Transmon-based simulator of nonlocal electron-phonon coupling: A platform for observing sharp small-polaron transitions

We propose an analog superconducting quantum simulator for a one-dimensional model featuring momentum-dependent (nonlocal) electron-phonon couplings of Su-Schrieffer-Heeger and “breathing-mode” types. Because its corresponding coupling vertex function depends on both the electron and phonon quasimomenta, this model does not belong to the realm of validity of the Gerlach-Löwen theorem that rules out any nonanalyticities in single-particle properties. The superconducting circuit behind the proposed simulator entails an array of transmon qubits and microwave resonators. By applying microwave driving fields to the qubits, a small-polaron Bloch state with an arbitrary quasimomentum can be prepared in this system within times several orders of magnitude shorter than the typical qubit decoherence times. We demonstrate that—by varying the externally tunable parameters—one can readily reach the critical coupling strength required for observing the sharp transition from a nondegenerate (single-particle) ground state corresponding to zero quasimomentum ($K_{\mathrm{gs}}=0$) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta $K_{\mathrm{gs}}$ and $-K_{\mathrm{gs}}$. Through exact numerical diagonalization of our effective Hamiltonian, we show how this nonanalyticity is reflected in the relevant single-particle properties (ground-state energy, quasiparticle residue, average number of phonons). We also show that the proposed setup provides an ideal testbed for studying the nonequilibrium dynamics of small-polaron formation in the presence of strongly momentum-dependent electron-phonon interactions.

Figure 1

Schematic of the simulator circuit containing transmon qubits $Q_n$, SC resonators $R_n$, and connecting circuits $B_n$ with three Josephson junctions. External magnetic fluxes ${\varphi }_n^b$ and ${\varphi }_n^t$ are threading the bottom and top loops of each connecting circuit, respectively. Note that the circuit elements are not drawn to scale.

Figure 2

Dimensionless parameters of the simulator: (a) Ratio of the hopping integral and the phonon energy, and (b) effective coupling strength $\lambda$, both shown as a function of the tunable dc-flux ${\varphi }_{b0}$, for two different values of the effective phonon frequency $\delta \omega$.

Figure 3

Ground-state energy, expressed in units of $\mathcal{E}\equiv {10}^{-3}\delta {\phi }_0^2{E}_J=2\pi \hslash \times 100$ MHz, as a function of the experimentally tunable parameter ${\varphi }_{b0}$. The solid curve corresponds to the effective phonon frequency $\delta \omega /2\pi =200$ MHz, while the dashed curve corresponds to $\delta \omega /2\pi =300$ MHz.

Figure 4

Small-polaron Bloch dispersions for two different values of the tunable parameter ${\varphi }_{b0}$, shown throughout the Brillouin zone. The solid curve corresponds to the $y$-axis scale marked on the left, while the dash-dotted curve corresponds to the scale on the right, as indicated by the arrows.

Figure 5

Characterization of the sharp transition between a quasifree excitation and a small polaron: (a) Ground-state quasiparticle residue $Z_{\mathrm{gs}}\equiv Z_{k=K_{\mathrm{gs}}}$, and (b) average number of phonons ${\overline{N}}_{\mathrm{ph}}$, versus the experimentally tunable parameter ${\varphi }_{b0}$. The solid and dashed curves correspond to $\delta \omega /2\pi =200$ MHz and $\delta \omega /2\pi =300$ MHz, respectively.

Figure 6

Illustration of the multiphononic nature of small-polaron excitations: Full ground-state phonon distribution at ${\varphi }_{b0}/\pi =0.98$, for $\delta \omega /2\pi =200$ MHz (squares) and $\delta \omega /2\pi =300$ MHz (circles). The dotted curve corresponds to the phonon distribution for ${\varphi }_{b0}/\pi =0.95$, i.e., below the critical coupling strength.