Superradiance-like Electron Transport Through a Quantum Dot

Abstract

In this chapter we theoretically show that intriguing features of coherent many-body physics can be observed in electron transport through a quantum dot (QD). We first derive a master equation based framework for electron transport in the Coulomb-blockade regime which includes hyperfine (HF) interaction with the nuclear spin ensemble in the QD.

2.9 Appendix to Chap. 2

2.1.1 2.9.1 Microscopic Derivation of the Master Equation

In this Appendix we provide some details regarding the derivation of the master equations as stated in Eqs. (2.2) and (2.30). It comprises the effect of the HF dynamics in the memory kernel of Eq. (2.13) and the subsequent approximation of independent rates of variation.

In the following, we will show that it is self-consistent to neglect the effect of the HF dynamics $\mathcal {L}_{1}\left( t\right)$ in the memory-kernel of Eq. (2.13) provided that the bath correlation time $\tau _{c}$ is short compared to the Rabi flips produced by the HF dynamics. This needs to be addressed as cooperative effects potentially drive the system from a weakly coupled into a strongly coupled regime. First, we reiterate the Schwinger-Dyson identity in Eq. (2.14) as an infinite sum over time-ordered nested commutators

$$\begin{aligned} e^{-i\left( \mathcal {L}_{0}+\mathcal {L}_{1}\right) \tau }=e^{-i\mathcal {L}_{0}\tau }\sum _{n=0}^{\infty }\left( -i\right) ^{n}\int _{0}^{\tau }d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\dots \int _{0}^{\tau _{n-1}}d\tau _{n}\,\tilde{\mathcal {L}}_{1}\left( \tau _{1}\right) \tilde{\mathcal {L}}_{1}\left( \tau _{2}\right) \dots \tilde{\mathcal {L}}_{1}\left( \tau _{n}\right) ,\end{aligned}$$

(2.49)

where for any operator X

$$\begin{aligned} \tilde{\mathcal {L}}_{1}\left( \tau \right) X=e^{i\mathcal {L}_{0}\tau }\mathcal {L}_{1}e^{-i\mathcal {L}_{0}\tau }X=\left[ e^{iH_{0}\tau }H_{1}e^{-iH_{0}\tau },X\right] =\left[ \tilde{H}_{1}\left( \tau \right) ,X\right] .\end{aligned}$$

(2.50)

More explicitly, up to second order Eq. (2.49) is equivalent to

$$\begin{aligned} e^{-i\left( \mathcal {L}_{0}\,+\,\mathcal {L}_{1}\right) \tau }X&= e^{-i\mathcal {L}_{0}\tau }X-ie^{-i\mathcal {L}_{0}\tau }\int _{0}^{\tau }d\tau _{1}\left[ \tilde{H}_{1}\left( \tau _{1}\right) ,X\right] \nonumber \\&-e^{-i\mathcal {L}_{0}\tau }\int _{0}^{\tau }d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\left[ \tilde{H}_{1}\left( \tau _{1}\right) ,\left[ \tilde{H}_{1}\left( \tau _{2}\right) ,X\right] \right] +\cdots \end{aligned}$$

(2.51)

Note that the time-dependence of $\tilde{H}_{1}\left( \tau \right)$ is simply given by

$$\begin{aligned} \tilde{H}_{1}\left( \tau \right) =e^{i\omega \tau }H_{+}+e^{-i\omega \tau }H_{-}+H_{\Delta \mathrm {OH}},\,\,\,\,\,\,\,\,\,\, H_{\pm }=\frac{g}{2}S^{\pm }A^{\mp },\end{aligned}$$

(2.52)

where the effective Zeeman splitting $\omega =\omega _{0}+g\left\langle A^{z}\right\rangle _{t}$ is time-dependent. Accordingly, we define $\tilde{\mathcal {L}}_{1}\left( \tau \right) =\tilde{\mathcal {L}}_{+}\left( \tau \right) \,+\,\tilde{\mathcal {L}}_{-}\left( \tau \right) \,+\,\tilde{\mathcal {L}}_{\Delta \mathrm {OH}}\left( \tau \right) =e^{i\omega \tau }\mathcal {L}_{+}\,+\,e^{-i\omega \tau }\mathcal {L}_{-}\,+\,\mathcal {L}_{\Delta \mathrm {OH}},$ where $\mathcal {L}_{x}\cdot =\left[ H_{x},\cdot \right]$ for $x=\pm ,\Delta \mathrm {OH}$. In the next steps, we will explicitly evaluate the first two contributions to the memory kernel that go beyond $n=0$ and then generalize our findings to any order n of the Schwinger-Dyson series.

