Nuclear Spin Noise in the Central Spin Model
Nina Fröhling Frithjof B. Anders Mikhail Glazov





Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund,
Otto-Hahn-Straße 4, 44227 Dortmund, Germany


Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund,
Otto-Hahn-Straße 4, 44227 Dortmund, Germany


Ioffe Institute, Polytechnicheskaya 26, 194021 St. Petersburg, Russia









We study theoretically the fluctuations of the
nuclear spins in quantum dots employing
the central spin model which accounts for the hyperfine interaction of the
nuclei with the electron spin. These
fluctuations are calculated
both with an analytical approach
using homogeneous hyperfine couplings (box model) and
with a numerical simulation
using a distribution of hyperfine coupling constants.
Both approaches are in good agreement. The box model serves
as a benchmark with low computational cost
that explains the basic features of the nuclear spin noise well.
We also demonstrate that  the nuclear spin noise spectra
comprise
a two-peak structure centered at the nuclear Zeeman frequency in  high magnetic fields
with the shape of the spectrum controlled by the distribution of the hyperfine constants.
This allows for a direct access to this distribution function
through nuclear spin noise spectroscopy. 





arabic


Nuclear spin noise in the central spin model
    Mikhail Glazov
    December 30, 2023
============================================






§ INTRODUCTION



The nuclear spin fluctuations <cit.> play an
important role in the spin dynamics in semiconductor nanosystems <cit.>.
In particular, these fluctuations provide an efficient mechanism of the electron spin decoherence in quantum dots <cit.>
due to the hyperfine interaction with electron spin.
The nuclear spin-spin correlation function also
controls the “warm-up” of the nuclear spin system in the alternating magnetic field <cit.>.


The spin noise technique <cit.> provides
a method for almost non-perturbative studies of the spin dynamics in thermal equilibrium or close to equilibrium conditions. 

In this technique a linearly polarized light beam propagating through the transparency region of the sample experiences a random fluctuation
of its polarization plane orientation caused by stochastic fluctuations of the magnetization of the media via the Faraday effect. 

First introduced 
to atomic systems <cit.>, this technique has 
successfully been
applied to semiconductors and semiconductor nanosystems <cit.>, see Refs. <cit.> for reviews. 
Since the optical response of a semiconductor is dominated, as a rule, by the electronic excitations, 
the spin noise spectroscopy technique has actively been used to study the electron spin fluctuations. 
The nuclear spin fluctuations 
determine the electron spin noise spectra in quantum dots <cit.>. 
The nuclear spin dynamics was revealed
in the specially designed <cit.> experiments:
The nuclear spins were first polarized, 
and afterwards the nuclear spin relaxation has been monitored via the electron spin noise <cit.>.

However, the nuclear spins also provide a sizable 
contribution to the Faraday effect in semiconductors <cit.>. 
Recently, their fluctuations have been observed by the spin noise technique in the GaAs sample with donor-bound electrons <cit.>. 
This motivates us to study theoretically the nuclear spin fluctuations for a system where a charge carrier, e.g., 
an electron, (i) is localized in a quantum dot or (ii) is bound to a neutral donor 
and interacts 
with a large number of nuclear spins in its vicinity. The coupled spin dynamics is accounted for by 
the central spin model that
has been studied previously  with the emphasis on the central spin 
(i.e., charge carrier)
dynamics both for systems in 
equilibrium <cit.> and for periodically pulsed systems <cit.>. 
While the interactions between the
nuclear spins are much smaller than the dominating hyperfine interaction and, therefore, are often neglected, 
the charge-carrier spin dephasing in a bath of interacting spins has also been investigated <cit.>. However, nuclear spin fluctuations have not been calculated within the central spin model to the best of our knowledge.  

In this paper, we present results for the nuclear spin correlation function and the nuclear spin noise 
within the central spin model including the Zeeman effect caused by the external magnetic field. We apply two complementary approaches based on (i) an analytical solution within the “box” model approach where all hyperfine coupling constants are assumed to be equal and (ii) a direct numerical procedure 
for a distribution 
of the hyperfine coupling constants. 

In the box model we can treat realistically large nuclear bath sizes at a 
relatively small computational expense. Furthermore,
simple and physically transparent results are obtained in the important limits of low and high magnetic fields, which can be readily compared with experimental data. The box model calculations also serve as a test-bed for the advanced numerical calculations which are essentially restricted by small sizes of the nuclear spin bath, but allow us to account for the 
distribution of the hyperfine coupling constants. Demonstrating the validity and good agreement between the approaches, we also demonstrate how the distribution of the hyperfine coupling constants can be determined from the nuclear spin noise spectrum.  

