Abstract
Validity conditions for the adiabatic approximation are useful tools to understand and predict the quantum dynamics. Remarkably, the resonance phenomenon in oscillating quantum systems has challenged the adiabatic theorem. In this scenario, inconsistencies in the application of quantitative adiabatic conditions have led to a sequence of new approaches for adiabaticity. Here, by adopting a different strategy, we introduce a validation mechanism for the adiabatic approximation by driving the quantum system to a noninertial reference frame. More specifically, we begin by considering several relevant adiabatic approximation conditions previously derived and show that all of them fail by introducing a suitable oscillating Hamiltonian for a single quantum bit (qubit). Then, by evaluating the adiabatic condition in a rotated noninertial frame, we show that all of these conditions, including the standard adiabatic condition, can correctly describe the adiabatic dynamics in the original frame, either far from resonance or at a resonant point. Moreover, we prove that this validation mechanism can be extended for general multiparticle quantum systems, establishing the conditions for the equivalence of the adiabatic behavior as described in inertial or noninertial frames. In order to experimentally investigate our method, we consider a hyperfine qubit through a single trapped Ytterbium ion ^{171}Yb^{+}, where the ion hyperfine energy levels are used as degrees of freedom of a twolevel system. By monitoring the quantum evolution, we explicitly show the consistency of the adiabatic conditions in the noninertial frame.
Introduction
The adiabatic theorem^{1,2,3} is a fundamental ingredient in a number of applications in quantum mechanics. Under adiabatic dynamics, a quantum system evolves obeying a sufficiently slowlyvarying Hamiltonian, which prevents changes in the populations of the energy eigenlevels. In particular, if the system is prepared in an eigenstate \({E}_{n}\mathrm{(0)}\rangle \) of the Hamiltonian H(t) at a time t = 0, it will evolve to the corresponding instantaneous eigenstate \({E}_{n}(t)\rangle \) at later times. The concept of adiabaticity plays a relevant role in a vast array of fields, such as energylevel crossings in molecules^{4,5}, quantum field theory^{6}, geometric phases^{7,8}, quantum computation^{9,10,11,12,13}, quantum thermodynamics^{14,15}, quantum games theory^{16}, among others. However, despite such a wide range of applications, both sufficiency and necessity of quantitative conditions for the adiabatic behavior have been challenged^{17}. In particular, inconsistencies in the application of the adiabatic theorem may appear for oscillating Hamiltonians as a consequence of resonant transitions between their energy levels^{18,19,20}. Such inconsistencies have led to a revisitation of the adiabatic theorem, yielding many new proposals of adiabatic conditions (ACs) and bounds for the energy gap in more general settings (see, e.g., refs^{21,22,23,24,25,26}).
The first experimental investigation on the comparison among these proposals has been considered by Du et al.^{27}, where the authors considered a single nuclear spin1/2 particle in a rotating magnetic field manipulated by nuclear magnetic resonance (NMR) techinques. It is then shown the violation of both the sufficiency and necessity of the traditional AC, with partial success via some other generalized ACs. It is remarkable such possible violations are already apparent for a single quantum bit (qubit) system. More specifically, the ACs analyzed in ref.^{27} can be cast in the form of adiabatic coefficients C_{n}(t) which, for a qubit, are given by
where \({E}_{n}(t)\rangle \) are eigenstates of H(t) with energies E_{n}(t), τ is the total evolution time, \({{\rm{\Delta }}}_{10}(t)=i{\gamma }_{1}(t)i{\gamma }_{0}(t)+\)\(\frac{d}{dt}{\rm{\arg }}[i{d}_{10}(t)]\), \({d}_{10}(t)=\frac{\langle {E}_{1}(t)\dot{H}(t){E}_{0}(t)\rangle }{{E}_{0}(t){E}_{1}(t)}\), and \({\gamma }_{n}(t)=\langle {E}_{n}(t){\dot{E}}_{n}(t)\rangle \), with the dot symbol denoting time derivative and · denoting the usual operator norm. The adiabaticity coefficient C_{1} is the wellknown standard (traditional) adiabatic condition^{3,20,28}, while the conditions C_{2}, C_{3} and C_{4} as shown above were derived by Tong et al.^{22}, Wu et al.^{23,25} and AmbainisRegev^{21}, respectively. In general, the adiabatic behavior in a quantum system is achieved when \({C}_{n}\ll 1\). In ref.^{27}, it is shown that the condition C_{1} is neither sufficient nor necessary for guaranteeing the adiabatic behavior, while the conditions C_{2}, C_{3}, and C_{4} seem to successfully indicate the resonant phenomena observed in their specific experiment (even though their validity is debatable for more general quantum systems).