2.1.1.1 First order correction

The first order contribution $n=1$ in Eq. (2.13) is given by

$$\begin{aligned} \Xi ^{\left( 1\right) }=i\int _{0}^{t}d\tau \int _{0}^{\tau }d\tau _{1}\mathsf {Tr_{B}}\left( \mathcal {L}_{T}e^{-i\mathcal {L}_{0}\tau }\left[ \tilde{H}_{1}\left( \tau _{1}\right) ,X\right] \right) .\end{aligned}$$

(2.53)

Performing the integration in $\tau _{1}$ leads to

$$\begin{aligned} \Xi ^{\left( 1\right) }= & {} \int _{0}^{t}d\tau \left\{ \frac{g}{2\omega }\left( 1-e^{-i\omega \tau }\right) \mathsf {Tr_{B}}\left( \mathcal {L}_{T}\left[ S^{+}A^{-},\tilde{X}_{\tau }\right] \right) \right. \nonumber \\&+\,\frac{g}{2\omega }\left( e^{i\omega \tau }-1\right) \mathsf {Tr_{B}}\left( \mathcal {L}_{T}\left[ S^{-}A^{+},\tilde{X}_{\tau }\right] \right) \nonumber \\&\left. +\,ig\tau \mathsf {Tr_{B}}\left( \mathcal {L}_{T}\left[ \left( A^{z}-\left\langle A^{z}\right\rangle _{t}\right) S^{z},\tilde{X}_{\tau }\right] \right) \right\} \end{aligned}$$

(2.54)

where, for notational convenience, we introduced the operators $X=\mathcal {L}_{T}\rho _{S}\left( t-\tau \right) \rho _{B}^{0}$ and $\tilde{X}_{\tau }=e^{-iH_{0}\tau }\left[ H_{T},\rho _{S}\left( t-\tau \right) \rho _{B}^{0}\right] e^{iH_{0}\tau }\approx \left[ \tilde{H}_{T}\left( \tau \right) ,\rho _{S}\left( t\right) \rho _{B}^{0}\right]$. In accordance with previous approximations, we have replaced $e^{-iH_{0}\tau }\rho _{S}\left( t-\tau \right) e^{iH_{0}\tau }$ by $\rho _{S}\left( t\right)$ since any additional term besides $H_{0}$ would be of higher order in perturbation theory [35, 36]. In particular, this disregards dissipative effects: In our case, this approximation is valid self-consistently provided that the tunneling rates are small compared to effective Zeeman splitting $\omega$. The integrand decays on the leads-correlation timescale $\tau _{c}$ which is typically much faster than the timescale set by the effective Zeeman splitting, $\omega \tau _{c}\ll 1$. This separation of timescales allows for an expansion in the small parameter $\omega \tau$, e.g. $\frac{g}{\omega }\left( e^{i\omega \tau }-1\right) \approx ig\tau$. We see that the first order correction can be neglected if the the bath correlation time $\tau _{c}$ is sufficiently short compared to the timescale of the HF dynamics, that is $g\tau _{c}\ll 1$. The latter is bounded by the total hyperfine coupling constant $A_{\mathrm {HF}}$ (since $\left| \left| gA^{x}\right| \right| \le A_{\mathrm {HF}}$) so that the requirement for disregarding the first order term reads $A_{\mathrm {HF}}\tau _{c}\ll 1$.

2.1.1.2 Second order correction

The contribution of the second term $n=2$ in the Schwinger-Dyson expansion can be decomposed into

$$\begin{aligned} \Xi ^{\left( 2\right) }=\Xi _{\mathrm {zz}}^{\left( 2\right) }+\Xi _{\mathrm {ff}}^{\left( 2\right) }+\Xi _{\mathrm {fz}}^{\left( 2\right) }.\end{aligned}$$

(2.55)

