Probing the Coherence of Solid-State Qubits at Avoided Crossings

Optically addressable paramagnetic defects in wide-band-gap semiconductors are promising platforms for quantum communications and sensing. The presence of avoided crossings between the electronic levels of these defects can substantially alter their quantum dynamics and be both detrimental and beneficial for quantum information applications. Here we present a joint theoretical and experimental study of the quantum dynamics of paramagnetic defects interacting with a nuclear spin bath at avoided crossings. We find that we can condition the clock transition of the divacancies in $\mathrm{Si}\mathrm{C}$ on multiple adjacent nuclear spins states. We suppress the effects of fluctuating charge impurities and demonstrate an increased coherence time at clock transition, which is limited purely by magnetic noise. Our results pave the way to designing single defect quantum devices operating at avoided crossings.

Popular Summary

Quantum technologies have the potential to create next-generation sensors and communication networks—and doing so requires building scalable platforms in which quantum bits (“qubits”) can be controlled individually and retain information for a long time. Optically addressable spin defects in wide-band-gap semiconductors are promising platforms for quantum technologies. However, controlling their operation is not a trivial task, and it requires understanding the coupling between the electronic defects and the surrounding bath of nuclear spins and how the electronic structure of spin defect influences their quantum dynamics. For example, the properties of spin defects and their interaction with nuclear spins can be substantially altered when avoided crossings emerge in the defect’s electronic structure. These specific electronic configurations can be both beneficial and detrimental to quantum technologies.

In this work, we address the effect of the strength of the nuclear coupling on the qubit dynamics at avoided crossings both theoretically and experimentally. We show how the emergence of avoided crossings can lead to changes in the primary source of errors in a qubit state and a transition from quantum to classical noise regimes.

For the first time, our results suggest possible ways to exploit nuclear spins in quantum devices operating close to avoided crossings. Combined with ab initio predictions of the spin Hamiltonian parameters, the theoretical framework utilized here allows one to predict and optimize the coherence properties of solid-state spin qubits yet to be explored experimentally.

Figure 1

(a) Schematic representation of the cluster-correlation expansion method used in this work (denoted as gCCE). Each cluster includes the central spin levels (denoted as $|+⟩$, $|-⟩$, and $|0⟩$ for spin 1 with nonzero longitudinal $D$ and transverse $E$ zero-field splitting). The interactions between clusters are treated at the mean-field level (${\hat{H}}_{\mathrm{mf}}$), using Monte Carlo sampling of bath states. (b) Schematic representation of the axial kk divacancy. (c) Schematic representation of the basal kh divacancy.

Figure 2

Ensemble Hahn-echo coherence of the axial kk divacancy in $4H$-$\mathrm{Si}\mathrm{C}$. (a) Coherence time $T_2$ (blue circles) as a function of the magnetic field $B_z$. The experimental results (red circles) and conventional CCE predictions (black circles) are from Ref. [5]. Gray shading denotes the range of $T_2$ shown in (b). (b) Experimental (red circles) and predicted (blue circles) coherence times $T_2$ as a function of energy splitting between qubit levels. The black line shows the pure dephasing results. (c) The oscillations in the ratio of the diagonal elements of the density matrix of the divacancy as a function of time for various values of the magnetic field. The dashed lines show the $\pm 2\mathrm{\%}$ range. (d) The off-diagonal elements of the density matrix for different magnetic fields computed using the gCCE method with Monte Carlo sampling of the bath states (color) and using the conventional CCE method (black).

Figure 3

Ramsey interferometry for three experimental kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ systems. (a)–(f) Ramsey precession and frequency spectrum for a defect (a),(b) with only weakly coupled nuclear spins (VVA), (c),(d) with one strongly coupled nuclear spin (VVB), and (e),(f) with three strongly coupled nuclear spins (VVC). For each defect, we show theoretical predictions and experimental results. White dashed lines show the positions of the hyperbolae [see Eq. (4)]. (g) Measured Ramsey precession of VVA at zero field with (black) or without (red) charge depletion (see the text), compared to the theoretical prediction (blue). The shaded area corresponds to the theoretically predicted decay. (h),(j) Distribution of $T_2^{\ast }$ for VVA (h) and VVB (j) as a function of the magnetic field ($B_z$). Shaded area in (j) shows the error of the fit. (i) Measured Ramsey precession of VVB at weak applied magnetic field ($B_z=12.5\mu \mathrm{T}$) compared to the theoretical prediction. Error bars correspond to 2SD.

Figure 4

Single defect Hahn-echo coherence time of the kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy. (a) Distribution of $T_2$ for VVA (left) and VVB (right) as a function of the magnetic field $B_z$ (VVA and VVB are the same defects represented in Fig. 3). (b) The Hahn echo decay for three different values of the magnetic field. Solid lines correspond to theoretical predictions, points to experimental measurements. Error bars correspond to 2SD.

