Stable quantum interference enabled by coexisting detuned and resonant STIRAPs^*

As represented in Fig. 1(a), the energy levels |g₀〉, |e_r,d1 〉 and |g_r,d〉 describe the initial ground state, two middle excited states and two ground hyperfine substates. The pump and Stokes lasers with Rabi frequencies Ω_p(t) and Ω_s(t) resonantly coupling states |g₀〉 → |e_r〉 and states |e_r〉 → |g_r〉, serve as the basis of a round-trip STIRAP transition. Typically speaking, such a five-level model contains two STIRAP paths, which are named as D-STIRAP and R-STIRAP, enabled by the round-trip optical pulses as displayed in Fig. 1(b). More concretely, R-STIRAP refers to the transition of |g₀ 〉 ⇄ |e_r 〉 ⇄ |g_r〉 with both one (or two)-photon detuning vanishing. Whereas D-STIRAP carries out among states |g₀〉 ⇄ |e_d1 〉 ⇄ |g_d〉, accompanied by the one (two)-photon detuning Δ₁ (δ) with respect to |e_r〉 (|g_r〉).^[29] As for a practical one-photon detuning Δ₁ which is usually orders of magnitude larger than the hyperfine energy difference δ caused by Zeeman splitting of an external magnetic field, we first safely ignore the detuned state |e_d1 〉 and pay attention to a four-level configuration. For example, in ⁸⁷Rb atoms the level spacing between |e_r〉 = |5² P_3/2,F = 2〉 and |e_d1 〉 = |5² P_3/2,F = 3〉 is about Δ₁ ≈ 267 MHz^[30] while the Zeeman splitting δ between |g_r〉 = |5² S_1/2,F = 2,m_F = 0〉 and |g_d〉 = |5² S_1/2,F = 2,m_F = 1〉 is only scaling of a few kHz typically.^[31] For a complete study, the impact of multi-excited states nearby will leave for discussion in Section 4.

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In the rotating-wave frame, the four-level scheme Hamiltonian reads (ℏ = 1)^[32]

$$\begin{eqnarray}&&\hat{ {\mathcal H} }=-\delta {\hat{\sigma }}_{{g}_{{\rm{d}}}{g}_{{\rm{d}}}}+\displaystyle \frac{1}{2}[{\varOmega }_{{\rm{p}}}{\hat{\sigma }}_{{g}_{0}{e}_{{\rm{r}}}}+{\varOmega }_{{\rm{s}}}({\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{r}}}}+{\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{d}}}})+{\rm{h}}{\rm{.c}}{\rm{.}}],\end{eqnarray} \tag{ 1 }$$

where ${\hat{\sigma }}_{ij}=|i\rangle \langle j|$ is the projection operator, and the two-photon detuning δ characterizes the energy difference between |g_r〉 and |g_d〉. For R-STIRAP with δ = 0, there exists a dark eigenstate with its energy E_d = 0. At the same time, the adjacent magnetic substate |g_d〉 detuned by δ [≠ 0] to the resonant level |g_r〉 permits the D-STIRAP, which supports a quasi-dark-state (not eigenstate) energy E_qd. Here the round-trip STIRAP pulses adopting a generalized form of^[33]

$$\begin{eqnarray}&&\begin{array}{l}{\varOmega }_{{\rm{p}}}(t)=A(t-\tau)+A(t-\tau -\Delta T-T),\\ {\varOmega }_{{\rm{s}}}(t)=A(t)+A(t-2\tau -\Delta T-T),\end{array}\end{eqnarray} \tag{ 2 }$$

are displayed in Fig. 1(b), where τ is the delayed time between two pulses Ω_p(s) and the amplitude function A(t) is given by

$$\begin{eqnarray}A(t)\equiv \left\{\begin{array}{ll}{\varOmega }_{0,{\rm{p}}({\rm{s}})}{\sin }^{4}(\displaystyle \frac{\pi t}{T}), & 0\leqslant t\leqslant T,\\ 0, & {\rm{otherwise}}.\end{array}\right.\end{eqnarray} \tag{ 3 }$$

Here T is the common pulse length, Ω_0,p(s) are the peak amplitudes of pump (Stokes) lasers.

