Finite time St\"uckelberg interferometry with nanomechanical modes

St\"uckelberg interferometry describes the interference of two strongly coupled modes during a double passage through an avoided energy level crossing. In this work, we experimentally investigate finite time effects in St\"uckelberg interference and provide an exact analytical solution of the St\"uckelberg problem. Approximating this solution in distinct limits reveals uncharted parameter regimes of St\"uckelberg interferometry. Experimentally, we study these regimes using a purely classical, strongly coupled nanomechanical two-mode system of high quality factor. The classical two-mode system consists of the in-plane and out-of-plane fundamental flexural mode of a high stress silicon nitride string resonator, coupled via electric gradient fields. The dielectric control and microwave cavity enhanced universal transduction of the nanoelectromechanical system allows for the experimental access to all theoretically predicted St\"uckelberg parameter regimes. We exploit our experimental and theoretical findings by studying the onset of St\"uckelberg interference in dependence of the characteristic system control parameters and obtain characteristic excitation oscillations between the two modes even without the explicit need of traversing the avoided crossing. The presented theory is not limited to classical mechanical two-mode systems but can be applied to every strongly coupled (quantum) two-level system, for example a spin-1/2 system or superconducting qubit.


I. INTRODUCTION
Strongly coupled nanomechanical resonators have proven themselves as prominent testbed for the investigation of various fundamental physical concepts. The recent studies of, for example, non-classical correlations 1,2 , quantum back-action 3 , quantum squeezing 4 and topological effects 5 in different nanomechanical systems demonstrate in an outstanding way the scientific impact of hybrid-mechanical systems. In addition, the high level of control over such coupled resonators allows for the realization of ultrasensitive vectorial force sensors 6,7 and Λ-type three level systems 8 .
Recently, this high level of control led to the demonstration of classical Stückelberg interference of two strongly coupled nanomechanical resonator modes 9 . This coherent transfer of energy has originally been studied in a broad range of quantum systems including, e.g., spin-1/2 systems 10-12 and superconducting qubits [13][14][15][16][17] , amongst many others. Typically, the coherent dynamics of a two-level system in the configuration proposed by Stückelberg 18 is theoretically modeled by an infinite time approach, the so-called adiabatic impulse model 16 .
Following this model, the interference of two quantum states during a double passage through an avoided level crossing solely relies on the mutual coupling and is independent of the exact time evolution of the two states in the vicinity of the avoided crossing. In this work, we go well beyond this simple approximation and show that the adiabatic impulse model represent just one particular limit, the infinite time limit, of the full Stückelberg problem 18 . We provide an exact analytical solution to the problem which captures the importance of finite time effects. By means of asymptotic approximations of the exact finite time solution, we identify up to six different parameter regimes of Stückelberg interferometry. Experimentally, we demonstrate that a classical strongly coupled nanomechanical two-mode system 9,19 allows for the investigation of all discussed asymptotic regimes due to high mechanical quality factors and hence lifetimes of the coherent mechanical modes in the millisecond regime 9,19 .
The manuscript is organized as follows. Following this introduction (I), the nanoelectromechancial system as well as the experimental techniques are introduced in the second part (II). In sections III A and III B, we derive an exact analytical solution of the Stückelberg problem, taking advantage of the conformity of classical and quantum interference in this particular problem 9 . Additionally, the asymptotic limits of the exact solution are derived (appendices A & B) which allows for a quantification of characteristic parameter regimes in Stückelberg interferometry. In chapter III C, we explicitely derive the asymptotic long time limit of the analytical solution. Chapter III D provides a brief summary of previous, approximative theoretical approaches of Stückelberg interferometry and establishes the link to the presented exact analytical solution. Section IV compares the different theories to the experimentally observed classical Stückelberg oscillations of a strongly coupled nanomechanical system. In the last part (V), we summarize the results.

