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arXiv:1506.05833 · Physical Review A

Spectroscopic probes of isolated nonequilibrium quantum matter: Quantum quenches, Floquet states, and distribution functions

Yunxiang Liao, Matthew S. Foster

Published Nov 19, 2015• 18 citations
arXivHTMLPDF

Abstract

We investigate radio-frequency (rf) spectroscopy, metal-to-superconductor tunneling, and angle-resolved photoemission spectroscopy (ARPES) as probes of isolated out-of-equilibrium quantum systems, and examine the crucial role played by the nonequilibrium distribution function. As an example, we focus on the induced topological time-periodic (Floquet) phase in a two-dimensional $p+ip$ superfluid, following an instantaneous quench of the coupling strength. The post-quench Cooper pairs occupy a linear combination of ``ground'' and ``excited'' Floquet states, with coefficients determined by the distribution function. While the Floquet band structure exhibits a single avoided crossing relative to the equilibrium case, the distribution function shows a population inversion of the Floquet bands at low energies. For a realization in ultracold atoms, these two features compensate, producing a bulk average rf signal that is well captured by a quasiequilibrium approximation. In particular, the rf spectrum shows a robust gap. The single crossing occurs because the quench-induced Floquet phase belongs to a particular class of soliton dynamics for the BCS equation. The population inversion is a consequence of this, and ensures the conservation of the pseudospin winding number. As a comparison, we compute the rf signal when only the lower Floquet band is occupied; in this case, the gap disappears for strong quenches. The tunneling signal in a solid-state realization is ignorant of the distribution function, and can show wildly different behaviors. We also examine rf, tunneling, and ARPES for weak quenches, such that the resulting topological steady state is characterized by a constant nonequilibrium order parameter. In a system with a boundary, tunneling reveals the Majorana edge states. However, the local rf signal due to the edge states is suppressed by a factor of the inverse system size, and is spatially deconfined throughout the bulk of the sample.

