Non-Markovian dynamics in chiral quantum networks with spins and photons

We study the dynamics of chiral quantum networks consisting of nodes coupled by unidirectional or asymmetric bidirectional quantum channels. In contrast to familiar photonic networks where driven two-level atoms exchange photons via 1D photonic nanostructures, we propose and study a setup where interactions between the atoms are mediated by spin excitations (magnons) in 1D $XX$ spin chains representing spin waveguides. While Markovian quantum network theory eliminates quantum channels as structureless reservoirs in a Born-Markov approximation to obtain a master equation for the nodes, we are interested in non-Markovian dynamics. This arises from the nonlinear character of the dispersion with band-edge effects, and from finite spin propagation velocities leading to time delays in interactions. To account for the non-Markovian dynamics we treat the quantum degrees of freedom of the nodes and connecting channel as a composite spin system with the surrounding of the quantum network as a Markovian bath, allowing for an efficient solution with time-dependent density matrix renormalization-group techniques. We illustrate our approach showing non-Markovian effects in the driven-dissipative formation of quantum dimers, and we present examples for quantum information protocols involving quantum state transfer with engineered elements as basic building blocks of quantum spintronic circuits.

Figure 1

(a) Photonic quantum network of two-level systems coupled to waveguides (optical fibers). (b) Quantum network with a spin chain representing a waveguide for spin excitations. To provide a systematic treatment of the quantum many-body dynamics beyond the Born-Markov approximation, we employ tDMRG techniques to treat the full dynamics of the two-level systems (nodes) and spin chains (quantum channels) within the region inside the dashed lines, thus defining an extended Markovian cut.

Figure 2

Two-level systems or “system spins” coupled to a photonic (a) or spin waveguide (b) as basic building block of a chiral quantum network. (a) System spins decay by emitting a photon into the left- and right-moving modes of the photonic waveguide. (b) System spins decay by emitting a magnon (spin excitation) into the spin waveguide (bath) in a flip-flop process. Chirality (${\gamma }_L\ne {\gamma }_R$) is achieved by introducing a complex hopping between neighboring system and bath spins. (c) Dispersion relation ${\omega }_k$ and asymmetric momentum coupling $g_k$. For $\tilde{\mathrm{\Delta }}=0$ and weak coupling ${(\tilde{J}/J)}^2\ll 1$, the system spins couple only to resonant waveguide modes around $k=\pm \pi /2a$ (see shaded region), such that for $\varphi =\pi /4$, a pure unidirectional system-bath coupling is achieved. (d) Demonstration of unidirectional emission of magnons with a single system spin coupled to the spin waveguide for $\varphi =\pi /4, \tilde{J}=0.3J$, and $\tilde{\mathrm{\Delta }}=0$. Dashed line corresponds to the exponential decay predicted by the Markovian theory. The lower panel shows the evolution of the bath spins occupation $\langle S_j^{+}{S}_j^{-}\rangle$, evidencing the unidirectional propagation of the emitted magnons through the spin waveguide.

Figure 3

A spin-optical interface converting right propagating spin excitations into optical photons propagating in a fiber. (a) The inclusion of such a finite waveguide with local losses at its ends, in the calculation of the network dynamics, extends the standard Markovian cut of quantum optics. (b) Possible experimental realization of the spin chain losses by a spin-photon interface obtained by coupling the edge bath spins to a cavity, decaying in an optical fiber.

