Non-Hermitian squeezed polarons

Recent experimental breakthroughs in non-Hermitian ultracold atomic lattices have dangled tantalizing prospects in realizing exotic, hitherto unreported, many-body non-Hermitian quantum phenomena. In this work, we propose and study an experimental platform for a radically different non-Hermitian phenomenon dubbed polaron squeezing. We find that it is marked by a dipolelike accumulation of fermions arising from an interacting impurity in a background of non-Hermitian reciprocity-breaking hoppings. We computed their spatial density and found that, unlike Hermitian polarons which are symmetrically localized around impurities, non-Hermitian squeezed polarons localize asymmetrically in the direction opposite to conventional non-Hermitian pumping and nonperturbatively modify the entire spectrum, despite having a manifestly local profile. We investigated their time evolution and found that, saliently, they appear almost universally in the long-time steady state, unlike Hermitian polarons which only exist in the ground state. In our numerics, we also found that, unlike well-known topological or skin localized states, squeezed polarons exist in the bulk, independently of boundary conditions. Our findings could inspire the realization of many-body states in ultracold atomic setups, where a squeezed polaron can be readily detected and characterized by imaging the spatial fermionic density.

Figure 1

Non-Hermitian polaron squeezing is distinct from other mechanisms for localized states, such as (a) topological localization and (b) the non-Hermitian skin effect, both single-body mechanisms requiring open boundaries. By contrast, squeezed polarons (c) are special asymmetric dipolelike accumulations across either side of an interacting impurity. They are many-body dressed states in the bulk, with charge density “squeezed” in the opposite direction from non-Hermitian pumping. Illustrative numerics are from ${\widehat{H}}_{\text{min}}$ of Eq. (1).

Figure 2

PBC spectrum $E$ [panels (a1)–(d1)] and spatial density $\rho \left(x\right)$ [panels (a2)–(d2)] for ${\widehat{H}}_{\text{min}}$ [Eq. (1)] with different nonreciprocities $\alpha$ and impurity interaction strengths $g$: (a1, a2) $\alpha =g=0$; (b1, b2) $\alpha =0, g=-100$; (c1, c2) $\alpha =1, g=0$; and (d1, d2) $\alpha =1, g=-100$. Energy eigenstates $|\psi \rangle$ are colored by their squeezing asymmetric parameter $\mathrm{\Gamma }$, which captures polaron squeezing: $\mathrm{\Gamma }$ is large only with both nonreciprocity and impurity interaction (d1). While the spatial polaron density $\rho \left(x\right)$ is symmetrically peaked about the impurity at $x_0=6$ when Hermitian (b2), it is asymmetrically squeezed in the non-Hermitian case (d2). (e1, e2) Dynamics of the spatial density for (e1) $\alpha =1, g=0$ and for (e2) $\alpha =1, g=-100$, with the dipolelike asymmetric profile as circled. All computations are with $N=6$ fermions in $L=12$ sites. The initial state $|{\psi }^R\left(0\right)\rangle$ is the ground state for $\alpha =0$ and is $\left(\right|101010101010\rangle +|010101010101\rangle )/\sqrt{2}$ for $\alpha =1$ [133].

Figure 3

(a1) Our effective interacting Hamiltonian for squeezed polarons [Eq. (5)] is based on a two-photon dissipative Raman process [29, 74] with impurity interactions $g$ from Feshbach resonance. The Rabi frequency is ${\mathrm{\Omega }}_0$ between hyperfine ground states $|A\rangle$ and $|B\rangle$ and the excited state $|e\rangle$ for ${}^{40}\mathrm{K}$ atoms; a phase difference ${\varphi }_0$ introduces nonreciprocity. $\mathrm{\Delta }$ is the single-photon detuning of the excited state $|e\rangle$, whose non-Hermitian decay rate $\gamma$ can be laser controlled. (a2) The impurity interaction $g$ is highly tunable through the magnetic field, with parameters given by Refs. [139, 140, 141]. (b1, b2) $g$ nonperturbatively modifies the PBC spectrum $E$ at half-filling $N=6$ and $2L=12$, such that all states $|\psi \rangle$ become squeezed with elevated squeezing asymmetry $\mathrm{\Gamma }$ when $\left|g\right|\ne 0$. Here, $t_2=\left(2\pi \right)\times 1000$ Hz [142] sets the energy scale.

Figure 4

(a) Squeezing expectation $\left|\mathrm{\Gamma }\right|$ in the $g-|\tilde{\gamma }|$ parameter space; note the vanishing squeezing at zero impurity interaction $g$ and the enhanced polaron squeezing at large effective decay rate $|\tilde{\gamma }|$ and very attractive ($g<0$) or repulsive ($g>0$) interactions. Here, $\mathrm{\Gamma }$ is computed from the long-time steady state evolved from the initial state $|{\psi }^R\left(0\right)\rangle =\left(\right|101010101010\rangle +|010101010101\rangle )/\sqrt{2}$. (b) Spatial density $\rho \left(x\right)$ as a function of $x$ and $g$ under PBCs. Note the very pronounced asymmetric profile across the impurity position $x_0=6$, particularly in the repulsive ($g>0$) case where $\rho \left(x_0\right)$ is strongly localized. Data are plotted at $\tilde{\gamma }/t_2=0.92-0.15i$, as indicated by the dashed red line in panel (a). (c1, c2) Simulated spatial density measurements. The density around the impurity ($x_0+1=7$) exhibits almost identical asymmetric profiles of squeezed polaron states (red dashed box) regardless of OBCs or PBCs, as long as the impurity interaction $g$ is nonzero, with a slight shift from sublattice effects. (d1, d2) Evolution of spatial densities from the initial state $|{\psi }^R\left(0\right)\rangle$ under PBCs, with an asymmetric steady-state squeezed polaron profile for (d2) $g\ne 0$. We used $\tilde{\gamma }/t_2=0.92-0.15i, t_1/t_2=1$, and $N=6$ fermions in $2L=12$ sites for all subfigures, and we used $g=0$ for panels (c1) and (d1) and $g/t_2=-20$ for panels (c2) and (d2).