Introduction

Nontrivial topology, mainly governed by non-Hermitian degeneracies1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19, has led to various counterintuitive and fascinating topological types of behavior by adiabatically encircling exceptional points (EPs) in parameter spaces, such as chiral phase accumulation20,21, non-adiabatic jumps22,23, and topological phonon transfer (TPT) or mode conversion24,25,26,27,28,29,30,31,32,33,34. These topological phenomena, however, are generally destroyed by dark modes, which are naturally decoupled from systems35,36,37,38,39,40, resulting in a complete malfunction of both EPs and topological operations. For example, TPT and mode switching can always happen when executing adiabatic closed paths enclosing an EP24,25. Surprisingly, if a system exhibits dark modes, both mode conversion and TPT between dark and bright modes are unfeasible, regardless of the adjustment of adiabatic trajectories and system parameters, due to the dark-mode-induced destruction of topological operations and EPs.

Previously established topological achievements in theories and experiments are focused exclusively on conventional scenarios20,21,22,23,24,25,26,27,28,29,30,31,32,33,34, where dark modes are entirely circumvented because topological responses are usually diminished by dark modes24,25. Nevertheless, the practical applicability of modern topological quantum technologies has challenged such progress by demonstrating that a universal approach must be explored for confronting this challenge posed by dark modes, rather than evading it41,42,43,44,45,46,47,48,49,50. In view of its significance, exploiting a fundamentally different topology immune to dark modes, as well as shielding both EPs and topological operations from dark-mode disturbances in practical devices, is highly desirable.

Here, we show how to address this long-standing challenge and achieve a profoundly different one-way TPT, unveiling its counterintuitive immunity against dark modes. This occurs because of the synergy of topological operations24,25 and synthetic magnetism51,52,53,54,55,56,57,58,59,60,61,62,63, resulting in an exceptional transition between dark-mode-nonbreaking (\({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\)) and dark-mode-breaking (\({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\)) regimes in a well-controlled manner. Note that a reconfigurable synthetic gauge field has recently been demonstrated in phase-dependent loop-coupled optomechanical configurations51,52,53,54,55,56,57,58,59,60,61,62,63.

We reveal that executing a topological operation in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime yields topological phonon blockade (TPB) between dark and bright modes; whereas performing it in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime allows TPT. This enables a versatile yet unique topological physics and provides an exciting possibility of bridging TPB and TPT at will, which is otherwise unattainable in previously established demonstrations20,21,22,23,24,25,26,27,28,29,30,31,32,33,34. Unlike previous approaches, where both EPs and topological operators are entirely malfunctioning due to dark modes20,21,22,23,24,25,26,27,28,29,30,31,32,33,34, our approach is entirely immune to this detrimental inactivation effect.

The dark-mode-engineering mechanism is universal, and it can be extended to break various dark-state effects in quantum physics and to beat the limitation40, where dark modes prevent conventional quantum control and ground-state preparation. Our findings could advance the control of quantum collective motion in macroscopic mechanical systems40 and the development of scalable quantum information processors utilizing excitations, with potential applications in state or energy manipulation, pulse generation, and the conversion of quantum information64,65,66,67. In a broader view, our study sheds light on the combination of dark-mode engineering, topological operations, and quantum networks, and offers an exciting prospect of revealing a unique quantum topology immune to dark modes (states).

Results

System and dark-mode engineering

We focus on a three-mode optomechanical system, consisting of two phonon modes coupled to a common photon mode through radiation-pressure interaction, as shown in Fig. 1a. The described system has already been implemented using state-of-the-art technology to achieve topological responses fully evading dark modes24. The system Hamiltonian reads (ℏ = 1)

$${{\mathcal{H}}}= \, {\omega }_{c}{a}^{{\dagger} }a+{\sum}_{j=1,2}\left[{\omega }_{j}{b}_{j}^{{\dagger} }{b}_{j}+{g}_{j}{a}^{{\dagger} }a({b}_{j}^{{\dagger} }+{b}_{j})\right]\\ +i\sqrt{{\kappa }_{{{\rm{in}}}}}{\epsilon }_{{{\rm{in}}}}({a}^{{\dagger} }{e}^{-i{\omega }_{L}t}-{\mbox{H.c.}})+{{{\mathcal{H}}}}_{\xi },\\ {{{\mathcal{H}}}}_{\xi }= \, \xi ({e}^{i\Theta }{b}_{1}^{{\dagger} }{b}_{2}+{e}^{-i\Theta }{b}_{2}^{{\dagger} }{b}_{1}),$$
(1)

where a (a†) and bj (\({b}_{j}^{{\dagger} }\)) are the annihilation (creation) operators of the photon and jth phonon modes with resonance frequencies ωc and ωj, respectively. The gj term is the light-motion interaction, and the \({\epsilon }_{{{\rm{in}}}}=\sqrt{P/(\hslash {\omega }_{L})}\) term denotes the laser driving with power P, frequency ωL, and input-coupling rate κin. Synthetic gauge fields51,52,53,54,55,56,57,58,59,60,61,62,63, employed for manipulating the dark mode, can be induced using a phase-dependent loop-coupling setup formed by the gj and \({{{\mathcal{H}}}}_{\xi }\) terms (with phonon-hopping coupling strength ξ and modulation phase Θ) (see Supplementary Information). Note that in the absence of synthetic magnetism (ξ = 0), our system naturally reverts to the conventional case24, associated with the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime.

