# Identification of quantum scars via phase-space localization measures

Saúl Pilatowsky-Cameo  Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, Apdo. Postal 70-543, C.P. 04510 CDMX, Mexico  Center for
Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  David Villaseñor  Instituto de Ciencias Nucleares, Universidad
Nacional Autónoma de México, Apdo. Postal 70-543, C.P. 04510 CDMX, Mexico
Miguel A. Bastarrachea-Magnani  Departamento de Física, Universidad Autónoma
Metropolitana-Iztapalapa, San Rafael Atlixco 186, C.P. 09340 CDMX, Mexico
Sergio Lerma-Hernández  Facultad de Física, Universidad Veracruzana, Circuito
Aguirre Beltrán s/n, C.P. 91000 Xalapa, Veracruz, Mexico  Lea F. Santos
Department of Physics, Yeshiva University, New York, New York 10016, USA
Jorge G. Hirsch  Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, Apdo. Postal 70-543, C.P. 04510 CDMX, Mexico

###### Abstract

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 the portion of the classical
phase space occupied by a quantum state. The measure is based on the
$\alpha$-moments of the Husimi function and is known as the Rényi occupation
of order $\alpha$. With this quantity and random pure states, we find a
general expression to identify states that are maximally delocalized in phase
space. Using this expression and the Dicke model, which is an interacting
spin-boson model with an unbounded four-dimensional phase space, we show that
the Rényi occupations with $\alpha>1$ are highly effective at revealing
quantum scars. Furthermore, by analyzing the high moments ($\alpha>1$) of the
Husimi function, we are able to identify qualitatively and quantitatively the
unstable periodic orbits that scar some of the eigenstates of the model.

##  1 Introduction

In the classical domain, typical trajectories of chaotic systems fill the
available phase space. Even if unstable periodic trajectories are present, one
does not find them by random sampling the phase space, because they form a
measure-zero set. In the quantum domain, however, their existence is revealed
through quantum scars. These are states that are not uniformly distributed,
but present high probabilities along the phase-space region occupied by
unstable periodic trajectories. The unexpected phenomenon of quantum scarring
was observed in early studies of the Bunimovich stadium billiard [McDonald and
Kaufman(1979), McDonald(1983)], although the term was coined later
[Heller(1984), Heller(1987), Berry(1989), Heller(1991), Kuś et al.(1991)Kuś,
Zakrzewski, and Życzkowski]. Recently, the subject has received boosted
attention due to the so-called “many-body quantum scars” [Turner et
al.(2018)Turner, Michailidis, Abanin, Serbyn, and Papić, Turner et
al.(2021)Turner, Desaules, Bull, and Papić], which still await for possible
links with the classical phase space.

Quantum scarring and localization are not synonyms [Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch], but both carry the idea of confined eigenstates. The
notion of localization in quantum systems presupposes a basis representation.
Anderson localization [Anderson(1958), Lee and Ramakhrishnan(1985)], for
example, refers to the suppression of the classical diffusion of particles in
real space due to quantum interferences. This phenomenon has a dynamical
counterpart originally studied in the context of the kicked rotor [Chirikov et
al.(1981)Chirikov, Izrailev, and Shepelyansky, Fishman et al.(1982)Fishman,
Grempel, and Prange, Izrailev(1990), Fishman(2010)]. Localization is an
extremely broad subject that extends to the hydrogen atom in a monochromatic
field [Casati et al.(1984)Casati, Chirikov, and Shepelyansky, Casati et
al.(1987)Casati, Chirikov, Shepelyansky, and Guarnesi], Rydberg atoms [Blümel
and Smilansky(1987)], quantum billiards [Borgonovi et al.(1996)Borgonovi,
Casati, and Li, Batistić and Robnik(2010), Batistić and Robnik(2013a),
Batistić and Robnik(2013b), Porter et al.(2017)Porter, Barr, Barr, and Reichl,
Batistić et al.(2019)Batistić, Lozej, and Robnik, Robnik(2020)], banded random
matrices [Casati et al.(1993)Casati, Chirikov, Guarneri, and Izrailev], and
interacting many-body quantum systems [Santos et al.(2004)Santos, Rigolin, and
Escobar, Basko et al.(2006)Basko, Aleiner, and Altshuler, Oganesyan and
Huse(2007), Lazarides et al.(2015)Lazarides, Das, and Moessner, Stanley et
al.()Stanley, Lin, Jaroszyński, and Popović], among others.

In finite spaces, the degree of delocalization of a quantum state is based on
how much it spreads on a chosen finite basis representation, as quantified by
participation ratios [Edwards and Thouless(1972), Izrailev(1990)] and Rényi
entropies [Atas and Bogomolny(2012), Atas and Bogomolny(2014)]. In unbounded
continuous spaces, such measures can reach arbitrarily large values. If one
wishes to investigate maximally delocalized states in these systems, one must
define a finite region as a reference volume. There are multiple ways to
select this bounded region, and each choice produces a different localization
measure [Villaseñor et al.(2021)Villaseñor, Pilatowsky-Cameo, Bastarrachea-
Magnani, Lerma-Hernández, and Hirsch, Wang and Robnik(2020)]. A natural volume
of reference is obtained by measuring the degree of localization of a quantum
state over the classical phase-space energy shell, as proposed in Ref.
[Villaseñor et al.(2021)Villaseñor, Pilatowsky-Cameo, Bastarrachea-Magnani,
Lerma-Hernández, and Hirsch]. This measure of localization has been named
Rényi occupation, because it is related to the exponential of the Rényi-Wehrl
entropies [Wehrl(1978), Gnutzmann and Zyczkowski(2001)]. To compute the Rényi
occupation, we represent the quantum states in the phase space through the
Husimi function [Husimi(1940)]. Thus, the Rényi occupations of order $\alpha$
refer to averages of normalized $\alpha$-moments of the Husimi functions.

In this work, we obtain a general analytical expression for the Rényi
occupations of order $\alpha$ using maximally delocalized random pure states.
We employ this expression to analyze the eigenstates of the Dicke model in the
chaotic regime and distinguish the eigenstates that are maximally delocalized
from those that are highly localized in phase space. The Dicke model
[Dicke(1954)] is an experimental spin-boson model that has a rich and
unbounded phase space.

By comparing the Rényi occupations of the high-energy eigenstates of the Dicke
model with the Rényi occupations of random states, as a function of $\alpha$,
we range the eigenstates according to their degree of delocalization. While
all eigenstates are scarred [Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo,
Villaseñor, Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch], the
most localized states are the ones scarred by unstable periodic orbits with
short periods. The analysis of high-moments ($\alpha>1$) of the Husimi
functions of these highly localized states allow us to identify more clearly
the classical unstable periodic orbits that cause their scarring. These orbits
are different from the ones found in Ref. [Pilatowsky-Cameo et
al.(2021a)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch] and their uncovering represents a step forward in the
difficult task of identifying the unstable periodic orbits underlying quantum
scars. In addition, we study the dynamical properties of the unveiled orbits
and use them to explain the structures of the eigenstates and their degree of
localization.

The paper is organized as follows. The Dicke model and its phase space are
presented in Sec. 2. In Sec. 3, we introduce a measure of localization of
quantum states based on the Rényi occupations of order $\alpha$ and provide an
analytical expression for maximally delocalized states. In Sec. 4 we discuss
quantum scarring in the light of the Rényi occupations and identify the
unstable periodic orbits that scar the most localized eigenstates. We also
interpret our localization measure in terms of the geometric and dynamical
properties of the classical periodic orbits. Our conclusions are presented in
Sec. 5.

