# Decoherence factor as a convolution:
an interplay between a Gaussian and an exponential coherence loss

Bin Yan  Theoretical Division, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  Center for Nonlinear Studies, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Wojciech H. Zurek  Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

###### Abstract

This paper identifies and investigates nature of the transition between
Gaussian and exponential forms of decoherence. We show that the decoherence
factor (that controls the time dependence of the suppression of the off-
diagonal terms when the density matrix is expressed in the pointer basis
representation) can be described by the convolution of Gaussian and
exponential functions, their contributions modulated by the strength of the
system-environment interaction. In the strong and weak coupling limits,
decoherence reduces to the familiar Gaussian and exponential forms,
respectively. The mechanism is demonstrated with two paradigmatic examples of
decoherence—a spin-bath model and the quantum Brownian motion.

_Introduction—_ Quantum systems lose quantum coherence when coupled to their
environments. Decoherence results in decay of the off-diagonal terms between
system’s pointer states—states that are immune to decoherence—in its density
matrix Zurek (1981, 1991); Schlosshauer (2019). It is an essential ingredient
responsible for the quantum-to-classical transition Zurek (2003, 2006) and the
main obstacle in the development of quantum information technologies Buluta
and Nori (2009); Preskill (2018); Nielsen and Chuang (2010).

Decoherence is caused by the flow of quantum information from the system $\cal
S$ into its environment $\cal E$. Decoherence factor turns the initial
superpositions of pointer states into their mixture. Decoherence factor is
often thought to decay exponentially, but this is not always the case. For
instance, power law decay may appear Schomerus and Lutz (2007); Cai and
Barthel (2013); Beau  _et al._ (2017); Pekola and Karimi (2020); Sarkar  _et
al._ (2017). Nonetheless, exponential decay is widely regarded as typical in
various settings Lindblad (1976); Gorini  _et al._ (1976); Benatti and
Floreanini (1998); Knight (1976); Knight and Milonni (1976); Burgarth and
Facchi (2017).

At short times, decoherence takes quadratic forms related to quantum Zeno
paradox von Neumann (1055); Misra and Sudarshan (1977); Facchi and Pascazio
(2008). This transient behavior is often quickly replaced by an exponential
decay, although sometimes quadratic decay initiates a Gaussian time dependence
Cucchietti  _et al._ (2005); Zurek  _et al._ (2007).

This interplay between Gaussian and exponential decoherebnce has not been—as
yet—understood or even investigated. We show that decoherence factor can be
described by a convolution $f*g$ of an exponential $f(t)=e^{-\Gamma|t|/2}$ and
a Gaussian $g(t)=e^{-\sigma^{2}t^{2}/2}$:

$(f*g)(t)\equiv\int_{-\infty}^{\infty}d\tau f(\tau)g(t-\tau)$ (1)

The Gaussian contribution originates from the spectrum: It is the Fourier
transform of the spectral density of the interaction Hamiltonian. Its origin
is easiest to understand for pure decoherence, when the self-Hamiltonians of
e.g. the spin-$\frac{1}{2}$ $\cal S$ and the $N$ spin-$\frac{1}{2}$ $\cal E$
can be neglected Cucchietti  _et al._ (2005); Zurek  _et al._ (2007). The
$2^{N}$ eigenvalues of the interaction Hamiltonian are then obtained by
summing the couplings of $\cal S$ to the environment spins with the sign (+ or
-) given by their relative orientations. Even for modest $N$ the distribution
of the eigenvalues approaches very quickly a Gaussian. Gaussian spectrum
arises in more general circumstances for physical systems with local
interactions Kota (2014).

The exponential factor accounts for the response of the environment to the
system-environment interactions: According the Fermi’s golden rule, the
environment decays from its initial state due to the coupling with the system
with the rate $\Gamma$ determined by the coupling strength.

The Fourier transform of the decoherence factor is then simply a product of
two Fourier images. One is the energy spectrum; the other ‘Breit-Wigner -
like’ represents the decay of the overlap of the environment states correlated
with different pointer states of the system.

These two contributions are responsible for the interplay between exponential
and Gaussian time-dependence of the decoherence factor. In the limit of weak
($\Gamma\ll\sigma$) and strong ($\Gamma\sim\sigma$) coupling, the
convolution—hence, the decoherence factor—exhibits predominantly exponential
or Gaussian decay, respectively. Below we derive the convolution form of the
decoherence factor in the spin-bath model, and then apply it to another
paradigmatic model of decoherence, the quantum Brownian motion.

_Spin-bath model—_ The model consists of a single spin linearly interacting
with a generic environment. The composite $\cal SE$ can be described by the
Hamiltonian:

$H=\lambda\sigma_{z}\otimes H_{\mathcal{I}}+H_{\mathcal{E}},$ (2)

where $\sigma_{z}$ is the Pauli operator for the spin,
$H_{\mathcal{I},\mathcal{E}}$ act on the environment, and $\lambda$ is a
dimensionless coupling strength. The self-Hamiltonian of the spin is
proportional to $\sigma_{z}$, and is absorbed into the interaction term. This
Hamiltonian is relevant for a large class of pure decoherence processes of
single-spin systems, and has been investigated in various contexts Zurek
(1982); Braun  _et al._ (2001); Cucchietti  _et al._ (2005); Zurek  _et al._
(2007) including quantum phase transitions Quan  _et al._ (2006).

Consider the composite $\cal SE$ that starts as a product:

$\rho_{\mathcal{SE}}(0)=\rho_{\mathcal{S}}(0)\otimes\rho_{\mathcal{E}}(0)$ (3)

and where the central spin is prepared in a pure state
$|\psi_{\mathcal{S}}(0)\rangle=a|0\rangle+b|1\rangle$. For the sake of
simplicity, in the following analysis, the environment initial state
$\rho_{\mathcal{E}}$ is assumed to be an eigenstate $|n\rangle$ of
$H_{\mathcal{E}}$ with energy $E_{n}$. The results can be generalized (e.g.,
to a thermal state) by averaging.

Figure 1: The overlap function of the spin-bath model with 14 bath spins.
Upper and lower panels correspond to different initial energy eigenstates and
coupling strengths, i.e., $\lambda=1$ and $2$, respectively. The first column
is the plots of the full overlap functions, with a zoom-in shown on the second
column and the third column (semi-log scale). Red solid (dashed) curves are
the best fit to the Lorentzian (exponential) function. Inset is the histogram
for the (Gaussian) global density of states.

Decoherence of the central spin is reflected in the decoherence factor $r(t)$
that suppresses off-diagonal element of its reduced density matrix $\rho(t)$:

$r(t)\equiv\rho_{12}(t)/(ab^{*}).$ (4)

Simple algebra shows that it is given by:

$\displaystyle r(t)=$ $\displaystyle\langle n|e^{i(H_{\mathcal{E}}+\lambda
H_{\mathcal{I}})t}e^{-i(H_{\mathcal{E}}-\lambda H_{\mathcal{I}})t}|n\rangle$
(5) $\displaystyle\equiv$ $\displaystyle\langle n|e^{i(H_{\mathcal{E}}+\lambda
H_{\mathcal{P}})t}e^{-iH_{\mathcal{E}}t}|n\rangle,$

where we have introduced an effective “perturbation” $H_{\mathcal{P}}$ using
the Baker-Campbell-Hausdorff formula, which appears in the same order as the
coupling $H_{\mathcal{I}}$ (see the Appendix). The second line of the above
equation can be interpreted as the survival probability of the initial state
after an imperfect echo loop, i.e., a forward evolution followed by a
perturbed backward evolution. This quantity is also known as the Loschmidt
echo Cucchietti  _et al._ (2003); Goussev  _et al._ (2012); Gorin  _et al._
(2006).

Denote the eigenenerges of the “perturbed” Hamiltonian,
$H_{\mathcal{E}}+\lambda H_{\mathcal{P}}$, as $E^{k}$, and the corresponding
eigenstate as $|k\rangle=\sum_{n}C^{k}_{n}|n\rangle$. The decoherence factor
(5) further reduces to a Fourier transformation,

$\displaystyle r(t)$
$\displaystyle=e^{iE_{n}t}\sum_{k}e^{-iE^{k}t}|C_{n}^{k}|^{2}$ (6)
$\displaystyle=e^{iE_{n}t}\int dE\ e^{-iEt}F(E,E_{n})\eta(E).$

Here, the discrete sum is replaced by a continuous integral with a “weighting
function”—the global density of states $\eta(E)$.
$F(E^{k},E_{n})\equiv\overline{|C^{k}_{n}|^{2}}$ is an averaged smooth
function in continuous energy $E^{k}$ of the squared overlap $|C_{n}^{k}|^{2}$
between the perturbed and unperturbed eigenfunctions, which we call the
_overlap_ (a.k.a. the _spread function_). The local density of states (LDOS),
a.k.a the strength function in the literature, is defined as

$P(E,E_{n})\equiv F(E,E_{n})\eta(E).$ (7)

Both the LDOS and the overlap have peaks around the resonance energy $E_{n}$.
The LDOS was introduced and intensively studied in the field of nuclear
physics Bohr and Mottelson (1998) and encountered elsewhere (see e.g. Gruss
_et al._ (2016). Using random matrix theory, it was derived to generally take
the form

$P(E,E_{n})=\frac{1}{2\pi}\frac{\Gamma(E)}{(E-E_{n})^{2}+(\Gamma(E)/2)^{2}},$
(8)

where $\Gamma(E)=2\pi V^{2}\eta(E)$ is given by Fermi’s golden rule; $V^{2}$
is the average of the square of the off-diagonal elements of the perturbation
matrix $\lambda H_{\mathcal{P}}$, when written in the basis of the unperturbed
Hamiltonian $H_{\mathcal{E}}$. It is known Flambaum and Izrailev (2000);
Santos  _et al._ (2012a, b) that in the weak perturbation regime, where the
width of the peak of the LDOS is much smaller than the width of the global
density of states $\eta(E)$, $\Gamma(E)\approx 2\pi V^{2}\eta(E_{n})$ can be
treated as a constant. In this case, the LDOS reduces to a Lorentzian (Breit-
Winger) distribution, whose Fourier transformation contributes to exponential
decay in the time domain. On the other hand, for large perturbations, the LDOS
converts to a Gaussian distribution Santos  _et al._ (2012a, b), with a width
bounded by the bandwidth of the global density of states.

To provide a unified treatment of the decoherence factor, including the cross-
over regime, we first note that the LDOS (8) is valid for an arbitrary
perturbation strength Bohr and Mottelson (1998). Comparing with definition
(7), we identify the overlap function that takes the form

$F(E,E_{n})=\frac{V^{2}}{(E-E_{n})^{2}+\left[\pi V^{2}\eta(E)\right]^{2}}.$
(9)

This is not a precise Lorentzian distribution since the ‘width’ term is
energy-dependent. Nevertheless, we will show that the overlap is insensitive
to this energy dependence, and has an approximate (unnormalized) Lorentzian
form, even for large perturbations that make the LDOS (7) far from a
Lorentzian shape.

