfigurable spectrum. We experimentally characterize static and time-modulated Stark shift in single silicon vacancy (V Si ) color centers in silicon carbide (SiC), and observe that the optical coherence properties of the V Si are preserved even under high-amplitude modulation. With this system, we demonstrate the proposed spectral engineering of single photons from a solid-state TLS, inducing photon emission from the V Si in a superposition of two and four discrete colors. Finally, we propose a pulse compression scheme using aperiodic modulation, capable of temporally compressing output photons by three orders of magnitude.

Continuous-wave scattering off a modulated two-level system
We study the modulated TLS via the quantum optics formalism of Markovian open quantum systems, with the goal of understanding the few-photon statistics and the single-photon spectra of scattered photons. We consider a modulated TLS driven by an arbitrary periodic modulation ∆(t) with period 2π/Ω, Figure 1 | Theoretical analysis of photon scattering from a modulated TLS. a Schematic depiction of single-photon scattering from a modulated TLS with ∆(t) = A sin(Ωt). A continuous-wave photon in the input optical channel is scattered into an output optical channel to produce a superposition of continuous-wave single-photons at frequencies ν + pΩ ∀ p ∈ Z. b The spectrum of the single-photon state in the output channel as a function of frequency ω on exciting a modulated two-level system (Ω = 2.5γ) with a Gaussian singlephoton wavepacket centered at ν with FWHM 0.625γ. c The total transmission in the output optical channel as a function of the frequency of the input photon. d The amplitude of the output twophoton wavepacket as a function of the initial time-instant t and the time-difference τ . e The two-photon correlation function as a function of the time-difference between the two photons in the output wavefunction. The dashed line indicates g (2) (τ ) for an unmodulated TLS. A = 5γ is used in all simulations.
where ω0 is the resonant frequency of the TLS and σ is the deexcitation operator. Furthermore, we impose a decay rate γ corresponding to the system lifetime. The Floquet eigenstates for this Hamiltonian can be computed analytically (see Supplementary Information). To study the form of the emitted photon wavepacket under excitation by a weak coherent state, we use single-and two-photon scattering matrices of the modulated TLS. While scattering matrices are traditionally computed only for time-independent systems, it was recently shown that they can be defined and computed for time-dependent systems. 24,25 As is shown in the Supplementary Information, the single-photon scattering matrix through the modulated TLS S(ω, ν), defined as the amplitude of producing an output photon at frequency ω when the system is excited with an input photon at frequency ν, can be expressed as: where where γi,o is the coupling rate into the input (output) channel, γ = γi + γo is the total decay rate of the two-level system, and αm is the Fourier-series coefficient of the phase exp −i t 0 ∆(t )dt accumulated by the excited state of the modulated TLS up to time t, corresponding to the harmonic exp(−imΩt). We note that the form of S(ω, ν) implies that the output photon state corresponding to an input photon state at frequency ν has frequencies ν + pΩ, p ∈ Z as a consequence of the periodic modulation of the emitter frequency (see Fig. 1a). In particular, a narrowband incident photon wavepacket would scatter from the modulated two-level system into different modulation sidebands (Fig. 1b). In the slow modulation regime (Ω0 γ), the total transmission from the TLS (T (ν)), computed as a sum of transmission into individual sidebands (|Sp(ν)| 2 ) , is simply a time-average of transmission spectra with different resonant frequencies. The fast modulation regime (Ω0 γ) is characterized by distinctive sidebands with high transmission at the resonant frequency (see Fig. 1c). The scattering amplitudes into different sidebands can be controlled by an appropriate choice of the time-dependent frequency modulation ∆(t).
For non-classical light generation, especially single-photon generation, it is of utmost importance to understand photon blockade through the modulated TLS. In this context, of particular interest is the scattering of a monochromatic twophoton pair at frequency ν from the modulated TLS. The output state |ψout(ν) can be described by its wavefunction Left panels: the target spectrum is denoted by red crosses (omitted for target value of zero). The optimized spectrum is shown in blue. Below, the corresponding ∆(t) is shown for two periods. Right panels: The resulting Floquet spectra are shown, with optical detuning on the x-axis and fundamental drive frequency Ω on the y-axis. The spectral harmonics of a Floquet eigenstate are dictated by the normalized drive amplitude A/Ω. By scaling ∆(t) with Ω, the separation of spectral peaks can be controlled while retaining their amplitude. ψout(t, t + τ ; ν), which represents the amplitude of one photon in the output being emitted at time t and the second photon being emitted after a delay of τ . A detailed calculation of ψout(t, t + τ ; ν) in terms of the modulation ∆(t) within the framework of the scattering theory is outlined in the Supplementary Information. Figure 1d shows G(t, τ ; ω0) = |ψout(t, t+ τ ; ω0)| 2 under different modulation regimes; in all cases, it can be seen that G(t, 0; ν) = 0. This implies that any modulated TLS shows perfect photon blockade. In the slow modulation regime, the amplitude of the two-photon wavepacket varies periodically with the delay τ as a consequence of the periodic drive. In the fast modulation regime, a continuous-wave exci-tation only significantly addresses one of the Floquet sidebands and consequently the scattering properties resemble that of an unmodulated TLS. Figure 1e shows the two-photon correlation function g (2) (τ ; ω0) = G(t, τ ; ω0) t/T 2 (ω0). As with G(t, τ ; ω0), we observe that g (2) (τ ; ω0) oscillates with τ in the slow modulation regime, and resembles the correlation function of an unmodulated TLS in the fast modulation regime.
Using a scattering matrix formalism, we have found that in the limit of fast drive a modulated TLS behaves very much like a conventional single-photon source but with a controllable spectrum, determined by the form of the drive ∆(t). Thus, a modulated TLS can serve as a source of photons whose spectral composition is amenable to standard numerical optimization techniques; one can optimize ∆(t) given a target single-photon spectrum. Figure 2 shows examples of single-photon spectra that can be obtained via such optimization (see Methods). We note that due to the periodicity of ∆(t), this approach is limited to producing photons with discrete, comb-like spectra. This restriction is lifted for aperiodic ∆(t), which we consider in a later section.