Fig. 1: Physical model and dark-mode control.
figure 1

a Schematic of synthetic-magnetism-engineered quantum devices, consisting of a photon mode a (with damping rate κ) optomechanically coupled to two phonon modes bj (with decay rates γj and coupling strengths gj), both of which are coupled to each other via a phase-dependent phonon-hopping interaction (ξ and Θ). b Effective coupling strengths \({\tilde{G}}_{\pm }\) versus Θ, when introducing synthetic magnetism. Simply tuning Θ = nπ and  ≠ nπ for an integer n leads to a dark mode (\({\tilde{G}}_{\pm }=0\), \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime) and its breaking (\({\tilde{G}}_{\pm }\ne 0\), \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime), respectively. We set ωj/ωm = 1 and Gj(ξ)/ωm = 0.1. c, d Mechanical linewidths and resonance frequencies versus the driving power P and detuning Δ ∈ [ − 1800kHz, 0], when the system operates in (c) the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime (without synthetic magnetism, i.e., ξ = 024) and (d) the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime (with synthetic magnetism, i.e., ξ/ωm = 5 × 10−4 and Θ/π = 1/9). The arrows describe the eigenvalue variation with increasing Δ for a fixed P.

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To explore how the dark mode is engineered by the synthetic gauge field, we derivate the linearized Hamiltonian, \({{{\mathcal{H}}}}_{{{\rm{lin}}}}=-\Delta \delta {a}^{{\dagger} }\delta a+ {\sum}_{j=1,2}[{\omega }_{j}\delta {b}_{j}^{{\dagger} }\delta {b}_{j}+{G}_{j}(\delta a\delta {b}_{j}^{{\dagger} }+{{\rm{H.c.}}})]+\xi ({e}^{i\Theta }\delta {b}_{1}^{{\dagger} }\delta {b}_{2}+{{\rm{H.c.}}})\), with Δ and Gj being the driving detuning and linearized photon-phonon coupling strength, respectively (see Supplementary Information for detailed derivations). This is achieved by applying a linearization procedure and expanding all operators o ∈ {a, bj, a†, \({b}_{j}^{{\dagger} }\)} as sums of their classical averages and quantum fluctuations, i.e., \(o=\bar{o}+\delta o\). Without synthetic magnetism (i.e., ξ = 0), the bright (\({{{\mathcal{B}}}}_{+}\)) and dark (\({{{\mathcal{B}}}}_{-}\)) modes emerge when ωj = ωm:

$${{{\mathcal{B}}}}_{+}=({G}_{1}\delta {b}_{1}+{G}_{2}\delta {b}_{2})/{G}_{0},\,bright,$$
(2a)
$${{{\mathcal{B}}}}_{-}=({G}_{2}\delta {b}_{1}-{G}_{1}\delta {b}_{2})/{G}_{0},\,dark,$$
(2b)

with \({G}_{0}=\sqrt{{G}_{1}^{2}+{G}_{2}^{2}}\). The dark mode is completely decoupled from the system, resulting in a complete blockade of both mode conversion and phonon transfer between the dark and bright modes24,25.

Surprisingly, this dark mode can be controlled at will by simply employing synthetic magnetism (i.e., ξ ≠ 0 and Θ ≠ 0). To elucidate the underlying physics behind this counterintuitive phenomenon, two normal modes associated with synthetic magnetism are introduced as: \({\tilde{{{\mathcal{B}}}}}_{\pm }={{\mathcal{F}}}\delta {b}_{1(2)}\mp {e}^{\pm i\Theta }{{\mathcal{K}}}\delta {b}_{2(1)}\), where \({{\mathcal{F}}}=| \delta {\tilde{\omega }}_{-}| /\sqrt{{(\delta {\tilde{\omega }}_{-})}^{2}+{\xi }^{2}}\) and \({{\mathcal{K}}}=\xi {{\mathcal{F}}}/\delta {\tilde{\omega }}_{-}\), with \(\delta {\tilde{\omega }}_{-}={\tilde{\omega }}_{-}-{\omega }_{1}\) and \({\tilde{\omega }}_{\pm }=({\omega }_{1}+{\omega }_{2}\pm \sqrt{{({\omega }_{1}-{\omega }_{2})}^{2}+4{\xi }^{2}})/2\). Subsequently, the linearized Hamiltonian becomes \({{{\mathcal{H}}}}_{{{\rm{lin}}}}=-\Delta \delta {a}^{{\dagger} }\delta a+{\sum}_{l=\pm }[{\tilde{\omega }}_{l}{\tilde{{{\mathcal{B}}}}}_{l}^{{\dagger} }{\tilde{{{\mathcal{B}}}}}_{l}+({\tilde{G}}_{l}{\tilde{{{\mathcal{B}}}}}_{l}\delta {a}^{{\dagger} }+{{\rm{H.c.}}})]\), with the effective coupling strengths:

$$\begin{array}{r}{\tilde{G}}_{\pm }={{\mathcal{F}}}{G}_{1(2)}\mp {e}^{\mp i\Theta }{{\mathcal{K}}}{G}_{2(1)}.\end{array}$$
(3)

We find from Eq. (3) that dark-mode engineering can be realized through two distinct mechanisms: (i) synthetic magnetism and (ii) asymmetric coupling. In the following, we provide a detailed elucidation of these two mechanisms.

(i) Synthetic magnetism enables dark-mode engineering. When G1 = G2, we plot \({\tilde{G}}_{\pm }\) versus Θ [see Fig. 1b and Supplementary Fig. S1 for more details], which reveals that for Θ = nπ, the dark mode decoupled from the system emerges, corresponding to the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime. Counterintuitively, an effective coupling of the dark mode to the system can be flexibly achieved just by tuning Θ ≠ nπ, resulting in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime. The fundamental physics driving these counterintuitive phenomena lies in the fact that a reconfigurable synthetic gauge field is built on demand just by steering Θ in the loop-coupled configuration, enabling an exceptional transition between the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes.