To establish this we perform a Taylor expansion of the energy-dependent width
$\Gamma(E)$ around the peak position $E_{n}$. The LDOS (8) is of a Lorentzian
form only at zeroth order of the expansion. However, the overlap $F(E,E_{n})$
has an exact (unnormalized) Lorentzian form up to the second order. Both first
and second-order energy-dependent terms can be absorbed into the quadratic
term $(E-E_{n})^{2}$. In this case, the effective width $\Gamma_{\rm eff}$ of
the overlap is

$\left(\frac{\Gamma_{\rm
eff}}{2}\right)^{2}=\frac{2\pi^{2}V^{4}\eta^{2}}{2+\pi^{2}V^{4}{\eta^{2}}^{\prime\prime}}-\left[\frac{\pi^{2}V^{4}{\eta^{2}}^{\prime}}{2+\pi^{2}V^{4}{\eta^{2}}^{\prime\prime}}\right]^{2},$
(10)

where $\eta(E)$ and its derivatives are all evaluated at $E_{n}$. This
expansion also shifts the effective peak position of the overlap by a small
energy residue, $E_{\rm eff}=E_{n}-E_{r}$, where
$E_{r}=(\pi^{2}V^{4}{\eta^{2}}^{\prime})/(2+\pi^{2}V^{4}{\eta^{2}}^{\prime\prime})$.

The corresponding Fourier transform of the overlap (9) is then proportional to
an exponential decay $e^{-\Gamma_{\rm eff}|t|/2}$ multiplied by a phase,
which, after absorbing the phase factor $e^{iE_{n}t}$ in the decoherence
factor (6), eventually contributes to a net oscillation $e^{iE_{r}t}$.

In physical systems, the Hamiltonian is in general given by few-body
interactions or local terms, whose global density of states satisfy universal
Gaussian distributions Kota (2014). This, upon Fourier transform, gives rise
to a Gaussian decay in the time domain, $e^{-\sigma^{2}t^{2}/2}$, where
$\sigma$ is the bandwidth (standard deviation) of the global density of
states. Here, without the loss of generality, we assume the spectral density
has a maximum at $E_{0}=0$. A non-zero $E_{0}$ only contributes a phase factor
$e^{iE_{0}t}$ to the decoherence factor.

The decoherence factor (6), up to an overall slow oscillation $e^{iE_{r}t}$,
is the Fourier transform of the product of two spectral images in the energy
domain—the Lorentzian overlap and the Gaussian spectral density. The Fourier
transform of such a product is a convolution of the two individual Fourier
transforms of these two spectral images. They are exponential and Gaussian
functions in the time domain, as derived above. Consequently, the decoherence
factor is

$\displaystyle r(t)\propto$ $\displaystyle\int_{-\infty}^{\infty}d\tau
e^{-\Gamma|\tau|/2}e^{-\sigma^{2}(t-\tau)^{2}/2}$ (11) $\displaystyle\propto$
$\displaystyle\sum_{\pm}e^{\pm\Gamma
t/2}\rm{Erfc}\left(\frac{\Gamma/2\pm\sigma^{2}t}{\sqrt{2}\sigma}\right),$

where Erfc is the complementary error function. The full derivation is
presented in the appendix.

For weak ($\Gamma\ll\sigma$) and strong ($\Gamma\sim\sigma$) couplings, this
expression reduces to exponential and Gaussian decays, respectively. When we
define the decoherence time as the time scale for the $\mathcal{O}(1)$ decay
of the decoherence factor, in the exponential and Gaussian regime, the
decoherence time scales as $\sim 1/\Gamma$ and $\sim 1/\sigma$, respectively.

Recall also that $\lambda$ quantifies the interaction strength between the
system and the environment. According to Fermi’s golden rule, for $\lambda\ll
1$, $\Gamma\propto\lambda^{2}$. For the other extreme case, when the
interaction eventually dominates the environment internal dynamics as studied
in Refs. Cucchietti  _et al._ (2005); Zurek  _et al._ (2007), the bandwidth of
the Hamiltonian, $\sigma\propto\lambda$.

We conclude that the decoherence time scales as $\sim\lambda^{-2}$ and
$\sim\lambda^{-1}$ in the weak and strong coupling limit, respectively. We
will verify these predictions in the study of the quantum Brownian motion.

Figure 2: Decay of the decoherence factor at various coupling strengths
$\lambda$ in semi-log scale, with the corresponding overlap function (second
column). The initial state of the environment is prepared as a pure energy
eigenstate in the middle of the spectrum. Triangles are the numerical data,
with the corresponding convolution functions (11) plotted in red curves. To
guide the eyes, we also plotted the best fit to Gaussian (dotted) and
exponential (dashed) functions.

To illustrate these results, we numerically simulate a specific bath term
consisting many interacting spin-$1/2$ particles. More precisely, the linear
coupling term and the bath internal Hamiltonian in Eq. (2) are constructed as

$H_{\mathcal{I}}=\sum_{i}a_{i}\sigma^{z}_{\mathcal{E},i},\quad
H_{\mathcal{E}}=\sum_{\alpha}\sum_{i\neq
j}b^{\alpha}_{ij}\sigma^{\alpha}_{\mathcal{E},i}\sigma^{\alpha}_{\mathcal{E},j},$
(12)

where $\alpha=x,y,z$, and $\sigma^{\alpha}_{\mathcal{E},i}$ are Pauli
operators vectors for the environment spins while $a_{i}$ and
$b^{\alpha}_{ij}$ are random numbers drawn from standard normal distribution.

Figure 3: Decoherence in the quantum Brownian motion at various values of
$\gamma_{0}$. The curves are plotted on the same time scale with the aid of a
numerical re-scaling parameter $t_{0}$. Black triangles, blue circles, and red
diamonds correspond to $\\{\gamma_{0},t_{0}\\}=\\{10^{-3},3.0\times
10^{-3}\\}$, $\\{10^{-6},1.0\\}$ and $\\{3\times 10^{-5},3.3\times
10^{-2}\\}$, and are fitted to Gaussian, exponential, and convolution
functions, respectively. The simulations are performed at temperature
$T=2.5\times 10^{4}$ with a high-energy cutoff $\Lambda=500$. Inset:
Decoherence time $\tau_{D}$ (in log-log scale) in the Gaussian (blue diamonds)
and exponential (red diamonds) regimes, which scales as $\sim
1/\gamma_{0}^{1/2}$ (blue dotted) and $\sim 1/\gamma_{0}$ (red dotted),
respectively.

In Fig. 1 (inset), the global density of states is shown to be well described
by a Gaussian distribution. The claim for the Lorentzian distribution of the
overlap is also justified. Note that the asymptotic tail of the overlap
becomes exponential. This agrees with Wigner’s derivation Wigner (1955, 1957)
for the tail of the strength function for banded matrices. The matrix of the
physical perturbation, represented in the basis of the unperturbed
Hamiltonian, is a banded matrix, i.e., its off-diagonal elements typically
decay exponentially fast Deutsch (2018) with the distance from the diagonal.

To visualize the change of the time dependence of the decoherence factor with
the growth of the coupling strength (and hence, with the width of the
overlap), here we focus on the case where the environment is initially
prepared in an energy eigenstate. The results can be generalized to a thermal
environment by averaging over a thermal distribution. This will be addressed
in the study of quantum Brownian motion. Fig. 2 depicts the decoherence factor
at various coupling strengths, clearly demonstrating the transition between
exponential and Gaussian decays, with a crossover well-described by their
convolution.

_Quantum Brownian motion—_ Encouraged by the above study of the spin-bath
model, we expect that the convolution paradigm is robust and applies to other
decoherence problems with similar structure. We now test this conjecture with
the familiar and extensively studied quantum Brownian motion (QBM) model
Feynman and Vernon (1963); Caldeira and Leggett (1983); Unruh and Zurek
(1989); Hu  _et al._ (1992). Surprisingly, the possibility of the exponential
to Gaussian transition has been so far overlooked.

QBM describes a single harmonic oscillator with a unit mass and a unit bare
frequency, linearly interacting with a thermal bath of non-interacting
harmonic oscillators. It can be solved exactly Hu  _et al._ (1992); Paz  _et
al._ (1993). To study the reduced dynamics of the system oscillator, one must
specify the spectral density function $I(\omega)$ of the bath. We assume:

$I(\omega)=\frac{2\gamma_{0}}{\pi}\omega\exp{\left(-\frac{\omega^{2}}{\Lambda^{2}}\right)}.$
(13)

It describes an Ohmic environment. $\Lambda$ is a high-energy cutoff, and
$\gamma_{0}$ quantifies the interaction strength (it is quadratic in the
linear coupling strength between the system and environment oscillators Hu
_et al._ (1992)).

We assume the system and the bath are initially uncorrelated. The bath is in a
thermal state with temperature $T$, and the system oscillator is prepared in a
“Schrödinger cat” superposition of two Gaussian wavepackets located at
different positions, i.e., $\psi(t=0)=\psi_{1}+\psi_{2}$, with

$\psi_{1,2}(x)=\mathcal{N}\exp{\left(-\frac{(x\mp
x_{0})^{2}}{2\delta^{2}}\right)},$ (14)

where $\mathcal{N}$ is a normalization factor. Due to couplings to the bath,
the system oscillator evolves into a mixed state, which can be written as

$\rho(t)=\rho_{1}(t)+\rho_{2}(t)+\rho_{\rm int}(t),$ (15)

where $\rho_{1,2}$ are the reduced states resulting from the evolution of
$\psi_{1,2}$ alone, and $\rho_{\rm int}$ is the interference term between them
Zurek (1986, 1991).

Decoherence of the system is reflected in the decay of $\rho_{\rm int}$. To
quantify this, we transform each term of the reduced state (15) into a
corresponding Wigner representation in the phase space, $W_{1,2}$ and $W_{\rm
int}$, and study the decay of the peak-to-peak ratio between the Wigner
functions:

$r_{B}(t)\equiv\frac{1}{2}\frac{W_{\rm int}|_{\rm peak}}{W_{1(\rm{or}\
2)}|_{\rm peak}}.$ (16)

This quantity is known to be a good measure of decoherence for QBM Paz  _et
al._ (1993). Figure 3 depicts the decay of $r_{B}$ at various values of
$\gamma_{0}$. Both the exponential and Gaussian regime are accurately
described by the convolution paradigm. In the appendix, we show that this
paradigm applies also to the case of low temperatures.

For QBM, we define the decoherence time $\tau_{D}$ through
$r_{B}(\tau_{D})=1/e$. Since $\gamma_{0}$ in the spectral density is quadratic
in the coupling strength between the system and environment oscillators, the
decoherence time $\tau_{D}$ is thus expected to scale as $\sim 1/\gamma_{0}$
and $\sim 1/\gamma_{0}^{1/2}$, in the exponential and Gaussian regime,
respectively. Both of these scaling behaviors are verified in the inset of
Fig. 3. We present in the appendix the full solution of the model and the
rationale for the convolution description.

_Discussion—_ Convolution provides a unified and robust description of the
interplay between the exponential to Gaussian decoherence. Their relative
import is controlled by the interaction strength between the system and the
environment as was seen in the two paradigmatic decoherence models, the spin-
bath and the quantum Brownian motion. In both of these models the transition
related to the echo operator in (5). We expect that the convolution paradigm
applies to a broader context than decoherence, whenever the same imperfect
echo process is involved. This may explain the Gaussian to exponential
transition in information scrambling Yan  _et al._ (2020), decay processes
Flambaum and Izrailev (2001), as well as recent experiments Sánchez  _et al._
(2020) showing similar crossover behaviors (although study of the convolution
paradigm to fields such as e.g. nuclear physics where LDOS has very different
origins than in decoherence is beyond the scope of our discussion).