Figure 4

Band-edge effects in the dynamics of a single strongly coupled system spin, in absence (a),(b) and presence (c),(d) of driving. (a) Occupation $\langle {\sigma }_1^{+}{\sigma }_1^{-}\rangle$ of an undriven (${\mathrm{\Omega }}_1=0$) and strongly coupled ($\tilde{J}/J=1$) system spin, showing a fast decay and a permanent oscillation due to the coupling to two bound states when $\varphi =\pi /4$ and $\mathrm{\Delta }=0$ [see Appendix pp3-s1]. The Markovian and WW predictions are shown as a dashed and dotted line, respectively. The non-Markovian behavior is detected via nonzero values of the BLP and RHP witnesses, represented by the blue and cyan regions, respectively. The BLP witness is calculated by comparing the evolution from the initial conditions ${\rho }_S^{\left(1\right)}=|e\rangle \langle e|$ and ${\rho }_S^{\left(2\right)}=|g\rangle \langle g|$, while the RHP witness is obtained by initially preparing the system and ancilla in the entangled state ${\rho }_{SA}\left(0\right)=|{\mathrm{\Psi }}_0{\rangle }_{SA}\langle {\mathrm{\Psi }}_0|$, with $|{\mathrm{\Psi }}_0{\rangle }_{SA}={\left(\right|e\rangle }_S{|e\rangle }_A+{|g\rangle }_S{|g\rangle }_A)/\sqrt{2}$. The WW prediction for the BLP witness is represented by a dotted line. (b) For the same situation as in (a), we plot the evolution of the bath spins occupation $\langle S_j^{+}{S}_j^{-}\rangle$, showing the localized bound-state oscillation around the system spin position. In addition, we observe a nearly unidirectional propagation of the magnons during the decay of the system spin. Despite that $\varphi =\pi /4$, there is also a small fraction of magnons propagating to the left as the strong coupling allows the system spin to decay to modes different than $k=\pm \pi /2a$ [see Fig. 2]. (c),(d) Dynamics of system and bath spins, under the same conditions as in (a) and (b), but with the inclusion of driving ${\mathrm{\Omega }}_1=\gamma /4$ in the system. The driving tends to wash out the band-edge oscillations, but they are still identified by the witnesses as the dominant non-Markovian behavior.

Figure 5

Retardation effects in the dynamics of two distant system spins, in the absence (a),(b) and presence (c),(d) of driving. (a) Occupations $\langle {\sigma }_1^{+}{\sigma }_1^{-}\rangle$ (red) and $\langle {\sigma }_2^{+}{\sigma }_2^{-}\rangle$ (blue) of the system spins sitting on the left and right end of the waveguide, respectively. They are separated by a distance $d=10a$, and are chirally coupled to the spin waveguide with $\tilde{J}=0.5J, \varphi =\pi /10$, and $\tilde{\mathrm{\Delta }}=0$ (such that $\tau =2.5/\gamma$ and ${\gamma }_L/{\gamma }_R\approx 0.26$). We observe the chiral decay of the initially excited left system spin, and the retarded excitation of the right system spin at $t\approx \tau$. This is well described by the WW prediction (dotted lines), but it strongly deviates from the Markovian prediction (dashed lines) as the reabsorptions are assumed to be instantaneous. Small deviations from the WW prediction stem from a weak coupling to the band-edge modes, which are not accounted for by the analytical expressions (15) and (16) (see Appendix pp3-s2). The non-Markovian witnesses BLP (blue) and RHP (cyan) identify the absorptions at times multiples of $\tau$, as well as the residual oscillations due to band-edge effects. The two initial conditions used to calculate the BLP witness are ${\rho }_{\mathrm{S}}^{\left(1\right)}=|eg\rangle \langle eg|$ and ${\rho }_{\mathrm{S}}^{\left(2\right)}=|gg\rangle \langle gg|$, whereas the RHP witness assumes an initial entangled state between system and ancilla given by ${\rho }_{SA}\left(0\right)=|{\mathrm{\Psi }}_0{\rangle }_{SA}\langle {\mathrm{\Psi }}_0|$, with $|{\mathrm{\Psi }}_0{\rangle }_{SA}={\left(\right|eg\rangle }_S{|eg\rangle }_A+{|gg\rangle }_S{|gg\rangle }_A)/\sqrt{2}$. (b) For the same situation as in (a), we plot the evolution of the bath spins occupation $\langle S_j^{+}{S}_j^{-}\rangle$, showing that the first emitted magnon reaches the right system spin at $t\approx \tau$, when it is absorbed and chirally reemitted. (c),(d) Dynamics of system and bath spins, under the same conditions as in (a),(b), but including driving on both system spins (${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_2=\gamma /2$). In this situation the analytical WW treatment cannot be applied, but the retarded absorptions are clearly identified as the dominant non-Markovian behavior.

