# Secular non-secular master equation

Le Tuan Anh Ho  Institute of Research and Development, Duy Tan University, Da
Nang, Viet Nam  Theory of Nanomaterials Group, Katholieke Universiteit Leuven,
Celestijnenlaan 200F, B-3001 Leuven, Belgium  Liviu F. Chibotaru  Theory of
Nanomaterials Group, Katholieke Universiteit Leuven, Celestijnenlaan 200F,
B-3001 Leuven, Belgium

(August 27, 2024)

###### Abstract

Redfield non-secular master equation governing relaxation of a spin in weak
interaction with a thermal bath is studied. Using the fact that the relaxation
follows the exponential law, we prove that in most cases the semi-secular
approximation is sufficient to find the system relaxation rate. Based on this,
a “secular” form of the non-secular master equation is for the first time
developed which correctly set up one of most fundamental equations in
relaxation investigation. This key secular form allows us to derive a general
formula of the phonon-induced quantum tunneling rate which is valid for the
entire range of temperature regardless of the basis. In incoherent tunneling
regime and localized basis, this formula reduces to the ubiquitous incoherent
tunneling rate. Meanwhile, in eigenstates basis, this tunneling rate is
demonstrated to be equal to zero. From this secular form, we end the
controversy surrounding the selection of basis for the secular approximation
by figuring out the conditions for using this approximation in localized and
eigenstates basis. Particularly, secular approximation in localized basis is
justified in the regime of high temperature and small tunnel splittings. In
contrast, a large ground doublet’s tunnel splitting is required for the
secular approximation in eigenstates basis. With these findings, this research
lays a sound foundation for any treatments of the spin-phonon relaxation under
any conditions provided that the non-secular master equation is relevant.

Suggested keywords

###### pacs:

33.15.Ta

††preprint: APS/123-QED

##  I Introduction

Real physical systems are always in contact with its surroundings, and spin
system is not an exception. In non-equilibrium state, a spin in thermal
contact with a reservoir will undergo a process of relaxation where its energy
is exchanged with environment of which quantized particles of the lattice
vibrations, phonons, is one of the most important contributions. Understanding
this spin-phonon relaxation process is thus crucial for many fundamental
problems and applications such as quantum coherence/decoherence in spin-based
qubit Zurek (2003); Stamp and Gaita-Ariño (2009); Golovach  _et al._ (2004);
Vincent  _et al._ (2012), implementation of spintronics devices Žutić  _et
al._ (2004); Bogani and Wernsdorfer (2008); Mannini  _et al._ (2010); Ardavan
_et al._ (2007), or high-density information storage/quantum computing using
single-molecule/single-atom magnets Sieklucka and Pinkowicz (2016);
Leuenberger and Loss (2001); Sessoli  _et al._ (1993); Zhang  _et al._ (2015);
Dunbar (2012).

Of all the possible scenarios, a spin in weakly coupling with the phonon bath
is one of the most often occurred. In this scenario, the equation of motion of
the spin reduced density matrix, so-called Redfield master equation, can be
derived using Born approximation of weakly coupling and Markov approximation
of short memoryBlum (2012); May and Kühn (2011). From this, characteristics of
the relaxation process can be figured out.

Formulated in the operators form, the Redfield master equation can be
represented by different specific matrix forms depending on the basis. In
molecular magnetism generally and single-molecule/single atom magnets
particularly, up to now existing works can be divided into two
groups:Leuenberger and Loss (2000); Mannini  _et al._ (2010); Garanin (2011);
Garanin and Chudnovsky (1997); Villain  _et al._ (1994); Gómez-Coca  _et al._
(2014); Hartmann-boutron  _et al._ (1996); Luis  _et al._ (1998) one uses the
localized (natural) basis and the other uses eigenstates (diagonal) basis . In
principle, the matrix forms of the Redfield master equation in these two bases
are equivalent Garanin (2011). However, due to its complexity in practice,
most of the theoretical works only used the secular approximation matrix form
of the master equation in either localized basis or eigenstates basis.
Unfortunately, these secular master equations are not equivalent to each
other. Moreover, the conditions for their application are also ambiguous. As a
consequence, the results and interpretations of involved physical quantities
are unreliable and varied between these bases.

Playing a vital role in single-molecule/single-atom magnets relaxation,
phonon-induced quantum tunneling effect has been rigorously investigated using
the master equation since the early days of single-molecule magnets (SMMs). By
treating SMM relaxation using the secular form in localized basis, a formula
for the _incoherent_ spin tunneling rate at (near) resonance has been proposed
Villain  _et al._ (1997); Garanin and Chudnovsky (1997); Leuenberger and Loss
(2000). From then, this formula is ubiquitously used in a wide range of
temperatureLiddle and van Slageren (2015); Atanasov  _et al._ (2015); Zadrozny
_et al._ (2013); Fort  _et al._ (1998), even when the incoherence condition is
unjustified. These will bias the interpretation of data and lead to inaccurate
explanation of the physical reasons causing fast/slow relaxation. Accordingly,
tactics for SMMs improvement is likely inadequate. In combination with the
mentioned weakness of the secular approximation, these urge for the necessity
of a new formula of the phonon-induced quantum tunneling rate derived from the
non-secular Redfield master equation which can accurately describe the effect
in any circumstance.

