All-optical noise spectroscopy of a solid-state spin

Noise spectroscopy elucidates the fundamental noise sources in spin systems, thereby serving as an essential tool toward developing spin qubits with long coherence times for quantum information processing, communication, and sensing. But existing techniques for noise spectroscopy that rely on microwave fields become infeasible when the microwave power is too weak to generate Rabi rotations of the spin. Here, we demonstrate an alternative all-optical approach to performing noise spectroscopy. Our approach utilizes coherent Raman rotations of the spin state with controlled timing and phase to implement Carr-Purcell-Meiboom-Gill pulse sequences. Analyzing the spin dynamics under these sequences enables us to extract the noise spectrum of a dense ensemble of nuclear spins interacting with a single spin in a quantum dot, which has thus far only been modeled theoretically. By providing spectral bandwidths of over 100 MHz, our approach enables the studies of spin dynamics and decoherence for a broad range of solid-state spin qubits.

An alternative approach for noise spectroscopy is to use optical fields. For example, optical measurements of resonance fluorescence 13,14 and Faraday rotation 15 have quantified the impact of noise on the dynamics of single spins without applying coherent spin control.
However, these approaches do not operate within the ground state energy manifold of the spin qubit and do not quantify the strength of interactions of different noise frequency components with the spin, as required for the full understanding of spin dynamics and decoherence. In addition, the bandwidth of noise spectroscopy techniques that measure resonance fluorescence or Faraday rotation on a single spin remained slower than the high frequencies (up to 100 MHz [16][17][18][19] ) associated with the noise sources that dictate spin dynamics and decoherence. Probing such high frequency noise of a single spin requires the application of sequences of all-optical coherent control pulses.
In this Letter, we demonstrate all-optical noise spectroscopy of a single solid-state spin utilizing coherent control. Our approach utilizes Raman rotation pulses with controllable timing, amplitude, and phase 20,21 to implement the Carr-Purcell-Meiboom-Gill (CPMG) control sequences. 22 With this approach, we probe the noise source of an ensemble of indium and arsenic nuclear spins (i.e., the Overhauser field) interacting with a single electron spin confined in an InAs/GaAs quantum dot, to extract its spectral density thus far only modeled theoretically. 16,17 Previously demonstrated microwave control of InAs/GaAs quantum dots has been orders of magnitude slower than required for noise spectroscopy due to the low g-factor (∼ 0.4) 23 of the electron confined in the quantum dot. In contrast, the high spectral bandwidths of noise spectroscopy (> 100 MHz) provided by our all-optical approach enable us to probe the Overhauser field, which features high frequencies due to the spread of the Larmor frequencies of the background nuclei at high magnetic fields. The extracted spectra verify a theoretical model that predicts two noise components (in-plane and outof-plane to the external field) caused by an inhomogeneous strain field in the quantum dot environment. 16,17 Understanding of such noise sources can quantify the achievable coherence times of spin qubits and predict the spin dynamics of such qubits for quantum information processing, communication, and sensing. The energy-level diagram of a negatively charged quantum dot under an external magnetic field in the Voigt geometry, including the optical pumping (solid red arrows) and Raman rotation (solid blue arrows) transitions used for noise spectroscopy, as well as the optical decay transitions from the excited state (dashed red arrows). Fluorescence collected from the |↑↓, ⇓ → |↑ optical decay transition indicates the quantum dot spin state. (c) The Ramsey experiment used to characterize our preliminary polarization of the nuclear ensemble. The resulting spin dynamics without (dashed red line) and with (solid blue line) polarization of the nuclear spin environment at an external field of B = 2.4 T fitted to the function ae −(T /T * 2 ) p shows the electron spin coherence time can be extended from T * 2 ≈ 2 ns (dashed red line) to T * 2 ≈ 48 ns (solid blue line). From the fits, we extract a ≈ 0.13 and a ≈ 0.5 with and without the nuclear polarization, respectively, and p ≈ 1.85 for both curves.
We perform the all-optical noise spectroscopy using a negatively charged quantum dot (Section I of the Supporting Information), 9,24-30 which acts as a quantum probe for the fluctuations of nuclear spins in its environment. Under an external magnetic field applied perpendicular to the sample growth direction (Voigt geometry), the electronic structure of these dots consists of an electron ground-state spin qubit ({|↑ , |↓ }) and two optically excited trion states ({|↑↓, ⇑ , |↑↓, ⇓ }). 9 Spontaneous decay from the excited states leads to fluorescence emission of single photons with high efficiency and indistinguishability. 25,26,28,29 The main noise source that leads to the decoherence of the quantum dot spin, S(ω), is the Overhauser field of indium and arsenic nuclear spins, consisting of frequencies of ∼ tens of MHz 16,17 (Section II of the Supporting Information). Here, we perform noise spectroscopy of the Overhauser field utilizing optical CPMG pulse sequences.
The CPMG sequences 22 are illustrated in Figure 1a. After a preliminary 4-µs long stage of nuclear spin polarization, 21 we initialize the spin in the |↑ +|↓ √ 2 state using an optical pumping pulse followed by a π 2 -rotation pulse (Section I of the Supporting Information). Then, we apply n ≥ 1 equally spaced π-pulses that modify the temporal dynamics of the spin. After the application of a second π 2 -pulse, a final optical pumping pulse induces a fluorescence signal that indicates the final spin state. We define the coherence function, C(T ), as the decay of the fluorescence signal as a function of the spin interrogation time, T (i.e., the time between the π 2 -pulses). To implement the all-optical coherent control pulses required for the CPMG sequences, we utilize detuned two-photon Raman excitations 20 (solid blue arrows in Figure 1b) enabled by a modulated laser. We perform this modulation using an arbitrary waveform generator, which introduces up to eight spin rotation π-pulses with precise timing, phase, and Rabi frequencies of up to ≈ 150 MHz (see Section III of the Supporting Information for the optimization of the parameters). Leveraging the arbitrary modulation capabilities with sampling rates of up to 65 Gs/s allows us to realize CPMG sequences at short times and desired intervals, which is essential for obtaining noise spectra with sufficiently high bandwidth and spectral resolution (Section IV of the Supporting Information).