The paper is organized as follows: In Sec. <ref> we briefly formulate the central spin model and introduce the notations. In Sec. <ref> we present the analytical solution of the central spin model within the “box” model. 
The results for the spin noise spectra are covered in Sec. <ref>. Section <ref> contains 
the numerical data for the central spin model with a distribution of 
the hyperfine coupling constants. A brief summary of results is given in Sec. <ref>.



§ MODEL




In order to describe the
nuclear spin fluctuations in electron doped quantum dots, we have to model all relevant interactions. The system is dominated by the Fermi contact hyperfine interaction between the electron spin S and the bath of N nuclear spins, I_k, k=1, …, N, with the characteristic time scale of several nanoseconds in typical epitaxial III-V quantum dots  <cit.>. The dipole-dipole interaction between the nuclear spins 
is five orders of magnitude weaker, only contributing to the dynamics on the time scale of 100 s <cit.>. The strength of quadrupolar interaction, caused by strain fields in the material, is highly dependent on the material. It was measured to be in the order of ∼ 100 ns <cit.> for InAs quantum dots. In this work we will focus on the dominant hyperfine interaction and exclude the much weaker quadrupolar and dipole-dipole interaction.
This simplification leads to the central spin model Hamiltonian H_CSM   <cit.> 

    Ĥ_CSM =    ∑_k=1^N  A_k I_k·S
    + ∑_k=1^N g_kμ_NB⃗·I⃗_k +g μ_BB⃗·S⃗.

Here A_k are the hyperfine coupling constants,
B is the external magnetic field, g is the charge carrier g-factor, (-g_k) are the nuclear 
g-factors, and μ_N and μ_ B are the nuclear and Bohr magnetons, respectively. 

The hyperfine coupling results in the intrinsic time scale T^∗ defined via

    1/T^∗^2= ∑_k=1^N 4I_k(I_k+1) /3 A_k^2.

In the following
we assume that all nuclei have the same spin I_k≡ I_0, put ħ=1 and use T^∗ as a unit of time. Correspondingly, we redefine the dimensionless Hamiltonian T^∗Ĥ_CSM→Ĥ_CSM, the dimensionless hyperfine coupling constants a_k = T^∗A_k and the dimensionless
external magnetic field b⃗ = T^∗g μ_BB. 
It is also convenient to introduce the ratio between the nuclear and the electron Zeeman splitting

    z_k = g_kμ_N/gμ_B.

Finally, the dimensionless Hamiltonian takes the form

    H_CSM=S⃗·(b⃗ +∑_k=1^N  a_k I_k)+ b⃗·∑_k=1^N  z_k I_k.

The parameter T^∗ in Eq. (<ref>) is on the order of several nanoseconds, and the hyperfine interaction strength
corresponds to the temperatures of the order 10 mK <cit.> in self-assembled InGaAs quantum dots.
The standard spin-noise experiments on epitaxial III-IV quantum dots are performed at temperatures of about 4… 6 K, which allows us to disregard the intricate physics of nuclear spin polaron formation and the nuclear self-polarization occurring at much lower temperatures <cit.>. Therefore, we employ the  
high-temperature limit and assume that all states are equally occupied. 


The strength of the hyperfine coupling A_k is proportional to the probability of the electron to be 
at the position of the kth nucleus, |ψ_e(R_k)|^2.
The envelope function of
the electron wave function in a d-dimensional  quantum dot can be written as

    ψ_e(r)= CL_0^-d/2exp(-|r|^m/2L_0^m) .

Here C is the dimensionless normalization constant, L_0 is the effective localization radius, and the parameter m determines the shape of the wave function: m=1 corresponds to a hydrogen-like electron wave function, while m=2 describes a Gaussian wave function.
Several different distributions have been used in simulations of the central spin model <cit.>. However, the exact distribution of the coupling constants does not influence the short time (on the scale of T^*) dynamics of the system.  




§ EXACT SOLUTION FOR THE BOX MODEL





 §.§ Energy spectrum and eigenstates


We start with the simplest possible case,
the so-called box model, where the hyperfine coupling constants A_k≡ A_0 are all the same.
Furthermore, we assume that the g-factors for all nuclei are identical, i.e., z_k≡ z
and assume that each nuclear spin is I_0=1/2. 
Then it follows from Eq. (<ref>) that the characteristic time scale is T^∗= 1/A_0√(N),
 and the dimensionless coupling constant a_0=1/√(N). 
Even though the number of nuclear spins is macroscopic, N→∞
in an experimental 
sample,  the number of nuclear spins effectively coupled to the central spin
with a significant coupling constants, N_eff, remains 
finite and can be estimated by the characteristic length scale of the localized electronic wave
function. This is because only the nuclei within the electron localization volume effectively interact with the charge carrier. In experiments N→ N_eff≈ 10^5 is a typical value for quantum dot samples.

Defining the total bath spin I=∑_k I_k results in the Hamiltonian

    H = hS_z + zhI_z + a_0 [I_zS_z+1/2(I^+S^-+I^-S^+)].

for an
external field 𝐛=h𝐞_z.