Figure 6

Strong hard-core effects in a spin waveguide, with unidirectional (a),(b) and bidirectional coupling (c),(d). (a) Two strongly driven system spins (${\mathrm{\Omega }}_{\alpha }=\gamma$), separated by a distance $d=60a$, and coupled unidirectionally to the waveguide with $\varphi =\pi /4, \tilde{J}=0.5J$, and $\tilde{\mathrm{\Delta }}=0$ (corresponding to $\tau =15/\gamma$). The evolution of the system spin occupations $\langle {\sigma }_{\alpha }^{+}{\sigma }_{\alpha }^{-}\rangle$ are shown in solid lines for a spin waveguide, and in dashed lines for a bosonic waveguide, with red and blue color corresponding to left ($\alpha =1$) and right ($\alpha =2$) system spins, respectively. The black dots represent the Markovian prediction of a single driven system spin ($N_S=1$), showing that both system spins reach the single-particle quasistate steady for $t<\tau$. The larger population of the right system spin for $t>\tau$ (blue solid line), compared to the bosonic case (blue dashed line), is due to the saturation of the spin waveguide. (b) For the same situation as in (a), we plot the evolution of the bath spins occupation $\langle S_j^{+}{S}_j^{-}\rangle$, showing the unidirectional emission of magnons for $t<\tau$. At $t>\tau$, the right system spin also emits into left-moving modes as an effect of the high density of the spin waveguide and the hard-core constraint. (c),(d) Dynamics of system and bath spins, under the same conditions as in (a) and (b), but with a bidirectional coupling to the waveguide $\varphi =0$. In addition to the saturation effect, the different behavior in the spin (solid) and a bosonic (dashed) case is due to collisions between the magnons. All these simulations were obtained using a MPS representation with a maximum bond dimension of $D=30$ and averaging over 4000 quantum trajectories, giving a statistical error of less than $0.01$ for the populations shown here. The bosonic waveguide case is obtained when replacing the spin operators $S_j^{-}$ in Eqs. (1, 2, 3) by bosonic operators $b_j$, and we restrict the number of excitations to a maximum of two per site.

Figure 7

Entanglement entropy $S\left({\rho }_m\right)$ across a bipartite splitting at the middle of the spin chain, for a representative sample of quantum trajectories. We assume the same physical situation and parameters as in Fig. 6 and check the convergence of the MPS approach by evolving the same quantum trajectory $|{\mathrm{\Psi }}_m\rangle$ for three different maximum bond dimensions $D=30,60,120$ (shown in red, green and blue lines, respectively), and compare their overlaps. For a unidirectional waveguide (a), the average entanglement entropy $\overline{S}$ (black line) remains low during the whole evolution, but at long times some trajectories with large entropy are not converged. In the bidirectional case (b), the progressive entanglement growth also limits the efficiency of the MPS approach at long times. In both cases (a) and (b), the dynamics for $\gamma t<25$ is well described by a bond dimension of $D=30$ (see Fig. 6).