Population initialized on state |g₀〉 individually undergoing R-STIRAP and D-STIRAP paths will interfere with each other. In this case, if we detect the final population on state |g₀〉 after all pulses, an oscillation pattern occurs as a function of the product of the hold time Δ T and the two-photon detuning δ. Such an interfering effect can serve as a new way for realizing a STIRAP atom interferometer. To qualitatively understand its essence, we first separately study them in decomposed three-level models, as shown in the insets of Figs. 2(a) and 2(b). The separated three-level Λ models that contain states |g₀〉, |e_r〉, |g_r〉 for R-STIRAP and states |g₀〉, |e_r〉, |g_d〉 for D-STIRAP, can be described by the reduced three-level Hamiltonians

$$\begin{eqnarray}&&{\hat{ {\mathcal H} }}_{{\rm{r}}}=\displaystyle \frac{1}{2}[{\varOmega }_{{\rm{p}}}{\hat{\sigma }}_{{g}_{0}{e}_{{\rm{r}}}}+{\varOmega }_{{\rm{s}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{r}}}}+{\rm{h}}{\rm{.c}}{\rm{.}}],\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\hat{ {\mathcal H} }}_{{\rm{d}}}\,=-\delta {\hat{\sigma }}_{{g}_{{\rm{d}}}{g}_{{\rm{d}}}}+\displaystyle \frac{1}{2}[{\varOmega }_{{\rm{p}}}{\hat{\sigma }}_{{g}_{0}{e}_{{\rm{r}}}}+{\varOmega }_{{\rm{s}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{d}}}}+{\rm{h}}{\rm{.c}}{\rm{.}}],\end{eqnarray} \tag{ 5 }$$

and by diagonalizing Eqs. (4) and (5), we analytically solve all eigenvalues, which are

$$\begin{eqnarray}&&{\epsilon }_{0}=0,\,\,\,{\epsilon }_{\pm }=\pm \sqrt{{\varOmega }_{{\rm{p}}}{}^{2}+{\varOmega }_{{\rm{s}}}{}^{2}}/2,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\epsilon }_{0}{^{\prime} }=\displaystyle \frac{\bigg(\delta +\tilde{\varOmega }\cos \displaystyle \frac{\zeta }{3}\bigg)}{3},\,\,\,{\epsilon }_{\pm }{^{\prime} }=\displaystyle \frac{\bigg(\delta +\tilde{\varOmega }\cos \displaystyle \frac{2\pi \mp \zeta }{3}\bigg)}{3},\end{eqnarray} \tag{ 7 }$$

with

$$\begin{eqnarray*}\begin{array}{lll}\tilde{\varOmega } & = & \sqrt{3({\varOmega }_{{\rm{p}}}{}^{2}+{\varOmega }_{{\rm{s}}}{}^{2})+4{\delta }^{2}},\\ \zeta & = & 2\pi -\arccos \displaystyle \frac{\delta }{{\tilde{\varOmega }}^{3}}(9{\varOmega }_{{\rm{s}}}{}^{2}-18{\varOmega }_{{\rm{p}}}{}^{2}+8{\delta }^{2}).\end{array}\end{eqnarray*}$$

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A qualitative representation of eigenvalues is comparably shown in Figs. 2(a)–2(b). It is clear that the dark energy E_d ≡ ε₀ persists zero due to the presence of an absolute zero-energy dark eigenstate $|{E}_{{\rm{d}}}\rangle =\displaystyle \frac{1}{\sqrt{{\varOmega }_{{\rm{p}}}^{2}+{\varOmega }_{{\rm{s}}}^{2}}}({\varOmega }_{{\rm{s}}}|{g}_{0}\rangle -{\varOmega }_{{\rm{p}}}|{g}_{{\rm{r}}}\rangle)$ in R-STIRAP. While as for D-STIRAP there does not exist such an eigenstate; however, we observe a quasi-dark state |E_qd〉 with its energy E_qd shifted by δ during the hold time in which Ω_p = Ω_s = 0. As displayed in Fig. 2(b), this quasi-dark energy E_qd is marked by a grey shaded curve which composes parts of eigenenergies ${\epsilon }_{0}{^{\prime} }$ and ${\epsilon }_{+}{^{\prime} }$ due to the emergence of two avoided crossings between them. Once the system evolves along with the quasi-dark state |E_qd〉, it will get an extra phase ΔΦ due to this shifted energy δ in E_qd. Finally, it leads to the population interference with that evolves along the dark eigenstate after the round-trip STIRAP pulses.

