Quantum-state transfer via resonant tunneling through local-field-induced barriers

Efficient quantum-state transfer is achieved in a uniformly coupled spin-$1/2$ chain, with open boundaries, by application of local magnetic fields on the second and last-but-one spins, respectively. These effective barriers induce the appearance of two eigenstates, bilocalized at the edges of the chain, which allow a high-quality transfer also at relatively long distances. The same mechanism may be used to send an entire e-bit (e.g., an entangled qubit pair) from one to the other end of the chain.

Figure 1

Sketch of the spin chain with sender and receiver located at the first and last sites, and with local field barriers of height $\omega$ applied to the second and last-but-one sites.

Figure 2

Spectrum of a spin chain with $N=17$ (upper panel) and $N=18$ sites (lower panel) versus $\omega$: In both cases, the two lowest eigenenergies move outside the band as $\omega$ increases, while the two positive eigenenergies closest to zero become quasidegenerate. The latter are represented by red, solid lines. A zero-energy eigenstate occurs in the odd chain, whose eigenvalue is represented by the green dot-dashed line. All energy values are reported in units of $J$.

Figure 3

IPR for the eigenstates ${|a\rangle }_k$ of a chain of $N=17$ (upper plot) and $N=18$ (lower plot) sites, sorted by increasing eigenvalues, versus $\omega$ (in units of $J$). The first two eigenstates, ${|a\rangle }_1$ and ${|a\rangle }_2$, rapidly reach $\text{IPR}\simeq 2$, becoming bilocalized on the barrier qubits. The states corresponding to $k=10,11$, which bilocalize on the sender and receiver qubits, reach the value $\text{IPR}\simeq 2$ for the even chain, whereas one of them remains slightly above that value for the odd chain.

Figure 4

(a) Fidelity (red thick line) and concurrence (blue thin line, starting from $C=0$ at $t=0$) for a chain of $N=100$ spins with $\omega =100J$. The dashed red and blue lines are, respectively, the maximum value of the fidelity and of the concurrence attainable for the homogeneous chain with $\omega =0$. Time is expressed in units of $J^{-1}$. (b) Maximum of fidelity (red thick line) and concurrence (blue thin line) versus the number of sites $N$ for $\omega =10J$ (solid line) and $\omega =0$ (dashed line).

Figure 5

Maximum fidelity in the time interval $Jt\in [0,4000]$ as a function of $\omega$ (in units of $J$) and $N$. Notice that for $\omega =0$ the fidelity is larger than $0.9$ only for short chains, while as $\omega$ increases, the fidelity is significantly enhanced.

Figure 6

(a) Maximum fidelity achievable in the time interval $Jt\in [0,4000]$. (b) Optimal times at which the best transmission is attained. The plots refer to chains of $N=22$ (red solid line) and $N=23$ sites (green dashed line). For odd (even) $N$, $t_{\text{MAX}}$ is linear (quadratic) in $\omega$.

Figure 7

Scheme of the setup used for entanglement transfer from the qubit pair $(0,1)$ to the pair $(0,N)$, where the qubit 0 is decoupled from the chain.

Figure 8

(Left panel) Averaged concurrence vs auxiliary field strength $\omega$ for different values of the disorder parameter $b$. (Right panel) Concurrence vs disorder strength $b$ for different values of $\omega$. The curves for $\omega >20$ are almost indistinguishable and no relevant change in the effects of the disorder is observed. In both panels, the length of the chain is $N=10$ and averages are performed over ${10}^5$ realizations of disorder. All energy values are expressed in units of $J$.

Figure 9

Averaged fidelity vs auxiliary field strength $\omega$ for different lengths of the chain. In this case a residual magnetic field is supposed to act on sites $3,N-2$ and $4,N-3$, with random values uniformly distributed between 0 and $\omega /10$ for the sites near the barriers, and between 0 and $\omega /40$ for the next to nearest sites, respectively. The plots show averages performed over ${10}^5$ realizations.

Figure 10

Sketch of the configuration for the transfer of an e-bit.

Figure 11

Concurrence transferred from the maximally initial entangled qubits pair $(1,2)$ to the pair $(N-1,N)$ for a chain of $N=33$ and $\omega =5,15,45$ (from left to right). Higher values of the local field, besides increasing the amount of transferred entanglement, regularize the dynamics.

Figure 12

(Upper panel) Average fidelity for a chain of $N=30$ sites with $K_1=60$ and $K_2=30$, where the three steps of the QST protocol are clearly visible from the time behavior of the two fields, which are switched according to the recipe of Eq. (11). (Lower panel) Same as in the upper plot, but with finite switching times for the fields. In this case, ${\omega }_2\left(t\right)$ and ${\omega }_{N-1}\left(t\right)$ are smoother versions of the step functions of Eq. (11), with exponential corrections: ${\omega }_2\left(t\right)=K_2/[\exp \left\{\alpha (t-t_2)\right\}+1]+(K_1-K_2)/[\exp \left\{\alpha (t-t_1)\right\}+1]$, and a similar behavior for ${\omega }_{N-1}\left(t\right)$. The three curves correspond to three values of $\alpha$ and it turns out that the achievable $\overline{F}$ decreases with the steps becoming smoother and smoother (that is, with increasing $\alpha$).