ESQPT2021 Seminar Series

Although positive  Lyapunov exponents are normally related to chaotic dynamics, it is possible to find positive Lyapunov exponents in completely regular systems at critical energies of classical  unstable stationary points.  These critical energies are also associated with singular behaviours of the density of states yielding   the so-called  excited-state quantum phase transitions (ESQPT).  In this seminar,   signatures of exponential instability are explored in the quantum domain by considering the fidelity out-of -time-order correlator (FOTOC) for initial coherent states with energies equal or close to ESQPT energies of the Lipkin-Meshkov-Glick model and Dicke model.  A classical-quantum correspondence is found, making the quantum Lyapunov exponent a good indicator of ESQPTs.

In this seminar I will show that Raman dressed Bose-Einstein condensates can be used as a flexible experimental platform to study dynamical and excited-state quantum phase transitions. In spinor Bose-Einstein condensates, spin-changing collisions are a remarkable proxy to coherently realize  macroscopic many-body quantum states. These processes have been employed to generate entanglement and to study dynamical quantum phase transitions in atomic condensates. Here, I will show that weakly-coupled Raman-dressed Bose-Einstein condensates are equivalent at the quantum level to artificial spinor gases with tunable spin-changing collisions. This equivalence allows us to explain the quantum phases and their dynamics in a simple unified framework. Furthermore, I will discuss how this mapping can be exploited to design a robust experimental preparation of the elusive supersolid-like stripe phase of the spin-orbit-coupled gas through an excited-state quantum phase transition.

Phase transitions are typically deemed as the ultimate embodiment of many-body physics in the limit of infinitely many degrees of freedom under competing interactions. Such notion also holds in the quantum realm, where abrupt changes have been identified in the ground-, excited-states, and even in the time-evolved state of a quantum system in the thermodynamic limit. Yet, it has been recently shown that phase transitions can take place even in a finite-component system, that is, without scaling up the number of degrees of freedom or the amount of interacting particles. In this context, the most prominent example is the quantum Rabi model, which I will use in this seminar as an example to illustrate and motivate the relevance that finite-component systems can have in the study of quantum critical phenomena. In doing so, I will show that the quantum Rabi model undergoes both ground- and excited-state quantum phase transitions, and discuss the differences and similarities with respect to standard many-body systems. In addition, I will show that the quantum Rabi model also exhibits the two types of dynamical quantum phase transitions, and thus, it can serve as a test-bed to explore critical behavior in the time domain. Finally, I will briefly discuss how to realize these critical phenomena in a single trapped-ion experiment.

We introduce a complex continuum level density of unbound systems and describe singularities in its real and imaginary parts caused by stationary points of a finite 1D multibarrier potential. We reveal the relation of the level density to a complex time shift of the transmitted wave and show that the real part of the level density and time shift describes the passage of the particle through classically allowed coordinate regions, whereas the imaginary part is connected with an instanton-like picture of the tunnelling through forbidden regions. Hence, the singularities in the complex level density represent a dual extension of ESQPTs from bound to continuum systems. We numerically demonstrate the predicted effects in several tunnelling potentials using the complex scaling method.

ESQPTs have an important drawback compared with quantum and thermal phase transitions: it is not easy to define two different phases, with different equilibrium properties, separated by the corresponding critical energy. In this talk, we propose a generic mechanism to cover this gap. Inspired by the classical constant energy surfaces of quantum collective systems, we find an operator which is a constant of motion only in one of these two phases. This means that we can define two different phases at both sides of the ESQPT: one in which equilibrium thermodynamics depends on the energy, other possible conserved charges (like parity) and this new constant of motion, and another one in which this observable plays no relevant role. Its main characteristic is that it acts like a discrete symmetry in the first of these phases, with just two possible eigenvalues. Hence, we propose that the trademark of this phase in many systems displaying ESQPTs, like the Dicke and Rabi models, is having two different discrete symmetries: parity and this new constant of motion. We show analytically that they cannot commute among themselves, and that this fact explains one of the typical features of this kind of ESQPTs: the transition from degenerate doublets to non-degenerate energy levels. By means of numerics in the Dicke model, we show that expected values of observables in equilibrium crucially depend on the expected value of the new constant of motion in the corresponding phase.

In many-body systems that admit a well defined classical (mea-field) limit, ESQPTs are asociated with singularities in the classical manifold of constant energy, and generically have the form of saddle points. When the techniques and concepts of semiclassical (periodic-orbit) quantization are lifted to the Fock space of many-body states, the quantum signatures of such phase space regions extend beyond the mean level density of relevance in the thermodynamic limit, entering the mesoscopic regime where finite size effcts are witnessed by the discrete spectrum. The many-body semiclassical quantization of hyperbolic regions around saddle points shows the emergence of an asymptotically unique local time scale with a characteristic logarithmic dependence with the total particle number. This unique time scale turns out to rule both the scrambling of quantum information due to local hyperbolicity, and its recovery due to coherent revivals.
In this talk, I will review the recent progress in extending semiclassical methods from their traditional application to sytems of particles, into the realm of quantum fields, and show how their application to the phenomena relted with ESQPTs leads to such detailed understanding of the role of interference in Fock space.

It has been proven in theory that the excited-state quantum phase transition (ESQPT) phenomena may widely occur in the interacting boson model (IBM) and most of them can be associated with the stationary points of the classical Hamiltonian, such as the saddle point. It was recently revealed in the IBM that the Jacobi-type transitions (JTTs) supported by experimental evidence can occur as a function of angular momentum, L, with the transitional point in some cases being close to the saddle point energy. In this talk, we will discuss the JTTs associated with the U(5)-SU(3) transition and try to answer whether or not the JTT could be consider as a manifestation of the related ESQPT in nuclear systems, since the JTT and ESQPT in a given system can both be connected with the saddle point.

Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.

Excited-state quantum phase transitions (ESQPTs) extend the notion of quantum phase transitions beyond the ground state. They are characterized by closing energy gaps amid the spectrum. Identifying order parameters for ESQPTs poses however a major challenge. We introduce a topological order parameter that distinguishes excited-state phases in a large class of mean-field models and can be accessed by interferometry in current experiments with spinor Bose-Einstein condensates. Our work opens a way for the experimental characterization of excited-state quantum phases in atomic many-body systems.

ESQPTs signatures have been clearly identified in molecular spectra. In this talk I will briefly outline the results obtained in different lines of research in our group, having as a driving line the pursuit of ESQPTs in molecular systems. The first part will report the results obtained modeling bending vibrations using a four-body 2D vibron model Hamiltonian. The improvement in the calculated spectra has allowed us to explain the HCN-HNC bond-breaking isomerization spectroscopic features within the ESQPT formalism, the second item presented in the talk. The third part of the talk comprises a detailed of anharmonicity in the vibron model, needed to understand our results. Finally, the seminar ends with a report on the use of quantum fidelity susceptibility, a fine probe on abrupt changes of excited states, to ascribe states to ESQPT phases.