Decoherence is the main obstacle in implementing quantum computing. The time
dependence of the decoherence factor will affect the error correction strategy
(e.g., selection of the time intervals between the error correcting
operations).

###### Acknowledgements.

The authors acknowledge Nikolai Sinitsyn and Lukasz Cincio for helpful
discussion, Jess Riedel and Michael Zwolak for constructive comments on the
manuscript. This research was supported by the U.S Department of Energy under
the LDRD program in Los Alamos. B.Y. also acknowledges support from U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division, Condensed Matter Theory Program, and the
Center for Nonlinear Studies. W.H.Z. acknowledges partial support by the
Foundational Questions Institute’s Consciousness in the Physical World
program, in partnership with the Fetzer Franklin Fund.

## References

  * Zurek (1981) W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?” Phys. Rev. D 24, 1516–1525 (1981).
  * Zurek (1991) W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36–44 (1991).
  * Schlosshauer (2019) M. Schlosshauer, “Quantum decoherence,” Phys. Rep. 831, 1–57 (2019).
  * Zurek (2003) W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
  * Zurek (2006) W. H. Zurek, “Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time,” Phys. Scr. 1998, 186 (2006).
  * Buluta and Nori (2009) I. Buluta and F. Nori, “Quantum simulators,” Science 326, 108–111 (2009).
  * Preskill (2018) J. Preskill, “Quantum computing in the NISQ era and beyond,” Quantum 2, 79 (2018).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2010).
  * Schomerus and Lutz (2007) H. Schomerus and E. Lutz, “Nonexponential decoherence and momentum subdiffusion in a quantum lévy kicked rotator,” Phys. Rev. Lett. 98, 260401 (2007).
  * Cai and Barthel (2013) Z. Cai and T. Barthel, “Algebraic versus exponential decoherence in dissipative Many-Particle systems,” Phys. Rev. Lett. 111, 150403 (2013).
  * Beau  _et al._ (2017) M. Beau, J. Kiukas, I. L. Egusquiza, and A. Del Campo, “Nonexponential quantum decay under environmental decoherence,” Phys. Rev. Lett. 119, 130401 (2017).
  * Pekola and Karimi (2020) J. P. Pekola and B. Karimi, “Qubit decay in circuit quantum thermodynamics,”  (2020), arXiv:2010.11122 [quant-ph] .
  * Sarkar  _et al._ (2017) S. Sarkar, S. Paul, C. Vishwakarma, S. Kumar, G. Verma, M. Sainath, U. D. Rapol, and M. S. Santhanam, “Nonexponential decoherence and subdiffusion in Atom-Optics kicked rotor,” Phys. Rev. Lett. 118, 174101 (2017).
  * Lindblad (1976) G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys. 48, 119–130 (1976).
  * Gorini  _et al._ (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of n‐level systems,” J. Math. Phys. 17, 821–825 (1976).
  * Benatti and Floreanini (1998) F. Benatti and R. Floreanini, “On the decay law for unstable open systems,” Phys. Lett. B 428, 149–156 (1998).
  * Knight (1976) P. L. Knight, “Ersaks’s regeneration hypothesis and deviations from the exponential decay law,” Phys. Lett. A 56, 11–13 (1976).
  * Knight and Milonni (1976) P. L. Knight and P. W. Milonni, “Long-time deviations from exponential decay in atomic spontaneous emission theory,” Phys. Lett. A 56, 275–278 (1976).
  * Burgarth and Facchi (2017) D. Burgarth and P. Facchi, “Positive hamiltonians can give purely exponential decay,” Phys. Rev. A 96, 010103 (2017).
  * von Neumann (1055) J. von Neumann, “Mathematical foundations of quantum mechanics,” (Princeton University Press, 1055) p. p.366.
  * Misra and Sudarshan (1977) B. Misra and E. C. G. Sudarshan, “The zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756–763 (1977).
  * Facchi and Pascazio (2008) P. Facchi and S. Pascazio, “Quantum zeno dynamics: mathematical and physical aspects,” J. Phys. A: Math. Theor. 41, 493001 (2008).
  * Cucchietti  _et al._ (2005) F. M. Cucchietti, J. P. Paz, and W. H. Zurek, “Decoherence from spin environments,” Phys. Rev. A. 72, 052113 (2005).
  * Zurek  _et al._ (2007) W. H. Zurek, F. M. Cucchietti, and J. P. Paz, “Gaussian decoherence and gaussian echo from spin environments,” Acta Phys. Pol. B. 38, 1685–1703 (2007).
  * Kota (2014) V. K. B. Kota,  _Embedded Random Matrix Ensembles in Quantum Physics_, 1st ed., Lecture Notes in Physics (Springer International Publishing, 2014).
  * Zurek (1982) W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862–1880 (1982).
  * Braun  _et al._ (2001) D. Braun, F. Haake, and W. T. Strunz, “Universality of decoherence,” Phys. Rev. Lett. 86, 2913–2917 (2001).
  * Quan  _et al._ (2006) H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, “Decay of loschmidt echo enhanced by quantum criticality,” Phys. Rev. Lett. 96, 140604 (2006).
  * Cucchietti  _et al._ (2003) F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H. Zurek, “Decoherence and the loschmidt echo,” Phys. Rev. Lett. 91, 210403 (2003).
  * Goussev  _et al._ (2012) A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. Wisniacki, “Loschmidt echo,” in  _Scholarpedia_ (Scholarpedia, 2012) p. 7(8):11687.
  * Gorin  _et al._ (2006) T. Gorin, T. Prosen, T. H. Seligman, and M. Žnidarič, “Dynamics of loschmidt echoes and fidelity decay,” Phys. Rep. 435, 33–156 (2006).
  * Bohr and Mottelson (1998) A. Bohr and B. R. Mottelson,  _Nuclear structure_ , Vol. 1 (World Scientific, 1998).
  * Gruss  _et al._ (2016) D. Gruss, K. A. Velizhanin, and M. Zwolak, “Landauer’s formula with finite-time relaxation: Kramers’ crossover in electronic transport,” Sci. Rep. 6, 24514 (2016).
  * Flambaum and Izrailev (2000) V. V. Flambaum and F. M. Izrailev, “Excited eigenstates and strength functions for isolated systems of interacting particles,” Phys. Rev. E 61, 2539–2542 (2000).
  * Santos  _et al._ (2012a) L. F. Santos, F. Borgonovi, and F. M. Izrailev, “Chaos and statistical relaxation in quantum systems of interacting particles,” Phys. Rev. Lett. 108, 094102 (2012a).
  * Santos  _et al._ (2012b) L. F. Santos, F. Borgonovi, and F. M. Izrailev, “Onset of chaos and relaxation in isolated systems of interacting spins: energy shell approach,” Phys. Rev. E 85, 036209 (2012b).
  * Wigner (1955) E. P. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math. 62, 548–564 (1955).
  * Wigner (1957) E. P. Wigner, “Characteristics vectors of bordered matrices with infinite dimensions II,” Ann. Math. 65, 203–207 (1957).
  * Deutsch (2018) J. M. Deutsch, “Eigenstate thermalization hypothesis,” Rep. Prog. Phys. 81, 082001 (2018).
  * Feynman and Vernon (1963) R. P. Feynman and F. L. Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Ann. Phys. 24, 118–173 (1963).
  * Caldeira and Leggett (1983) A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum brownian motion,” Physica A 121, 587–616 (1983).
  * Unruh and Zurek (1989) W. G. Unruh and W. H. Zurek, “Reduction of a wave packet in quantum brownian motion,” Phys. Rev. D 40, 1071–1094 (1989).
  * Hu  _et al._ (1992) B. L. Hu, J. P. Paz, and Y. Zhang, “Quantum brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise,” Phys. Rev. D 45, 2843–2861 (1992).
  * Paz  _et al._ (1993) J. P. Paz, S. Habib, and W. H. Zurek, “Reduction of the wave packet: Preferred observable and decoherence time scale,” Phys. Rev. D. 47, 488–501 (1993).
  * Zurek (1986) W. H. Zurek, “Reduction of the wavepacket: How long does it take?” in  _Frontiers of Nonequilibrium Statistical Physics_, edited by Gerald T Moore and Marlan O Scully (Springer US, Boston, MA, 1986) pp. 145–149.
  * Yan  _et al._ (2020) B. Yan, L. Cincio, and W. H. Zurek, “Information scrambling and loschmidt echo,” Phys. Rev. Lett. 124, 160603 (2020).
  * Flambaum and Izrailev (2001) V. V. Flambaum and F. M. Izrailev, “Unconventional decay law for excited states in closed many-body systems,” Phys. Rev. E 64, 026124 (2001).
  * Sánchez  _et al._ (2020) C. M. Sánchez, A. K. Chattah, K. X. Wei, L. Buljubasich, P. Cappellaro, and H. M. Pastawski, “Perturbation independent decay of the loschmidt echo in a Many-Body system,” Phys. Rev. Lett. 124, 030601 (2020).

##  Appendix A Effective perturbation

The echo operator in the decoherence factor, Eq. (5) in the main text, is
written in term of an effective perturbation $H_{\mathcal{P}}$, i.e.,

$e^{i(H_{\mathcal{E}}+H_{\mathcal{I}})t}e^{-i(H_{\mathcal{E}}-H_{\mathcal{I}})t}=e^{i(H_{\mathcal{E}}+H_{\mathcal{P}})t}e^{-iH_{\mathcal{E}}t}.$
(17)

In practice, the interaction $H_{\mathcal{I}}$, as well as the effective
perturbation $H_{\mathcal{P}}$, are much smaller compared to the environment
Hamiltonian $H_{\mathcal{E}}$. This allows to expand echo operator as a power
series. Intuitively, the effective perturbation $H_{\mathcal{P}}$ should be
twice as large as the interaction $H_{\mathcal{I}}$, as it only appears in the
forward evolution loop. This can be verified by expanding the logarithm of the
echo operator. Using the Baker-Campbell-Hausdorff formula, up to first order,
we have

$\displaystyle\log{\left(e^{i(H_{\mathcal{E}}+H_{\mathcal{I}})t}e^{-i(H_{\mathcal{E}}-H_{\mathcal{I}})t}\right)}$
(18) $\displaystyle=$ $\displaystyle
2itH_{\mathcal{I}}+\frac{1}{2}t^{2}\left[2H_{\mathcal{I}},H_{\mathcal{E}}\right]+\frac{1}{12}2it^{3}\left[H_{\mathcal{E}},\left[2H_{\mathcal{I}},H_{\mathcal{E}}\right]\right]-\frac{1}{24}2t^{4}\left[H_{\mathcal{E}}\left[H_{\mathcal{E}},2\left[2H_{\mathcal{I}},H_{\mathcal{E}}\right]\right]\right]+\cdots$
$\displaystyle\log{\left(e^{i(H_{\mathcal{E}}+H_{\mathcal{P}})t}e^{-iH_{\mathcal{E}}t}\right)}$
$\displaystyle=$ $\displaystyle
itH_{\mathcal{P}}+\frac{1}{2}t^{2}\left[H_{\mathcal{P}},H_{\mathcal{E}}\right]+\frac{1}{12}2it^{3}\left[H_{\mathcal{E}},\left[H_{\mathcal{I}},H_{\mathcal{P}}\right]\right]-\frac{1}{24}2t^{4}\left[H_{\mathcal{E}}\left[H_{\mathcal{E}},\left[H_{\mathcal{P}},H_{\mathcal{E}}\right]\right]\right]+\cdots$

Hence, to the first order, $H_{\mathcal{P}}=2H_{\mathcal{I}}$.