In order to elucidate these problems, in this work, we start from the most
general equation governing the relaxation process in the system, Redfield non-
secular master equation, then develop an unique unprecedented secular form for
it. Having this form, we apply it to the localized and eigenstates basis to
find the conditions for the secular approximation in these bases. During this
process, the issue concerning the general formula for the phonon-induced
quantum tunneling rate is also tackled. The paper is then organized into 5
sections. In particular, we devote Section II for derivation of the “secular”
form of the non-secular master equation regardless of the basis. A general
formula for the phonon-induced quantum tunneling rate is also introduced in
this section. Equipped with these, Section III concentrates on the detailed
form of the developed “secular” non-secular master equation and accordingly
quantum tunneling rate in the localized basis. Meanwhile, the master equation
in eigenstates basis and related quantities are investigated in Section IV. As
usual, implications and conclusions are given in the last section.

##  II Secular form of non-secular master equation

Relaxation of a spin system with Hamiltonian $\mathcal{H}$ in a thermal bath
can be described by the Redfield master equation Blum (2012); Garanin (2011):

$\frac{d\rho}{dt}=-i\left[\mathcal{H},\rho\right]+\hat{R}\rho,$ (1)

where $\hat{R}$ is the Redfield super-operator. In a specific basis
$\left\\{\ket{m}\right\\}$, the above equation becomes

$\displaystyle\frac{d\rho_{mn}}{dt}$
$\displaystyle=\sum_{k,l}R^{\prime}_{mn,kl}\rho_{kl},$ (2)

where

$R_{mn,kl}^{\prime}\equiv
R_{mn,kl}+i\left(\delta_{mk}\mathcal{H}_{ln}-\mathcal{H}_{mk}\delta_{ln}\right).$
(3)

Here $R_{mm,mm}=-\sum_{n\neq m}R_{nn,mm}$ is the rate of population loss from
state $\ket{m}$ to all other $\ket{n}\neq\ket{m}$;
$R_{mm,nn}\equiv\Gamma_{mn}\,\forall n\neq m$ is the population transition
rate from $\ket{n}$ to $\ket{m}$; $R_{mn,mn}\equiv-\gamma_{mn}$ is the
coherence dephasing rate of $\rho_{mn}$; and other $R_{mn,kl}$ represents the
coherence transfer rate at which the varied amplitude of the density matrix
$\rho_{kl}$ influencing the element $\rho_{mn}$.

It is clear that solution of equation Eq. (2) is a linear combination of time
exponentials where exponents $\lambda_{i}$ are eigenvalues of the matrix
$\hat{R}^{\prime}$. Since this system of equation describes the relaxation of
a physical system, there exists one eigenvalue, so-called $\lambda_{0}$,
corresponding to the equilibrium states and thus equal to 0. Meanwhile, one of
its eigenvalue, so-called $\lambda$, describes the “slow” relaxation of the
spin system. Other eigenvalues are usually large and fast damped. Without loss
of generality, we can assume that the solution of Eq. (2) is approximately of
the form:

$\rho_{mn}=A_{mn}e^{-\lambda t}+B_{mn}.$ (4)

Insert this form into Eq. (2) then identify those terms with the exponent
$\lambda$ and solve for $A_{mn}$, we obtain

$A_{mn}=\frac{1}{1-\frac{\left|R_{mn,nm}^{\prime}\right|^{2}}{\left|R_{mn,mn}^{\prime}+\lambda\right|^{2}}}\frac{1}{R_{mn,mn}^{\prime}+\lambda}\sum_{\left(kl\right)\notin\left\\{mn,nm\right\\}}\left(\frac{R^{\prime}_{mn,nm}R^{\prime}_{nm,kl}}{R_{mn,mn}^{\prime*}+\lambda}-R^{\prime}_{mn,kl}\right)A_{kl},$
(5)

where we have taken into account the fact $A_{nm}=A_{mn}^{*}$ and
$R_{mn,kl}^{*}=R_{nm,lk}$ (see Appendix A).

In a basis where $\ket{m}$ and $\ket{n}$ mainly reside in the subspace of
different $m^{\mathrm{th}}$ and $n^{\mathrm{th}}$ doublet/singlet,
$R_{mn,mn}^{\prime}$ can be approximated as:

$R_{mn,mn}^{\prime}=R_{mn,mn}+i\omega_{nm},$ (6)

where $\omega_{mn}=\varepsilon_{m}-\varepsilon_{n}$ is the energy gap between
these two doublets/singlets. Since $\left|\omega_{nm}\right|$ is effectively
much larger than any $R_{mn,kl}$, $A_{mn}$ thus becomes negligible. The only
noticeable components are $A_{mm^{\prime}}$ where $\ket{m}$ and
$\ket{m^{\prime}}$ mainly resides in the same $m^{\mathrm{th}}$ doublet’s
subspace. A critical consequence of this property is that the semi-secular
approximation is _always sufficient_ in finding the dynamics of $\rho_{mm}$
and $\rho_{mm^{\prime}}$ as long as the above condition of the basis is
satisfied, i.e. we can approximate:

$\displaystyle\frac{d\rho_{mm}}{dt}$
$\displaystyle\approx\sum_{n}\left(R_{mm,nn}^{\prime}\rho_{nn}+R_{mm,nn^{\prime}}^{\prime}\rho_{nn^{\prime}}\right),$
(7) $\displaystyle\frac{d\rho_{mm^{\prime}}}{dt}$
$\displaystyle\approx\sum_{n}\left(R_{mm^{\prime},nn}^{\prime}\rho_{nn}+R_{mm^{\prime},nn^{\prime}}^{\prime}\rho_{nn^{\prime}}\right).$
(8)

These form a closed system of equations. In order to have a closed “secular”
system of equations of the diagonal density matrix elements only, we simplify
$A_{mm^{\prime}}$ using the semi-secular approximation version of Eq. (5) to
obtain