To ensure that spin dephasing is minimal during the application of the pulses, we use the Raman coherent control to polarize the nuclear spin ensemble prior to the application of any CPMG sequence. 21 This step suppresses the uncertainty in the total magnitude of the Overhauser field at the beginning of an experiment, thereby reducing the inhomogeneous dephasing of the quantum dot spin. As shown by a Ramsey experiment (Figure 1c; see Section I of the Supporting Information), such a 4-µs long nuclear polarization step increases the inhomogeneous dephasing time of the quantum dot spin from T * 2 ≈ 2 ns (dashed red line in Figure 1c) to T * 2 ≈ 48 ns (solid blue line in Figure 1c), consistent with previous Ramsey experiments. 21 After completing the nuclear polarization step, the inhomogeneous dephasing time is an order of magnitude longer than the spin rotation π-pulses, which allows us to use such pulses for the realization of CPMG sequences for high-frequency noise spectroscopy.
We first apply the simplest form of the CPMG sequence, namely the Hahn-echo experiment consisting of a single π-pulse ( Figure 2a). Consistent with previous Hahn-echo measurements on quantum dots, 17,31,32 the decay timescale of the spin dynamics (i.e., the coherence time) increases as a function of the external magnetic field, B, up to T 2 ≈ 1 µs (blue dots in Figure 2a). This increase is associated with the Zeeman terms of the indium and arsenic nuclei dominating over the inhomogeneous broadening of these nuclei at high magnetic fields (Section II of the Supporting Information). Furthermore, the spin exhibits a two stage decay in its coherence that consists of a fast drop of the signal contrast (at T ≈ 30 ns), followed by a second decay (starting at T ≈ 100 ns). This behavior is analogous to previously observed Hahn-echo spin dynamics of quantum dots. 17,32 Intuitively, the two stages of decoherence suggest that separate spectral components of the noise affect the spin dynamics at separate timescales. Simulation results (the lines in Figure 2a), which consider such separate noise components associated with the strained nuclear environment of the quantum dot 16,17 (Section II of the Supporting Information), agree with the experimental results, thereby confirming this hypothesis. To experimentally extract these spectral noise components, we apply CPMG sequences with increasing numbers of pulses.
The temporal dynamics of the quantum dot spin under the application of such sequences are presented in Figure 2b and Figure 2c for the external magnetic fields of 1.2 T and 2 T, respectively. As shown in Figure 2b for B = 1.2 T, the measured spin coherence times  Table 1 of the Supporting Information) that dominate the interaction with the electron spin due to their spin-9/2 nature. Meanwhile, the amplitude of the noise (at the central frequency) decreases with the magnetic field ( Figure   3e) as nuclear Zeeman interactions dominate over the broadening of the nuclei due to strain fields (Section II of the Supporting Information). 16,17 The extracted noise spectra verify a previously established theoretical model of the Overhauser field. 16,17 Using this model, we simulate the noise spectra that represent the hyperfine coupling of the quantum dot spin to indium and arsenic nuclear spins experiencing quadrupolar coupling to strain fields (insets of Figure 3a-c), which exhibit sharp peaks that correspond to the different nuclear spin numbers. To compare the experimental results of noise spectroscopy with theory, we first use the theoretical spectra to calculate CPMG coherence functions with n = 1, 2, 4 and 8 π-pulses under ideal conditions. We then apply the algo- prolonging spin coherence times and for the preservation of arbitrary spin states (e.g., XY8based sequences 33 ) for quantum information processing. Furthermore, the implementation of multi-pulse sequences with ultra high spectral resolutions (e.g., the "DYSCO" sequence 4 ) may enable the identification of individual Larmor frequencies associated with nuclear species (e.g., the peaks in the insets of Figure 3), as well as the ultrahigh resolution probing of external fluctuating magnetic fields with a single spin.
To conclude, we introduce an all-optical approach for noise spectroscopy and implement it to study the environment of InAs quantum dots, for which the application of microwave control is challenging. The high Rabi frequencies and precise control capabilities of the Raman approach provide spectral bandwidths (> 100 MHz) that enable the identification of high frequency noise spectra. Leveraging the diffraction-limited spatial resolution provided by the optical fields could enable the measurement of noise correlations within neighboring regions of a given sample. Performing all-optical noise spectroscopy while modifying material properties (e.g., concentrations of dopants, layer thicknesses) could lead to the design of novel semiconducting heterostructures for quantum information processing in a variety of materials that host optically-active spins. The availability of optical coherent control in the solid-state community could enable such studies of spins in direct narrow bandgap semiconductors such as AlGaAs, 10 InP 11 and ZnSe, 37 as well as color center spins with large spin-orbit couplings in wide bandgap semiconductors such as diamond. 12 In addition, noise spectroscopy of spins coupled to photonic structures could quantify the impact of the structures on spin coherence, thereby leading to the design of optimal photonic platforms for spin-photon interfaces. The experimentally extracted spectral densities, S(ω), can be plugged into equations of spin dynamics under the application of protocols for quantum sensing, communication, and information processing, thereby evaluating the potential of optically-active spin qubits for quantum technologies.

Supporting Information
Experimental methods, theoretical modeling of the noise, optimization of experimental parameters, spectral decomposition algorithm, and studies of laser-induced tunneling.