##  Appendix B The convolution function

For the standard Fourier transform $\hat{\mathcal{F}}$,

$\hat{\mathcal{F}}[f(x)]\equiv\int dx\ f(x)e^{-2\pi ixt},$ (19)

the convolution theorem asserts

$\hat{\mathcal{F}}[fg]=\hat{\mathcal{F}}[f]*\hat{\mathcal{F}}[g],$ (20)

where the convolution $*$ is defined as

$f*g\equiv\int_{-\infty}^{\infty}d\tau\ f(\tau)g(x-\tau).$ (21)

For convenience, we adapt the Fourier transform for the angular frequency
$2\pi x$, namely,

$\hat{\mathcal{F}}[f(x)]\equiv\int dx\ f(x)e^{-ixt},$ (22)

with the corresponding inverse Fourier transform normalized by a pre-factor
$1/2\pi$. Under this convention, the convolution theorem reads

$\displaystyle\hat{\mathcal{F}}[fg]$ $\displaystyle=2\pi\int dx\ e^{-2\pi
ixt}f(2\pi x)g(2\pi x),$ (23) $\displaystyle=2\pi\int dx\ e^{-i2\pi xt}f(2\pi
x)*\int dx\ e^{-i2\pi xt}g(2\pi x)$
$\displaystyle=\frac{1}{2\pi}\hat{\mathcal{F}}[f(x)]\hat{\mathcal{F}}[g(x)].$

This allows to evaluate the decoherence factor in the main text as

$\displaystyle r(t)=$ $\displaystyle e^{iE_{n}t}\int dE\
e^{-iEt}F(E,E_{n})\eta(E)$ (24) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}e^{iE_{n}t}\hat{\mathcal{F}}\left[F(E,E_{n})\right]*\hat{\mathcal{F}}\left[\eta(E)\right].$

Assuming that the global spectrum density has a Gaussian shape with a band-
width (standard deviation) $\sigma$, its Fourier transform is a Gaussian decay

$\hat{\mathcal{F}}\left[\eta(E)\right]=e^{-\sigma^{2}t^{2}/2}.$ (25)

As derived in the main text, the overlap takes the form

$F(E,E_{n})=\frac{V^{2}}{(E-E_{n}+E_{r})^{2}+(\Gamma_{\rm eff}/2)^{2}},$ (26)

with an effective width $\Gamma_{\rm eff}$ and a small shift $E_{r}$ of the
peak position $E_{n}$. Its Fourier transform is given by

$\hat{\mathcal{F}}\left[F(E,E_{n})\right]\propto
e^{-i(E_{n}-E_{r})t}e^{-\Gamma_{\rm eff}|t|/2}.$ (27)

Here we omitted the normalization factor since the decoherence factor always
start from unity.

Evaluating the convolution of the above two Fourier transforms of the overlap
and the spectral density, we arrive at the final expression of the decoherence
factor

$\displaystyle r(t)\propto$ $\displaystyle\
e^{-\Gamma|t|/2}*e^{-\sigma^{2}t^{2}/2}$ (28) $\displaystyle=$
$\displaystyle\int\ d\tau\ e^{-\Gamma|\tau|/2}e^{-\sigma^{2}(t-\tau)^{2}/2}$
$\displaystyle\propto$ $\displaystyle\ e^{-\Gamma
t/2}\rm{Erfc}\left(\frac{\Gamma/2-\sigma^{2}t}{\sqrt{2}\sigma}\right)+e^{\Gamma
t/2}\rm{Erfc}\left(\frac{\Gamma/2+\sigma^{2}t}{\sqrt{2}\sigma}\right),$

Note that the above solution gives the unnormalized magnitude of the
decoherence factor, which starts from unity at the initial time. The
decoherence factor also has a residue oscillation $e^{-iE_{r}t}$ induced by
the small shift $E_{r}$ of the resonance peak of the overlap function.

##  Appendix C Quantum Brownian motion

Quantum Brownian (QBM) has been extensively studied in the literature. In this
section, we present the model, and, to make the paper self-contained, collect
the essential ingredients for solving it. We then discuss the convolution
function description for decoherence of QBM.

###  C.1 The model

The model contains a one-dimensional harmonic oscillator (the system) with
unit mass and bare frequency $\Omega_{0}$, and a collection of harmonic
oscillators (the bath) with mass $\\{m_{k}\\}$ and frequency
$\\{\omega_{k}\\}$. The system oscillator interacts linearly with the bath
oscillators through

$H_{\mathcal{I}}=-\sum_{k}c_{k}\hat{x}_{\mathcal{S}}\hat{x}_{\mathcal{B}}^{k},$
(29)

with coupling strengths $\\{c_{k}\\}$. It is well-known that the effect of the
bath oscillators on the reduced dynamics of the system oscillator can be fully
characterized with a spectral density function, defined as

$I(\omega)=\sum_{k}\frac{c^{2}_{k}}{2m_{k}\omega_{k}}\delta(\omega-\omega_{k}).$
(30)

Here, we consider the particular form of the spectral density describing the
Ohmic environment, as is widely used in the literature:

$I(\omega)=\frac{2\gamma_{0}}{\pi}\omega\exp{\left(-\frac{\omega^{2}}{\Lambda^{2}}\right)},$
(31)

where $\Lambda$ is a high energy cut-off, and $\gamma_{0}$ is a constant that
quantifies the linear coupling strength (it is quadratic in the coupling
strength).

As considered in Ref. Paz  _et al._ (1993), we assume the bath is prepared in
a thermal state with temperature $T$, and the initial state of the system
oscillator is in a superposition of two Gaussian wave packages,

$\psi(x,t=0)=\psi_{1}(x)+\psi_{2}(x),$ (32)

where $\psi_{1,2}$ are Gaussian wavefunctions localized at separated
locations, i.e.,

$\psi_{1,2}=\mathcal{N}\exp{\left(-\frac{(x\mp
x_{0})^{2}}{2\delta^{2}}\right)},$ (33)

with $\mathcal{N}$ a normalization constant. Due to coupling to the bath of
oscillators, the state of the system oscillator will become mixed during time
evolution, which can be factorized into

$\rho(t)=\rho_{1}(t)+\rho_{2}(t)+\rho_{\rm int}(t).$ (34)

Here $\rho_{1,2}$ are the reduced density matrices from the evolution of
$\psi_{1,2}$, and $\rho_{\rm int}$ is the interference part. The state can be
visualized using the Wigner function in the phase space, defined for the
density matrix as

$W(x,p)\equiv\int\frac{dz}{2\pi}e^{ipz}\rho(x-z/2,x+z/2).$ (35)

Represented as a Winger function, the reduced density matrix (34) has three
corresponding terms

$W(t)=W_{1}(t)+W_{2}(t)+W_{\rm int}(t).$ (36)

As shown in Ref. Paz  _et al._ (1993), $W_{1,2}$ remain as separated Gaussian
packages in the phase space, while the interference term $W_{\rm int}$ decays
in time. To quantify the decay of interference, we study the ratio between the
peak values of $W_{\rm int}$ and $W_{1(2)}$:

$r_{B}(t)\equiv\frac{1}{2}\frac{W_{\rm int}|_{\rm peak}}{W_{1(\rm{or}\
2)}|_{\rm peak}},$ (37)

This quantity has been demonstrated Paz  _et al._ (1993) to be a good measure
for decoherence.

###  C.2 The solution

In this section, we collect the essential results of the solution that allow
us to perform numerical simulations. More details of the derivations can be
found in Refs. Hu  _et al._ (1992); Paz  _et al._ (1993).

The total reduced density matrix (34) can be solved using path integral Hu
_et al._ (1992)

$\rho(x,y,t)=\int\int
dx^{\prime}dy^{\prime}J(x,y,t;x^{\prime},y^{\prime},t^{\prime})\rho(x^{\prime},y^{\prime},t^{\prime}),$
(38)

where the propagator, written in terms of the variables $X=x+y$ and $Y=x-y$,
is given by

$J=(X,Y,t;X^{\prime},Y^{\prime},t^{\prime})=\frac{b_{3}}{2\pi}\exp\left({-a_{11}Y^{2}-a_{12}YY^{\prime}-a_{22}Y^{\prime
2}}\right)\exp\left({ib_{1}XY+ib_{2}X^{\prime}Y-ib_{3}XY^{\prime}-ib_{4}X^{\prime}Y^{\prime}}\right).$
(39)

Here, the functions $a_{ij}(t)$ and $b_{i}(t)$ can be determined by the
equations

$\displaystyle\ddot{u}(s)+\Omega_{0}^{2}u(s)+2\int_{0}^{s}ds^{\prime}\
\eta(s-s^{\prime})u(s^{\prime})=0,$ (40) $\displaystyle
2b_{1}(t)=\dot{u}_{2}(t),\quad 2b_{3}(t)=\dot{u}_{2}(0),$ $\displaystyle
2b_{2}(t)=\dot{u}_{1}(t),\quad 2b_{4}(t)=\dot{u}_{1}(0),$ $\displaystyle
a_{ij}=\frac{1}{1+\delta_{ij}}\int_{0}^{\infty}\int_{0}^{\infty}dsds^{\prime}\
u_{i}(s)u_{j}(s^{\prime})\mu(s-s^{\prime}),$

with boundary conditions $u_{1}(0)=u_{2}(t)=1$ and $u_{1}(t)=u_{2}(0)=0$, and
$\mu(s),\eta(s)$ are the noise and dissipation kernels determined by the
spectral density:

$\displaystyle\mu(s)$ $\displaystyle=\int_{0}^{\infty}d\omega\
I(\omega)\coth\left(\frac{\omega}{2T}\right)\cos\left(\omega s\right)$ (41)
$\displaystyle\eta(s)$ $\displaystyle=-\int_{0}^{\infty}d\omega\
I(\omega)\sin\left(\omega s\right).$

We solve the above integrodifferential equation for $u(t)$ with a shooting
method incorporating the imposed boundary conditions. With these solutions, we
can then compute the peak-to-peak ratio (37) between the Wigner functions:

$\displaystyle r_{B}(t)$ $\displaystyle=\frac{1}{2}\frac{W_{\rm int}|_{\rm
peak}}{W_{1(\rm{or}\ 2)}|_{\rm peak}}$ (42)
$\displaystyle=\exp{\left(-x_{0}^{2}/\delta^{2}+\kappa_{p}^{2}/\delta_{2}^{2}+\delta_{1}^{2}\kappa_{x}^{2}\right)},$

where

$\displaystyle\delta_{1}^{2}=$
$\displaystyle\left(a_{22}+\frac{1}{4\delta^{2}+\delta^{2}b_{4}^{2}}\right)b_{3}^{-2},$
(43) $\displaystyle\delta_{2}^{2}=$
$\displaystyle\frac{1}{4}\left[a_{11}+\delta^{2}b_{2}^{2}-\frac{1}{4\delta_{1}^{2}}\left(\frac{a_{12-2\delta^{2}b_{2}b_{4}}}{b_{3}}^{2}\right)\right]^{-1},$
$\displaystyle\kappa_{x}=$
$\displaystyle\frac{x_{0}}{2\delta_{1}^{2}\delta^{2}b_{3}},\quad\kappa_{p}=\frac{x_{0}\delta_{2}^{2}(a_{12}-2\delta^{2}b_{2}b_{4})}{2\delta^{2}\delta_{1}^{2}b_{3}^{2}}.$

Here, the peak values for $W_{1,2}$ are the same, so we do not distinguish
between them.