Equation (<ref>) can also be  recast into the form

    H = hS_z + zh L + a_0/2(F^2-I^2-S^2),

where the total spin F=I+S and L is the z component of the total nuclear spin. 

In the special case of h=0, the Hamiltonian (<ref>) can be diagonalized via 
addition of the angular momenta
resulting in an energy spectrum of
the form <cit.>:

    E_F,F_z, I = a_0/2I,    if F=I+1/2, 
    
          -a_0/2(I+1),    if F=I-1/2.
 
In order to obtain the solution 
for a finite magnetic field, 
it is convenient to introduce the nuclear spin states |I,L⟩, where L=-I,…, I is the z-component of the nuclear spin and use the basis set 

    B =    {|I, -I⟩|↓⟩,  |I, -I⟩|↑⟩, |I, -I+1⟩|↓⟩,
       |I, -I+1⟩|↑⟩,..., |I, I⟩|↓⟩,  |I, I⟩|↑⟩},
  
where |↑⟩, |↓⟩ denote the electron spin states with S_z=± 1/2, respectively. In the basis (<ref>) the Hamiltonian matrix can be expressed in the block-diagonal form as <cit.>

    H  = [  E_+(I, -I)                                   …                       0;                M_I^-I+1                       …                        ;           ⋮    M_I^-I+2                                   ⋮;           ⋮                       ⋱           ⋮;           ⋮                               M_I^I           ⋮;           0                       …             E_-(I, I+1) ].

The states |I, -I⟩|↓⟩  and |I, I⟩|↑⟩ are decoupled since electron-nuclear flips are impossible. Correspondingly, the formally defined quantities E_+(I, -I) and E_-(I,I+1) in the matrix (<ref>) are scalars of the value

    E_+(I, -I)= ⟨I, -I|⟨↓| H |↓⟩|I, -I⟩ = a_0 I - h/2-zhI,
       E_-(I, I+1) = ⟨I, I|⟨↑| H |↑⟩|I, I⟩ = a_0 I + h/2+zhI,

and M_I^L are 2×2 matrices in the basis {|I,L⟩|↓⟩, |I,L-1⟩|↑⟩},

    M_I^L = 
    [         -h+a_0L/2+zhL                 ζ_I^L;                 ζ_I^L h+a_0(L-1)/2+zh(L-1), ]

with

    ζ_I^L = a_0/2√(I(I+1)-L(L-1)).

Diagonalization of matrix (<ref>) results in the eigenvalues 

    E_±(I, L) = -a_0/4±1/2√((a_0/2(2L-1) + h(1-z))^2+4(ζ_I^L)^2).

and the eigenstates

    |I,    L, -⟩ = 
       𝒜_I, L, -|I, L⟩|↓⟩+ℬ_I, L, -|I, L-1⟩|↑⟩,
    
    |I,    L, +⟩ = 
       𝒜_I, L, +|I, L⟩|↓⟩+ℬ_I, L, +|I, L-1⟩|↑⟩.

Here, the pre-factors 𝒜_I, L, ± and ℬ_I, L, ±, 

    𝒜_I, L, ± = 2ζ_I^L/√((h+a_0L+2E_L, ±-2zhL)^2+(2ζ_I^L)^2)  ,
    ℬ_I, L, ± = h+a_0L+2E_L, ±-2zhL/√((h+a_0L+2E_L, ±-2zhL)^2+(2ζ_I^L)^2),

describe the mixing of the Ising states (<ref>) constituting the eigenstates.

The first and the last eigenstate are the Ising states with extremal values of the total spin 
component F_z

    |I, -I, +⟩   = |I, -I⟩|↓⟩
    |I, I+1, -⟩   = |I, I⟩|↑⟩

and the corresponding eigenvalues presented in Eq. (<ref>). 
It is convenient to extend the
Eqs. (<ref>) to take into account
L=-I, I+1 and define 𝒜_I, -I, -=ℬ_I, -I, ±=𝒜_I, I+1, ±=ℬ_I, I+1, +=0 as well as 𝒜_I, -I, +=ℬ_I, I+1, -=1.

Equations (<ref>), (<ref>) and (<ref>) describe the 
energy spectrum and the eigenfunctions 
of the box model accounting for Zeeman splitting of both the electron and the nuclei.
 Eq. (<ref>) reduces to Eq. (<ref>)
for h=0.
In the case of a
negligible hyperfine coupling, a_0→ 0, the system separates 
into decoupled non-interacting nuclei and a single electron
with the energy spectrum given by the corresponding Zeeman splittings.