A. Experimental set-up
We study self-interference of a classical nanomechanical two-mode system using two samples of the same basic design. Sample A is investigated in a pulse-tube cryostat at a temperature of 10 K which serves solely for temperature stabilization. The experiments on sample B are conducted at room temperature. Independent of the ambient temperature, both samples operate deeply in the classical regime and do not exhibit quantum mechanical properties 9,19 . The samples consist of freely-suspended and doubly clamped silicon nitride (SiN) string resonators, fabricated in a top-down approach from a high-stress silicon nitride film on a fused silica substrate. The 100 nm thick and 270 nm wide silicon nitride strings exhibit a high tensile pre-stress of 1.46 GPa resulting from the LPCVD deposition process of the SiN atop the fused silica wafer. The high tensile pre-stress translates into high mechanical quality factors up to Q ≈ 500, 000 at mechanical resonance frequencies of ω m /2π ≈ 6.5 MHz at room temperature. Sample A consists of a 50 µm long string resonator, whereas on sample B we study a 55 µm long string. As depicted in Fig. 1 a and Fig. 1 b, the string resonators exhibit two fundamental flexural vibration modes with orthogonal mode polarizations, namely one perpendicular to the sample plane (out-of-plane) and one parallel to the sample plane (in-plane). For dielectric control and transduction of the string resonators (cf. Fig. 1 c) we process two gold electrodes adjacent to the SiN strings, which form a capacitor and are connected to a microwave cavity 20 via a bond wire. The oscillation of the dielectric silicon nitride string between the gold electrodes periodically modulates the capacitance. This change in capacitance in turn modulates the λ/4 microstrip cavity signal with resonance frequency at approximately Ω c /2π ≈ 3.6 GHz by producing sidebands on the cavity signal at Ω ± = Ω c ± ω m , where ω m /2π ≈ 6.5 MHz denotes the mechanical resonance frequency. The modulation induced sidebands are not resolved but can be demodulated via a heterodyne in-phase-quadrature mixing technique 20 before subsequent low-pass filtering and amplification. Finally, the demodulated signal is captured using a spectrum analyzer.
In addition to the described microwave cavity enhanced heterodyne dielectric detection, Figure 1. Schematic experimental set-up. a, False color scanning electron micrograph of a 50 µm long, 270 nm wide and 100 nm thick silicon nitride string resonator (green) in oblique view. The mechanical resonator is flanked by two 1 µm wide gold electrodes (yellow), which are processed on top of the silicon nitride and form a capacitor providing dielectric drive, tuning and detection as well as mode coupling. b, Schematic illustration of the two orthogonally polarized fundamental flexural vibration modes of the silicon nitride string resonator. The oscillation in z-direction, perpendicular to the sample plane, is referred to as out-of-plane oscillation, whereas the oscillation in y-direction, parallel to the sample plane, is referred to as in-plane oscillation. c, Schematic equivalent circuit diagram of the electrical drive, tuning and heterodyne detection scheme. The voltage ramp is added to the DC tuning voltage by a summation amplifier and combined with the resonant sinusoidal RF drive tone at a bias tee. The combined voltages are applied to one of the gold electrodes versus the ground of the microwave cavity. The bypass capacitor acts as a ground for the microwave cavity. The microwave cavity is driven on resonance and the signal is read-out via a heterodyne IQ-mixing technique, demodulating the sidebands induced by the oscillation of the nanomechanical resonator. the gold electrodes are used at the same time for dielectric actuation and control of the mechanical resonance 21 . Applying a DC bias to one of the electrodes induces an electric polarization in the dielectric silicon nitride string, which, in turn, couples to the gradient of the inhomogeneous electric field, generating a gradient force. Adding a resonant sinusoidal RF drive tone with frequency ω m /2π to the DC voltage at a bias tee results in a periodic force which drives the vibrational resonance of the nanomechanical silicon nitride string resonator 22 . Approximating the induced electrical polarization by a dipole moment 21,22 , its magnitude scales linearly with the applied DC voltage. Since the electric field gradient is also directly proportional to the DC voltage, the resonance frequency of the nanomechanical string resonator shifts quadratically with the applied DC bias 21 . By means of careful sample design, the in-plane polarized vibration mode can be engineered to shift downwards in resonance frequency with increasing DC bias, whereas the out-of-plane polarized resonance tunes towards higher resonance frequencies 21 . Thereby, the inherent resonant frequency off-set between the two orthogonally polarized vibration modes, which arises from the rectangular cross-section of the nanomechanical string, can be compensated. Near resonance, the two modes hybridize into normal modes 6,7,23 of the strongly coupled system, diagonally polarized along ±45°with respect to the sample plane. The strong coupling, mediated by the inhomogeneous electric field 19,23 , is reflected by the pronounced avoided crossing of the two mechanical modes with level splitting ∆/2π as depicted in Fig. 2 a.

