Abstract
Dynamical phase transitions, characterized by non-analytic behaviors in time domain, extend the equilibrium phase transitions to far-from-equilibrium situations 1, 2 . Furthermore, it's predicted that the dynamical phase transitions can be precisely identified by discontinuities of Pancharatnam geometric phase during the time evolution, even in a general nonadiabatic and non-cyclic process [3] [4] [5] . Here, we report the observation of dynamical phase transitions via directly measuring Pancharatnam geometric phase in a quenched topological nanomechanical system. We present a flexible strategy based on eight strong-coupled high-quality-factor nanomechanical oscillators to realize an one-dimensional reconfigurable lattice described by the Su-Schrieffer-Heeger Hamiltonian. Due to the chiral symmetry, the dynamical phase of the time-evolved state quenching from an initial topological edge state is 1 arXiv:1807.04483v2 [quant-ph]

naturally eliminated, so that we can directly read the Pancharatnam geometric phase of the state. We find that, the Pancharatnam geometric phase jumps π when a dynamical phase transition takes place, which is detected by the normalized amplitude of edge oscillator. This work not only provides a quantitative method to identify the dynamical phase transitions, but also opens the door for studying non-equilibrium topological dynamics with a well-controlled nanomechanical system.
In recent years, non-equilibrium phenomena beyond the Ginzburg-Landau paradigm have attracted tremendous attentions and interests 1, 2, [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] . In analog to equilibrium phase transitions, dynamical phase transitions (DPTs) are a kind of outstanding non-equilibrium phenomena which manifest non-analytic behaviors of the rate function (i.e. the logarithm function of returning probability to initial state) in the time domain 1, 2 . So far, DPTs in sudden quenches have been found
by measuring the rate function or observing the dynamical vortices etc [13] [14] [15] [16] . Theoretically, in a DPT, the non-analytic behavior occurs and the Pancharatnam geometric phase (PGP) simultaneously jumps π at the critical time when the returning probability goes through zero 5 . This means that the PGP provides an exact quantitative hallmark of DPTs. To the best of our knowledge, as the PGP in a quenched system always accompanies and mixes with the dynamical phase 2, 5, 16, 18 , there is still no experiment observation of DPTs by directly measuring PGP.
We present a novel scheme for eliminating the dynamical phase in sudden quenches from a topological edge state, which is a prominent signature of nontrivial topology [19] [20] [21] [22] [23] . Considering the Su-Schrieffer-Heeger (SSH) model -a typical topological model 24 , due to its chiral symmetry 25 , the initial edge state equally populates all symmetrical pairs of final eigenstates and so that the dynamical phase is naturally eliminated. The scheme is different from the previous studies that quenching from the ground state 1, 13-15 or a bulk state 5, [16] [17] [18] 26 .
To investigate the topological dynamics, we developed a method to fabricate the high-qualityfactor nanomechanical array of eight oscillators with strong couplings. The quality factor can reach about 1 × 10 5 in vacuum at 77K temperature, which is further better than the previous mechanical lattice at 4K temperature 27 . In this Letter, we report an experimental observation of DPTs via directly measuring PGP in this nanomechanical lattice. The coupled oscillators can be flexibly engineered by capacitive coupling 28 . Sudden quenches between different structures are easily configured by tuning the different coupling voltages between oscillators. The initial state preparation and high-resolution measurement are enable with the standard magneto-motive technique 29 . Since the dynamical phase is naturally eliminated, we directly observe the PGP and its jumps at critical times by demodulating the motion of the edge oscillator. We also observe the non-analytic behaviors of the rate function, which provides another signature of the DPTs. This is the first time exploring the relation between PGP and DPTs in experiment.
We study the quench dynamics in a one-dimensional (1D) topological model -the SSH chain 24 . This model describes a 1D super-lattice with two sites per unit cell, and its Hamiltonian readsĤ
where |j is the state of exciting the j-th site, J A and J B are intracell and intercell hopping respectively showed in the top of Fig. 1a . Its topological aspects can be characterized by a topological invariant -the winding number 25 . In momentum space, the Hamiltonian is given asĤ and 60Hz coupling strength alternately to realize a topological phase in SSH model. On the contrary, we realize a trivial phase by choosing 60Hz and 20Hz alternately. These two structures are determined by measuring the stabilized frequency response of the edge oscillator (see Methods).
We study the quench dynamics of a edge state fromĤ i :
The initial state is prepared at the edge site,
which is the eigenstate of a large dimeric topological HamiltonianĤ i . The first quench cross the underlying topological phase transition, while the second quench stays in the same topological phase. The evolution of system follows |ψ(t) = e −iĤ f t |ψ(0) after quenches. The Loschmidt amplitude is given as
where r(t) is the modulus of Loschmidt amplitude, φ(t) is the phase containing a dynamical phase
ψ(s)|Ĥ f |ψ(s) ds and a PGP φ P 5 , and G n (t) = | ψ n |ψ(0) | 2 e −iEnt is partial Loschmidt amplitude with eigenstates |ψ n and eigenvalues E n of the finial HamiltonianĤ f .
Due to the chiral symmetry, the initial edge state equally populates all symmetrical pairs of final eigenstates so that the occupation is always symmetrical part for G n at each moment because of E −m = −E +m . Thus, Loschmidt amplitude G(t) ∈ R and the phase φ(t) of Loschmidt amplitude is 0 or π all the time.