###  C.3 Convolution in QBM

As shown in the main text, the convolution function fits the decoherence of
QBM very well. In this section, we explore the mechanism behind this. Instead
of solving the reduced dynamics of the system harmonic oscillator alone, here
we need to include the bath degrees of freedom into consideration.

The bath is prepared in a thermal state. However, for simplicity, in the
following discussion we assume the bath is initially in a pure energy
eigenstate. The result can be generalized to thermal states by doing a simple
thermal average. Under this condition, the initial total wavefunction of the
system oscillator and the bath is

$\Psi(t=0)=(\psi_{1}+\psi_{2})\otimes\psi_{\mathcal{B}},$ (44)

which results in two branches of the total wavefunction at any time instant
$t$, i.e.,

$\Psi(t)=\Psi_{1,\mathcal{B}}+\Psi_{2,\mathcal{B}}.$ (45)

The reduced states of the system oscillator from each branch, e.g.,
$\rho_{1,2}=\rm{Tr}|\Psi_{1,2,\mathcal{B}}\rangle\langle\Psi_{1(2),\mathcal{B}}|$,
are mixed states, whose Wigner representations (the first two terms in Eq.
(36)) in the phase space take Gaussian forms. As shown in Ref. Paz  _et al._
(1993), during time evolution, these two Gaussian packages are always
separated and are approximately orthogonal to each other, as are the system’s
reduced density matrices $\rho_{1,2}(t)$ – they are (approximately) supported
on orthogonal subspaces of the total Hilbert space. Therefore, we can
decompose the total wavefunction at any time instant into:

$\Psi(t)\approx\sum_{k}^{N}\psi_{1}^{k}(t)\otimes\psi^{k}_{\mathcal{B}1}(t)+\sum_{k}^{N}\psi_{2}^{k}(t)\otimes\psi^{k}_{\mathcal{B}2}(t).$
(46)

In this expansion, the basis states for the first branch,
$\\{\psi_{1}^{k}(t)\\}_{k}$, are orthogonal to those of the second branch,
$\\{\psi_{2}^{k}(t)\\}_{k}$. Here, we truncate the infinite dimensional
Hilbert space to a finite (but large) $N$-dimensional Hilbert space within the
relevant energy scale. By choosing a sufficiently large $N$, this can
approximate the exact wave function to any degree of accuracy.

The above assumptions are specifically made for the QBM model and are
justified by the known solutions. From now on, we will keep the discussion in
a more formal and abstract manner beyond the QBM model. We write the linear
interaction in a generic form

$H_{\mathcal{I}}=H_{\mathcal{I}}^{\mathcal{S}}\otimes
H_{\mathcal{I}}^{\mathcal{B}}.$ (47)

As a matter of convention, we also choose the basis
$\\{\psi_{1,2}^{k}(t)\\}_{k}$ as an eigenbasis of the interaction Hamiltonian
$H_{\mathcal{I}}^{\mathcal{S}}$ – this is always possible since for any basis
of the truncated Hilbert space we can diagonalize
$H_{\mathcal{I}}^{\mathcal{S}}$ and use the resulting eigenbasis to expand the
wavefunction into the form (46). The corresponding eigenvalues of the basis
are denoted as $\\{\lambda_{1(2)}^{k}(t)\\}_{k}$.

As introduced in the previous sections, the reduced state of the system
oscillator can be factorized into

$\rho(t)=\rho_{1}(t)+\rho_{2}(t)+\rho_{\rm int}(t),$ (48)

where $\rho_{1}$ ($\rho_{2}$) is reduced from the first (second) branch of the
total wavefunction, and $\rho_{\rm int}$ comes from the interference part. In
our $2N$-dimensional truncated total Hilbert space, $\rho_{\rm int}$ has
matrix element

$\rho_{\rm
int}^{1i,2j}=\rm{Tr}|\psi^{i}_{\mathcal{B}1}(t)\rangle\langle\psi^{j}_{\mathcal{B}2}(t)|.$
(49)

With this, we can build up the evolution trajectories for the bath
wavefunctions $\\{\psi_{\mathcal{B}1}^{k}(t)\\}_{k}$ and
$\\{\psi_{\mathcal{B}2}^{k}(t)\\}_{k}$. Consider the evolution of the total
wavefunction from the given state $\Psi(t)$ at time $t$ to time $t+\Delta t$
with an infinitesimal $\Delta t$. The evolution reads

$\displaystyle\Psi(t+\Delta t)=e^{-iH\Delta
t}\Psi(t)=\sum_{k}^{N}\psi_{1}^{k}(t)\otimes
e^{-i\left(H_{0}+\lambda_{1}^{k}(t)H_{I}^{\mathcal{B}}\right)\Delta
t}\psi^{k}_{\mathcal{B}1}(t)+\sum_{k}^{N}\psi_{2}^{k}(t)\otimes
e^{-i\left(H_{0}+\lambda_{2}^{k}(t)H_{I}^{\mathcal{B}}\right)\Delta
t}\psi^{k}_{\mathcal{B}2}(t),$ (50)

where $H_{0}$ is the internal Hamiltonian (non-interacting part) of the total
system. Hence, we conclude that the trajectory $\psi_{\mathcal{B}1}^{k}(t)$
(and similarly for the second branch $\psi_{\mathcal{B}2}^{k}(t)$) can be
interpreted as an evolution with a time-dependent Hamiltonian, i.e.,

$\psi_{\mathcal{B}1}^{k}(t)=\overleftarrow{\mathcal{P}}e^{-i\int
d\tau\left(H_{0}+\lambda_{1}^{k}(\tau)H_{I}^{\mathcal{B}}\right)\tau}\psi^{k}_{\mathcal{B}1}(0),$
(51)

where the time ordering $\overleftarrow{\mathcal{P}}$ is imposed. Plugging the
above equation into Eq. (49), we get the matrix element of the interference
part of the reduced density matrix,

$\displaystyle\rho_{\rm int}^{1i,2j}=$ $\displaystyle\rm{Tr}\left[e^{-i\int
d\tau\left(H_{0}+\lambda_{1}^{i}(\tau)H_{I}\right)\tau}|\psi^{i}_{\mathcal{B}1}(0)\rangle\langle\psi^{j}_{\mathcal{B}2}(0)|e^{i\int
d\tau\left(H_{0}+\lambda_{2}^{j}(\tau)H_{I}\right)\tau}\right]$ (52)
$\displaystyle=$ $\displaystyle\langle\psi^{j}_{\mathcal{B}2}(0)|e^{i\int
d\tau\left(H_{0}+\lambda_{2}^{j}(\tau)H_{I}\right)\tau}e^{-i\int
d\tau\left(H_{0}+\lambda_{1}^{i}(\tau)H_{I}\right)\tau}|\psi^{i}_{\mathcal{B}1}(0)\rangle$
$\displaystyle=$ $\displaystyle\langle\psi_{\mathcal{B}}|e^{i\int
d\tau\left(H_{0}+\lambda_{2}^{j}(\tau)H_{I}\right)\tau}e^{-i\int
d\tau\left(H_{0}+\lambda_{1}^{i}(\tau)H_{I}\right)\tau}|\psi_{\mathcal{B}}\rangle.$

Note that in the above equation we used a boundary condition imposed by the
initial state of the bath, i.e.,
$\psi_{\mathcal{B}1,2}^{k}(0)=\psi_{\mathcal{B}}$. We also omitted the time
ordering operator, but the integral is time ordered and is interpreted as an
evolution generated by the Hamiltonian $H_{0}$ with small time-dependent
perturbations $\lambda_{1,2}(t)H_{\mathcal{I}}^{\mathcal{B}}$.

Hence, the interference part of the reduced density matrix is related to the
echo operators, which then contribute to the convolution function in the same
manner as in the qubit model discussed in the main text. It is worth
emphasizing that the above derivation is by no means a method for solving the
reduced density matrix from the original model Hamiltonian, since the
effective perturbations $\lambda_{1,2}(t)$ have to be extracted from the
solution (46) known in the first place. The purpose for this transformation is
to extract the structure in terms of the echo operators.

###  C.4 QBM at low temperatures

In the main text, we have considered decoherence of QBM at high temperatures,
and decoherence of the spin-bath model for a single energy eigenstate at the
middle of the spectrum. In both of these cases, well defined overlap can be
established, whose peaks are far from the edge of the spectrum. At extremely
low temperatures, the overlap cannot extend below the ground state energy.
This low-energy cutoff may modify the shape of the overlap, and hence reduce
the accuracy of our analysis in terms of convolution functions.

To test this situation, we simulated $r_{B}(t)$ as defined in the main text
for the QBM model at zero temperature. At early times, $r_{B}(t)$ can still be
accurately described by the convolution function (Fig. C1 left). This further
demonstrated the robustness of the convolution paradigm. The low energy cutoff
of the overlap mainly affects the long time behavior of its Fourier transform.
Indeed, at long times, as shown in Fig. C1 (right), the decay of $r_{B}(t)$ is
suppressed by a power-law.

Figure C1: Decoherence in the quantum Brownian motion at zero temperature with
$\gamma_{0}=0.01$, and frequency cut-off $\Lambda=500$. Red circles are the
numerical data of $r_{B}$. Left: At early times, decay of $r_{B}$ is well-
described by the convolution function (red solid curve). To guide the eyes, we
also plotted the best Gaussian (black dotted) and exponential (blue dashed)
fit. Right (log-log scale): At late times, decay of $r_{B}$ is suppressed by a
power law. The blue dashed line is fitted to $0.017t^{-0.63}+0.02$.


# On Fridman invariant, injectivity radius function and squeezing function

Akhil Kumar  Department of Mathematics, University of Delhi, Delhi–110 007,
India<EMAIL_ADDRESS>and  Sanjay Kumar Pant  Department of
Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Delhi–110 078,
India<EMAIL_ADDRESS>
###### Abstract.

We give a class of domains for which Fridman invariant and injectivity radius
function coincide with respect to Carathéodory metric. We give explicit
expressions of the squeezing functions for these domains and investigate some
of their properties.

###### Key words and phrases:

Squeezing function; Injectivity radius function; Fridman invariants.

###### 2020 Mathematics Subject Classification:

32F45, 32H02

##  1\. Introduction

In [19], Ng, Tang and Tsai raised a question: “For which $c$-hyperbolic
complex manifold $X$ we have $H_{X}^{c}(z)\leq i_{X}^{c}(z)$ for all $z\in
X$?” In one complex dimension, they have already shown that
$H_{X}^{k}(z)=i_{X}^{k}(z)$ for all $z\in X$, where $X$ is any 1-dimensional
$k$-hyperbolic complex manifold. In the context of this question, we state our
main result

######  Theorem 1.1.

Let $\Omega=\mathbb{D}\setminus K$ be a domain in $\mathbb{C}$, where
$\mathbb{D}=\\{\xi\in\mathbb{C}:|\xi|<1\\}$ and $K$ is a removable set in
$\mathbb{D}$. Then, we have

$H_{\Omega}^{c}(z)=i_{\Omega}^{c}(z)=\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}\ \mbox{for all}\
z\in\Omega.$

Here, $H_{\Omega}^{c}(z)$ and $i_{\Omega}^{c}(z)$ refer to the Fridman
invariant and the injectivity radius function with respect to the Carathéodory
metric.

