# Electronic statistics-on-demand: bunching, anti-bunching, positive and
negative correlations in a molecular spin-valve

Nicholas S. Davis  College of Science and Engineering, James Cook University,
Townsville, QLD, 4811, Australia  Samuel L. Rudge  Physikalisches Institut,
Albert-Ludwigs Universität Freiburg, Freiburg, 79100, Germany  Daniel S. Kosov
College of Science and Engineering, James Cook University, Townsville, QLD,
4811, Australia

###### Abstract

One of the long-standing goals of quantum transport is to use the noise,
rather than the average current, for information processing. However,
achieving this requires on-demand control of quantum fluctuations in the
electric current. In this paper, we demonstrate theoretically that transport
through a molecular spin-valve provides access to many different statistics of
electron tunneling events. Simply by changing highly tunable parameters, such
as electrode spin-polarization, magnetization angle, and voltage, one is able
to switch between Poisson behavior, bunching and anti-bunching of electron
tunnelings, and positive and negative temporal correlations. The molecular
spin-valve is modeled by a single spin-degenerate molecular orbital with local
electronic repulsion coupled to two ferromagnetic leads with magnetization
orientations allowed to rotate relative to each other. The electron transport
is described via Born-Markov master equation and fluctuations are studied with
higher-order waiting time distributions. For highly magnetized parallel-
aligned electrodes, we find that strong positive temporal correlations emerge
in the voltage range where the second transport channel is partially open.
These are caused by a spin-induced electron-bunching, which does not manifest
in the stationary current alone.

##  I Introduction

The current generation of electronic technology has progressed rapidly through
constant, iterative improvements to compactness. With smaller components,
devices occupy less space within a circuit and dissipate less energy, allowing
the construction of densely-packed circuits with improved computational power
[1]. This miniaturization process, however, is soon approaching the molecular
size limit of a few nanometers. On this scale, the well-understood
deterministic nature of electron transport is overturned by the probabilistic
nature of quantum effects, opening new possibilities for information
processing and corresponding research avenues. Among the most promising is the
field of molecular spintronics [2], which explores how electron spin can be
used in application to molecular-sized devices, as opposed to using just the
electron charge [3].

A molecular spintronic device typically consists of a molecule connecting two
macroscopic electrodes. The transition from electronics to spintronics demands
that there be a level of magnetization within the system; some components must
possess a dominant spin-orientation. Although one can achieve this through
nonmagnetic leads and a single-molecule magnet [4, 5], in this paper, we focus
exclusively on spin-valves created from a nonmagnetic molecule coupled to two
ferromagnetic electrodes [6, 7, 8, 9, 10, 11]. Such devices, commonly called
magnetic tunnel junctions, have been experimentally realized with quantum dots
[12, 13, 14, 15], carbon nanotubes [16, 17, 18, 19, 20], and even C60
molecules [21, 22].

The main idea of a molecular spin-valve is to control the flow of electrons
via the magnetic orientation of the electrodes. Through magnetic properties
alone, the valve can shift between high and low conductance states by rotation
of the electrodes [23, 8]. Beyond this basic function, however, spin-valves
also produce numerous regimes with interesting effects that, combined with the
size advantage of a molecular device, make these devices preferable over
conventional electronics. Two such effects that have already been demonstrated
prototypically are spin-blockade induced negative differential resistance
(NDR) [8, 24] and spin filtering [5, 25]. Spin filtering describes a situation
in which a particular spin orientation is preferentially allowed to flow
through the system, creating spin-polarized current. The utility of spin
filtering is expansive as it can be used to transfer large magnetic moments
through a system, thus providing a means of flipping magnetic states, known as
spin crossover [26].

Figure 1: Examples of different statistics for electronic transport. Each peak
represent electron detection event in the drain. In addition to the standard
Poisson process, one can also observe: (a) electron bunching (super-Poisson
statistics), (b) electron anti-bunching (sub-Poisson statistics), (c) positive
temporal correlations, and (d) negative temporal correlations. All these
different statistics can be observed in a molecular spin-valve.

In this paper, we show that, additionally to these interesting effects, a
molecular spin-valve provides the possibility of controlling quantum current
fluctuations on a single electron level. By adjusting highly tunable
parameters, one can switch between different statistics of electron transport:
Poisson, electron bunching (super-Poisson statistics), electron anti-bunching
(sub-Poisson statistics), and positive and negative temporal correlations, all
of which are shown in Fig.(1). We explore these different fluctuation
statistics using the waiting time distribution (WTD).

While a relatively recent addition to the analysis of charge fluctuations in
open quantum systems [27], waiting time theory has developed rapidly in the
last $10$ years. Consequently, it has been used to investigate a wide variety
of transport scenarios, such as tunneling through molecules with electron-
electron [28, 27, 29, 30] and electron-phonon [31, 32, 33, 34] interactions,
telegraphic switching [35], double [36, 37] and triple [38, 39] quantum dots,
superconducting junctions [40, 41, 42, 43, 44], coherent conductors [45, 46,
47], non-Markovian transport [48, 49], periodically driven transport [50, 51],
and transport in the transient regime [52, 53, 54, 55]. As opposed to the full
counting statistics (FCS), which is the most prevalent method for analyzing
charge fluctuations, the WTD provides information on transport at short
timescales, particularly via correlations between successive waiting times
[28, 50, 56, 57, 51, 58]. Although there exist finite-frequency and time-
dependent FCS capable of analysis at similar timescales [59, 60, 61, 62], as
well as discussions of correlation behavior with current cumulants [63, 64,
65], most FCS treatments work in the long-time or zero-frequency limit [66].
Measuring waiting times experimentally is a difficult task, requiring
sophisticated single-electron detection methods [67, 68, 69, 70]; however,
there are several novel proposals to measure waiting times, such as using a
quantum ‘clock’ [71] and matrix product states [72].

In the context of quantum dot spin-valves, finite-frequency FCS have
previously been used to analyze how the interplay between Larmor precession
and a spin blockade affect the current noise [73, 74], while the zero-
frequency noise has been used to investigate the spin torque [75] and
dynamical blockade [76, 77, 78]. The WTD and time-dependent FCS have also been
previously used to analyze the coherent dynamics present in this system,
finding that Larmor precession causes coherent oscillations in the WTD and
time-dependent charge cumulants [6, 79]. Several groups have also worked with
spin-dependent fluctuation statistics, rather than just using the electron
charge. Tang et al., for example, have recently developed the theory of spin-
resolved waiting times in a quantum dot spin-valve [10], while Sánchez et al.
have developed a correlation coefficient for the spin-resolved current [80].
Our analysis, however, focuses on correlations between successive waiting
times, and we find that, surprisingly, the strongest correlations appear when
the coherent dynamics of the system are diminished.