(ii) Asymmetric coupling provides an alternative pathway for dark-mode engineering. Specifically, we find that for Θ = 0, the symmetric coupling G1 = G2 creates a dark mode \({\tilde{{{\mathcal{B}}}}}_{-}\), entering the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime; whereas broken symmetry (i.e., asymmetrical coupling G1 ≠ G2) fully quenches this dark mode, establishing the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime.

Rise and fall of an EP

An adiabatic elimination of the photon mode yields an effective Hamiltonian for two phonon modes (see Supplementary Information for detailed derivations):

$${{{\mathcal{H}}}}_{{{\rm{eff}}}}=\left(\begin{array}{cc}{\omega }_{1}-\frac{i{\gamma }_{1}}{2}-i{g}_{1}^{2}\sigma &\xi {e}^{i\Theta }-i{g}_{1}{g}_{2}\sigma \\ \xi {e}^{-i\Theta }-i{g}_{1}{g}_{2}\sigma &{\omega }_{2}-\frac{i{\gamma }_{2}}{2}-i{g}_{2}^{2}\sigma \end{array}\right),$$
(4)

where the laser driving induces a complex motional susceptibility σ, defined as

$$\sigma=\frac{P{\kappa }_{{{\rm{in}}}}[\, \chi ({\omega }_{0})-{\chi }^{*}(-{\omega }_{0})]}{\hslash {\omega }_{L}[{(\kappa /2)}^{2}+{\Delta }^{2}]},$$
(5)

with the driving detuning Δ = ωL − ωc and an optical susceptibility \(\chi ({\omega }_{0})={[\kappa /2-i({\omega }_{0}+\Delta )]}^{-1}\), for ω0 = (ω1 + ω2)/2. By tuning σ, an EP is easily reached, needing to control over both Im(σ) and Re(σ). Physically, the imaginary and real parts of the corresponding complex eigenvalues correspond to mechanical spectral linewidths and resonance frequencies, respectively. It is enough to tune P and Δ for reaching and encircling this EP, because these parameters are easily manipulated in situ with high precision, timing accuracy, and dynamic range24.

For elucidating the effect of the dark mode on the EP, the mechanical spectra are plotted as functions of Δ and P in both \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes, as depicted in Fig. 1c, d and Supplementary Fig. S2. In the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, we show that despite the continuous evolution of the mechanical spectra with system parameters, the EP vanishes at both low and high laser powers, owing to the emergence of the dark mode [see Fig. 1c]. Counterintuitively, in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, for a lower laser power, each eigenvalue follows an enclosed path, beginning and ending at a same point; while for a higher laser power, both eigenvalues follow open trajectories, each of which ends at the starting point of the other, indicating the emergence of the EP because of breaking the dark mode [see Fig. 1d]. By adjusting Δ and P, the EP [marked by the yellow star in Fig. 1d], where the eigenstates coalesce, appears when the system operates in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, but not in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime.

Efficiency of topological phonon transfer

The TPT efficiency F+, which quantifies the phonon transfer from the dark (\({{{\mathcal{B}}}}_{-}\)) to bright (\({{{\mathcal{B}}}}_{+}\)) modes, is defined by24

$${{{\bf{F}}}}_{+}=| {{{\mathcal{B}}}}_{+}(\tau ){| }^{2}/\left[| {{{\mathcal{B}}}}_{+}(\tau ){| }^{2}+| {{{\mathcal{B}}}}_{-}(\tau ){| }^{2}\right],$$
(6)

indicating the fraction of the remaining energy in the bright mode after executing the adiabatic closed control loops, with \(| {{{\mathcal{B}}}}_{\pm }(\tau )| \) denoting the amplitudes of the bright and dark modes at the end of the control loops (see Supplementary Information for detailed derivations). This definition of F+ must satisfy the property that before performing the control loops, all energy is preserved in the dark mode. Equation (6) clearly shows that (i) a perfect TPT from the dark to bright modes happens when F+ = 1, and (ii) a TPB from the dark to bright modes is observed when F+ = 0. For ensuring the system stability, the following experimentally feasible parameters are chosen in our simulations24: g1(2)/2π = 1.03 (2.84) Hz, κ(κin)/2π = 177 (70) kHz, ωj/2π = 788.024 kHz, and γ1(2)/2π = 0.6 (1.4) Hz.

Inactivation and activation of topological operations

When the system operates in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes, we display the resonance frequencies and decay rates of mechanical normal modes versus a narrow range of the laser power P (see Fig. 2 and Supplementary Fig. S3). In the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, only the bright mode evolves with P, while the dark mode remains invariant, irrespective of the adjustment of system parameters, resulting in a completely deactivation of both the EP and topological operations [Fig. 2a]. This effect enables a complete blockade of the mode conversion and phonon transfer between the dark and bright modes. However, in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, both modes evolve simultaneously with P, which leads to the emergence of characteristic features of the EP, enabling the activation of the EP and topological operations [Fig. 2b]. This offers an exciting opportunity for the revival of the mode conversion and phonon transfer between the dark and bright modes.

Fig. 2: Disappearance and appearance of an EP.
figure 2

a, b Resonance frequencies and (c, d) linewidths of mechanical normal modes versus the laser power P in (a, c) the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime (ξ = 024) and (b, d) the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime (ξ/ωm = 5 × 10−4 and Θ/π = 1/9), when Δ/2π = −847.74 kHz.