Reconfiguring photon spectra using optimized periodic modulation
To realize the proposal experimentally, it is essential to identify a single-photon source amenable to fast, broadband modulation. A suitable solid-state defect must possess a widely tunable optical transition while displaying minimal spectral diffusion in a stationary state and under rapid modulation. Here, we study the Stark tuning and the spectral stability of single V Si color centers in SiC, and find the V Si to be an excellent candidate for spectral engineering via time modulation. The V Si is a color center with promise for quantum information processing technologies due to its long spin coherence time, 27 unique 3/2 spin system, 28,29 and compatibility with scalable photonic architectures. 30 The V Si spectrum comprises two optical transitions, which are separated by 1 GHz via spin-spin coupling (corresponding to spin ±1/2 and ±3/2 sublevels). The V Si occurs at two inequivalent lattice sites in the SiC lattice, denoted by h and k, with the zero-phonon lines at 861 nm and 916 nm, respectively. The properties of the defects are largely similar; 31, 32 one distinction is that the optical coherence of k-V Si is more robust against dephasing caused by acoustic phonons (characterized by narrower linewidths at elevated temperatures). 33 We first probe the static Stark shift of the V Si by applying a voltage across gold electrodes fabricated on the surface of SiC (see Supplemental Information), and measure the single-defect spectrum (see Methods). All measurements are performed at 5 K via resonant absorption spectroscopy, i.e., photoluminescence excitation (PLE). As shown in Fig. 3a, we observe that the zero-phonon line of the V Si can be tuned by 200 GHz without degradation of spectral properties: in other solid-state emitters, the degradation manifests as blinking, charge conversion, or carrier tunneling. 34,35 The measurement is performed by incrementing the applied voltage in steps of 0.5 V and sweeping a tunable laser, programmed to track the frequency of the V Si as it shifts. From electrostatic simulation of the electrodes and the Lorentz local field approximation, 36 we calculate the local electric field strength at the V Si location (assuming the defect position is accurate within 1 µm 3 ), and deduce a strong Stark shift of 3.65 ± 0.09 GHz/(MV/m). This corresponds to an electric dipole moment of 0.72 ± 0.02 Debye, in disagreement with the theoretical prediction of 0.2 Debye. 37 We note that a recent experimental study in V Si ensembles estimated the dipole moment to be 0.18 Debye. 38 Nevertheless, the wide-range, high resolution characterization of the Stark shift of single color centers presented in this work gives us confidence in the dipole moment magnitude we report. A further investigation of spectral diffusion properties and Stark shift for k-and h-V Si is presented in the Supplementary Information.
We then proceed to characterize the time-modulation properties of the V Si . Applying sinusoidal modulation ∆(t) for a range of frequencies and amplitudes, we observe that the V Si spectrum matches the prediction of the scattering matrix theory (Fig. 3b). Crucially, we see that the V Si spectral stability is not impacted by the periodic drive. The strongest modulation, at Ω/2π = 10 GHz (Fig. 3c), corresponds to a maximum optical transition slew rate of 1 GHz ps −1 . At this slew rate, the V Si transition traverses its 30 MHz transform-limited linewidth 1.8×10 5 times within the optical lifetime of 5.5 ns. Even under such rapid modulation, the optical transitions remain as narrow as in the unmodulated V Si , and only three times broader than the transform-limited linewidth.
To generate spectrally-engineered Floquet states, we drive the V Si periodically with optimized harmonics. To demon-strate the range of achievable spectra, we prepare the V Si in Floquet states that emit photons in an equal superposition of two and four colors. We restrict the optimization to a bandwidth of 6 GHz, limited by the external microwave losses. Figure 4a(d) shows the theoretical spectrum resulting from the optimization of a two-(four-) color photon spectrum. The parameters of the corresponding microwave drive ∆(t) are shown in Fig. 4b(e). The experimentally-generated two-(four-) color state is shown in Fig. 4c(f). We note that the PLE measurement is performed in the weak-excitation regime, and thus probes the spectral composition of the photons spontaneously emitted by the system.
We have thus far considered a modulated TLS under continuous-wave optical excitation. However, it is also important to understand the behavior of the modulated TLS under pulsed optical excitation; this would determine whether a modulated emitter could be used as a deterministic source of shaped single photons. In a traditional TLS single-photon source, ondemand generation of highly-indistinguishable photons relies on coherent excitation with a short π-pulse, which prepares the system in the excited state with high fidelity, thus generating very nearly one-and-only-one photon per excitation. 39 The πpulse must be resonant with the TLS, as a detuned pulse would induce off-resonant Rabi oscillations, which cannot efficiently transfer population.
To determine the feasibility of high-fidelity single-photon generation from a modulated quantum emitter, we investigate numerically the behavior of the modulated V Si under pulsed optical drive using experimental parameters corresponding to the high amplitude modulation in Fig. 3c. We model the excited state population dynamics upon excitation with a short Gaussian optical pulse centered on one of the sidebands (Fig. 5a). We find that, unsurprisingly, the population dynamics are strongly dependent upon the phase of the periodic modulation ∆(t) relative to the arrival time of the optical pulse, as shown in Fig. 5b. Clearly, the simple π-pulse prescription used in an unmodulated TLS for complete population transfer from the ground to the excited state does not apply. Instead, the phase and the pulse area are two free pa-rameters that determine the single-photon generation fidelity (incidentally, both can be precisely controlled in experiment). For a range of experimentally-relevant pulse widths, we search the two-parameter space to compare the optimal single-photon generation fidelity of a modulated TLS to that of the unmodulated TLS: we evaluate the pulse-wise two-photon correlation function g (2) [0] as well as the expected number of emitted photons per pulse. Crucially, we find that the single-photon generation fidelity for a modulated TLS is very similar to that of the time-independent TLS, 40 reinforcing the potential utility of the modulated TLS as a deterministic single-photon source.