##  2 Dicke Model

The Dicke model [Dicke(1954)] was initially introduced to explain the
phenomenon of superradiance [Hepp and Lieb(1973a), Wang and Hioe(1973),
Garraway(2011), Kirton et al.(2019)Kirton, Roses, Keeling, and Dalla Torre]
and has since fostered a wide variety of theoretical studies, including the
behavior of out-of-time-ordered correlators [Chávez-Carlos et al.(2019)Chávez-
Carlos, López-del-Carpio, Bastarrachea-Magnani, Stránský, Lerma-Hernández,
Santos, and Hirsch, Lewis-Swan et al.(2019)Lewis-Swan, Safavi-Naini,
Bollinger, and Rey, Pilatowsky-Cameo et al.(2020)Pilatowsky-Cameo, Chávez-
Carlos, Bastarrachea-Magnani, Stránský, Lerma-Hernández, Santos, and Hirsch],
manifestations of quantum scarring [de Aguiar et al.(1992)de Aguiar, Furuya,
Lewenkopf, and Nemes, Furuya et al.(1992)Furuya, de Aguiar, Lewenkopf, and
Nemes, Bakemeier et al.(2013)Bakemeier, Alvermann, and Fehske, Pilatowsky-
Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-
Hernández, Santos, and Hirsch, Pilatowsky-Cameo et al.(2021a)Pilatowsky-Cameo,
Villaseñor, Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch], non-
equilibrium dynamics [Altland and Haake(2012), Kloc et al.(2018)Kloc,
Stránský, and Cejnar, Lerma-Hernández et al.(2018)Lerma-Hernández, Chávez-
Carlos, Bastarrachea-Magnani, Santos, and Hirsch, Lerma-Hernández et
al.(2019)Lerma-Hernández, Villaseñor, Bastarrachea-Magnani, Torres-Herrera,
Santos, and Hirsch, Kirton et al.(2019)Kirton, Roses, Keeling, and Dalla
Torre, Villaseñor et al.(2020)Villaseñor, Pilatowsky-Cameo, Bastarrachea-
Magnani, Lerma, Santos, and Hirsch], and measures of quantum localization with
respect to phase space [Wang and Robnik(2020), Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch]. The model is also of great interest to experiments with
trapped ions [Cohn et al.(2018)Cohn, Safavi-Naini, Lewis-Swan, Bohnet,
Gärttner, Gilmore, Jordan, Rey, Bollinger, and Freericks, Safavi-Naini et
al.(2018)Safavi-Naini, Lewis-Swan, Bohnet, Gärttner, Gilmore, Jordan, Cohn,
Freericks, Rey, and Bollinger], superconducting circuits [Jaako et
al.(2016)Jaako, Xiang, Garcia-Ripoll, and Rabl], and cavity assisted Raman
transitions [Baden et al.(2014)Baden, Arnold, Grimsmo, Parkins, and Barrett,
Zhang et al.(2018)Zhang, Lee, Kumar, Arnold, Masson, Grimsmo, Parkins, and
Barrett].

The Dicke Hamiltonian describes a single mode of the electromagnetic field
interacting with a set of $\mathcal{N}$ two-level atoms [Dicke(1954)]. Setting
$\hbar=1$, the Hamiltonian is given by

$\hat{H}_{D}=\omega\hat{a}^{\dagger}\hat{a}+\omega_{0}\hat{J}_{z}+\frac{\gamma}{\sqrt{\mathcal{N}}}(\hat{a}^{\dagger}+\hat{a})(\hat{J}_{+}+\hat{J}_{-}),$
(1)

where $\hat{a}^{\dagger}$ ($\hat{a}$) is the bosonic creation (annihilation)
operator of the field mode and $\hat{J}_{+}$ ($\hat{J}_{-}$) is the raising
(lowering) operator defined by $\hat{J}_{\pm}=\hat{J}_{x}\pm i\hat{J}_{y}$,
where $\hat{J}_{x,y,z}=(1/2)\sum_{k=1}^{\mathcal{N}}\hat{\sigma}_{x,y,z}^{k}$
are the collective pseudo-spin operators satisfying the SU(2) algebra and
$\hat{\sigma}_{x,y,z}$ are the Pauli matrices. The Hamiltonian parameters are
the radiation frequency of the electromagnetic field $\omega$, the transition
frequency of the two-level atoms $\omega_{0}$, and the atom-field coupling
strength $\gamma$. Since the Hamiltonian commutes with the total pseudo-spin
operator
$\hat{\textbf{J}}^{2}=\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}$, the
Hilbert space is separated into different invariant subspaces for each
eigenvalue $j(j+1)$ of $\hat{\textbf{J}}^{2}$. We work within the totally
symmetric subspace that includes the ground-state of the system and is defined
by the maximum value of the pseudo-spin length $j=\mathcal{N}/2$. The model
also possesses a discrete parity symmetry,
$\hat{\Pi}=e^{i\pi(\hat{a}^{\dagger}\hat{a}+\hat{J}_{z}+j)}$, which leads to
an additional subspace separation according to the eigenvalues of the parity
operator $\Pi_{\pm}=\pm 1$.

The model has been used in studies of the quantum phase transition from a
normal ($\gamma<\gamma_{c}$) to a superradiant ($\gamma>\gamma_{c}$) phase,
which arises when the coupling strength reaches the critical value
$\gamma_{c}=\sqrt{\omega\omega_{0}}/2$ [Hepp and Lieb(1973a), Hepp and
Lieb(1973b), Wang and Hioe(1973), Emary and Brandes(2003)]. The model also
exhibits regular or chaotic behavior depending on the Hamiltonian parameters
and excitation energies [Chávez-Carlos et al.(2016)Chávez-Carlos,
Bastarrachea-Magnani, Lerma-Hernández, and Hirsch]. In this paper, we choose
the resonant frequency case $\omega=\omega_{0}=1$, system size
$j=\mathcal{N}/2=100$, and $\gamma=2\gamma_{c}$ to work in the superradiant
phase. We focus on high energies, where the system displays hard-chaos
behavior.

###  2.1 Classical Limit

The classical Hamiltonian of the Dicke model is obtained by taking the
expectation value of the quantum Hamiltonian $\hat{H}_{D}$ under the tensor
product of bosonic Glauber and atomic Bloch coherent states, that is
$|\bm{x}\rangle=|q,p\rangle\otimes~{}|Q,P\rangle$ [de Aguiar et al.(1991)de
Aguiar, Furuya, Lewenkopf, and Nemes, de Aguiar et al.(1992)de Aguiar, Furuya,
Lewenkopf, and Nemes, Bastarrachea-Magnani et al.(2014a)Bastarrachea-Magnani,
Lerma-Hernández, and Hirsch, Bastarrachea-Magnani et al.(2014b)Bastarrachea-
Magnani, Lerma-Hernández, and Hirsch, Bastarrachea-Magnani et
al.(2015)Bastarrachea-Magnani, López-del-Carpio, Lerma-Hernández, and Hirsch,
Chávez-Carlos et al.(2016)Chávez-Carlos, Bastarrachea-Magnani, Lerma-
Hernández, and Hirsch, Villaseñor et al.(2020)Villaseñor, Pilatowsky-Cameo,
Bastarrachea-Magnani, Lerma, Santos, and Hirsch], and dividing it by the
system size,

$\displaystyle h_{\text{cl}}(\bm{x})$
$\displaystyle=\frac{\langle\bm{x}|\hat{H}_{D}|\bm{x}\rangle}{j}=\frac{\omega}{2}\big{(}p^{2}+q^{2}\big{)}+$
(2) $\displaystyle+\frac{\omega_{0}}{2}\big{(}P^{2}+Q^{2}\big{)}+2\gamma
qQ\sqrt{1-\frac{P^{2}+Q^{2}}{4}}-\omega_{0}.$

The bosonic Glauber and atomic Bloch coherent states are, respectively,

$\displaystyle|q,p\rangle$
$\displaystyle=e^{-(j/4)\left(q^{2}+p^{2}\right)}\exp(\sqrt{\frac{j}{2}}\left(q+ip\right)\hat{a}^{\dagger})|0\rangle,$
(3) $\displaystyle|Q,P\rangle$
$\displaystyle=\bigg{(}1-\frac{P^{2}+Q^{2}}{4}\bigg{)}^{j}\\!\\!\exp(\frac{\left(Q+iP\right)\hat{J}_{+}}{\scriptstyle\sqrt{4-P^{2}-Q^{2}}})|j,-j\rangle,$

with $|0\rangle$ being the photon vacuum and $|j,-j\rangle$ the state with all
the atoms in their ground state.

