# Truncated Wigner approximation for the bosonic model of large spin baths

Mohsen Yarmohammadi<EMAIL_ADDRESS>Department of Physics,
The University of Texas at Dallas, Richardson, Texas 75080, USA  Condensed
Matter Theory, TU Dortmund University, Otto-Hahn Straße 4, 44221 Dortmund,
Germany  Katrin Bolsmann<EMAIL_ADDRESS>Condensed Matter
Theory, TU Dortmund University, Otto-Hahn Straße 4, 44221 Dortmund, Germany
Yvonne Ribbeheger<EMAIL_ADDRESS>Condensed Matter Theory,
TU Dortmund University, Otto-Hahn Straße 4, 44221 Dortmund, Germany  Timo
Gräßer<EMAIL_ADDRESS>Condensed Matter Theory, TU Dortmund
University, Otto-Hahn Straße 4, 44221 Dortmund, Germany  Götz S. Uhrig
<EMAIL_ADDRESS>Condensed Matter Theory, TU Dortmund University,
Otto-Hahn Straße 4, 44221 Dortmund, Germany

###### Abstract

The central spin model has a wide applicability, it is ideally suited to
describe a small quantum system, for instance a quantum bit, in contact to a
bath of spins, e.g., nuclear spins, or other small quantum systems in general.
According to previous work [Röhrig et al., Phys. Rev. B 97, 165431 (2018)], a
large bath of quantum spins can be described as a bath of quantum harmonic
oscillators. But the resulting quantum model is still far from being
straightforward solvable. Hence we consider a chain representation for the
bosonic degrees of freedom to study how well a truncated Wigner approximation
of the effective model of harmonic oscillators works in comparison with other
approximate and exact methods. Numerically, we examine the effect of the
number of bath spins and of the truncation level, i.e., the chain length.

##  I Introduction

The central spin model (CSM) is a well-known model describing the interaction
of a single “central” spin with surrounding spins Gaudin (1976, 1983), for
instance, the interaction of the spin of a localized electron with nuclear
spins in quantum dots Loss and DiVincenzo (1998); Koppens  _et al._ (2008);
Coish and Loss (2004). In view of the intense search for physical realizations
of quantum bits Nielsen and Chuang (2010), a localized electron in a quantum
dot can be seen as a two-level system and thus as a promising candidate for
quantum bits Greilich  _et al._ (2009); Economou  _et al._ (2006); Carter  _et
al._ (2009). The CSM is a quantum many-body system and major progress has been
made to understand its properties in its applications for phenomena in
material science and quantum information technology Schachenmayer  _et al._
(2015); Smith  _et al._ (2019); Liu  _et al._ (2010); Witzel and Das Sarma
(2006). Polarization recovery in a longitudinal field Braun  _et al._ (2005);
Smirnov  _et al._ (2020), nuclei-induced frequency focusing Schering  _et al._
(2018); Greilich  _et al._ (2007); Kleinjohann  _et al._ (2018), spin
precession mode locking Schering  _et al._ (2020); Yugova  _et al._ (2012),
the effect of spin inertia Smirnov  _et al._ (2018); Schering  _et al._
(2019), spin noise Glazov and Ivchenko (2012); Jäschke  _et al._ (2018);
Fröhling  _et al._ (2018); Glasenapp  _et al._ (2016); Hackmann and Anders
(2014), and many other effects belong to the particularly rich physics of the
CSM. Furthermore, the CSM is also used to understand the dynamics of quantum
sensors Wojciechowski  _et al._ (2018) which helps to reach high
sensitivities.

For a finite, not too large number of bath spins Coish and Baugh (2009), it is
possible to use the Bethe ansatz Gaudin (1976); Faribault and Schuricht
(2013a, b) to diagonalize the CSM Hamiltonian and to analyze rigorous
restrictions of the central spin dynamics stemming from conserved quantities
Uhrig  _et al._ (2014); Seifert  _et al._ (2016). If all couplings are equal
the CSM reduces to the so called box model allowing one to compute the spin
dynamics for large spin baths essentially analytically Melikidze  _et al._
(2004); Kozlov (2007); Bortz and Stolze (2007). However, the complexity of the
CSM in practical applications is related mainly to the electron spin
decoherence when interacting with an (almost) infinite number of nuclei spins
Yang  _et al._ (2016); Merkulov  _et al._ (2002); Khaetskii  _et al._ (2002);
Coish and Loss (2004); Yao  _et al._ (2006); Barnes  _et al._ (2012); Witzel
and Das Sarma (2007). In this scenario, the initial polarization and
information on the spin state is quickly and irreversibly lost.

To describe this decoherence of the central spin and to conceive strategies
against it, various approaches have been conceived. Density-matrix
renormalization group (DMRG) deals with up to 1000 spins, but only up to
relatively short times Stanek  _et al._ (2013, 2014) due to the fast growth of
entanglement. The linked-cluster and cluster-correlation expansions Scarola
_et al._ (2004); Saikin  _et al._ (2007); Yang and Liu (2008); Lindoy and
Manolopoulos (2018) investigate the long-time spin decoherence, but of finite,
relatively small spin baths. Moreover, considering the nuclear-electric
quadrupolar interactions for a few spins, the spin-noise spectrum at various
timescales has been calculated using Chebyshev polynomials Hackmann  _et al._
(2015); Hackmann and Anders (2014); Smirnov  _et al._ (2021). Furthermore, a
coherent interface between electron and nuclear spins was recently developed
Gangloff  _et al._ (2019) with the vision to realize long-lived quantum
memory.

Although a classical description of CSM with a large-enough number of nuclear
spins can be justifed over a long time, it neglects all quantum mechanical
aspects Stanek  _et al._ (2013, 2014) which are vital for quantum bits. This
originates from the fact that the central spin is a truly quantum mechanical
object and its back-action on the bath spins is not classical. The truncated
Wigner approximation (TWA) Polkovnikov (2010) is a general semi-classical
approach in which quantum fluctuations are partly taken into account through
random initial conditions for the classical equations of motion. Although the
equations of motion themselves are still purely classical, correlations and
the probabilities of quantum measurements can be simulated to a certain
degree. The TWA has often been used to simulate the dynamics of the CSM
Davidson and Polkovnikov (2015); Schachenmayer  _et al._ (2015); Fauseweh  _et
al._ (2017). The spins are taken as classical vectors precessing around local
classical fields. We abbreviate this semi-classical approach to spins sTWA. It
can be implemented for moderate numbers of spins ($N\approx 200$) if one has
to simulate long times. Experimentally, the bath sizes range from $10^{4}$ to
$10^{6}$ still exceeding numerical resources by far even though a hierarchical
chain representation based on generalized Overhauser fields helps to reconcile
large spin baths and long-time simulations Fauseweh  _et al._ (2017).