Aperiodic modulation for bandwidth expansion
Fast periodic modulation of a TLS restricts the photon to a discrete, comb-like spectrum, a restriction that is lifted for aperiodic modulation. We explore numerically an application of We analyze numerically a V Si modulated with ∆(t) = (16 GHz) sin(2πt(10 GHz)) and excited by a Gaussian laser pulse (green) centered at the first sideband. b. Numerical simulation of the population dynamics of a two-level system (initially in the ground state) shows a strong dependence on the relative phase between the microwave drive and the optical pulse (FWHM of 166 ps, pulse area of 10π). c Expected number of photons emitted for different pulse widths. d Pulse-wise two-photon correlation function for different pulse widths, indicating the contribution of multi-photon emission events. g (2) [0] approaching zero with σ † σ remaining near unity indicates a single-photon source that emits one-and-only-one photon with high fidelity. aperiodic modulation to bandwidth expansion of single photons, enabled by the high slew rate attainable with Stark shift modulation. The approach is illustrated in Fig. 6. In the absence of modulation, the TLS emits narrowband Lorentzian photons (Fig. 6a). By frequency-chirping the TLS in the process of photon emission (Fig. 6b), it is possible to match the spectral composition of the single-photon wavepacket to that of a much broader Gaussian pulse (Fig. 6c) (see Supplementary Information). However, the emitted photon is chirped (Fig. 6d). Through dispersion correction (e.g., by passing through a chirped fiber Bragg grating 41 ), the photon can be converted into a short, nearly-Guassian pulse (Fig. 6e,f). Using the V Si tuning parameters obtained in experiment, we find that bandwidth expansion by three orders of magnitude is achievable. Such bandwidth control is particularly beneficial for connecting quantum nodes that operate at very different timescales, such as quantum dots, superconducting qubits, and rare earth ions. Temporal compression would also ease the power requirements for quantum frequency conversion in a waveguide, where the critical pump power for unity conversion is inversely proportional to the signal pulse duration.

