Quantum Correlations and Entanglement in Generalized Dicke-Ising Models

Abstract

Quantum systems inside high-Q cavities offer an excellent testbed for the control of emergent symmetries induced by light and their interplay with quantum matter. Recently several developments in cavity experiments with neutral atoms and other quantum objects such as ions motivate the study of their quantum correlated properties and their entanglement to tailor and control the behavior of the system. Using the enhanced coupling between light and interacting matter we explore the properties of emergent superradiant modes using our newly developed Light-Matter DMRG algorithm with strongly interacting spin chains. We explore a experimentally viable generalization of the transverse Ising chain coupled to the cavity light where it is possible to induce multimode structures tailored by the light pumped into the system. We find a plethora of scenarios can be explored with clear and accesible measurable signatures. This allows to study the physics of emergent orders and strong quantum correlations with quantum spins where the local and long range coupling can be efficiently simulated. We find that quantum spin nematic states with long range order and magnon pairs emerge as the transitions to superradiant phases take place. Notably, we show the cavity field allows the optimization of entanglement between spins for different light induced modes which can be used for quantum state engineering of quantum correlated states. Our methods can be used to model other hybrid quantum systems efficiently.

Figures (10)

• Figure 1: Light mode structure induced to the spin chains in the cavity(a)-(b), Experimental scheme for the light. Panel (a): Spins along the chain, sphere size is proportional to the light mode coupling strength at each site. The diffraction maxima configuration ($\phi=0$), a ferromagnetic mode is induced []. The diffraction minima configuration ($\phi=\pi/2$), generates a staggered spin field []. The golden ratio (RBBRG) configuration ($\phi=\arccos(1/5)=\tilde{\varphi}$) inducing a three mode staggered spin structure (red, blue, green) every 5 sites []. The arrows point the direction of the spin in the quantization axis "$x$". Panel (b): The light setup, the angle $\phi$ between the cavity axis $\hat{e}_c$ and the light pumped into the system with direction $\hat{e}_p$ ($-\hat{e}_p$) for a standing wave cavity and retro-reflected pump beams in the pump axis from the side (only one is shown, the other one is parallel). The light polarization unit vector is $\hat{e}_z$ and the spins are along the spatial axis $\hat{e}_x$.
• Figure 2: Phase diagrams for different light induced mode structures and the behavior of the mean bond-nematic order parameter ($\tilde{\mathcal{Q}}^B$). When the system becomes superradiant and the quantization axis of the spins changes there is the formation of bond nematic quantum states. These can persists deeper into the superradiant state depending on the short-range spin interaction strength $J$ and the light induced configuration. Panels (a,b,c): correspond to $J<0$, normal FM state (I) and superradiant configurations (III-VII). Panels (d,e,f): correspond to $J>0$ normal AFM state (II) and superradiant configurations (V-VII). The light induced mode structures are: (a,d) for $\phi=0$ diffraction maxima [], (b,e) for $\phi=\pi/2$ diffraction minima [] and (c,f) for $\phi=\arccos(1/5)$, the golden ratio configuration [], as shown in Fig.\ref{['fig:Fig1']} (a). The order parameters of different phases are in Table \ref{['tab:phases']}. Parameters in the simulations are: $N=400$ spins, $\omega_0/|J|=0.1$, $\kappa/|J|=10$ with $\Delta_c$ and $V_p$ in units of $|J|$.
• Figure 3: Behavior with the golden ratio mode $\varphi$ induced by light. Main Panels: Local bond nematic order parameter $\mathcal{Q}^B_{nm}$, local magnon pair amplitude $\mathcal{P}_{nm}$. Inset Panels: Entanglement entropies $\mathcal{S}_E$ for different mode structures in the system. Main panels (a,b) show the amplitude variation of $\mathcal{Q}^B_{nm}$ near the transition to the SR state and how long range structured correlations emerge across the whole lattice. Main panels (c,d) show the amplitude variation of the magnon pairs with long range coherent oscillations in the SR. The Ising coupling is $J>0$ for (a, c) and $J<0$ for (b, d). Inset panels (a.1,b.1) show the entanglement entropy of the sites with R modes (red,[- - -],), the B modes (blue) [ - - -], and the G mode (green) [- - - - ], across the whole lattice, following Fig.\ref{['fig:Fig1']}. Inset panels (c.1,d.1) show the entanglement entropy of the sites AFM$_x$ singlet sites ($1+k$ and $2+k$) [solid purple, - - -] and ($3+k$ and $4+k$) [dashed purple, - - -] across the whole lattice with $k=5\times (n-1)$, $n=1,\dots,N/5$; the entanglement entropy of half the chain is shown in black. The parameters are: $n=25$ the middle of the chain for $\mathcal{Q}^B_{nm}$ and $\mathcal{P}_{nm}$; $N=50$ sites, $\omega_0/|J|=0.1$, $\kappa/|J|=10$ with $\Delta_c$ and $V_p$ in units of $|J|$ for all plots.
• Figure S1: Order parameters for normal Ferromagnetic ordering and Superradiant Ferromagnetic setup. (a) Cavity light, (b) magnetization along quantization axis $"z"$, (c) mean magnon pairing amplitude, (d) magnetization along quantization axis $"x"$. The Ising coupling is $J<0$ FM.
• Figure S2: Order parameters for normal Anti-ferromagnetic ordering and Superradiant Ferromagnetic setup. (a) Cavity light, (b) staggered magnetization along quantization axis $"z"$, (c) mean magnon pairing amplitude, (d) magnetization along quantization axis $"x"$. The Ising coupling is $J>0$ AFM.