Adiabatic quantum transport in a spin chain with a moving potential

Many schemes to realize quantum state transfer in spin chains are not robust to random fluctuations in the spin-spin coupling strength. In efforts to achieve robust quantum state transfer, an adiabatic quantum population transfer scheme is proposed in this study. The proposed scheme makes use of a slowly moving external parabolic potential and is qualitatively explained in terms of the adiabatic following of a quantum state with a moving separatrix structure in the classical phase space of a pendulum analogy. Detailed aspects of our adiabatic population transfer scheme, including its robustness, is studied computationally. Applications of our adiabatic scheme in quantum information transfer are also discussed, with emphasis placed on the usage of a dual spin chain to encode quantum phases. The results should also be useful for the control of electron tunneling in an array of quantum dots.

Figure 1

Phase space portrait for the classical pendulum Hamiltonian associated with Eq. (5), with $C=2$ and $J=1$. The closed curve enclosing the island and intersecting with both $p-p_0=0$ and $x=\pm \pi$ is the separatrix. Variables $p-p_0$ and $x$ plotted here take dimensionless values. Note that if the separatrix is moving up slowly by increasing $p_0$ gradually, an initial state enclosed by the separatrix is expected to adiabatically follow the movement of the separatrix. This suggests that a slowly moving parabolic potential applied to a spin chain can be used to adiabatically transfer spin excitation along a spin chain.

Figure 2

Excitation probability transferred to the last spin in adiabatic quantum transport along a chain of 101 spins, as a function of the amplitude of the external moving magnetic potential characterized by $C$ [in units of $J$, see the text below Eq. (1)]. The moving speed of the control field is $S=0.005$.

Figure 3

Adiabatic transfer of spin excitation initially localized exclusively at the $n=0\text{th}$ site along a chain of 101 spins, for a field amplitude given by $C=8$ and its moving speed given by $S=0.005$. Panels (a), (b), and (c) are for times $t=0$, $10000$, and $20000$. Note that the final peak excitation probability remains as high as 0.97.

Figure 4

Adiabatic quantum transport along a chain of 101 sites with an initial Gaussian excitation profile, at (a) $t=0$, (b) $t=10000$, (c) $t=20000$, and (d) $t=21000$. The amplitude of the parabolic field is given by $C=2$, and the moving speed of the control field is given by $S=0.005$. Note that the final excitation probability transferred to the last spin is as high as 0.997.

Figure 5

Adiabatic transfer of spin excitation for an initial state exclusively localized at the $n=0\text{th}$ site along a chain of 101 spins. The amplitude of the moving parabolic potential is given by $C=8$, and the moving speed is given by $S=0.025$. Panels (a), (b), and (c) show the excitation profile at times $t=1000$, 2600, and 4000.

Figure 6

Adiabatic transport of an initial Gaussian profile of spin excitation (same as in Fig. 4) along a chain of 101 spins. The amplitude of the moving parabolic potential is given by $C=2$, and the moving speed is given by $S=0.1$. Panels (a), (b), and (c) are for $t=300$, 600, and 1000.

Figure 7

Adiabatic transfer of spin excitation for an initial state exclusively localized at the $n=0\text{th}$ site along a chain of 101 spins, in the presence of static disorder with the noise amplitude given by $\Delta =0.5$. Other parameters are the same as in Fig. 3. The excitation profile at time $t=7000$, $14000$, and $20000$ are shown in panels (a), (b), and (c). Results here are very similar to those shown in Fig. 3.

Figure 8

Adiabatic transfer of an initial Gaussian profile of spin excitation along a chain of 101 spins, in the presence of static disorder characterized by $\Delta =0.5$. Other parameters are the same as in Fig. 4. Panels (a), (b) and (c) are for $t=6000$, $12000$, and $21500$. Results here are very similar to those shown in Fig. 4.

Figure 9

Adiabatic transport of spin excitation for an initial state localized exclusively at $n=0\text{th}$ site along a chain of 101 spins, in the presence of dynamic disorder characterized by $A=0.025$ and ${\omega }_{\text{max}}=0.1$. Other parameters are the same as in Fig. 7. Panels (a), (b), and (c) are for $t=7000$, $14000$, and $20000$.

Figure 10

Adiabatic transfer of spin excitation for an initial state exclusively localized at the $n=0\text{th}$ site along a chain of 101 spins. ${\omega }_{\text{max}}=1.0$, and other parameters are the same as in Fig. 9.