$A_{mm^{\prime}}=C_{mm^{\prime}}\left(A_{mm}-A_{m^{\prime}m^{\prime}}\right)+\sum_{k}D_{mm^{\prime},kk}A_{kk}+\sum_{k\neq
m,m^{\prime}}D_{mm^{\prime},kk^{\prime}}A_{kk^{\prime}},$ (9)

where

$\displaystyle
C_{mm^{\prime}}\equiv-\frac{1}{1-\frac{\left|R_{mm^{\prime},m^{\prime}m}^{\prime}\right|^{2}}{\left|R_{mm^{\prime},mm^{\prime}}^{\prime}+\lambda\right|^{2}}}\frac{i}{R_{mm^{\prime},mm^{\prime}}^{\prime}+\lambda}\left(\frac{R_{mm^{\prime},m^{\prime}m}\mathcal{H}_{m^{\prime}m}}{R_{mm^{\prime},mm^{\prime}}^{\prime*}+\lambda}+\mathcal{H}_{mm^{\prime}}\right),$
(10) $\displaystyle
D_{mm^{\prime},kl}\equiv\frac{1}{1-\frac{\left|R_{mm^{\prime},m^{\prime}m}^{\prime}\right|^{2}}{\left|R_{mm^{\prime},mm^{\prime}}^{\prime}+\lambda\right|^{2}}}\frac{1}{R_{mm^{\prime},mm^{\prime}}^{\prime}+\lambda}\left(\frac{R_{mm^{\prime},m^{\prime}m}R_{m^{\prime}m,kl}}{R_{mm^{\prime},mm^{\prime}}^{\prime*}+\lambda}-R_{mm^{\prime},kl}\right),$
(11) $\displaystyle
R_{mm^{\prime},mm^{\prime}}^{\prime}=R_{mm^{\prime},mm^{\prime}}+i\left(\mathcal{H}_{m^{\prime}m^{\prime}}-\mathcal{H}_{mm}\right),$
(12)

Apparently, Eq. (9) is self-consistent. Drawing a similar expression for
$A_{kk^{\prime}}$ and inserting back, finally we reach

$A_{mm^{\prime}}=\sum_{k}\left[F_{mm^{\prime},kk^{\prime}}\left(A_{kk}-A_{k^{\prime}k^{\prime}}\right)+G_{mm^{\prime},kk}A_{kk}\right],$
(13)

where

$\displaystyle F_{mm^{\prime},kk^{\prime}}\equiv
C_{mm^{\prime}}\delta_{mk}\delta_{m^{\prime}k^{\prime}}+\sum_{l\neq
m,m^{\prime}}D_{mm^{\prime},ll^{\prime}}C_{ll^{\prime}}\delta_{lk}\delta_{l^{\prime}k^{\prime}}+\sum_{l\neq
m,m^{\prime}}\sum_{p\neq
l,l^{\prime}}D_{mm^{\prime},ll^{\prime}}D_{ll^{\prime},pp^{\prime}}C_{pp^{\prime}}\delta_{pk}\delta_{p^{\prime}k^{\prime}}+\ldots,$
(14) $\displaystyle G_{mm^{\prime},kk}\equiv D_{mm^{\prime},kk}+\sum_{l\neq
m,m^{\prime}}D_{mm^{\prime},ll^{\prime}}D_{ll^{\prime},kk}+\sum_{l\neq
m,m^{\prime}}\sum_{p\neq
l,l^{\prime}}D_{mm^{\prime},ll^{\prime}}D_{ll^{\prime},pp^{\prime}}D_{pp^{\prime},kk}+\ldots.$
(15)

Denoting $\varLambda$ as the expression of $A_{nn}$ and $A_{nn^{\prime}}$
corresponding to the right-hand side of Eq. (7) and simplifying with the help
of Eq. (13), we obtain

$\displaystyle\varLambda$
$\displaystyle=\sum_{k^{\mathrm{th}}}\Gamma_{m}^{k}\left(A_{kk}-A_{k^{\prime}k^{\prime}}\right)+\sum_{k}\left[R_{mm,kk}+R_{mm,kk}^{\mathrm{\left(corr\right)}}\right]A_{kk},$
(16)

where

$\displaystyle\Gamma_{m}^{k}\equiv
i\left[\mathcal{H}_{m^{\prime}m}\left(F_{mm^{\prime},kk^{\prime}}-F_{mm^{\prime},k^{\prime}k}\right)-\mathcal{H}_{mm^{\prime}}\left(F_{m^{\prime}m,kk^{\prime}}-F_{m^{\prime}m,k^{\prime}k}\right)\right],$
(17) $\displaystyle
R_{mm,kk}^{\mathrm{\left(corr\right)}}\equiv\Gamma_{mk}^{\mathrm{\left(corr\right)}}\equiv
i\left(\mathcal{H}_{m^{\prime}m}G_{mm^{\prime},kk}-\mathcal{H}_{mm^{\prime}}G_{m^{\prime}m,kk}\right)+\sum_{n}R_{mm,nn^{\prime}}\left(G_{nn^{\prime},kk}+F_{nn^{\prime},kk^{\prime}}-F_{nn^{\prime},k^{\prime}k}\right).$
(18)

Since $R_{mm,mm}=-\sum_{k\neq m}R_{kk,mm}$ and
$R_{mm,mm}^{\mathrm{\left(corr\right)}}=-\sum_{k\neq
m}R_{kk,mm}^{\mathrm{\left(corr\right)}}$ (see proof in Appendix A), Eq. (16)
can be rewritten as:

$\displaystyle\varLambda$
$\displaystyle=\sum_{k^{\mathrm{th}}}\Gamma_{m}^{k}\left(A_{kk}-A_{k^{\prime}k^{\prime}}\right)+\sum_{k\neq
m}\left[\left(\Gamma_{mk}+\Gamma_{mk}^{\mathrm{\left(corr\right)}}\right)A_{kk}-\left(\Gamma_{km}+\Gamma_{km}^{\mathrm{\left(corr\right)}}\right)A_{mm}\right].$
(19)

Converting back to $\rho_{mm}$ yields the most important result of this work:

$\displaystyle\frac{d\rho_{mm}}{dt}$
$\displaystyle=\sum_{k^{\mathrm{th}}}\Gamma_{m}^{k}\left(\rho_{kk}-\rho_{k^{\prime}k^{\prime}}\right)+\sum_{k\neq
m}\left[\left(\Gamma_{mk}+\Gamma_{mk}^{\mathrm{\left(corr\right)}}\right)\rho_{kk}-\left(\Gamma_{km}+\Gamma_{km}^{\mathrm{\left(corr\right)}}\right)\rho_{mm}\right]+C,$
(20)

where $C$ is a constant dependent on the initial and equilibrium value of the
diagonal density matrix elements. Since this constant could be removed by a
trivial change of variable and does not affect to the relaxation rate, its
detailed form is neglected.

The non-secular density matrix equation now reduces to the form of a self-
consistent “secular” one containing only the diagonal density matrix elements.
From these diagonal density matrix elements, off-diagonal elements can be
easily calculated using Eq. (13).

As can be seen from Eq. (20),
$\Gamma_{m}^{k}=-\Gamma_{m}^{k^{\prime}}=-\Gamma_{m^{\prime}}^{k}=\Gamma_{m^{\prime}}^{k^{\prime}}$
plays the role of the transition rate of population difference from
doublet/singlet $k^{\mathrm{th}}$ representative by $\ket{k}$ (or
$\ket{k^{\prime}}$) to state $\ket{m}$. In particular,
$\Gamma_{m}^{m^{\prime}}=\Gamma_{m^{\prime}}^{m}$ can be considered as the
_general_ quantum tunneling rate between two states $\ket{m}$ and
$\ket{m^{\prime}}$. Meanwhile,
$\Gamma_{kl}^{\mathrm{\left(corr\right)}}\,\forall l\neq k$ is a correction to
the transition rate $\Gamma_{kl}$ resulting from the semi-secular/non-secular
approximation where the time-dependence of off-diagonal density matrix
elements is taken into account.

It is worthy to note that due to $C_{m^{\prime}m}=-C_{mm^{\prime}}^{*}$ and
$D_{m^{\prime}m,lk}=D_{mm^{\prime},kl}^{*}$, we have
$G_{m^{\prime}m,kk}=G_{mm^{\prime},kk}^{*}$ and
$F_{m^{\prime}m,k^{\prime}k}=-F_{mm,kk^{\prime}}^{*}$. As a corollary, both
$\Gamma_{m}^{k}$ and $\Gamma_{mk}^{\mathrm{\left(corr\right)}}$ are real as
expected for any quantity representing a population transition rate.

Since $A_{mm^{\prime}}\,\forall m$ and $A_{kk}\,\forall k$ are definite, it is
obvious from Eq. (13) that the series $F_{mm^{\prime},kk^{\prime}}$ and
$G_{mm^{\prime},kk}$ must converge. Consequently, terms in those series should
monotonically decrease. Zeroth-order approximation of those series then are
$F_{mm^{\prime},kk^{\prime}}\approx
C_{mm^{\prime}}\delta_{mk}\delta_{m^{\prime}k^{\prime}}$ and
$G_{mm^{\prime},kk}\approx D_{mm^{\prime},kk}$. Accordingly, the zeroth-order
approximation of $d\rho_{mm}/dt$ takes the form:

$\displaystyle\frac{d\rho_{mm}}{dt}=\Gamma_{m}^{m^{\prime}\left(0\right)}\left(\rho_{m^{\prime}m^{\prime}}-\rho_{mm}\right)+\sum_{k\neq
m}\left[\left(\Gamma_{mk}+\Gamma_{mk}^{\mathrm{\left(corr\right)\left(0\right)}}\right)\rho_{kk}-\left(\Gamma_{km}+\Gamma_{km}^{\mathrm{\left(corr\right)\left(0\right)}}\right)\rho_{mm}\right],$
(21)

where we have ignored the constant $C$ and denoted

$\displaystyle\Gamma_{m}^{m^{\prime}\left(0\right)}=-i\left(\mathcal{H}_{m^{\prime}m}C_{mm^{\prime}}+\mathcal{H}_{mm^{\prime}}C_{m^{\prime}m}\right),$
(22)
$\displaystyle\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}=\left(i\mathcal{H}_{m^{\prime}m}D_{mm^{\prime},kk}+R_{mm,kk^{\prime}}C_{kk^{\prime}}+\sum_{n^{\mathrm{th}}}R_{mm,nn^{\prime}}D_{nn^{\prime},kk}\right)+\mathrm{h.c.}\,\forall
k\neq m,$ (23)

where h.c. is the Hermitian conjugate. As is clear, in zeroth-order
approximation, only the quantum tunneling rate between $\ket{m}$ and
$\ket{m^{\prime}}$ enters the rate equation of $\rho_{mm}$ whereas other
$\Gamma_{m}^{k\left(0\right)}\,\forall k^{\mathrm{th}}\neq m^{\mathrm{th}}$ is
zero. This fact underlines the important role of the doublet internal
tunneling in relaxation of a spin system compared to the contribution from the
population difference of other doublets.