B. Measurement scheme
In this work, we study the effects of finite times in classical Stückelberg interferometry.
In general, Stückelberg interference 18 occurs during a double passage through an avoided energy level crossing within the coherence time of the strongly coupled system. Both energy branches accumulate phase during the double passage, giving rise to self-interference. This brings about interference fringes depending on the difference in the accumulated phase. The probability to find the system either in the upper or the lower energy branch after the double passage oscillates in dependence of the level splitting, the traversal time as well as the initialization and turning point 16,18 .
Experimentally, we realize the double passage of the avoided crossing using fast triangular voltage ramps 9 . The voltage ramps are provided by an arbitrary function generator (AFG) and combined with the fixed DC tuning voltage at a summation amplifier. A detailed description of the ramps can be found in appendix C. In the following, we focus solely on the measurement principle which is equivalent for the investigation of sample A at 10 K and sample B at room temperature. The detailed experimental parameters of the respective samples are discussed in chapter IV. Note that the presented voltage ramp sequence 9 is analogous to the one employed by Sun et al. 24 and differs from the frequently performed periodic driving schemes in Stückelberg interferometry experiments 16 . The schematic sequence of the applied voltage ramp is depicted in Fig. 2 b. The system is initialized in the lower frequency branch at ω 1 (U i )/2π by the application of a resonant sinusoidal RF drive tone. Hereby, U i denotes the initialization voltage to which the DC tuning voltage is set during a Stückelberg experiment. Note that this voltage corresponds to a sweep voltage of zero. The sweep voltage defines the additional ramp voltage provided by the AFG. At t = t start , the fast voltage ramp is turned on and detunes the system from the resonant drive at ω 1 (U i )/2π. From this time on, the mechanical resonator is not driven any more and its oscillation decays exponentially (green dashed line in Fig. 2 b). Note that the mechanical energy decays on a larger timescale than the duration of the fast voltage ramp. The sweep voltage ramps the system from U i through the avoided crossing at voltage U a , up to the absolute peak voltage U p = U i + U p and back to the read-out voltage U f during time ϑ. At time t = t ϑ+ε , we start to measure the exponential decay of the mechanical oscillation at frequency ω 1 (U f )/2π in the lower branch at the read-out voltage U f . The return signal needs to be measured at U f since the drive at ω 1 (U i )/2π cannot be turned off during the experiment. Hence, a measurement at U i would lead to another excitation of the mode and therefore destroy the interference.
Additionally, the exponential decay of the return signal power needs to be measured with a temporal off-set ε to avoid transient effects. The exponential decay is extrapolated back to the time t ϑ where the voltage ramp ended via a fit and the resulting return signal power is normalized to the signal power at the time of initialization of the resonance (t = t start ).
This normalization process can lead to return probabilities exceeding a value of unity due to experimental scatter and different characteristic signal power heights at the initialization and read-out voltage. Consequently, we use the term normalized squared return amplitude for the experimental data instead of return probability.
For each particular measurement, the voltage ramp has a fixed voltage sweep rate β and fixed peak voltage U p . The experiment is repeated for a set of different voltage sweep rates at a fixed peak voltage. Subsequently, the peak voltage is changed and the measurement procedure is repeated. In this way, we investigate classical Stückelberg interferometry as a function of sweep speed and sweep distance which can be absorbed into a single variable, namely time.
