Spectroscopic signatures of emergent elementary excitations in a kinetically constrained long-range interacting two-dimensional spin system

Abstract

Lattice spin models featuring kinetic constraints constitute a paradigmatic setting for the investigation of glassiness and localization phenomena. The intricate dynamical behavior of these systems is a result of the dramatically reduced connectivity between many-body configurations. This truncation of transition pathways often leads to a fragmentation of the Hilbert space, yielding highly collective and therefore often slow dynamics. Moreover, this mechanism supports the formation of characteristic elementary excitations, which we investigate here theoretically in a two-dimensional Rydberg lattice gas. We explore their properties as a function of interaction strength and range, and illustrate how they can be experimentally probed via sideband spectroscopy. Here, we show that the transition rate to certain delocalized superposition states of elementary excitations displays collective many-body enhancement.

Figures (4)

• Figure 1: Elementary excitations in a facilitated two-dimensional Rydberg square lattice.(a) Level scheme of two atoms separated by one lattice constant $a$. The atoms are modelled by two states, a ground state $\ket \downarrow\xspace$ (beige) and a Rydberg state $\ket \uparrow\xspace$ (red). Both states are coupled by a laser with Rabi frequency $\Omega$ and detuning $\Delta$. The latter is chosen to compensate the nearest-neighboring Rydberg interaction, i.e. , $V_1+\Delta=0$, which is the so-called facilitation condition. (b) Some near-resonant processes in a 3$\times$3-lattice, assuming that $\Omega$ is much larger than the next-nearest-neighbor interaction. (c) Structure and energy of some elementary excitations. Here, $V_2$ and $V_3$ are the interaction energies between next-nearest and next-next-nearest neighbors, respectively. The grey box marks a subspace $\mathcal{H}_{(\leftrightarrow,\updownarrow)}$, where elementary excitations are formed by chains of Rydberg atoms. These carry energy $r\Delta$ from their $r$ Rydberg excitations and energy $(r-1)V_1$ from the $r-1$ nearest-neighbor pairs. Under facilitation condition, this implies a constant energy shift $-V_1$ for these elementary excitations. For $V_3\ll\Omega$, they are near-resonantly coupled by the laser, as shown in panel (b).
• Figure 2: Emergence of the kinetic constraint. Correlation function $C^{(x)}(d_x,d_y,t)$ in a $4\times 4$ lattice with an initial single excitation positioned at site $(2,2)$. (a,b) Data obtained by evolving the system under the full Hamiltonian \ref{['eq:exact Hamiltonian']}, for weak interaction $V_1=\Omega$ and strong interaction $V_1=50\Omega$, respectively. In the latter case, a single Rydberg excitation initiates the growth of chains of Rydberg excitations along the rows and columns of the lattice [see Fig. \ref{['fig:Overview']}(b)]. (c) Correlations calculated from the reduced model \ref{['eq:reduced-Hamiltonian']}, for $V_1=50\Omega$. Rows (b,c) are in excellent agreement, which demonstrates that the reduced model description becomes appropriate when the interaction strength is sufficiently large. The snapshots are taken at times $\Omega t=0.5,1,2$.
• Figure 3: Spectroscopy and collective enhancement.(a,b) Excitation spectrum for a $2\times2$ and $3\times3$ lattice as a function of the interaction strength $V_1$ and modulation frequency $\omega_\text{s}$. In red we show the time-averaged number of Rydberg atoms $\bar{n}$, which is calculated considering the full dynamics under the Hamiltonian (\ref{['eq:Ham spectroscopy']}), with $\Omega_\text{s}=0.2\,\Omega$. To evaluate $\bar{n}$ we discretize the integral with sampling points $\Omega t\in\{1,2,\cdots,30\}$. The grey scale shows the transition matrix element $M_E$ at the expected resonance energy, computed with the spectrum of the reduced model Hamiltonian (\ref{['eq:reduced-Hamiltonian']}). Note, that the matrix elements in a $2\times2$ lattice are constant ($M_E=2$), in contrast to the matrix elements in a $3\times3$ lattice. This is a result of the next-next-nearest neighbor interaction $V_3$, which does not contribute for a $2\times2$ lattice. The inset of panel (a) illustrates the idea behind the sideband spectroscopy: the Rabi frequency modulation at frequency $\omega_\text{s}$ probes excitations, which are formed by (kinetically constrained) many-body states that are near-resonantly coupled by the laser (Rabi frequency $\Omega$). For intermediate interaction strength we observe excitations of many-body configurations that are not contained in the subspace $\mathcal{H}_{(\leftrightarrow,\updownarrow)}$, e.g. triangular patterns of Rydberg atoms with energy $-V_1+V_2$. With increasing interaction strength only excitations contained in $\mathcal{H}_{(\leftrightarrow,\updownarrow)}$ are excited. (c) Spectrum of an $11\times11$ lattice, calculated with the reduced model (\ref{['eq:reduced-Hamiltonian']}). Here, we used a logarithmic scale $\log_{10}(M_E)$ to increase the contrast of the spectral lines. The grey dashed lines correspond to the energies $-V_1+(r-2)V_3$ of Rydberg chains with length $r\ge2$, see End Matter. (d) Collectively enhanced spectral line in the $11\times11$ lattice. The inset depicts the transition matrix element $M_{E_\text{res}}$ evaluated with the corresponding eigenstate at energy $E_\text{res}$ for different lattice sizes $N=N_x\times N_y$ with $N_x=N_y\in\{2,3,\cdots,20\}$.
• Figure B1: Time-resolved spectroscopy signal of two facilitated atoms.(a) Time evolution of the Rydberg density $\langle n\rangle_t$ under a sideband modulated laser of two facilitated atoms for varying modulation frequency $\omega_\text{s}$ and $\Omega_\text{s}=0.2\,\Omega$, and $V_1=10\,\Omega$. (b) Time-averaged Rydberg density $\bar{n}$ in the time window, indicated by the grey shading in panel (a). The average $\bar{n}$ is evaluated for $\tau=2/\Omega$, using $200$ equidistant sampling points of the integral. The red line depicts $\bar{n}(\omega_\text{s})$ and the dashed black lines indicate the theoretical eigenvalues at $\omega_\text{s}=|E_\pm|$, Eq. \ref{['eq:eigenenergy two facilitated atoms']}. (c) Time-averaged Rydberg density $\bar{n}$ for the full time window ($\tau=10/\Omega$). The integral is evaluated with $1000$ equidistant sampling points. We observe that the spectral lines become only visible for sufficiently long coherent evolution times.