The Hamiltonian in Eq. (7) can be diagonalized with three instaneous eigenstates

$$\begin{aligned} \vert +\rangle =& \frac{\sqrt{2}}{2} \bigl(\sin (\theta ) \vert 1 \rangle + \vert 2\rangle + \cos (\theta )\vert 3\rangle \bigr); \end{aligned}$$

(10)

$$\begin{aligned} \vert d\rangle = & \cos (\theta ) \vert 1\rangle - \sin (\theta ) \vert 3\rangle ; \end{aligned}$$

(11)

$$\begin{aligned} \vert -\rangle =& \frac{\sqrt{2}}{2} \bigl(\sin (\theta ) \vert 1 \rangle - \vert 2\rangle + \cos (\theta )\vert 3\rangle \bigr), \end{aligned}$$

(12)

corresponding to eigenvalues \(\Omega (t)\), 0, and \(-\Omega (t)\) respectively. In the ideal three-level STIRAP, one adiabatically changes \(\theta (t)\) from 0 to \(\pi /2\) by tuning the ratio \(\Omega _{P}(t) / \Omega _{S}(t)\) such that once the initial state is chosen to be \(\vert 1\rangle \), it will always remain in \(\vert d\rangle \) (therefore it will eventually be in \(\vert 3\rangle \)). The unitary matrix *W* to diagonalize \(\hat{H}(t)\) can be written as

$$\begin{aligned} W(\theta ) = \begin{bmatrix} \frac{\sqrt{2}}{2} \sin (\theta ) & \cos (\theta ) & \frac{\sqrt{2}}{2}\sin (\theta ) \\ \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \cos (\theta ) & -\sin (\theta ) & \frac{\sqrt{2}}{2}\cos (\theta ) \end{bmatrix}. \end{aligned}$$

(13)

To gain better insight into the time evolution, we go to the adiabatic picture, in which the Eq. (3) becomes [26]

$$\begin{aligned} \frac{d\hat{\rho }_{a}}{dt} = -{\mathrm{i}}[\hat{H}_{a}, \hat{\rho }_{a}] - [M, \hat{\rho }_{a}] + \mathcal{D}_{a}(\hat{\rho }_{a}). \end{aligned}$$

(14)

Here \(\hat{\rho }_{a}\), \(\hat{H}_{a}\), \(\mathcal{D}_{a}\) are the corresponding operators to *ρ̂*, *Ĥ*, \(\mathcal{D}\) respectively in the adiabatic basis \(\{\vert +\rangle , \vert d\rangle , \vert -\rangle \}\), which can be written as

$$\begin{aligned} &\hat{\rho }_{a} = W^{\dagger}\hat{\rho }W = \begin{bmatrix} \rho _{++} & \rho _{+d} & \rho _{+-} \\ \rho _{d+} & \rho _{dd} & \rho _{d-} \\ \rho _{-+} & \rho _{-d} & \rho _{--} \end{bmatrix}; \end{aligned}$$

(15)

$$\begin{aligned} &\hat{H}_{a}(t) = W^{\dagger} \hat{H}W = \Omega (t) \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}; \end{aligned}$$

(16)

$$\begin{aligned} &\mathcal{D}_{a}(\hat{\rho }_{a}) = \gamma \bigl( \hat{F}_{a} \hat{\rho }_{a} \hat{F}^{\dagger }_{a} - \hat{\rho }_{a} \bigr), \end{aligned}$$

(17)

with

$$\begin{aligned} \hat{F}_{a} = W^{\dagger} \hat{F}W = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}. \end{aligned}$$

(18)

Here we have used \(\rho _{uv}\) with \(u, v \in \{+, d, -\}\) to denote the element \(\langle u\vert \hat{\rho }_{a}\vert v\rangle \). The gauge matrix *M* satisfies

$$\begin{aligned} M = W^{\dagger} \dot{W} = \frac{\sqrt{2}}{2} \dot{ \theta} \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 1 \\ 0 & -1 & 0 \end{bmatrix}, \end{aligned}$$

(19)

which results from the time dependence of the adiabatic basis. Now substituting Eqs. (16), (17), (19) into Eq. (14), we get the following set of equations

$$\begin{aligned}& \dot{\rho}_{++} = \frac{\sqrt{2}}{2}\dot{\theta} ( \rho _{d+} + \rho _{+d} ) + \gamma (\rho _{--} - \rho _{++} ); \end{aligned}$$