In previous approaches 16 , Stückelberg interferometry has been investigated in the limit of infinite times. That means the initialization and turning point on the left and the right hand side of the avoided crossing are far away from the point of maximum coupling, which is at voltage U a where the level splitting is ∆/2π, compared to the characteristic time-scale of the system. This infinite time approximation is referred to as the adiabatic impulse model 16 and is summarized in chapter III D. In this work, we go beyond this approximation via the investigation of finite time effects. Experimentally, we interface this regime by turning points, i.e. peak voltages, close to or even before the avoided crossing, which still results in characteristic Stückelberg oscillations.

A. Theory of strongly coupled modes
In order to theoretically model the two modes in the strong coupling regime, we follow the work of Novotny et al. 25 and write the system as two coupled differential equations: where m = m 0 /2 denotes the effective mass of the resonator with physical mass m 0 , u j (j = 1, 2) the displacement of mode j, ω j = k j /m the respective angular resonance frequency, k j the spring constant of mode j, and κ the coupling constant between the two modes. Using the ansatz u j (t) = u 0,j exp(−iω ± t) in Eq. (1) yields the resonance frequencies of the two normal modes in the coupled system: Here, we define the level splitting  The ramp starts at t = t start . The sweep voltage is increased from zero to peak voltage U p at voltage sweep rate β, which increases the absolute voltage from U i to U p = U i + U p . At the apex of the triangular voltage ramp (peak voltage U p ), the sweep voltage is decreased at the same rate to the read-out voltage U f , which is approached at time t = t ϑ . Hence, the complete triangular voltage ramp has a duration of ϑ. Note that the read-out voltage U f is off-set from the initialization voltage U i as explained in the text. As a consequence, the sweep voltage does not return to zero.
The ring-down of the mechanical signal power (green dashed line) is measured after a delay ε (at time t = t ϑ+ ), and a fit (black dotted line) is used to extract its magnitude at time t = t ϑ . The measured return signal is normalized to the mechanical signal power at t = t start .
where the coupling λ, in general, can be complex valued. If the level splitting exceeds the dissipation in the system, namely the linewidth of the mechanical resonances, the modes can coherently exchange energy on a faster timescale than the energy decay. This strong coupling regime allows for the investigation of time dependent phenomena, like non-adiabatic Landau-Zener tunneling 23,25 in the classical regime, coherent dynamics of classical two-mode systems 19,26,27 and classical state interferometry 9 .

B. Finite-time Stückelberg interferometry
We look for a solution of Eq. (1) in the experimentally relevant limit where κ/k j 1, a normalized amplitude, i.e. |c 1 (t)| 2 + |c 2 (t)| 2 = 1, and we defineω j = (k j + κ)/m. By replacing our ansatz for u j (t) in Eq. (1), we find Since the amplitudes c j (t) are slowly varying in time compared to the oscillatory function Thus, the evolution of the normalized amplitudes is described by where we have defined c(t) = (c 1 (t) c 2 (t)) T and with λ = κ/(mω 1 ). To obtain Eq. (6), we have used that in the vicinity of the avoided crossingω 2 ω 1 . This yields (ω 2 2 −ω 2 1 )/(2ω 1 ) ω 2 −ω 1 and we assume that the difference in frequency is changed in time such thatω 2 −ω 1 αt, where α denotes the frequency sweep rate. Note that we employ the frequency sweep rate α in the theory which is converted to the experimentally accessible voltage sweep rate β using the conversion factor ζ from frequency to voltage (cf. section IV A and Ref. 9): By applying the time-dependent unitary transformation to Eq. (5), with 1 2 the two-dimensional identity operator, we find that the transformed amplitudes obey the differential equation where the dynamical matrix with c LZ (t i ) the initial condition of the system and We have and where we have introduced dimensionless quantities by defining τ = √ αt and η = λ/ √ α.