###  2.2 Phase space

The classical Hamiltonian $h_{\text{cl}}(\bm{x})$ has a four-dimensional phase
space $\mathcal{M}$ in the coordinates $\bm{x}=(q,p;Q,P)$, where
$P^{2}+Q^{2}\leq 4$, because the pseudo-spin degree of freedom is bounded.
This space can be partitioned into a family of classical energy shells

$\mathcal{M}(\epsilon)=\\{\bm{x}\in\mathcal{M}\,\mid\,h_{\text{cl}}(\bm{x})=\epsilon\\}$

in terms of the rescaled classical energy $\epsilon=E/j$. Due to energy
conservation, for a given initial condition $\bm{x}\in\mathcal{M}(\epsilon)$,
the classical evolution given by $h_{\text{cl}}$ remains in
$\mathcal{M}(\epsilon)$, that is, $\bm{x}(t)\in\mathcal{M}(\epsilon)$ for all
times $t$.

The classical energy shells are bounded with respect to the three-dimensional
surface measure
$\operatorname{d\\!}\bm{x}\,\delta(h_{\text{cl}}(\bm{x})-\epsilon)$. The
finite volume of $\mathcal{M}(\epsilon)$ is given by

$\int_{\mathcal{M}}\operatorname{d\\!}\bm{x}\,\delta(h_{\text{cl}}(\bm{x})-\epsilon)=(2\pi\hbar_{\text{eff}})^{2}\nu(\epsilon),$
(4)

where $\hbar_{\text{eff}}=1/j$ is the effective Planck constant [Ribeiro et
al.(2006)Ribeiro, de Aguiar, and de Toledo Piza], which determines the volume
of the Plank cell $(2\pi\hbar_{\text{eff}})^{2}$, and $\nu(\epsilon)$ is the
semiclassical density of states obtained by taking only the first term in the
Gutzwiller trace formula [Gutzwiller(1971), Gutzwiller(1990), Bastarrachea-
Magnani et al.(2014a)Bastarrachea-Magnani, Lerma-Hernández, and Hirsch] (see
App. A for more details on this quantity).

To calculate the average value, $\left\langle f\right\rangle_{\displaystyle}$,
of a phase-space function $f(\bm{x})$ over the classical energy shell
$\mathcal{M}(\epsilon)$, we integrate with respect to the surface measure
$\operatorname{d\\!}\bm{x}\,\delta(h_{\text{cl}}(\bm{x})-\epsilon)$ and divide
the result by the total volume $(2\pi\hbar_{\text{eff}})^{2}\nu(\epsilon)$,

$\displaystyle\left\langle f\right\rangle_{\displaystyle}$
$\displaystyle=\left\langle f(\bm{x})\right\rangle_{\in\mathcal{M}()}$ (5)
$\displaystyle\equiv\frac{1}{(2\pi\hbar_{\text{eff}})^{2}\nu(\epsilon)}\int_{\mathcal{M}}\\!\\!\operatorname{d\\!}\bm{x}\,\delta(h_{\text{cl}}(\bm{x})-\epsilon)f(\bm{x}).$

##  3 Rényi Occupations

The Rényi entropy of order $\alpha\geq 0$ [Rényi(1961)],

$S_{\alpha}(\mathcal{B},\hat{\rho})=\frac{1}{1-\alpha}\log\left(\sum_{i=1}^{\infty}\expectationvalue{\hat{\rho}}{\phi_{i}}^{\alpha}\right),$
(6)

is a common tool to measure the degree of delocalization of a quantum state
$\hat{\rho}$ over a given basis $\mathcal{B}=\\{\ket{\phi_{i}}\mid
i\in\mathbb{N}\\}$ that forms a countable set of states. The quantity
$L_{\alpha}=\exp(S_{\alpha})$ is the generalized participation ratio to the
power $1/(\alpha-1)$, and it counts the effective number of states
$\ket{\phi_{i}}$ that compose the state $\hat{\rho}$. The fact that
$\mathcal{B}$ is an orthonormal basis ensures that
$\sum_{i=1}^{\infty}\expectationvalue{\hat{\rho}}{\phi_{i}}=1$. If
$\mathcal{B}$ is an arbitrary countable set of states, the measure
$L_{\alpha}$ can be generalized by performing an additional normalization as
follows:

$L_{\alpha}(\mathcal{B},\hat{\rho})=\left(\frac{\sum_{i=1}^{\infty}\expectationvalue{\hat{\rho}}{\phi_{i}}^{\alpha}}{\left(\sum_{i=1}^{\infty}\expectationvalue{\hat{\rho}}{\phi_{i}}\right)^{\alpha}}\right)^{1/(1-\alpha)}.$
(7)

The same idea can be extended to the case of non-countable sets of states.
Consider the set

$\mathcal{C}_{\epsilon}=\\{\ket{\bm{x}}\mid\bm{x}\in\mathcal{M}(\epsilon)\\}$
(8)

of coherent states over a classical energy shell $\mathcal{M}(\epsilon)$ at
energy $\epsilon$. In this case, a measure similar to that in Eq. (7) can be
defined, written in terms of the Husimi function $\mathcal{Q}_{\hat{\rho}}$.
This function is a quasi-distribution used to represent a quantum state in the
phase space $\mathcal{M}$, and is given by

$\mathcal{Q}_{\hat{\rho}}(\bm{x})=\expectationvalue{\hat{\rho}}{\bm{x}}\geq 0$
(9)

for each point $\bm{x}\in{\mathcal{M}}$.

For a non-countable set of states we can no longer perform a discrete sum over
the states of $\mathcal{C}_{\epsilon}$, but we can instead use the three-
dimensional surface measure of $\mathcal{M}(\epsilon)$ to perform an average
over $\mathcal{M}(\epsilon)$, as defined in Eq. (5),

$\langle\mathcal{Q}_{\hat{\rho}}^{\alpha}\rangle_{\displaystyle}=\langle\expectationvalue{\hat{\rho}}{\bm{x}}^{\alpha}\rangle_{\in\mathcal{M}()}.$
(10)

Substituting the sums
$\sum_{i=1}^{\infty}\expectationvalue{\hat{\rho}}{\phi_{i}}^{\alpha}$ in Eq.
(7) by these averages, we obtain the Rényi occupations of order $\alpha\geq 0$
[Villaseñor et al.(2021)Villaseñor, Pilatowsky-Cameo, Bastarrachea-Magnani,
Lerma-Hernández, and Hirsch],

$\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho})=\left(\frac{\big{\langle}\mathcal{Q}_{\hat{\rho}}^{\alpha}\big{\rangle}_{\displaystyle}}{\left\langle\mathcal{Q}_{\hat{\rho}}\right\rangle_{\displaystyle}^{\alpha}}\right)^{1/(1-\alpha)}.$
(11)