 §.§ Calculation of the spin correlation functions


Having diagonalized the system, it is possible to evaluate the dynamics of any desired observable that can be expressed in the basis B, Eq. (<ref>). Here, we are specifically interested in the
autocorrelation functions of the form C(t)=⟨O(t)O(0)|$⟩ withO(t)being the operator of the electron or nuclear spin taken in the Heisenberg representation. 
Expressing the time-evolution operator of the system via the eigenvalues and eigenstates (<ref>), (<ref>) and (<ref>) we arrive at

    C(t) = 1/Z∑_I=0^N/2N_w(I)∑_σ = ±
    σ' = ±∑_L=-I
    L'=-I^I+1 |⟨I, L, σ| O |I,  L', σ'||⟩^2
    ×exp{i[E_σ(I, L)-E_σ'(I, L')]t}.

HereZis the partition function,N_w(I)denotes
the number of configurations ofNbath spins  that have the total bath spin length ofI, and it is given by

    N_w(I) = (N/2-I)!(2I+1)/N/2+I+1.

We made use of the high temperature limit whereZ=2^N+1for1/2bath spins. 
In the next section we present analytical and numerical results for the autocorrelation functions and spin-noise power spectra defined as

    (O^2)_ω = ∫_-∞^∞ d t e^iω t C(t),

for the electron and nuclear spin components. We consider both fluctuations of the longitudinal and transversal spin components with respect to the external magnetic field orientation.




§ SPIN NOISE SPECTRA


 §.§ Box model


For the autocorrelation function of the electronic spin transverse to the magnetic field, one arrives at

    ⟨S_x(t)S_x(0)|=⟩   1/4Z∑^N/2_I=0 N_w(I)∑_σ, σ'=±∑_L=-I^I𝒜_I, L, σ^2 ℬ_I, L+1, σ'^2
       ×cos{[E_σ(I, L)-E_σ'(I, L')]t},

and for the autocorrelation function of the electron spin longitudinal to the magnetic field
we obtain

    ⟨ S_z   (t)S_z(0)⟩=
    1/4Z∑^N/2_I=0 N_w(I)∑_σ, σ'=±∑_L=-I^I+1 |ℬ_I, L, σℬ_I, L, σ'
       -𝒜_I, L, σ𝒜_I, L, σ'|^2
     exp{i[E_σ(I, L)-E_σ'(I, L')]t}.

The autocorrelation function of the total bath spin can be computed analogously, both
for the transverse spin component to the magnetic field

    ⟨ I_x(t)I_x(0)⟩=1/4Z∑^N/2_I=0N_w(I)∑_σ, σ'=±×
       ∑_L=-I^I[𝒜_I, L, σ𝒜_I, L+1, σ'ℬ_I, L, σℬ_I, L+1, σ'
       ×√(I(I+1)-(L-1)L)
       ×√(I(I+1)-(L+1)L)]^2
       ×cos[(E_σ'(I, L+1)-E_σ(I, L))t]

and the longitudinal spin component to the magnetic field

    ⟨ I_z(t)I_z(0)⟩=1/4Z∑^N/2_I=0N_w(I)∑_σ, σ'=±∑_L=-I^I+1 |L𝒜_I, L, σ𝒜_I, L, σ'
       -(L+1)ℬ_I, L, σℬ_I, L, σ'|^2
    exp{i[E_σ(I, L)-E_σ'(I, L')]t}.


We note that for the macroscopic number of nuclei the electron spin autocorrelation functions reduce to the expressions derived within the semi-classical model where the nuclear spins are considered as frozen <cit.>: 
For instance,  
spin fluctuations are isotropic ath=0, and
Eq. (<ref>) reduces to

    ⟨ S_z   (t)S_z(0) ⟩ = 1/2Z∑_I=0^N/2N_w(I)/(2I+1)^2∑_L=-I^I{(2L-1)^2
       +4(I+L)(I-L+1)cos[a_0(2I+1)t/2] }.

The direct calculation of the sum <cit.> atN→∞yields

    ⟨ S_z(t)S_z(0) ⟩ = 1/4[1/3 + 2/3(1-t^2/4)exp(-t^2)]

in full agreement with Refs. <cit.>. 

Since the total spin𝐅 = 𝐒 + 𝐈is conserved in the absence of an external magnetic field,
the nuclear spin𝐈remains constant within the tolerance of1/√(N)→0leading to the constant correlation function

    ⟨ I_z(t) I_z(0)⟩ = N/4

in the limitN→∞.






The nuclear spin noise 
is given by the Fourier transformation of the expression (<ref>), 
resulting
in a sum ofδ-function peaks centered atω_L, σ, σ' = ± [E_σ'(L+1)-E_σ(L)].Introducing additional dephasing rateγ, 
we replaceδ-functions by a broadened Lorentzian

    Δ_γ(ω) = 1/πγ/ω^2+γ^2

where we setγT^*=0.001.