In this framework, a fully quantum mechanical approach Röhrig  _et al._ (2018)
based on iterated equations of motion (iEoM) has been suggested for large spin
baths. The asset of this approach is that it is particularly suited to capture
very large or even infinitely large spin baths. The bath of spins is mapped to
a bath of hierarchically coupled bosons and the central spin is mapped to a
four-dimensional impurity. But the fully quantum mechanical evaluation of the
dynamics of the effective bosonic model for long times represents still a
tremendous challenge. Hence, it is interesting to study approximate ways to
treat this effective bosonic model.

In this work, we study the application of the TWA to the mapped effective
bosonic model resulting from iEoM Röhrig  _et al._ (2018), i.e., to the
harmonic oscillators. The impurity is described by two spins with $S=1/2$
which, in turn, are treated as classical vectors. In order to distinguish this
TWA from the one resulting from the classical treatment of the spins we call
it bosonic TWA (bTWA). Clearly, the bTWA would remove the restrictions on the
maximum number of bosonic modes which can be simulated. The immediate aim is
to describe the experimental spin noise spectra Glasenapp  _et al._ (2016);
Zapasskii  _et al._ (2013); Crooker  _et al._ (2010). To benchmark the bTWA,
we compare our data to data from some of the above-mentioned techniques under
the same conditions.

This paper is organized as follows. In Sec. II, we review the CSM and in Sec.
III, we present its bosonic formulation. In Sec. IV, we present our results
and compare them with results from other techniques. Finally, the paper is
summarized in Sec. V.

##  II Initial Model

In this section, we briefly introduce the CSM. For our proof-of-principle
study, we restrict ourselves to the paradigmatic isotropic version of the CSM.
This implies that we neglect dipole-dipole interaction Merkulov  _et al._
(2002); Schliemann  _et al._ (2003), quadrupolar couplings Bulutay (2012);
Bulutay  _et al._ (2014); Chekhovich  _et al._ (2012); Sinitsyn  _et al._
(2012); Hackmann  _et al._ (2015), and spin-orbit couplings Nowack  _et al._
(2007); Rančić and Burkard (2014); Khaetskii and Nazarov (2000, 2001);
Golovach  _et al._ (2004) of the nuclear spins which usually become relevant
on very long timescales. We start with the CSM comprising a central spin
$\hat{\vec{S}}_{0}$ with $S=1/2$ interacting through the hyperfine coupling
with a bath of $N$ spins $\hat{\vec{S}}_{i}$. The Hamiltonian reads

$\hat{\mathcal{H}}={}\sum^{N}_{i=1}\,J_{i}\,\hat{\vec{S}}_{0}\cdot\hat{\vec{S}}_{i}\,,$
(1)

where $J_{i}$ denotes the hyperfine coupling of the $i$-th spin in the bath.
In electronic quantum dots, the coupling constants $J_{i}$ are proportional to
the probability that the electron is present at the site of the nucleus $i$
Merkulov  _et al._ (2002); Schliemann  _et al._ (2003) which is given by the
modulus squared of the electronic wave function. It is convenient to define a
composite field for the effect of the bath spins,
$\hat{\vec{B}}={}\sum^{N}_{i=1}\,J_{i}\,\hat{\vec{S}}_{i}$, which is called
the Overhauser field. With its help, the Hamiltonian can simply be rewritten
as $\hat{\mathcal{H}}=\hat{\vec{S}}_{0}\cdot\hat{\vec{B}}$.

Let us consider an infinite spin bath ($N\to\infty$) with decreasing
couplings. We consider the generic parametrization $J_{i}=C\exp(-i\gamma)$
Faribault and Schuricht (2013a, b, c); Seifert  _et al._ (2016); Schering  _et
al._ (2018); Fauseweh  _et al._ (2017) with $i\in[1,N]$, where the prefactor
$C$ sets the energy scale. For $\gamma>0$, the exponential term is decreasing
with $i$. The meaning of $\gamma$ is elucidated by the following argument.
Even if $N\to\infty$, there is only a finite number of bath spins which is
appreciably coupled to the central spin. We denote this number by
$N_{\text{eff}}$ and define it by the ratio of the squared sum of all
couplings and the sum of all squared couplings Merkulov  _et al._ (2002);
Schliemann  _et al._ (2003); Stanek  _et al._ (2013, 2014); Gravert  _et al._
(2016); Fauseweh  _et al._ (2017), i.e., $N_{\rm
eff}:={}\Big{(}\sum^{N}_{i=1}J_{i}\Big{)}^{2}/J^{2}_{\mathrm{Q}}$, where
$J^{2}_{\mathrm{Q}}:=\sum^{N}_{i=1}J^{2}_{i}$. Inserting our parametrization
$J_{i}$ into $N_{\rm eff}$ in the limit $N\to\infty$, we find for small values
$\gamma$

$\displaystyle N_{\text{eff}}=\frac{2}{\gamma}+\mathcal{O}(\gamma)\,.$ (2)

So $\gamma$ is about twice the inverse number of effectively coupled spins.
The electron spin in quantum dots is coupled to a very large number of bath
spins Merkulov  _et al._ (2002); Schliemann  _et al._ (2003); Lee  _et al._
(2005); Petrov  _et al._ (2008), $N_{\text{eff}}\approx 10^{4}$ to $10^{6}$,
so, $\gamma\approx 10^{-4}$ to $10^{-6}$ is a realistic estimate. Moreover, we
set the energy scale for all simulations by requiring $J_{\mathrm{Q}}=1$. This
results in $C\simeq\sqrt{2\gamma}\approx 10^{-2}$ to $10^{-3}$, which is a
very small number implying that the contribution of an individual bath spin is
negligible. Only suitable sums over all spins have a sizable impact. In
contrast, for large $\gamma$, we deal with a small number of bath spins, see
Eq. (2), and the dynamics of the central spin can be determined using fully
quantum mechanical descriptions Hackmann  _et al._ (2015); Hackmann and Anders
(2014); Lindoy and Manolopoulos (2018).

##  III Effective model and semi-classical approach

In what follows, we sketch the mapping of the spin bath on a bosonic bath
(iEoM Röhrig  _et al._ (2018)). Then, we introduce the semi-classical approach
bTWA based on a hierarchical chain representation to describe the long-time
spin dynamics.