In the limit $\alpha\to 1$, Eq. (11) gives

$\mathfrak{L}_{1}(\epsilon,\hat{\rho})=\left\langle\mathcal{Q}_{\hat{\rho}}\right\rangle_{\displaystyle}\exp(-\frac{\left\langle\mathcal{Q}_{\hat{\rho}}\log\mathcal{Q}_{\hat{\rho}}\right\rangle_{\displaystyle}}{\left\langle\mathcal{Q}_{\hat{\rho}}\right\rangle_{\displaystyle}}).$
(12)

The name “Rényi occupations” is inspired by the relationship of these measures
with the exponential of the Rényi entropies. The values of
$\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho})$ range from 0 to 1, and they
indicate the percentage of the classical energy shell that is occupied by the
quantum state. The Rényi occupation of a quantum state equals 1 if the
corresponding Husimi function is constant in the classical energy shell, which
means that the state is uniformly delocalized, covering the classical energy
shell homogeneously. Notice, however, that this value is not reached by any
realistic pure state and not even by random pure states, as explained in the
subsection below.

In this work, we explore how the Rényi occupation depends on $\alpha$ and
which information about the degree of localization of the quantum state can be
extracted from different $\alpha$’s. In general, $\alpha<1$ makes the Husimi
distribution look more homogeneous over the classical energy shell. In the
limit $\alpha=0$, one has

$\mathfrak{L}_{0}(\epsilon,\hat{\rho})=\frac{\left\langle\mathcal{Q}_{\hat{\rho}}^{0}\right\rangle_{\displaystyle}}{\left\langle\mathcal{Q}_{\hat{\rho}}\right\rangle_{\displaystyle}^{0}}=\frac{\left\langle
1\right\rangle_{\displaystyle}}{1}=1$ (13)

for any state, because the Husimi function can be zero only on a zero-measure
set of points. In contrast, for $\alpha>1$, the Rényi occupation becomes less
sensitive to the regions of the classical energy shell where the Husimi
function is relatively small. As $\alpha$ increases, these regions cease to
contribute to $\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho})$, leaving only those
regions where the Husimi function attains relatively large values, and thus
decreasing the occupation value. We return to this discussion at the end of
Sec. 3.3 and on how to make use of the $\alpha$-moments of the Husimi function
in the analysis of scarring in Sec. 4.

To benchmark the Rényi occupations as a measure of localization, we describe
below the behavior of $\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho}_{R})$ for
random pure states $\hat{\rho}_{R}=\ket{\psi_{R}}\bra{\psi_{R}}$, which are
the most delocalized states over the phase space. For this, we need to
investigate the Husimi moments
$\langle\mathcal{Q}_{\hat{\rho}_{R}}^{\alpha}\rangle_{\displaystyle}$, which
we do first for systems with a finite Hilbert space, before proceeding with
the Dicke model.

###  3.1 Maximally delocalized states in finite Hilbert spaces

It has been shown that for random pure states $\ket{\psi_{R}}$ completely
spread in a Hilbert space of dimension $N$, the statistical averages
$\expectationvalue{\,\cdot\,}_{\psi_{R}}$ of the projections of
$\ket{\psi_{R}}$ into an arbitrary state $\ket{\phi}$ gives [Kuś et
al.(1988)Kuś, Mostowski, and Haake, Jones(1990), Gnutzmann and
Zyczkowski(2001)]

$\expectationvalue{\absolutevalue{\innerproduct{\psi_{R}}{\phi}}^{2\alpha}}_{\psi_{R}}=\frac{\Gamma(N)\Gamma(1+\alpha)}{\Gamma(N+\alpha)}$
(14)

where $\Gamma$ is the gamma function. Applying this result to coherent states
$\ket{\phi}=\ket{\bm{x}}$ and performing an additional average over all points
$\bm{x}$ of the phase space $\mathcal{M}$, we obtain

$\expectationvalue{\expectationvalue{\absolutevalue{\innerproduct{\psi_{R}}{\bm{x}}}^{2\alpha}}_{\bm{x}\in{\mathcal{M}}}}_{\psi_{R}}=\frac{\Gamma(N)\Gamma(1+\alpha)}{\Gamma(N+\alpha)}.$
(15)

As $N$ increases, one expects that the variance of the averages
$\expectationvalue{\,\cdot\,}_{\psi_{R}}$ will decrease [Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch]. Thus, for sufficiently large $N$, a single, but typical,
random state $\ket{\psi_{R}}$ will satisfy

$\expectationvalue{\absolutevalue{\innerproduct{\psi_{R}}{\bm{x}}}^{2\alpha}}_{\bm{x}\in{\mathcal{M}}}=\frac{\Gamma(N)\Gamma(1+\alpha)}{\Gamma(N+\alpha)}.$
(16)

Moreover, in the limit of large $N$, we can do the approximation
$\Gamma(N+\alpha)/\Gamma(N)\approx{N^{\alpha}}$, which leads to the result

$\left(\frac{\Big{\langle}\absolutevalue{\innerproduct{\psi_{R}}{\bm{x}}}^{2\alpha}\Big{\rangle}_{\bm{x}\in{\mathcal{M}}}}{\Big{\langle}\absolutevalue{\innerproduct{\psi_{R}}{\bm{x}}}^{2}\Big{\rangle}_{\bm{x}\in{\mathcal{M}}}^{\alpha}}\right)^{1/(1-\alpha)}\approx\;\;\;\Gamma(1+\alpha)^{1/(1-\alpha)},$
(17)

that is independent of the dimension $N$.

###  3.2 Maximally delocalized states in the Dicke model

In analogy to Eq. (17), we say that an arbitrary state
$\hat{\rho}=\outerproduct{\psi}{\psi}$ of the Dicke model is maximally
delocalized if

$\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho})\approx\mathfrak{L}_{\alpha}^{\text{max}}\equiv\Gamma(1+\alpha)^{1/(1-\alpha)}.$
(18)

This maximum level of delocalization is attained, on average, by random pure
states, although several eigenstates of the Dicke model are maximally
delocalized as well.

The reason why we can extrapolate the result in Eq. (17), valid for large
finite-dimensional Hilbert spaces, to the infinite-dimensional Hilbert space
of the Dicke model is because we perform averages over individual classical
energy shells, which are bounded. Their finite volume in phase space induces a
large but finite effective dimension that allows us to treat the Hilbert space
of the Dicke model as if it were finite dimensional [Pilatowsky-Cameo et
al.()Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández, and
Hirsch].

In our studies of the Dicke model in Ref. [Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch], we found numerically that
$\mathfrak{L}_{2}^{\text{max}}=1/2$ for pure random states. In Ref. [Batistic
et al.(2020)Batistic, Lozej, and Robnik], an approximate maximum value of
$0.7$ was found numerically in billiards for a measure similar to
$\mathfrak{L}_{1}$. These values are obtained directly from Eq. (18), which
gives $\mathfrak{L}_{2}^{\text{max}}=\Gamma(3)^{-1}=1/2$ and
$\mathfrak{L}_{1}^{\text{max}}=\lim_{\alpha\to
1}\Gamma(1+\alpha)^{1/(1-\alpha)}\approx 0.66$. These results reinforce the
validity of the expression (18) and suggest that it may be applicable to other
models.