(20a)

$$\begin{aligned}& \dot{\rho}_{--} = \frac{\sqrt{2}}{2}\dot{\theta} ( \rho _{d-} + \rho _{-d} ) + \gamma (\rho _{++} - \rho _{--} ); \end{aligned}$$

(20b)

$$\begin{aligned}& \dot{\rho}_{dd} = -\frac{\sqrt{2}}{2}\dot{\theta} (\rho _{+d} + \rho _{d+} + \rho _{d-} +\rho _{-d} ); \end{aligned}$$

(20c)

$$\begin{aligned}& \dot{\rho}_{+d} = -{\mathrm{i}}\Omega \rho _{+d} - \frac{\sqrt{2}}{2} \dot{\theta} (-\rho _{dd} + \rho _{++}+\rho _{+-} ) + \gamma (\rho _{-d} - \rho _{+d} ); \end{aligned}$$

(20d)

$$\begin{aligned}& \dot{\rho}_{d-} = -{\mathrm{i}}\Omega \rho _{d-}- \frac{\sqrt{2}}{2} \dot{\theta} (- \rho _{dd} + \rho _{+-} + \rho _{--} ) + \gamma (\rho _{d+} - \rho _{d-} ); \end{aligned}$$

(20e)

$$\begin{aligned}& \dot{\rho}_{+-} = -2{\mathrm{i}}\Omega \rho _{+-}+ \frac{\sqrt{2}}{2} \dot{\theta} (\rho _{d-}+\rho _{+d} ) + \gamma (\rho _{-+} - \rho _{+-} ), \end{aligned}$$

(20f)

We note that in Ref. [26], the second term in Eq. (14) is neglected since it depends on *θ̇* which is assumed to be small. However in our case if this term is neglected, we will arrive at a solution where the population is trapped in \(\vert d\rangle \), since it is a dark state of the dissipator \(\mathcal{D}_{a}\).

The set of Eqs. (20a)–(20f) are difficult to solve analytically in general. However, they can be significantly simplified with several reasonable assumptions. First, in the context of STIRAP, the adiabatic condition in Eq. (1) requires *θ̇* to be smaller compared to other relevant parameters. Second, we assume that in the adiabatic basis the off-diagonal terms of \(\hat{\rho }_{a}\) are small, that is \(\rho _{uv} \ll 1\) if \(u \neq v\). Now we subtract Eq. (20b) from Eq. (20a) and get an equation for \(g = \rho _{++} - \rho _{--} \) as

$$\begin{aligned} \dot{g} = \frac{\sqrt{2}}{2} \dot{\theta} (\rho _{d+} + \rho _{+d} - \rho _{d-} - \rho _{-d} ) - 2\gamma g. \end{aligned}$$

(21)

The first term on the right-hand side of Eq. (21) contains *θ̇* and \(\rho _{d+} + \rho _{+d} - \rho _{d-} - \rho _{-d}\) which are both small numbers. Thus we neglect this term and get \(\dot{g} = - 2\gamma g\). Since \(g(t)\) is initially 0, and get \(g(t)=0\) for all *t*, namely

$$\begin{aligned} \rho _{++}(t) = \rho _{--}(t). \end{aligned}$$

(22)

Similarly, subtracting Eq. (20e) from Eq. (20d), we get an equation for \(h = \rho _{+d} - \rho _{d-}\) as

$$\begin{aligned} \dot{h} = -{\mathrm{i}}\Omega h - \gamma \bigl(h + h^{\ast}\bigr), \end{aligned}$$

(23)

where we have used \(\rho _{++}(t) = \rho _{--}(t)\). Now since \(h(t)\) is initially 0, from Eq. (23) we have \(h(t)=0\) for all *t*, namely

$$\begin{aligned} \rho _{+d}(t) = \rho _{d-}(t). \end{aligned}$$

(24)

Finally, the second term on the right-hand side of Eq. (20f) can be neglected for the same reason, and we get

$$\begin{aligned} \dot{\rho}_{+-} = -2{\mathrm{i}}\Omega \rho _{+-} + \gamma (\rho _{-+} - \rho _{+-} ). \end{aligned}$$

(25)