###  III.1 Objective

We begin with the application of the Heisenberg equation of motion to the CSM,
$\partial_{t}\hat{\mathcal{A}}={\tt i}[\hat{\mathcal{H}},\hat{\mathcal{A}}]$
(throughout the present work, $\hbar$ is set to unity), where
$\hat{\mathcal{A}}$ are operators of the CSM forming a suitable operator basis
for the products of all components of spin operators at all sites Röhrig  _et
al._ (2018). In the end, we are interested in the $\alpha$ component of the
spin-spin autocorrelation function of the central spin at infinite temperature

$S^{\alpha}(t)={}\langle\hat{S}^{\alpha}_{0}(t)\hat{S}^{\alpha}_{0}(0)\rangle\,,$
(3)

for small values of the parameter $\gamma$ corresponding to large spin baths.
In particular, the long-term behavior of $S^{z}(t)$ provides information about
the fate of state with the central spin aligned along the $z$-axis initially,
i.e., at $t=0$. Assuming infinite temperature is well justified because the
thermal energy in the bath at temperatures of a few Kelvin is at least one
order of magnitude larger than the individual couplings in a quantum dot
Urbaszek  _et al._ (2013).

For motivation, we provide the autocorrelation if a constant external or
internal magnetic field is applied Merkulov  _et al._ (2002); Stanek  _et al._
(2013)

$\displaystyle\hat{\vec{S}}_{0}(t)={}$
$\displaystyle{\vec{n}}\big{[}{\vec{n}}\cdot\hat{\vec{S}}_{0}(0)\big{]}+\big{\\{}\hat{\vec{S}}_{0}(0)-{\vec{n}}\big{[}{\vec{n}}\cdot\hat{\vec{S}}_{0}(0)\big{]}\big{\\}}\cos(Bt)$
(4)
$\displaystyle-\big{[}{\vec{n}}\times\hat{\vec{S}}_{0}(0)\big{]}\sin(Bt)\,,$

where ${\vec{n}}$ points in the direction of the magnetic field. This formula
is identical to the classical one since ${\vec{B}}$ is a classical vector and
the equations of motion are linear in the spin operators. Obviously, powers of
$B$ up to infinite order occur so that a suitable operator basis needs
operators including high powers of the Overhauser field if we want to capture
its intrinsic quantum character and the ensuing dynamics.

Figure 1: Sketch of the CSM described by Eqs. (9a) and (9b). The central spin
and the bosons in the chain are shown by the black and light gray solid
spheres, respectively. The solid two-sided arrows inside the boxes illustrate
the couplings $\chi_{j}/2$ between bosons of different flavors at the same
site of the chain, while the dotted ones indicate the couplings $\eta_{j}/2$
between bosons on adjacent sites.

If one neglects the dynamics of the Overhauser field completely, the frozen
Overhauser field approximation is retrieved for which one averages over all
random directions and random strengths of the Overhauser field Merkulov  _et
al._ (2002); Stanek  _et al._ (2013) yielding

$S^{\alpha}(t)=\frac{1}{12}\left[2e^{-J_{\mathrm{Q}}^{2}t^{2}/8}(1-J_{\mathrm{Q}}^{2}t^{2}/4)+1\right]\,.$
(5)

This analytic result is convenient as reference, see the figures below.

###  III.2 Effective model with higher powers of the Overhauser field

The orthogonal Hermite polynomials of the Overhauser field and similar
composite weighted sums of the bath spin have been introduced by Röhrig et al.
Röhrig  _et al._ (2018) as suitable operator basis. These polynomials are
orthogonal for a Gaussian weight function Abramowitz and Stegun (1965), and
can be applied to different components of generalized Overhauser fields by the
recursive relation

$\displaystyle
G^{\alpha}_{j}H_{n}(G^{\alpha}_{j})=\sqrt{n}H_{n-1}(G^{\alpha}_{j})+\sqrt{n+1}H_{n+1}(G^{\alpha}_{j}),$
(6)

where $\alpha=\\{x,y,z\\}$ and $H_{0}(G^{\alpha}_{j})=1$ by definition. The
polynomials $H_{n}(G^{\alpha}_{j})$ are the Hermite polynomials of degree $n$
in the generalized Overhauser field vectors $\vec{G}_{j}$. These fields are
defined by

$G^{\alpha}_{j}:=2\sum_{i=1}^{N}\mathcal{P}_{j}(J_{i})\hat{S}^{\alpha}_{i}\,,$
(7)

where the real orthogonal polynomials $\mathcal{P}_{j}(x)$ are defined such
that they comply with the orthogonality relation Röhrig  _et al._ (2018);
Fauseweh  _et al._ (2017)

$\delta_{j,m}=\sum_{i=1}^{N}\mathcal{P}_{j}(J_{i})\mathcal{P}_{m}(J_{i})\,.$
(8)

The polynomials $\mathcal{P}_{j}(J_{i})$ describe the weight of each bath spin
$\vec{S}_{i}$. The established EoM for this basis of operators tells us that a
single $H_{n}(G^{\alpha}_{j})$ is transformed into the terms
$\sqrt{n}H_{n-1}(G^{\alpha}_{j})$ and $\sqrt{n+1}H_{n+1}(G^{\alpha}_{j})$.
This is identical to the effect of an annihilation ($\hat{a}$) and creation
($\hat{a}^{\dagger}$) bosonic operator, respectively, applied to the
eigenstates $\ket{n}$ of an harmonic oscillator.

Eventually, a quantum mechanical representation of large spin baths by means
of the iEoM for the generalized Overhauser fields including an external
magnetic field has been obtained and developed, see Ref. Röhrig  _et al._ ,
2018 for further details. It is shown that in the limit $N\to\infty$ the
isotropic CSM can be mapped onto a four-dimensional impurity coupled to a non-
interacting bosonic bath yielding the effective Hamiltonian
$\hat{\mathcal{H}}_{\text{eff}}=\hat{\mathcal{H}}^{\text{CS}}_{\text{eff}}+\hat{\mathcal{H}}^{\text{ch}}_{\text{eff}}+\hat{\mathcal{H}}^{\rm
Z}_{\rm eff}$ in the presence of an external Zeeman magnetic field $h$ along
the $z$-direction. It is given by

