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arXiv:2310.20525

Extracting spectral properties of small Holstein polarons from a transmon-based analog quantum simulator

Vladimir M. Stojanović

Published Oct 31, 2023• 0 citations
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Abstract

The Holstein model, which describes purely local coupling of an itinerant excitation (electron, hole, exciton) with zero-dimensional (dispersionless) phonons, represents the paradigm for short-range excitation-phonon interactions. It is demonstrated here how spectral properties of small Holstein polarons -- heavily phonon-dressed quasiparticles, formed in the strong-coupling regime of the Holstein model -- can be extracted from an analog quantum simulator of this model. This simulator, which is meant to operate in the dispersive regime of circuit quantum electrodynamics, has the form of an array of capacitively coupled superconducting transmon qubits and microwave resonators, the latter being subject to a weak external driving. The magnitude of $XY$-type coupling between adjacent qubits in this system can be tuned through an external flux threading the SQUID loops between those qubits; this translates into an {\em in-situ} flux-tunable hopping amplitude of a fictitious itinerant spinless-fermion excitation, allowing one to access all the relevant physical regimes of the Holstein model. By employing the kernel-polynomial method, based on expanding dynamical response functions in Chebyshev polynomials of the first kind and their recurrence relation, the relevant single-particle momentum-frequency resolved spectral function of this system is computed here for a broad range of parameter values. To complement the evaluation of the spectral function, it is also explained how -- by making use of the many-body version of the Ramsey interference protocol -- this dynamical-response function can be measured in the envisioned analog simulator.

DOI
10.48550/arxiv.2310.20525

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The Holstein model, which describes purely local coupling of an itinerant excitation (electron, hole, exciton) with zero-dimensional (dispersionless) phonons, represents the paradigm for short-range excitation-phonon interactions. It is demonstrated here how spectral properties of small Holstein polarons -- heavily phonon-dressed quasiparticles, formed in the strong-coupling regime of the Holstein model -- can be extracted from an analog quantum simulator of this model. This simulator, which is meant to operate in the dispersive regime of circuit quantum electrodynamics, has the form of an array of capacitively coupled superconducting transmon qubits and microwave resonators, the latter being subject to a weak external driving. The magnitude of XYXYXY-type coupling between adjacent qubits in this system can be tuned through an external flux threading the SQUID loops between those qubits; this translates into an {\em in-situ} flux-tunable hopping amplitude of a fictitious itinerant spinless-fermion excitation, allowing one to access all the relevant physical regimes of the Holstein model. By employing the kernel-polynomial method, based on expanding dynamical response functions in Chebyshev polynomials of the first kind and their recurrence relation, the relevant single-particle momentum-frequency resolved spectral function of this system is computed here for a broad range of parameter values. To complement the evaluation of the spectral function, it is also explained how -- by making use of the many-body version of the Ramsey interference protocol -- this dynamical-response function can be measured in the envisioned 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.
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-L\"owen 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 (Kgs=0{K}_{\mathrm{gs}}=0Kgs​=0) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta Kgs{K}_{\mathrm{gs}}Kgs​ and $\ensuremath{-}{K}_{\mathrm{gs}}$. 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.
Strong coupling between mobile charge carriers or excitons and lattice vibrations gives rise to polarons: quasiparticles formed by charge carriers dressed by a cloud of phonon excitations. Polaron formation determines transport properties of systems ranging from transition-metal oxides and organic semiconductors to light-harvesting complexes. In excitonic systems that admit a hard-core boson description, mapping the excitations to pseudospins provides a natural framework for studying polaron physics, including the nonperturbative regime in which hopping amplitudes, phonon energies, and coupling strengths are comparable. Here, we propose a trapped-ion blueprint for realizing spin-phonon Hamiltonians with programmable spatial coupling between excitation density and phonon displacements, encompassing local Holstein and long-range Fröhlich-type interactions. To access strongly dressed quasiparticles whose bare spectral weight is suppressed, we introduce displaced Green's functions implemented through spin-dependent phonon displacements. Their spectral response reveals the polaron band spectrum, including binding energies and phonon-excitation sidebands, while spatially resolved measurements probe the structure of the phonon dressing cloud, providing a route to polaron tomography. We further complement this spectroscopy with direct and ancilla-assisted measurements of spin-phonon correlations that track the coupled dynamics of the excitation and its environment. These approaches connect the emergent quasiparticle spectrum to the microscopic buildup of its dressing cloud, enabling controlled studies of how the spatial range of coupling shapes polaron formation and mobility.
We propose a scheme for investigating the nonequilibrium aspects of small-polaron physics using an array of superconducting qubits and microwave resonators. This system, which can be realized with transmon or gatemon qubits, serves as an analog simulator for a lattice model describing a nonlocal coupling of a quantum particle (excitation) to dispersionless phonons. We study its dynamics following an excitation-phonon (qubit-resonator) interaction quench using a numerically exact approach based on a Chebyshev-moment expansion of the time-evolution operator of the system. We thereby glean heretofore unavailable insights into the process of the small-polaron formation resulting from strongly momentum-dependent excitation-phonon interactions, most prominently about its inherent dynamical timescale. To further characterize this complex process, we evaluate the excitation-phonon entanglement entropy and show that initially prepared bare-excitation Bloch states here dynamically evolve into small-polaron states that are close to being maximally entangled. Finally, by computing the dynamical variances of the phonon position and momentum quadratures, we demonstrate a pronounced non-Gaussian character of the latter states, with a strong antisqueezing in both quadratures.
We propose an analog quantum simulator for the Holstein molecular-crystal model based on a superconducting circuit QED system in the dispersive regime. By varying the driving field on the superconducting resonators, one can access both the adiabatic and antiadiabatic regimes of this model, and the strong electron-phonon coupling required for small-polaron formation can be readily reached. We show that a small-polaron state of arbitrary quasimomentum can be generated by applying a microwave pulse to the resonators. We also show that significant squeezing in the resonator modes can be achieved in the polaron-crossover regime through a measurement-based scheme.
Gate-based digital quantum simulations offer an exciting new paradigm for studying the many-body physics of strongly correlated systems. In this context, electron-phonon models are challenging for qubit-only quantum simulators, as bosonic degrees of freedom require costly finite-dimensional encodings. Here, we elaborate on an alternative approach based on a digital-analog circuit QED architecture, where fermions are encoded in transmon qubits while bosons are represented directly by microwave resonators. The central building block of this framework is a qubit-resonator Rabi gate that emulates strong electron-phonon coupling and can be implemented through a sequence of resonant Jaynes-Cummings gates interleaved with layers of single-qubit rotations. Using this Rabi gate as the fundamental unitary operation, we construct quantum circuits for the Hubbard-Holstein and Yukawa-Sachdev-Ye-Kitaev models, which describe, respectively, strongly correlated electrons coupled to phonons and phonon-mediated interactions among Majorana fermions. We further demonstrate how nonclassical phonon physics and signatures of quantum chaos in these models can be probed through circuit simulations, and develop measurement and variational protocols tailored to near-term superconducting quantum hardware.