Phonon-induced frequency shift in semiconductor spin qubits

TL;DR

This work develops a phonon-coupled framework for semiconductor spin qubits in silicon quantum dots, deriving an effective low-energy Hamiltonian via Schrieffer-Wolff transformation that captures spin-phonon, valley-phonon, and spin-valley interactions. By analyzing a single phonon mode and then a thermal phonon bath, the authors show that phonons induce a temperature-dependent frequency shift of the spin qubit that is positive at low temperature and can become negative at higher temperatures, producing a non-monotonic dependence with a peak whose position scales with the Zeeman energy $E_z$, orbital splitting $\hbar\omega_{0,x}$, and spin-orbit coupling strength $b_{SL}$. They quantify contributions from three energy scales—spin, valley, and orbital—and find that while the effect can reproduce qualitative features observed experimentally, the magnitude is typically smaller, with peaks shifting in temperature according to the control parameters. The results emphasize the role of phonons in qubit frequency shifts and offer experimentally testable predictions by varying magnetic fields, dot size, and material parameters. The study also discusses limitations and alternative mechanisms, such as electric dipoles from two-level fluctuators, that could contribute to the measured shifts.

Abstract

Spin qubits have proven to be a feasible candidate for quantum computation, and some realizations of spin qubits already benefit from advanced device manufacturing in the semiconductor industry. Compared to superconducting platforms, spin qubits can operate at higher temperatures from tens of millikelvin up to a few kelvin. However, recent experiments show a non-trivial and often non-monotonic dependence of the spin qubit frequency on the temperature, featuring a region of decreased sensitivity to temperature fluctuations. In this work, we aim to gain insight into the physics behind such temperature shifts in the low-temperature limit. Investigating the spin qubits' interaction with phonon modes of the host material, we can explain some of the key features of the observed behavior and estimate the temperature sweet spot for the qubit frequency shift.

Figures (10)

• Figure 1: Schematic of an electron spin qubit in a silicon quantum dot with thermally activated acoustic waves in the material.
• Figure 2: (a) Schematic of the energy scales relevant for the electron confined in a Si quantum dot. The orbital energy splitting $\hbar \omega_{0,x}$ is determined by the harmonic confinement potential (grey) in $x$-direction. The two low-lying conduction band minima of the silicon heterostructure are located at $\pm z$ in $\mathbf{k}$-space, yielding two valleys split by energy $E_{v}$. A magnetic field splits the spin states by $E_{z}$. A micromagnet induces a slanting magnetic field at the dot position and gives rise to an effective spin-orbit interaction with strength $b_{\rm SL}$. Imperfections at the heterostructure interface can lead to a valley-orbit interaction $g_{\rm vo}$. Finally, the electron-phonon interaction $C_{x,\lambda, \mathbf{k}}$ induces transitions between the orbital levels under phonon emission or absorption. The light-blue box indicates the low-energy subspace relevant for the spin qubit. (b) Results of the Schrieffer-Wolff transformation to derive an effective Hamiltonian including the spin-phonon coupling $g_{\rm sp}$, valley-phonon coupling $g_{\rm vp}$, and spin-valley coupling $g_{\rm sv}$. (c) Effective treatment using a two-level system (e.g., $\ket{v_0} = \ket{\downarrow}$ and $\ket{v_1} = \ket{\uparrow}$) split by $\varepsilon$ and its coupling to a phonon bath with strength $g_{\lambda, \mathbf{k}}$. In case of a spin coupling to phonons, this corresponds to $\ket{v_0} = \ket{\downarrow}$ and $\ket{v_1} = \ket{\uparrow}$, $g_{\lambda, \mathbf{k}} = g_{{\rm sp}, \lambda, \mathbf{k}}$, and $\varepsilon = E_{z}$.
• Figure 3: Eigenenergies obtained from diagonalizing the blocks in Eq. \ref{['Eq:GeneralSingleModeHamiltonian']} similar to the block in the red box, as a function of the phonon energy and for different phonon numbers $n$. At $\hbar \omega \approx 0$, the states $\ket{n,v_0}$ and $\ket{n,v_1}$ (depicted in blue and gray) are well separated by approximately the energy $\varepsilon$ (shown as green double arrows). Without the interaction between spin and phonons, $g=0$, one obtains the black dashed lines with a crossing at the resonance. The presence of an interaction yields an anticrossing of width $2\sqrt{n} g$ at the energy $\hbar \omega = \varepsilon$ leading to a role change of the energy levels. An example for the definition of the two qubit levels with phonon occupation number $n=2$ is shown as red dashed lines featuring a jump at the resonance. The green double arrows show the resulting abrupt change in frequency at the resonance.
• Figure 4: Temperature-dependent qubit energy shift calculated as in Eq. \ref{['Eq:GeneralPhononShift']} where we subtract the splitting at $T=0.02$ K. Shifts for various Zeeman fields $E_{z}$ at orbital splitting (a) $\hbar \omega_{0,x}=0.15$ meV$~\approx 36.3$ GHz and $b_{\rm SL} = 2$ GHz, (b) $\hbar \omega_{0,x}=0.4$ meV$~\approx 96.8$ GHz and $b_{\rm SL} = 2$ GHz. In (a) only results up to $E_{z} = 35$ GHz are shown as the approximation $E_{z} \ll \hbar \omega_{0,x}$ is violated with increasing $E_{z}$. (c) Plots from (a) for $E_{z}=20$ GHz (turquoise) and 40 GHz (blue) with different values for $b_{\rm SL} = 2$ GHz and 4 GHz. For the $y$-axis scale we have used the scaling function "SignedLog" of Mathematica.
• Figure 5: Temperature-dependent shift of the valley splitting where the shift at $T=0.1$ K is subtracted. We have used $\hbar \omega_{0,x}=1$ meV $\approx 242$ GHz and $g_{\rm vo} = 10$ GHz. We find a non-monotonous shift where the maximum occurs at higher temperatures compared to the spin shift. For the $y$-axis scale we have used the "SignedLog" scaling function of Mathematica.