$\displaystyle\hat{\mathcal{H}}^{\text{CS}}_{\text{eff}}={}$
$\displaystyle\frac{1}{2}\sum_{\alpha=1}^{3}\hat{K}_{\alpha}\big{(}\hat{a}^{\dagger}_{1,\alpha}+\hat{a}_{1,\alpha}\big{)}\,,$
(9a) $\displaystyle\hat{\mathcal{H}}^{\text{ch}}_{\text{eff}}={}$
$\displaystyle\frac{\texttt{i}}{2}\sum_{j=1}^{N_{\text{tr}}}\sum_{\alpha,\beta,\delta=1}^{3}\epsilon_{\alpha\beta\delta}\hat{M}_{\beta}\big{[}\chi_{j}\hat{a}^{\dagger}_{j,\delta}\hat{a}_{j,\alpha}$
$\displaystyle\qquad\qquad+\eta_{j}(\hat{a}^{\dagger}_{j+1,\delta}\hat{a}_{j,\alpha}-\hat{a}^{\dagger}_{j,\alpha}\hat{a}_{j+1,\delta})\big{]}\,,$
(9b) $\displaystyle\hat{\mathcal{H}}^{\rm Z}_{\rm eff}={}$
$\displaystyle-h\hat{K}_{z}\,,$ (9c)

where $\hat{\mathcal{H}}^{\text{CS}}_{\text{eff}}$ refers to the central spin
located at the head of a bosonic chain, whereas
$\hat{\mathcal{H}}^{\text{ch}}_{\text{eff}}$ acts on a bosonic chain with
flavors $\alpha$ as depicted in Fig. 1. In the above equations,
$\epsilon_{\alpha\beta\delta}$ is the Levi-Civita tensor. The couplings
$\eta_{j}$ and $\chi_{j}$ result from the recursion of the orthogonal
polynomials $\mathcal{P}_{j}$ which can be expressed in the matrix form

$\hat{\mathcal{T}}=\begin{pmatrix}\chi_{1}&\eta_{1}&0&0&\cdots\\\
\eta_{1}&\chi_{2}&\eta_{2}&0&\cdots\\\ 0&\eta_{2}&\chi_{3}&\eta_{3}&\cdots\\\
\vdots&\vdots&\ddots&\ddots&\ddots\end{pmatrix}\,,$ (10)

with
$J_{i}\underline{\mathcal{P}}_{j}(J_{i})=\hat{\mathcal{T}}\underline{\mathcal{P}}_{j}(J_{i})$
using the vector of polynomials
$\underline{\mathcal{P}}_{j}(J_{i})=\left[\mathcal{P}_{1}(J_{i}),\mathcal{P}_{2}(J_{i}),\cdots\mathcal{P}_{n}(J_{i})\right]^{\top}$.
By definition, we have $\eta_{0}=0$. While the chain is half-infinite for an
infinite bath, it is truncated at $j_{\rm max}$ in practical calculations
Fauseweh  _et al._ (2017); Röhrig  _et al._ (2018) so that we also have
$\eta_{j_{\text{max}}}=0$. (In Ref. Fauseweh  _et al._ , 2017, the truncation
level was denoted by $N_{\rm tr}=j_{\text{max}}$.)

The commutation and anticommutation of the operators of the central spin with
$\hat{\sigma}_{\alpha}$ (Pauli matrices) in the chain are expressed by the
matrices $\hat{K}_{\alpha}$ and $\hat{M}_{\alpha}$, respectively, with matrix
elements $\langle\langle
n|\hat{K}_{\alpha}|m\rangle\rangle={}\frac{1}{2}\langle\langle\hat{\sigma}_{n}|[\hat{\sigma}_{\alpha},\hat{\sigma}_{m}]\rangle\rangle$
and $\langle\langle
n|\hat{M}_{\alpha}|m\rangle\rangle={}\frac{1}{2}\langle\langle\hat{\sigma}_{n}|\\{\hat{\sigma}_{\alpha},\hat{\sigma}_{m}\\}\rangle\rangle$
for $\\{m,n\\}\in\\{x,y,z\\}$. The notation
$\langle\langle\dots\rangle\rangle$ is used for the scalar product of
operators for which we use
$\langle\langle\hat{A}|\hat{B}\rangle\rangle:=\langle\hat{A}^{\dagger}\hat{B}\rangle_{T=\infty}$,
i.e., the expectation value at infinite temperature. Straightforwardly, we
find

$\displaystyle\hat{K}_{\alpha}={}$ $\displaystyle{\tt
i}\begin{pmatrix}0&0&0&0\\\ 0&0&\delta_{\alpha,z}&-\delta_{\alpha,y}\\\
0&-\delta_{\alpha,z}&0&\delta_{\alpha,x}\\\
0&\delta_{\alpha,y}&-\delta_{\alpha,x}&0\end{pmatrix}\,,$ (11a)
$\displaystyle\hat{M}_{\alpha}={}$
$\displaystyle\begin{pmatrix}0&\delta_{\alpha,x}&\delta_{\alpha,y}&\delta_{\alpha,z}\\\
\delta_{\alpha,x}&0&0&0\\\ \delta_{\alpha,y}&0&0&0\\\
\delta_{\alpha,z}&0&0&0\end{pmatrix}\,.$ (11b)

We emphasize that the chain Hamiltonian
$\hat{\mathcal{H}}^{\text{ch}}_{\text{eff}}$ induces only a slow dynamics
because the coupling between the head of the chain and its next site is
$J_{\mathrm{Q}}=1$, while the coupling between different chain sites as well
as the hopping processes between different flavors at each site is of order
$\sqrt{\gamma}J_{\mathrm{Q}}\thickapprox 10^{-2}$ to $10^{-3}$. Therefore, the
quantum effects such as the dynamics in the bath and eventually dephasing and
relaxation of the polarization of the central spin due to the presence of the
bath of spins is slow.

Finally, we state that the autocorrelation expressed by the derived effective
model reads

$S^{\alpha}(t)=\frac{1}{4}\langle e_{\alpha},\textbf{0}|e^{-{\tt
i}\hat{\mathcal{H}}_{\rm eff}}|e_{\alpha},\textbf{0}\rangle\,,$ (12)

with
$e_{\alpha}=(0,\delta_{\alpha,x},\delta_{\alpha,y},\delta_{\alpha,z})^{\top}$
and 0 being the vacuum of all bosons. The autocorrelation (12) can be
reformulated with the help of the matrix $\hat{M}_{\alpha}$ and
$e_{1}=(1,0,0,0)^{\top}$

$\displaystyle S^{\alpha}(t)$ $\displaystyle=\frac{1}{4}\langle
e_{1},\textbf{0}|\hat{M}_{\alpha}e^{-{\tt i}\hat{\mathcal{H}}_{\rm
eff}}\hat{M}_{\alpha}|e_{1},\textbf{0}\rangle\,,$ (13a)
$\displaystyle=\frac{1}{4}\langle e_{1},\textbf{0}|e^{{\tt
i}\hat{\mathcal{H}}_{\rm eff}}\hat{M}_{\alpha}e^{-{\tt
i}\hat{\mathcal{H}}_{\rm eff}}\hat{M}_{\alpha}|e_{1},\textbf{0}\rangle\,,$
(13b) $\displaystyle=\frac{1}{4}\langle
e_{1},\textbf{0}|\hat{M}_{\alpha}(t)\hat{M}_{\alpha}(0)|e_{1},\textbf{0}\rangle\,,$
(13c)

where we used the fact that $\hat{\mathcal{H}}_{\rm
eff}|e_{1},\textbf{0}\rangle=0$ since $\hat{K}_{\alpha}e_{1}=0$ and all
bosonic terms in the chain part annihilate the bosonic vacua.

