Cavity-induced superconducting and $4k_F$ charge-density-wave states

We propose two experimental setups for fermionic atoms in a high-finesse optical resonator in which either a superconducting state with $s$-wave symmetry of the pairs or a $4k_F$ charge-density wave can self-organize. In order to stabilize the $s$-wave pairing, a two component attractively interacting fermionic gas is confined to a one-dimensional chain structure by an optical lattice. The tunneling of the atoms along the chains is suppressed initially by an energy offset between neighboring sites. A Raman transition using the cavity mode and a transversal pump laser then reintroduces a cavity-assisted tunneling. The feedback mechanism between the cavity field and the atoms leads to a spontaneous occupation of the cavity field and of a state of the fermionic atoms which is dominated by $s$-wave pairing correlations. Extending the setup to a quasi-one-dimensional ladder structure where the tunneling of atoms along the rungs of the ladder is cavity assisted, the repulsively interacting fermionic atoms self-organize into a $4k_F$ charge-density wave. We use adiabatic elimination of the cavity field combined with state-of-the-art density-matrix renormalization-group methods in finite systems in order to identify the steady-state phases of the system.

Figure 1

(a) Two different internal states of fermionic atoms represented by spin $\uparrow$ and $\downarrow$ are loaded into decoupled one-dimensional structures formed by optical lattices and coupled to the dynamical field of an optical cavity. Tunneling along the chain ($y$ direction) is strongly suppressed by a potential offset $\mathrm{\Delta }$ between neighboring lattice sites. (b) Cavity-assisted Raman processes induced by two running-wave pump beams, restore tunneling along the $y$ direction for each spin state of the atoms. The ground states are $|g_{\uparrow }\rangle$ and $|g_{\downarrow }\rangle$ and the intermediate states of the Raman process are $|e_{\uparrow }\rangle$ and $|e_{\downarrow }\rangle$, respectively. The two pump laser beams are coupled differently to spin $\sigma =\uparrow ,\downarrow$ with Rabi frequencies ${\mathrm{\Omega }}_{p_1\sigma }\text{and}{\mathrm{\Omega }}_{p_2\sigma }$, and $g_0$ is the vacuum-Rabi frequency of the cavity. The cavity-pump detuning is denoted by ${\delta }_{\text{CP}}$.

Figure 2

Phase diagram of the Hubbard chain with doping. The $s$-wave superconductor is the dominant phase for the attractive interaction ($U<0$). The sketch shows the onsite singlet pairs which then condense. The charge-density wave with wave vector $2k_F$ is the dominant phase for repulsive interaction ($U>0$). It is characterized by oscillating density-density correlations which are depicted in the sketch.

Figure 3

The expectation value of the tunneling $\langle K\rangle /L$ for an attractive Hubbard chain $U<0$ vs the rescaled tunneling amplitude $J/\left|U\right|$. Shown are two system sizes $L=192\text{and}384$ with filling $n=0.9375$ which lie on top of each other. When the tunneling is very large compared to the interaction the value of the tunneling approaches the noninteracting value ($U=0$) shown by the red arrow. The green solid line is the linear curve with slope ${\left(\frac{AL}{\left|U\right|}\right)}^{-1}\approx 3.87$. The crossing of the two curves gives the self-consistent solution with $J=\frac{2}{8}\left|U\right|$.

Figure 4

The density-density and $s$-wave pairing correlations of the attractive Hubbard chain with $J=\frac{2}{8}\left|U\right|$ calculated from DMRG for a system size $L=384$ and $N=360$ particles ($n=0.9375$). Whereas both correlations decay algebraically, the $s$-wave pairing correlation is the dominant correlation. The density-density correlation has oscillations with period of $2\pi n=2k_F$ on top and is fitted with the function $l^{-\gamma }\{a+bcos\left[2\pi n(l-l_0)\right]\}$ (dashed line). Fits with algebraic functions $\sim l^{-\gamma }$ are also shown (solid lines).

Figure 5

A balanced mixture of fermionic atoms in two internal states is loaded into a structure of decoupled ladders formed by optical lattices. Tunneling along the rung of the ladder is strongly suppressed by a potential offset $\mathrm{\Delta }$ between neighboring lattice sites and restored by a cavity-assisted Raman process.

Figure 6

Phase diagram of the Hubbard ladder at small doping away from half filling. The unconventional superconducting state with a singlet pairing on the rungs is the dominant phase for the intermediate values of the ratio of the rung tunneling to the tunneling within a leg ($J_{\perp }/J_{\parallel }$). The charge-density wave with wave vector $4k_F$ is the dominant phase for small and large $J_{\perp }/J_{\parallel }$.

Figure 7

Graphical interpretation of the self-consistency condition. The dependence of the rung tunneling (blue circles) on different ratios of the tunneling amplitudes $J_{\perp }/J_{\parallel }$ is shown for the repulsive Hubbard ladder with $U=8J_{\parallel }$ in a system of $L=192$ rungs and $N=360$ particles ($n=0.9375$) and for the noninteracting Hubbard ladder (red dashed line). The intersections with the linear function (green solid line) ${\left(\frac{AL}{J_{\parallel }}\right)}^{-1}\approx 0.64$ give the solutions of the self-consistency condition. For $U=8J_{\parallel }$, the intersection at $J_{\perp }=2J_{\parallel }$ indicates a stable nontrivial self-consistent solution. The intersection around $J_{\perp }=1.5J_{\parallel }$ is not stable. The green shaded region marks the regime of possible stable solutions.

Figure 8

The density-density correlations (orange squares) and the unconventional pair correlations (blue circles) vs distance $l$ computed starting from site $j=20$ in a Hubbard ladder with $U=8J_{\parallel },J_{\perp }=2J_{\parallel }$ for systems of $L=192$ rungs and $N=352$ particles ($n=0.9375$). Both correlations decay algebraically with distance. The curves are fitted with the function $l^{-\beta }\{a+bcos\left[2\pi n(l-l_0)\right]\}$ with the period of $2\pi n=4k_F$ (dashed lines). The fits with only the algebraic function are also shown as a guide to the eye (solid lines). The density-density correlations decay slower than the superconducting correlations and, thus, are dominating, which is the signature of the charge-density wave ${\mathrm{CDW}}^{4k_F}$.