In [6, 7], Fridman introduced a holomorphic invariant known as the Fridman
invariant. Let $X$ be an $n$-dimensional $d$-hyperbolic complex manifold. For
$z\in X$, the Fridman invariant denoted by $h_{X}^{d}$, is defined as

$h_{X}^{d}(z)=\inf\left\\{\dfrac{1}{r}:B_{X}^{d}(z,r)\subseteq
f(\mathbb{B}^{n}),f:\mathbb{B}^{n}\to X\right\\},$

where $f:\mathbb{B}^{n}\to X$ is an injective holomorphic map and
$B_{X}^{d}(z,r)$ is a ball centred at $z$ with radius $r$ with respect to
$d$-metric. The Fridman invariant is very useful in decoding some geometrical
properties of the manifold. For example, Fridman [7] has shown that if Fridman
invariant with respect to Kobayashi metric $h_{X}^{k}(z)=0$ for some $z\in X$,
where $X$ is any connected $k$-hyperbolic complex manifold, then $X$ is
biholomorphic to $\mathbb{B}^{n}$. For comparison purposes, Nikolov and Verma
[20] considered a modification in $h_{X}^{d}$, which is denoted by $H_{X}^{d}$
and defined as

$H_{X}^{d}(z)=\sup\\{\tanh r:B_{X}^{d}(z,r)\subseteq
f(\mathbb{B}^{n}),f:\mathbb{B}^{n}\to X\\},$

where $f:\mathbb{B}^{n}\to X$ is an injective holomorphic map.

Let $X$ be an $n$-dimensional $d$-hyperbolic complex manifold. For $z\in X$,
the injectivity radius function with respect to $d$-metric denoted by
$i_{X}^{d}$, is defined as

$i_{X}^{d}(z)=\sup\\{\tanh r:B_{X}^{d}(z,r)\ \mbox{is simply connected}\\}.$

For Carathéodory and Kobayashi metrics, we denote injectivity radius functions
by $i_{X}^{c}$ and $i_{X}^{k}$ respectively. The injectivity radius with
respect to $d$-metric is defined as

$i_{X}^{d}=\inf_{z\in X}i_{X}^{d}(z).$

A Riemann Surface $X$ is said to be uniformly thick if the injectivity radius
function $i_{X}^{k}(z)$ on it admits a positive lower bound. In [19], Ng, Tang
and Tsai extended the definition of uniformly thickness to higher dimension.
An $n$-dimensional $d$-hyperbolic complex manifold is said to be $d$-uniformly
thick if Fridman invariant $H_{X}^{d}(z)$ on it, admits a positive lower
bound. For example, all simply connected bounded domains in $\mathbb{C}$ are
both $c$-uniformly thick and $k$-uniformly thick. For other important
properties of injectivity radius function, one can refer to the following
papers: [11, 12, 25].

Let us recall the definition of a removable set. A compact set $K$ in
$\mathbb{C}$ is said to be removable for bounded holomorphic functions, or
simply removable, if for every open superset $U$ of $K$ in $\mathbb{C}$, each
bounded and holomorphic function on $U\setminus K$ extends holomorphically on
the whole of $U$. One can see that a removable set $K$ in $\mathbb{C}$ is
totally disconnected.

######  Remark 1.2.

Theorem 1.1 implies that for the class of domains
$\Omega=\Omega^{\prime}\setminus K$, where $\Omega^{\prime}$ is any simply
connected domain other than $\mathbb{C}$ and $K$ is any removable set in
$\Omega^{\prime}$, the Fridman invariant and the injectivity radius function
coincide with respect to the Carathéodory metric.

######  Remark 1.3.

In Theorem 1.1, we take the domain obtained by deleting a removable set from
the unit disc. It is natural to ask that what happens to the Fridman invariant
and the injectivity radius function if we replace the unit disc by an
arbitrary domain in Theorem 1.1? In Theorem 1.4, we show that one side of the
inequality holds.

######  Theorem 1.4.

Let $\Omega$ be a domain in $\mathbb{C}$ and $K$ be a removable set in
$\Omega$. Then for $D=\Omega\setminus K$, we have

$i_{D}^{c}(z)\leq H_{D}^{c}(z)\leq\min_{w\in K}\tanh[c_{\Omega}(z,w)].$

######  Remark 1.5.

Theorem 1.4 implies that the domain $D=\Omega\setminus K$, where $\Omega$ is
any domain in $\mathbb{C}$ and $K$ is any removable set in $\Omega$, is not
$c$-uniformly thick. Theorem 1.6 is about a domain where the above inequality
does not hold in general.

######  Theorem 1.6.

Let $D=A_{r}\setminus\\{p\\}$ be a domain in $\mathbb{C}$, where
$A_{r}=\\{\xi\in\mathbb{C}:r<|\xi|<1,r>0\\}$ and $p\in(-1,-\sqrt{r})$. Then
for $z\in(r,1)$

$i_{D}^{c}(z)\neq\tanh[c_{A_{r}}(z,p)]\ \mbox{and}\
H_{D}^{c}(z)\neq\tanh[c_{A_{r}}(z,p)].$

Our next result gives the explicit expression of the squeezing function for
the domain given in the Theorem 1.1.

######  Theorem 1.7.

Let $\Omega=\mathbb{D}\setminus K$ be an open connected set. Then the
squeezing function on $\Omega$ is given by

$S_{\Omega}(z)=\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

The $S_{\Omega}(z)$ refer squeezing function which is defined as follows. In
2012, Deng, Guan and Zhang in their paper [2], developed the idea of squeezing
function by following the work of Liu, Sun and Yau [15], [16] and Yeung [26].

Let $\Omega\subseteq\mathbb{C}^{n}$ be a bounded domain. For $z\in{\Omega}$
and a injective holomorphic map $f:\Omega\to\mathbb{B}^{n}$ with $f(z)=0$,
define

$S_{\Omega}(z,f)=\sup\\{r:\mathbb{B}^{n}(0,r)\subseteq f(\Omega)\\},$

where $\mathbb{B}^{n}$ denotes the unit ball in $\mathbb{C}^{n}$ and
$\mathbb{B}^{n}(0,r)$ denotes the ball entered at origin with radius $r$ in
$\mathbb{C}^{n}$. The squeezing function on $\Omega$, denoted by $S_{\Omega}$,
is defined as

$S_{\Omega}(z)=\sup_{f}\\{S_{\Omega}(z,f)\\},$

where supremum is taken over all injective holomorphic maps
$f:\Omega\to\mathbb{B}^{n}$ with $f(z)=0.$

It is clear that the Fridman invariant, the injectivity radius function and
the squeezing function are biholomorphic invariants. It is interesting to note
that any domain in class $\Omega^{\prime}\setminus K$, where $\Omega^{\prime}$
is any simply connected domain other than $\mathbb{C}$ and $K$ is any
removable set in $\Omega^{\prime}$, is not a holomorphic homogeneous regular
domain. A domain is said to be a holomorphic homogeneous regular (HHR) domain
if the squeezing function on it admits a positive lower bound. For other
important properties of squeezing functions, see, for example: [4, 13, 17, 18,
21, 22, 23] and references therein.

######  Remark 1.8.

Now the same question arises for the squeezing function: what about the
squeezing function if we replace an arbitrary domain instead of the unit disc
in Theorem 1.7? In Theorem 1.9, we see that the one-sided inequality still
holds.

######  Theorem 1.9.

Let $\Omega$ be a bounded domain in $\mathbb{C}$ and $K$ be a removable set in
$\Omega$. Then for $D=\Omega\setminus K$, we have

$S_{D}(z)\leq\min_{w\in K}\tanh[c_{\Omega}(z,w)].$

######  Remark 1.10.

Theorem 1.9 implies that the domain $D=\Omega\setminus K$, where $\Omega$ is
any bounded domain in $\mathbb{C}$ and $K$ is any removable set in $\Omega$,
is not a HHR domain. In Theorem 1.11, we give an example of domain to
illustrate that the above inequality does not hold in general.

######  Theorem 1.11.

Let $D=A_{r}\setminus\\{-\sqrt{r}\\}$ be a domain, where
$A_{r}=\\{\xi\in\mathbb{C}:r<|\xi|<1,r>0\\}$. Then for $z\in(r,\sqrt{r})$

$S_{D}(z)\neq\tanh[c_{A_{r}}(z,-\sqrt{r})].$

##  2\. Some notations and Observations

The Carathéodory pseudo distance between $\xi_{1}$ and $\xi_{2}$ on a domain
$\Omega\subseteq\mathbb{C}^{n}$, denoted by $c_{\Omega}(\xi_{1},\xi_{2})$, is
defined as

$c_{\Omega}(\xi_{1},\xi_{2})=\displaystyle\sup_{f}\\{\tanh^{-1}|\alpha|:f:\Omega\to\mathbb{D}\
\text{holomorphic},f(\xi_{1})=0,f(\xi_{2})=\alpha\\}.$

Let $d_{\rho}(\xi_{1},\xi_{2})$ denote the Poincaré distance between $\xi_{1}$
and $\xi_{2}$ in $\mathbb{D}$. Let $\mu:[0,1)\to\mathbb{R}_{\geq 0}$ be a
real-valued function defined as $\mu(x)=\dfrac{1}{2}\log\dfrac{1+x}{1-x}$. It
is clear that $\mu$ is a strictly increasing function, and for
$\xi\in\mathbb{D}$, $\mu(|\xi|)$ is the Poincaré distance between 0 and $\xi$.
One can observe that the inverse of this function can be expressed as
$\tanh(x)$.

In [24], Simha given the explicit formula for Carathéodory distance for the
annulus $D=\\{\xi\in\mathbb{C}:1/R<|\xi|<R\\}$. Let $\xi_{1},\xi_{2}\in D$,
then

(2.1) $c_{D}(\xi_{1},\xi_{2})=\tanh^{-1}c^{\prime}_{D}(\xi_{1},\xi_{2}),$

where

(2.2)
$c^{\prime}_{D}(\xi_{1},\xi_{2})=\dfrac{1}{R|\xi_{2}|}|f(\xi_{1},\xi_{2})|\cdot|f(|\xi_{1}|^{-1},-|\xi_{2}|)|,$

and for $\xi_{1}>0$,

$f(\xi_{1},\xi_{2})=\left(1-\dfrac{\xi_{2}}{\xi_{1}}\right)\dfrac{\displaystyle\prod_{n=1}^{\infty}\left(1-\dfrac{\xi_{2}}{R^{4n}\xi_{1}}\right)\left(1-\dfrac{\xi_{1}}{R^{4n}\xi_{2}}\right)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\dfrac{\xi_{1}\xi_{2}}{R^{4n-2}}\right)\left(1-\dfrac{1}{R^{4n-2}\xi_{1}\xi_{2}}\right)}.$

After some computation, the expression can be reformulated in terms of the
Schottky-Klein prime function $\omega(\xi_{1},\xi_{2})$ for annulus. Let
$\xi_{1}>0,\xi_{2}\in A_{r}=\\{\xi\in\mathbb{C}:r<|\xi|<1,r>0\\}$, then the
Carathéodory distance between $\xi_{1}$ and $\xi_{2}$ on $A_{r}$ is given by

(2.3)
$c_{A_{r}}(\xi_{1},\xi_{2})=\tanh^{-1}c^{\prime}_{A_{r}}(\xi_{1},\xi_{2}),$

where

(2.4)
$c^{\prime}_{A_{r}}(\xi_{1},\xi_{2})=\dfrac{1}{\xi_{1}}\left|\dfrac{\omega(\xi_{2},\xi_{1})}{\xi_{1}\omega(\xi_{2},\xi_{1}^{-1})}\right|\left|\dfrac{\omega\left(\xi_{1},-\dfrac{r}{|\xi_{2}|}\right)}{\dfrac{r}{|\xi_{2}|}\omega\left(\xi_{1},-\dfrac{|\xi_{2}|}{r}\right)}\right|,$

and $\omega(\xi_{1},\xi_{2})$ is expressed as

$\omega(\xi_{1},\xi_{2})=(\xi_{1}-\xi_{2})\displaystyle\prod_{n=1}^{\infty}\dfrac{(\xi_{1}-r^{2n}\xi_{2})(\xi_{2}-r^{2n}\xi_{1})}{(\xi_{1}-r^{2n}\xi_{1})(\xi_{2}-r^{2n}\xi_{2})}\
\mbox{for}\ \xi_{1},\xi_{2}\in\mathbb{C}\setminus\\{0\\}.$

The Schottky-Klein prime function satisfies the following identities see [1,
Section 5.3].

