Spectral features of polaronic excitations in a superconducting analog simulator

We investigate the spectral properties of polaronic excitations within the framework of an analog quantum simulator based on inductively coupled superconducting transmon qubits and microwave resonators. This system emulates a lattice model that describes a nonlocal coupling of an itinerant spinless-fermion excitation to dispersionless (Einstein-type) phonons through the Peierls and breathing-mode interaction mechanisms. The model is characterized by a sharp, level-crossing transition at a critical value of the effective excitation-phonon coupling strength; above the transition point, the ground state of this model corresponds to a heavily dressed (small-polaron) excitation. Using the kernel-polynomial method, we evaluate the momentum-frequency resolved spectral function of this system for a broad range of parameters. In particular, we underscore the ramifications of the fact that the zero-quasimomentum Bloch state of a bare excitation represents the exact eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon coupling strength. We also show that—based on the numerically evaluated spectral function and its well-known relation with the survival probability of the initial, bare-excitation Bloch state (the Loschmidt echo)—one can make predictions about the system dynamics following an excitation-phonon interaction quench. To make contact with anticipated experimental realizations, we utilize a previously proposed method for extracting dynamical-response functions in systems with local (single-qubit) addressability using the multiqubit (many-body) version of the Ramsey interference protocol.

Figure 1

Layout of the envisioned analog simulator [23], which consists of SC qubits $Q_n$ (with single-qubit charging and Josephson energies $E_C^s$ and $E_J^s$, respectively), resonators $R_n$, and coupler circuits $B_n$. Here ${\varphi }_n^l$ and ${\varphi }_n^u$ are the respective total magnetic fluxes threading the lower and upper loops of $B_n$.

Figure 2

Momentum-frequency resolved spectral function $A(k,\omega )$ for six different quasimomenta $k$ in the Brillouin zone, evaluated for a range of frequencies. The parameter values used are $\delta \omega /\left(2\pi \right)=200\mathrm{MHz}$ and ${\varphi }_{\text{dc}}/\pi =0.95$; the corresponding effective coupling strength is ${\lambda }_{\text{eff}}=0.51$. The obtained spectral function satisfies the sum rule in Eq. (6).

Figure 3

Momentum-frequency resolved spectral function $A(k,\omega )$ for six different quasimomenta $k$ in the Brillouin zone, evaluated for a range of frequencies. The parameter values used are $\delta \omega /\left(2\pi \right)=200\mathrm{MHz}$ and ${\varphi }_{\text{dc}}/\pi =0.97$; the corresponding effective coupling strength is ${\lambda }_{\text{eff}}=1.41$. The obtained spectral function satisfies the sum rule in Eq. (6).

Figure 4

Momentum-frequency resolved spectral function $A(k,\omega )$ for six different quasimomenta $k$ in the Brillouin zone, evaluated for a range of frequencies. The parameter values used are $\delta \omega /\left(2\pi \right)=200\mathrm{MHz}$ and ${\varphi }_{\text{dc}}/\pi =0.98$; the corresponding effective coupling strength is ${\lambda }_{\text{eff}}=3.17$. The obtained spectral function satisfies the sum rule in Eq. (6).

Figure 5

Time dependence of the Loschmidt echo for four different initial free-excitation quasimomenta ($k=2\pi /5,3\pi /5,4\pi /5$, and $\pi$). The parameter values used are the same as in Fig. 3.