Coherent tunneling by adiabatic passage of an exchange-only spin qubit in a double quantum dot chain

A scheme based on coherent tunneling by adiabatic passage (CTAP) of exchange-only spin qubit quantum states in a linearly arranged double quantum dot chain is demonstrated. Logical states for the qubit are defined by adopting the spin state of three electrons confined in a double quantum dot. The possibility to obtain gate operations entirely with electrical manipulations makes this qubit a valuable architecture in the field of quantum computing for the implementation of quantum algorithms. The effect of the external control parameters as well as the effect of the dephasing on the coherent tunneling in the chain is studied. During adiabatic transport, within a constant energy degenerate eigenspace, the states in the double quantum dots internal to the chain are not populated, while transient populations of the mixed states in the external ones are predicted.

Figure 1

Schematic of the tunneling in the quantum channel with $N$ double quantum dots. The black dots in (a) and (b) denote the electrons before and after the tunneling. The quantities indicated by ${\omega }_i$, $\omega$, and ${\omega }_f$ are the pulses that have to be furnished by external gates in order to make the tunneling possible.

Figure 2

Gaussian pulses as a function of $t$, for $t_{\max }=10\pi /{\omega }^{\max }$, $\sigma =t_{\max }/8$, and ${\omega }_S^{\max }=10{\omega }^{\max }$.

Figure 3

Probability ${\rho }_{ff}$ of finding the three electrons at the tail of the chain in the double quantum dot $N$ as a function of $t_{\max }$ for $\sigma =t_{\max }/8$ and ${\omega }^{\max }=10$. Each family of curves represents a different value of the dephasing parameter $\Gamma$ for a different number of double quantum dots composing the chain.

Figure 4

Eigenvalues of three double quantum dots system as a function of time. The state at zero energy is triple degenerate.

Figure 5

Populations of the density matrix $\rho$ as a function of time for $N=3$, ${\omega }^{\max }=1$, $t_{\max }=25\pi /{\omega }^{\max }$, $\sigma =t_{\max }/8$, and $\Gamma =0$.

Figure 6

Probability ${\rho }_{P_3{S}_3}$ of finding the three electrons at the tail of the chain for $N=3$ as a function of $t_{\max }$ and $\Gamma$ for $\sigma =t_{\max }/8$ and ${\omega }^{\max }=10$.

Figure 7

Probability ${\rho }_{P_3{S}_3}$ of finding the three electrons at the tail of the chain for $N=3$ as a function of $t_{\max }$ and ${\omega }^{\max }$ for $\sigma =t_{\max }/8$. Each plot illustrates a different value of the dephasing $\Gamma$.

Figure 8

Absolute value of the difference between ${\rho }_{ff}$ and ${\rho }_{\mathrm{ideal}}$ as a function of $t_{\max }$ for two different variations (+1% and +10%) on the amplitude of ${\omega }_i\left(t\right)$ and ${\omega }_f\left(t\right)$ for $N=3$.

Figure 9

Absolute value of the difference between ${\rho }_{ff}$ and ${\rho }_{\mathrm{ideal}}$ as a function of $t_{\max }$ for two different variations (+1% and +10%) on the amplitude of ${\omega }_i\left(t\right)$, $\omega \left(t\right)$ and ${\omega }_f\left(t\right)$ for $N=5$.

Figure 10

Same as Fig. 9 with $N=7$.

Figure 11

Same as Fig. 9 with $N=9$.

Figure 12

Absolute value of the difference between ${\rho }_{ff}$ and ${\rho }_{\mathrm{ideal}}$ as a function of $t_{\max }$ for two different variations (+1% and +10%) on the peak time of ${\omega }_i\left(t\right)$ and ${\omega }_f\left(t\right)$ for $N=3$.

Figure 13

Absolute value of the difference between ${\rho }_{ff}$ and ${\rho }_{\mathrm{ideal}}$ as a function of $t_{\max }$ for two different variations (+1% and +10%) on the peak time of ${\omega }_i\left(t\right)$, $\omega \left(t\right)$, and ${\omega }_f\left(t\right)$ for $N=5$.

Figure 14

Same as Fig. 13 with $N=7$.

Figure 15

Same as Fig. 13 with $N=9$.

Figure 16

(a) Scheme of the qubit chain where performing successive SWAP operations. Note that every qubit must be filled with three electrons. (b) Comparison between the transfer time for CTAP and successive SWAPs as a function of the chain length $N$ for three different values of the tunneling rate ${\omega }^{\max }$.