Figure 5

Single defect coherence times $T_2^{\ast }$ and $T_2$ of the kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy at zero and high magnetic fields for different bath couplings. (a),(b) The coherence time $T_2^{\ast }$ for zero (a) and high (b) fields as a function of the square root of the sum of squares of the hyperfine couplings with bath spins $\sqrt{\sum _i{A}_{iz}^2}$. Significant deviations from the least-squares fit (solid lines) are present for systems containing single nuclear spins with high hyperfine coupling, and the coherence decay may not be approximated by a single exponential [39]. (c) The coherence time $T_2$ for zero (blue) and high (red) magnetic fields, and for a hypothetical system with $E=0$ MHz at zero field (black). All calculations are performed for $\mathrm{Si}\mathrm{C}$ with natural isotope concentration for the weakly coupled nuclear bath.

Figure 6

Dependence of the coherence time on the transverse zero-field splitting. (a),(b) Coherence times $T_2^{\ast }$ (a) and $T_2$ (b) of the kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy with different hypothetical values of transverse ZFS $E$. (c),(d) The heat map of the single defect $T_2^{\ast }$ ($T_2$) for 120 different nuclear spin configurations of both the hk-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ (red) and kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ (blue) divacancies as a function of the magnetic field. The color corresponds to the number of configurations with a given $T_2^{\ast }$ ($T_2$) at the given magnetic field. The circles (squares) indicate the ensemble averaged coherence time.

Figure 7

Single defect coherence of the basal kh divacancy in $4H$-$\mathrm{Si}\mathrm{C}$ predicted by different theoretical approximations. (a),(b) The absolute value of the off-diagonal elements of the density matrix $|{\rho }_{0+}|=|⟨0|\hat{\rho }|+⟩|$ for one random nuclear configuration corresponding to Ramsey (a) and Hahn-echo (b) experiments. The results of the gCCE at different orders (from 1 to 3) without mean-field (MF) interactions (left) and with mean-field corrections (right). The exact solution for a bath of nine nuclear spins for the Ramsey decay is shown as a black line.

Figure 8

Off-diagonal element of the density matrix of the qubit at zero magnetic field for various decoupling pulses $N$.

Figure 9

Coherence time of the basal divacancy at zero magnetic field as a function of the number of decoupling pulses $N$.

Figure 10

The convergence of the off-diagonal element ${\rho }_{+0}$ of the density matrix in Hahn-echo measurements of the kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy as a function of the number of random bath states.

Figure 11

The convergence of diagonal elements ${\rho }_{-1-1}$ and ${\rho }_{11}$ of the density matrix in Hahn-echo measurements of the kh-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy as a function of the number of random bath states. Calculations are performed for five random nuclear spin configurations at zero magnetic field. Gray dashed lines correspond to the $\pm 10\mathrm{\%}$ range. We set ${\rho }_{00}=0.5$ for all calculations.

Figure 12

Density matrix elements of the $kh$-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy for two random nuclear spin spatial configurations of a weakly coupled nuclear bath (in columns) as a function of the magnetic field. The lines with brighter color correspond to the exact solution for a smaller bath (nine nuclear spins). The dotted white line corresponds to the initial value of ${\rho }_{ii}$. We set ${\rho }_{00}=0.5$ for all calculations.

Figure 13

Density matrix elements of the $kh$-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy for a nuclear configuration containing a strongly coupled nuclear spin with $A_{iz}=8.93$ MHz at low magnetic fields. The color indicates the ratio of $|a|$ and $|b|$ in the initial $|+⟩=a|-1_z⟩+b|+1_z⟩$ state of the qubit. In each case, the qubit is initially prepared in the state $|\psi ⟩=(1/\sqrt{2})(|+⟩+|0⟩)$. The lines with brighter color correspond to the exact solution with the same initial state for a smaller bath (nine nuclear spins). The dotted gray line corresponds to the initial value of the density matrix diagonal element. We set ${\rho }_{00}=0$ for all calculations.

Figure 14

The distribution of coherence times $T_2^{\ast }$ (top) and $T_2$ (bottom) for theoretical nuclear configurations obtained for VVA (red) and VVB (orange). Blue lines correspond to configurations shown in the main text.

Figure 15

Ramsey oscillations in the system with the strongly coupled nuclear spin. Left: ${\rho }_{0+}$ at the magnetic field of 12.5 $\mu \mathrm{T}$ in the presence of the nuclear spin with $A_{iz}=0.75$ MHz initialized in the spin-up ($⟨{\uparrow }_n|$, orange), spin-down ($⟨{\downarrow }_n|$, blue), or thermalized (green) state. Right: $T_2^{\ast }$ against the magnetic field for a system with differently initialized strongly coupled nuclear spins. All other nuclear spins are completely randomized.

Figure 16

Coherence time against the magnetic field for the theoretical configurations containing strongly coupled nuclear spins. Numbers (1–12) correspond to different bath configurations with one or several strongly coupled nuclear spins (Table 1).

Figure 17

Coherence time of the $kh$-${\mathrm{V}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{Si}}$ divacancy for Ramsey (top) and Hahn-echo (bottom) experiments as a function of the transverse ZFS E for five random nuclear spin configurations without strongly coupled nuclear spins.