Spectroscopy of the soliton lattice formation in quasi-one-dimensional fermionic superfluids with population imbalance

Motivated by recent experiments in low-dimensional trapped fermionic superfluids, we study a quasi-one-dimensional (quasi-1D) superfluid with a population imbalance between two hyperfine states using an exact mean-field solution for the order parameter. When an effective “magnetic field” exceeds a critical value, the superfluid order parameter develops spatial inhomogeneity in the form of a soliton lattice. The soliton lattice generates a band of quasiparticle states inside the energy gap, which originate from the Andreev bound states localized at the solitons. Emergence of the soliton lattice is accompanied by formation of a spin-density wave, with the majority fermions residing at the points in space where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) order parameter vanishes. We discuss possibilities for experimental detection of the quasi-1D FFLO state using elastic and inelastic optical Bragg scattering and radiofrequency spectroscopy. We show that these measurements can provide necessary information for unambiguous identification of the spatially inhomogeneous quasi-1D FFLO state and the soliton lattice formation.

Figure 1

(a) An array of 1D tubes confining cold atoms, where $n$ is the tube number and $t_{\perp }$ is the tunneling matrix element between adjacent tubes. (b) Fermi surfaces for the majority ($\uparrow$) and minority ($\downarrow$) atoms in the quasi-1D limit $t_{\perp }\ll E_F$ (where $E_F$ is the Fermi energy of 1D motion along the tube). The momentum components $p$ and $p_{\perp }$ are parallel and perpendicular to the tubes, respectively.

Figure 2

The 1D energy dispersion $E\left(p\right)$ of the fermionic atoms in the hyperfine states $|1\rangle$ and $|2\rangle$ with a population imbalance described by the chemical potentials ${\mu }_{\uparrow }$ and ${\mu }_{\downarrow }$ and the Fermi momenta $p_F^{\uparrow }$ and $p_F^{\downarrow }$. The third, unpopulated state $|3\rangle$ can be used for detection purposes. The atomic energy differences between the states $|2\rangle$ and $|1\rangle$ and the states $|3\rangle$ and $|2\rangle$ are denoted as ${\omega }_{12}$ and ${\omega }_{23}$, respectively.

Figure 3

Schematic plot of the quasiparticle energy dispersion $E\left(p\right)$ in the inhomogeneous superfluid phase. Here $\xi \left(p\right)$ is the normal-state dispersion, $2\mu ={\mu }_{\uparrow }+{\mu }_{\downarrow }$, and $Q=p_F^{\uparrow }-p_F^{\downarrow }$.

Figure 4

Schematic plot of the superfluid pairing for the LO order parameter. In each panel, the centers of the three pairs of arrows, representing pairs of atoms on the Fermi surface, are located at the point with the momentum either $+Q/2$ or $-Q/2$, which is a half of the Cooper pair momentum in the LO phase.

Figure 5

Schematic plots of the density of states (top row), superconducting order parameter $\Delta \left(x\right)$ (middle row), and the spin density ${\rho }_s\left(x\right)$ (bottom row) for different values of the effective magnetic field $h$. (a) For $h<h_c$, the ground state of the system is uniform, as in the BCS theory. (b) For one unpaired atom $h=h_c+\delta h$, $\Delta \left(x\right)$ has a kink soliton. (c) For a small spin imbalance $h>h_c$, $\Delta \left(x\right)$ forms a soliton lattice. (d) For a large imbalance $h\gg h_c$ corresponding to the LO limit, $\Delta \left(x\right)$ is sinusoidal. Note that ${\rho }_s\left(x\right)$ peaks at the points in space where $\Delta \left(x\right)$ vanishes.

Figure 6

The single-particle energy dispersion relation $E\left(p\right)$ in the quasi-1D FFLO state described by Eq. (32). The spectrum is obtained by solving Eqs. (45) and (47) for given values of the momentum $p$ and the parameter $b=\pm$. This plot is a zoom into Fig. 3 near the Fermi momentum $p_F$.

Figure 7

The pairing potential $\Delta \left(x\right)$ and the spin density ${\rho }_s\left(x\right)$ calculated for the quasi-1D FFLO state. (a) The soliton-lattice regime for $h$ close to the critical value $h_c$. The spikes of ${\rho }_s\left(x\right)$ are aligned with the kink solitons, where $\Delta \left(x\right)$ changes sign. (b) The Larkin-Ovchinnikov regime for $h\gg {\Delta }_0$. The spin-density modulation is small compared with the uniform spin background.