Since \(\rho _{+-}\) is initially 0, from Eq. (25) we get

$$\begin{aligned} \rho _{+-}(t)=0 \end{aligned}$$

(26)

for all *t*. Substituting Eqs. (22), (24), (26) back into Eqs. (20a)–(20f), and assuming \(\rho _{+d} = a + {\mathrm{i}}b\) with \(a(t)\) and \(b(t)\) real functions, we get the following closed set of equations for \(\rho _{dd}\), *a*, *b*

$$\begin{aligned}& \dot{\rho}_{dd} = -2\sqrt{2}\dot{\theta} a; \end{aligned}$$

(27a)

$$\begin{aligned}& \dot{a} = \Omega b - \frac{\sqrt{2}}{4}\dot{\theta} (1-3\rho _{dd} ); \end{aligned}$$

(27b)

$$\begin{aligned}& \dot{b} = -\Omega a - 2\gamma b , \end{aligned}$$

(27c)

where we have used \({\mathrm{tr}}(\hat{\rho }_{a}) = 1 \). Equations (27a)–(27c) are still difficult to solve analytically since their coefficients are time-dependent in the general case.

In the following, we consider a specific driving protocol as follows:

$$\begin{aligned} &\Omega _{P}(t)= \Omega _{0} \sin \biggl(\frac{\pi t}{2 T_{0}}\biggr); \end{aligned}$$

(28a)

$$\begin{aligned} &\Omega _{S}(t)= \Omega _{0} \cos \biggl(\frac{\pi t}{2 T_{0}}\biggr), \end{aligned}$$

(28b)

where \(\Omega _{0}\) denotes the strength of the laser coupling and \(T_{0}\) is the total duration of it. Such a choice of driving protocol is more of theoretical convenience than of experimental relevance. Later we will also consider the more commonly used Gaussian driving protocol. We can see that \(\Omega _{P}(0)/\Omega _{S}(0) = 0\) and \(\Omega _{P}(T_{0})/\Omega _{S}(T_{0}) = \infty \). The advantage of the protocol in Eq. (28a)–(28b) is that we have \(\Omega (t) = \Omega _{0}\), \(\theta (t) = \frac{\pi t}{2T_{0}}\) and \(\dot{\theta} = \frac{\pi}{2T_{0}}\). We further assume that in Eqs. (27a)–(27c) \(a(t)\) and \(b(t)\) are slowly varying variables compared to \(\rho _{dd}(t)\) (This approximation is partially due to the fact that in the adiabatic limit \(\rho _{dd}(t) \approx 1\) while the off diagonal terms \(a(t) \approx b(t) \approx 0\).). As a result we can set \(\dot{a}=\dot{b}=0\) and then Eqs. (27a) can be solved as

$$\begin{aligned} \rho _{dd}(t) = \frac{1}{3} + \frac{2}{3}e^{-3 \chi t}, \end{aligned}$$

(29)

with

$$\begin{aligned} \chi = \frac{2\gamma \dot{\theta}^{2}}{\Omega ^{2}}. \end{aligned}$$

(30)

To check the validity of Eq. (29), we compared it with the exact numerical solutions from Eq. (14). In Fig. 2(a) we show an instance of the driving protocol in Eq. (28a)–(28b). In Fig. 2(b), (c), (d) we show \(\rho _{dd}\) predicted by our analytical solution in Eq. (29) and the exact solution by Eq. (14) as functions of time *t* versus different values of *γ*, \(\Omega _{0}\), \(T_{0}\) respectively. We can see that our analytic prediction agrees very well with the exact solution in the wide parameter range we have considered.

It would be more clear to directly look at the dynamics in the original basis \(\{\vert 1\rangle , \vert 2\rangle , \vert 3\rangle \}\). Here we show the exact time evolution of the occupations \(\rho _{11}\), \(\rho _{22}\) and \(\rho _{33}\) in Fig. 3 by directly solving the Lindblad master equation (Eq. (3)) in the original picture. From Fig. 3(a), (c), (e), we can see that for smaller *γ*, and larger \(\Omega _{0}\) or \(T_{0}\), the intermediate state \(\vert 2\rangle \) is indeed seldomly occupied during the time evolution, and from Fig. 3(b), (d), (f) we can see that most of the population indeed transfers from state \(\vert 1\rangle \) to state \(\vert 3\rangle \) when these conditions are satisfied.