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Specifically, the eigenstates coalesce at a particular value of the control parameters, and in the vicinity of this EP, they display a structure analogous to the Riemann surfaces of a complex square-root function. When executing a closed-loop path by adiabatically varying P and Δ, the generating smooth evolution on the eigenvalue manifold can return to its starting point only if the EP is not enclosed by the loop. In contrast, adiabatically encircling the EP in a closed-loop trajectory induces a counterintuitive path starting on one sheet but ending on the other, giving rise to TPT. Our study sheds light on the synergy of dark-mode engineering and topological operations, offers an unconventional tool for tasks that cannot be executed by conventional topological mechanisms, and benefits for implementing dark-mode-free topological physics.

TPB and TPT enabled by dark-mode engineering

The TPT efficiency F+ is plotted versus ΔMax and PMax in both \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) ( g2 = g1, ξ/ωm = 5 × 10−4, and Θ = 0) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) ( g2 = 2.76g1, ξ/ωm = 5 × 10−4, and Θ/π = 1/9) regimes, as shown in Figs. 3a, b and Supplementary Fig. S4. We reveal that in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, TPB occurs (F+ = 0, blue dashed horizontal lines); while in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, an excellent TPT emerges (F+ = 1, red solid curves). Specifically, in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, thermal phonons concealed in the dark mode that is decoupled from the system cannot be transferred to the bright mode regardless of the tuning of the system parameters, giving rise to TPB [Fig. 3c]. In stark contrast to this, in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, an efficient extraction of thermal phonons is achieved, yielding TPT [Fig. 3d]. These findings demonstrate that leveraging the dark-mode control enabled by synthetic magnetism not only establishes a flexible switch between TPB and TPT, but also provides the possibility of immunizing all topological quantum resources against various dark-mode disturbances in practical devices.

Fig. 3: Dark-mode-engineered topological phonon transfer and blockade.
figure 3

a, b TPT efficiency F+ versus (a) the maximum driving detuning ΔMax for PMax = 750 μW and (b) the maximum driving power PMax for ΔMax = −290 kHz, by executing a closed control loop. Two insets show how the loops vary along the horizontal axis of each panel, and the black crosses indicate the location of the EP. c, d Imposing topological operations in (c) \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and (d) \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes enables TPB and TPT, respectively.

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Physically, in conventional schemes (i.e., without synthetic magnetism), topological behavior is inherently fragile to dark modes decoupled from the system, leading to a complete malfunction of both EPs and topological operations24,25. However, by employing synthetic magnetism, all topological responses are immune to these dark modes, resulting in the function of topological operations. Our approach offers a way of enabling practical dark-mode-sensitive quantum setups to be effectively ideal, beneficial for achieving dark-mode-immune topological resources.

Dark-mode-engineered nonreciprocal topology

To study the dependence of one-way topological dynamics on dark-mode engineering, we display the TPT efficiency versus the duration τ of the closed control loops, when the system operates in both \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) ( g2 = g1, ξ/ωm = 5 × 10−4, and Θ = 0) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) (g2 = 2.76g1, ξ/ωm = 5 × 10−4, and Θ/π = 1/9) regimes, as shown in Fig. 4 and Supplementary Fig. S5. In the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, TPB always happens (i.e., F+ = 0) no matter how to execute the control loops in parameter spaces (see lower blue and red solid horizontal lines). In the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, by rapidly winding around the EP (i.e., τ → 0), TPB (i.e., F+ → 0) is observed; while with adiabatically encircling this EP (i.e., τ ≫ 1 ms), an excellent TPT is achieved (i.e., F+ → 1). These findings demonstrate that the vanishing TPT, corresponding to the emergence of TPB, results from either the dark mode in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime or a rapid encirclement of the EP in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime. By adiabatically winding around the EP, the TPT limiting behavior is contingent upon both the direction of the control loop and the mode initially excited. For example, by executing a clockwise (counterclockwise) loop, the blue (red) curves represent conventional adiabaticity (i.e., the efficiency becomes 1 with τ); while the red (blue) curves show the opposite behavior (i.e., the efficiency becomes 0 with τ). These results prove that, in general, the dark-mode engineering can achieve the nonreciprocity of each topological operation for an anti-clockwise or clockwise control loop enclosing an EP.

Fig. 4: Nonreciprocal topological dynamics via dark-mode engineering.
figure 4

TPT efficiency versus the duration τ of a closed control loop, in both \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes. The control-loop shape enclosing an EP is the same for all data series, and this EP is encircled in (a) clockwise and (b) counterclockwise directions. Blue or red curves correspond to the data where the bright or dark mode is initially excited, respectively.

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For example, when the dark mode is initially excited and the system operates in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, we reveal that adiabatically encircling an EP in the counterclockwise direction gives rise to an excellent TPT [F+ → 1, see the red dashed curve in Fig. 4b], but not in the clockwise sense [F+ → 0, see the red solid curve and symbols in Fig. 4a]. Physically, adiabatic behavior occurs when the system operates only in the less-damped eigenmode. This is because when working in the more-damped mode, the fierce competition between the differential-loss effect (that is exponentially large in τ) and the nonadiabatic transfer (which is exponentially small in τ) results in a breakdown of adiabaticity, which causes the system to eventually relax to the less-damped mode. Such a process can also be interpreted as a manifestation of the Stokes phenomenon of asymptotics.