(2.5) $\omega(\xi_{1},\xi_{2})=-\omega(\xi_{2},\xi_{1})$ (2.6)
$\omega(\overline{\xi_{1}},\overline{\xi_{2}})=\overline{\omega(\xi_{1},\xi_{2})}$
(2.7)
$\omega(1/\xi_{1},1/\xi_{2})=-\dfrac{\omega(\xi_{1},\xi_{2})}{\xi_{1}\xi_{2}}$
(2.8)
$\omega(r^{2}\xi_{1},\xi_{2})=-\dfrac{\xi_{2}\omega(\xi_{1},\xi_{2})}{\xi_{1}}$

######  Lemma 2.1.

Fix $z>0\in A_{r}$. Then we have $c_{A_{r}}(z,\overline{w})=c_{A_{r}}(z,w)\
\mbox{for all}\ w\in A_{r}.$

###### Proof.

Since by (2.4)

$\displaystyle
c^{\prime}_{A_{r}}(z,\overline{w})=\dfrac{1}{z}\left|\dfrac{\omega(\overline{w},z)}{z\omega(\overline{w},z^{-1})}\right|\left|\dfrac{\omega\left(z,-\dfrac{r}{|\overline{w}|}\right)}{\dfrac{r}{|\overline{w}|}\omega\left(z,-\dfrac{|\overline{w}|}{r}\right)}\right|=\dfrac{1}{z}\left|\dfrac{\omega(\overline{w},\overline{z})}{z\omega(\overline{w},\overline{z}^{-1})}\right|\left|\dfrac{\omega\left(z,-\dfrac{r}{|w|}\right)}{\dfrac{r}{|w|}\omega\left(z,-\dfrac{|w|}{r}\right)}\right|.$

By (2.6)

$\displaystyle
c^{\prime}_{A_{r}}(z,\overline{w})=\dfrac{1}{z}\left|\dfrac{\overline{\omega(w,z)}}{z\overline{\omega(w,z^{-1})}}\right|\left|\dfrac{\omega\left(z,-\dfrac{r}{|w|}\right)}{\dfrac{r}{|w|}\omega\left(z,-\dfrac{|w|}{r}\right)}\right|=c^{\prime}_{A_{r}}(z,w)$

It follows that $c_{A_{r}}(z,\overline{w})=c_{A_{r}}(z,w)\ \mbox{for all}\
w\in A_{r}.$ ∎

######  Lemma 2.2.

[19, Lemma 1] Let $r<z<1$ and $-1<w<-r$. Then we have the following.

  1. (1)

$c_{A_{r}}(z,w)=c_{A_{r}}\left(z,\dfrac{r}{w}\right)$.

  2. (2)

Fix $z$ and let $c_{A_{r}}(z,w)$ be a function of $w$, where $w$ varies
between $(-1,-r)$. Then the minimum value of $c_{A_{r}}(z,w)$ is attained at
$w=-\sqrt{r}$ i.e.

$c_{A_{r}}(z,-\sqrt{r})<c_{A_{r}}(z,w)\ \mbox{for all}\
w\in(-1,-r)\setminus\\{-\sqrt{r}\\}.$

##  3\. Proof of theorems

###### Proof of Theorem 1.1.

Let us assume $p=\displaystyle\min_{w\in K}c_{\mathbb{D}}(z,w)$. First we
prove that the ball
$B_{\Omega}^{c}(z,p)=\\{\xi\in\Omega:c_{\Omega}(z,\xi)<p\\}$ is simply
connected. Since $K$ is removable set, it is easy to see that
$c_{\Omega}(\xi_{1},\xi_{2})=c_{\mathbb{D}}(\xi_{1},\xi_{2})$ for all
$\xi_{1},\xi_{2}\in\Omega$. This implies
$B_{\Omega}^{c}(z,p)=\\{\xi\in\Omega:c_{\mathbb{D}}(z,\xi)<p\\}$. We claim
that $B_{\Omega}^{c}(z,p)=B_{\mathbb{D}}^{c}(z,p)$. It is obvious that
$B_{\Omega}^{c}(z,p)\subseteq B_{\mathbb{D}}^{c}(z,p)$. For other side
containment, let us assume there is some $\xi^{\prime}\in
B_{\mathbb{D}}^{c}(z,p)$ with $\xi^{\prime}\notin B_{\Omega}^{c}(z,p)$. This
implies $\xi^{\prime}\in K$ such that $c_{\mathbb{D}}(z,\xi^{\prime})<p$. This
contradict the fact that $p=\displaystyle\min_{w\in K}c_{\mathbb{D}}(z,w)$.
Therefore, $B_{\Omega}^{c}(z,p)=B_{\mathbb{D}}^{c}(z,r)$ which is just a
Poincaré ball in the unit disc. Hence $B_{\Omega}^{c}(z,p)$ is simply
connected. Thus

(3.1) $i_{\Omega}^{c}(z)\geq\tanh p=\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

For $z\in\Omega$, let $f:\mathbb{D}\to\Omega$ be any injective holomorphic
function with $B_{\Omega}^{c}(z,r)\subseteq f(\mathbb{D})\subseteq\Omega$.
This implies

(3.2) $B_{\Omega}^{c}(z,r)=\\{\xi\in\Omega:c_{\mathbb{D}}(z,\xi)<r\\}\subseteq
f(\mathbb{D})\subseteq\Omega.$

Since $f(\mathbb{D})$ is simply connected domain in $\Omega$, we have
$B_{\Omega}^{c}(z,r)=B_{\mathbb{D}}^{c}(z,r)$. By (3.2)

(3.3)
$B_{\mathbb{D}}^{c}(z,r)=\\{\xi\in\mathbb{D}:c_{\mathbb{D}}(z,\xi)<r\\}\subseteq
f(\mathbb{D})\subseteq\Omega.$

Consider a conformal map $g:\mathbb{D}\to\mathbb{D}$ such that

$g(\xi)=\dfrac{\xi-z}{1-\overline{z}\xi}.$

Note that $g$ is an isometry, it maps $B_{\mathbb{D}}^{c}(z,r)$ onto the ball
with center zero and radius $r$ in the unit disc. Therefore, by (3.3)

$B_{\mathbb{D}}^{c}(0,r)=\\{\xi\in\mathbb{D}:c_{\mathbb{D}}(0,\xi)<r\\}\subseteq
g(\Omega).$

Thus, we get

(3.4) $H_{\Omega}^{c}(z)\leq\tanh r\leq\displaystyle\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

Hence by [19, Theorem 1(1)], (3.1) and (3.4)

$\min_{w\in K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}\leq
i_{\Omega}^{c}(z)\leq H_{\Omega}^{c}(z)\leq\displaystyle\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

∎

###### Proof of Theorem 1.4.

For $z\in D$, let $H_{D}^{c}(z)=c$. Then by definition there exist a sequence
of positive numbers $\\{r_{n}\\}$ converging to $\tanh^{-1}c$ and a sequence
of injective holomorphic functions $f_{n}:\mathbb{D}\to D$ such that
$B_{D}^{c}(z,r_{n})\subseteq f_{n}(\mathbb{D})\subseteq D$ for all $n$. It is
easy to see that $c_{D}(\xi_{1},\xi_{2})=c_{\Omega}(\xi_{1},\xi_{2})$ for all
$\xi_{1},\xi_{2}\in D$. This implies

(3.5) $B_{D}^{c}(z,r_{n})=\\{\xi\in D:c_{\Omega}(z,\xi)<r_{n}\\}\subseteq
f_{n}(\mathbb{D})\subseteq D\ \mbox{for all}\ n.$

Since $f_{n}(\mathbb{D})$ is simply connected domain in $D$, we have
$B_{D}^{c}(z,r_{n})=B_{\Omega}^{c}(z,r_{n})$ for all $n$. By (3.5)

(3.6)
$B_{\Omega}^{c}(z,r_{n})=\\{\xi\in\Omega:c_{\Omega}(z,\xi)<r_{n}\\}\subseteq
f_{n}(\mathbb{D})\subseteq D\ \mbox{for all}\ n.$

If $c>\displaystyle\min_{w\in K}\tanh[c_{\Omega}(z,w)]$, then there is some
$w^{\prime}\in K$ such that $w^{\prime}\in B_{\Omega}^{c}(z,r_{n})$ for some
large $n$. This is not possible because $B_{\Omega}^{c}(z,r_{n})\subseteq D$
for all $n$. Thus

(3.7) $H_{\Omega}^{c}(z)\leq\min_{w\in K}\tanh[c_{\Omega}(z,w)].$

Hence the result follows from [19, Theorem 1(1)] and (3.7). ∎

###### Proof of Theorem 1.6.

For $z\in(r,1)$, let us assume $i_{D}^{c}(z)=\tanh[c_{A_{r}}(z,p)]$. Then by
definition, there exists a sequence of positive numbers $\\{r_{n}\\}$
converging to $c_{A_{r}}(z,p)$ such that $B_{D}^{c}(z,r_{n})$ are simply
connected for all $n$. By Lemma 2.2, $c_{A_{r}}(z,-\sqrt{r})<c_{A_{r}}(z,\xi)$
for all $\xi\in(-1,-r)\setminus\\{-\sqrt{r}\\}$, this implies that
$-\sqrt{r}\in B_{D}^{c}(z,r_{n})$ for large $n$. Observe that
$B_{D}^{c}(z,r_{n})=B_{A_{r}}^{c}(z,r_{n})$ for large $n$. By [5],
$B_{A_{r}}^{c}(z,r_{n})$ is connected. Hence $B_{A_{r}}^{c}(z,r_{n})$ is path
connected in $\mathbb{C}$. Also, by Lemma 2.1, $B_{A_{r}}^{c}(z,r_{n})$ is
symmetric about $x$\- axis. This implies there exist two paths $\gamma_{1}$
and $\gamma_{2}$ in $B_{A_{r}}^{c}(z,r_{n})$ connecting $z$ and $-\sqrt{r}$,
lying on the upper half plane and lower half plane respectively for large $n$.
Let $\gamma$ denote the closed curve in $B_{A_{r}}^{c}(z,r_{n})$ induced by
$\gamma_{1}$ and $\gamma_{2}$ for large $n$. Then $\gamma$ is a closed curve
in $A_{r}$. Note that $\gamma$ is not null-homotopic in $A_{r}$. This is a
contradiction because $B_{D}^{c}(z,r_{n})=B_{A_{r}}^{c}(z,r_{n})$ is simply
connected.