###  III.3 The bosonic truncated Wigner approximation

In order to apply a TWA to the effective model defined in the previous section
we need to represent the four-dimensional impurity by objects which have
classical counterparts. Here we choose two spins with $S=1/2$ which together
span a four dimensional Hilbert space. We denote their singlet state by
$\ket{s}$ and their three triplet states by $\ket{t_{\alpha}}$ for
$\alpha\in\\{x,y,z\\}$, identified with the four-dimensional Cartesian vectors
$\ket{s}=\begin{pmatrix}1&0&0&0\end{pmatrix}^{\top}$ and
$\ket{t_{\alpha}}=\begin{pmatrix}0&\delta_{\alpha x}&\delta_{\alpha
y}&\delta_{\alpha z}\end{pmatrix}^{\top}$. Elementary linear algebra Sachdev
and Bhatt (1990) yields the action of the spin operators on these states

$\displaystyle\hat{S}_{\nu,\alpha}\ket{s}={}$
$\displaystyle-\frac{(-1)^{\nu}}{2}\sum_{\beta}\delta_{\alpha\beta}\ket{t_{\beta}}\,,$
(14a) $\displaystyle\hat{S}_{\nu,\alpha}\ket{t_{\beta}}={}$
$\displaystyle-\frac{1}{2}\big{[}2(-1)^{\nu}\delta_{\alpha\beta}\ket{s}-{\tt
i}\sum_{\delta}\epsilon_{\alpha\beta\delta}\ket{t_{\delta}}\big{]}\,,$ (14b)

where $\nu=\\{1,2\\}$ labels the spin $\hat{S}_{1}$ and $\hat{S}_{2}$,
respectively. With these definitions, the matrices $\hat{K}$ and $\hat{M}$ in
Eqs. (11a) and (11b) can be expressed in terms of these spin operators

$\displaystyle\hat{K}_{\alpha}={}$
$\displaystyle-(\hat{S}_{1,\alpha}+\hat{S}_{2,\alpha})\,,$ (15a)
$\displaystyle\hat{M}_{\alpha}={}$
$\displaystyle\hat{S}_{1,\alpha}-\hat{S}_{2,\alpha}\,.$ (15b)

The annihilation and creation operators of the harmonic oscillators can be
expressed by position and momentum operators in the standard way

$\displaystyle\hat{r}_{j,\alpha}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{j,\alpha}+\hat{a}_{j,\alpha})\,,$
(16a) $\displaystyle\hat{p}_{j,\alpha}=\frac{{\tt
i}}{\sqrt{2}}(\hat{a}^{\dagger}_{j,\alpha}-\hat{a}_{j,\alpha})\,.$ (16b)

With these relations, the Hamiltonian in Eq. (9) can be rewritten into

$\displaystyle\hat{\mathcal{H}}^{\text{CS}}_{\text{eff}}=$
$\displaystyle{}-\frac{1}{\sqrt{2}}(\hat{\vec{S}}_{1}+\hat{\vec{S}}_{2})\cdot\hat{\vec{r}}_{1}\,,$
(17a) $\displaystyle\hat{\mathcal{H}}^{\text{ch}}_{\text{eff}}=$
$\displaystyle\frac{1}{2}\sum^{N\rm
tr}_{j=1}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\cdot[(\chi_{j}\hat{\vec{r}}_{j}+\eta_{j-1}\hat{\vec{r}}_{j-1}+\eta_{j}\hat{\vec{r}}_{j+1})\times\hat{\vec{p}}_{j}],$
(17b) $\displaystyle\hat{\mathcal{H}}^{\text{Z}}_{\text{eff}}=$
$\displaystyle{}h(\hat{\vec{S}}_{1,z}+\hat{\vec{S}}_{2,z})\,.$ (17c)

The ensuing time evolution of the operators $\hat{\vec{r}}$, $\hat{\vec{p}}$,
$\hat{\vec{S}}_{1}$, and $\hat{\vec{S}}_{2}$ according to the Heisenberg
equation of motion reads

$\displaystyle\frac{\rm d}{{\rm d}t}\hat{\vec{r}}_{1}={}$
$\displaystyle\frac{\chi_{1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{r}}_{1}+\frac{\eta_{1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{r}}_{2}\,,$
(18a) $\displaystyle\frac{\rm d}{{\rm d}t}\hat{\vec{p}}_{1}={}$
$\displaystyle\frac{\chi_{1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{p}}_{1}+\frac{\eta_{1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{p}}_{2}$
$\displaystyle+\frac{1}{\sqrt{2}}(\hat{\vec{S}}_{2}+\hat{\vec{S}}_{1})\,,$
(18b)

for $j=1$ while for general $j>1$ we obtain

$\displaystyle\frac{\rm d}{{\rm d}t}\hat{\vec{r}}_{j}={}$
$\displaystyle\frac{\chi_{j}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{r}}_{j}+\frac{\eta_{j}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{r}}_{j+1}$
$\displaystyle+\frac{\eta_{j-1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{r}}_{j-1}\,,$
(19a) $\displaystyle\frac{\rm d}{{\rm d}t}\hat{\vec{p}}_{j}={}$
$\displaystyle\frac{\chi_{j}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{p}}_{j}+\frac{\eta_{j}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{p}}_{j+1}$
$\displaystyle+\frac{\eta_{j-1}}{2}(\hat{\vec{S}}_{2}-\hat{\vec{S}}_{1})\times\hat{\vec{p}}_{j-1}\,,$
(19b) $\displaystyle\frac{\rm d}{{\rm d}t}\hat{\vec{S}}_{\nu}={}$
$\displaystyle\frac{1}{\sqrt{2}}\hat{\vec{S}}_{\nu}\times\hat{\vec{r}}_{1}+\frac{3-2\nu}{2}\hat{\vec{S}}_{\nu}\times\sum^{N_{\rm
tr}}_{j=1}\big{[}\chi_{j}(\hat{\vec{r}}_{j}\times\hat{\vec{p}}_{j})$
$\displaystyle+\eta_{j}(\hat{\vec{r}}_{j+1}\times\hat{\vec{p}}_{j})+\eta_{j-1}(\hat{\vec{r}}_{j-1}\times\hat{\vec{p}}_{j})\big{]}-h\hat{\vec{S}}_{\nu}\times{\vec{z}}\,,$
(19c)

where we use ${\vec{z}}=\begin{pmatrix}0&0&1\end{pmatrix}^{\top}$ in the last
term of Eq. (19c). The sought autocorrelation (3) has been expressed for the
effective model in (13c) which implies

$S^{\alpha}(t)={}\frac{1}{4}\langle(\hat{S}^{\alpha}_{1}(t)-\hat{S}^{\alpha}_{2}(t))(\hat{S}^{\alpha}_{1}(0)-\hat{S}^{\alpha}_{2}(0))\rangle\,,$
(20)

where the expectation value is taken with respect to the singlet state of spin
1 and 2 and the bosonic vacua.

