Subharmonic transitions and Bloch-Siegert shift in electrically driven spin resonance

We theoretically study coherent subharmonic (multiphoton) transitions of a harmonically driven spin. We consider two cases: magnetic resonance (MR) with a misaligned, i.e., nontransversal, driving field, and electrically driven spin resonance (EDSR) of an electron confined in a one-dimensional, parabolic quantum dot, subject to Rashba spin-orbit interaction. In the EDSR case, we focus on the limit where the orbital level spacing of the quantum dot is the greatest energy scale. Then, we apply time-dependent Schrieffer-Wolff perturbation theory to derive a time-dependent effective two-level Hamiltonian, allowing us to describe both MR and EDSR using the Floquet theory of periodically driven two-level systems. In particular, we characterize the fundamental (single-photon) and the half-harmonic (two-photon) spin transitions. We demonstrate the appearance of two-photon Rabi oscillations, and analytically calculate the fundamental and half-harmonic resonance frequencies and the corresponding Rabi frequencies. For EDSR, we find that both the fundamental and the half-harmonic resonance frequencies change upon increasing the strength of the driving electric field, which is an effect analogous to the Bloch-Siegert shift known from MR. Remarkably, the drive-strength-dependent correction to the fundamental EDSR resonance frequency has an anomalous, negative sign, in contrast to the corresponding Bloch-Siegert shift in MR which is always positive. Our analytical results are supported by numerical simulations, as well as by qualitative interpretations for simple limiting cases.

Figure 1

Electrically driven spin resonance in a 1D quantum dot. (a) An electron occupying the ground state ${\psi }_0\left(z\right)$ of a parabolic confinement potential is excited by an ac electric field of amplitude $E_{\mathrm{ac}}$ and frequency $\omega$. The electron is subject to a homogeneous magnetic field $\mathbf{B}$ and spin-orbit interaction characterized by the direction ${\mathbf{n}}_{\mathrm{SO}}$. (b) Left: Orbital levels labeled by the oscillator quantum number $n$, separated by the level spacing $\hslash {\omega }_0$. Right: Diagram representing one of the many fifth-order virtual processes contributing to the half-harmonic resonance. Horizontal lines represent the energy eigenstates of the harmonic oscillator Hamiltonian. Arrows labeled by $E, B$, and SO correspond to matrix elements of the ac electric field, magnetic field, and spin-orbit Hamiltonians, respectively. Note that the spin-orbit Hamiltonian provides both spin-flip and spin-conserving matrix elements only if both $cos\theta$ and $sin\theta$ are nonzero. [See Eq. (36).]

Figure 2

Magnetic resonance in a misaligned ac field: structure of the Floquet Hamiltonian at the fundamental resonance. Panels show cases when the ac field is perpendicular to the static field (a), is parallel to the static field (b), has finite perpendicular and parallel components (c). Horizontal lines (blue arrows) correspond to diagonal (off-diagonal) matrix elements of the Floquet Hamiltonian $\mathcal{F}$. The vertical position of each horizontal line corresponds to the value of the diagonal matrix element.

Figure 3

Magnetic resonance in a misaligned ac field: structure of the Floquet Hamiltonian at the half-harmonic resonance. Panels show cases when the ac field is perpendicular to the static field (a), is parallel to the static field (b), has finite perpendicular and parallel components (c). Horizontal lines (blue arrows) correspond to diagonal (off-diagonal) matrix elements of the Floquet Hamiltonian $\mathcal{F}$. The vertical position of each horizontal line corresponds to the value of the diagonal matrix element.

Figure 4

Electrically driven Rabi spin dynamics at half-harmonic resonance. The excited-state occupation probability $P_E$ is shown as a function of the drive frequency $\omega$ and time $t$; the numerical data reveal the chevron pattern characteristic of magnetic resonance. Parameters: $\theta =\pi /4, \tilde{\alpha }/\tilde{B}={\tilde{E}}_{\mathrm{ac}}/\tilde{B}=1, \hslash {\omega }_0/\tilde{B}=5$. The analytical results for the half-harmonic resonance frequency ${\omega }_{\mathrm{res}}^{\left(2\right)}$ and the Rabi frequency ${\mathrm{\Omega }}_{\mathrm{res}}^{\left(2\right)}$ are also displayed. For the above parameter values, the latter one is related to the time period of the oscillation via $2\pi /{\mathrm{\Omega }}_{\mathrm{res}}^{\left(2\right)}=2\pi \times 625\hslash /\tilde{B}$.

Figure 5

Bloch-Siegert shift and power broadening of the half-harmonic resonance. The maximal excited-state occupation probability $P_E^{\max }$ is shown as a function of the amplitude ${\tilde{E}}_{\mathrm{ac}}$ of the driving ac electric field and drive frequency $\omega$. Parameters: $\theta =\pi /4, \tilde{\alpha }/\tilde{B}=1$, and $\hslash {\omega }_0/\tilde{B}=8$. The red line indicates the analytical result for the resonance frequency as the function of electric field based on Eq. (13).

Figure 6

Anisotropy of the half-harmonic resonance. The maximal excited-state occupation probability $P_E^{\max }$ is shown as a function of the angle $\theta$ characterizing the spin-orbit interaction and the drive frequency $\omega$ measured from ${\omega }_{\mathrm{res}}^{\left(1\right)}/2$; see Eq. (8). The solid line corresponds to the analytically obtained half-harmonic resonance frequency; i.e., it shows ${\omega }_{\mathrm{res}}^{\left(2\right)}-\frac{{\omega }_{\mathrm{res}}^{\left(1\right)}}{2}$. [See Eq. (13)]. Parameters: $\tilde{E}/\tilde{B}=1, \tilde{\alpha }/\tilde{B}=1$, and $\hslash {\omega }_0/\tilde{B}=5$.

Figure 7

Schematic form of the matrices we encounter during the SW perturbation theory. The label $\epsilon$ refers to blocks with matrix elements that are much smaller than the energy difference between the relevant and irrelevant subspaces. In the EDSR problem, the energy scale of the those blocks is $\tilde{B}\sim \epsilon \hslash {\omega }_0$.