Figure 8

Driven-dissipative preparation of pure dimer states, in the Markovian regime (a) Level scheme and couplings, in the Markovian limit, and under the conditions for the dimer formation between two system spins. (b) Minimal configuration for dimer formation, using $N_{\mathrm{S}}=2, N_{\mathrm{B}}=4, d=2a, \tilde{\mathrm{\Delta }}=0$, and ${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_2=\mathrm{\Omega }$. (c) Dynamical formation of dimer steady state illustrated by the evolution of the purity ${\mathcal{P}}_S$ of the system density matrix and its overlap with the dimer state ${\mathcal{F}}_{\mathrm{D}}$. Both approach unity, in good agreement with the Markovian prediction (dashed lines). The imperfect decoupling between system and waveguide is detected by a nonzero mutual information $I_{SB}$ (grey line). The purity $\mathcal{P}$ of the whole $2+4$ spin network also approaches unity, signaling the formation of a composite dark state between system and bath spins. Other parameters are $\tilde{J}=0.1J, \mathrm{\Omega }/\gamma =1$, and $\varphi =\pi /10$. (d) For the same situation as in (c), we show snapshots of the bath spin occupations $\langle S_j^{+}{S}_j^{-}\rangle$, evidencing the formation of a constant magnon flux between the system spins, as $\gamma t\to \infty$. (e) Dissipative formation of a dimerized phase of $N_S=12$ system spins, illustrated by the singlet correlations $C_{\alpha ,\alpha +1}$, which build up in pairs, as predicted in the Markovian limit (dashed lines). Other parameters are $\tilde{J}=0.3J, \mathrm{\Omega }/\gamma =1, \tilde{\mathrm{\Delta }}=0, \varphi =\pi /4$, and $N_B=24$. (f) For the same situation as in (e), we show the evolution of the bath spin occupations $\langle S_j^{+}{S}_j^{-}\rangle$, showing the decoupling between dimers in steady state, which occurs successively from left to right due to the unidirectional coupling. The dynamics shown in (e) and (f) was calculated using an MPS representation with a maximum bond dimension of $D=10$ and averaging over 400 quantum trajectories, giving a statistical error of less than $0.01$ for the occupations shown here.

Figure 9

Robustness of steady-state correlations in the non-Markovian regime. (a) Quantum mutual information $I_{12}$ between the two system spins in steady state, as a function of $\gamma \tau =(d/a){(\tilde{J}/J)}^2$, for fixed ratio $\mathrm{\Omega }/\gamma =1$, and two values of chirality: $\varphi =\pi /4$ (unidirectional, red symbols), and $\varphi =\pi /16$ (asymmetric bidirectional, blue symbols). The various symbols correspond to different combinations of distance and couplings given by $d/a=[2,4,6]$ and $\tilde{J}/J=[0.1,0.18,0.25,0.35,0.6]$, where the different values of $\tilde{J}/J$ are represented by squares, crosses, circles, triangles, and diamonds, with increasing value. The Markovian predictions are represented as dashed lines, showing that the maximal correlations are obtained in this limit ($\tau =0^{+}$). The linear scaling of $I_{12}$ with $\gamma \tau$ in the region $\gamma \tau \ll 1$ identifies retardation effects as the main source of deviation from the Markovian prediction. (b) Mutual information $I_{12}$ as a function of $\mathrm{\Omega }/\gamma$, for $\varphi =\pi /4$ (red) and $\varphi =\pi /16$ (blue) cases, and two values of retardation $\gamma \tau =[0.065,0.245]$ (increasing for darker color), corresponding to $\tilde{J}/J=[0.18,0.35]$ and $d/a=2$. There is an optimal $\mathrm{\Omega }/\gamma$, at which the correlations are maximal, which reduces with $\gamma \tau$ and ${\gamma }_L/{\gamma }_R$. The dashed lines show the Markovian predictions for each chirality, which saturate to the maximum value $I_{12}\to 2$ for $\mathrm{\Omega }/\gamma \to \infty$, as the dimer state coincides with the singlet.