Finally, the flow describing the evolution of c 1 (t) and c 2 (t) is given by The flow ϕ(t, t i ) allows us to write in a simple way the state of the system after multiple passages through the avoided crossing. In particular, for a double passage we have with ϕ b (t, t i ) = σ x ϕ(t, t i )σ x describing the evolution of the system during the back sweep, σ x denotes the Pauli matrix in x-direction, and t p labels the time at which the forward (backward) sweep stops (starts). The fact that ϕ b (t, t i ) = σ x ϕ(t, t i )σ x can be understood by noticing that during the back sweep the frequency of mode 1 (2) decreases (increases) while it increases (decreases) during the forward sweep (see Fig. 2 a).
From Eq. (16), one obtains the return probability to mode 1, C. Asymptotic solution for the long-time limit

Long time limit
In this section, we show how to obtain an approximate form of Eq. (17) in the long-time limit, i.e. τ, |τ i | , τ p 1. Using the respective asymptotic expansion of the parabolic cylinder function (see appendix A), we find and The functions cos[θ(τ )], sin[θ(τ )], and ξ(τ ) are defined in appendix A and Γ(z) is the gamma function.

Infinite time limit
If one further assumes that η/τ, η/ |τ i | , η/τ p 1, then cos[θ(τ )] and sin[θ(τ )] can be expanded in powers of η/τ . We find cos[θ(τ )] = 1 + O(η 2 /τ 2 ) and sin[θ(τ )] = O(η/τ ). In this limit, which we refer to as the infinite-time limit, the return probability becomes where Majorana) non-adiabatic transition probability 18,31-33 and we have defined the phase acquired during the double passage As we will show below, Eq. (20) can also be found using the so-called adiabatic impulse model. While the latter model allows one to easily find an expression of the return probability in the limit η/τ, η/ |τ i | , η/τ p 1, it is very hard to extend the adiabatic impulse model to other parameter regimes. Another drawback of the adiabatic impulse model is that the leading order corrections to Eq. (20) cannot be found. In appendix B, we give an expression for the leading order correction to Eq. (20), which demonstrates that even in the infinite-time limit the return probability depends explicitly on τ i and τ .

D. Adiabatic impulse model
In this section, we briefly recapitulate a previous theoretical approach to Stückelberg interferometry known as the adiabatic impulse model 16 . The main assumptions of the adiabatic impulse model are that all non-adiabatic transitions happen at τ = 0 and that the system follows perfect adiabatic evolution from τ i → 0 − and from 0 + → τ p , where |τ i | η with τ i < 0 and τ p η. Given the assumptions of the model, it is convenient to work in the basis of instantaneous eigenstates of Eq. (10).
The non-adiabatic part of the evolution is described with a scattering matrix that relates the probability amplitudes right before the avoided crossing at t = 0 − and right after the avoided crossing at t = 0 + . The scattering matrix (in the basis of instantaneous eigenstates) Here, The adiabatic part of the evolution is described by the unitary evolution operator with σ z the Pauli matrix in the z-direction and we have defined the dynamical phase Within this formalism, the state of the system after a double passage is given by Here, we have chosen 0 − and 0 + to represent fixed times along the time axis. Note that we employ the scattering matrix N instead of its Hermitian conjugate in the back sweep.
Since the scattering matrix is expressed in the basis of instantaneous eigenstates, there is no difference in which direction the non-adiabatic transition is performed.
In general, all four adiabatic evolution operators in Eq. (25) contribute to the acquired dynamical phase of the system. For the particular case of the presented experiment, we initialize the system in an eigenstate of the coupled system, which is the out-of-plane mode.
In this scenario, the first and the last adiabatic evolution operators in Eq. (25) turn into global phases, which do not contribute to the two-mode interference. Hence, the interference of the two modes is solely governed by the phase evolution in between the two scattering matrices. Finally, the adiabatic impulse model yields the return probability where we have used the fact that in the infinite-time limit (η/τ 1) the instantaneous eigenstates and diabatic states of Eq. (10) coincide with each other.