Figure <ref> shows the nuclear spin noise calculated neglecting any nuclear Zeeman splitting (z≡0) for different magnitudes of the transversal magnetic field. 
Since the spectra are even functions ofω, we plot only the regionω⩾0. 


Ath→0a single Lorentzian peak is observed atω=0in agreement with Eq. (<ref>): the electron impact on nuclear spins is negligible. With increasingh, 
the peak shifts away from zero frequency. 
The total spin𝐅is no longer conserved in the presence of magnetic field,
and the Knight field generated by the precessing electron spin
results in the nuclear spin precession. At high magnetic fieldh≫1the peak in the spectrum approachesω^* = a_0/2[ω^* T^∗=1/(2√(N))],
originating
in the Knight field created by the electron spin.



The effect of the nuclear Zeeman splitting is illustrated in Fig. <ref>,  which shows the nuclear spin noise 
obtained from the Fourier transformation of Eq. (<ref>). We assume that the nuclear Zeeman splitting is three orders of magnitude smaller (z=0.001) than the electron Zeeman splitting. This ratio is typical for the experimentally studied systems <cit.>. For relatively weak magnetic fields,zh ≪a_0, 
the nuclear spin noise spectra are independent
of the nuclear Zeeman splitting. The influence of the nuclear Zeeman term 
becomes apparent forzh>1/√(N)as a shift of the peak. Ath=10, a double peak structure aroundω=a_0/2emerges 
with the distance between the peaks governed by nuclear Zeeman splitting. 
Upon further increasing of the field,zh ≫a_0, 
the spin noise spectrum exhibits two closely lying peaks centered atω= zh, where the peak splitting is given by the hyperfine coupling constanta_0.


The nuclear spin noise dependency on the different bath sizes 
is illuminated in Fig. <ref>. 
Here, the Fourier transformation of Eq. (<ref>) is shown for 
a constant combination of the external magnetic field and the bath size,h√(N)=300.  

Since the integral of the bath noise spectra results inN/4, the spin noise is scaled withNin order to make the plots comparable. The peak location given by
the scale of1/2T^∗√(N)is not affected by the bath size. As the energy spectrum broadens with an increasing number of spins, so does the bath noise spectrum. Note that the characteristic time scale,T^∗, depends onN.

The expression in Eq. (<ref>) can be evaluated in𝒪(N^2)computation time. This is a massive advantage to a full exact diagonalization, which is computed in𝒪(2^2N+2)or the Lanczos method, 
which computes the autocorrelation function in𝒪(2^N+1). 
Replacing the exponential
runtime by a quadratic one
allows for simulation of much larger systems. 
The disadvantage of this method is that it cannot 
include a distribution of coupling constants,
and it is impossible to extend the system to additional interactions if necessary. 
Below we briefly discuss the role of the distribution of the hyperfine coupling constants within the high magnetic field limit and 
in Sec. <ref> we will present the results within the numerical approach for arbitraryhwith account for the distribution of the coupling constantsa_k.









 §.§ High magnetic field limit



We have shown in Eq. (<ref>) that the total nuclear spin correlation function does not decay in the box model in the largeNlimit
and the absence of an external magnetic field. Decoherence is caused by a electric quadruple nuclear
interaction with random easy axis <cit.>
as well as dipole-dipole interactions <cit.>
between neighboring nuclei. 
Here we
focus on the influence of the distribution 
of hyperfine coupling constants, which is clearly the dominating factor.
Before presenting general results, it is instructive to analyze 
first the limit of large magnetic fields, whereh≫a_0, but
with the requirement of the validity of the high-temperature limit:
the thermal orientation of electron and nuclear spins can be disregarded. 
At large external magnetic fields, electron-nuclear spin flips are suppressed, leading to the 
effective Hamiltonian

    H_hmf = hS_z + h∑_k  z_k I_z + ∑_k a_k I_zS_z.

This Ising-like coupling is also applicable to some extent 
in the case of quantum dots with heavy holes <cit.> in an arbitrary magnetic field.
Without the spin-flip term, the Hamiltonian (<ref>) is already diagonal. In this case,⟨I_z(t) I_z(0)⟩is given by Eq. (<ref>) and 

    ⟨I_x(t)I_x(0)|=⟩   1/4Z∑^N_k=1,
     L_k=↑, ↓ , σ=↑, ↓⟨σ, {L_k}|I_k^x(t)I_k^x(0)|σ, {L_k}|
    ⟩
    =   N/4cos(a_0t/2)cos(zht).

Thus the nuclear spin noise spectrum contains four peaks at±(a_0/2±zh)=±a_0(1/2±zh√(N)). In the absence of a nuclear Zeeman splitting, the nuclear spin bath precesses in the magnetic field induced by the electron, the Knight field of the strengtha_0. This distinguishes the characteristic frequency of the nuclear spin noise from those of 
the electron spin noise, which precesses in the Overhauser field with the characteristic
frequency1/T^∗.