Scalable network-based TPT and quantum collective ground-state preparation

The proposed dark-mode-engineering mechanism is very general, and can be generalized to quantum optomechanical networks, where N≥3 phonon modes are coupled to a shared photon mode through optomechanical coupling, \({{{\mathcal{H}}}}_{{{\rm{omc}}}}={\sum }_{j=1}^{N}{g}_{j}{a}^{{\dagger} }a({b}_{j}+{b}_{j}^{{\dagger} })\), and the nearest-neighbor phonon modes are coupled to each other via phase-dependent phonon-hopping interactions, \({{{\mathcal{H}}}}_{{{\rm{phi}}}}={\sum }_{j=1}^{N-1}{\xi }_{j}({e}^{i{\Theta }_{j}}{b}_{j}^{{\dagger} }{b}_{j+1}+{{\rm{H.c.}}})\) (see Supplementary Information). It has been demonstrated that these modulation phases are governed by the term \({\sum }_{\nu=1}^{j-1}{\Theta }_{\nu }\) (j ∈ [2, N]) (see Supplementary Information for detailed derivations), and hence we can safely assume Θ1 = π and Θj∈[2, N−1] = 0.

In the absence of synthetic magnetism (i.e., ξj = 0), only a single bright mode coupled to the system is induced,

$${{\mathcal{B}}}={\sum }_{j=1}^{N}\delta {b}_{j}/\sqrt{N},\,\,bright\,{{\rm{mode}}},$$
(7)

and N − 1 dark modes decoupled from the system emerge. Surprisingly, synthetic gauge fields (i.e., ξj ≠ 0 and Θ1 ≠ 2nπ), induced by the phase-dependent loop-coupling quantum networks, can lead to a simultaneous breaking of all N − 1 dark modes (see Supplementary Information for detailed derivations), offering an exciting opportunity of switching quantum networks between the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes. Therefore, thermal phonons from the dark to bright modes are blockaded in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, but transferred in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, making TPT feasible in quantum networks. These findings demonstrate that the scalable network-based TPT, with immunity against dark modes, can be achieved just by applying dark-mode engineering to quantum networks.

The framework developed here introduces a versatile strategy with wide-ranging implications in quantum physics. Specifically, it can be broadly applied to steer quantum collective motion in macroscopic mechanical systems. Quantum control of collective phonon modes in large-scale mechanical resonators represents an emerging research frontier40. The proposed dark-mode engineering can contribute a powerful and widely applicable tool for this growing field. For example, the recent breakthrough experiment40 highlights a key limitation in observing quantum collective motion in macroscopic mechanical resonators: Only a single bright mode can be cooled to its quantum ground state, while all N − 1 dark modes remain entirely uncooled. This inherent decoupling of all dark modes from the system renders the dark modes inaccessible to conventional quantum control and ground-state preparation. Building on this pioneering work40, our study fundamentally overcomes this constraint by introducing a dark-mode engineering mechanism that enables simultaneous ground-state preparation of both bright and dark modes. By simply activating synthetic magnetism, our scheme unlocks the full quantum potential of collective phononic dynamics. It establishes a paradigm in quantum optomechanics that is no longer bound by the limitations imposed by dark modes.

In the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) regime, all dark modes remain uncooled (\({n}_{{{\mathcal{D}}}}=1{0}^{3}\)); whereas in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, they are simultaneously cooled to their quantum ground states (\({n}_{{{\mathcal{D}}}} < 1\)), as shown in Fig. 5a. Note that for clarity and brevity, detailed analytical derivations of the final phonon occupations of all dark and bright modes are presented in Section V of Supplementary Information. Physically, thermal phonons trapped in dark modes that are naturally decoupled from the system cannot be extracted through optomechanical sideband cooling68,69,70,71, making ground-state preparation of all dark modes unattainable40. In stark contrast, upon transitioning into the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regime, simultaneous ground-state preparation of all dark and bright modes becomes achievable near the red-sideband resonance (Δ/ωm = − 1). These results suggest that dark-mode engineering offers flexible control and effective protection of fragile quantum collective ground states.

Fig. 5: Quantum collective ground-state preparation and general dark-state control.
figure 5

a Effective mean phonon number \({n}_{{{\mathcal{D}}}}\) of all N − 1 dark modes versus the driving detuning Δ in the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) (ξj = 0)40 and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) (ξj/ωm = 0.05, Θ = Θ1 = π/2, Θj∈[2, N−1] = 0) regimes. Here we consider the case of N = 4 and set ωj/ωm = 1, Gj/ωm = 0.1, κ/ωm = 0.2, γj/ωm = 10−6, and \({\bar{n}}_{{{\rm{th}}}}=1{0}^{3}\). b Schematic diagram of a three-level system with states \(\left\vert g\right\rangle \), \(\left\vert \, f\right\rangle \), and \(\left\vert e\right\rangle \), corresponding to their energies Eg, Ef, and Ee. Λ-type coupling configuration involving the transition processes \(\left\vert g\right\rangle \to \left\vert e\right\rangle \) and \(\left\vert \, f\right\rangle \to \left\vert e\right\rangle \), with detunings Δj=1,2 and coupling strengths Ωj=1,2. A phase-dependent resonance interaction with strength ΩbeiΘ between two lower states (\(\left\vert g\right\rangle \) and \(\left\vert \, f\right\rangle \)) is utilized to induce synthetic magnetism, which engineers the dark-state effect, exhibited by the Λ-type three-level system in the two-photon resonance regime (Δ1 = Δ2 = Δ). c Probability \({P}_{e}^{[s]}\) of the excited state \(\left\vert e\right\rangle \) in the eigenstates \(\left\vert {\lambda }_{s}\right\rangle \) versus the modulation phase Θ when ξ = 0.5. Clearly, the excited-state probability of one of the three eigenstates is zero (\({P}_{e}^{[s]}=0\)) at Θ = nπ, indicating the emergence of a dark state, and this dark-state effect can be flexibly broken (\({P}_{e}^{[s]}\ne 0\)) by tuning Θ ≠ nπ.