By the same reason, the dynamical phase keeps zero so that φ dyn is naturally eliminated and the phase φ(t) we measure is directly give the PGP (see Supplementary Information) . According to the original definition of DPTs 1, 2 , the rate function derived from Loschmidt amplitude as
will become nonanalytic at the critical time when DPTs take place. Thus, when Loschmidt amplitude G(t) goes through zeros in quenched dynamics, the DPTs take place while the PGP jumps The experimental results (circles and triangles) in Fig. 3b are measured by averaging 500 times, and they are in great agreement with the theoretical results.
To diagnose the DPTs, we also measure the rate function in these two quenches. We assume that all oscillators have the same decay rates in the short time of evolution, due to the strong coupling strengths much larger than the decay rates, the high quality factors and the small differences between the oscillators. Therefore, all amplitudes can be normalized every moment. We obtain the rate function from the normalized amplitude of the first oscillator in two quenches, and observe non-analytic behaviors of the rate function when the system is quenched across an underlying topological phase transition, see the red circles with error bar in the Fig. 3c . The non-analytic behaviors directly verify the DPTs. For the quench in the same topological phase, the rate function is always analytic, which suggests that there is no appearance of DPTs, see the blue triangle with error bar. In summary, by using a reconfigurable nanomechanical array, we direct observe PGP and rate function after a sudden quench from the topological edge state in the SSH model. In the quench from underlying topological nontrivial to trivial phase, we observe π phase jumps of PGP in company with nonanalytic of rate function, indicating the occurrence of DPTs. In the quench from underlying topological nontrivial to nontrivial phase, we observe that PGP keeps the same while the rate function is analytic. We explore the relation between the PGP and the rate function for diagnosing the dynamical phase transition.
For topological system, there exists bulk-edge correspondence in the equilibrium where the number of edge states corresponds to the bulk topological invariants 30 . Theoretical results have also shown that the bulk-boundary correspondence for dynamical phase transitions 31 . Experimentally exploring the bulk-edge relations in quenched dynamics will be an interesting and important problem deserved further study. In addition, our system can be used to study more complicated phenomena including interplay between topology, dissipation and nonequilibrium dynamics 32 .
Methods
The chain coupled oscillators. In our study, eight coupled oscillators are realized by nearestneighboring parametric couplings. The vibrational amplitudes of the oscillators follow the equations of motion,
where
is the time-dependent coupling between the j-th and (j + 1)-th beams. η j is the coupling strength which can be controlled by the sum of the applied V DC and V AC between the j-th and (j + 1)-th beams. Using slowly varying complex amplitudes ψ j as
) and rotating wave approximation in Supplementary Information (or see 33, 34 ), the equations of motion can be rewritten as
where ψ j = j|ψ is made to be normalized as j |ψ j | 2 = 1. H can be made as SSH model by
Apart from equation (6) , the coherent energy always dissipates into the environment in the the actual physics systems. If the decay rate γ j of each beam is far less than the coupling strength, we can still observe coherent dynamics long before the system decays into the environment. On the other hand, if the beams almost have the same decay rate γ j = γ (j = 1, 2, ..., 8), we can separate the complex amplitude as exponential decay term multiple the coherent term, |ψ = e −γt |ϕ . The dynamics of coherent term |ϕ is still governed by the SSH model. Based on the two considerations, we try to decrease the decay rates and make a approximation that the decay rates is almost same in short evolution time. Table S1 .
Parametric Couplings. To generate the electrostatic forces for parametrical couplings between beams, an AC voltage V j AC (t) and a DC voltage V DC are applied between j-th and (j + 1)-th beam, the coupling as the following form can be generated 28 .
Here, C j (δx j ) is the effective capacity between the oscillators and δx j = x j − x j+1 . The form of
The DC voltage and AC voltage are combined by a bias-tee, as showed in Fig. S1 and Fig.   S2 where V P is a sum of the neighborhood AC coupling voltage. The typical value of AC voltage used are listed in Table S2 and Table S3 .
The effective coupling strength is confirmed by measuring frequency spectrum of the response. As the Fig. S3 a shows, we measure the response of one oscillator coupled with another adjacent one. The split of peak directly describes coupling strength.
The frequency response of entire coupled chain is measured at edge oscillators, which is
showed Fig. S3 c and d , correspond to topological phase and trivial phase of SSH model. Although eight peaks are not distinguishable in these spectrums because of oscillators dissipation, the appearance of zero mode is remarkable distinction between topological and trivial. We also measure the response under the same coupling strengths, showed in Fig. S4 .
Measurement Scheme. The sample is placed in a vacuum of 3×10 −6 Pa and cooled to 77K for the stable frequency and high quality factor. The other measurement circuit at room temperature and all measurements followed the standard magneto-drive method 29 . The standard lock-in amplifiers A sine wave sequence generated by waveform generator (Keysight 33522B) is an excited source before dynamical quenching. At the same time, the sequence is the reference signal for lock-in amplifier after quenching when measuring the edge oscillator. In this way, the phase demodulated by lock-in keeps the same at every measurement and the accuracy of PGP is ensured. 

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