Francisco Pérez-Bernal

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Despite its importance to experiments, numerical simulations, and the development of theoretical models, self-averaging in many-body quantum systems out of equilibrium remains underinvestigated. Usually, in the chaotic regime, self-averaging is taken for granted. The numerical and analytical results presented here force us to rethink these expectations. They demonstrate that self-averaging properties depend on the quantity and also on the time scale considered. We show analytically that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices. We also analyze the participation ratio, Rényi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiments with ion traps. We find that self-averaging holds at short times for the quantities that are local in space, while at long times, self-averaging applies for quantities that are local in time. Various behaviors are revealed at intermediate time scales.

This work is concerned with the excited state quantum phase transitions (ESQPTs) defined in Ann.Phys. 323, 1106 (2008). In many-body models that exhibit such transitions, the ground state quantum phase transition (QPT) occurs in parallel with a singularity in the energy spectrum that propagates to higher energies as the control parameter increases beyond the QPT critical point. The analysis of the spectrum has been a main tool for the detection of these ESQPTs. Studies of the effects of this transition on the system dynamics are more limited. Here, we extend our previous works and show that the evolution of an initial state with energy close to the ESQPT critical point may be extremely slow. This result is surprising, because it may take place in systems with long-range interactions, where the dynamics is usually expected to be very fast. A timely example is the one-dimensional spin-1/2 model with infinite-range Ising interaction studied in experiments with ion traps. Its Hamiltonian has a U(2) algebraic structure. More generally, the slow dynamics described here occurs in two-level bosonic or fermionic models with pairing interactions and a U(v+1) Hamiltonian exhibiting a QPT between its limiting U(v) and SO(v+1) dynamical symmetries. In this work, we compare the results for v=1, 2, and 3.

Excited state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions (QPTs) to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-1/2 system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.

We study the structure of the eigenstates and the dynamics of a system that undergoes an excited state quantum phase transition (ESQPT). The analysis is performed for two-level pairing models characterized by a U(n+1) algebraic structure. They exhibit a second order phase transition between two limiting dynamical symmetries represented by the U(n) and SO(n+1) subalgebras. They are, or can be mapped onto, models of interacting bosons. We show that the eigenstates with energies very close to the ESQPT critical point, E_{ESQPT}, are highly localized in the U(n)-basis. Consequently, the dynamics of a system initially prepared in a U(n)-basis vector with energy close to E_{ESQPT} may be extremely slow. Signatures of an ESQPT can therefore be found in the structures of the eigenstates and in the speed of the system evolution after a sudden quench. Our findings can be tested experimentally with trapped ions.