The paper is organized as follows. Section II describes the theory; the model
Hamiltonian, master equations, and the WTD, along with its main statistical
tools, are introduced. Results are presented and discussed in Section III,
which discusses the average current, a measure of the noise, and temporal
correlations. Section IV presents the conclusions, while the explicit form of
the Liouvillian and jump superoperators are given in Appendix A and B.

We use natural units throughout the paper: $\hbar=e=1$.

Figure 2: Schematic of a molecular spin-valve, which consists of an Anderson
model coupled to ferromagnetic electrodes. The spin quantization axis of the
source electrode is aligned with the $z$-axis, whereas the drain electrode is
allowed to rotate along the $x-z$ plane, enclosing an angle $\phi$ with the
$z-$axis.

##  II Theory

###  II.1 Model

To model the spin-valve, we couple a molecule to two ferromagnetic leads,
which has the total Hamiltonian

$\displaystyle H_{\text{tot}}$ $\displaystyle=H_{\text{mol}}+H_{\text{el}}+V.$
(1)

Here, $H_{\text{mol}}$ is the molecular Hamiltonian, $H_{\text{el}}$ is the
Hamiltonian of the source and drain electrodes, and $V=V_{S}+V_{D}$ enables
tunneling between these two subsystems.

For the molecule, we use an Anderson model, with Hamiltonian

$\displaystyle H_{\text{mol}}$ $\displaystyle=\sum_{\sigma}\varepsilon
d^{{\dagger}}_{\sigma}d_{\sigma}+Ud^{{\dagger}}_{\uparrow}d_{\uparrow}d^{{\dagger}}_{\downarrow}d_{\downarrow},$
(2)

where $\sigma\in\\{\uparrow,\downarrow\\}$ denotes spin in the molecule,
$\varepsilon=\varepsilon_{\uparrow}=\varepsilon_{\downarrow}$ is a spin-
degenerate molecular orbital energy, and $U$ is the Coulomb interaction
between two electrons occupying the same orbital. We choose the spin
quantization axis of the Anderson model to be along the $z$-direction and
refer to it as the ‘global’ spin orientation; the ‘local’ quantization
direction of the ferromagnetic electrodes will be taken relative to this.

The source and drain ferromagnetic electrodes are reservoirs of non-
interacting electrons

$\displaystyle H_{\text{el}}$
$\displaystyle=\sum_{\alpha\in\\{S,D\\}}\sum_{\mathbf{k}_{\alpha},\sigma_{\alpha}}\varepsilon_{\mathbf{k}_{\alpha}\sigma_{\alpha}}c_{\mathbf{k}_{\alpha}\sigma_{\alpha}}^{\dagger}c_{\mathbf{k}_{\alpha}\sigma_{\alpha}},$
(3)

where the subscript $\mathbf{k}_{\alpha}$ denotes the energy states of lead
$\alpha\in\\{S,D\\}$. The electrodes are held at local thermal equilibrium
with the same temperature, $T_{\alpha}=T$, but different chemical potentials,
$\mu_{\alpha}$. In all calculations, we adjust the chemical potentials
symmetrically around the Fermi level, which is kept at zero:
$\mu_{S}=V_{SD}/2=-\mu_{D}$, where $V_{SD}=\mu_{S}-\mu_{D}$ is the source-
drain bias voltage.

To explicitly include the ferromagnetic nature of the leads, we apply an
asymmetry to the spin-dependent density of states,
$\rho_{\alpha\sigma_{\alpha}}(\omega)$, for majority, $\sigma_{\alpha}=+$, and
minority, $\sigma_{\alpha}=-$, spins; see Fig.(2). The spin-polarization,

$\displaystyle p_{\alpha}$
$\displaystyle=\frac{\rho_{\alpha+}-\rho_{\alpha-}}{\rho_{\alpha+}+\rho_{\alpha-}},$
(4)

thus describes the level of magnetization of lead $\alpha$. The ‘majority
spin’ and ‘minority spin’ terminology is used here to reflect the fact that
the electrodes may have their spin quantized along different axes to the
global spin orientation in the molecule. In our treatment, we always choose
the quantization axis of electron spin in the source electrode to be along the
$z$-axis, thus keeping it aligned with the molecule so that
$\sigma_{S}\in\\{\uparrow,\downarrow\\}$. The drain electrode, however, is
allowed have spin quantized in any direction in the $x$-$z$ plane, which
encloses an angle $\phi$ with the $z$-axis.

The different magnetization angles are also reflected in the source and drain
tunneling Hamiltonians, $V=V_{S}+V_{D}$, which take the form

$\displaystyle V_{S}$
$\displaystyle=\sum_{\mathbf{k}_{S}}t_{S}\left(c^{{\dagger}}_{\mathbf{k}_{S}+}d_{\uparrow}+c^{{\dagger}}_{\mathbf{k}_{S}-}d_{\downarrow}\right)+\text{h.c.},$
(5) $\displaystyle V_{D}$
$\displaystyle=\sum_{\mathbf{k}_{D}}t_{D}\left\\{\left[c^{{\dagger}}_{\mathbf{k}_{D}+}\cos\left(\frac{\phi}{2}\right)-c^{{\dagger}}_{\mathbf{k}_{D}-}\sin\left(\frac{\phi}{2}\right)\right]d_{\uparrow}\right.$
$\displaystyle\>\>\>\>\>\>\left.+\left[c^{{\dagger}}_{\mathbf{k}_{D}+}\sin\left(\frac{\phi}{2}\right)+c^{{\dagger}}_{\mathbf{k}_{D}-}\cos\left(\frac{\phi}{2}\right)\right]d_{\downarrow}\right\\}+\text{h.c.}.$
(6)

In Eqs.(5)-(6), we have assumed that the tunneling matrix elements,
$t_{\alpha}$, are real and independent of electrode energy and spin. Since we
further assume that the spin-dependent density of states in each electrode is
constant,
$\rho_{\alpha,\sigma_{\alpha}}(\omega)=\rho_{\alpha,\sigma_{\alpha}}$, the
tunneling matrix elements are related to the spin-dependent electrode-system
coupling strength via
$\Gamma_{\alpha\sigma_{\alpha}}=2\pi|t_{\alpha}|^{2}\rho_{\alpha\sigma_{\alpha}}$,
where we can define the corresponding electrode-dependent coupling strength
and total coupling strength as
$\Gamma_{\alpha}=\left(\Gamma_{\alpha,+}+\Gamma_{\alpha,-}\right)/2$ and
$\Gamma=\left(\Gamma_{\alpha}+\Gamma_{\alpha}\right)/2$, respectively.