As mentioned earlier, we have P aip 1→1 = P inf 1→1 . The main difficulty in using the adiabatic impulse model to get the return probability in regimes other than η/τ 1 lies in finding an appropriate scattering matrix N that explicitly depends on time.   Note that in the definition of t p , the absolute peak voltage U p appears instead of the peak voltage of the applied voltage ramp U p (see Fig. 2  In order to investigate the validity of the different theoretical approaches, we perform Stückelberg interferometry experiments for a large set of peak voltages U p at room temperature using a second sample. In particular, we study the finite time dynamics of the system for absolute peak voltages, i.e. turning points, close to the avoided crossing and even observe interference without traversing the latter. The sample measured at room temperature with P 1→j the return probability to mode j = 1 or j = 2. Note that Eq. (28) is applied to all three different theoretical approaches in the following.
In Fig. 4 b, the self-interference of the two-mode system clearly extends beyond the dotted line which represents the position of the avoided crossing at voltage U B,a . Additionally, the theory exhibits distinct features in the interference pattern which are not reproduced by the experiment. In order to experimentally resolve these features, the system needs to interfere precisely with the same constant parameters in every particular measurement pixel from Fig. 4 a. Since the experiments are performed at room temperature, the system parameters vary strongly from measurement to measurement due to temperature fluctuations. Experimentally, we partially account for this effect by the implementation of an initialization voltage feedback loop 9 , which ensures the initialization of the system at the same resonance frequency, at least within one horizontal line from Fig. 4 a. Nevertheless, the fluctuations and uncertainties prevent the system from interfering with the precise same parameters in every particular measurement. Further information on the experimental uncertainties is provided elsewhere 9 . Note that the experimental data in Fig. 4 a is taken in a non-consecutive way over a timespan of approximately 6 months which clearly demonstrates the validity of the data.
The results for the asymptotic theory (appendix A) are depicted in Fig. 4

B. Interference visibility
In order to study the crossover from the infinite time limit to the finite time domain in more detail, we extract the interference visibility in dependence of the peak voltage from the experimental data and the different theoretical approaches. Note that by interference visibility we refer to the original definition of interference contrast 35 and not to the single-shot read-out-visibility as frequently referred to in e.g. spin systems 36 . The interference visibility from the experimental data for a given peak voltage is calculated from the corresponding horizontal line-cut in Fig. 4 a by the difference of the maximum and minimum normalized squared return amplitude divided by their sum 35 . As exemplarily illustrated in Fig. 5  interference visibility drops down close to zero for peak voltages smaller than U p ≈ 2.0 V, To obtain a more intuitive understanding of the interference visibility, we replace the parameter U p by introducing the ratio between two characteristic dimensionless scales, which are the dimensionless time τ p and the dimensionless coupling η (see top x-axis in Fig. 5 b).
As explained above [Eqs. (27)], τ p corresponds to the distance from the avoided crossing to the turning point of the double sweep and hence becomes negative for sweeps where the avoided crossing is not passed (cf. top axis in Fig. 5 b). The dimensionless coupling In order to asymptotically expand the parabolic cylinder functions, we define a critical dimensionless time τ crit . This critical dimensionless time serves as a measure for the employed dimensionless times τ i , τ p and τ f in the theory. Those parameters can either be bound by τ crit (−τ crit ≤ τ ≤ τ crit ) or unbound (|τ | > τ crit ). Here, one should keep in mind that τ i is defined as smaller than zero. As further discussed in appendix A, the parabolic cylinder functions can mathematically be approximated by a power series. Hereby, the magnitude of τ crit specifies up to which order the power series is expanded. For the following calculations we defined τ crit = 2. Figure 6 depicts the theoretical return probability calculated from the asymptotic theory as in Fig. 4 c for an extended peak voltage range. The layover in Fig. 6 displays the boundary lines of the different parameter regimes from which the asymptotic solution is calculated. We recover six different parameter regimes, labeled by roman numerals from I to VI, which are specified in Table I. As can be seen immediately from Table I Table I. all three characteristic times are above threshold, which is regime IV. Since the characteristic times are not bound in this regime it is considered as the long time limit, which includes the there is no sharp border between the long time limit and the infinite time limit. Whereas the former requires the dimensionless times to be much larger than one, the latter exhibits the additional constraint that the dimensionless times are large compared to the dimensionless coupling. In this infinite time limit, the dynamics of the strongly coupled two-mode system is governed by the coupling strength of the two modes since the exact evolution in terms of dimensionless time plays a minor role. In fact, this is the only regime which, to the best of our knowledge, has been considered in the past in the framework of Landau-Zener type Accordingly, the two vertical border-lines in Fig. 6, which are independent of the peak voltage Note that for τ i = −τ f , i.e., if the system could be read-out at the initialization point after a symmetric voltage ramp, the right vertical border-line in Fig. 6 would vanish. Even though regime II and regime V are exclusively observed in our particular measurement scheme, the importance of finite time effects in Stückelberg interferometry is definitely pointed out by the presence of regimes I, III and VI. Especially regime I and III are of great interest since they reveal the dynamics of Stückelberg interferometry between two strongly coupled modes without the explicit need of traversing the avoided energy level crossing.