The proof-of-principle STIRAP atom interferometer can also be indicated with the help of STIRAP interfering effect. As demonstrated in Fig. 1(c), starting from the initialized state |g₀〉 that the forward STIRAP pulse (Stokes exceeds pump) acts as the first "beam splitter" (BS) that coherently splits into two STIRAP paths, giving rise to a superposition state between |g_r〉 and |g_d〉. However, before the arrival of the second inverse pulse pair, there exists a hold time Δ T enabling a free evolution along with the absolute dark or quasi-dark states, accordingly. Until reaching the second BS (pump exceeds Stokes), the population will converge at state |g₀〉 again. If detecting the final population of |g₀〉 it reveals a clear interference pattern that strongly depends on the relative phase ΔΦ accumulated between the two paths. This behavior is analogous to the undergoing of different optical paths, which is the so-called STIRAP atom interferometer. By detecting the interference fringe, we can precisely obtain a relative phase ΔΦ and other information.

In order to know the real population dynamics in reduced models that could help us understand the observation of interference more clearly, we begin with a study of population transfer and eigenstate evolution in individual STIRAP paths. Numerical results for the realistic time-dependent population dynamics come from solving a master equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\hat{\rho }}_{{\rm{r}}({\rm{d}})}}{{\rm{d}}t}={\rm{i}}[{\hat{\rho }}_{{\rm{r}}({\rm{d}})},{\hat{ {\mathcal H} }}_{{\rm{r}}({\rm{d}})}(t)]+{\hat{ {\mathcal L} }}_{{\rm{r}}({\rm{d}})},\end{eqnarray} \tag{ 8 }$$

where the subscript r(d) refers to R (or D)-STIRAP, and the density matrix ${\hat{\rho }}_{{\rm{r}}({\rm{d}})}$ is of 3 × 3 type. ${\hat{ {\mathcal L} }}_{{\rm{r}}({\rm{d}})}$ representing the influence of spontaneous decay from state |e_r〉, takes forms of

$$\begin{eqnarray}&&{\hat{ {\mathcal L} }}_{{\rm{r}}}=\Gamma \displaystyle \sum _{j\in \{{g}_{0},{g}_{{\rm{r}}}\}}\bigg[{\hat{\sigma }}_{j{e}_{{\rm{r}}}}{\hat{\rho }}_{{\rm{r}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}j}-\displaystyle \frac{1}{2}({\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}}{\hat{\rho }}_{{\rm{r}}}+{\hat{\rho }}_{{\rm{r}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}})\bigg],\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\hat{ {\mathcal L} }}_{{\rm{d}}}=\Gamma \displaystyle \sum _{j\in \{{g}_{0},{g}_{{\rm{d}}}\}}\bigg[{\hat{\sigma }}_{j{e}_{{\rm{r}}}}{\hat{\rho }}_{{\rm{d}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}j}-\displaystyle \frac{1}{2}({\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}}{\hat{\rho }}_{{\rm{d}}}+{\hat{\rho }}_{{\rm{d}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}})\bigg],\end{eqnarray} \tag{ 10 }$$

with Γ the decay rate. P_g0 , P_er , P_gr(d) are defined by the diagonal elements of the density matrix representing the state population. We show the real population dynamics in Fig. 3(a1) for R-STIRAP that reveals a well-expected transfer perfectly coinciding with the ideal dark-eigenstate |s_d〉 evolution as shown in Fig. 3(a2). That fact confirms the coherent adiabatic population transfer in R-STIRAP.