###  II.2 Born-Markov master equation

Although the total dynamics are described by the total density matrix,
$\rho_{\text{tot}}$, we actually only need the reduced density matrix for the
molecular degrees of freedom, $\rho_{\text{mol}}$, since the electrodes are
reservoirs of non-interacting electrons. The corresponding Hilbert space is
spanned by the four orthonormal eigenstates of $H_{\text{mol}}$: the dot can
be empty, $\,|{0}\rangle$, occupied by a single spin-up,
$\,|{\uparrow}\rangle$, or spin-down, $\,|{\downarrow}\rangle$, electron, or
two electrons of opposite spin, $\,|{2}\rangle$. The coupled elements of the
reduced density matrix are

$\displaystyle\rho_{\text{mol}}$ $\displaystyle=\left[\begin{array}[]{c c c
c}\rho_{00}&0&0&0\\\ 0&\rho_{\uparrow\uparrow}&\rho_{\uparrow\downarrow}&0\\\
0&\rho_{\downarrow\uparrow}&\rho_{\downarrow\downarrow}&0\\\
0&0&0&\rho_{22}\end{array}\right],$ (11)

where most off-diagonal elements of $\rho_{\text{mol}}$ describe coherences
between states of different charge occupancy, and so decouple from the
transport. The two single electron coherences, $\rho_{\sigma\bar{\sigma}}$,
are necessary in this treatment, as the spin accumulated within the system
extends into the $x$ and $y$ directions.

The reduced system density matrix time-evolves according to a generalized
quantum master equation (GME). Although most previous treatments of the
quantum dot spin-valve have used a real-time diagrammatic approach to
calculate the required transition rates [8, 73, 6], in the weak-coupling limit
a Born-Markov master equation (BMME) approach is equivalent. Here, the time-
evolution of $\rho_{\text{mol}}$ amounts to solving the Liouville-von Neumann
equation in the interaction picture, after tracing out the electrode degrees
of freedom:

$\displaystyle\frac{d}{dt}\rho_{\text{mol},I}(t)$
$\displaystyle=\text{Tr}_{\text{el}}\left[V_{I}(t),\rho_{\text{tot},I}(t)\right],$
(12)

where operators in the interaction picture are
$A_{I}(t)=e^{i\left(H_{\text{mol}}+H_{\text{el}}\right)t}Ae^{-i\left(H_{\text{mol}}+H_{\text{el}}\right)t}$.

Under the Born approximation, the influence of the Anderson-type molecule on
the evolution of the electrodes is negligible and the total density matrix
factorizes as $\rho_{\text{tot}}=\rho_{\text{mol}}\rho_{\text{el}}$. If one
further assumes that correlations in the electrodes decay faster than the
timescale of system dynamics, that is, that the transport is Markovian, then
one can apply standard steps to Eq.(12) and obtain a Redfield-type master
equation in the basis of eigenstates of $H_{\text{mol}}$ [81, 82]:

$\displaystyle i\dot{\rho}_{mn}$
$\displaystyle=\omega_{mn}\rho_{mn}+\sum_{\alpha}\sum_{\sigma\sigma^{\prime}}\sum_{kl}\>\Sigma_{\alpha,\sigma\sigma^{\prime}}^{>}(\omega_{lk})\langle
m|d^{{\dagger}}_{\sigma}|k\rangle\langle
k|d_{\sigma^{\prime}}|l\rangle\rho_{ln}+\Sigma_{\alpha,\sigma\sigma^{\prime}}^{<}(\omega_{kl})^{*}\langle
m|d_{\sigma}|k\rangle\langle
k|d^{{\dagger}}_{\sigma^{\prime}}|l\rangle\rho_{ln}$
$\displaystyle\>\>\>\>+\Sigma_{\alpha,\sigma\sigma^{\prime}}^{<}(\omega_{nl})\langle
m|d^{{\dagger}}_{\sigma}|k\rangle\rho_{kl}\langle
l|d_{\sigma^{\prime}}|n\rangle+\Sigma_{\alpha,\sigma\sigma^{\prime}}^{>}(\omega_{ln})^{*}\langle
m|d_{\sigma}|k\rangle\rho_{kl}\langle
l|d^{{\dagger}}_{\sigma^{\prime}}|n\rangle-(m\leftrightarrow
n,k\leftrightarrow l)^{*},$ (13)

where the last term indicates that the remaining four terms of the master
equation are obtained by switching the positions of $\langle{m}|\,$ and
$\,|{n}\rangle$, as well as $\langle{k}|\,$ and $\,|{l}\rangle$, then taking
the Hermitian conjugate of the result. The exact forms of the lesser,
$\Sigma_{\alpha,\sigma\sigma^{\prime}}^{<}(\omega)$, and greater,
$\Sigma_{\alpha,\sigma\sigma^{\prime}}^{>}(\omega)$, self-energies are in
Appendix A.

In Liouville space, Eq.(13) becomes a time-local first-order differential
equation,

$\displaystyle\dot{\bm{\rho}}(t)$ $\displaystyle=\mathbf{L}\bm{\rho}(t),$ (14)

where $\mathbf{L}$ is the time-independent Liouvillian superoperator
containing all system dynamics, which is also written explicitly in Appendix
B, and the system density matrix has been mapped to a vector in Liouville
space, $\rho_{\text{mol}}(t)\rightarrow\bm{\rho}(t)$, with

$\displaystyle\bm{\rho}(t)$
$\displaystyle=\left[\rho_{00},\rho_{\uparrow\uparrow},\rho_{\uparrow\downarrow},\rho_{\downarrow\uparrow},\rho_{\downarrow\downarrow},\rho_{22}\right]^{T}.$
(15)

The original formulation of Braun et al. [8], which many subsequent treatments
also use [73, 74, 6], rewrites the GME as a master equation for the system
charge states coupled to Bloch-like equations for the $x$, $y$, and $z$
components of the dot spin:
$\bm{\rho}=\left[\rho_{00},\rho_{11},\rho_{22},S_{x},S_{y},S_{z}\right]^{T}$,
with $\rho_{11}=\rho_{\uparrow\uparrow}+\rho_{\downarrow\downarrow}$ and

$\displaystyle\left\\{S_{x},S_{y},S_{z}\right\\}=\left\\{\frac{\rho_{\uparrow\downarrow}+\rho_{\downarrow\uparrow}}{2},i\frac{\rho_{\uparrow\downarrow}-\rho_{\downarrow\uparrow}}{2},\frac{\rho_{\uparrow\uparrow}-\rho_{\downarrow\downarrow}}{2}\right\\}.$
(16)

Although our master equation in Eq.(13) and the corresponding Liouvillian in
Eqs.(40)-(54) are not in this form, we note that they describe exactly the
same dynamics as long as the molecule-electrode coupling is weak: $\Gamma\ll
k_{B}T$, which all of our results conform to.