For $z\in(r,1)$, let us assume $H_{D}^{c}(z)=\tanh[c_{A_{r}}(z,p)]$. Then by
definition, there exist a sequence of positive numbers $\\{r_{n}\\}$
converging to $c_{A_{r}}(z,p)$ and a sequence of injective holomorphic
functions $f_{n}:\mathbb{D}\to D$ such that $B_{D}^{c}(z,r_{n})\subseteq
f_{n}(\mathbb{D})\subseteq D$ for all $n$. Observe that
$B_{D}^{c}(z,r_{n})=B_{A_{r}}^{c}(z,r_{n})$ for large $n$. Similarly as above,
we can find a closed curve $\gamma$ in $B_{A_{r}}^{c}(z,r_{n})$ which is not
null-homotopic in $A_{r}$ for large $n$. This contradict the fact that
$f_{n}(\mathbb{D})$ is simply connected for all $n$.

∎

###### Proof of Theorem 1.7.

For $z\in\Omega$, by using the automorphism of the unit disc, it is easy to
see that

(3.8) $S_{\Omega}(z)\geq\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

By [2, Theorem 2.1], there is an injective holomorphic function
$f:\Omega\to\mathbb{D}$ with $f(z)=0$ such that
$\mathbb{D}(0,S_{\Omega}(z))\subseteq f(\Omega)$. Since $K$ is removable set,
there exists a holomorphic function $\tilde{f}:\mathbb{D}\to\mathbb{C}$ with
$\tilde{f}(\xi)=f(\xi)$ for all $\xi\in\Omega$. Note that $K$ have no
interior. This implies $\tilde{f}:\mathbb{D}\to\overline{\mathbb{D}}$ and by
the maximum modulus principle $\tilde{f}:\mathbb{D}\to\mathbb{D}$. It is easy
to see that $\tilde{f}(w)\notin f(\Omega)$ for all $w\in K$. This implies

(3.9) $S_{\Omega}(z)\leq|\tilde{f}(w)|\ \mbox{for all}\ w\in K.$

By distance decreasing property of Poincaré metric

$\displaystyle
d_{\rho}(0,\tilde{f}(w))=d_{\rho}(\tilde{f}(z),\tilde{f}(w))\leq
d_{\rho}(z,w)=d_{\rho}\left(0,\dfrac{w-z}{1-\overline{z}w}\right).$

Therefore,

$\displaystyle\mu(|\tilde{f}(w)|)$
$\displaystyle\leq\mu\left(\left|\dfrac{w-z}{1-\overline{z}w}\right|\right).$

Thus, we get

(3.10) $|\tilde{f}(w)|\leq\left|\dfrac{w-z}{1-\overline{z}w}\right|\ \mbox{for
all}\ w\in K.$

By (3.9) and (3.10),

(3.11) $S_{\Omega}(z)\leq\min_{w\in
K}\left\\{\left|\dfrac{w-z}{1-\overline{z}w}\right|\right\\}.$

Hence the result follows from (3.8) and (3.11). ∎

###### Proof of Theorem 1.9.

For $z\in\Omega$, by [2, Theorem 2.1], there exists an injective holomorphic
function $f:D\to\mathbb{D}$ with $f(z)=0$ such that
$\mathbb{D}(0,S_{D}(z))\subseteq f(D)$. Similar to the Proof of Theorem 1.7,
we can find a holomorphic function $\tilde{f}:\Omega\to\mathbb{D}$ with
$\tilde{f}(\xi)=f(\xi)$ for all $\xi\in D$ and $\tilde{f}(w)\notin f(D)$ for
all $w\in K$. By decreasing property of Carathéodory metric

(3.12) $c_{\mathbb{D}}(\tilde{f}(z),\tilde{f}(w))\leq c_{\Omega}(z,w)\
\mbox{for all}\ w\in K.$ $\displaystyle
c_{\mathbb{D}}(0,\tilde{f}(w))=\tanh^{-1}|\tilde{f}(w)|\leq c_{\Omega}(z,w)\
\mbox{for all}\ w\in K.$

Therefore,

$\displaystyle S_{\Omega}(z)\leq|\tilde{f}(w)|\leq\tanh[c_{\Omega}(z,w)]\
\mbox{for all}\ w\in K.$

Hence

$\displaystyle S_{D}(z)\leq\min_{w\in K}\tanh[c_{\Omega}(z,w)].$

∎

###### Proof of Theorem 1.11.

For $z\in(r,\sqrt{r})$, suppose that $S_{D}(z)=\tanh[c_{A_{r}}(z,-\sqrt{r})]$.
This implies there exists a injective holomorphic function $f:D\to\mathbb{D}$
with $f(z)=0$ such that $\mathbb{D}(0,\tanh c_{A_{r}}(z,-\sqrt{r}))\subseteq
f(D)$. Therefore, there is an injective holomorphic function
$\tilde{f}:A_{r}\to\mathbb{D}$ with $\tilde{f}(\xi)=f(\xi)$ for all $\xi\in
D$. It follows that

(3.13) $\displaystyle S_{A_{r}}(z)\geq\tanh c_{A_{r}}(z,-\sqrt{r}).$

By (2.3) and (2.4)

(3.14) $\displaystyle\tanh
c_{A_{r}}(z,-\sqrt{r})=\dfrac{1}{z}\left(\dfrac{\omega(z,-r^{1/2})}{\sqrt{r}\omega(z,-r^{-1/2})}\right)^{2}.$

Consider

$f(z)=\dfrac{\omega(z,-r^{1/2})}{\sqrt{r}\omega(z,-r^{-1/2})}$

for $z\in A_{r}$. This is a conformal map from $A_{r}$ onto circularly slit
disc of radius $\sqrt{r}$ with $f(-\sqrt{r})=0$ and
$f(\partial\mathbb{D})=\partial\mathbb{D}$. By (2.6), it is easy to see that
$f(\overline{z})=\overline{f(z)}$. It follows that $f(z)\in(-1,\sqrt{r})$ for
$z\in(-1,-r)$ and $f(z)\in(\sqrt{r},1)$ for $z\in(r,1)$. Therefore, by (3.14)

$\tanh c_{A_{r}}(z,-\sqrt{r})>\dfrac{r}{z}.$

Hence, by (3.13)

$\displaystyle S_{A_{r}}(z)>\dfrac{r}{z},$

which is not possible by the expression of squeezing function on annulus. In
[18], the expression of squeezing function on annulus is given by

$S_{A_{r}}(z)=\max\left\\{|z|,\dfrac{r}{|z|}\right\\}.$

∎

## Acknowledgement

We are thankful to Prof. Kaushal Verma for raising the problems related to the
removable sets.

## References

  * [1] D. Crowdy, _Solving problems in multiply connected domains_ ,SIAM, Philadelphia (2020).
  * [2] F. Deng, Q. Guan, L. Zhang, _Some properties of squeezing functions on bounded domains_ , Pac. J. Math., 257(2) (2012), 319–341.
  * [3] F. Deng, Q. Guan, L. Zhang, _Properties of squeezing functions and global transformations of bounded domains_ , Trans. Am. Math. Soc., 368(4) (2016), 2679–2696.
  * [4] F. Deng, X. Zhang, _Fridman’s invariants, squeezing functions and exhausting domains_ , Acta. Math. Sin., 35(10) (2019), 1723–1728.
  * [5] L. Frerick, G. Schmieder, _Connectedness of the Carathéodory discs for doubly connected domains_ , Ann. Pol. Math., 85 (2005), 281–282.
  * [6] B. L. Fridman, _On the imbedding of a strictly pseudoconvex domain in a polyhedron_ , Dokl. Akad. Nauk SSSR, 249(1) (1979), 63–67.
  * [7] B. L. Fridman, _Biholomorphic invariants of a hyperbolic manifold and some applications_ , Trans. Am. Math. Soc., 276(2) (1983), 685–698.
  * [8] P. Gumenyuk, O. Roth, _On the squeezing function for finitely connected planar domains_ , Math. Ann., 384 (2022), 551–574.
  * [9] N. Gupta, S. K. Pant, _Squeezing function corresponding to polydisk_ , Complex Anal. Synerg., 8(12) (2022), 1–8.
  * [10] N. Gupta, S. K. Pant, _d-balanced squeezing function_ , Complex Var. Elliptic Equ., 68(4) (2023), 619–631.
  * [11] J. M. Hwang, W. K. To, _Injectivity radius and gonality of a compact Riemann surface_ , Am. J. Math., 134(1) (2012), 259–283.
  * [12] L. Keen, N. Lakic, _Hyperbolic Geometry from a Local Viewpoint,_ , vol. 68 Cambridge University Press, Cambridge (2007).
  * [13] K. T. Kim, L. Zhang, _On the uniform squeezing property of bounded convex domains in $\mathbb{C}^{n}$_, Pac. J. Math., 282(2) (2016), 341–358.
  * [14] S. G. Krantz., _Function theory of several complex variables_ , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
  * [15] K. Liu, X. Sun, S. T. Yau, _Canonical metrics on the moduli space of Riemann surfaces I_ , J. Differ. Geom., 68(3) (2004), 571–637.
  * [16] K. Liu, X. Sun, S.T. Yau, _Canonical metrics on the moduli space of Riemann surfaces II_ , J. Differ. Geom., 69(1) (2005), 163–216.
  * [17] P. Mahajan, K. Verma, _A comparison of two biholomorphic invariants_ , Int. J. Math., 30(1) (2019), 195–212.
  * [18] T. W. Ng, C. C. Tang, J. Tsai, _The squeezing function on doubly-connected domains via the Loewner differential equation_ , Math. Ann., 380 (2021), 1741–1766.
  * [19] T.W. Ng, C.C. Tang, J. Tsai, Fridman Function, Injectivity Radius Function and Squeezing Function, J. Geom. Anal. 32(21) (2022), 1–27.
  * [20] N. Nikolov, K. Verma, _On the squeezing function and Fridman invariants_ , J. Geom. Anal., 30 (2019), 1218–1225.
  * [21] F. Rong, S. Yang, _On Fridman invariants and generalized squeezing functions_ , Chin. Ann. Math. Ser. B, 43(2) (2022), 161–174.
  * [22] F. Rong, S. Yang, _On the comparison of the Fridman invariant and the squeezing function_ , Complex Var. Elliptic Equ., 67(4) (2022), 932–937.
  * [23] F. Rong, S. Yang, _On the generalized squeezing functions and Fridman invariants of special domains_ , Ann. Mat. Pura Appl., 201 (2022), 1801–1815.
  * [24] R. R. Simha, _The Carathéodory metric of the annulus_ , Proc. Am. Math, Soc. 50 (1975), 162–166.
  * [25] T. Sugawa, _Various domain constants related to uniform perfectness_ , Complex Variable Theory Appl., 36(4) (1998), 311–345.
  * [26] S. K. Yeung, _Geometry of domains with the uniform squeezing property_ , Adv. Math., 221(2) (2009), 547–569.