Noteworthy is that neglecting the spin-flip terms 
allows to extend the theory to arbitrary distributions of the hyperfine coupling constantsa_kand nuclear magnetic moments, 
so that we generalize Eq. (<ref>) to

    ⟨I_x(t)I_x(0)|=⟩1/4∑_k=1^N cos(a_kt/2)cos(z_kh t).



While the short-time dynamics of the spin system is not influenced by the particular form of the distribution of the hyperfine coupling constants, the distribution ofA_kplays an important role in the spin decoherence on the longer time scale,t≳10T^∗. 
In the high magnetic field limit
the summation overkin Eq. (<ref>) can be performed analytically in the
largeNlimit, and  nuclear spin noise spectrum acquires the form:

    (I_x^2)_ω = πN/4∫ da 𝒫(a) [Δ_γ(ω-a/2)+Δ_γ(ω+a/2)],

where𝒫(a)is the distribution function of the coupling constantsa_k. For 
a
quantum dot with an
envelope ind-dimensional space (d=2for self-assembled quantum dots where the size quantization along the growth axis is much stronger than that in the quantum dot plane,d=3for isotropic quantum dots) given by Eq. (<ref>),𝒫(a)can be written as by <cit.>
    𝒫(a) =d/m1/r_0^d a[ln(a_ max/a)]^d/m-1.

Here,r_0=R_0/Lis the dimensionless cut-off parameter which characterizes the effective radiusR_0where the hyperfine coupling becomes unimportant,a_maxis the maximal hyperfine coupling constant. 
We recall thatmis the power in the exponent of the  wave function stated in Eq. (<ref>):m=1for hydrogenic envelope function relevant for donor-bound electrons,m=2for the Gaussian envelope functions. In our simulations we use the following set of parameters

    m=1:     r_0=2
       m=2:     r_0=1.5,

in agreement with previous works <cit.>.


Figure <ref> shows the nuclear spin noise without the 
nuclear Zeeman-splitting, derived from the Fourier transformation 
of Eq. (<ref>). In actual calculations it is convenient to generate first
a configuration ofA_kvia

    A_k(x) = A_maxe^-r_0^m x^m/3,

wherexis the random real in the range[0,1], 
and secondly normalize the couplings toa_k=T^∗A_k. 
The time scaleT^∗, and thereforea_max, can vary slightly between different 
sets of the hyperfine couplings.

To stress that the box model including all spin flips 
perfectly matches the high magnetic field limit 
of the CSM with 
constant hyperfine couplings, both results are plotted. 
It follows from Eqs. (<ref>) and (<ref>) that the nuclear spin noise spectrum
approaches

    (I_x^2)_ω = πN/2𝒫(2ω),   ω>0
 
forγ→0.
To justify the use of the Ising model in the high magnetic field limit, the nuclear
spin noise calculated within the box model with (light blue dashed curve) and without (green curve) spin flips are included in Fig. <ref> demonstrating perfect agreement. 
Analogous to the approach in the box model, theδ-peaks 
of the nuclear spin noise was
broadened to a  Lorentzian withγT^*=0.001.




Figure <ref> also shows a comparison of the nuclear spin noise between the continuum limit of Eq. (<ref>) and the Fourier transformation of Eq. (<ref>) using 
the finite broadening 
forN=1000 nuclear spins in the absence of the nuclear Zeeman-splitting. 
The corresponding analytical curves calculated via Eq. (<ref>) 
are plotted
in Fig. <ref> as dashed lines of the same color and fully agree with numerical calculations forω>0.01. 
The singularity  ata=0of the distribution𝒫(a), Eq. (<ref>),
is cut off by the dimensionless radiusr_0defining the smallesta<cit.>.
Therefore, the analytical and numerical solutions agree perfectly for the same𝒫(a)at high frequencies.
The difference for→0arises from the hard cutoff of𝒫(a)versus finite Lorenzian broadening
of the numerical solution
for a large but finite number of nuclei. 



In Fig. <ref> the nuclear spin noise is depicted  in the high magnetic field limit including the nuclear Zeeman term, again derived through the Fourier transformation of Eq. (<ref>). 
The spin noise is strongly concentrated aroundzh, (z_k=z ∀ k), for large external magnetic fields similar to the plot forh=1000in Fig. <ref>.

Again, the box model results with and without spin flips agree perfectly, 
which further underlines the validity of the use of the Ising model for 
high magnetic fields. 
The two peak structure centered at the nuclear Zeeman frequencyhzis clearly visible, similarly to the results plotted in Fig. <ref>. 
The curves  for the two distributions ofa_kare shown to illustrate
the additional broadening 
effect of the hyperfine coupling 
distribution onto the noise spectra and, consequently, the modified shape of peaks.

These results indicate the possibility of measuring the distribution of the hyperfine coupling constants directly via spin noise spectroscopy, as seen in Eq. (<ref>). 