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General dark-state control in quantum optics

Our dark-mode engineering mechanism is highly versatile, offering a robust framework for controlling a wide range of physical phenomena associated with dark states. To clarify this, we reveal the physical mechanism responsible for breaking the dark-state effect in Λ-type three-level systems, by introducing a phase-dependent transition (i.e., the phase in a loop-coupling configuration leads to synthetic magnetism) between the two lower levels [see Fig. 5(b)]. It is well established that in a Λ-type three-level system under two-photon resonance, a dark state emerges, characterized by the zero value of the superposition coefficient associated with the excited state.

In a typical natural atom, direct transitions between the two lower states in a Λ-type three-level system are forbidden by selection rules. However, such transitions become feasible in artificial cyclic three-level systems or through indirect coupling mechanisms. Accordingly, the Hamiltonian of the system is expressed as

$${{\mathcal{H}}}= \, {E}_{e}\left\vert e\right\rangle \left\langle e\right\vert+{E}_{f}\left\vert \, f\right\rangle \left\langle f\right\vert+{E}_{g}\left\vert g\right\rangle \left\langle g\right\vert+{\Omega }_{1}\left(\left\vert e\right\rangle \left\langle g\right\vert {e}^{-i{\omega }_{1}t}\right.\\ \left.+\left\vert g\right\rangle \left\langle e\right\vert {e}^{i{\omega }_{1}t}\right)+{\Omega }_{2}(\left\vert e\right\rangle \left\langle f\right\vert {e}^{-i{\omega }_{2}t}+\left\vert \, f\right\rangle \left\langle e\right\vert {e}^{i{\omega }_{2}t})\\ +{\Omega }_{b}(\left\vert \, f\right\rangle \left\langle g\right\vert {e}^{i\Theta }{e}^{-i{\omega }_{b}t}+\left\vert g\right\rangle \left\langle f\right\vert {e}^{-i\Theta }{e}^{i{\omega }_{b}t}),$$
(8)

where Eg, Ef, and Ee are the energy of the three energy levels, i.e., \(\left\vert g\right\rangle \), \(\left\vert \, f\right\rangle \), and \(\left\vert e\right\rangle \), respectively. The two monochromatic fields with the driving frequencies ω1 and ω2 are, respectively, coupled to the atomic transitions \(\left\vert g\right\rangle \to \left\vert e\right\rangle \) and \(\left\vert \, f\right\rangle \to \left\vert e\right\rangle \), with Ω1 and Ω2 being the corresponding real transition amplitudes. In such a system, we introduce the transition detunings as Δ1 = Ee − Eg − ω1 and Δ2 = Ee − Ef − ω2, which are related to the two transitions \(\left\vert g\right\rangle \to \left\vert e\right\rangle \) and \(\left\vert \, f\right\rangle \to \left\vert e\right\rangle \), respectively.

Clearly, we see that a dark state is possessed by the Λ-type coupling system, when the two-photon resonance (Δ1 = Δ2 = Δ) is fulfilled for the transitions. To illustrate the dark-state breaking, we introduce a resonance field that couples the two lower states, i.e., \(\left\vert \, f\right\rangle \) and \(\left\vert g\right\rangle \). Notably, this coupling features a phase-dependent strength, which induces synthetic magnetism, serving as the critical factor that enables the breaking of the dark state. In a rotating frame with respect to \({H}_{0}=({E}_{g}+{\omega }_{1})\left\vert e\right\rangle \left\langle e\right\vert+{E}_{f}\left\vert \, f\right\rangle \left\langle f\right\vert+{E}_{g}\left\vert g\right\rangle \left\langle g\right\vert \), the the system Hamiltonian becomes

$$\begin{array}{rcl}{{{\mathcal{V}}}}_{I}=\Delta \left\vert e\right\rangle \left\langle e\right\vert+{\Omega }_{1}(\left\vert e\right\rangle \left\langle g\right\vert+\left\vert g\right\rangle \left\langle e\right\vert )+{\Omega }_{2}\left(\left\vert e\right\rangle \left\langle f\right\vert \right.\\ \qquad\left.+\left\vert \, f\right\rangle \left\langle e\right\vert \right)+{\Omega }_{b}(\left\vert \, f\right\rangle \left\langle g\right\vert {e}^{i\Theta }+\left\vert g\right\rangle \left\langle f\right\vert {e}^{-i\Theta }).\end{array}$$
(9)

By defining three basis states via the following vectors:

$$\left\vert e\right\rangle={(1,0,0)}^{T},\left\vert \, f\right\rangle={(0,1,0)}^{T},\left\vert g\right\rangle={(0,0,1)}^{T},$$
(10)

the interaction Hamiltonian in Eq. (9) becomes

$${{{\mathcal{V}}}}_{I}=\Omega \left(\begin{array}{ccc}0&1&1\\ 1&0&\xi {e}^{i\Theta }\\ 1&\xi {e}^{-i\Theta }&0\end{array}\right),$$
(11)

where ξ = Ωb/Ω. Without loss of generality, we have considered the symmetric coupling case (Ω1 = Ω2 = Ω) as well as the conditions of single-photon and two-photon resonance (Δ1 = Δ2 = Δ = 0). Then we obtain the eigenequation \(\frac{1}{\Omega }{{{\mathcal{V}}}}_{I}\left\vert {\lambda }_{s}\right\rangle={\lambda }_{s}\left\vert {\lambda }_{s}\right\rangle \), where λs are the eigenvalues, determined by the secular (cubic) equation

$${\lambda }^{3}-(2+{\xi }^{2})\lambda -2\xi \cos \Theta=0.$$
(12)

