Quantum model for mode locking in pulsed semiconductor quantum dots

Quantum dots in GaAs/InGaAs structures have been proposed as a candidate system for realizing quantum computing. The short coherence time of the electronic quantum state that arises from coupling to the nuclei of the substrate is dramatically increased if the system is subjected to a magnetic field and to repeated optical pulsing. This enhancement is due to mode locking: oscillation frequencies resonant with the pulsing frequencies are enhanced, while off-resonant oscillations eventually die out. Because the resonant frequencies are determined by the pulsing frequency only, the system becomes immune to frequency shifts caused by the nuclear coupling and by slight variations between individual quantum dots. The effects remain even after the optical pulsing is terminated. In this work, we explore the phenomenon of mode locking from a quantum mechanical perspective. We treat the dynamics using the central-spin model, which includes coupling to 10–20 nuclei and incoherent decay of the excited electronic state, in a perturbative framework. Using scaling arguments, we extrapolate our results to realistic system parameters. We estimate that the synchronization to the pulsing frequency needs time scales in the order of $1\mathrm{s}$.

Figure 1

(a) Spectrum of the longitudinal Overhauser field $O^x={\sum }_j{a}_j{I}_j^x$ after 200, 2000, and $20000$ pulses, with realistic couplings $a_j$, chosen such that $T^{*}=\sqrt{10}\mathrm{ns}\approx 3.16\mathrm{ns}$. (b), (c) A similar plot for couplings multiplied by factors $\sqrt{10}$ and 10, respectively, i.e., with $T^{*}=1\mathrm{ns}$ and $T^{*}=\frac{1}{10}\sqrt{10}\mathrm{ns}\approx 0.316\mathrm{ns}$. In (a)–(c), the distributions are normalized to an integral of 1. (d)–(f) Relative probability distributions ${\rho }_{\mathrm{rel}}\left(O^x\right)=\rho {\left(O^x\right)}_t/\rho {\left(O^x\right)}_0$. In (d) and in the insets of (e) and (f), the black vertical lines indicate the values where the resonance condition is fulfilled, according to Eqs. (16) and (20). The insets of (e) and (f) span the same horizontal and vertical range as panel (d).

Figure 2

Behavior of the transverse components $O^{\mu }$ $\left(\mu =y,z\right)$ after many pulses $\left(t=19901T_{\mathrm{pulse}}\right)$. (a), (b) Probability distributions of the observables in their respective eigenbasis. (c), (d) Relative differences ${\rho }_{\mathrm{rel}}$ of the distribution at $t=19901T_{\mathrm{pulse}}$ with the initial one. (e), (f) Time evolution $O^{\mu }\left(t\right)$ after $19901T_{\mathrm{pulse}}$, with $t_{\mathrm{red}}\equiv t-19901T_{\mathrm{pulse}}$. The shaded regions indicate fast oscillations, with frequency close to the Larmor frequency $\lambda$. The nuclear couplings have been chosen such that $T^{*}=1\mathrm{ns}$.

Figure 3

(a) Growth rate ${\eta }_t={\rho }_{\mathrm{rel}}/t$ for couplings scaled such that $T^{*}=3.16\mathrm{ns}$ (a) or $T^{*}=1\mathrm{ns}$ (b). (c) Dependence of peak heights of ${\rho }_{\mathrm{rel}}$ on $\mathcal{A}$ for different $N$. The lower horizontal axis shows $\mathcal{A}$, and the upper horizontal axis the equivalent dephasing times $T^{*}$. For each $N$, we fit power laws $\propto {\mathcal{A}}^{\alpha }$ with $\alpha \approx 1$, depicted as straight lines. In the inset, we zoom in on the regime around $T^{*}=2\mathrm{ns}$. The axes of the inset are linear.

Figure 4

(a) Effect of the distribution of couplings on the peak structure. (b) Dependence of the peak growth rate on the normalized cutoff radius ${\tilde{r}}_{\mathrm{cutoff}}$. The squared sum of the couplings and the number of nuclei are fixed by $T^{*}=2\mathrm{ns}$ and $N=15$, respectively.

Figure 5

Amplitude of the central-spin Larmor oscillations after termination of periodic pulsing. Here, pulses (indicated by the vertical solid lines and the label P) are applied periodically every $13.2\mathrm{ns}$ until $t=132\mathrm{ns}$. The vertical dashed lines indicate $t=(132+13.2l)\mathrm{ns}$ $\left(l=1,2,...\right)$ where the revivals are located. The three panels differ in the amount of mode locking: (a) no mode locking, (b) weak mode locking (Lorentzian peaks with $w=0.05)$, and (c) strong mode locking $\left(w=0.005\right)$. The insets show the initial distribution of $O={\sum }_j{a}_j{I}_j^x$. In all cases, $\mathcal{A}=2{\mathrm{ns}}^{-2}$ $\left(T^{*}=2\mathrm{ns}\right)$ and $N=15$. The shaded regions indicate fast oscillations.

Figure 6

Errors between the exact and perturbative results for Hamiltonian (B1) with $A=\tilde{A}=0.05$ and some arbitrary initial state $\left|\psi \right(0)\rangle$. (a) Overlap errors $1-|\langle {\psi }_{\mathrm{ex}}\left(t\right)|{\psi }_{\mathrm{pert}}\left(t\right)\rangle {|}^2$. (b) Spin errors $\parallel \langle {\vec{S}}_{\mathrm{ex}}\rangle \left(t\right)-\langle {\vec{S}}_{\mathrm{pert}}\rangle \left(t\right)\parallel$. The colors distinguish the perturbation orders, red for $(o_{\mathrm{eigenvalues}},o_{\mathrm{eigenvectors}})=(0,0)$, green for $(1,1)$, blue for $(2,1)$, and yellow for $(2,2)$. For the latter two, the error values have been magnified by the factors ${10}^5$ in (a) and 100 in (b). The red curve coincides with the green one.

Figure 7

(a) Typical logarithmic histogram of the absolute value $z=|{\rho }_{pq;\sigma \tau }|$ of the matrix elements of the density matrix. The binning on the horizontal axis is given by $[2^{-(k+1)},2^{-k}],k=0,1,2,...$. The different curves distinguish the distance $d$ to the diagonal (number of different spins between $p$ and $q)$. (b) Weight of the matrix elements, defined as number $N_{\mathrm{el}}$ times value $z$. The dashed curve indicates the accumulated weight $W\left(z\right)$ of matrix elements up to $z$ [see Eq. (C1)]. For both plots, $N=15$ and $T^{*}=1\mathrm{ns}$. The vertical dotted lines show the standard truncation value $\theta =4^{-(N+1)}=2^{-32}$.