§ FULL CENTRAL SPIN MODEL




The full CSM cannot be easily solved analytically  at arbitrary magnetic fields
and with inhomogeneous hyperfine couplings although it is formally integrable via the Bethe ansatz <cit.>.

A quantum-mechanical, numerical approach based on the Lanczos method with restart (see the Appendix for details), can address this problem and 
also allows to incorporate additional interactions, such as the 
nuclear quadrupolar electric coupling <cit.>.
In such an approach, however, the bath size is severely limited, since the Hilbert-space dimension increases exponentially with the number of nuclear spins. In this paper,N=17spins are simulated, and thea_kdistribution introduced in Eq. (<ref>)
is used with a cut-off radius ofr_0=1.5andm=2. 
We perform a Fourier transformation
of the real-time dynamics to obtain the corresponding nuclear spin noise spectra;
a detailed description of the method can be found in the Appendix. 
Here, the nuclear spin correlation function is first evaluated in real time, and 
converted into a spectral function via a fast Fourier transformation. 
This differs from the previous approaches where the Fourier transformation was performed analytically.
In order to mimic larger bath sizes with the limited number of nuclear spins, we generated 25 different 
configurations ofNhyperfine couplings,{ a_k }, and perform a configuration average of the individual spectra.
We maintain the definition of the characteristic time scaleT^*, Eq. (<ref>), and state all
parameter in the previously introduced dimensionless units.






Figures <ref> (<ref>) show the 
nuclear spin noise spectra obtained by the Lanczos method without (with) nuclear Zeeman splitting in a transversal magnetic field. 

In Fig. <ref>(a) the spectrum for a small magnetic field (h=0.1) is depicted, along with the box model spectrum generated from Eq. (<ref>) for comparison.
At high magnetic fields (Fig. <ref>(b))
the spectra qualitatively resemble the previous analytical results obtained in high magnetic field limit
forh>1andωT^∗<0.3.
To illustrate that point, we added the analytical spectra 
computed analogously to the spectra in Fig. <ref>
as dashed black curve using the samea_kdistribution.

As shown in Eq. (<ref>), the bath spin noise spectra center around the frequency of the Knight field in the absence of a nuclear Zeeman splitting. 
 Since the relative strength of the Knight field compared to 
 Overhauser field shrinks with increasing number of nuclear spins at a fixedT^*, 
the main spectral weight of the spin noise spectra
is located at a smaller frequency forN=1000than the spin noise spectra forN=17.
We take this into account by rescaling the frequency axis with the factor2T^∗√(N).

The full numerical solution shows a much broader frequency spectrum than the analytical 
results for the box model in high magnetic fields.
But forωT^∗<0.2andh>1, the qualitative form of the 
nuclear spin noise in the full model is very well matched  by the high magnetic field limit.

Note that the finite number of nuclear spins in the Lanczos approach
 limits the number of discrete eigenvalues of the systems. 
Furthermore the fast Fourier transform does not include an additional 
Lorentz broadening. Both effects
lead to noisy spectra in comparison to the smooth spectra of the high magnetic field limit and the box model from Sec. <ref>.



A significantly different picture emerges forh>10when the nuclear Zeeman term is included
in the Hamiltonian. 
The resulting transversal nuclear spin noise is shown in Fig. <ref>(a) and (b)
for the same parameters as in Fig. <ref> but with a finitez=10^-3.
Ath=100, in Fig. <ref>(a), a clear drop becomes apparent athz=0.1whose origin becomes clear by the comparing the spectra with the
box model result added as a dashed brown line. 
The nuclear Zeeman splitting introduces a
two peak structure analytically illustrated by Eq. (<ref>). For a
low and an intermediate field, the 
shape of the spin noise spectrum is strongly influenced by the distribution function 𝒫(a), and therefore, significantly
broader than the corresponding box model spectra.
The spectrum, however, approaches the analytical high magnetic field limit 
forh=1000and  the samea_k-distribution – 
plotted as a dashed black line for comparison in in Fig. <ref>(b).



Comparing the results for the full CSM with the two analytical approximations,
the box model or the high magnetic field limit,
we can conclude that both analytical techniques give a good understanding of the basic features of the 
nuclear spin noise.
The analytical results can be obtained at very little computational costs and 
allow for bath sizes that are several orders of magnitudes larger than the bath sizes that can be treated in the quantum mechanical
evaluation of the general CSM.





§ CONCLUSION




We studied the nuclear spin fluctuations,

both analytically and numerically, using the central spin model. 

This model includes the hyperfine interaction of nuclear spins with electron spin 
as well as the Zeeman effects for the electrons and the nuclei. 