All calculations are performed in the nonequilibrium steady state,
$\bar{\bm{\rho}}$, which is the null vector of the full Liouvillian:

$\displaystyle\mathbf{L}\bar{\bm{\rho}}$ $\displaystyle=0.$ (17)

###  II.3 WTD

Let us begin to monitor time delays between successive quantum tunnelings in
the nonequilibrium steady state. Throughout the paper, we are solely concerned
with tunnelings to the drain, which are contained in the quantum jump operator
$\mathbf{J}$. To identify the exact form of $\mathbf{J}$, one could formally
rewrite Eq.(13) as an $n$-resolved master equation [83] and include all terms
associated with $\rho^{(n-1)}(t)$ [39]. However, one could also just directly
identify those terms in Eq.(13) that lower the system’s charge occupation by
$1$ via a tunneling to the drain, in this case including all single particle
coherences; this is the approach we take and the resulting quantum jump
operator is shown explicitly in Appendix B.

The WTD for $n$ consecutive waiting times, $w_{n}(\tau_{n},\dots,\tau_{1})$,
is defined as the joint probability distribution that, given an initial
tunneling to the drain, the next electron waits a time $\tau_{1}$ before
tunneling to the drain, the electron after that then waits a time $\tau_{2}$
before tunneling to the drain, and so on up to the $n$th successive electron:

$\displaystyle w_{n}(\tau_{n},\dots,\tau_{1})$
$\displaystyle=\frac{\text{Tr}\left[\prod\limits_{m=1}^{n}\left(\mathbf{J}e^{\left(\mathbf{L}-\mathbf{J}\right)\tau_{m}}\right)\mathbf{J}\bar{\bm{\rho}}\right]}{\text{Tr}\left[\bm{J}\bar{\bm{\rho}}\right]},$
(18)

where one can obtain Eq.(18) either from an extension of Brandes’
phenomenological formula [27] or the idle-time approach defined by van Kampen
[84, 58].

To compute expectation values and analyze the fluctuations, we introduce the
moment-generating function (MGF) for the joint WTD,

$\displaystyle K_{n}(\vec{\chi})$
$\displaystyle=\int_{0}^{\infty}d\vec{\tau}\>e^{i\vec{\chi}\cdot\vec{\tau}}w_{n}(\vec{\tau}),$
(19)

which contains vectors of waiting times,
$\vec{\tau}=\left(\tau_{n},\dots,\tau_{1}\right)$, and counting fields,
$\vec{\chi}=\left(\chi_{n},\dots,\chi_{1}\right)$. From the MGF in Eq.(19), we
can calculate the moments of the joint WTD as

$\displaystyle\langle\tau_{n}^{k_{n}}\dots\tau_{1}^{k_{1}}\rangle$
$\displaystyle=\frac{\partial^{k_{n}}}{\partial(i\chi_{n})^{k_{n}}}\dots\frac{\partial^{k_{1}}}{\partial(i\chi_{n})^{k_{1}}}K_{n}(\vec{\chi})\Big{|}_{\chi_{n}=\dots\chi_{1}=0};$
(20)

this is a relatively simple task given that the integral over $\vec{\tau}$ can
be computed analytically:

$\displaystyle
K_{n}(\vec{\chi})=(-1)^{n}\frac{\text{Tr}\left[\prod\limits_{m=1}^{n}\left(\mathbf{J}\left(\mathbf{L}-\mathbf{J}+i\chi_{m}\right)^{-1}\right)\mathbf{J}\bar{\bm{\rho}}\right]}{\text{Tr}\left[\mathbf{J}\bar{\bm{\rho}}\right]}.$
(21)

Our analysis will focus on the first and second moments of the first-order
WTD,

$\displaystyle\langle\tau^{k}\rangle$
$\displaystyle=\int_{0}^{\infty}d\tau_{1}\;\tau^{k}w_{1}(\tau_{1})$ (22)
$\displaystyle=k!(-1)^{k+1}\frac{\text{Tr}\left[\mathbf{J}\left(\mathbf{L}-\mathbf{J}\right)^{-(k+1)}\mathbf{J}\bar{\bm{\rho}}\right]}{\text{Tr}\left[\mathbf{J}\bar{\bm{\rho}}\right]},$
(23)

as well as the two-time correlations of the n-order WTDs, which are

$\displaystyle\langle\tau_{i}\tau_{j}\rangle$
$\displaystyle=\int^{\infty}_{0}d\vec{\tau}\tau_{i}\tau_{j}w_{n}(\vec{\tau})=(-1)^{i-j-1}\frac{\text{Tr}\left[\mathbf{J}\left(\mathbf{L}-\mathbf{J}\right)^{-2}\left(\mathbf{J}\left(\mathbf{L}-\mathbf{J}\right)^{-1}\right)^{i-j-1}\mathbf{J}\left(\mathbf{L}-\mathbf{J}\right)^{-2}\mathbf{J}\bar{\bm{\rho}}\right]}{\text{Tr}\left[\mathbf{J}\bar{\bm{\rho}}\right]}.$
(24)

####  II.3.1 WTD fluctuation statistics

We now introduce the randomness parameter [57, 58],

$\displaystyle
R=\frac{\langle\tau^{2}\rangle-\langle\tau\rangle^{2}}{\langle\tau\rangle^{2}},$
(25)

to provide a measure of variance relative to the mean and characterize the
Poissonian nature of the transport in the same manner as the widely used Fano
factor, $F=\frac{\langle I^{2}\rangle-\langle I\rangle^{2}}{\langle
I\rangle}$. If the counting of detected electrons is a Poissonian process,
then $F=1$, as the variance of a Poisson process is equal to its mean. Super-
Poissonian noise, $F>1$, therefore, indicates a variance greater than the
mean, which is produced by electron bunching. The opposite scenario of sub-
Poissonian noise, $F<1$, then indicates electron ‘anti-bunching’. These
processes are shown in Fig.(1).

The associated WTD of a Poisson process is exponential:
$w_{\text{Po}}(\tau)=\lambda e^{-\lambda\tau}$, with corresponding average
$\langle\tau\rangle=1/\lambda$ and variance
$\langle\tau^{2}\rangle-\langle\tau\rangle^{2}=1/\lambda^{2}$. As a result,
the randomness parameter is also $R=1$ for a Poisson process, with the same
interpretations for super- and sub-Poissonian noise as well.