Figure 10

Non-Markovian effects in the dimer steady state via two-time correlations. (a),(b) Two-time singlet correlation function ${\tilde{C}}_{12}(\infty ,t^{\prime })$ in steady state, for different delay times $\tau \approx 0.19/\gamma$ and $\tau \approx 0.39/\gamma$ (increasing for darker color), in the unidirectional $\varphi =\pi /4$ (a), and in an asymmetric bidirectional case $\varphi =\pi /16$ (b). (a) For ${\gamma }_L=0$, the correlations are rigidly displaced by $t^{\prime }=\tau$ as compared to the Markovian prediction (dashed), showing that the effects of retardation can be compensated by a simple time shift of $\tau$. (b) For ${\gamma }_L/{\gamma }_R\approx 0.45$, there is a strong overall reduction of the two-time correlations with $\tau$, relative to the Markovian prediction (dashed). Other parameters are $d/a=[6,12], \tilde{J}/J=0.18, \mathrm{\Omega }/\gamma =1$, and $\tilde{\mathrm{\Delta }}=0$. (c),(d) “Two-time” mutual information $I_{12}(\infty ,t^{\prime })$ in steady state, for the same parameters as in (a) and (b). (c) In addition to the expected rigid shift by $\tau$, we observe a small decrease in the maximum of $I_{12}(\infty ,t^{\prime })$ due to residual dispersion effects. (d) Besides showing the overall reduction of correlations with $\tau , I_{12}(\infty ,t^{\prime })$ is also sensitive to two peaks at $t^{\prime }=\pm \tau$, showing that correlations can be restored by a time shift in either direction, the maximum of the two obtained in the direction of chirality. (e),(f) Deviation of the maximal two-time mutual information $\delta {\tilde{I}}_{12}=I_{12}^{\mathrm{mark}}-{\tilde{I}}_{12}(\infty ,t_{\max }^{\prime })$ and of the equal time one $\delta I_{12}=I_{12}^{\mathrm{mark}}-I_{12}$, with respect the Markovian prediction $I_{12}^{\mathrm{mark}}$ obtained for $\tau =0^{+}$. (e) We plot $\delta {\tilde{I}}_{12}$ (circles) and $\delta I_{12}$ (crosses), as a function of $\gamma \tau =(d/a){(\tilde{J}/J)}^2$, for the same parameters as in Fig. 9. In particular, red symbols correspond to ${\gamma }_L=0$, and blue symbols to ${\gamma }_L/{\gamma }_R\approx 0.45$. For ${\gamma }_L=0, {\tilde{I}}_{12}(\infty ,t_{\max }^{\prime })$ allows to extract correlations much closer to the Markovian prediction than $I_{12}$, whereas for ${\gamma }_L/{\gamma }_R\approx 0.45$ the correlations are only slightly increased. (f) The same conclusions from (e) can be drawn when plotting $\delta {\tilde{I}}_{12}$ (solid) and $\delta I_{12}$ (dashed), as a function of driving $\mathrm{\Omega }/\gamma$, for the same parameters and color code as in Fig. 9.

Figure 11

Steady-state entropy between system and waveguide (a) Relevant part of the network composed by the system spins and only the bath spins connecting them (enclosed by dashed lines). (b) Steady-state entropy $S\left({\rho }_{ss}^{\prime }\right)$ of the subsystem indicated in (a), as a function of chirality ${\gamma }_L/{\gamma }_R$ and retardation $\gamma \tau$. The vanishing entropy in the limit of weak coupling ($\tilde{J}/J\ll 1$) and unidirectional interactions (${\gamma }_L/{\gamma }_R\to 0$), signals the formation of a composite dark state (with no output) despite the finite time delay $\tau$. At larger coupling ($\tilde{J}/J\gtrsim 1$), the composite dark state is strongly degraded due to dispersion effects, which is evidenced by an increase of the entropy. Calculations are done for $N_B=6$, but $S\left({\rho }_{ss}^{\prime }\right)$ depends insignificantly on the waveguide size. Other parameters are $d/a=2, \mathrm{\Omega }=\gamma /4$, and $\tilde{\mathrm{\Delta }}=0$.

Figure 12

State transfer via chiral spin chain. (a) Schematic representation of the quantum state transfer with chiral coupling to the spin chain. (b) Numerical simulation of state transfer between two system spins (qubits), separated by $d=68a$. The qubit populations are represented in the upper panel (solid lines), together with the pulse shaping of the coupling strengths ${\tilde{J}}_{\alpha }\left(t\right)$ to achieve a symmetric spin-wave packet (dashed lines). The lower panel shows the propagation of the Gaussian wave packet in the waveguide. (c) Fidelity of the transfer $\mathcal{F}$ as a function of distance $d$ for a cosine $\omega \left(k\right)=-2Jcos\left(ka\right)$ (yellow line) and an exactly linear dispersion relation $\omega \left(k\right)=2Ja\left|k\right|$ (red line). (d) Robustness of the state transfer protocol in the presence of disorder in the nearest-neighbor hopping of the spin chain $J_j$. We plot the state transfer fidelity $\mathcal{F}$ as a function of $d$ and the variance ${\sigma }^2$ of the random distribution, showing the destructive impact of disorder at large spatial separations between the qubits.