Figure 8

Schematic plot of the elastic Bragg diffraction experiment. Photons with the momentum $\left|\mathit{k}\right|=2\pi /\lambda$ are elastically scattered to the momentum ${\mathit{k}}^{\prime }$ by the spin-density modulation with the period $l/2$. The Bragg condition for constructive interference is $l\sin \theta =m\lambda$ (or $2k_x=2Qm$), where $m$ is an integer.

Figure 9

Transition rate $W_{{\mathit{k}}^{\prime },\mathit{k}}^{\mathrm{el}}$ for the elastic Bragg scattering vs. the momentum transfer $q_x=k_x^{\prime }-k_x$, calculated from Eq. (66) for $L=12l$. The scattering rate has peaks at $q_x=2mQ$. The effective magnetic fields $h=1.005h_c$ (solid line) and $h=1.03h_c$ (dashed line) correspond to the spin polarizations $P\approx 4.1\%$ and $P\approx 4.2\%$ in Eq. (1). Here we used ${\Delta }_0/E_F=0.2$.

Figure 10

Contour plot of the dynamical spin structure factor $S(\Omega ,q)$ vs. the momentum transfer $q$ and the energy-transfer $\Omega$. Line A represents the threshold for the type IIa processes creating two quasiparticles in the upper band in Fig. 6. The minimal energy transfer threshold $\Omega =2{\mathcal{E}}_3$ is achieved at $q=Q$, as indicated by the dashed lines. Line B represents the threshold for the type I and IIb processes, which involve one quasiparticle in the midgap band and another in the upper band in Fig. 6. The minimal energy-transfer threshold $\Omega ={\mathcal{E}}_3-{\mathcal{E}}_2$ is achieved at $q=0$. The spin structure factor $S(\Omega ,q)$ is maximal at the threshold lines A and B, as indicated by the bright colors. The plot is calculated for the effective magnetic field $h=1.05h_c$ corresponding to the dimensionless spin polarization $P\approx 4.3\%$ at ${\Delta }_0/E_F=0.2$.

Figure 11

Absorption spectrum $W_{\sigma \to 3}^{\mathrm{FFLO}}\left({\omega }_{d\sigma }\right)$, Eq. (89), vs. the frequency detuning ${\omega }_{d\sigma }$, Eq. (82). For the minority fermions, $\sigma =\downarrow$, the spectrum is shown by the dashed curve and labeled as $W_{\mathrm{bot}}\left({\omega }_{d\sigma }\right)$, because it originates from the bottom band in Fig. 3. It is blue-shifted similarly to the absorption spectrum (87) for the BCS state. For the majority fermions, $\sigma =\uparrow$, the spectrum consists of two terms $W_{\mathrm{bot}}\left({\omega }_{d\sigma }\right)$ and $W_{\mathrm{mid}}\left({\omega }_{d\sigma }\right)$. The latter term is shown by the solid curve and originates from the middle band in Fig. 3 with $|E_{\lambda }|<h$ in Fig. 6. The width of this peak is of the order of ${\Delta }_2$, and some spectral density is red-shifted to lower frequencies. The plots are obtained for $h=1.25h_c$, which corresponds to the dimensionless spin polarization $P\approx 5.1\%$ assuming ${\Delta }_0/E_F=0.2$.

Figure 12

Plots of the Fourier components of the Bogoliubov amplitudes $|{\tilde{u}}_{\lambda b}\left(m\right)|$ and $|{\tilde{v}}_{\lambda b}\left(m\right)|$ from Eq. (78) for $m=0,\pm 1$, branch $b=+$, and the right Fermi point ($\alpha =1$). The amplitudes for the branch $b=-$ can obtained from the ones shown above using $|{\tilde{u}}_{\lambda ,b}\left(m\right)|=|{\tilde{v}}_{\lambda ,-b}(-m)|$. The amplitudes for the left Fermi point ($\alpha =2$) can be obtained using Eqs. (24). Energy dependence of these functions reflects the subtle structure of the wave function describing the quasi-1D FFLO state. For the presentation purposes, we have chosen a large effective magnetic field $h=1.95h_c$, along with $k\simeq 0.7$ and ${\Delta }_2\simeq 1.4{\Delta }_0$.

Figure 13

Plots of the absolute values of the Fourier amplitudes $|{\tilde{v}}_{\lambda ,b}\left(m\right)|$ vs. $h/h_c$ for $E_{\lambda }=1.2{\mathcal{E}}_3$. The higher-order Fourier amplitudes decrease fast with the increase of the effective magnetic field $h$.