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However, as for D-STIRAP there does not exist an absolute single eigenstate for achieving the adiabatic transfer since $|{\epsilon }_{0}{^{\prime} }\rangle$, $|{\epsilon }_{\pm }{^{\prime} }\rangle$ are all related to the lossy excited state |e_r〉. Because |e_r〉 suffers from an inevitable loss Γ that may break the adiabaticity of STIRAP. Fortunately, thanks to the emergence of avoided crossings by shift δ where the eigenenergies ${\epsilon }_{0}{^{\prime} }$ and ${\epsilon }_{+}{^{\prime} }$ become degenerate. The expected population transfer labeled by the grey shaded curves as presented in Figs. 3(b2)–3(b3), first obeys |g₀〉 of $|{\epsilon }_{0}{^{\prime} }\rangle$, and then jumps to |g_d〉 of $|{\epsilon }_{+}{^{\prime} }\rangle$ at the first avoided crossing. Subsequently, it returns back to |g₀〉 of $|{\epsilon }_{0}{^{\prime} }\rangle$ at the second avoided crossing. Benefiting from such a coherent population transfer between $|{\epsilon }_{0}{^{\prime} }\rangle$ and $|{\epsilon }_{+}{^{\prime} }\rangle$, Fig. 3(b1) shows a perfect population dynamics for D-STIRAP under δ = 0.05. Although the system does not contain a dark eigenstate, we find that the population dynamics agrees well with the evolution of quasi-dark state |E_qd〉, as shown by grey shaded curves in Figs. 3(b2)–3(b3). Therefore, our results indicate that the coherent population transfer can be well kept even in D-STIRAP as long as δ is suitable. Nevertheless, once δ is increased to 0.5 that is comparable to Ω₀ [= 1.0] which may break the coherent transfer between $|{\epsilon }_{0}{^{\prime} }\rangle$ and $|{\epsilon }_{+}{^{\prime} }\rangle$ at the avoided crossing point, the final population P_g0 (∞) on state |g₀〉 becomes lower, see Figs. 3(c1)–3(c3). This is made by a larger energy gap at the avoided crossing when δ is increased, which leads to the decoherence effect by the excited-state loss in the process. To avoid this effect, we will adopt Ω₀ = 5.0 in the calculation.

So far, we have clearly shown the real population transfer as well as the ideal eigenstate evolution. Our results verify that both R-STIRAP and D-STIRAP can enable a perfect coherent population transfer in its individual three-level configurations as long as δ is appropriate.

To obtain the STIRAP interference pattern under the influence of the relative phase ΔΦ, we resort to the full system and numerically solve the master equation^[34]

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }}{{\rm{d}}t}={\rm{i}}[\hat{\rho },\hat{ {\mathcal H} (t)}]+\hat{ {\mathcal L} }(\hat{\rho }),\end{eqnarray} \tag{ 11 }$$

with $\hat{\rho }$ a 4 × 4 density matrix referring to the basis of |g₀〉,|e_r〉,|g_r〉,|g_d〉, and the Lindblad operator $\hat{ {\mathcal L} }$ is given by{^[35]}

$$\begin{eqnarray}&&\hat{ {\mathcal L} }(\hat{\rho })=\Gamma \displaystyle \sum _{j\in \{{g}_{0},{g}_{{\rm{r}}},{g}_{{\rm{d}}}\}}\bigg[{\hat{\sigma }}_{j{e}_{{\rm{r}}}}\hat{\rho }{\hat{\sigma }}_{{e}_{{\rm{r}}}j}-\displaystyle \frac{1}{2}({\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}}\hat{\rho }+\hat{\rho }{\hat{\sigma }}_{{e}_{{\rm{r}}}{e}_{{\rm{r}}}})\bigg].\end{eqnarray} \tag{ 12 }$$

Note that Γ is the spontaneous decay rate from the resonant excited state |e_r〉 determined by its lifetime, and the population on state |g₀〉 defined by P_g0(t) = ρ_{g0 g0} (t) serves as the main observable quantity in detection. For t → ∞ it means the final time. Here ρ_ij (t) (i,j ∈ (g₀,e_r,g_r,d) stands for the element of density matrix $\hat{\rho }$, and ρ_jj (t) means the |j〉 state population. As predicted by the experiment,^[21] we numerically study the relationship between P_g0(∞) and Δ T, and confirm its strongly oscillating behavior.