Applying the standard TWA Polkovnikov (2010), the leading quantum corrections
are recovered by averaging classical trajectories over distributions of
initial conditions. The equations of motions (18) and (19) are viewed as
differential equations for classical vectors starting from random initial
conditions. For this purpose, normal distributions have turned out to be
particularly suitable for the initial conditions. Their asset is that only the
mean value and the variance are needed to determine the distribution fully. We
choose a normal distribution for spin $\vec{S}_{1}$ with vanishing mean value
and variance $1/4$ for each component because $(\hat{S}^{\alpha})^{2}=1/4$ for
$S=1/2$ Davidson and Polkovnikov (2015). Since we mimic a singlet state
$\vec{S}_{2}$ is always chosen to be $-\vec{S}_{1}$ initially.

The position and momentum components are also drawn from a normal distribution
with vanishing means. The variances are straightforwardly computed considering
(16) yielding
$\langle\hat{r}_{j,\alpha}^{2}\rangle=1/2=\langle\hat{p}_{j,\alpha}^{2}\rangle$.
In practice, the time-evolution of the central spin $S^{\alpha}(t)$ in Eq.
(20) is calculated for configuration average over $\mathcal{M}$ classical
trajectories with $\mathcal{M}$ being of the order of $10^{6}-10^{7}$ to keep
statistical errors low.

##  IV Numerical results

Here we show results of the two TWAs which are the protagonists of this study.
The sTWA relies on the classical equations of motion for the spin operators of
original CSM. Either each spin is tracked individually or a hierarchical chain
representation is used. This does not make any discernible difference. In
contrast, the bTWA solves the classical equations of motion for the effective
model obtained by mapping the large spin bath to a bath of bosons.

Since $J_{\mathrm{Q}}$ is the energy unit in the numerical calculations, all
times are measured in units of $1/J_{\mathrm{Q}}$ having set $\hbar$ to unity.
The equations of motion have no lower or upper validity cutoff in time and,
thus, can be applied to discuss the spin-spin correlation from $t=0$ to
$t\to\infty$. The effective number of coupled spins $N_{\rm eff}$ can also be
chosen arbitrarily, but we keep in mind that the mapping to the effective
model becomes exact in the limit of large spin baths. Further details of the
effect of $N_{\rm eff}=2/\gamma$ in the bTWA are provided in App. A.

Figure 2 shows the autocorrelation of the central spin in absence of external
magnetic fields. This is the central result of this paper. Clearly, we see
that both approaches, sTWA and bTWA, are converged with respect to the
truncation level $j_{\rm max}$ (for further details of the effect of $j_{\rm
max}$ in the bTWA, see App. B). The curves for $j_{\rm max}=16$ do not differ
discernibly from those for $j_{\rm max}=32$. In the inset, we focus on the
behavior on short to moderate times. Here the agreement between both
approaches is very good. Since we know from previous studies Stanek  _et al._
(2014) that the sTWA represents the quantum mechanical result very well we
deduce that the bTWA also works well in this temporal regime.

Figure 2: Comparison of $S^{z}(t)$ obtained by sTWA and by bTWA for the
truncation levels $j_{\rm max}=16$ and 32, fixed number of bath spins
$N=1000$, $\gamma=0.01$ ($N_{\rm eff}=200$), and zero external magnetic field.

In the main panel of Fig. 2 we discern a significant discrepancy between the
sTWA and the bTWA. This is quite surprising in view of the nice agreement up
to $t\approx 30/J_{\rm Q}$. The convincing results obtained previously with
sTWA Stanek  _et al._ (2014) agrees with rigorous bounds Uhrig  _et al._
(2014); Seifert  _et al._ (2016) indicating a very slow decay of the
autocorrelation. Thus, the conclusion is indicated that the bTWA does not
approximate the long-time behavior of the CSM well. Still, it is (i) desirable
to corroborate this conclusion further and (ii) important to understand
whether the mapping to the effective bosonic model introduces the observed
difference or whether it is the TWA applied to the bosonic model which induces
this discrepancy.

Figure 3:  Comparison of $S^{z}(t)$ from various approaches (BA, sTWA, DMRG,
and iEoM) for fixed number of bath spins $N=36$ and two different (a)
$\gamma=1/18$ ($N_{\rm eff}=N=36$) and (b) $\gamma=1/12$ ($N_{\rm eff}=24$),
see App. A for further details of the effect of $\gamma$ in the bTWA. The
analytic data for random static (frozen) Overhauser field (fOver) from Eq. (5)
is included for comparison as well. In both iEoM and the TWAs, we use the
truncation level $j_{\rm max}=3$, see App. B for further details of the effect
of $j_{\rm max}$ in the bTWA.

Among the other approaches we employ the Bethe ansatz (BA) from which we use
the data published in Ref. Seifert  _et al._ , 2016. The BA works perfectly
for long times, but only for a moderate number of bath spins. Second, in
systematically controlled numerical DMRG calculations we consider 4096 states
Stanek  _et al._ (2013) with a threshold of 0.001 for the accumulated
discarded weight with second-order Trotter-Suzuki decomposition. The DMRG is
not able to follow the dynamics for long times due to the rapid growth of
entanglement. But up to $t\approx 50/J_{\rm Q}$ it is reliable. The quantum
mechanical evaluation of the iEoM up to $j_{\rm max}=3$ with {181,8,1} number
of bosons, respectively, yields reliable data as well up to $t\approx
50/J_{\rm Q}$ Röhrig  _et al._ (2018). Data from these methods are depicted in
Fig. 3 for two different sets of $N$ and $N_{\rm eff}$. The results from BA
and DMRG agree very nicely for all times except for a tiny discrepancy at the
minimum which we attribute to numerical inaccuracies. Note that the BA is
evaluated based on Monte Carlo importance sampling implying small statistical
fluctuations Faribault and Schuricht (2013a, b).