For homogeneous coupling constants, i.e., within the box model approximation, the system is exactly solvable, and the auto-correlation function of the total bath polarization can be obtained
analytically for an arbitrary number of nuclear spins. 
Including the nuclear Zeeman splitting  in the model, we observe the emergence of two peaks in the 
nuclear spin noise spectrum centered at the nuclear spin precession frequency.
The distance between the peaks is given by the hyperfine coupling constant. 

The effect of the hyperfine coupling distribution function
constants was studied analytically
in the high magnetic field limit. 
We demonstrate that the shape of the nuclear spin noise spectrum is controlled by 
this distribution function
of the hyperfine coupling constants, i.e., by the shape of the electron envelope function.
A Lanczos algorithm with restarts was employed
to numerically obtain the bath correlation 
for inhomogeneous couplings and arbitrary magnetic fields. 
The agreement between the numerical and analytical solutions is particularly good at high magnetic fields. 
Therefore, the analytical results 
provide a good description 
of the system, but with significantly reduced computational costs.



We thank D.S. Smirnov for valuable discussions.
We acknowledge the financial support by the
Deutsche Forschungsgemeinschaft and the Russian Foundation of Basic
Research through the transregio TRR 160. M.M.G. is grateful to RFBR projects 17-02-00383 and 15-52-12012 and Russian President grant MD-1555.2017.2 for partial support.








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§ LANCZOS METHOD WITH RESTARTS


We used the Lanczos time evolution method with restarts <cit.> in order to keep stability and stochastic evaluation of the trace
<cit.>
by averaging over a small numberR ≪Dof randomly chosen states|r⟩. The relative error made by the stochastic evaluation of the trace

    Tr(O) ≈1/R∑_r=0^R-1⟨r|O|r|⟩

is of the order𝒪(1/√(DR))<cit.>. 
Using this technique, we approximate the second order nuclear spin auto-correlation function by

    ⟨I_x(t)I_x(0)|=⟩2/RD∑_r^R⟨1_r(t)|I_x|2_r(t)|⟩eq:IxIx_lan

where

    |1_r(t)⟩   =    e^-iH t|r⟩
    |2_r(t)⟩   =    e^-iH t I_x |r⟩.

The nuclear spin dynamics happen on a time scale much longer than the electron spin dynamics, we need to be able to simulate up to large times without losing numerical accuracy.
Therefore we discretize the timet_n=nτ, n=0,..., N_tto to calculate the evolution of the state as

    |ψ(t_n)⟩ = e^-iH τ|ψ(t_n-1)⟩

in smaller time stepsτ.


The Lanczos-Krylov Algorithm <cit.> is used here for calculating
the real-time propagation. The vectors

    |ψ⟩, H|ψ⟩, H^2|ψ⟩, …, H^M|ψ⟩,

define the Krylov space, aMdimensional subspace of the Hilbert space.
Each time, the state|ψ(t_n-1)⟩ from the previous time step is used
as new starting vector. Via the Lanczos algorithm, we gain theM×Mdimensional tridiagonal matrix representation of the HamiltonianH(M)in the Krylov space. We diagonalizeH(M)and obtain the approximate eigenstates|ν^n⟩H(M)|ν^n⟩= ϵ_ν^n|ν^n⟩

and the corresponding eigenvaluesϵ_ν^n. The time evolution can then be approximately written as

    |1_r(t_n)⟩ = e^-iHτ|1_r(t_n-1)⟩=∑_ν^M e^-iϵ_ν^nτ
    c^1_n(t_n-1)  |ν^n⟩.
 
wherec^1_n(t_n-1)  = ⟨ν^n|1_r(t_n-1)|$⟩. The result becomes exact for τ→ 0.
To arrive at |1_r(t_N_t)⟩, N_t Lanczos time evolution steps have to be computed.

The same time evolution is needed for the vector |2_r(t)⟩,

    |2_r(t_n)⟩ = e^-iHτ|2_r(t_n-1)⟩=∑_ν^M e^-iϵ_ν^nτ
    c^2_n(t_n-1)
    |ν^n⟩,
 
where c^2_n(t_n-1)=⟨ν^n|2_r(t_n-1)|$⟩. Having gained both|1_r(t_n)⟩and|2_r(t_n)⟩, we can calculate the nuclear spin auto-correlation function⟨I_x(t)I_x(0)|$⟩ using Eq. (<ref>). The nuclear spin noise is then computed via fast Fourier-transformation. For n time steps 2N_t Lanczos iterations are needed. A finer time resolution can be achieved by substituting τ with Δ t (0<Δ t<τ) in Eq. (<ref>) and (<ref>), calculating the nuclear spin correlation function at t_n-1+Δ t. This way, several time steps between t_n-1 and t_n can be calculated without having to do the Lanzcos algorithm with a different starting vector, thus saving computation time. This gives us the flexibility to chose τ smaller or larger depending on the dynamics of the system, while keeping the same time resolution.  