Before quantitatively understanding this relationship, we first solve for an analytical expression of the relative phase ΔΦ, which is

$$\begin{eqnarray}&&\Delta \Phi ={\Phi }_{{\rm{d}}}-{\Phi }_{{\rm{r}}}\approx \phi (\delta)+\delta \times \Delta T,\end{eqnarray} \tag{ 13 }$$

with Φ_d and Φ_r the accumulated phases in each STIRAP path, given by^[36,37]

$$\begin{eqnarray}&&\begin{array}{l}{\Phi }_{{\rm{d}}}=\displaystyle {\int }_{0}^{t}{E}_{{\rm{qd}}}({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }\approx \phi (\delta)+\delta \times \Delta T,\\ {\Phi }_{{\rm{r}}}=\displaystyle {\int }_{0}^{t}{E}_{{\rm{d}}}({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }=0.\end{array}\end{eqnarray} \tag{ 14 }$$

Here the total evolution time before the measurement is t = 2(T + τ) + Δ T. E_qd(t) and E_d(t) in integrals stand for the quasi-dark state and dark eigenstate energies, respectively; see Fig. 2 and texts below. Note that no phase is accumulated in R-STIRAP due to E_d = 0. However, for D-STIRAP we could get a non-zero phase accumulation. By replacing E_qd(t), a more concrete expression for calculating Φ_d is

$$\begin{eqnarray}\begin{array}{lll}{\Phi }_{{\rm{d}}} & = & \displaystyle {\int }_{0}^{T}{\epsilon }_{0}{^{\prime} }({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }+\displaystyle {\int }_{T}^{T+2\tau +\Delta T}{\epsilon }_{+}{^{\prime} }({\rm{d}}{t}{^{\prime} }){\rm{d}}{t}{^{\prime} }\\ & & +\displaystyle {\int }_{T+2\tau +\Delta T}^{2T+2\tau +\Delta T}{\epsilon }_{0}{^{\prime} }({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }.\end{array}\end{eqnarray} \tag{ 15 }$$

We can define $\phi (\delta)=\displaystyle {\int }_{0}^{T}{\epsilon }_{0}{^{\prime} }({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }+\displaystyle {\int }_{T+2\tau +\Delta T}^{2T+2\tau +\Delta T}{\epsilon }_{0}{^{\prime} }({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }=2\displaystyle {\int }_{0}^{T}{\epsilon }_{0}{^{\prime} }({t}{^{\prime} }){\rm{d}}{t}{^{\prime} }$ due to the symmetry of ${\epsilon }_{0}{^{\prime} }(t)$ by round-trip pulses, which describes the phase accumulated during the atom-light interaction region. The second term $\displaystyle {\int }_{T}^{T+2\tau +\Delta T}{\epsilon }_{+}{^{\prime} }({\rm{d}}{t}{^{\prime} }){\rm{d}}{t}{^{\prime} }\approx \delta \times \Delta T$ comes from a steady shift δ of quasi-dark energy ${\epsilon }_{+}{^{\prime} }$ during the hold time Δ T region. Therefore for a given δ, the phase difference ΔΦ between two paths is only determined by δ × Δ T since φ(δ) is unvaried. By adjusting Δ T, we observe a high-contrast interfering pattern, as plotted in Fig. 4(a), where the peaks from constructive interference locate exactly at ΔΦ = 2nπ (n ∈ integers). Note that the constructive interference could be enabled by the least phase difference with n = 1, which is ΔΦ = 2π = δ × (1/f), resulting in the oscillating frequency f of the interference pattern, analytically expressed as

$$\begin{eqnarray}&&f=\delta /2\pi .\end{eqnarray} \tag{ 16 }$$

It is clear that f increases linearly with δ.

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A numerical verification as presented in Fig. 4(b) also shows a robust linear tendency between f and δ. In Fig. 4(b), the analytical expression of Eq. (16) is plotted by a solid line, while numerical results (black dots) are obtained from the frequency spectrum analysis when transforming into the frequency domain by Fourier transform. For every δ, it is possible to get a steady interference oscillation pattern within a hold time Δ T, like in Fig. 4(a). That fact, on one hand, provides a reliable determination of the Zeeman splitting energy δ under an adjustable external magnetic field; on the other hand, it interactively returns a high-contrast quantum interference effect for further practical applications. This visual STIRAP interference pattern can also be observed when transferring to the frame with respect to δ since ΔΦ is also δ-dependent. By scanning the hyperfine energy δ, a high-contrast interfering pattern can also be observed as shown in Fig. 4(c). However, this pattern turns to be a quick amplitude-damping behavior with the increase of |δ| due to the breakup of coherence between R-STIRAP and D-STIRAP. So only around the two-photon resonance, the interference pattern is best.