The first term $\Xi _{\mathrm {zz}}^{\left( 2\right) }$ contains contributions from $H_{\Delta \mathrm {OH}}$ only

$$\begin{aligned} \Xi _{\mathrm {zz}}^{\left( 2\right) }= & {} \int _{0}^{t}d\tau \int _{0}^{\tau }d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\mathsf {Tr_{B}}\left( \mathcal {L}_{T}e^{-i\mathcal {L}_{0}\tau }\left[ \tilde{H}_{\Delta \mathrm {OH}}\left( \tau _{1}\right) ,\left[ \tilde{H}_{\Delta \mathrm {OH}}\left( \tau _{2}\right) ,X\right] \right] \right) \end{aligned}$$

(2.56)

$$\begin{aligned}= & {} -\int _{0}^{t}d\tau \left( g\tau \right) ^{2}\mathsf {Tr_{B}}\left[ \mathcal {L}_{T}\left( \delta A^{z}S^{z}\tilde{X}_{\tau }\delta A^{z}S^{z}-\frac{1}{2}\left\{ \delta A^{z}S^{z}\delta A^{z}S^{z},\tilde{X}_{\tau }\right\} \right) \right] \end{aligned}$$

(2.57)

Similarly, $\Xi _{\mathrm {ff}}^{\left( 2\right) }$ which comprises contributions from $H_{\mathrm {ff}}$ only is found to be

$$\begin{aligned} \Xi _{\mathrm {ff}}^{\left( 2\right) }= & {} \frac{g^{2}}{4\omega ^{2}}\int _{0}^{t}d\tau \left\{ \left( 1+i\omega \tau -e^{i\omega \tau }\right) \mathsf {Tr_{B}}\left[ \mathcal {L}_{T}\left( S^{+}S^{-}A^{-}A^{+}\tilde{X}_{\tau }+\tilde{X}_{\tau }S^{-}S^{+}A^{+}A^{-}\right) \right] \right. \nonumber \\&\left. +\left( 1-i\omega \tau -e^{-i\omega \tau }\right) \mathsf {Tr_{B}}\left[ \mathcal {L}_{T}\left( S^{-}S^{+}A^{+}A^{-}\tilde{X}_{\tau }+\tilde{X}_{\tau }S^{+}S^{-}A^{-}A^{+}\right) \right] \right\} .\end{aligned}$$

(2.58)

Here, we have used the following simplification: The time-ordered products which include flip-flop terms only can be simplified to two possible sequences in which $\mathcal {L}_{+}$ is followed by $\mathcal {L}_{-}$ and vice versa. This holds since

$$\begin{aligned} \mathcal {L}_{\pm }\mathcal {L}_{\pm }X=\left[ H_{\pm },\left[ H_{\pm },X\right] \right] =H_{\pm }H_{\pm }X+XH_{\pm }H_{\pm }-2H_{\pm }XH_{\pm }=0.\end{aligned}$$

(2.59)

Here, the first two terms drop out immediately since the electronic jump-operators $S^{\pm }$ fulfill the relation $S^{\pm }S^{\pm }=0$. In the problem at hand, also the last term gives zero because of particle number superselection rules: In Eq. (2.13) the time-ordered product of superoperators acts on $X=\left[ H_{T},\rho _{S}\left( t-\tau \right) \rho _{B}^{0}\right]$. Thus, for the term $H_{\pm }XH_{\pm }$ to be nonzero, coherences in Fock space would be required which are consistently neglected; compare Ref. [36]. This is equivalent to ignoring coherences between the system and the leads. Note that the same argument holds for any combination $H_{\mu }XH_{\nu }$ with $\mu ,\nu =\pm$.

Similar results can be obtained for $\Xi _{\mathrm {fz}}^{\left( 2\right) }$ which comprises $H_{\pm }$ as well as $H_{\Delta \mathrm {OH}}$ in all possible orderings. Again, using that the integrand decays on a timescale $\tau _{c}$ and expanding in the small parameter $\omega \tau$ shows that the second order contribution scales as ${\sim }\left( g\tau _{c}\right) ^{2}$. Our findings for the first and second order correction suggest that the n-th order correction scales as ${\sim }\left( g\tau _{c}\right) ^{n}$. This will be proven in the following by induction.