Since the vast majority of the relaxation studies uses the localized and
eigenstates basis, detailed form of the quantum tunneling rate as well as the
correction to other transition rate in these bases are indispensable.
Therefore, in the following, we will consecutively apply our developed
formulas specifically for these bases. However, for the sake of simplicity,
hereinafter we only retain the zeroth-order approximation of $d\rho_{mm}/dt$.

##  III “Secular” non-secular master equation in localized basis

Localized basis vectors $\ket{m^{*}}$ and $\ket{m^{\prime*}}$ corresponding to
$m^{\mathrm{th}}$ doublet of a spin system are defined asGatteschi  _et al._
(2006):

$\displaystyle\ket{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{+_{m}^{\left(0\right)}}-\ket{-_{m}^{\left(0\right)}}\right),$
(24) $\displaystyle\ket{m^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{+_{m}^{\left(0\right)}}+\ket{-_{m}^{\left(0\right)}}\right),$
(25)

where $\ket{\pm_{m}^{\left(0\right)}}$ are eigenstates of the spin Hamiltonian
in zero external magnetic field corresponding to the $m^{\mathrm{th}}$
doublet. In the presence of an external magnetic field, the Hamiltonian in the
subspace of this doublet can take the form:

$\mathcal{H}_{\left\\{\ket{m},\ket{m^{\prime}}\right\\}}=\frac{W_{mm^{\prime}}}{2}\left(\ket{m}\bra{m}-\ket{m^{\prime}}\bra{m^{\prime}}\right)+\frac{\Delta_{mm^{\prime}}}{2}\left(\ket{m}\bra{m^{\prime}}+\ket{m^{\prime}}\bra{m}\right),$
(26)

where $W_{mm^{\prime}}$ is the energy bias between $\ket{m}$ and
$\ket{m^{\prime}}$ and $\Delta_{mm^{\prime}}$ is the tunnel splitting of
$m^{\mathrm{th}}$ doublet. Without loss of generality, these two quantities
can be assumed to be real. From this, we can easily have

$\displaystyle
C_{mm^{\prime}}=\frac{i\Delta_{mm^{\prime}}}{2}\frac{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)-i\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)-R_{mm^{\prime},m^{\prime}m}}{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{mm^{\prime},m^{\prime}m}\right|^{2}},$
(27) $\displaystyle
D_{mm^{\prime},kl}=\frac{R_{mm^{\prime},m^{\prime}m}R_{m^{\prime}m,kl}+R_{mm^{\prime},kl}\left[\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)-i\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)\right]}{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{mm^{\prime},m^{\prime}m}\right|^{2}}.$
(28)

where $\gamma_{mm^{\prime}}^{\prime}$ and
$\gamma_{mm^{\prime}}^{\prime\prime}$ denote the real and imaginary part of
$\gamma_{mm^{\prime}}$ respectively . Substituting these into Eq. (22) and
(23) results in

$\displaystyle\Gamma_{m}^{m^{\prime}\left(0\right)}$
$\displaystyle=\frac{\Delta_{mm^{\prime}}^{2}}{2}\frac{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)-\left(R_{mm^{\prime},m^{\prime}m}+R_{m^{\prime}m,mm^{\prime}}\right)/2}{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{mm^{\prime},m^{\prime}m}\right|^{2}},$
(29)

and

$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}=\left\\{\frac{i\Delta_{mm^{\prime}}}{2}\frac{R_{mm^{\prime},m^{\prime}m}R_{m^{\prime}m,kk}+R_{mm^{\prime},kk}\left[\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)-i\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)\right]}{\left(\gamma_{mm^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{mm^{\prime}}+\gamma_{mm^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{mm^{\prime},m^{\prime}m}\right|^{2}}\right.\\\
+\frac{i\Delta_{kk^{\prime}}}{2}\frac{R_{mm,kk^{\prime}}\left[\left(\gamma_{kk^{\prime}}^{\prime}-\lambda\right)-i\left(W_{kk^{\prime}}+\gamma_{kk^{\prime}}^{\prime\prime}\right)-R_{kk^{\prime},k^{\prime}k}\right]}{\left(\gamma_{kk^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{kk^{\prime}}+\gamma_{kk^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{kk^{\prime},k^{\prime}k}\right|^{2}}\\\
\left.\sum_{n^{\mathrm{th}}}R_{mm,nn^{\prime}}\frac{R_{nn^{\prime},n^{\prime}n}R_{n^{\prime}n,kk}+R_{nn^{\prime},kk}\left[\left(\gamma_{nn^{\prime}}^{\prime}-\lambda\right)-i\left(W_{nn^{\prime}}+\gamma_{nn^{\prime}}^{\prime\prime}\right)\right]}{\left(\gamma_{nn^{\prime}}^{\prime}-\lambda\right)^{2}+\left(W_{nn^{\prime}}+\gamma_{nn^{\prime}}^{\prime\prime}\right)^{2}-\left|R_{nn^{\prime},n^{\prime}n}\right|^{2}}\right\\}+\mathrm{h.c.}\,\forall
k\neq m.$ (30)

As all Redfield operator elements $R_{mn,kl}$ are functions of temperature,
both $\Gamma_{m}^{m^{\prime}\left(0\right)}$ and
$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}$, and accordingly
$\Gamma_{m}^{m^{\prime}}$ and $\Gamma_{mk}^{\mathrm{\left(corr\right)}}$, also
depend on temperature.