Notice that the upper bound for the degree of delocalization of pure states
given by $\mathfrak{L}_{\alpha}^{\text{max}}$ is not equal to one, instead
$\mathfrak{L}_{\alpha}^{\text{max}}<1$ for $\alpha>0$. This happens because
quantum interference effects prevent a quantum pure state from homogeneously
covering the classical energy shell and cause its Husimi function to be zero
in some points of the phase space [Korsch et al.(1997)Korsch, Müller, and
Wiescher]. Only mixed states, which can be obtained by performing infinite-
time averages, can homogeneously cover the classical energy shells
[Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-
Magnani, Lerma-Hernández, Santos, and Hirsch].

###  3.3 Eigenstates

Figure 1: Localization measure $\Lambda_{\alpha}(\epsilon_{k},\hat{\rho}_{k})$
for the $2437$ eigenstates $\hat{\rho}_{k}=\outerproduct{E_{k}}{E_{k}}$ with
eigenenergies $\epsilon_{k}$ inside the chaotic energy interval $[-0.6,-0.4]$.
The 8 representative eigenstates labeled A-H are further analyzed in Fig. 2.
The system size is $j=100$.

To quantify how close the degree of delocalization of a quantum state is to
the maximal value attained by random states, we define the following measure:

$\Lambda_{\alpha}(\epsilon,\hat{\rho})=\frac{\mathfrak{L}_{\alpha}^{\text{max}}}{\mathfrak{L}_{\alpha}(\epsilon,\hat{\rho})}.$
(19)

A state that is as delocalized as a random pure state gives
$\Lambda_{\alpha}\approx 1$, which is the lower bound for this measure. As the
state $\hat{\rho}$ becomes more localized in the classical energy shell at
$\epsilon$, $\Lambda_{\alpha}(\epsilon,\hat{\rho})$ increases. For a given
$\alpha$, the value $\Lambda_{\alpha}(\epsilon,\hat{\rho})=n$ indicates that
the state $\hat{\rho}$ is $n$ times more localized than a random state.

In Fig. 1, we study $\Lambda_{\alpha}(\epsilon_{k},\hat{\rho}_{k})$ as a
function of $\alpha$ for all the eigenstates
$\hat{\rho}_{k}=\outerproduct{E_{k}}{E_{k}}$ of the Dicke model with
eigenenergies $\epsilon_{k}=E_{k}/j$ inside the energy interval $[-0.6,-0.4]$,
where chaos is predominant. Each eigenstate is represented by a thin gray
line. One sees that most eigenstates cluster slightly above
$\Lambda_{\alpha}=1$, indicating that they are nearly as delocalized as random
pure states. Values of $\Lambda_{\alpha}<1$ are possible, but disappear as $j$
increases [Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch]. A portion of the
eigenstates has values of $\Lambda_{\alpha}$ that are much higher than 1, some
reaching $\Lambda_{4}=5$. As we show next, these high values of localization
are caused by strong quantum scarring. It is worth noting, however, that the
lines in Fig. 1 are not parallel and rather have multiple crossings. This
tells us that there is indeed information to be gained by analyzing
$\Lambda_{\alpha}$ for different values of $\alpha$.

The thick colored lines in Fig. 1 mark 8 eigenstates, labeled A-H, that we
select for further analysis. These eigenstates constitute a representative
sample of the different behaviors we see in Fig. 1. States A and B have the
highest value of $\Lambda_{\alpha}$ for $\alpha>2$, D has the highest value of
$\Lambda_{\alpha}$ for $\alpha<1$, H has the lowest value of
$\Lambda_{\alpha}$ for all $\alpha$, and $\Lambda_{\alpha}(G)=1$ for all
$\alpha$. States C, E, and F were selected to have evenly spread values of
$\Lambda_{4}$ between B and G. We focus on these 8 eigenstates henceforth, but
we have indeed verified that the analysis we perform describes other
eigenstates with similar values of $\Lambda_{\alpha}$, and hence the
conclusions we reach are general.

Figure 2: Projected moments of the Husimi functions,
$\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)$, for the eigenstates labeled A-H
in Fig. 1 and
$\widetilde{\mathcal{Q}^{\alpha}_{\hat{\rho}_{R}}}(Q,P,\epsilon)$ for a pure
random state (R) from a Gaussian orthogonal ensemble obtained by weighting the
positive parity eigenstates inside an energy window of width $1.5$ centered at
the energy $\epsilon=-0.5$. Each column corresponds to a different value of
$\alpha\in\\{0.5,1,2,3,4\\}$. The solid red line in (A2)-(D2) represent the
projections in $(Q,P)$ of the unstable periodic orbits
$\mathcal{O}_{\text{A}}$, $\mathcal{O}_{\text{B}}$, $\mathcal{O}_{\text{C}}$,
$\mathcal{O}_{\text{D}}$, and the dashed red line in (D2) is for the mirrored
orbit $\overline{\mathcal{O}}_{\text{D}}$. The system size is $j=100$.

Visualizing the moments of the four-dimensional Husimi function comes to our
aid for explaining the different degrees of localization of the eigenstates
shown in Fig. 1. We consider the projection into the atomic $(Q,P)$ plane of
the $\alpha$-moments of the Husimi function $\mathcal{Q}_{k}$ of eigenstates
$\hat{\rho}_{k}$ intersected with the classical energy shell at
$\epsilon_{k}$, so that we obtain two-dimensional pictures, that is,

$\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)=\iint\operatorname{d\\!}q\operatorname{d\\!}p\,\delta(h_{\text{cl}}({\bm{x}})-\epsilon_{k})\,\mathcal{Q}_{k}({\bm{x}})^{\alpha}.$
(20)

In Refs. [Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch, Pilatowsky-Cameo et
al.(2021a)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch], we studied $\widetilde{\mathcal{Q}^{1}_{k}}(Q,P)$ and
verified that it allows for the visual identification of quantum scars. In
Ref. [Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch], we concluded that
all the eigenstates of the Dicke model are scarred, even though most of them
are almost as delocalized as random states. This is corroborated again by the
representative eigenstates shown in Fig. 2.

In Fig. 2, we compare $\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)$ for the 8
representative eigenstates A-H from Fig. 1 and also
$\widetilde{\mathcal{Q}^{\alpha}_{\hat{\rho}_{R}}}(Q,P,\epsilon)=\iint\operatorname{d\\!}q\operatorname{d\\!}p\,\delta(h_{\text{cl}}({\bm{x}})-\epsilon)\,\mathcal{Q}_{\hat{\rho}_{R}}({\bm{x}})^{\alpha}$
for a random state $\hat{\rho}_{R}$ centered at energy $\epsilon$. This state
is obtained by weighting the positive parity eigenstates inside a rectangular
energy window with real coefficients normally distributed as in the Gaussian
orthogonal ensembles of random matrix theory. For each state, we show the
distributions for five values of $\alpha\in\\{0.5,1,2,3,4\\}$.

In contrast to the random state [Fig. 2 (R0)-(R4)], the projected moments of
the Husimi functions of the eigenstates [Fig. 2 (A0)-(H4)] always display
structures that are related with underlying unstable periodic orbits, and
therefore imply that the eigenstates are scarred [Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch]. When examining the figure, one should keep in mind that
values of $\alpha<1$ tend to homogenize the distribution
$\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)$, up to the limiting case where
$\widetilde{\mathcal{Q}^{0}_{k}}(Q,P)=1$ independently of $(Q,P)$. On the
other hand, values of $\alpha>1$ tend to erase the small contributions from
the Husimi function, so that regions where
$\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)$ is large get enhanced. The high
$\alpha$-moments of the Husimi function are therefore useful tools in the
analysis of quantum scarring, as discussed next.