DOI
10.1103/physreva.92.053620

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In this thesis, we consider two different topics: the probes of out-of-equilibrium topological matter and many-body-localization (MBL) in higher dimensions. The first part focuses on the two-dimensional (2D) topological Floquet p+ip superfluid induced by an instantaneous quench of the interaction strength. We investigate radio-frequency (rf) spectroscopy, metal-to-superconductor tunneling, and angle-resolved photoemission spectroscopy (ARPES) as probes of this out-of-equilibrium quantum system in both the cold-atomic and solid-state realizations. We notice a single avoided crossing in the Floquet band structure and a population inversion in the occupation of these bands. In addition, the rf spectrum is well captured by a quasiequilibrium approximation and shows a robust gap. We also compute the rf signal when only the lower Floquet band is occupied as well as the tunneling signal. In both cases, the spectra show wildly different behaviors compared with the rf signal from the post-quench state, and the gap disappears for strong quenches. We attribute this to the fact that the distribution function influences the rf but not the tunneling signal. Finally, we look into the ARPES spectrum which exhibits a clear series of Floquet copies. The second part is devoted to interacting fermions in the presence of quenched disorder. We derive a finite-temperature Keldysh response theory for such a system in the delocalized phase, in the form of the Finkel'stein nonlinear sigma model (FNLσM). Applicable to any symmetry class with at least a U(1) symmetry, our formulation automatically incorporates the correct infrared cutoffs for quantum corrections to transport, and provides a framework to study the real and virtual scattering processes. In particular, we are able to rederive from the FNLσM the Altshuler-Aronov-Khmelnitsky equations for dephasing. We then propose a strategy to approach the 2D ergodic-to-MBL transition from the ergodic side. Such a transition is explored as a dephasing catastrophe in an isolated system with short-range interactions. In this case, the self-generated heat bath responsible for dephasing exhibits diffusive (non-Markovian) density fluctuations. We recast the dephasing of quantum interference corrections as a self-interacting polymer loop, and study its critical behavior using renormalization group approach. In dimensions d=4-ε, we identify a nontrivial fixed point corresponding to a finite temperature where the dephasing time diverges. This fixed point can be associated with a toy version of 2D ergodic-to-MBL transition, provided that it survives to ε=2.
Quantum Fisher information in a quenched p+ipp + ipp+ip superfluid
The quantum Fisher information (QFI) is widely used to characterize quantum phases and transitions, but its diagnostic power sometimes relies on selecting special generators based on prior knowledge of the underlying physics. We ask whether this requirement can be relaxed in nonequilibrium systems by studying the QFI of the long-time asymptotic state of a two-dimensional p+ip superfluid following an instantaneous quench of the coupling strength, using the particle number within a large subextensive subsystem as the generator. In equilibrium, the ground-state QFI is continuous across the topological transition between the weak-pairing BCS and the strong-pairing BEC phases, showing no direct signature of the transition. After the quench, however, the QFI associated with the same generator distinguishes the three dynamical phases and can encode the topology of the pre-quench state. In phase I, with a vanishing order parameter, the QFI Fourier spectrum consists of a single zero-frequency spike determined by the nonequilibrium distribution function. In phase II, with a constant nonzero order parameter, the QFI spectrum contains a zero-frequency spike and two continua separated by a gap set by the minimum asymptotic quasiparticle energy. The continuum edge behavior reveals whether this minimum occurs at zero or finite momentum. In the former case, the spectral weight vanishes at the edge, with the sign just inside the continuum encoding the pre-quench topology for large subsystem. In phase III, with a time-periodic order parameter, the QFI spectrum exhibits discrete peaks at integer multiples of the oscillation frequency, together with continua associated with the Floquet quasienergy spectrum. Our results show that driving the system out of equilibrium can enhance the diagnostic power of the QFI for a standard physical observable, revealing information inaccessible in equilibrium.
Periodically driven quantum systems provide a versatile platform for engineering and exploring non-equilibrium states of matter that have no equilibrium counterparts– a field commonly re- ferred to as Floquet engineering. The research presented in this thesis addresses the microscopic interpretation of experimental spectroscopic signatures, the stability of these driven states in the presence of strong many-body interactions, and the emergence of novel forms of non-equilibrium criticality encoded in quantum entanglement. The first part of the thesis focuses on the interpretation of time- and angle-resolved photo- emission spectroscopy (tr-ARPES), the primary experimental tool for detecting Floquet–Bloch sidebands (FBs). The measured signal reflects interference between processes occurring both inside the material and in the vacuum. In particular, Floquet-dressed initial states coexist and interfere with Volkov-dressed final states arising from the laser-assisted photoelectric effect (LAPE). Their experimental identification requires careful comparison with theoretical predic- tions. Using graphene as a model Dirac material, we perform a systematic comparison between two theoretical frameworks: the first-order perturbative Born approximation (PB1) and time- dependent nonequilibrium Green’s functions (tdNEGF), and compare them with experimental results. We disentangle intrinsic Floquet states, LAPE, and their interference, and quantify how polarization, incidence angle, and near-surface screening shape the observed spectra. Our findings show that, once matrix elements are included, both methods capture qualitative trends, but tdNEGF is necessary to describe the full energy–momentum-resolved features, including hybridization gaps. While non-interacting systems exhibit well-defined FBs, interacting systems absorb energy and can evolve toward an infinite-temperature state. To study this competition between coherent driving and many-body heating, in the second part we consider a one-dimensional chain of interacting spinless fermions using exact diagonalization (ED) and matrix product state (MPS) methods. We compute time-dependent spectral functions and correlation measures, and compare the spectral width of the many-body excitation continuum with the driving frequency to define low- and high-frequency regimes. In our parameter regime, low-frequency driving leads to strong overlap between Floquet sectors, resulting in rapid heating and loss of coherence. For weakly populated continua, we find a transient timescale for the suppression of FBs. When the driving frequency exceeds the continuum width, FBs are well separated and long-lived. Thefinalpartofthethesisinvestigatesnon-equilibriumphenomenafromaninformation-theoretic perspective. Wereportanewclassofdynamicalphasetransitions,termedtemporalentanglement transitions (TET) characterized by non-analytic reorganizations of the entanglement spectrum, in a periodically driven transverse-field Ising chain. These transitions occur in the spectrum of the entanglement Hamiltonian and are signaled by dynamical spontaneous symmetry breaking, marked by the closure of the Schmidt gap and parity flips. These transitions remain invisible to conventional local observables. Finite-size scaling reveals that TET belong to a universality class with a correlation-length exponent ν = 1, mirroring equilibrium Ising universality despite their purely dynamical origin.
The interaction of time-dependent electromagnetic fields with electrons in quantum materials can give rise to many both fundamentally interesting as well as technologically relevant phenomena like transient band structure manipulations or the emergence of prethermal states of matter, which are not a result of heating. This thesis contributes theoretical insights to both these topics: We model the interference of Floquet and Volkov side bands, which are a result of the dressing of electron states by light, in a time-resolved angle-resolved photoemission spectroscopy (ARPES) experiment. Furthermore, we analyze the emergence of a band-like feature in the non-equilibrium spectral function of a one-dimensional charge-density wave insulator upon periodic driving. Secondly, we use the fermionic truncated Wigner approximation (fTWA), a phase space method for the time evolution of interacting fermions, to study the light-induced order parameter dynamics in SU(N)-symmetric fermionic lattice models. By comparison with exact analytical results for an interaction quench in the Hubbard model, we find that dephasing-induced prethermalization is correctly described by fTWA. We discuss photoinduced prethermal transitions between competing phases of matter in the large-N Hubbard-Heisenberg model and find a pronounced frequency-dependence of the transition. This work contributes to a better understanding of photoinduced order parameter dynamics from a microscopic perspective.
We determine the radio-frequency (rf) spectra for nonstationary states of a fermionic condensate produced by a rapid switch of the scattering length. The rf spectrum of the nonequilibrium state with constant BCS order parameter has two features in contrast with equilibrium where there is a single peak. The additional feature reflects the presence of excited pairs in the steady state. In the state characterized by periodically oscillating order parameter, the rf-absorption spectrum contains two sequences of peaks spaced by the frequency of oscillations. Satellite peaks appear due to a process where a rf photon in addition to breaking a pair emits or absorbs oscillation quanta.
We study how time- and angle-resolved photoemission (tr-ARPES) reveals the dynamics of BCS-type, sss-wave superconducting systems with time-varying order parameters. Approximate methods are discussed, based on previous approaches to either optical conductivity or quantum dot transport, to enable computationally efficient prediction of photoemission spectra. One use of such predictions is to enable extraction of the underlying order parameter dynamics from experimental data, which is topical given the rapidly growing use of tr-ARPES in studying unconventional superconductivity. The methods considered model the two-time lesser Green's functions with an approximated lesser self-energy that describes relaxation by coupling of the system to two types of baths. The approach primarily used here also takes into consideration the relaxation of the excited states into equilibrium by explicitly including the level-broadening of the retarded and advanced Green's functions. We present equilibrium and nonequilibrium calculations of tr-ARPES spectrum from our model and discuss the signatures of different types of superconducting dynamics.