Correlations between successive tunneling events will be measured using a
linear Pearson correlation coefficient,

$\displaystyle PC_{\tau_{n},\tau_{1}}$
$\displaystyle=\frac{\langle\tau_{n}\tau_{1}\rangle-\langle\tau\rangle^{2}}{\langle\tau^{2}\rangle-\langle\tau\rangle^{2}},$
(26)

where we are explicitly only considering the Pearson coefficient of the first,
$\tau_{1}$, and last, $\tau_{n}$, waiting times in a sequence, as it contains
nontrivial information about possible correlations in
$w_{n}(\tau_{n},\dots,\tau_{1})$. Other two-time correlations can always be
reduced to the averages performed over the lower order WTDs. For example,
consider the second-order mean

$\displaystyle\langle\tau_{n-k}\tau_{l}\rangle$
$\displaystyle=\int_{0}^{\infty}\prod\limits_{m=1}^{n}d\tau_{m}\;\tau_{n-k}\tau_{l}w_{n}(\tau_{n},\dots,\tau_{1})$
(27)
$\displaystyle=\int_{0}^{\infty}\prod\limits_{m=l}^{n-k}d\tau_{m}\;\tau_{n-k}\tau_{l}w_{n-l-k+1}(\tau_{n-k},\dots,\tau_{l}).$
(28)

The Pearson correlation coefficient is widely used in statistics as a measure
of the average linear correlation between two stochastic variables. In the
waiting time context, a positive value, $1>PC_{\tau_{n},\tau_{1}}>0$,
indicates that if the first waiting time, $\tau_{1}$, is longer (shorter) than
average, then it is probable that the waiting time between the $n^{\text{th}}$
and $(n+1)^{\text{th}}$ electron tunnelings, $\tau_{n}$, will also be longer
(shorter) than average. Negative correlations, $0>PC_{\tau_{n},\tau_{1}}>-1$,
in contrast, indicate that a longer (shorter) than average $\tau_{1}$ is
likely to be followed by a shorter (longer) than average $\tau_{n}$.

We will also use a function that measures the correlation between each
possible pair of the first two successive waiting times $\tau_{1}$ and
$\tau_{2}$:

$\displaystyle C(\tau_{2},\tau_{1})$
$\displaystyle=\frac{w_{2}(\tau_{2},\tau_{1})}{w_{1}(\tau_{2})w_{1}(\tau_{1})}-1.$
(29)

$C(\tau_{2},\tau_{1})$ is interpreted similarly to the Pearson coefficient,
although it provides a measure of the correlation between each pair of
successive waiting times, rather than an average over the entire space of
$\tau_{1}$ and $\tau_{2}$.

##  III Results

###  III.1 Stationary current

In Ref.[8], Braun et al. present plots of the stationary current as a function
of bias voltage in the same parameter regimes that we will probe with waiting
times. We will not reproduce the plots here, which are included in the
supplemental material, but will outline a brief explanation of the physics as
a way of contextualizing our later results. In the following, we will refer to
three magnetization angles: the parallel case is when the leads magnetization
is along the same axis, $\phi=0$; the perpendicular case is when the drain is
magnetized at an angle of $\phi=\frac{\pi}{2}$ to the source electrode; and
the anti-parallel case corresponds to a relative magnetization angle of
$\phi=\pi$. We also note that all discussions are for an equal spin-
polarization of the leads: $p_{S}=p_{D}=p$.

The parallel scenario shows, somewhat counter-intuitively, the same current
profile as transport through a standard Anderson model with no lead
magnetization. One might expect that, as the lead magnetization increases and
tunneling is dominated by majority spin electrons, only one transport channel
will be activated, even at high voltages when double occupancy is
energetically allowed. However, the small percentage of minority spin
electrons still play an active role in the transport; although they only
rarely tunnel into the dot, they also only rarely tunnel out, spending a much
greater time on the dot than majority spin electrons, thus creating a spin-
blockade. Since $\langle I\rangle$ is measured in the long-time limit, and the
original probability for a minority spin electron to tunnel in from the source
is small, it cannot distinguish between these time-scales for majority and
minority spins, and so ‘sees’ the same transport as through an Anderson-type
molecule coupled to nonmagnetic electrodes.

The average current of anti-parallel leads, in contrast, displays a
significant dependence on the strength of the magnetization within the leads.
One still observes the double step behavior associated with two distinct
transport channels, but for $p\rightarrow 1$ the current is severely
diminished. Here, majority spins from the source electrode are treated as
minority spins in the drain electrode, and vice versa, which means that total
tunneling through the system, and thus the current, is highly suppressed.

Besides this behavior, the most interesting current feature discussed by Braun
et al. occurs in the perpendicular case, which displays NDR between
$\varepsilon\leq V_{SD}/2\leq\varepsilon+U$; that is, when a single transport
channel is open. Braun et al. explain that, in this regime, increasing voltage
tends to align the accumulated spin of the dot in the opposite direction to
the magnetization in the drain, which strongly suppresses the current. The
effect is modified by the inclusion of the virtual exchange field, which
produces Larmor-like spin precession.

###  III.2 Bunching and anti-bunching

(a)

(b)

(c)

Figure 3: Randomness parameter, $R$, as a function of bias voltage for
different spin-polarizations and relative magnetization angles: (a) $\phi=0$,
(b) $\phi=\frac{\pi}{2}$, (c) $\phi=\pi$. Both electrodes have the same spin-
polarization: $p_{S}=p_{D}=p$. Other parameters are $\varepsilon=10k_{B}T$,
$U=30k_{B}T$, $k_{B}T=0.1\text{meV}$, $\gamma=0.01\text{meV}$.

In this section, we will analyze the transport using the randomness parameter
instead of the stationary current, as shown in Figs.(3(a))-(3(c)). We note
that we get similar results to that found by Braun et al. in Ref.[73] for the
Fano factor. First, as one can see in Fig.(3(a)), parallel lead magnetizations
and strong electrode spin-polarization produce highly super-Poissonian
transport, as $R\gg 1$ when a single transport channel is open in
$\varepsilon\leq V_{SD}/2\leq\varepsilon+U$. This originates directly from the
effect we discussed before; although transport is dominated by majority spin
electrons, when minority spin electrons tunnel into the dot, they tend to stay
there for a long time, due to a lack of available states in the drain. This
evidently causes electron bunching in tunnelings to the drain, and the
transport becomes strongly super-Poissonian.

Fig.(3(b)) also displays super-Poissonian transport, although the underlying
physical reason now originates from the NDR discussed above. As electrons
tunnel through the system, the accumulated spin tends to align in a direction
anti-parallel to the drain magnetization, so that after a series of tunnelings
eventually a spin-down electron becomes stuck in the dot: it cannot tunnel
back to the source due to the bias voltage; no other electrons can tunnel in
because double occupancy is energetically forbidden, and it cannot tunnel to
the drain because there is a low density of states for that spin-polarization.
Again, this produces electron bunching and $R>1$, although we note that that
magnitude is much reduced compared to Fig.(3(a)).

In the case of anti-parallel magnetization angles, shown in Fig.(3(c)), the
transport is at all voltages sub-Poissonian. At $p=0$, the $R$ is that of
transport through a normal Anderson model, with behavior that is well-known in
the literature [85, 86, 87]. As the spin-polarization increases, however,
electron tunneling through the system is almost completely suppressed and the
time between tunnelings to the drain exponentially increases, pushing the
transport towards Poissonian behavior.

Overall, Figs.(3(a))-(3(c)) demonstrate that a quantum dot spin-valve is able
to produce statistics on demand; simply by changing the relative magnetization
angle one is able to produce super-Poissonian, sub-Poissonian, and Poissonian
noise.

