Geometrical frustration, power law tunneling and non-local gauge fields from scattered light

Abstract

Designing the amplitude and range of couplings in quantum systems is a fundamental tool for exploring a large variety of quantum mechanical effects. Here, we consider off-resonant photon scattering processes on a geometrically shaped molecular cloud. Our analysis shows that such a setup is properly modeled by a Bose-Hubbard Hamiltonian where the range, amplitude and sign of the tunneling processes of the scattered photonic modes can be accurately tuned. Specifically, by varying the molecular distribution, we demonstrate that different configurations characterized by geometrical frustration, long-range power law hopping processes, and non-local gauge fields can be achieved. Our results thus represent a powerful and alternative approach to perform an accurate Hamiltonian engineering of quantum systems with non trivial coupling structures.

Figures (4)

• Figure 1: Physical setup. (a) A Gaussian beam represented by blue and red patterns illuminates a geometrically shaped molecular cloud in a cavity with molecule density $\rho(\vec{r})$. The beam frequency is tuned so that off-resonant scattering is the dominant light-matter interaction. Off-resonant light scattering onto the molecules induces a dynamic evolution of the beam’s mode occupations $f_{n}(\vec{r})$, e.g., in the Laguerre-Gaussian basis. (b) To second-order perturbation theory, the effective bosonic Hamiltonian features tunable long-range hoppings $t_{n,n'}$ and density-density interactions $U_{n,n'}$. The geometry of the molecular cloud controls the sign and phase of the hopping and interaction amplitudes.
• Figure 2: Effective geometries. (a) Purely 1D geometry induced by fixing $c_1=1$, $\varphi_1 = 0.9\pi$, and $c_i=0$ for $\forall i>1$. (b) Effective frustrated triangular geometry obtained by fixing $c_1=3/4$, $c_2=1/4$, $\varphi_1 = 0.9\pi$, $\varphi_2 = 1.1\pi$ and $c_i=0$ for $\forall i>2$. (c) Effective frustrated triangular geometry with hopping amplitudes connecting up to three sites obtained by fixing $c_1=1/3$, $c_2=1/3$, $c_3=1/3$, $\varphi_1 = 0.9\pi$, $\varphi_2 = 1.1\pi$, $\varphi_3 = 1.02\pi$ and $c_i=0$ for $\forall i>3$.
• Figure 3: Power-law decay of the hopping amplitudes The matrix $t_{i,j}$ representing the hopping amplitudes in the kinetic Hamiltonian as in Eq. \ref{['eq:H_power_law']} for three different values of the exponent: (a) $\beta = 0.5$, (b) $\beta = 1$ and (c) $\beta = 2$. The first seven hopping amplitudes ($i-7\leq j \leq i+7$) are tuned to be nonzero by switching on the corresponding $c$ coefficients in the density of the scatterers.
• Figure 4: Classical gauge fields. (a) The complex phase of the nearest ($t_1$) and next-nearest ($t_2$) neighbor hopping can be tuned so that going around a plaquette on the triangular ladder, the bosonic particle acquires a phase of $\Theta_1 = \pi$. (b) By switching on the next-to-next-nearest neighbor hopping ($t_3$), in addition to $\Theta_1$, another plaquette, defined by $t_1,t_2$ and $t_3$ can be identified. Choosing the complex phase of $t_3$ accordingly, a phase shift of $\Theta_2 = \pi/2$ by going around this second plaquette can be induced.