Photon-assisted quantum state transfer and entanglement generation in spin chains

We propose a protocol for state transfer and entanglement generation between two distant spin qubits (sender and receiver) that have different energies. The two qubits are permanently coupled to a far-off-resonant spin chain, and the qubit of the sender is driven by an external field, which provides the energy required to bridge the energy gap between the sender and the receiver. State transfer and entanglement generation are achieved via virtual single-photon and multiphoton transitions to the eigenmodes of the channel.

Figure 1

Schemetic representation of the two equivalent systems under consideration. (a) Two distant detuned spin qubits, the sender and the receiver, are permanently coupled to a far-off-resonant spin chain, which is initially prepared in its ground (vacuum) state. The qubit of the sender is initially prepared in a qubit state $|\psi \rangle$ and it is driven by an external field. (b) Employing the Jordan-Wigner transformation, the system pertains to the transfer of a single excitation between two detuned discrete states, via a far-off-resonant array of $N$ coupled empty states. Note the energy difference between the sender and the receiver (i.e., ${\omega }_{\mathrm{s}}\ne {\omega }_{\mathrm{r}}$) for both (a) and (b).

Figure 2

Schematic representation of the system under consideration. (a) After diagonalization of the Hamiltonian of the channel, the system of Fig. 1 reduces to the transfer of an excitation between to detuned levels, via virtual excitation of a far-off-resonant band of empty states, pertaining to the eigenmodes of the channel. In general, the coupling of the sender to the $k\mathrm{th}$ eigenmode involves absorption or emission of $n$ photons, for all possible $n=0,1,2,...$. Adjusting the strength of the driving $z_0$ and the photon frequency $\omega$, one can control the $n\text{−}$photon transition(s) that dominate the dynamics. (b) Eliminating the modes of the channel adiabatically, one obtains an effective two-level system pertaining to the shifted states of the sender and the receiver. The standard $\nu \text{−}$photon Rabi model is obtained when the RWA is valid.

Figure 3

Logarithmic plot of $|{\mathcal{J}}_n\left(z\right)|$ as a function of $a$, for different values of the order $z$. For Bessel functions with negative $n$, one may recall the identity ${\mathcal{J}}_{-n}\left(z\right)={(-1)}^n{\mathcal{J}}_n\left(z\right)$.

Figure 4

Performance of three different faithful state-transfer schemes for increasing detuning between the sender and the receiver, for $N=3$. The plotted fidelity is ${\mathcal{F}}_{\min }$ (see main text). Scheme A refers to the protocol of Refs. [7, 8], scheme B to the protocol of Ref. [15], and scheme C to the protocol of Ref. [13]. In all of the cases, the sender is on resonance with the chain, while the receiver is detuned by ${\omega }_{\mathrm{r},\mathrm{s}}$. The horizontal dashed line marks the classical threshold of 2/3. Similar behavior is obtained for other values of $N\ge 2$, while for a given value of the ratio ${\omega }_{\mathrm{r},\mathrm{s}}/g_{\mathrm{s}}$, the fidelity drops slowly with increasing $N$.

Figure 5

State transfer and entanglement generation mediated by a chain of $N=2$ spin qubits. Numerical solution of Eqs. (10) for continuous $\left[f\right(t)=1]$ harmonic driving (a) and pulsed harmonic driving [(b), (c)]. Parameters of the system: ${\omega }_{\mathrm{c}}-{\omega }_{\mathrm{s}}=22.0g_{\mathrm{s}}$, ${\omega }_{\mathrm{r}}-{\omega }_{\mathrm{s}}=2.0g_{\mathrm{s}}$, $g_{\mathrm{r}}=1.0g_{\mathrm{s}}$, $\kappa =6.0g_{\mathrm{s}}$, $\omega =2g_{\mathrm{s}}$, and $z_0=2.0$. Parameters of the pulse: $g_{\mathrm{s}}{t}_0=300$ and $\mathrm{\Theta }\left(T\right)=\pi /4$; $g_{\mathrm{s}}\tau =25.45$ (b); $\mathrm{\Theta }\left(T\right)=\pi /2$; $g_{\mathrm{s}}\tau =50.90$. The dashed green curve refers to Eq. (17b). Time is in units of $g_s^{-1}$.

Figure 6

State transfer and entanglement generation mediated by a chain of $N=3$ spin qubits. Numerical solution of Eqs. (10) for continuous $\left[f\right(t)=1]$ harmonic driving (a) and pulsed harmonic driving [(b), (c)]. Parameters of the system: ${\omega }_{\mathrm{c}}-{\omega }_{\mathrm{s}}=32.0g_{\mathrm{s}}$, ${\omega }_{\mathrm{r}}-{\omega }_{\mathrm{s}}=2.0g_{\mathrm{s}}$, $g_{\mathrm{r}}=1.0g_{\mathrm{s}}$, $\kappa =14.0g_{\mathrm{s}}$, $\omega =2g_{\mathrm{s}}$, and $z_0=2.0$. Parameters of the pulse: $g_{\mathrm{s}}{t}_0=350$ and $\mathrm{\Theta }\left(T\right)=\pi /4$; $g_{\mathrm{s}}\tau =32.66$ (b); $\mathrm{\Theta }\left(T\right)=\pi /2$; $g_{\mathrm{s}}\tau =65.11$. The dashed green curve refers to Eq. (17b). Time is in units of $g_s^{-1}$.

Figure 7

State transfer and entanglement generation mediated by a chain of $N$ spin qubits. The average fidelity $\overline{\mathcal{F}}$ (red), the minimum fidelity ${\mathcal{F}}_{min}$ (black), and the concurrence (blue) are plotted as functions of time for $N=2$ [(a), (c)] and $N=3$ [(b), (d)]. (a) Single-photon resonance ($\nu =1$) and other parameters as in Fig. 5. (b) Single-photon resonance ($\nu =1$), and other parameters as in Fig. 6. (c) Two-photon resonance ($\nu =2$), $z_0=3$, $g_{\mathrm{s}}\tau =41.42$ for $\mathrm{\Theta }\left(T\right)=\pi /4$, and $g_{\mathrm{s}}\tau =82.85$ for $\mathrm{\Theta }\left(T\right)=\pi /2$. Other parameters as in Fig. 5. (d) Two-photon resonance ($\nu =2$), $z_0=3$, $g_{\mathrm{s}}\tau =53.1$ for $\mathrm{\Theta }\left(T\right)\simeq \pi /4$, and $g_{\mathrm{s}}\tau =106.2$ for $\mathrm{\Theta }\left(T\right)\simeq \pi /2$. Other parameters as in Fig. 6. The fidelities correspond to $\mathrm{\Theta }\left(T\right)\simeq \pi /2$ and the concurrence for $\mathrm{\Theta }\left(T\right)\simeq \pi /4$. Time is in units of $g_s^{-1}$.