Here the pattern contrast or so-called visibility during a finite hold time Δ T given by^[38]

$$\begin{eqnarray}&&vis(\Delta T)=\displaystyle \frac{{P}_{g0}^{\max }-{P}_{g0}^{\min }}{{P}_{g0}^{\max }+{P}_{g0}^{\min }},\end{eqnarray} \tag{ 17 }$$

can serve as a key parameter for describing the interference quality. ${P}_{g0}^{\max (\min)}$ stands for the maximal (minimal) population during the hold time. The amplitude damping observed in Fig. 4(c) is due to the competing effect between R-STIRAP and D-STIRAP. When |δ| is small and appropriate, the population can be transferred along both paths efficiently as displayed in Figs. 3(a1) and 3(b1), arising a high-contrast interfering pattern. However, if |δ| is far from resonance, only the R-STIRAP will play a dominant role rather than the inefficient D-STIRAP, which would deeply lower the interfering amplitude as well as its visibility. The limit case is the atomic population would transfer along single three-level R-STIRAP path and finally return back to state |g₀〉 without showing a visible quantum interference pattern. A detailed discussion for coherent population transfer of individual R- and D-STIRAP has been presented in Subsection 2.2.

To verify that, we add significant fluctuations to the peak intensity of laser pulses by adopting the expression of

$$\begin{eqnarray}&&{\varOmega }_{{\rm{p}}({\rm{s}})}{^{\prime} }(t)={\varOmega }_{{\rm{p}}({\rm{s}})}(t)+\delta {\varOmega }_{{\rm{p}}({\rm{s}})}(t),\end{eqnarray} \tag{ 18 }$$

with the perturbation term δΩ_p(s) adopted from a range of [–δΩ,δΩ] and δΩ is chosen to be the maximal modulation amplitude. For comparison, we numerically study two ways of adding this laser intensity noise: i.e., δΩ_p(s) is a time-dependent stochastic value within [–δΩ,δΩ]; or δΩ_p(s) is modified to be a regular sinusoidal function which is δΩ_ps)(t) = δΩsinω t,^[42] and the frequency ω is arbitrary. Here ω = π/5. In the calculation, the maximal modulation amplitudes are δΩ = 0.2Ω₀ and 0.5Ω₀ ^[43] in cases (b) and (c) for comparing the variation with stronger fluctuations.

In Fig. 5, case (a) [the first column] represents the original laser pulse profile (Fig. 5(a1)), the realistic population dynamics under different hold times (Figs. 5(a2)–5(a5)), and the expected interference pattern (Fig. 5(a6)) under the condition of no fluctuation. Due to the change of hold time that gives rise to different accumulated phases in D-STIRAP, the final population on state P_g0(∞)(red-solid in Figs. 5(a2)–5(a5)) reveals a Δ T-dependent character, agreeing with the oscillating behavior in Fig. 5(a6). Furthermore, even under strong stochastic fluctuations to Ω_p(s)(t), see the second and fourth columns of Fig. 5, where δΩ = (0.2,0.5)Ω₀, our results confirm that the population dynamics as well as the interference pattern do not change at all, which are the same as the findings in Figs. 5(a2)–5(a6) with no fluctuations. The reason for that remarkable preservation of high-quality interference (black-solid) as displayed in Figs. 5(b6) and 5(c6) should be understood by the phase accumulation during the hold time Δ T, which critically depends on the quasi-dark energy E_qd. If δΩ_s(p) = 0, the phase difference shows a precise relation with respect to the splitting δ. However, as δΩ_s(p) ≠ 0, it will lead to an extra perturbed phase. Thanks to a stochastic fluctuation that arises a complete compensation of the phase change on average, the system could sustain the usual unperturbed phase difference.

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Nevertheless, if the perturbation is replaced by a regular sinusoidal modulation, as plotted in Figs. 5(b12) and 5(c12) with the same amplitudes δΩ/Ω₀ = 0.2 and 0.5, we observe that the population dynamics is strongly impacted by the sinusoidal modulation, resulting in a thorough breakdown of the interference pattern (blue-dashed curves in Figs. 5(b6) and 5(c6)). Because it is difficult to exactly overcome the phase accumulation for a sinusoidal modulation during the evolution. We could expect a more visible interference output if the modulation frequency ω is increased. Since a larger ω leads to a fast modulation to the pulses that is more similar to a stochastic perturbation, the resulting interference quality will be improved.

