Adiabatic electron charge transfer between two quantum dots in presence of $1/f$ noise

Controlled adiabatic transfer of a single electron through a chain of quantum dots has been recently achieved in GaAs and Si/SiGe based quantum dots, opening prospects for turning stationary spin qubits into mobile ones and solving in this way the problem of long-distance communication between quantum registers in a scalable quantum computing architecture based on quantum dots. We consider theoretically the process of such an electron transfer between two tunnel-coupled quantum dots, focusing on control by slowly varying the detuning of energy levels in the dots. We take into account the fluctuations in detuning caused by $1/f$-type noise that is ubiquitous in semiconductor nanostructures and analyze their influence on probability of successful transfer of an electron in a spin eigenstate. With numerical and analytical calculations we show that probability of electron not being transferred due to $1/f^{\beta }$ noise in detuning is $\propto {\sigma }^2{t}^{\beta -1}/v$, where $\sigma$ characterizes the noise amplitude, $t$ is the interdot tunnel coupling, and $v$ is the detuning sweep rate. Interestingly, this means that the noise-induced errors in charge transfer are independent of $t$ for $1/f$ noise. For realistic parameters taken from experiments on silicon-based quantum dots, we obtain the minimal probability of charge transfer failure between a pair of dots is limited by $1/f$ noise in detuning to be the on the order of 0.01. This means that in order to reliably transfer charges across many quantum dots, charge noise in the devices should be further suppressed, or tunnel couplings should be increased, in order to allow for faster transfer (and less exposure to noise), while not triggering the deterministic Landau-Zener excitation.

Figure 1

Schematic graph of detuning dependence of four lowest energy levels of a spin-split electron tunneling between the two quantum dots. On the far left side, the order of states, from the lowest-energy one, is $|L_{-}\rangle ,|L_{+}\rangle ,|R_{-}\rangle$, and $|R_{+}\rangle$, where $L/R$ denotes the state localized in left/right dot, and $s=\pm$ denotes the lower/higher energy spin state. The tunnel coupling $t$ corresponds to spin-conserving transition between the dots, while $t^{\prime }\ll t$ coupling is the spin-flip transition allowed by finite spin-orbit interaction. ${\mathrm{\Delta }}_{L/R}$ are spin splittings in the left/right dot. Their difference, which leads to spin $+$ and $-$ anticrossing occurring at distinct values of $\epsilon$, is exaggerated compared to realistic situation in order to show its role. The colors and symbols ${\cap }_{-},m,w$, and ${\cup }_{+}$ mark the energy levels that would be followed by a state initialized in one of eigenstates at negative $\epsilon$, if the change to positive $\epsilon$ was done very slowly, so that the electron is dragged adiabatically through all the level anticrossings. The ${\cap }_{+}$ path (dashed line) corresponds to evolution of the state initialized in the $|L_{+}\rangle$ state when detuning is changes at such a rate that the $t^{\prime }$ transitions are diabatic, while $t$ anticrossing is traversed adiabatically. Interference angle $\varphi$ is given by the area between lines corresponding to energies $w$ and $m$ states.

Figure 2

Transition probability in the noiseless case as a function of sweep rate for $t=5,10\mu \mathrm{eV},t^{\prime }=0.05,0.1\mu \mathrm{eV}$, and $B=1$ T. The oscillations of the probability are too fast to be discerned.

Figure 3

Envelope of transfer probability oscillations as a function of sweep velocity $v$ in a presence of quasistatic fluctuations of ${\mathrm{\Delta }}_{L/R}$, for three different concentration of nuclear spins in silicon, with $\sqrt{{\delta }^2}$ from [53], $t^{\prime }=0.1\mu \mathrm{eV}$ and $B=1$ T. Dashed line corresponds to completely decohered result given by Eq. (14).

Figure 4

Top: probability of successful electron transfer $p_R=1-p_L$ (i.e., electron ending in right dot); bottom: probability of nonadiabatic excitation, the error $p_L$ (i.e., the probability of leaving the electron in the left dot). Results of simulations for $1/f$ noise in detuning characterized by $\sqrt{S\left(1\mathrm{Hz}\right)}=0.2$, 0.5, and $1\mu \mathrm{eV}/{\mathrm{Hz}}^{1/2}$ are marked by symbols, which are compared with Eq. (26), shown as dashed line, and a re-exponentiation of this result (see text), shown as solid lines, which agrees very well with numerical simulations in the whole range of results. Bottom panel corresponds to the region marked by a black rectangle on top. The results for tunneling $t=5$, 15, and $25\mu \mathrm{eV}$ converge at low $v$, when the value of $p_R<1$ is determined only by noise, and it is independent of $t$. For larger $v$, the transition to Landau-Zener dependence due to deterministic nonadiabatic effects described by Eq. (2) occurs at $v\propto t^2$. For considered noise power the error always exceeds ${10}^{-3}$.

Figure 5

Simulated probability of nonadiabatic excitation (i.e., leaving the electron behind in the left dot) for $1/f^{\beta }$ noise with $\beta =1.5$, characterized by $\sqrt{S\left(1\mathrm{Hz}\right)}=0.5\mu \mathrm{eV}/{\mathrm{Hz}}^{1/2}$. The simulated results (symbols) for tunneling $t=5$, 15, and $25\mu \mathrm{eV}$ do not converge to the same dependence at low $v$ and fit the exponential curve (solid lines).

Figure 6

Probability of successful electron transfer calculated for electron initialized in higher energy state (spin up) for $t^{\prime }=0,0.1\mu \mathrm{eV}$ and $1/f$ noise (with parameters as in Fig. 4). Symbols correspond to numerical simulation of influence of noise for $t^{\prime }=0.1\mu \mathrm{eV}$. Solid lines are analytical result for $t^{\prime }=0$, which are the same as results for the ground state shown in Fig. 4). Dashed line is a product of Eqs. (14) and $1-|c_{-}{\left(\infty \right)|}^2$ where $|c_{-}{\left(\infty \right)|}^2$ is given by Eq. (26), i.e., it corresponds to analytical formula accounting approximately for all the effects of noise in presence of $t^{\prime }\ne 0$.