###  III.3 Temporal correlations

Beyond just the bunching and anti-bunching available in plots of the
randomness parameter, the spin-valve also displays temporal correlations in
certain regimes, which one can see in Figs.(4(a))-(4(c)).

(a)

(b)

(c)

Figure 4: Pearson correlation coefficient, $PC_{\tau_{2},\tau_{1}}$, between
the first two successive waiting times, $\tau_{1}$ and $\tau_{2}$, as a
function of bias voltage for different spin-polarizations and relative
magnetization angles: (a) $\phi=0$, (b) $\phi=\frac{\pi}{2}$, (c) $\phi=\pi$.
All parameters are the same as in Figs.(3(a))-(3(c)).

At all magnetization angles and for small lead spin-polarization, one sees
correlation behavior between the first two successive waiting times similar to
that already reported for a standard Anderson model [28, 30]. At low voltages,
when zero or one transport channel is open, successive waiting times are
completely uncorrelated. In the double occupancy regime, however, the dot can
fill with two electrons. A short initial waiting time, $\tau_{1}$, therefore,
likely means that both electrons tunnel to the drain in quick succession. The
dot then has to be filled with an electron from the source before another
tunneling to the drain can occur, making the second waiting time, $\tau_{2}$,
much longer; $\tau_{1}$ and $\tau_{2}$ become negatively correlated. The
linear strength of this correlation is evidently not very large, as
$\max\\{|PC|\\}=0.1$, and it completely disappears for highly polarized leads.
In the highly spin-polarized anti-parallel case, it is now unlikely that the
dot will be doubly occupied and also unlikely that both of those electrons
will tunnel to the drain in quick succession. In the perpendicular case, the
reasoning is similar, but the magnetization angles are closer so spin-
precession can somewhat mitigate the correlation suppression.

In Fig.(4(a)), however, one can see that, beside the already discussed
correlation behavior at high voltages, parallel magnetization and high lead
spin-polarization produces strong positive correlations between successive
waiting times. This occurs in the small voltage window corresponding to the
second transport channel opening, as one can see by direct comparison with the
stationary current. A more comprehensive picture of these correlations is
shown in Fig.(5), which plots the two-time correlation function defined in
Eq.(29) at the voltage corresponding to the Pearson coefficient maximum.

Here, one can see that the large Pearson coefficient originates from positive
correlations between two long waiting times:
$\tau_{1},\tau_{2}>3\langle\tau\rangle$. Although in the plot we have
constricted the range to values between $-1<C(\tau_{1},\tau_{2})<1$, so as to
see the fine detail, the actual values go much higher. The correlation between
two extremely long successive waiting times, for example, is
$C(6\langle\tau\rangle,6\langle\tau\rangle)=20$. Correspondingly, a short
initial waiting time and a long secondary waiting time, or vice versa, are
strongly negatively correlated.

Figure 5: Relative correlation between successive waiting times,
$C(\tau_{2},\tau_{1})$, plotted for parallel magnetizations: $\phi=0$. Spin-
polarization within the leads is $p=0.99$ and the voltage is
$V_{SD}=75k_{B}T$, corresponding to the peak in the positive correlations from
Fig.(4(a)). All other parameters are the same as in Figs.(3(a))-(3(c)).

To understand the nature of these temporal correlations for pairs of long
waiting times, consider Fig.(6(a)). Here, we have plotted the $z$-component of
the dot spin averaged over the quantum measurement projected density matrix,
$\langle
S_{z}\rangle=(\rho_{\uparrow\uparrow}-\rho_{\downarrow\downarrow})/2$, as a
function of the initial waiting time, $\tau_{1}$. Note that, since the lead
magnetizations are parallel, the $x$\- and $y$-components of the average spin
are zero. The dashed line corresponds to the projected density matrix of a
system that starts in the stationary state, undergoes a jump to the drain,
$\mathbf{J}$, and then evolves for some time without a jump to the drain,
$e^{\left(\mathbf{L}-\mathbf{J}\right)\tau_{1}}$:

$\displaystyle\bm{\rho}(\tau_{1})$
$\displaystyle=\frac{e^{(\mathbf{L}-\mathbf{J})\tau_{1}}\mathbf{J}\bar{\bm{\rho}}}{\text{Tr}\left[e^{(\mathbf{L}-\mathbf{J})\tau_{1}}\mathbf{J}\bar{\bm{\rho}}\right]}.$
(30)

For $\tau_{1}<2\langle\tau\rangle$, the accumulated spin in the dot is in the
positive $z$ direction, so that it is likely to be occupied by majority spin
electrons from the source. Fig.(6(a)) shows, however, that when this majority
spin electron tunnels out to the drain, modeled now with

$\displaystyle\bm{\rho}(\tau_{1})$
$\displaystyle=\frac{\mathbf{J}e^{(\mathbf{L}-\mathbf{J})\tau_{1}}\mathbf{J}\bar{\bm{\rho}}}{\text{Tr}\left[\mathbf{J}e^{(\mathbf{L}-\mathbf{J})\tau_{1}}\mathbf{J}\bar{\bm{\rho}}\right]},$
(31)

then the accumulated spin in the dot returns to zero and there is no
preference for a tunneling of either spin, apart from that imposed by lead
spin-polarization.

In contrast, if the initial waiting time is longer,
$\tau_{1}>2\langle\tau\rangle$, then the accumulated spin in the dot tends to
be negative, and this accumulated spin remains after the next tunneling to the
drain. Since this is within the voltage regime where the second transport
channel is beginning to open, which one can see by direct comparison with
$\langle I\rangle$, it is likely that this second jump to the drain is not due
to the minority spin electron tunneling out, but rather a majority spin
tunneling in from the source and out quickly to the drain. Consequently, the
system remains likely to be occupied by a minority spin electron and this low
probability mechanism is again required for the next jump, ensuring that
$\tau_{2}$ is long as well.

Indeed, one can see this directly in Fig.(6(b)), where we have plotted
$\langle S_{z}\rangle$ during $\tau_{2}$, after setting
$\tau_{1}=5\langle\tau\rangle$. The dashed-line indicates the dot is highly
likely to possess a minority spin during the $\tau_{2}$. At small $\tau_{2}$,
$\langle S_{z}\rangle$ is close to zero, which demonstrates the relatively
small chance that this jump was a minority spin electron leaving the system.
Note that we have not shown the case of a short $\tau_{1}$, as the resulting
plot is extremely similar to Fig.(6(a)); the system empties itself with a
majority spin tunneling to the drain, and so the dynamics in $\tau_{1}$
repeat.

Evidently, these correlations are highly dependent on the leads being strongly
polarized, as they disappear even for $p=0.5$. They are also limited to a
narrow voltage and magnetization angle regime; this is illustrated in Fig.(7),
which maps $PC_{\tau_{2},\tau_{1}}$ and $R$ as a function of both $V_{SD}$ and
$\phi$.

