Macroscopic entanglement between localized domain walls inside a cavity

TL;DR

The paper presents a scheme to generate stable, tunable entanglement between two localized Bloch domain walls (DWs) in nanomagnetic strips placed inside a single-mode chiral cavity, mediated by an optomechanical-like interaction via the inverse Faraday effect. By linearizing the driven-dissipative dynamics and analyzing the resulting dissipative phases, robust DW–DW entanglement is shown to emerge near phase boundaries, with the ability to switch entanglement between DWs and photon–DW pairs by adjusting detuning and pinning. In the dispersive regime, adiabatic elimination of the cavity field yields an effective inter-DW coupling $G^{ m eff}_{ij}$ that drives two-mode squeezing of the macroscopic domains, enabling macroscopic entanglement without strong intrinsic nonlinearities. The work also links entanglement to observable optical spectra and exceptional-point physics, and demonstrates thermal robustness up to a few kelvin with suitably high DW frequencies, pointing toward practical quantum-information and sensing applications using stationary spin textures.

Abstract

We present a scheme for generating stable and tunable entanglement between two localized Bloch domain walls in nanomagnetic strips kept inside a chiral optical cavity. The entanglement is mediated by the effective optomechanical interaction between the cavity photons and the two macroscopic, collective modes of the pinned domain walls. By controlling the pinning potential and optical driving frequency, the robust, steady-state entanglement between the two macroscopic domain walls can survive beyond the typical milli-Kelvin temperature range.

Figures (7)

• Figure 1: Experimental setup of domain walls inside a chiral cavity. The illustration shows linearly polarized light generated by a laser, which is passed through a Faraday isolator and a quarter-wave plate (QWP) to excite a circularly polarized optical mode. A right-circularly-polarized standing wave is created inside the chiral cavity, which then couples to the two ferromagnetic strips. The Bloch walls in the strips are pinned where the cavity field is present. The reflected optical mode is then collected at one port of a differential photodiode (Diff. PD) and the reference laser light is collected at the other port to obtain the spectrum via homodyne detection. The output and input are separated and merged with the use of a beam splitter (BS).
• Figure 2: Dissipative phases and two-mode entanglement of the system as a function of the detuning $\tilde{\Delta}$ and $G_{\rm{eff}}=G |g_1|/\omega_1$ for $\omega_2=\omega_1$ (a-d) and $\omega_2=10~\omega_1$ (e-h). The plots (a) and (e) show the number of stable solutions, with the distinct dissipative phases $p\in\{0,1,...,6\}$ shown in (b) and (f). The entanglement between the two DWs $\mathcal{E}_{1|2}$ and between the photon and DW modes $\mathcal{E}_{a|1(2)}$ are shown in plots (c) and (d). Finally, the plots (g) and (h) show the entanglement between the photon and DW modes $\mathcal{E}_{a|1}$ and $\mathcal{E}_{a|2}$ for $\omega_2=10~\omega_1$. The blue and green dashed lines highlight two interesting regions of phase transition. The former indicates the jump from a region with two to three real solutions, while the latter exhibits a line of transcritical bifurcation where $\mathcal{E}_{1|2}$ in plot (c) is optimal. The parameters taken are $\kappa_a=2$ MHz, $\kappa_{1(2)}=1$ MHz, $\omega_1=1$ GHz, $g_{1(2)}=1$ MHz and temperature $T=2$ mK.
• Figure 3: Entanglement switching and two-mode squeezing. The figure shows (top) bipartite entanglement $\mathcal{E}_{i|j}$ and (bottom) lowest eigenvalue $\nu^{\rm{min}}_{i|j}$ of the covariance matrix $\tilde{V}^{(k)}_{i|j}$, between modes $i$ and $j$ (where $i,j\in\{a,1,2\}$). The plots are for $\omega_1=\omega_2$ (a,c) and $\omega_2=10~\omega_1$ (b,d). All other parameters are same as in Fig. \ref{['fig:stability']}.
• Figure 4: Output optical spectrum. The plots (a,b) show the spectrum at $\omega_2=\omega_1$ for the roots $k=1$ and $k=2$, which exhibit level attraction and repulsion, respectively. The bottom (c,d) shows the spectrum at $\omega_2=10~\omega_1$ for $k=1$ and $k=2$, respectively. The green dotted line shows the associated eigenvalues.
• Figure 5: Thermal stability of macroscopic entanglement between the two domain walls. (a) Plot shows entanglement as a function of the ratio $\kappa_a/\kappa_1$ for different temperatures $T$. Here $\omega_{1(2)}=\omega_0$ = 1 GHz, $\tilde{\Delta}=-40~\omega_{1(2)}$ and $\kappa_1=1$ MHz. The vertical black dotted line highlights the maximum entanglement. (b) The plot shows the temperature dependence of $\mathcal{E}_{1|2}$ at fixed $\kappa_a/\kappa_1=0.2$ and varying $\omega_{1(2)}/\omega_0=1,10,100$ with solid blue, orange dashed, and green dotted curves. Now, for fixed $\omega_{1(2)}=\omega_0$, the dissipation is varied, such that $\kappa_a/\kappa_1$ = 1 and 0.01 with blue dot-dashed and dotted curves, respectively. The vertical black dotted lines mark the cutoff temperature $T_c \approx 26$ mK, $0.26$ K, and $2.6$ K for $\omega_{1(2)}$ = 1 GHz, 10 GHz, and 100 GHz, respectively.