2.1.1.3 n-th order correction

The scaling of the n-th term in the Dyson series is governed by the quantities of the form

$$\begin{aligned} \xi _{+-\cdots }^{\left( n\right) }\left( \tau \right) =g^{n}\int _{0}^{\tau }d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\ldots \int _{0}^{\tau _{n-1}}d\tau _{n}e^{i\omega \tau _{1}}e^{-i\omega \tau _{2}}\dots , \end{aligned}$$

(2.60)

where the index suggests the order in which $H_{\pm }$ (giving an exponential factor) and $H_{\Delta \mathrm {OH}}$ (resulting in a factor of 1) appear. Led by our findings for $n=1,2$, we claim that the expansion of $\xi _{+-\cdots }^{\left( n\right) }\left( \tau \right)$ for small $\omega \tau$ scales as $\xi _{+-\cdots }^{\left( n\right) }\left( \tau \right) \sim \left( g\tau \right) ^{n}$. Then, the $\left( n+1\right)$-th terms scale as

$$\begin{aligned} \xi _{-(\Delta \mathrm {OH})+-\cdots }^{\left( n+1\right) }\left( \tau \right)&= g^{n+1}\int _{0}^{\tau }d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\dots \int _{0}^{\tau _{n-1}}d\tau _{n}\int _{0}^{\tau _{n}}d\tau _{n+1}\left( \begin{array}{c} e^{-i\omega \tau _{1}}\\ 1\end{array}\right) e^{+i\omega \tau _{2}}\cdots \end{aligned}$$

(2.61)

$$\begin{aligned}&= g\int _{0}^{\tau }d\tau _{1}\left( \begin{array}{c} e^{-i\omega \tau _{1}}\\ 1\end{array}\right) \xi _{+-\dots }^{\left( n\right) }\left( \tau _{1}\right) \end{aligned}$$

(2.62)

$$\begin{aligned}&{ \sim }\left( g\tau \right) ^{n+1}. \end{aligned}$$

(2.63)

Since we have already verified this result for $n=1,2$, the general result follows by induction. This completes the proof.

2.1.2 2.9.2 Adiabatic Elimination of the QD Electron

For a sufficiently small relative coupling strength $\varepsilon$ the nuclear dynamics are slow compared to the electronic QD dynamics. This allows for an adiabatic elimination of the electronic degrees of freedom yielding an effective master equation for the nuclear spins of the QD.

Our analysis starts out from Eq. (2.35) which we write as

$$\begin{aligned} \dot{\rho }=\mathcal {W}_{0}\rho +\mathcal {W}_{1}\rho , \end{aligned}$$

(2.64)

where

$$\begin{aligned} \mathcal {W}_{0}\rho= & {} -i\left[ \omega _{0}S^{z},\rho \right] +\gamma \left[ S^{-}\rho S^{+}-\frac{1}{2}\left\{ S^{+}S^{-},\rho \right\} \right] +\Gamma \left[ S^{z} \rho S^{z}-\frac{1}{4}\rho \right] \end{aligned}$$

(2.65)

$$\begin{aligned} \mathcal {W}_{1}\rho= & {} -i\left[ H_{\mathrm {HF}},\rho \right] . \end{aligned}$$

(2.66)

Note that the superoperator $\mathcal {W}_{0}$ only acts on the electronic degrees of freedom. It describes an electron in an external magnetic field that experiences a decay as well as a pure dephasing mechanism. In zeroth order of the coupling parameter $\varepsilon$ the electronic and nuclear dynamics of the QD are decoupled and SR effects cannot be expected. These are contained in the interaction term $\mathcal {W}_{1}$.

Formally, the adiabatic elimination of the electronic degrees of freedom can be achieved as follows [61]. To zeroth order in $\varepsilon$ the eigenvectors of $\mathcal {W}_{0}$ with zero eigenvector $\lambda _{0}=0$ are

$$\begin{aligned} \mathcal {W}_{0}\mu \otimes \rho _{SS}=0, \end{aligned}$$

(2.67)

where $\rho _{SS}=\left| \downarrow \right\rangle \left\langle \downarrow \right|$ is the stationary solution for the electronic dynamics and $\mu$ describes some arbitrary state of the nuclear system. The zero-order Liouville eigenstates corresponding to $\lambda _{0}=0$ are coupled to the subspaces of “excited” nonzero (complex) eigenvalues $\lambda _{k}\ne 0$ of $\mathcal {W}_{0}$ by the action of $\mathcal {W}_{1}$. Physically, this corresponds to a coupling between electronic and nuclear degrees of freedom. In the limit where the HF dynamics are slow compared to the electronic frequencies, i.e. the Zeeman splitting $\omega _{0}$, the decay rate $\gamma$ and the dephasing rate $\Gamma$, the coupling between these blocks of eigenvalues and Liouville subspaces of $\mathcal {W}_{0}$ is weak justifying a perturbative treatment. This motivates the definition of a projection operator P onto the subspace with zero eigenvalue $\lambda _{0}=0$ of $\mathcal {W}_{0}$ according to