##  4 Quantum Scarring and Unstable Periodic Orbits

Quantum scarring corresponds to the concentration of a quantum state around
the phase-space region occupied by a periodic orbit

$\mathcal{O}=\\{\bm{x}(t)\,\mid\,t\in[0,T]\\}\subseteq\mathcal{M}(\epsilon)$
(21)

with a periodic initial condition $\bm{x}(T)=\bm{x}(0)$ that is unstable, that
is, has a positive Lyapunov exponent $\lambda$. These orbits are always
present in chaotic systems and are deeply connected with the quantum spectrum
of such systems [Gutzwiller(1971), Gutzwiller(1990)].

###  4.1 Identifying Unstable Periodic Orbits

By tracking the peaks of the projected fourth-moment of the Husimi function
displayed in Figs. 2 (A4), (B4), (C4), and (D4), we can uncover the unstable
periodic orbits that significantly scar the states A-D. The procedure goes as
follows. Those peaks give a set of possible initial conditions for periodic
orbits on the variables $(Q_{i},P_{i})$. By varying the bosonic coordinate $p$
and selecting $q$ according to the fixed energy, we identify the whole set of
coordinates $\bm{x}_{i}\in\mathcal{M}(\epsilon_{k})$ where the Husimi function
attains a maximum, thus yielding a set of possible initial conditions for a
periodic orbit on all variables $\bm{x}_{i}$. We evolve these initial
conditions and select those that roughly follow the green lines in Figs. 2
(A4), (B4), (C4), and (D4). The evolution is halted when the evolved point
$\bm{x}_{i}(T_{i})$ approximately returns to the initial condition
$\bm{x}_{i}(0)$. This gives an approximate period $T_{i}$ and an initial
condition $\bm{x}_{i}$, which can be converged to a truly periodic condition
by means of an algorithm known as the monodromy method [Baranger et
al.(1988)Baranger, Davies, and Mahoney, de Aguiar and Malta(1988),
Simonović(1999), Pilatowsky-Cameo et al.(2021a)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch]111Code available at
github.com/saulpila/DickeModel.jl.. This procedure leads to the five unstable
periodic orbits labeled $\mathcal{O}_{\text{A}}$, $\mathcal{O}_{\text{B}}$,
$\mathcal{O}_{\text{C}}$, $\mathcal{O}_{\text{D}}$, and
$\overline{\mathcal{O}}_{\text{D}}$ in Figs. 2 (A2), (B2), (C2), and (D2). The
mirrored orbit $\overline{\mathcal{O}}_{\text{D}}$ is obtained by changing the
signs of the coordinates $Q$ and $q$ of $\mathcal{O}_{\text{D}}$. We only
mirror $\mathcal{O}_{\text{D}}$, because the other orbits are symmetric under
the parity transformation $(Q,q)\mapsto(-Q,-q)$. We stress that these unstable
periodic orbits are different from those that we found in Ref. [Pilatowsky-
Cameo et al.(2021a)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-
Hernández, Santos, and Hirsch], whose origin could be traced down to the
fundamental excitations around the ground state configuration. Further studies
of these new orbits may eventually reveal the properties and origin of these
new families of unstable periodic orbits for the Dicke model.

The Lyapunov times $1/\lambda$ of the unstable periodic orbits identified here
are smaller but of the same order of their respective periods $T$. Table 1
gives the values of the Lyapunov exponents, $\lambda$, and the periods divided
by the Lyapunov times, $T\lambda$, for each of the five unstable periodic
orbits that we found. The competition between the period of the orbit and the
Lyapunov time is important for scarring. As originally discussed in Ref.
[Heller(1984)], a non-stationary state will be affected by a neighboring
periodic orbit if the Lyapunov time of the orbit is larger than its period. If
instead the Lyapunov time is shorter than the period, the periodic orbit does
not have time to develop and scar the non-stationary state. Here, we observe
that even if this condition is slightly broken, $T\lambda\gtrsim 1$, the
periodic orbits still cause a significant localization of the eigenstates.

| $\mathcal{O}_{\text{A}}$ | $\mathcal{O}_{\text{B}}$ | $\mathcal{O}_{\text{C}}$ | $\mathcal{O}_{\text{D}},\overline{\mathcal{O}}_{\text{D}}$
---|---|---|---|---
$\lambda$ | 0.143 | 0.191 | 0.253 | 0.153
$T\lambda$ | 1.86 | 2.97 | 3.43 | 2.21
$\mathcal{P}_{k}$ | 33.2 | 31.1 | 28.3 | 22.5
Table 1:  Lyapunov exponent $\lambda$, its product with the period $T$, and
the scarring measure $\mathcal{P}_{k}(\mathcal{O})$ (22) for the orbits A-D.

To verify that the states A-D are indeed scarred by the unstable periodic
orbits $\mathcal{O}_{\text{A}}$-$\mathcal{O}_{\text{D}}$, we employ the
following scarring measure [Pilatowsky-Cameo et al.(2021a)Pilatowsky-Cameo,
Villaseñor, Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch]:

$\mathcal{P}_{k}(\mathcal{O})=\frac{\tr(\hat{\rho}_{k}\hat{\rho}_{\mathcal{O}})}{\tr(\hat{\rho}_{\epsilon}\hat{\rho}_{\mathcal{O}})},$
(22)

where

$\hat{\rho}_{\mathcal{O}}=\frac{1}{T}\int_{0}^{T}\operatorname{d\\!}t\,\outerproduct{\bm{x}(t)}{\bm{x}(t)}$
(23)

is a tubular state composed of all coherent states $\ket*{\bm{x}(t)}$ whose
centers lie in the periodic orbit $\mathcal{O}=\\{\bm{x}(t)\mid t\in[0,T)\\}$
and
$\hat{\rho}_{\epsilon}=\langle\outerproduct{\bm{x}}{\bm{x}}\rangle_{\in\mathcal{M}()}$
is a mixed state completely delocalized in the classical energy shell at
$\epsilon=h_{\text{cl}}(\mathcal{O})$, that is, a mixture of all coherent
states whose centers lie within this classical energy shell (see details in
Ref. [Pilatowsky-Cameo et al.(2021a)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch]). A value
$\mathcal{P}_{k}(\mathcal{O})=n$ indicates that the state
$\hat{\rho}_{k}=\outerproduct{E_{k}}{E_{k}}$ is $n$ times more likely to be
found near the orbit $\mathcal{O}$ than a completely delocalized state, for
which $\mathcal{P}_{k}(\mathcal{O})=1$. For eigenstates not scarred by the
periodic orbit $\mathcal{O}$ and also random states,
$\mathcal{P}_{k}(\mathcal{O})\leq 1$ [Pilatowsky-Cameo et
al.(2021a)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch]. The results of the scarring measure obtained with Eq.
(22) for the states A-D are shown in Table 1 and confirm that these states are
significantly scarred by the unstable periodic orbits
$\mathcal{O}_{\text{A}}$-$\mathcal{O}_{\text{D}}$.

One observes secondary structures in Figs. 2 (A0), (B0), (C0), and (D0) that
are not directly generated by the unstable periodic orbits delineated in Figs.
2 (A2), (B2), (C2), and (D2). These structures may be related to the self-
interferences of the unstable periodic orbits [Berry(1989)] or to other
secondary scars associated with other unstable periodic orbits that we have
not identified. We leave this question for future investigations.

