Nuclear frequency focusing in periodically pulsed semiconductor quantum dots described by infinite classical central spin models

The coherence of an electronic spin in a semiconductor quantum dot decays due to its interaction with the bath of nuclear spins in the surrounding isotopes. This effect can be reduced by subjecting the system to an external magnetic field and by applying optical pulses. By repeated pulses in long trains the spin precession can be synchronized to the pulse period $T_{\mathrm{R}}$. This drives the nuclear spin bath into states far from equilibrium leading to nuclear frequency focusing. In this paper, we use an efficient classical approach introduced in Fauseweh et al. [Fauseweh, Schering, Hüdepohl, and Uhrig, Phys. Rev. B 96, 054415 (2017)] to describe and to analyze this nuclear focusing. Its dependence on the effective bath size and on the external magnetic field is elucidated in a comprehensive study. We find that the characteristics of the pulse as well as the nuclear Zeeman effect influence the behavior decisively.

Figure 1

Representative result for the spin-spin correlation (14) at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for a spin bath with $\gamma ={10}^{-2}$ using pulse model I from (16). The solid orange line indicates the envelope (18b).

Figure 2

Relative prepulse signal as a function of time at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various spin baths using pulse model I from (16). Inset: Excellent data collapse when the time axis is scaled with $\sqrt{\gamma }$.

Figure 3

Upper panel: Distribution of the effective Larmor frequency $|\vec{h}-\vec{B}|$, shifted by $|\vec{h}|$, at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for $\gamma ={10}^{-2}$ after $n_{\mathrm{p}}$ pulses using pulse model I from (16). The vertical dashed lines indicate the values which satisfy the odd resonance condition (20). Lower panel: Same as upper panel, but after $n_{\mathrm{p}}=3000$ pulses. Additionally, the distributions of the Overhauser field components $B^{\alpha }\left(\alpha \in \{x,y,z\}\right)$ are shown.

Figure 4

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition (21) as a function of the scaled time $t\sqrt{\gamma }$ at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various spin baths using pulse model I from (16).

Figure 5

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition (21) as a function of time for $\gamma ={10}^{-2}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model I from (16). Inset: Excellent data collapse when the time axis is scaled with $1/h$.

Figure 6

Representative result for the spin-spin correlation (14) at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for a spin bath with $\gamma ={10}^{-2}$ using pulse model II from (17). The solid orange line indicates the envelope (18b).

Figure 7

Relative prepulse signal as a function of time at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various spin baths using pulse model II from (17).

Figure 8

Upper panel: Distribution of the effective Larmor frequency $|\vec{h}-\vec{B}|$, shifted by $|\vec{h}|$, at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for $\gamma ={10}^{-2}$ after $n_{\mathrm{p}}$ pulses using pulse model II from (17). The vertical dashed lines indicate the values which satisfy the even resonance condition (21). Lower panel: Same as upper panel, but after $n_{\mathrm{p}}=3000$ pulses. Additionally, the distributions of the Overhauser field components $B^{\alpha }\left(\alpha \in \{x,y,z\}\right)$ are shown.

Figure 9

Upper panel: Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ attributed to the even resonance (21) in the Overhauser field distribution as a function of the scaled time $t\sqrt{\gamma }$ at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various sizes of the spin baths using pulse model II from (17). Lower panel: Same as upper panel, but for $h=80J_{\text{Q}}$ and adjusted sizes of the spin bath [see Eq. (23)].

Figure 10

Various pairs of magnetic field $h$ and inverse spin bath size $\gamma$ fulfilling $P=h{\gamma }^2=9\times {10}^{-5}{J}_{\text{Q}}$ to test the conjecture (23).

Figure 11

Upper panel: Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser fields generating even resonance as a function of time for $\gamma ={10}^{-2}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model II from (17). Lower panel: Same as upper panel, but for $\gamma ={10}^{-4}$.

Figure 12

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser fields generating even resonance as a function of the scaled time $t/h^2$ for $\gamma ={10}^{-2}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model II from (17). Lower panel: Same as upper panel, but for $\gamma ={10}^{-4}$.

Figure 13

Relative prepulse signal as a function of time at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various sizes $2/\gamma$ of the spin bath using pulse model I from (16) including the nuclear Zeeman effect. Inset: Excellent data collapse when the time axis is scaled with $\gamma$.

Figure 14

Upper panel: Distribution of the effective Larmor frequency $|\vec{h}-\vec{B}|$, shifted by $|\vec{h}|$, at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for $\gamma ={10}^{-2}$ after $n_{\mathrm{p}}$ pulses using pulse model I from (16) and including the nuclear Zeeman effect. The vertical dashed lines indicate the values which satisfy the even resonance condition (21). Lower panel: Same as upper panel, but after $n_{\mathrm{p}}=3000$ pulses. Additionally, the distributions of the Overhauser field components $B^{\alpha }\left(\alpha \in \{x,y,z\}\right)$ are shown.

Figure 15

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the even resonances in the Overhauser field distribution as a function of scaled time $t\gamma$ at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various sizes of the spin bath using pulse model I from (16) including the nuclear Zeeman effect.

Figure 16

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition (21) as a function of time for $\gamma ={10}^{-2}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model I from (16) and including the nuclear Zeeman coupling. Inset: Time axis scaled with $1/h^2$.

Figure 17

Relative prepulse signal as a function of time at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various spin baths using pulse model II from (17) and including the nuclear Zeeman coupling. Inset: Time axis scaled with $\gamma$.

Figure 18

Upper panel: Distribution of the effective Larmor frequency $|\vec{h}-\vec{B}|$, shifted by $|\vec{h}|$, at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for $\gamma ={10}^{-2}$ after $n_{\mathrm{p}}$ pulses using pulse model II from (17) and including the nuclear Zeeman effect. The vertical dashed lines indicate the values which satisfy the even resonance condition (21). Lower panel: Same as upper panel, but after $n_{\mathrm{p}}=3000$ pulses. Additionally, the distributions of the Overhauser field components $B^{\alpha }\left(\alpha \in \{x,y,z\}\right)$ are shown.

Figure 19

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition as a function of the scaled time $t\gamma$ at $h=40J_{\text{Q}}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various spin baths using pulse model II from (17) and including the nuclear Zeeman coupling.

Figure 20

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition as a function of time for $\gamma ={10}^{-2}$ at pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model II from (17) and including the nuclear Zeeman coupling.

Figure 21

Weight ${\mathrm{\Sigma }}_{\mathrm{even}}$ of the Overhauser field distribution fulfilling the even resonance condition as a function of the scaled time $t/h^2$ for $\gamma ={10}^{-2}$ and pulse repetition time $T_{\text{R}}=5\pi /J_{\text{Q}}$ for various external magnetic fields $h$ using pulse model II from (17) and including the nuclear Zeeman coupling. Taking the scale on the time axis into account one notes the significantly slower dynamics for lower values of $\gamma$ in the lower panel.

Figure 22

Example of the fit to obtain the phase jump $\mathrm{\Delta }\phi$; see main text and Eqs. (A1) and (A2).