$$\begin{aligned} P\rho =\mathsf {Tr_{el}}\left[ \rho \right] \otimes \rho _{SS}=\mu \otimes \left| \downarrow \right\rangle \left\langle \downarrow \right| , \end{aligned}$$

(2.68)

where $\mu =\mathsf {Tr_{el}}\left[ \rho \right]$ is a density operator for the nuclear spins, $\mathsf {Tr_{el}}\dots$ denotes the trace over the electronic subspace and by definition $\mathcal {W}_{0}\rho _{SS}=0$. The complement of P is $Q=1-P$. By projecting the master equation on the P subspace and tracing over the electronic degrees of freedom we obtain an effective master equation for the nuclear spins in second order perturbation theory

$$\begin{aligned} \dot{\mu }=\mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}P\rho -P\mathcal {W}_{1}Q\mathcal {W}_{0}^{-1}Q\mathcal {W}_{1}P\rho \right] . \end{aligned}$$

(2.69)

Using $\mathsf {Tr_{el}}\left[ S^{z}\rho _{SS}\right] =-1/2$, the first term is readily evaluated and yields the Knight shift seen by the nuclear spins

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}P\rho \right] =+i\frac{g}{2}\left[ A^{z},\mu \right] . \end{aligned}$$

(2.70)

The derivation of the second term is more involved. It can be rewritten as

$$\begin{aligned} -\mathsf {Tr_{el}}&\left[ P\mathcal {W}_{1}Q\mathcal {W}_{0}^{-1}Q\mathcal {W}_{1}P\rho \right] \nonumber \\&= -\mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}\left( 1-P\right) \mathcal {W}_{0}^{-1}\left( 1-P\right) \mathcal {W}_{1}P\rho \right] \end{aligned}$$

(2.71)

$$\begin{aligned}&= \int _{0}^{\infty }d\tau \,\mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}e^{\mathcal {W}_{0}\tau }\mathcal {W}_{1}P\rho \right] -\int _{0}^{\infty }d\tau \,\mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}P\mathcal {W}_{1}P\rho \right] . \end{aligned}$$

(2.72)

Here, we used the Laplace transform $-\mathcal {W}_{0}^{-1}=\int _{0}^{\infty }d\tau \, e^{\mathcal {W}_{0}\tau }$ and the property $e^{\mathcal {W}_{0}\tau }P=Pe^{\mathcal {W}_{0}\tau }=P$.

Let us first focus on the first term in Eq. (2.72). It contains terms of the form

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\left[ A^{+}S^{-},e^{\mathcal {W}_{0}\tau }\left[ A^{-}S^{+},\mu \otimes \rho _{SS}\right] \right] \right]= & {} \mathsf {Tr_{el}}\left[ S^{-}e^{\mathcal {W}_{0}\tau }\left( S^{+}\rho _{SS}\right) \right] A^{+}A^{-}\mu \end{aligned}$$

(2.73)

$$\begin{aligned}&-\,\mathsf {Tr_{el}}\left[ S^{-}e^{\mathcal {W}_{0}\tau }\left( S^{+}\rho _{SS}\right) \right] A^{-}\mu A^{+}\end{aligned}$$

(2.74)

$$\begin{aligned}&+\,\mathsf {Tr_{el}}\left[ S^{-}e^{\mathcal {W}_{0}\tau }\left( \rho _{SS}S^{+}\right) \right] \mu A^{-}A^{+}\end{aligned}$$

(2.75)

$$\begin{aligned}&-\,\mathsf {Tr_{el}}\left[ S^{-}e^{\mathcal {W}_{0}\tau }\left( \rho _{SS}S^{+}\right) \right] A^{+}\mu A^{-} \end{aligned}$$

(2.76)

