Superradiant Topological Peierls Insulator inside an Optical Cavity

We consider a spinless ultracold Fermi gas tightly trapped along the axis of an optical resonator and transversely illuminated by a laser closely tuned to a resonator mode. At a certain threshold pump intensity, the homogeneous gas density breaks a ${\mathbf{Z}}_2$ symmetry towards a spatially periodic order, which collectively scatters pump photons into the cavity. We show that this known self-ordering transition also occurs for low field seeking fermionic particles when the laser light is blue detuned to an atomic transition. The emergent superradiant optical lattice in this case is homopolar and possesses two distinct dimerizations. Depending on the spontaneously chosen dimerization, the resulting Bloch bands can have a nontrivial topological structure characterized by a nonvanishing Zak phase. In the case where the Fermi momentum is close to half of the cavity-mode wave number, a Peierls-like instability here creates a topological insulator with a gap at the Fermi surface, which hosts a pair of edge states. The topological features of the system can be nondestructively observed via the cavity output: the Zak phase of the bulk coincides with the relative phase between laser and cavity field, while the fingerprint of edge states can be observed as additional broadening in a well-defined frequency window of the cavity spectrum.

Figure 1

Schematic view of fermionic atoms trapped in a one-dimensional elongated tube along the axis of an optical resonator and driven by a transverse blue-detuned (${\mathrm{\Delta }}_a>0$ with respect to an atomic transition $g\leftrightarrow e$) laser with Rabi frequency $\mathrm{\Omega }$. ${\mathrm{\Delta }}_c$ denotes detuning with respect to a standing-wave cavity mode. The atoms are low field seekers trapped at the light intensity minima generated by the interference of the cavity field and the plane-wave transverse pump laser. The relative phase $\mathrm{\Delta }\phi$ of the cavity output and the pump laser corresponds to the Zak phase of the lattice bands in real time (see Fig. 2). (Inset) The coupling of momentum states via pump and cavity fields.

Figure 2

Spontaneous ${\mathbf{Z}}_2$-symmetry breaking at the superradiant self-ordering transition in the configuration of Fig. 1 seen via the modulus of coherent cavity-field amplitude (the solid line) as a function of the pump strength $\eta$. The blue dashed line shows the cavity phase $\mathrm{\Delta }\phi$ relative to the driving laser, for which we have either $\mathrm{\Delta }\phi =0$ or $\mathrm{\Delta }\phi =\pi$ depending on the lattice dimerization. (Insets) Corresponding atomic densities (the black solid line) and lattice potential (rescaled by 0.5, the blue solid line) within a unit cell, for $\hslash \eta \sqrt{N}=1.55E_R$. $\mathrm{\Delta }\phi$ coincides with the Zak phase ${\phi }_{\text{Zak}}$ of the lowest Bloch band. The gray-shaded area denotes an unstable region of the system (see Fig. 3). Results are obtained for spin-polarized fermions with the parameters $k_F=k_c/2$, $k_{B}T=0.01E_R$, $\hslash {\mathrm{\Delta }}_c=12E_R$, $U_{0}N=32.35E_R$.

Figure 3

Phase diagram in the $\eta -k_F$ plane. For a strong enough pump $\eta$ (above the blue line), the system self-orders into a superradiant pattern. In the self-ordered phase, the system is in either a SRM phase or an insulator state. The superradiant insulating phase, in turn, is either a SRBI or a SRTI, characterized by their Zak phases (see Fig. 2). At the half filling $k_F=k_c/2$, we have a direct phase transition between the uniform Fermi gas and the SRBI or SRTI. Otherwise, this phase transition occurs only through the SRM phase. For an even stronger drive (above the red line) the self-ordered phase becomes unstable. The parameters are the same as in Fig. 2.

Figure 4

Signature of the edge states in the absorption spectrum of the SRTI phase with respect to the propagation of cavity photons. The imaginary part of the polarization [Eq. (6)] as a function of frequency is shown for a finite system encompassing 50 unit cells at half filling. The total absorption $\chi$ (the black solid line) has two contributions: ${\chi }_{\mathrm{ed}\to 2}$ (the red dashed line) is due to the presence of edge states, while ${\chi }_{1\to 2}$ (the blue dashed line) is present also without edge states. (Inset) The atomic energy spectrum, where filled (empty) circles indicate occupied (empty) states. The two isolated eigenenergies in the midgap correspond to the two localized edge states. The parameters are the same as in Fig. 2, with $\hslash \eta \sqrt{N}=1.55E_R$.