The iEoM approach, i.e., the quantum mechanical evaluation of the effective
bosonic model also agrees well with the BA and DMRG data, in particular for
the slow decay beyond $t\approx 6/J_{\rm Q}$. Only the wiggles at $t\approx
50/J_{\rm Q}$ indicate that the evaluation with the limited number of bosons
is at the verge of its validitiy at this time. The discrepancies of the iEoM
data to BA and DMRG data can be attributed to the fact that the mapping to the
effective model is valid for large spin baths only, see the discussion in Ref.
Röhrig  _et al._ , 2018. The sTWA data does not capture the minimum
particularly well, but it agrees with the other approaches (BA, DMRG, iEoM)
for longer times. The frozen Overhauser data from Eq. (5) is characterized by
the constant plateau for long times because no dynamics of the Overhauser
field is included.

What is the behavior of the data from bTWA? As we have already seen in Fig. 2
for short and moderate times the agreement with sTWA and thus with the other
data is good. In view of the long-time discrepancy observed in Fig. 2 we focus
on the longer times beyond $20/J_{\rm Q}$. We discern that the data from bTWA
clearly lies below the other data which coincide very well (except for the
frozen Overhauser curve). This observation corroborates our finding in Fig. 2
that the TWA applied to the effective bosonic model does not approximate the
long-time behavior reliably. In addition, we learn that the iEoM data, i.e.,
the quantum mechanical evaluation of the effective bosonic model, works fine
at these times. Hence, Fig. 3 provides evidence that it is not the mapping to
the effective bosonic bath which is responsible for the discrepancy, but the
bTWA. Hence, the two questions posed above are answered.

Figure 4: Comparison of the $x$-component of the spin-spin autocorrelation
obtained from DMRG, iEoM, and the bTWA at finite magnetic field
$h/J_{\mathrm{Q}}=10$ along the $z$-direction. The parameters are $N=500$,
$j_{\rm max}=3$ and $N_{\rm eff}=200$. The period of the Larmor precession is
given by $T_{\rm L}=2\pi/\sqrt{h^{2}+J^{2}_{\mathrm{Q}}/2}\approx
0.63/J_{\mathrm{Q}}$. The envelope function shown as black line is given by
$S_{\rm env.\,\,func.}(t)=\frac{1}{4}\,\exp(-J^{2}_{\mathrm{Q}}t^{2}/8)$. For
the effect of $h$ as well as the $z$-component of the spin-spin
autocorrelation obtained from the bTWA, see App. C.

This raises the question why the TWA is not as efficient as it is when applied
directly to the spins. We do not yet have a concluding answer but the
hypothesis suggesting itself is that the conserved quantities of the quantum
effective bosonic model and its classical counterparts are not the same. In
the CSM, the conserved quantities of the quantum and of the classical model
are the same which makes their dynamics very similar Stanek  _et al._ (2014).

Finally, we address the CSM in a finite magnetic field which has been well
investigated both theoretically and experimentally Beugeling  _et al._ (2016,
2017). Data from DMRG, iEoM, and bTWA is depicted in Fig. 4 for a magnetic
field in $z$-direction. In the main panel, all data sets agree very well. All
of them show the clear signature of Larmor precession with a period $T_{\rm
L}=2\pi/\sqrt{h^{2}+J^{2}_{\mathrm{Q}}/2}\approx 0.63/J_{\mathrm{Q}}$, cf.
Refs. Hackmann and Anders, 2014; Stanek  _et al._ , 2014. The envelope
function of the Larmor precession is given by $S_{\rm
env.\,\,func.}(t)=\frac{1}{4}\,\exp(-J^{2}_{\mathrm{Q}}t^{2}/8)$ Merkulov  _et
al._ (2002).

If we zoom far into the behavior at longer times after the signal has
dephased, only minor discrepancies occur. This behavior is not unexpected
since we learned already in the previous figures that the bTWA works well for
times below $30/J_{\rm Q}$. Hence the Larmor precessions and the Gaussian
dephasing as shown by the black envelope function are retrieved reliably. Only
the small discrepancies at later times indicate that the approximate treatment
is not perfect at long times. But in a magnetic field the signal has
essentially vanished anyway in the long-time regime.

##  V Summary and discussion

In this article, we theoretically studied the spin dynamics of the central
spin in the central spin model (CSM). The CSM describes a so-called central
spin coupled to spins in its environment in a star-like topology, i.e.,
without coupling between pairs of bath spins. This model is relevant for a
plethora of physical systems where a small quantum system is coupled to a bath
of other small quantum systems. A particularly interesting framework is the
realization of quantum bits and their decoherence mechanisms due to their
interaction with spin baths.

For many phenomena the long-time dynamics of large spin baths needs to be
described reliably which poses an insurmountable challenge to brute force
numerical approaches because of the exponential growth of the quantum Hilbert
space. Hence, accurate, systematically controlled approximative approaches are
needed. One of them is the mapping of the CSM with a large spin bath to a bath
of bosons, i.e., to an effective bosonic model, including a four-dimensional
impurity at the head of the chain. The bosonic degrees of freedom can be
represented in a star topology or in a chain topology Röhrig  _et al._ (2018).
The latter has the advantage that one can add site by site of the chain in
order to reach a reliable description up to longer and longer times. Thus, we
employed this representation here. Still, the quantum mechanical evaluation of
the resulting central spin dynamics is a great numerical challenge. For this
reason, we studied in the present article how well a truncated Wigner
approximation (TWA) for the bosonic effective model, dubbed bTWA, captures the
sought dynamics. This kind of approximation averages correlations of classical
trajectories over distributions of initial conditions and describes leading
quantum correlations in this way Polkovnikov (2010).

We found that the bTWA works very nicely for short and moderate times if the
spin bath is large. This condition on the size of the spin bath does not
result from the TWA, but from the mapping of the CSM to the effective bosonic
model. Only a few bosonic sites in the chain representation of the bosonic
bath are necessary.

Much to our surprise, however, we found a qualitative discrepancy of the bTWA
results compared to other approaches at long times. In this regime, the bTWA
results display a significantly faster decay than the results by a direct
application of the TWA to the CSM, dubbed sTWA. This discrepancy does not stem
from the sTWA, but from the bTWA. Inspecting and comparing the behavior at
moderate times where results from other approaches such as Bethe ansatz and
DMRG are available indicates clearly that the correlations from bTWA are the
deviating ones which are decaying too fast. Although the origin of this
unexpected discrepancy is still unclear, we presume that the _classical_
effective bosonic model, from which the trajectories are derived, that are
averaged in bTWA over initial conditions, has different, probably less,
conserved quantities than the quantum effective bosonic model or the original
CSM. Note that the quantum and the classical CSM share the same conserved
quantities Stanek  _et al._ (2014); Uhrig  _et al._ (2014); Seifert  _et al._
(2016) so that their very similar behavior is plausible.