This can be simplified using the following relations: Since $\rho _{SS}=\left| \downarrow \right\rangle \left\langle \downarrow \right|$, we have $S^{-}\rho _{SS}=0$ and $\rho _{SS}S^{+}=0$. Moreover, $\left| \uparrow \right\rangle \left\langle \downarrow \right|$ and $\left| \downarrow \right\rangle \left\langle \uparrow \right|$ are eigenvectors of $\mathcal {W}_{0}$ with eigenvalues $-\left( i\omega _{0}+\alpha /2\right)$ and $+\left( i\omega _{0}-\alpha /2\right)$, where $\alpha =\gamma +\Gamma$, yielding

$$\begin{aligned} e^{\mathcal {W}_{0}\tau }\left( S^{+}\rho _{SS}\right)= & {} e^{-\left( i\omega _{0}+\alpha /2\right) \tau }\left| \uparrow \right\rangle \left\langle \downarrow \right| \end{aligned}$$

(2.77)

$$\begin{aligned} e^{\mathcal {W}_{0}\tau }\left( \rho _{SS}S^{-}\right)= & {} e^{+\left( i\omega _{0}-\alpha /2\right) \tau }\left| \downarrow \right\rangle \left\langle \uparrow \right| . \end{aligned}$$

(2.78)

This leads to

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\left[ A^{+}S^{-},e^{\mathcal {W}_{0}\tau }\left[ A^{-}S^{+},\mu \otimes \rho _{SS}\right] \right] \right] =e^{-\left( i\omega _{0}+\alpha /2\right) \tau }\left( A^{+}A^{-}\mu -A^{-}\mu A^{+}\right) . \end{aligned}$$

(2.79)

Similarly, one finds

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\left[ A^{-}S^{+},e^{\mathcal {W}_{0}\tau }\left[ A^{+}S^{-},\mu \otimes \rho _{SS}\right] \right] \right] =e^{+\left( i\omega _{0}-\alpha /2\right) \tau }\left( \mu A^{+}A^{-}-A^{-}\mu A^{+}\right) . \end{aligned}$$

(2.80)

Analogously, one can show that terms containing two flip or two flop terms give zero. The same holds for mixed terms that comprise one flip-flop and one Overhauser term with ${\sim } A^{z}S^{z}$. The term consisting of two Overhauser contributions gives

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\left[ A^{z}S^{z},e^{\mathcal {W}_{0}\tau }\left[ A^{z}S^{z},\mu \otimes \rho _{SS}\right] \right] \right] =-\frac{1}{4}\left[ 2A^{z}\mu A^{z}-\left[ A^{z}A^{z},\mu \right] \right] . \end{aligned}$$

(2.81)

However, this term exactly cancels with the second term from Eq. (2.72). Thus we are left with the contributions coming from Eqs. (2.79) and (2.80). Restoring the prefactors of $-ig/2$, we obtain

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}Q\left( -\mathcal {W}_{0}^{-1}\right) Q\mathcal {W}_{1}P\rho \right]= & {} \frac{g^{2}}{4}\int _{0}^{\infty }d\tau \left[ e^{-\left( i\omega _{0}+\alpha /2\right) \tau }\left( A^{-}\mu A^{+}-A^{+}A^{-}\mu \right) \right. \nonumber \\&\left. +\,e^{+\left( i\omega _{0}-\alpha /2\right) \tau }\left( A^{-}\mu A^{+}-\mu A^{+}A^{-}\right) \right] . \end{aligned}$$

(2.82)

Performing the integration and separating real from imaginary terms yields

$$\begin{aligned} \mathsf {Tr_{el}}\left[ P\mathcal {W}_{1}Q\left( -\mathcal {W}_{0}^{-1}\right) Q\mathcal {W}_{1}P\rho \right] =c_{r}\left[ A^{-}\mu A^{+}-\frac{1}{2}\left\{ A^{+}A^{-},\mu \right\} \right] +ic_{i}\left[ A^{+}A^{-},\mu \right] , \end{aligned}$$

(2.83)

where $c_{r}=g^{2}/\left( 4\omega _{0}^{2}+\alpha ^{2}\right) \alpha$ and $c_{i}=g^{2}/\left( 4\omega _{0}^{2}+\alpha ^{2}\right) \omega _{0}$. Combining Eq. (2.70) with Eq. (2.83) directly gives the effective master equation for the nuclear spins given in Eq. (2.3) in the main text.