# Effective dimensions of infinite-dimensional Hilbert spaces: A phase-space approach.

@article{PilatowskyCameo2022EffectiveDO, title={Effective dimensions of infinite-dimensional Hilbert spaces: A phase-space approach.}, author={Sa{\'u}l Pilatowsky-Cameo and David Villase{\~n}or and Miguel Angel Bastarrachea-Magnani and Sergio Lerma-Hern'andez and Jorge G. Hirsch}, journal={Physical review. E}, year={2022}, volume={105 6-1}, pages={ 064209 } }

By employing Husimi quasiprobability distributions, we show that a bounded portion of an unbounded phase space induces a finite effective dimension in an infinite-dimensional Hilbert space. We compare our general expressions with numerical results for the spin-boson Dicke model in the chaotic energy regime, restricting its unbounded four-dimensional phase space to a classically chaotic energy shell. This effective dimension can be employed to characterize quantum phenomena in infinite…

## One Citation

### Identification of quantum scars via phase-space localization measures

- PhysicsQuantum
- 2022

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies…

## References

SHOWING 1-10 OF 45 REFERENCES

### Particle-Spin Coupling in a Chaotic System: Localization-Delocalization in the Husimi Distributions

- Physics
- 1991

The wave functions of the Dicke Hamiltonian, describing a spin coupled to a bosonic mode, are studied via Husimi distributions. A classical analogue of this system is also obtained. For several…

### Quantum vs classical dynamics in a spin-boson system: manifestations of spectral correlations and scarring

- PhysicsNew Journal of Physics
- 2020

We compare the entire classical and quantum evolutions of the Dicke model in its regular and chaotic domains. This is a paradigmatic interacting spin-boson model of great experimental interest. By…

### Quantum localization measures in phase space.

- PhysicsPhysical review. E
- 2021

A general scheme to define localization in measure spaces, based on what is called Rényi occupations, is presented, from which any measure of localization can be derived, and is applied to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model.

### Quantum suppression of chaos in the spin-boson model.

- PhysicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1996

It is found that the transition to chaos in the classical system has no apparent effect on any of these variables, in the spin-1/2 case, in disagreement with some claims made in earlier studies of this system.

### Identification of quantum scars via phase-space localization measures

- PhysicsQuantum
- 2022

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies…

### Dynamical Fluctuations of Observables and the Ensemble of Classical Trajectories

- Physics
- 1993

The long term behavior of non-integrable quantum systems is investigated to find a classical counterpart that describes the statistical properties of dynamical fluctuation of quantum observables. On…

### Quantum scarring in a spin-boson system: fundamental families of periodic orbits

- Physics
- 2020

As the name indicates, a periodic orbit is a solution for a dynamical system that repeats itself in time. In the regular regime, periodic orbits are stable, while in the chaotic regime, they become…

### Statistical properties of the localization measure of chaotic eigenstates in the Dicke model.

- PhysicsPhysical review. E
- 2020

The findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

### Coherent quantum dynamics: What fluctuations can tell

- Physics
- 2015

Coherent states provide a natural connection of quantum systems to their classical limit and are employed in various fields of physics. Here we derive general systematic expansions, with respect to…

### Chaos and the quantum phase transition in the Dicke model.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

A semiclassical Dicke model is derived that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos, and it is demonstrated that the system undergoes a transition from quasi-integrability to quantum chaotic.