Since the dynamics of the strongly coupled classical two-mode system can be mapped onto the dynamics of a quantum mechanical two-level system in Stückelberg interferometry 9 , the same regimes are existent in every quantum mechanical two-level system such as e.g.
superconducting qubits [13][14][15]17 or spin-1/2 systems 10-12 . To the best of our knowledge, such regimes have so far not been investigated in the framework of Stückelberg interferometry and might be a prominent candidate for future investigations of quantum two-level systems.

V. CONCLUSION
In conclusion, we have demonstrated the importance of finite time effects in Stückelberg interferometry. Providing a complete and exact theoretical solution to the double passage Stückelberg problem, we have shown that the commonly employed adiabatic impulse model 16 does not address the full complexity of the problem 18 . In particular, the adiabatic impulse model solely describes one single parameter regime, where the dynamics of the system is completely governed by the coupling of the two modes corresponding to an infinite time limit.
We have been able to asymptotically expand the provided exact finite time theoretical model and have hereby classified previously undiscovered parameter regimes in Stückelberg interferometry. The theoretical findings have been confirmed in remarkably good agreement by a detailed experimental study of the dynamics of a classical two mode system 19,26 realized by two strongly coupled high quality factor nanomechanical string resonator modes. All theoretically predicted parameter regimes have been demonstrated experimentally by a thorough investigation of classical Stückelberg interferometry 9 . We observed clear oscillations in the experimentally accessible normalized squared return amplitude, even without traversing the avoided crossing in excellent agreement with the provided exact theory. These findings have been supported by a detailed study of the interference visibility over a huge parameter range.
Interestingly, the dynamics of the investigated classical two-mode system can be mapped to the dynamics of quantum mechanical two-level systems, as has recently been demonstrated by the authors 9 . As a consequence, the above theoretical findings can be applied one-to-one to quantum mechanical two-level systems.
If we define (see Eq. (20) in the main text) then the return probability to leading order in η/τ, η/τ i , η/τ p is given by P 1→1 = P from which we deduce the inverse sweep rate 1/β. The corresponding ramp time is hence given by The right hand side flank of the triangular voltage ramp decreases the absolute voltage from U p to the read-out voltage U f , which is off-set from the initialization voltage U i by U offset = 0.5 V. As described in the main text, the exponential decay of the returning excitation has to be measured at a different read-out frequency since the resonant sinusoidal drive tone at fixed frequency ω 1 (U i )/2π cannot be turned off during the voltage ramp. Hence, the above introduced voltage off-set is employed. It is important to note that the voltage off-set has to be adjusted in such a way, that the mechanical resonance at the read-out voltage ω 1 (U f )/2π is not excited by the resonant drive tone at ω 1 (U i )/2π. The exponential decay of the mechanical resonance after the triangular voltage ramp is measured in region three using a spectrum analyzer in a timespan of t readout = t ramp × Total Samples 500 Sa − 2 × t ramp = U p × 1/β 500 kSa 500 Sa − 2 . (C3) After the measurement, the absolute voltage is ramped back from U f to the initialization voltage U i (region four) by decreasing the sweep voltage from U offset to zero, which takes 100 samples of the total sample number of 500 kSa.