To this end we discuss the implications of our analytic solution in Eq. (29). From Eq. (11) we have \(\rho _{33}(T_{0}) = \rho _{dd}(T_{0})\). Therefore the final occupation of \(\rho _{dd}(T_{0})\) represents the population transfer efficiency. Then from Eq. (29) we have

$$\begin{aligned} \rho _{33}(T_{0}) = \frac{1}{3} + \frac{2}{3}e^{- \frac{3\pi \gamma \dot{\theta}}{\Omega ^{2}}}, \end{aligned}$$

(31)

where we have used \(\dot{\theta} T_{0} = \pi /2\). We can see that \(\rho _{33}(T_{0})\) decreases exponentially with *γ*. However, the effect of dephasing can be made arbitrarily small if we increase laser coupling strength Ω or increase the laser duration (such that *θ̇* will be smaller). Interestingly, based on Eq. (31) we can define an additional adiabatic condition on top of Eq. (1) which takes the dephasing strength into account. The additional adiabatic condition is simply \(-3\chi T_{0} \ll 1\), which is

$$\begin{aligned} \dot{\theta} \ll \frac{\Omega ^{2}}{3\pi \gamma}. \end{aligned}$$

(32)

Complete population transfer can still be achieved as long as Eqs. (1), (32) are both satisfied. Now for comparison, when using the same driving protocol as in Eq. (28a)–(28b), we will obtain \(\eta = 3T_{0}/8\) in Eq. (2), that is, the population transfer efficiency is independent of Ω, but decreases exponentially both with \(\gamma _{13}\) and \(T_{0}\). Therefore complete population transfer can never be achieved as long as \(\gamma _{13} \neq 0\), since we can neither tune *θ̇* to be very small (\(T_{0}\) will be very large and increase the damping) or very large (which breaks the adiabatic condition in Eq. (1)). The reason for the qualitatively different predictions between our result and that of Ref. [26] can be seen from Eq. (14). In our case, the absence of dephasing on the first and third spin results in the missing of the term \(\gamma _{13}\) as defined in Ref. [26]. As a result, the state \(\vert d\rangle \) forms a dark space that is decoupled from the states \(\vert \pm \rangle \) in the adiabatic limit (which means that \(\dot{\theta}\rightarrow 0\) and the second term on the right hand side of Eq. (14) can be neglected) and perfect population transfer can still be achieved.

Our analytic solution in Eq. (29) does not hold for general laser driving protocols. To show the validity of the physical picture we obtained based on the specific driving protocol in Eq. (28a)–(28b), we numerically study the effect of local dephasing under the commonly used Gaussian driving protocol as follows

$$\begin{aligned} \Omega _{P}(t) &= \Omega _{0} \exp \biggl(- \frac{ (t - \tau /2 )^{2}}{T^{2}} \biggr); \end{aligned}$$

(33)

$$\begin{aligned} \Omega _{S}(t) &= \Omega _{0} \exp \biggl(- \frac{ (t + \tau /2 )^{2}}{T^{2}} \biggr). \end{aligned}$$

(34)

Here *T* denotes the width of the Gaussian laser coupling, *τ* is the delay between the two lasers. We choose the duration of the lasers, denoted as \(T_{0}\), to be \(T_{0} = 8T\), namely the time-dependent driving will start from \(t = -T_{0}/2\) and end at \(t=T_{0}/2\). The population \(\rho _{dd}\) on the dark state \(\vert d\rangle \) as functions of *γ*, \(\Omega _{0}\) and \(T_{0}\), are shown in Fig. 4. From Fig. 4(a), we can clearly see that dephasing can strongly suppress the population transfer efficiency. While from Fig. 4(c), (e), we can see that near-perfect population transfer can be restored by increasing \(\Omega _{0}\) or \(T_{0}\). This demonstrates that the physical picture obtained from our analytic solution is still valid for other laser driving protocols. Additionally, from Fig. 4(b), (d), (f), we can see that the simplified set of equations in Eqs. (27a)–(27c) agree very well with the exact solutions in the wide parameter range considered in Fig. 4, which again validates our approximations made to derive it.

To this end, we discuss about possible physical implementation for our theoretical model. The coupled cavities system could be a good candidate for us, for example it has already been shown that coupled cavities system can be effectively described as coupled spins [31, 32], whose Hamiltonian is almost identical to the one we used in Eq. (4) and the coupling strength can be tuned up to 100 MHz [33]. Moreover, dephasing is a common source of noise for cavities, with the typical decoherence time (inverse of the dephasing strength) \(T_{c}\) ranging from hundreds of nano seconds to a few micro seconds [34, 35]. Therefore, it is possible to realize our theoretical model using two perfect cavities which are both coupled to an intermediate bad cavity under dephasing noise. We believe this work could significantly enhance the theoretical understanding of population transfer through noisy quantum channels and could be useful for practical quantum information processing.