At low voltages, in the Coulomb blockade regime, the second spin-channel is
completely closed and the mechanism of another majority spin electron
tunneling in is not allowed. At high voltages, however, the correlations also
disappear because the dot can now easily be doubly occupied and the
accumulated spin does not tend to negative values at long times. The narrow
range of magnetization angles, on the other hand, is understood by considering
what happens as the leads move away from parallel alignment. As
$\phi\rightarrow\pi/2$, spin-precession allows the minority spin electron to
couple to states in the drain and thus escape the dot at earlier times, and as
$\phi\rightarrow\pi$, the current as a whole is diminishes, which directly
suppresses the correlations as well.

(a)

(b)

Figure 6: $\langle S_{z}\rangle$, the $z$-component of the measurement-
averaged spin, (a) as a function of the initial waiting time, $\tau_{1}$, and
(b) as a function of the second waiting time, $\tau_{2}$, given a long initial
waiting time: $\tau_{1}=5\langle\tau_{1}\rangle$. All parameters are the same
as in Fig.(5).

(a)

(b)

Figure 7: Transport characteristics of the spin-valve across the relevant
voltage range, $V_{SD}$, and half-rotation of the drain electrode, $\phi$. In
(a) the successive waiting time correlation, $PC_{\tau_{2},\tau_{1}}$, is
plotted, while (b) contains the randomness parameter, $R$. The spin
polarization is $p=0.99$ and other parameters are the same as in Fig.(5).

In an experimental setup, it is unlikely that one would have a completely
efficient detector capable of measuring each electron tunneling event. A
natural question, then, is whether these correlations survive into later
tunnelings; that is, if the detector misses a series of waiting times after
the first one, $\tau_{2},\dots,\tau_{n-1}$, but captures the $n^{\text{th}}$
waiting time, $\tau_{n}$, are $\tau_{1}$ and $\tau_{n}$ still correlated?

Fig.(8) shows that, for this model, this is indeed the case. The initial
waiting time is non-negligibly correlated with many successive waiting times,
even up to the $w_{1}(\tau_{10})$, as shown in the plot. As one would expect,
however, as $n$ increases, the Pearson coefficient between $\tau_{1}$ and
$\tau_{n}$ decreases. The peak in each curve also occurs over a similar
voltage range, indicating that the same processes discussed previously also
produce the correlations in higher-order waiting times.

Figure 8: Higher-order temporal correlations between $\tau_{1}$ and
$\tau_{n}$, $PC_{\tau_{n},\tau_{1}}$, as a function of bias voltage $V_{SD}$.
All parameters are the same as in Figs.(4(a)).

##  IV Conclusions

Molecular spin-valves are experimentally realizable transport junctions that
offer the possibility of harnessing the quantum spin, rather than just the
electric charge, of tunneling electrons. In this paper, we explore the extent
to which current fluctuations can be controlled in a spin-valve system, using
the WTD and its associated statistics, alongside a Born-Markov master equation
approach to describe the system dynamics.

First, we analyzed the randomness parameter, $R$, as a function of spin-
polarization and relative magnetization angle, finding similar behavior to
that of the Fano factor in the same regimes. For parallel magnetizations, and
in the single occupancy voltage regime, $R\gg 1$ when $p\rightarrow 1$; the
small percentage of minority spin tunnelings cause severe electron bunching.
This is also evident for perpendicular orientations, as the NDR in the
current-voltage plot manifests as $R>1$ too. Here, though, the level of
electron bunching is much lower, as Larmor-like spin precession somewhat
mitigates the effect. Sub-Poissonian transport is observed for the case of
anti-parallel leads and low magnetization, in accordance with known behavior
in an Anderson model, while high magnetization strongly suppresses the current
and thus drives transport towards Poissonian behavior. Most importantly, it is
observed that all three transport regimes are accessible simply by changing
only the applied voltage and relative orientation of lead magnetization.

Temporal correlations are absent for any system arrangement allowing only a
single transport channel, while voltages well above the double occupation
threshold yield weak negative correlations. The most interesting region occurs
within a small voltage band around the opening of the second transport channel
and small angular deviation from the parallel orientation; strong temporal
correlations emerge for highly magnetic leads within this region. From the
scaled correlation map between all possible pairs of successive waiting times,
one can see that, while two short waiting times are uncorrelated, two long
waiting times are strongly positively correlated. For this to occur, minority
spin accumulates in the molecule during the first long waiting time and is
retained after the next tunneling event, making it highly likely that a second
long waiting time follows the first. For this mechanism to function, the
minority spin electron must become thoroughly trapped within the molecule; the
leads must be highly spin-polarized to severely limit availability of minority
electrons. This is assisted by a significant Coulomb repulsion, which allows
trapped minority electrons to impede majority electrons. Increasing the
voltage bias well into the double occupancy regime diminishes the effect by
allowing electrons to easily overcome repulsion, while deviating from the
parallel electrode orientation aids the mobility of minority spin electrons by
means of greater Lamor-like precession.

While the focus is largely on exploring correlations between successive
waiting times, a surprising observation is made: significant correlations also
extend to higher-order waiting time distributions. Driven by the same
processes as those in successive waiting times, correlations between the
$1^{\text{st}}$ and the $n^{th}$ waiting time decreases with $n$. These
correlations are significant even up to tenth-order: the maximum value of
$PC_{\tau_{10},\tau_{1}}=0.1998$.

Not only are the aforementioned temporal correlations relatively large, the
various regimes are accessible within a single device by manipulation of both
$V_{SD}$ and $\phi$, each over a short range. This demonstrates how the spin-
valve device can controllably switch between regimes to provide versatile
information transfer. It should be noted that these functions are highly
sensitive to electron detection errors. Specifically, if the function of the
device is to encode information in the duration of the waiting time, failing
to detect a single electron will cause significant processing errors. However,
the observation of notable correlations in higher-order waiting times does
offer some robustness, as the third waiting time is still correlated with the
first even if the second is undetected.

##  V Acknowledgments

The authors thank Julian Lawn for creating Figs.(1)-(2) as well as his helpful
discussions.