Conclusion
Here, we have proposed and demonstrated spectral optimization of a quantum emitter and showed that a simple modulated two-level system can be used as a spectrally reconfigurable deterministic single-photon source. Using a scattering matrix formalism, we develop a rigorous model of this system, and use the model to engineer unconventional photon states. Using color centers modulated via the Stark shift, we experimentally demonstrate spectral shaping and substantiate the feasibility of extreme temporal manipulation of emitted photons. As this approach to spectral control of single photon emission requires only a rapidly modulated optical transition, it should be applicable to other solid-state defects 42 either via the Stark effect or via acoustic modulation. 43 TLS modulation based on the Stark effect is a flexible technique, compatible with integrated nanophotonic cavity systems which enhance atom-photon interactions. 44,45 Cavity integration would enable a solid-state implementation of the fast timemodulated Jaynes-Cummings system, which has received extensive theoretical investigation. 46,47 Furthermore, such a system would enable direct temporal photon shaping via timemodulation of the sideband amplitude, and would enable efficient spectrally-filtered resonant excitation-detection schemes.
Lastly, we note that the experimental demonstration of coherence-preserving rapid spectral modulation of an optical transition may have applications beyond spectral shaping. In atomic and superconducting qubit systems, frequency modulation has been proposed for simulating topological phase transitions, 18 overcoming dephasing, 20 and implementing quantum gates using resolved sidebands. 48 In spin-based solid-state systems, where inhomogeneous broadening plagues the indistinguishability of photons emitted from different quantum nodes, optical transitions that are widely-tunable both statically and dynamically can open pathways toward scalable integrated quantum photonic systems. An appropriately-chosen ∆(t) (not exceeding the maximum demonstrated slew rate of 1 GHz ps −1 ) applied to the V Si initially in the excited state causes the V Si to emit a chirped photon with a broad, near-Gaussian spectrum, as shown in c. d Although the frequency composition of the emission is Gaussian, the pulse is chirped, as seen from the relative phases of the constituent frequencies. e The chirped photon can be phase-corrected by dispersion engineering to be transformed into a short pulse with a near-Gaussian temporal profile. f The resulting dispersion-corrected pulse, now only 13 ps long, deviates only slightly at its center from the Gaussian profile, a nonideality that originates from the fine oscillations in the spectrum, shown in the insets in c.

Floquet spectrum optimization
When the emitter linewidth is much smaller than the modulation period, which is the case for our optimizations, the shape of the spectrum is dictated solely by the Fourier components of exp −i t 0 ∆(t )dt . Thus, we can define our desired Fourier series decomposition of our Floquet state and optimize the Fourier components of exp −i t 0 ∆(t )dt to match those of the desired state. To do this we use the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Our optimization parameters are the real and imaginary parts of all Fourier series components of ∆(t) within the defined bandwidth, and our cost function is the mean squared error of the Fourier series components calculated with the discrete Fourier transform.

Sample Preparation
The experiments were performed using a 100 µm-thick 4H-28 Si 12 C epilayer grown by chemical vapour deposition on a ntype (0001) 4H-SiC substrate. Color centers are generated via electron irradiation. In order to investigate whether the spectral stability of the V Si is influenced by the electron irradiation energy, one sample was irradiated with an average energy of 2 MeV (at QST, Japan) and another at an average energy of 23 MeV (at Stanford SLAC, USA), with a dose of 1 · 10 13 cm -2 and 5 · 10 12 cm -2 , respectively. Samples were annealed for 30 minutes at 300 • C after irradiation. Samples were diced and their edges were polished (DAG 810 from Disco Corp.). Then, 3 µm were removed from the surface with reactive ion etching (using SF6), to minimize the presence of defects that arise from mechanical processing. Gold electrodes were patterned on the sample edge via e-beam lithography and liftoff. No difference in the properties of single V Si was observed between the two samples; however, as expected, the higher-energy irradiation produced a greater fraction of optically-active defects of unknown origin.

Experimanetal Setup
The measurements are performed in a closed-cycle cryostat (Montana Instruments) at a temperature of 5 K. The sample is mounted onto a custom-built circuit board with a microwave stripline optimized for high-frequency operation. The signal is delivered onto the sample with aluminum wirebonds. The cut-off frequency of the microwave setup was measured to be 10.5 GHz. Optical spectra of the V Si are measured via the PLE technique: by scanning a weak resonant laser (power at the objective lens ranging between 50 and 150 nW) across the transition, and detecting only the emission into the phonon side-band via a tunable long-pass filter (Semrock). To control the charge state of the emitter, a 1 µs above-resonant (740 nm) repump pulse is applied at a 1 kHz repetition rate. For DC Stark tuning characterization, voltage is applied to the gold electrodes via a programmable voltage source (Keithley). Singlefrequency microwave drive is delivered via a continuous-wave signal generator with 3.3 GHz bandwidth (Rhode-Shwartz). Engineered multi-frequency microwave drives are generated by an arbitrary waveform generator (Keysight) with amplification (MiniCircuits). A diagram of the experimental setup is shown in Supplemental Figure S1.

Data availability
All data relevant to the current study are available from the corresponding author on request. [49][50][51]

Code availability
The code used for scattering matrix simulations can be accessed at https://github.com/rahultrivedi1995/oqs_scattering