To our knowledge, a typical external noise is stochastic not regular. Therefore, our protocol can deservedly exhibit robust stability towards arbitrary stochastic fluctuations from the laser intensity noise, preserving a high-contrast interference output.

In fact, the Zeeman splitting δ can also be disturbed due to the instability of the external magnetic field, probably suffering from a stochastic shift δ' to the splitting level. In that case, for the R-STIRAP path, the two-photon resonance condition is perturbed by a stochastic detuning δ', leading to the Hamiltonian ${\hat{ {\mathcal H} }}_{{\rm{r}}}$ described by

$$\begin{eqnarray}&&{\hat{ {\mathcal H} }}_{{\rm{r}}}=-{\delta }{^{\prime} }{\hat{\sigma }}_{{g}_{{\rm{r}}}{g}_{{\rm{r}}}}+\displaystyle \frac{1}{2}[{\varOmega }_{{\rm{p}}}{\hat{\sigma }}_{{g}_{0}{e}_{{\rm{r}}}}+{\varOmega }_{{\rm{s}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{r}}}}+{\rm{h}}{\rm{.c}}{\rm{.}}],\end{eqnarray} \tag{ 19 }$$

and for D-STIRAP the level shift turns to be δ + δ', and its Hamiltonian ${\hat{ {\mathcal H} }}_{{\rm{d}}}$ becomes

$$\begin{eqnarray}&&{\hat{ {\mathcal H} }}_{{\rm{d}}}=-(\delta +{\delta }{^{\prime} }){\hat{\sigma }}_{{g}_{{\rm{d}}}{g}_{{\rm{d}}}}+\displaystyle \frac{1}{2}[{\varOmega }_{{\rm{p}}}{\hat{\sigma }}_{{g}_{0}{e}_{{\rm{r}}}}+{\varOmega }_{{\rm{s}}}{\hat{\sigma }}_{{e}_{{\rm{r}}}{g}_{{\rm{d}}}}+{\rm{h}}{\rm{.c}}{\rm{.}}].\end{eqnarray} \tag{ 20 }$$

Intuitively such fluctuation will influence the population transfer along R-STIRAP and D-STIRAP paths simultaneously, lowering the conversion efficiency. However, we show that due to the randomness of fluctuations that can be almost overcome on average, our scheme is able to maintain an observable quantum interference pattern even under very strong energy-level disturbance. Relevant numerical results are summarized in Fig. 6, where the time-dependent population transfer P_g0(t) along each STIRAP path as well as the final interference pattern are separately displayed. In the calculation, the random number δ' is created within the range of [–0.5,0.5]δ by using the random number generator in Matlab. Figures 6(a1) and 6(a2) show two sets of random numbers δ'(t) for δ = 0.05 and 0.5 respectively. When the Zeeman splitting δ = 0.05(small), our results verify that such fluctuation δ' arises no visible changes, as represented in Figs. 6(b1)–6(d1) and 6(b2)–6(d2). Our findings show that the population dynamics for both R-STIRAP and D-STIRAP are perfectly agreeable, giving rise to a high-contrast quantum interference, see Figs. 6(d1) and 6(d2).

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{Moreover, if the original splitting energy is relatively large, e.g., δ = 0.5, yet keeping δ < Ω₀, the time-dependent population dynamics P_g0(t) for individual R-STIRAP and D-STIRAP paths still does not change much under the influence of stronger fluctuations. However, the interference is totally destroyed. This fact can be understood by combining with Fig. 4(c), in which the interference visibility emerges a clear reduction if |δ| is enhanced. Because the coherence between R-STIRAP and D-STIRAP decreases as δ increases. A limitation lies in if δ is far off-resonance that two STIRAP paths separately perform an individual population transfer. To this end, no interference will emerge between them.

All in all, our protocol shows robust stability against stochastic fluctuations from the unstable energy shifts. Even if the two-photon resonance can not be preserved perfectly, the production of quantum interference pattern is still robust. The only way for breaking such quantum interference is making two-path decoherence, e.g., via a bigger ground energy splitting δ. Therefore, once δ is appropriately chosen, the stability of our protocol against fluctuations is robust.