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arXiv:1401.4783 · Physical Review B

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

Vladimir M. Stojanović, Mihajlo Vanević A, Eugene Demler, Lin Tian

Published Apr 21, 2014• 38 citations
arXivHTMLPDF

Abstract

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 (${K}_{\mathrm{gs}}=0$) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta ${K}_{\mathrm{gs}}$ 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.

DOI
10.1103/physrevb.89.144508

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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−Kgs\ensuremath{-}{K}_{\mathrm{gs}}\ensuremath−Kgs​. 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.
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 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.
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.
The polaron concept captures physical situations involving an itinerant quantum particle (excitation) that interacts strongly with bosonic degrees of freedom and becomes heavily boson-dressed. While the Gerlach-L\"owen theorem rules out the occurrence of nonanalyticities of ground-state-related quantities for a broad class of polaron models, examples were found in recent years of sharp transitions pertaining to strongly momentum-dependent interactions of an excitation with dispersionless (zero-dimensional) phonons. On the example of a lattice model with Peierls-type excitation-phonon interaction, such level-crossing-type small-polaron transitions are analyzed here through the prism of the entanglement spectrum of the excitation-phonon system. By evaluating this spectrum in a numerically exact fashion it is demonstrated that the behavior of the entanglement entropy in the vicinity of the critical excitation-phonon coupling strength chiefly originates from one specific entanglement-spectrum eigenvalue, namely the smallest one. While the discrete translational symmetry of the system implies that those eigenvalues can be labeled by the bare-excitation quasimomentum quantum numbers, here it is shown numerically that they are predominantly associated with the quasimomenta 0 and $\ensuremath{\pi}$, including cases where a transition between the two takes place deeply in the strong-coupling regime.
While the concept of the entanglement spectrum has heretofore been utilized to address various many-body systems, the models describing an itinerant spinless-fermion excitation coupled to zero-dimensional bosons (e.g. dispersionless phonons) have as yet not received much attention in this regard. To fill this gap, the ground-state entanglement spectrum of a model that includes two of the most common types of short-ranged, nonlocal excitation-phonon interaction -- the Peierls- and breathing-mode couplings -- is numerically evaluated here. This model displays a sharp, level-crossing transition at a critical coupling strength, which signifies the change from a nondegenerate ground state at the quasimomentum Kgs=0K_{\textrm{gs}}=0Kgs​=0 to a twofold-degenerate one corresponding to a symmetric pair of nonzero quasimomenta. Another peculiarity of this model is that in the special case of equal Peierls- and breathing-mode coupling strengths the bare-excitation Bloch state with the quasimomentum 000 or πππ is its exact eigenstate. Moreover, below a critical coupling strength this state is the ground state of the model. Thus, the sharp transition between a bare excitation and a heavily phonon-dressed (polaronic) one can be thought of as a transition between vanishing and finite entanglement. It is demonstrated here that the smallest ground-state entanglement-spectrum eigenvalue to a large extent mimics the behavior of the entanglement entropy itself and vanishes in this special case of the model; by contrast, all the remaining eigenvalues diverge in this case. The implications of excitation-phonon entanglement for WWW-state engineering in superconducting and neutral-atom-based qubit arrays serving as analog simulators of this model are also discussed.
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.