For excited doublet, since the conditions
$\gamma_{mm^{\prime}}^{\prime}\gg\lambda,\gamma_{mm^{\prime}}^{\prime}\gg\left|R_{mm^{\prime},m^{\prime}m}\right|$,
and
$\gamma_{mm^{\prime}}^{\prime}\gg\left(R_{mm^{\prime},m^{\prime}m}+R_{m^{\prime}m,mm^{\prime}}\right)$
are usually satisfied over nearly entire temperature range, a simpler version
of Eq. (29) can be used to find the quantum tunneling rate inside these
excited doublet

$\displaystyle\Gamma_{m}^{m^{\prime}\left(0\right)}$
$\displaystyle=\frac{\Delta_{mm^{\prime}}^{2}}{2}\frac{\gamma_{mm^{\prime}}^{\prime}}{\gamma_{mm^{\prime}}^{\prime
2}+\left(\gamma_{mm^{\prime}}^{\prime\prime}+W_{mm^{\prime}}\right)^{2}},$
(31)

which is basically similar to the well-known incoherent tunneling rate Garanin
and Chudnovsky (1997); Leuenberger and Loss (2000).

However, for the ground doublet at low temperature, since those mentioned
conditions are now hardly fulfilled, a direct application of Eq. (31) can lead
to an inaccurate result. In that circumstance, a full treatment of the quantum
tunneling rate $\Gamma_{m}^{m^{\prime}\left(0\right)}$ is essential. At high
temperature though, the validity of the incoherent tunneling rate for the
ground doublet is subject to the relative magnitude between the dephasing rate
$\gamma_{mm^{\prime}}^{\prime}$ and $\lambda$.

Similarly to the quantum tunneling rate, we can also approximate
$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}$ at high temperature
by neglecting $\lambda$ in Eq. 30. However, it becomes interesting when the
temperature is sufficiently high such that the coherence transfer rates are
much smaller than the dephasing rates, i.e.
$R_{nn^{\prime},n^{\prime}n},R_{n^{\prime}n,kk}\ll\left|\gamma_{nn^{\prime}}\right|\,\forall
n,k$. Under this condition,
$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}$ becomes

$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}\approx\frac{i}{2}\left(\frac{\Delta_{mm^{\prime}}R_{mm^{\prime},kk}}{\gamma_{mm^{\prime}}+iW_{mm^{\prime}}}+\frac{\Delta_{kk^{\prime}}R_{mm,kk^{\prime}}}{\gamma_{kk^{\prime}}+iW_{kk^{\prime}}}\right)+\mathrm{h.c.}\,\forall
k\neq m.$ (32)

As is clear, even at high temperature, this correction may still not be small
compared to $\Gamma_{mk}$ except for the case when the spin system is out of
resonance or $\Delta_{nn^{\prime}}/\gamma_{nn^{\prime}}\,\forall n$ is large
enough. This fact indicates that the correction to the transition rate
$\Gamma_{mk}^{\mathrm{\left(corr\right)}\left(0\right)}$, besides the quantum
tunneling rate $\Gamma_{m}^{m^{\prime}\left(0\right)}$, must be taken into
account even at high temperature. Moreover, from the above formula we can also
deduce the condition for applying the secular approximation in localized
basis, i.e. all the corrections can be ignored. Specifically, Eq. 32 shows
that the secular approximation in localized basis is valid only at high
temperature and small tunneling splitting. Intriguingly, the latter condition
is quite opposite to the one for the secular approximation in eigenstates
basis which will be derived in the next section.

As a limiting case, out of resonance, both quantum tunneling rate and the
corrections to the transition rates becomes negligible. Accordingly, the
“secular” non-secular master equation reduces to the non-tunneling secular
master equation as expected.

##  IV “Secular” non-secular master equation in eigenstates basis

In eigenstates basis, the Hamiltonian of the spin system is diagonal:

$\mathcal{H}=\sum_{\alpha^{\mathrm{th}}}\left(\varepsilon_{\alpha}\ket{\alpha}\bra{\alpha}+\varepsilon_{\alpha^{\prime}}\ket{\alpha^{\prime}}\bra{\alpha^{\prime}}+\right),$
(33)

where $\ket{\alpha}$ and $\ket{\alpha^{\prime}}$ belong to the same doublet
$\alpha^{\mathrm{th}}$. In this basis,
$\gamma_{\alpha\alpha^{\prime}}=\gamma_{\alpha^{\prime}\alpha}$ is real
Garanin (2011) and

$\displaystyle C_{\alpha\alpha^{\prime}}=0,$ (34) $\displaystyle
D_{\alpha\alpha^{\prime},\beta\gamma}=\frac{R_{\alpha\alpha^{\prime},\alpha^{\prime}\alpha}R_{\alpha^{\prime}\alpha,\beta\gamma}+R_{\alpha\alpha^{\prime},\beta\gamma}\left(\gamma_{\alpha\alpha^{\prime}}-\lambda-i\omega_{\alpha\alpha^{\prime}}\right)}{\left(\gamma_{\alpha\alpha^{\prime}}-\lambda\right)^{2}+\omega_{\alpha\alpha^{\prime}}^{2}-\left|R_{\alpha\alpha^{\prime},\alpha^{\prime}\alpha}\right|^{2}},$
(35)