The solutions to Eq. (12) are then obtained using the Cardano formula, and the eigenstates can be expressed as:

$$\left\vert {\lambda }_{s}\right\rangle={c}_{g}^{[s]}\left\vert g\right\rangle+{c}_{f}^{[s]}\left\vert \, f\right\rangle+{c}_{e}^{[s]}\left\vert e\right\rangle,\,s=1,2,3.$$
(13)

The dark state can be verified by calculating the probability amplitude \({P}_{e}^{[s]}\) of the excited state \(\left\vert e\right\rangle \) within these eigenstates:

$${P}_{e}^{[s]}={\left\vert \langle e| {\lambda }_{s}\rangle \right\vert }^{2}={\left\vert {c}_{e}^{[s]}\right\vert }^{2}\left\{\begin{array}{c} \!=0,\,Dark\,{{\rm{state}}},\hfill \\ \ne 0,\, Bright\,{{\rm{state}}},\end{array}\right.$$
(14)

where \({P}_{e}^{[s]}=0\) and \({P}_{e}^{[s]}\ne 0\) correspond to the emergence of the dark and bright states, respectively. Specifically, we present the probability \({P}_{e}^{[s]}\) of the excited state \(\left\vert e\right\rangle \) in the three eigenstates \(\left\vert {\lambda }_{s}\right\rangle \) versus the modulation phase Θ, for a fixed value of ξ, as depicted in Fig. 5c. We observe that one of the eigenstates becomes a dark state (\({P}_{e}^{[s]}=0\)) when Θ = nπ, while no dark states emerge in other cases (\({P}_{e}^{[s]}\ne 0\)). It demonstrates that by tuning Θ ≠ nπ, the dark-state effect can be broken on demand. Therefore, the phase-dependent resonant transition \(\left\vert g\right\rangle \leftrightarrow \left\vert \, f\right\rangle \), yielding synthetic magnetism, can be utilized to break the dark-state effect in the Λ-type three-level system. Our findings could advance the development of scalable quantum information processors utilizing photons and phonons, with potential applications in state or energy manipulation, photon or phonon pulse generation, quantum repeaters, and the conversion of information between excitations64,65,66,67.

Proposed experimental implementations

The proposed physical model is general and, in principle, can be implemented using standard optomechanical platforms. Realizing the TPT while maintaining immunity against dark modes requires two key ingredients: in-parallel optomechanical couplings between multiple phonon modes and a shared photon mode, and phase-dependent phonon-hopping interactions between the nearest-neighbor phonon modes. Both types of interactions must be accessible within viable experimental platforms. While each type of coupling has been demonstrated independently in previous experiments, their simultaneous implementation within the same experimental setup has not yet been reported. Nevertheless, under current state-of-the-art experimental conditions, integrating both kinds of interactions into a unified system appears fully feasible.

Recent advances have enabled the experimental realization of in-parallel optomechanical couplings between multiple phonon modes and a shared photon mode in both optical24,72,73 and microwave74,75,76,77,78 domains. In the optical regime, these in-parallel optomechanical couplings are implemented via “membrane-in-the-middle" optomechanical architectures24,72,73; whereas in the microwave domain, they are realized using circuit electromechanical platforms74,75,76,77,78. Simultaneously, the phase-dependent phonon-hopping interaction between the nearest-neighbor phonon modes can be implemented using photonic-crystal optomechanical platforms54 or circuit electromechanical systems74,75,76,77,78,79. In photonic-crystal-based implementations, this phase-dependent coupling arises from the mediation of two auxiliary cavity fields54; whereas in circuit electromechanical systems, it emerges indirectly through the coupling of two mechanical resonators to a charge qubit.

(i) Building on these state-of-the-art experimental advances in both parallel optomechanical couplings and phase-dependent phonon-hopping interactions, the proposed model can be easily realized using photonic-crystal optomechanical architectures featuring optical and mechanical couplings between two optomechanical cavities54. In this scheme, each cavity is driven by a distinct phase-correlated laser field, and the effective implementation of our model emerges through the adiabatic elimination of any one cavity-field mode under the large-detuning regime39. Specifically, we consider a multimode physical system featuring both optical and mechanical interactions between the two optomechanical cavities. In the large-detuning regime of the cavity mode, this mode can be adiabatically eliminated, yielding an effective three-mode phase-dependent loop-coupled Hamiltonian39, where both the proposed model and the resulting synthetic magnetism can be easily realized under state-of-the-art experimental conditions using photonic-crystal optomechanical-cavity systems54.