Figure 3: Top panels: in green, the Husimi projection
$\widetilde{\mathcal{Q}}_{\mathcal{O}}$ for the orbits
$\mathcal{O}_{\text{A}}$ (A1), $\mathcal{O}_{\text{B}}$ (B1),
$\mathcal{O}_{\text{C}}$ (C1), and added projections
$(\widetilde{\mathcal{Q}}_{\mathcal{O}_{\text{D}}}+\widetilde{\mathcal{Q}}_{\overline{\mathcal{O}}_{\text{D}}})/2$
for orbit $\mathcal{O}_{\text{D}}$ and the mirrored orbit
$\overline{\mathcal{O}}_{\text{D}}$ (D1). The red arrows are placed at
constant time intervals along the unstable periodic orbits
$\mathcal{O}_{\text{A}}$ (A1), $\mathcal{O}_{\text{B}}$ (B1),
$\mathcal{O}_{\text{C}}$ (C1), and
$\mathcal{O}_{\text{D}},\overline{\mathcal{O}}_{\text{D}}$ (D1). Bottom
panels: Three-dimensional plots of the periodic orbits
$\mathcal{O}_{\text{A}}$ (A2), $\mathcal{O}_{\text{B}}$ (B2),
$\mathcal{O}_{\text{C}}$ (C2), and $\mathcal{O}_{\text{D}}$ (opaque)
$\overline{\mathcal{O}}_{\text{D}}$ (translucent) (D2). In these panels, the
variables $Q$, $P$, and $p$ correspond to each of the three axis, as marked,
and the color represents the fourth variable $q$. The system size is $j=100$.

###  4.2 Highly localized eigenstates and scarring

The dynamical properties of the unstable periodic orbits that scar states A-D
allow us to explain the structures of the distributions in Figs. 2 (A0)-(D4).
To do this, we consider the Husimi function of the tubular state
$\hat{\rho}_{\mathcal{O}}$,

$\mathcal{Q}_{\mathcal{O}}(\bm{x})=\expectationvalue{\hat{\rho}_{\mathcal{O}}}{\bm{x}}=\frac{1}{T}\int_{0}^{T}\operatorname{d\\!}t\,\absolutevalue{\innerproduct{\bm{x}}{\bm{y}(t)}}^{2},$
(24)

where $\bm{y}(t)\in\mathcal{O}$, and compute the projection of this function
into the $(Q,P)$ plane,

$\widetilde{\mathcal{Q}}_{\mathcal{O}}(Q,P)=\iint\operatorname{d\\!}q\operatorname{d\\!}p\,\delta(h_{\text{cl}}(\bm{x})-\epsilon)\,\mathcal{Q}_{\mathcal{O}}(\bm{x}),$
(25)

at energy $\epsilon=h_{\text{cl}}(\mathcal{O})$. In Figs. 3 (A1), (B1), and
(C1) we plot in green the projections $\widetilde{\mathcal{Q}}_{\mathcal{O}}$
for the orbits $\mathcal{O}_{\text{A}}$, $\mathcal{O}_{\text{B}}$, and
$\mathcal{O}_{\text{C}}$, respectively, and in Fig. 3 (D1) we plot the added
projections
$(\widetilde{\mathcal{Q}}_{\mathcal{O}_{\text{D}}}+\widetilde{\mathcal{Q}}_{\overline{\mathcal{O}}_{\text{D}}})/2$
for orbit $\mathcal{O}_{\text{D}}$ and the mirrored orbit
$\overline{\mathcal{O}}_{\text{D}}$. One sees that the projections
$\widetilde{\mathcal{Q}}_{\mathcal{O}}(Q,P)$ of the Husimi function of the
tubular state in Figs. 3 (A1), (B1), (C1), and (D1) are similar to the
projections $\widetilde{\mathcal{Q}^{\alpha}_{k}}(Q,P)$ of the Husimi function
of the eigenstates A-D shown in Figs. 2 (A2)-(D2), Figs. 2 (A3)-(D3), Figs. 2
(A4)-(D4).

In Figs. 3 (A2)-(D2), we show three-dimensional plots of the four-dimensional
periodic orbits $\mathcal{O}_{\text{A}}$, $\mathcal{O}_{\text{B}}$,
$\mathcal{O}_{\text{C}}$, and both $\mathcal{O}_{\text{D}}$ and
$\overline{\mathcal{O}}_{\text{D}}$. The plots display the variables $Q$, $P$,
and $p$, and the color represents the fourth variable $q$. The shapes seen in
Figs. 3 (A1)-(D1) are precisely the orbits in Figs. 3 (A2)-(D2) projected into
the $(Q,P)$-plane.

Figure 3 contains only semiclassical information, which helps us understand
specific features and the degree of localization of the eigenstates A-D. For
example, when the classical dynamics is fast, the scars are less intense. In
Figs. 3 (A1)-(D1), we place red arrows at constant time intervals along the
corresponding periodic orbits. The closer those arrows are, the slower the
classical dynamics is. One sees that the brighter green regions correspond to
the regions with a high density of red arrows. These are regions of longer
permanence of the classical orbit, which produce deeper imprinted scars
[Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-
Magnani, Lerma-Hernández, Santos, and Hirsch, Pilatowsky-Cameo et
al.(2021a)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch, Kelly et al.(2020)Kelly, Timmermans, and Tsai].

The dynamical properties of the orbits also explain why $\Lambda_{\alpha}$
increases with $\alpha$ for some eigenstates. The faster the dynamics is in a
given region of the phase space, the less probable it is to find a point
following the orbit there. This gets reflected in the high moments
($\alpha>1$) of the Husimi function, where small contributions tend to be
erased, increasing the value $\Lambda_{\alpha}$. We can see this effect by
comparing Figs. 2 and 3. As $\alpha$ grows, some colored regions in the
leftmost columns of Fig. 2 are not visible in the rightmost columns. These are
regions of low probability, which have faster dynamics (more separated arrows
in Fig. 3).

The general features discussed in the two previous paragraphs are described
below in more detail for the eigenstates A-D.

  * •

The unstable periodic orbit $\mathcal{O}_{\text{A}}$ [Fig. 3 (A1)], which
heavily scars state A, has slower dynamics near the small oval region in the
center of the orbit, where most red arrows are concentrated. Comparing
$\mathcal{O}_{\text{A}}$ with $\mathcal{O}_{\text{B}}$, one sees that
$\mathcal{O}_{\text{A}}$ is larger in phase space, so state A is less
localized than state B and $\Lambda_{\alpha}(A)<\Lambda_{\alpha}(B)$ for
$\alpha<3$, as shown in Fig. 1. Nevertheless, $\alpha=4$ is sufficiently large
to erase the less bright regions of $\mathcal{O}_{\text{A}}$, where the
classical dynamics is fast, leaving only the contributions from the central
oval region and thus leading to a higher value of localization and to
$\Lambda_{\alpha}(A)>\Lambda_{\alpha}(B)$ for $\alpha>3$.

  * •

The unstable periodic orbit $\mathcal{O}_{\text{B}}$ covers only a small
portion of the classical energy shell and has a relatively constant speed in
the phase space, as seen by the constant density of red arrows in Fig. 3 (B1),
so the state B maintains a high value of localization in Fig. 1 for all values
of $\alpha\in[0,4]$.

  * •

The unstable periodic orbit $\mathcal{O}_{\text{C}}$ presents two bright
regions in Fig. 3 (C1), at $Q=0$ and $P\approx\pm 0.8$, where the dynamics is
slow. As seen in Fig. 1, state C has a lower value of $\Lambda_{\alpha}$ than
state B, because $\mathcal{O}_{\text{C}}$ is larger in phase-space than
$\mathcal{O}_{\text{B}}$. However, the slope of $\Lambda_{\alpha}$ at
$\alpha=4$ is larger for state C than for state B, because the less bright
loops of $\mathcal{O}_{\text{C}}$, where the classical dynamics is fast,
disappear for $\alpha=4$ [Fig. 2 (C4)], leaving only the two bright spots at
$Q=0$.