But clearly, further studies are called for to (i) identify unambiguously the
origin of the discrepancy and (ii) to conceive reliable and efficient
evaluation techniques for the effective bosonic model. One idea suggesting
itself is to use numerical renormalization group techniques to evaluate its
dynamics. Surely, this will enhance our understanding of decoherence and
relaxation of small quantum systems suitable for realizing quantum bits or
quantum sensors.

###### Acknowledgements.

We would like to thank P Schering for useful discussions and for providing
data of other approximate and exact approaches. This study has been supported
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) and
the Russian Foundation for Basic Research in the International Collaborative
Research Centre TRR 160 (GSU), by the DFG in project UH 90/14-1 (TG and GSU),
and by the Studienstiftung des Deutschen Volkes (KB). In addition, we also
thank for the computing time provided on the Linux HPC cluster LiDO3 at TU
Dortmund University. M.Y. greatly acknowledges the financial support by the
National Science Foundation through award numbers DMR-1945529, PHY-1607611 and
NSF PHY1748958 as well as from the Welch Foundation through award number
AT-2036-20200401.

##  Appendix A Effect of the effective number of bath spins $N_{\text{eff}}$
in the bTWA

Figure A.1: The effect of the effective number of bath spins characterized by
$\gamma=2/N_{\rm eff}$ in the bTWA on the spin-spin correlation at fixed
$j_{\rm max}=3$ and $N=500$. The dotted fitting function in the inset is
$f(\gamma)=a\sqrt{\gamma}+b$ with $a=0.285\pm 0.005$ and $b=S^{z}_{\rm
min}(t_{\rm min}J_{\rm Q}=\sqrt{12})$, which confirms the square root
proportionality of the minimum value of the correlation on $\gamma$.

The effective number of bath spins $N_{\text{eff}}$ is one of the parameters
influencing the minimum autocorrelation at intermediate time scales as well as
the decoherence rates at long time scales. So, in the bTWA, it is important to
investigate a range of $N_{\rm eff}$ for fixed $j_{\rm max}=3$ and $N=500$ as
depicted in Fig. A.1, namely $N_{\rm eff}=200$, 100, 40, 25, and 20,
respectively, corresponding to $\gamma=0.01$, 0.02, 0.05, 0.08, and 0.10. We
obtain a square root behavior of $S^{z}_{\rm min}(t_{\rm min})$ as shown in
the inset of Fig. A.1 for increasing $\gamma$ (decreasing $N_{\text{eff}}$).
The coefficients $a=0.285\pm 0.005$ and $b=S^{z}_{\rm min}(t_{\rm min}J_{\rm
Q}=\sqrt{12})$ in the fitting function $f(\gamma)=a\sqrt{\gamma}+b$ depend on
the set of the other parameters. The spin-spin autocorrelation for $\gamma=0$
equals the one for the frozen Overhauser field with $S^{z}_{\rm min}(t_{\rm
min}J_{\rm Q}=\sqrt{12})\simeq 0.009$ as a benchmark, see Eq. (5). This fact
stems from the hyperfine coupling to the $i$-th bath that is proportional to
the square root of $\gamma$.

For larger values of $\gamma$ beyond $\simeq 0.08$ we observe that the further
changes of $\gamma$ do not change the curves anymore at least up to moderate
times. This observation agrees with what was found by sTWA Fauseweh  _et al._
(2017).

##  Appendix B Effect of the truncation level $j_{\rm max}$ in the bTWA

Figure B.1: The effect of truncation level characterized by $j_{\rm max}$ in
the bTWA on the spin-spin autocorrelation at fixed $N=500$ and $\gamma=0.01$
($N_{\rm eff}=200$).

Here we study the effect of the maximum number of bosonic modes $j_{\rm max}$
in the bTWA, see Fig. B.1, at fixed number of bath spins $N=500$ and
$\gamma=0.01$ (corresponding to $N_{\rm eff}=200$). The curve for $j_{\rm
max}=0$ shows the result for the frozen Overhauser field in Eq. (5). The curve
for $j_{\rm max}=1$ induces only a very small temporal evolution of the
Overhauser bath because the central spin is coupled only to a single harmonic
oscillator which has a small effect on the position of the minimum. But the
long-time plateau value of the autocorrelation stays close to the frozen
Overhauser field one for the studied times.

Taking into account a larger number of bosonic modes $j_{\rm max}\geq 2$, the
difference between the static, frozen Overhauser result and the dynamic
autocorrelations further increases. The frozen Overhauser curve (dashed line)
is always below the other curves at short timescales. Clearly, the decay of
the autocorrelation sets in only for $t>\tau$ after a specific time
$\tau\simeq 10/J_{\rm Q}$ which is almost independent of the set of
parameters. For the shown time interval, the curves do not change
significantly anymore for $j_{\rm max}\geq 3$ in accordance with previous
results Röhrig  _et al._ (2018).

##  Appendix C Effect of the external magnetic field on the spin-spin
autocorrelation in the bTWA

In this appendix, we address the role of a longitudinal magnetic field in the
bTWA with the parameters $j_{\rm max}=3$, $N=500$, and $\gamma=0.01~{}(N_{\rm
eff}=200)$ in Fig. C.1. In this case, the solution of Eq. (19c) displays the
precession of the central spin about the effective magnetic field, i.e., the
Overhauser field plus the external magnetic field. Depending on the considered
spin component the Zeeman effect implies different behavior. For the
$z$-autocorrelation of the central spin, Fig. C.1(a), one finds that the
decoherence rate is strongly suppressed by the magnetic field in a way that it
approaches zero at strong fields where the spin-spin autocorrelation becomes
almost time-independent and tends to take the initial value of 1/4. This
implies that the central spin polarization parallel to the external magnetic
field is stabilized for $h\gg J_{\rm Q}$.

Figure C.1: The effect of the external longitudinal magnetic field on the (a)
$z$-component and (b) $x$-component of the spin-spin autocorrelation at fixed
$j_{\rm max}=3$, $N=500$, and $\gamma=0.01$. The longitudinal central spin
polarization is stabilized with the external longitudinal field and the
amplitude of the oscillations is damped for increasing $h$ such that at strong
enough magnetic fields an almost time-independent autocorrelation function
$S^{z}(t)\to 1/4$ results. In contrast, the transversal spin-spin
autocorrelation displays prominent Larmor precession which quickly dephase due
to the random Overhauser field.

Upon increasing magnetic field, the minimum of the longitudinal
autocorrelation occurs earlier and earlier before it is reduced to small
oscillations and eventually to an almost constant plateau. In contrast to the
longitudinal dynamics of the central spin, the transversal dynamics, Fig.
C.1(b), displays pronounced Larmor precessions with fast decreasing amplitude
due to the dephasing induced by the fluctuations of the Overhauser field.

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