(ii) Moreover, the proposed model can be readily implemented using circuit electromechanical platforms74,75,76,77,78,79, which comprise N micromechanical resonators (MRs) (i.e., bj=1⋯N) coupled to a microwave cavity characterized by an equivalent inductance L and capacitance C. In this quantum setup, the displacement xj=1⋯N of each MR independently modulates the total capacitance via Cj=1⋯N(xj), thereby tuning the cavity resonance frequency ωc. This modulation interaction gives rise to an electromechanical coupling described by gj=1⋯N = (ωc/2C)∂Cj(xj)/∂xj, enabling the precise quantum control over the system’s dynamics. Meanwhile, an effective phase-dependent phonon-hopping interaction between the two nearest-neighbor MRs is induced by the coupling of the two MRs to a superconducting charge qubit39. It reveals that an effective phase-dependent phonon-hopping interaction emerges between the two nearest-neighbor MRs and then, the modulation phase in the loop-coupling configuration induces synthetic magnetism. These findings highlight the direct relevance of the proposed phenomena to state-of-the-art experiments in circuit electromechanical systems, suggesting that current experimental capabilities are sufficient to realize the proposed scheme and that the predicted effects are observable with cutting-edge implementations.

Building on the insightful observation that synthetic magnetic fields provide a powerful route to engineering gauge potentials for photons, we highlight the growing interest in photonic synthetic dimension platforms60. These platforms not only enable strong photon-photon interactions but also naturally facilitate the realization of synthetic magnetic fields60,61,62,63. Specifically, the synthetic magnetic field significantly aids the generation of gauge potentials for photons. Recent advances have generated intense interest in photonic platforms with synthetic dimensions60, which enable strong photon-photon interactions and synthetic magnetic fields60,61,62,63. These developments highlight the significant potential for realizing our theoretical proposal within synthetic-dimensional photonic systems. Importantly, the proposed dark-mode-engineering mechanism could be easily transferred to chip-based platforms, marking a significant step toward scalable quantum information processing. This prospect is particularly exciting, as it offers a practical pathway to integrate topological photonic or phononic functionalities into compact, chip-scale architectures.

Discussion

Optomechanical dark modes, analogous to the coherent-population trapped state or dark state in atomic physics80,81, can be broadly classified into two categories: optical dark modes and mechanical dark modes. Specifically, optical dark modes arise from the coupling of two optical modes to a shared mechanical mode65,66,67, while mechanical dark modes emerge from the coupling of two mechanical modes to a common optical mode35,36,37,38,39,40. Notably, prior demonstrations65,66,67 focus on optical dark modes that enable optical mode conversion, whereas our work focuses on mechanical dark modes that suppress mechanical mode conversion.

The phonon-mediated and photon-mediated processes, which respectively correspond to the optical and mechanical mode conversions, can be pursued using multimode optomechanical systems. However, the influence of optomechanical dark modes on these two conversions differs fundamentally. In the previous studies65,66,67, an optical dark (bright) mode, formed by a special coherent superposition of two optical modes, is decoupled from (coupled to) the mechanical mode. Consequently, the formation of the optical dark mode serves as a shield, protecting the system from mechanical dissipation. Despite being decoupled, the optical dark mode can still mediate an effective interaction between the two optical modes, thereby facilitating the optical-field conversion between them. Moreover, the presence of the optical dark mode enables a phonon-mediated coupling (which is immune to thermal mechanical motion) for various quantum applications, thereby eliminating the need for quantum ground-state cooling of the mechanical resonator. In our work, the mechanical dark mode emerges as a distinct coherent superposition of two mechanical modes coupled to a common cavity mode. The interference-induced cancellation in coupling effectively decouples the mechanical dark mode from the system. While this decoupling still permits an effective interaction between the two mechanical modes, both mechanical mode conversion and topological phonon transfer between mechanical dark and bright modes become fundamentally suppressed, regardless of adiabatic trajectory design or system parameter tuning. The underlying physical mechanism of this counterintuitive phenomenon is due to dark-mode-induced disruption of both topological operations and EPs. Here, we address this long-standing challenge posed by mechanical dark modes and demonstrate a fundamentally distinct one-way topological phonon transfer, revealing an unexpected immunity to dark-mode-induced obstruction. This robustness arises from the interplay between topological operations and synthetic magnetism, enabling a controlled transition between the \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{N}}}\) and \({{\mathcal{D}}}{{\mathcal{M}}}{{\mathcal{B}}}\) regimes.

In a general topological system, the emergence of an EP signals a transition in the system’s topological phase as a function of the parameter space. This transition is often accompanied by a change in the system’s topological invariants, such as the conversion from a topologically trivial to a topologically nontrivial state82. However, these behaviors always occur in the traditional energy-band topology, involving non-Hermitian gap structures82. In this work, we focus on a three-mode optomechanical system consisting of two mechanical modes coupled to a shared cavity-field mode. By adiabatically eliminating the cavity-field mode, we obtain an effective non-Hermitian Hamiltonian for the two mechanical modes. In this system, topological operations encircling an EP enable nonreciprocal phonon transfer between two mechanical normal modes. Note that our system is a unique non-Hermitian gapless structure, and the traditional concept of energy-band topology is not directly applicable to our studied system. This indicates that the system is fundamentally different from the one investigated in our work, as our focus is exclusively on the non-Hermitian gapless structure rather than on the gap structure. Therefore, the transition from a topologically trivial to a nontrivial state, which happens in the traditional energy-band topology, lies beyond the scope of our system.

In conclusion, we demonstrated a versatile switch between TPB and TPT arising from a general dark-mode engineering, without which it vanishes. Our study differs from what is known in previously established demonstrations, mainly because we are focused on overcoming the challenge from dark-mode contamination in topological responses, but not on deliberately circumventing it. The proposed physical mechanism is highly universal, thereby enabling the control of a wide range of physical effects associated with dark states. Our work presents a comprehensive approach to engineering and protecting topological resources from both dark modes and dark states, providing a perspective on the construction of an unconventionally nonreciprocal topology with immunity to dark modes.