Chiral Light-Matter Interaction beyond the Rotating-Wave Approximation

I introduce and analyze chiral light-matter interaction in the ultrastrong coupling limit where the rotating-wave approximation cannot be made. Within this limit, a two-level system with a circularly polarized transition dipole interacts with a copolarized mode through rotating-wave terms. However, the counterrotating terms allow the two-level system to couple to a counterpolarized mode with the same coupling strength, i.e., one that is completely decoupled within the rotating-wave approximation. Although such a Hamiltonian is not particle number conserving, the conservation of angular momentum generates a $U(1)$ symmetry which allows constructing an ansatz. The eigenstates and dynamics of this novel model are computed for single-cavity interactions and for a many-mode system. The form of the ansatz provides significant analytic insight into the physics of the ground state and the dynamics; e.g., it indicates that the ground states are two-mode squeezed. This work has significant implications for engineering light-matter interaction and novel quantum many-body dynamics beyond the rotating-wave approximation.

Figure 1

(a) Schematic of the single-cavity chiral Rabi model. A two-level system with a circularly polarized dipole moment (red circle) interacts with the copolarized mode $a$ through rotating-wave terms (blue) and with a counterpolarized mode through counterrotating terms (green). (b) A many-mode open system is composed by introducing an array of cavities with nearest-neighbor coupling. The $a$ and $b$ modes are orthogonal and only couple through the two-level system.

Figure 2

(a) First several eigenenergies of the chiral Rabi Hamiltonian for different angular momentum quantum numbers $l$ for ${\omega }_0={\omega }_c$. (b) Observables of the lowest energy eigenstates in the $l=0$ (red) and $l=1$ (blue) manifolds. The plot shows the excited-state population $⟨{\widehat{\sigma }}_{ee}⟩$ (solid lines), $⟨{\widehat{a}}^{†}\widehat{a}⟩$ (dashed lines), and $⟨{\widehat{b}}^{†}\widehat{b}⟩$ (dotted lines). (c) Normally ordered variance $⟨:(\mathrm{\Delta }{\widehat{X}}_a+\mathrm{\Delta }{\widehat{X}}_b{)}^2:⟩$ (solid lines) and $⟨:(\mathrm{\Delta }{\widehat{X}}_a-\mathrm{\Delta }{\widehat{X}}_b{)}^2:⟩$ (dashed lines). Squeezing occurs for values below zero.

Figure 3

(a) Ground state energy of the many-body Hamiltonian for the $l=0$ (red) and $l=1$ (blue) manifolds vs coupling coefficient $g$ computed by truncating the ansatz at $n=2$ with $L=20$ sites. Here, $J=0.2{\omega }_0$ and ${\omega }_0={\omega }_c$. The dashed blue line shows the single excitation bound states computed within the RWA and the region within the horizontal cyan lines is the photon band. The symbols I, II, III show the different phases of the $l=1$ subspace (see main text). (b) Observables for the $l=0$ (red) and $l=1$ (blue) ground states.

Figure 4

Many-body time dynamics starting in the $|e⟩|00⟩$ state for ${\omega }_0={\omega }_c$ and $J=0.2{\omega }_0$ with (a)–(c) $g=0.1{\omega }_0$, (d)–(f) $g=0.5{\omega }_0$, and (g)–(i) $g={\omega }_0$, computed using MPS (see SM [29] for details). Left (center) column shows the number of photons in the $a$ ($b$) mode vs site index $i$ and normalized time $gt$. The right column shows observables vs time: the excited-state population of the TLS $⟨{\widehat{\sigma }}_{ee}⟩$ (solid black), photons in the $a$ mode ${\sum }_i⟨{\widehat{a}}_i^{†}{\widehat{a}}_i⟩$ (dashed black), in the $b$ mode ${\sum }_i⟨{\widehat{b}}_i^{†}{\widehat{b}}_i⟩$ (dotted black), and the number of photons in cavity $i=0$ for mode $a⟨{\widehat{a}}_0^{†}{\widehat{a}}_0⟩$ (solid gray) and mode $b⟨{\widehat{b}}_0^{†}{\widehat{b}}_0⟩$ (dotted gray).