Thus,

$\displaystyle\Gamma_{\alpha}^{\alpha^{\prime}\left(0\right)}$
$\displaystyle=0=\Gamma_{\alpha}^{\beta}\,\forall\beta,$ (36)
$\displaystyle\Gamma_{\alpha\beta}^{\mathrm{\left(corr\right)}\left(0\right)}$
$\displaystyle=\sum_{\delta^{\mathrm{th}}}R_{\alpha\alpha,\delta\delta^{\prime}}\frac{R_{\delta\delta^{\prime},\delta^{\prime}\delta}R_{\delta^{\prime}\delta,\beta\beta}+R_{\delta\delta^{\prime},\beta\beta}\left(\gamma_{\delta\delta^{\prime}}-\lambda-i\omega_{\delta\delta^{\prime}}\right)}{\left(\gamma_{\delta\delta^{\prime}}-\lambda\right)^{2}+\omega_{\delta\delta^{\prime}}^{2}-\left|R_{\alpha\alpha^{\prime},\alpha^{\prime}\alpha}\right|^{2}}+\mathrm{h.c.}\,\forall\beta\neq\alpha.$
(37)

Strikingly, the phonon-induced quantum tunneling rate is _non-existent_ when
working in the eigenstates basis.

Since for excited doublets
$\sqrt{\gamma_{\delta\delta^{\prime}}^{2}+\omega_{\delta\delta^{\prime}}^{2}}$
often dominates over $\lambda$ and other coherence transfer rates,
$\Gamma_{\alpha\beta}^{\mathrm{\left(corr\right)}\left(0\right)}$ can be
further approximated as:

$\displaystyle\Gamma_{\alpha\beta}^{\mathrm{\left(corr\right)}\left(0\right)}$
$\displaystyle=R_{\alpha\alpha,gg^{\prime}}\frac{R_{gg^{\prime},g^{\prime}g}R_{g^{\prime}g,\beta\beta}+R_{gg^{\prime},\beta\beta}\left(\gamma_{gg^{\prime}}-\lambda-i\omega_{gg^{\prime}}\right)}{\left(\gamma_{gg^{\prime}}-\lambda\right)^{2}+\omega_{gg^{\prime}}^{2}-\left|R_{\alpha\alpha^{\prime},\alpha^{\prime}\alpha}\right|^{2}}+\mathrm{h.c.}\,\forall\beta\neq\alpha,$
(38)

where only contribution from the ground doublet $g^{\mathrm{th}}$ survives in
the expression of
$\Gamma_{\alpha\beta}^{\mathrm{\left(corr\right)}\left(0\right)}$. Moreover,
in the case the tunnel splitting of the ground doublet $\Delta_{gg^{\prime}}$
is large, this contribution from the ground doublet also becomes negligible.
As a consequence, the “secular” non-secular master equation reduces to the
secular one. From this, we can conclude that a preeminent ground doublet’s
tunnel splitting in comparison to the coherence transfer rates is the
_sufficient_ condition for using the secular master equation in eigenstates
basis at a particular temperature.

At low temperature where only the ground doublet is populated, it is likely
that the secular approximation for the Redfield master equation in eigenstates
basis is also a good approximation. This results from the fact that in this
regime, all the coherence transfer rates entering the expression of
$\Gamma_{\alpha\beta}^{\mathrm{\left(corr\right)}\left(0\right)}$ appears to
be miniature compared to $\left|\omega_{gg^{\prime}}\right|$. In contrast,
when localized basis is considered in the same regime, the semi-secular
approximation has to be used instead. This stresses the difference between
using these two common bases in relaxation problem.

##  V Conclusions

In summary, the present study has demonstrated that the semi-secular
approximation for the Redfield master equation is mostly sufficient to deduce
the relaxation rate. In order to find this rate, a self-consistent secular
form of the non-secular/semi-secular master equation has also been developed.
Containing much less number of equations, this secular form of the semi-
secular approximation allows to reduce considerably the amount of effort in
calculating the relaxation rate as well as facilitates different
approximations to the master equation depending on the investigated regimes.
Based on this finding, for the first time, a general formula of the phonon-
induced quantum tunneling rate and correction to other transition rates
regardless of working basis are proposed. Surprisingly, applying the general
“secular” form of the non-secular/semi-secular approximation for localized
basis, we found that a proper treatment of the spin relaxation must take into
account the corrections to both familiar incoherent quantum tunneling rate and
population transition rates between localized states. Meanwhile, in
eigenstates basis, our “secular” non-secular master equation shows that the
phonon-induced quantum tunneling rate in this basis is always equal to zero.
Moreover, the conditions for applying secular approximation in both localized
and eigenstates basis are also elucidated. In particular, secular
approximation in localized basis is justified in the regime of high
temperature and small tunnel splittings. On the contrary, a large ground
doublet’s tunnel splitting is required for the secular approximation in
eigenstates basis. At intermediate conditions, the semi-secular approximation
(in any basis) is essential for extracting the relaxation rate. This finding
thus settles the debate over which basis should be used in the secular master
equation. On the whole, by establishing an accurate “secular” non-secular
master equation, our research serves as a sturdy base for the exploration of
the relaxation phenomenon in any condition provided that the Redfield non-
secular master equation is relevant. Last but not least, it is important to
stress that the “secular” non-secular master equation is also valid for
describing the relaxation of not just spin but also any general system in weak
interaction with a thermal bath as well.