Figure 13

Time-reversal and state transfer resilient to dispersion. (a) Schematic illustration of time reversal of a wave packet. At $t=t_0$, the sign of the hopping $J\left(t\right)$ is reversed, which reverses the sign of the dispersion relation (right) and the direction of propagation of the wave packet (left). (b) Application of the time-reversal protocol to the spontaneous emission of a system spin into the spin waveguide. Lower panel: The excitation in the waveguide clearly shows the inversion of propagation direction at $t=t_0$, without a change of the wave-packet shape. Upper panel: As the system spin population shows, the wave packet is perfectly reabsorbed at $t=2t_0$. Parameters are $\tilde{J}=0.4J_0, J_0{t}_0=190$, and $d=398a$. (c) Application of the time reversal to a state-transfer protocol that is insensitive to dispersive effects. At time $t=t_0$, the emitted wave packet is time reversed and then reflected due to the presence of a broken link $j_a,j_a+1$. (d) Numerical simulation of the protocol with $\tilde{J}=0.25J_0, J_0{t}_0=190$, and $d=398a$. Upper panel: Population of the system spins. Lower panel: Occupation of the bath spins in the waveguide, showing the trajectory of the time-reversed wave packet.

Figure 14

Two-qubit quantum gate via spin-spin collision. (a) Schematic representation of the entangling gate protocol. Two system spins with opposite chirality emit in opposite directions into the waveguide. Due to the spin nature of the waveguide, the magnonic wave packets pick up a phase of $\pi$ in the collision, allowing for the realization of a phase gate, after the reabsorption of the excitations. (b) Numerical simulation of the gate for an initial state ${|e\rangle }_1{|e\rangle }_2$ between two system spins, separated by $d=58a$, and the rest of the parameters as in Fig. 12 for the time-dependent couplings. The spin populations are represented in the upper panel. The lower panel shows the propagation of the two wave packets in the spin waveguide. (c) Transfer matrix ${\chi }_{j,i}$ and (d) the error of the gate as a function of interspin distance $d$, showing the resilience of the protocol to dispersion effects.

Figure 15

General solution of a single emitter chirally coupled to the spin waveguide. (a) Contour integral used in the calculation of $c_1\left(t\right)$ [see Eq. (C11)]. (b),(c) Phase diagram of the chiral bound state problem as a function of ${(\tilde{J}/J)}^2$ and $\mathrm{\Delta }$, for $\varphi =\pi /4$ (b) and $\varphi =0$ (c).

Figure 16

(a),(b) Entropies $S\left({\rho }_m\right)$ for a partition in the middle of a bosonic waveguide, for a representative sample of MPS quantum trajectories, and calculated for three different bond dimensions $D=30,60,120$ (shown as red, green, and blue lines, respectively). Parameters are the same as in Figs. 7 and 7, except for the bosonic nature of the waveguide $S_j^{-}\to b_j$. The average entropy $\overline{S}\left(t\right)$ is represented by a black solid line. For the unidirectional bosonic waveguide (a), the behavior is analogous to the spin case in Fig. 7 with the majority of the trajectories staying at low entropy, but with some of them reaching high values at long times. In the bidirectional case (b), the trajectories have a larger average entropy compared to (a), but a much smaller one with respect to its spin counterpart in Fig. 7. (c),(d) Comparison of the average entropy $\overline{S}\left(2\tau \right)$ for spin and boson waveguides, and as a function of $\gamma \tau$. We assume the same situation and parameters as in Fig. 6, with $D=30$ and $\tilde{J}=0.5J$, but we vary $\gamma \tau$ by choosing the distance between system spins as $d/a=30,40,50,60$. The average entropy $\overline{S}\left(2\tau \right)$ increases with $\tau$ (up to an oscillation with $\overline{k}d$), and is larger for a spin than for a boson waveguide due to the hard-core constraint.