Simulation of two-boson bound states using arrays of driven-dissipative coupled linear optical resonators

We present a strategy based on two-dimensional arrays of coupled linear optical resonators to investigate the two-body physics of interacting bosons in one-dimensional lattices. In particular, we want to address the bound pairs in topologically nontrivial Su-Schrieffer-Heeger arrays. Taking advantage of the driven-dissipative nature of the resonators, we propose spectroscopic protocols to detect and tomographically characterize bulk doublon bands and doublon edge states from the spatially resolved transmission spectra, and to highlight Feshbach resonance effects in two-body collision processes. We discuss the experimental feasibility using state-of-the-art devices, with a specific eye on arrays of semiconductor micropillar cavities.

Figure 1

(a) Sketch of a dimerized lattice with superlattice unit cell $d$, providing the 1D Su-Schrieffer-Heeger (SSH) model described by the Bose-Hubbard Hamiltonian (1). (b) Two-dimensional classical setup emulating the two-particle quantum physics in the one-dimensional SSH model shown in (a). (c) Sketch of the polariton micropillar cavities that may be used to realize the two-dimensional lattice shown in (b) implementing the mapping of the one-dimensional two-body system shown in (a).

Figure 2

(a), (b) Two-body excitation spectra in the SSH model for a system with periodic boundary conditions as a function of the center-of-mass momentum $K$, displaying scattering bands (green region) and bulk bound states (red bands), calculated for interaction strength (a) $U/J_1=5$ and (b) $U/J_1=10$. (c) Energy bands of two-boson excitations versus interaction strength $U$ for a system with open boundary conditions: scattering states (green), bulk bound states (red), single-particle edge states (blue), and doublon edge states (magenta and black). (d) Integrated steady-state intensity from the pillars on the three main diagonals $m=n, m=n\pm 1$ obtained in a driven-dissipative system of $21\times 21$ sites, pump position in $(m,n)=(2,2)$, for two values of dissipation $\gamma /J_1=0.1$ (blue) and $\gamma /J_1=0.2$ (brown). Peaks in the spectrum are found in correspondence of bulk and edge doublon modes. To represent the bulk doublon bands in (a)–(c), we considered the results for periodic boundary conditions. In such case, as shown in Figs. 2 and 2, doublon states for a given value of $K$ are well defined if no scattering states are available at the same energy. As soon as the doublon energy band enters the scattering continuum (or the single-particle edge state band), eigenstates acquire a hybridized nature and the distinction between doublon and scattering states is washed out. Physically, this mixing is responsible for a finite lifetime of the doublon states, possibly followed by revivals in a spatially finite system [62].

Figure 3

(a)–(c) Spectroscopic analysis of the near-to-diagonal steady-state intensity $I_{D3}$ as a function of the pumping frequency $\omega$ for $U/J_1=5, \gamma /J_1=0.1$, and $L=21$. The relevant energy ranges are selected in panels (a)–(c) to highlight peak $I$, peaks $II, III, IV$, and peak $V$, respectively. The different line colors indicate different pumping positions, as detailed in the legend inset. The vertical magenta dotted (black dashed) lines indicate the energy of the doublon edge states at the weak-link (strong-link) edge in the nondissipative 1D array. The red shaded areas indicate the energy range of the bulk doublon bands in the nondissipative system. (d)–(k) Steady-state intensity profile for pumping in site (2,2) [(d)–(h)] and for pumping in site $(L,L)$ [(i)–(k)]. The pumping frequency is indicated in the different panels.

Figure 4

Localization properties of the steady-state intensity distributions as a function of $\gamma$; the localization properties are quantified by $r_p=\sqrt{\langle {(x-m)}^2+{(y-n)}^2\rangle }$, depending on the pumping position $(m,n)$. (a) Weak-link localization obtained when pumping in $(m,n)=(2,2)$ at the same frequencies as in Fig. 3: d, yellow diamonds; e and g, green and blue triangles; f, red squares; h, purple circles. (b) Strong-link localization obtained when pumping in $(m,n)=(L,L)$ at the same frequencies as in Fig. 3: i, yellow diamonds; j, red squares; k, purple circles. For comparison, we plot also results for ${\mathrm{k}}^{\prime }$ (black empty triangles) obtained when pumping at energy $\omega /J_1=12.86$ corresponding to the edge bound state of the nondissipative system.

Figure 5

Weight of the driven-dissipative system stationary state on bulk doublons (red thick solid lines), doublon edge states (magenta thin solid lines), single-boson edge states (blue dashed lines) for pumping frequency in the range $3<\omega /J_1<7$. The weight on the rest of the states is shown by the dot-dashed gray lines and turns out to be negligible for those parameters. The calculations are performed for a system of $21\times 21$ sites, $U/J_1=5, J_2/J_1=5, \gamma /J_1=0.1$ and pumping positions (a) $(m,n)=(1,1)$; (b) $(m,n)=(1,11)\text{and}(11,1)$; (c) $(m,n)=(11,21)\text{and}(21,11)$; (d) $(m,n)=(21,21)$.

Figure 6

State tomography in an array of $21\times 21$ pillars for $J_2/J_1=5, U/J_1=5, \gamma /J_1=0.1$ versus actual eigenmode profiles. The color maps represent the total integrated steady-state intensity as a function of pumping point $(m,n)$. Depending on the driving frequency, one can visualize different types of two-boson states: (a) single-boson edge state for $\omega /J_1=-5$; (b) scattering state of two bosons for $\omega /J_1=0$; (c) doublon edge state for $\omega /J_1=3.68$; (d) bulk doublon state for $\omega /J_1=13.25$. (e)–(h) Eigenmodes of the structure representing single-boson edge state, scattering state, doublon edge state, and bulk doublon state, respectively. Eigenmode frequencies ${\omega }^{\left(i\right)}/J_1$ are equal to $-4.908$, 0.0026, 3.698, and 13.23, respectively.

Figure 7

Demonstration of Feshbach resonance in a driven-dissipative system. (a) Total field intensity on the diagonal pillars $(m,m)$ as a function of pumping frequency for three different values of interaction strength $U/J_1=9,10,11$. (b) Steady-state intensity profile calculated for $J_2/J_1=5, U/J_1=10, \omega /J_1=U, \gamma /J_1=0.1$ and pumping position $(16,25)\text{and}(25,16)$ shown by the white squares. Excitation of Feshbach molecules is observed through a sizable population on the diagonal pillars. The calculations are performed for an array of $41\times 41$ sites in order to minimize finite-size effects.