###### Acknowledgements.

L. T. A. H. would like to acknowledge financial support from the Flemish
Science Foundation (FWO).

##  Appendix A Properties of the relaxation matrix elements in an arbitrary
basis

  * •

Relaxation matrix elements $R_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}$ in
the eigenstates basis can be calculated from the spin-phonon Hamiltonian
$V$Garanin (2011):

$\displaystyle R_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}$
$\displaystyle=\frac{\pi}{\hbar
Z_{\mathrm{b}}}\sum_{ww^{\prime}}\left\\{-\sum_{\gamma}e^{-E_{w}/kT}\delta\left(\varepsilon_{\alpha^{\prime}}-\varepsilon_{\gamma}+E_{w}-E_{w^{\prime}}\right)V_{\alpha
w,\gamma w^{\prime}}V_{\gamma
w^{\prime},\alpha^{\prime}w}\delta_{\beta\beta^{\prime}}\right.$ (39)
$\displaystyle\quad-\sum_{\gamma}e^{-E_{w}/kT}\delta\left(\varepsilon_{\beta^{\prime}}-\varepsilon_{\gamma}+E_{w}-E_{w^{\prime}}\right)V_{\beta^{\prime}w,\gamma
w^{\prime}}V_{\gamma w^{\prime},\beta w}\delta_{\alpha\alpha^{\prime}}$ (40)
$\displaystyle\left.\quad+e^{-E_{w^{\prime}}/kT}\left[\delta\left(\varepsilon_{\beta}-\varepsilon_{\beta^{\prime}}+E_{w}-E_{w^{\prime}}\right)+\delta\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}+E_{w}-E_{w^{\prime}}\right)\right]V_{\alpha
w,\alpha^{\prime}w^{\prime}}V_{\beta^{\prime}w^{\prime},\beta w}\right\\},$
(41)

where $\ket{w}$ and $\ket{w^{\prime}}$ designate the eigenstates of the
thermal bath and $Z_{\mathrm{b}}$ is the bath partition function.

  * •

Relaxation matrix elements $R_{mn,kl}$ in an arbitrary basis
$\left\\{\ket{m}\right\\}$ are related to the elements
$R_{\alpha\beta,\gamma\delta}$ in the eigenstates basis as followsGaranin
(2011):

$R_{mn,kl}=\sum_{\alpha,\beta,\gamma,\delta}\braket{m}{\alpha}\braket{\beta}{n}\braket{\gamma}{k}\braket{l}{\delta}R_{\alpha\beta,\gamma\delta},$
(42)

and vice versa,

$R_{\alpha\beta,\gamma\delta}=\sum_{m,n,k,l}\braket{\alpha}{m}\braket{n}{\beta}\braket{k}{\gamma}\braket{\delta}{l}R_{mn,kl}.$
(43)

  * •

$R_{mn,kl}^{*}=R_{nm,kl}$ – Proof: this property can be easily derived from
the relation between $R_{mn,pq}$ and $R_{\alpha\beta,\gamma\delta}$ above.
Indeed,

$\displaystyle R_{mn,kl}^{*}$
$\displaystyle=\sum_{\alpha,\beta,\gamma,\delta}\braket{m}{\alpha}^{*}\braket{\beta}{n}^{*}\braket{\gamma}{k}^{*}\braket{l}{\delta}R_{\alpha\beta,\gamma\delta}^{*}$
$\displaystyle=\sum_{\beta,\alpha,\delta,\gamma}\braket{n}{\beta}\braket{\alpha}{m}\braket{\delta}{l}\braket{k}{\gamma}R_{\beta\alpha,\delta\gamma}=R_{nm,kl}.$
(44)

From this, it is straightforward to show that
$R_{mn,kl}^{\prime*}=R_{nm,kl}^{\prime}$,
$C_{m^{\prime}m}=-C_{mm^{\prime}}^{*}$, and
$D_{m^{\prime}m,lk}=D_{mm^{\prime},kl}^{*}$.

  * •

$\sum_{m}R_{mm,kl}=0$ – Proof: from the property
$\sum_{\gamma}R_{\alpha\alpha,\gamma\delta}=0$, we have

$\displaystyle\sum_{m}R_{mm,kl}$
$\displaystyle=\sum_{m}\sum_{\alpha,\beta,\gamma,\delta}\braket{m}{\alpha}\braket{\beta}{m}\braket{\gamma}{k}\braket{l}{\delta}R_{\alpha\beta,\gamma\delta}$
$\displaystyle=\sum_{\alpha,\beta,\gamma,\delta}\delta_{\alpha\beta}\braket{\gamma}{k}\braket{l}{\delta}R_{\alpha\beta,\gamma\delta}$
$\displaystyle=\sum_{\gamma,\delta}\braket{\gamma}{k}\braket{l}{\delta}\left(\sum_{\alpha}R_{\alpha\alpha,\gamma\delta}\right)=0.$
(45)

As a result, $R_{mm,mm}=-\sum_{k\neq m}R_{kk,mm}$.

  * •

$\sum_{k}R_{kk,mm}^{\mathrm{\left(corr\right)}}=0$ – Proof:

$\displaystyle\sum_{k}R_{kk,mm}^{\mathrm{\left(corr\right)}}$
$\displaystyle=i\sum_{k}\left(\mathcal{H}_{k^{\prime}k}G_{kk^{\prime},mm}-\mathcal{H}_{kk^{\prime}}G_{k^{\prime}k,mm}\right)+\sum_{n}\left(\sum_{k}R_{kk,nn^{\prime}}\right)\left(G_{nn^{\prime},mm}+F_{nn^{\prime},mm^{\prime}}-F_{nn^{\prime},m^{\prime}m}\right)$
(46)
$\displaystyle=i\sum_{k^{\mathrm{th}}}\left[\left(\mathcal{H}_{k^{\prime}k}G_{kk^{\prime},mm}-\mathcal{H}_{kk^{\prime}}G_{k^{\prime}k,mm}\right)+\left(\mathcal{H}_{kk^{\prime}}G_{k^{\prime}k,mm}-\mathcal{H}_{k^{\prime}k}G_{kk^{\prime},mm}\right)\right]=0$
(47)

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