##  Appendix A Spin-resolved self-energies

The self-energies in Eq.(13) are Fourier transforms of bath-correlation
functions, obtained using standard methods [81], and can be written as

$\displaystyle\Sigma^{<}_{\alpha,\sigma\sigma^{\prime}}(\omega)$
$\displaystyle=\Delta^{<}_{\alpha,\sigma\sigma^{\prime}}(\omega)+\frac{i}{2}\gamma^{\alpha}_{\sigma\sigma^{\prime}}f^{+}_{\alpha}(\omega)$
(32) $\displaystyle\Sigma^{>}_{\alpha,\sigma\sigma^{\prime}}(\omega)$
$\displaystyle=-\Delta^{>}_{\alpha,\sigma\sigma^{\prime}}(\omega)-\frac{i}{2}\gamma^{\alpha}_{\sigma\sigma^{\prime}}f^{-}_{\alpha}(\omega).$
(33)

Here, $f^{\pm}_{\alpha}(\omega)$ are the relevant Fermi-Dirac functions for
electrode $\alpha$,

$\displaystyle f^{\pm}_{\alpha}(\omega)$
$\displaystyle=\frac{1}{1+e^{\pm\left(\omega-\mu_{\alpha}\right)/k_{B}T}},$
(34)

and $\Delta^{<,>}_{\alpha,\sigma\sigma^{\prime}}(\omega)$ and
$\gamma^{\alpha}_{\sigma\sigma^{\prime}}$ describe the renormalization and
broadening of the system energy levels due to interaction with the electrodes,
respectively.

The energy-independent $\gamma^{\alpha}_{\sigma\sigma^{\prime}}$ terms are

$\displaystyle\gamma^{S}_{\sigma\sigma}$ $\displaystyle=\Gamma_{S}\left(1\pm
p_{S}\right)$ (35) $\displaystyle\gamma^{S}_{\sigma\bar{\sigma}}$
$\displaystyle=0$ (36) $\displaystyle\gamma^{D}_{\sigma\sigma}$
$\displaystyle=\Gamma_{D}\left[1\pm p_{D}\cos(\phi)\right]$ (37)
$\displaystyle\gamma^{D}_{\sigma\bar{\sigma}}$
$\displaystyle=\Gamma_{D}\sin(\phi)p_{D},$ (38)

where $\uparrow$ and $\downarrow$ correspond to signs of $+$ and $-$,
respectively. In the wideband limit, the renormalizations can be calculated
analytically using the digamma function, $\psi(x)$ [39]:

$\displaystyle\Delta^{<}_{\alpha,\sigma\sigma^{\prime}}(\omega)$
$\displaystyle=\frac{\gamma^{\alpha}_{\sigma\sigma^{\prime}}}{2\pi}\Re\left\\{\psi\left(\frac{1}{2}+\frac{i}{2\pi
k_{B}T}\left(\omega-\mu_{\alpha}\right)\right)\right\\},$ (39)

with
$\Delta^{>}_{\alpha,\sigma\sigma^{\prime}}(\omega)=-\Delta^{<}_{\alpha,\sigma\sigma^{\prime}}(\omega)$.

##  Appendix B Spin-valve BMME in Liouville space

To move from the GME in Hilbert space to a time-local differential equation in
Liouville space, the system dynamics are collected into one superoperator,
$\mathbf{L}$, which has the form

$\displaystyle\mathbf{L}=\sum_{\alpha}(\mathbf{A}_{\alpha}-i\mathbf{D}_{\alpha}),$
(40)

where

$\displaystyle\mathbf{A}_{\alpha}$ $\displaystyle=\left[\begin{array}[]{c c c
c c
c}-2\Gamma_{\alpha}f_{\alpha}^{+}&\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha}^{-}&\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha}^{-}&\gamma^{\alpha}_{\downarrow\uparrow}f_{\alpha}^{-}&\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha}^{-}&0\\\
\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha}^{+}&-\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha,U}^{+}-\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha}^{-}&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&0&\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha,U}^{-}\\\
\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha}^{+}&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&\Gamma_{\alpha}(-f_{\alpha,U}^{+}-f_{\alpha}^{-})&0&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&-\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha,U}^{-}\\\
\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha}^{+}&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&0&\Gamma_{\alpha}(-f_{\alpha,U}^{+}-f_{\alpha}^{-})&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&-\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha,U}^{-}\\\
\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha}^{+}&0&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&\frac{1}{2}\gamma^{\alpha}_{\uparrow\downarrow}(f_{\alpha,U}^{+}-f_{\alpha}^{-})&-\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha,U}^{+}-\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha}^{-}&\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha,U}^{-}\\\
0&\gamma^{\alpha}_{\downarrow\downarrow}f_{\alpha,U}^{+}&-\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha,U}^{+}&-\gamma^{\alpha}_{\uparrow\downarrow}f_{\alpha,U}^{+}&\gamma^{\alpha}_{\uparrow\uparrow}f_{\alpha,U}^{+}&-2\Gamma_{\alpha}f_{\alpha,U}^{-}\end{array}\right]\text{
and}$ (47) $\displaystyle\mathbf{D}_{\alpha}$
$\displaystyle=\left[\begin{array}[]{c c c c c c}0&0&0&0&0&0\\\
0&0&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0&0\\\
0&\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0&0&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0\\\
0&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0&0&\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0\\\
0&0&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&-\gamma^{\alpha}_{\uparrow\downarrow}\delta_{\alpha}&0&0\\\
0&0&0&0&0&0\end{array}\right].$ (54)

In defining the matrices in Eq.(47) and Eq.(54), we have introduced the new
notation

$\displaystyle\delta_{\alpha}$
$\displaystyle=\frac{1}{2\pi}\Re\left\\{\psi\left(\frac{1}{2}+\frac{i}{2\pi
k_{B}T}(\varepsilon+U-\mu_{\alpha})\right)-\right.$ $\displaystyle\hskip
42.67912pt\left.\psi\left(\frac{1}{2}+\frac{i}{2\pi
k_{B}T}(\varepsilon-\mu_{\alpha})\right)\right\\},$ (55) $\displaystyle
f_{\alpha}^{\pm}$ $\displaystyle=f_{\alpha}^{\pm}(\varepsilon),\hskip
14.22636ptf_{\alpha U}^{\pm}=f_{\alpha}^{\pm}(\varepsilon+U).$ (56)

Following the procedure outlined at the start of Section II.3, the jump
superoperator is defined as

$\displaystyle\mathbf{J}$ $\displaystyle=\left[\begin{array}[]{c c c c c
c}0&\gamma^{D}_{\uparrow\uparrow}f_{D}^{-}&\gamma^{D}_{\uparrow\downarrow}f_{D}^{-}&\gamma^{D}_{\uparrow\downarrow}f_{D}^{-}&\gamma^{D}_{\downarrow\downarrow}f_{D}^{-}&0\\\
0&0&0&0&0&\gamma^{D}_{\downarrow\downarrow}f_{D,U}^{-}\\\
0&0&0&0&0&-\gamma^{D}_{\uparrow\downarrow}f_{D,U}^{-}\\\
0&0&0&0&0&-\gamma^{D}_{\uparrow\downarrow}f_{D,U}^{-}\\\
0&0&0&0&0&\gamma^{D}_{\uparrow\uparrow}f_{D,U}^{-}\\\
0&0&0&0&0&0\end{array}\right].$ (63)

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