  * •

The behavior of $\Lambda_{\alpha}$ for state D in Fig. 1 is peculiar in that
it initially grows for $\alpha<2$, but flattens after $\alpha>2$. The
localization measure increases as $\alpha$ grows to 2, because the
contribution of the fast outer loops strongly diminish. But then, for
$\alpha>2$, since the dynamics in the loops of intermediate sizes has a
relatively constant speed, $\Lambda_{\alpha}$ flattens. In fact, Fig. 1
suggests that the flattening for large $\alpha$ holds also for state B, while
$\Lambda_{\alpha}$ for states A and C keeps growing at least up to $\alpha=4$.

We close this subsection by discussing state E, which in terms of localization
is midway between the highly localized eigenstates A-D and the maximally
delocalized eigenstates F-H described in the next subsection. The projected
Husimi moment distributions depicted in Figs. 2 (E0)-(E4) show localization
around some regions that must be associated with a classical unstable periodic
orbit with slow dynamics at $Q\approx 0$ and $P\approx\pm 0.9$. However, the
numerical method used to identify the orbits that scar the eigenstates A-D
fails for state E. This suggests that the ratio between the period and the
Lyapunov time, $T\lambda$, of the unstable periodic orbit that scars this
eigenstate should be much larger than one.

###  4.3 Maximally delocalized eigenstates

As seen in Fig. 1, the eigenstates F, G and H have values of
$\Lambda_{\alpha}\approx 1$, similar to those of random states. However,
contrary to Figs. 2 (R0)-(R4), the panels (F0)-(H4) show that these
eigenstates still display orbit-like patterns, that is, they exhibit closed
loops. These patterns must belong to a set of periodic orbits that scar the
eigenstates [Pilatowsky-Cameo et al.(2021b)Pilatowsky-Cameo, Villaseñor,
Bastarrachea-Magnani, Lerma-Hernández, Santos, and Hirsch], but we have not
yet been able to identify these orbits with the numerical method that we have
implemented. They must have periods that are much larger than their
corresponding Lyapunov times, which makes them more difficult to find. Since
these states are scarred by unstable periodic orbits of larger periods, they
are more delocalized than the states A-D, which explains why
$\Lambda_{\alpha}(F),\Lambda_{\alpha}(G),\Lambda_{\alpha}(H)<\Lambda_{\alpha}(A),\Lambda_{\alpha}(B),\Lambda_{\alpha}(C),\Lambda_{\alpha}(D)$.

The results in this work support our claims in Ref. [Pilatowsky-Cameo et
al.(2021b)Pilatowsky-Cameo, Villaseñor, Bastarrachea-Magnani, Lerma-Hernández,
Santos, and Hirsch] that all eigenstates are scarred by unstable periodic
orbits, yet some of them are as delocalized in phase space as random states.
Identifying the unstable periodic orbits that scar the states E-H would bring
further confirmation to this conjecture.

##  5 Summary and Conclusions

Generalized inverse participation ratios and Rényi entropies of order $\alpha$
quantify the degree of delocalization of quantum states in Hilbert space. Our
interest in this work was instead in the structure of quantum states in phase
space. For this analysis, we employed a measure of localization called Rényi
occupation of order $\alpha$, which is defined over classical energy shells
and is based on the $\alpha$-moments of the Husimi function. We showed that
the Rényi occupations of random states have a simple analytical form that is a
function of $\alpha$ only. We used this result to define a localization
measure given by the ratio of the energy-shell occupation of random states to
that of the state under investigation. This measure was then used to analyze
the eigenstates of the Dicke model.

As $\alpha$ increases beyond 1, the smaller contributions from the Husimi
function are erased and only the larger peaks survive. By examining these
peaks in the fourth-moment of the Husimi function of highly localized
eigenstates, we were able to visually identify the classical unstable periodic
orbits that scar those states.

Our study of the classical dynamics of the identified orbits assisted our
understanding of the structure and the degree of localization of the
eigenstates. In regions of the phase space where the classical dynamics is
slow, the scars are deeper imprinted, so the values of the $\alpha$-moments of
the Husimi functions are larger and persist as $\alpha$ increases.

In conclusion, we have shown that quantum states supply valuable information
about the non-linear dynamics of the classical limit of the model. Highly
localized states are scarred by orbits of relatively short period and the
peaks in the high-moments ($\alpha>1$) of their Husimi distributions provide a
powerful tool to identify classical unstable periodic orbits. This is a major
accomplishment given the difficulty in identifying the unstable periodic
orbits underlying scarred states. Our analysis shows that we can learn about
the classical dynamics by studying the quantum realm.

The localization measure $\Lambda_{\alpha}$ introduced in this work applies to
other models with unbounded phase space, such as quantum billiards, and our
technique to find classical unstable periodic orbits could be extended to
those systems as well.

## Acknowledgments

We acknowledge the support of the Computation Center - ICN, in particular to
Enrique Palacios, Luciano Díaz, and Eduardo Murrieta. SP-C, DV, and JGH
acknowledge financial support from the DGAPA- UNAM project IN104020, and SL-H
from the Mexican CONACyT project CB2015-01/255702. LFS was supported by the
NSF Grant No. DMR-1936006.

##  Appendix A Appendix: Semiclassical Density of States

The semiclassical approximation of the density of states of a quantum system
is obtained by taking the first term of the well-known Gutzwiller trace
formula [Gutzwiller(1971), Gutzwiller(1990)]. This semiclassical density of
states is the ratio of the three-dimensional volume of the classical energy
shell to the four-dimensional volume of the Planck cell,

$\nu(\epsilon)=\frac{1}{(2\pi\hbar_{\text{eff}})^{2}}\int_{{\mathcal{M}}}\operatorname{d\\!}{\bm{x}}\,\delta(h_{\text{cl}}({\bm{x}})-\epsilon).$
(26)

The explicit expression of $\nu(\epsilon)$ for the Dicke model was derived in
Ref. [Bastarrachea-Magnani et al.(2014a)Bastarrachea-Magnani, Lerma-Hernández,
and Hirsch]. Here, this expression is slightly modified to be consistent with
the notation used in this article, and it reads

$\displaystyle\nu(\epsilon)=\frac{2j^{2}}{\omega}\left\\{\begin{array}[]{ll}\frac{1}{\pi}\int_{y_{-}}^{y_{+}}dyf(y,\epsilon)&\text{if
}\epsilon_{0}\leq\epsilon\leq-\omega_{0},\\\
\frac{1+\epsilon/\omega_{0}}{2}+\frac{1}{\pi}\int_{\epsilon/\omega_{0}}^{y_{+}}dyf(y,\epsilon)&\text{if
}|\epsilon|<\omega_{0},\\\ 1&\text{if
}\epsilon\geq\omega_{0},\end{array}\right.$ (30)

where

$f(y,\epsilon)=\arccos\left(\sqrt{\frac{2\gamma_{c}^{2}(y-\epsilon/\omega_{0})}{\gamma^{2}(1-y^{2})}}\right),$
(31)
$y_{\pm}=-\frac{\gamma_{c}}{\gamma}\left(\frac{\gamma_{c}}{\gamma}\mp\sqrt{\frac{2(\epsilon-\epsilon_{0})}{\omega_{0}}}\right),$
(32)

and

$\epsilon_{0}=-\frac{1}{2}\left(\frac{\gamma_{c}^{2}}{\gamma^{2}}+\frac{\gamma^{2}}{\gamma_{c}^{2}}\right).$
(33)

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