Here, instead of looking for new proposals of ACs, we adopt a different strategy to analyze the conditions in Eq. (1). More specifically, we consider the dynamics of the system in a noninertial reference frame, where it is possible to show that all the conditions in Eq. (1), including the traditional AC, work with no violations with respect to the exact solution of Schödinger equation. We emphasize that this approach implies in a validation mechanism already applicable for currently adopted adiabatic conditions, yielding a general yet simple solution for analyzing adiabaticity, including in a resonance setting. Therefore, the approach put forward here generalizes that presented originally in ref.^{17}, where the reference frame change has been used as a tool to reveal inconsistencies in the framework of the standard AC. To carryout our approach, we consider a Hamiltonian that leads to a failure of all coefficients in Eq. (1), which means that all the ACs in Eq. (1) are neither necessary nor sufficient to describe the adiabaticity of the system. Then, by implementing a change of reference frame, all of ACs are shown to become both necessary and sufficient conditions for the example considered. Remarkably, we can generalize our results to a generic manybody Hamiltonian, where we provide conditions for the equivalence of the adiabatic behavior in both inertial and noninertial reference frames These theoretical results are realized in a single trapped Ytterbium ion ^{171}Yb^{+} system, with excellent experimental agreement.
Results
Oscillating hamiltonian for a single qubit in trapped ions
Let us begin by considering the Hamiltonian
where we assume \({\omega }_{0}\gg {\omega }_{{\rm{T}}}\). The set of eigenvectors of H(t) is given by \({E}_{n}(t)\rangle ={{\mathscr{N}}}_{n}^{1}(t)[{(\mathrm{1)}}^{n}{\alpha }_{n}(t\mathrm{)0}\rangle +\mathrm{1}\rangle ]\), \({\alpha }_{n}(t)=\frac{1}{2}\,\cos \,\theta \,\csc (\omega t)[\mathrm{2(}\,\,{\mathrm{1)}}^{n}\,\cos \,\theta +{\rm{\Sigma }}]\), and \({{\mathscr{N}}}_{n}^{2}(t)=1+{\alpha }_{n}^{2}(t)\), with \({{\rm{\Sigma }}}^{2}=3+\,\cos (2\theta )\)\(2\,\cos (2\omega t)\,{\sin }^{2}\,(\theta )\), \(\theta =\arctan ({\omega }_{0}/{\omega }_{{\rm{T}}})\), and \(n\in \{0,1\}\). The energies are \({E}_{n}(t)={(1)}^{n}{\omega }_{0}\sec (\theta ){\rm{\Sigma }}/4\). Here we describe the system dynamics by
The experiment is performed using a single Ytterbium ion ^{171}Yb^{+}, which is trapped in a six needle Paul trap, with the experimental setup schematically shown in Fig. 1. The qubit is encoded in the hyperfine energy levels of ^{171}Yb^{+} (a hyperfine qubit), represented as \(\mathrm{0}\rangle \equiv {}^{2}S_{\mathrm{1/2}}\,F\mathrm{=0,}\,{m}_{F}\mathrm{=0}\rangle \) and \(1\rangle \equiv {}^{2}S_{1/2}\,F=1,\,{m}_{F}=0\rangle \)^{29}. We coherently drive the hyperfine qubit with a programmable arbitrary waveform generator (AWG)^{30} after Doppler cooling and standard optical pumping process. A 369.5 nm laser is used for fluorescence detection to measure the population of the \(\mathrm{1}\rangle \) state. Observation of more than one photon implies population in \(\mathrm{1}\rangle \).
The system is initialized in the state \(\mathrm{0}\rangle \) with optical pumping, so that the adiabatic dynamics is achieved if the system evolves as \({\psi }_{{\rm{ad}}}(t)\rangle ={E}_{1}(t)\rangle \), up to a global phase. It is possible to show that the Hamiltonian presents a neartoresonance situation when we set \(\omega {\omega }_{0}\ll {\omega }_{{\rm{T}}}\). Thus, to study the adiabaticity validity conditions in our Hamiltonian in Eq. (2) we compute the coefficients in Eq. (1) for different values of the \(\omega \). In our experiment, we set the detuning \({\omega }_{0}=2\pi \times 1.0\,{\rm{MHz}}\), the coupling strength \({\omega }_{{\rm{T}}}=2\pi \times 20.0\,{\rm{KHz}}\), and \(\omega =a\times {\omega }_{0}\) (\(a=10.0,1.0173,1.0,0.9827\,{\rm{and}}\,0.1\), respectively).
In Fig. 2, we experimentally compute the fidelity of obtaining the system in the \({\psi }_{{\rm{ad}}}(t)\rangle \), where we use the fidelity as \( {\mathcal F} (t)={\rm{Tr}}[\rho (t){\rho }_{{\rm{ad}}}(t)]\), where \(\rho (t)\) is solution of the Eq. (3) and \({\rho }_{{\rm{ad}}}(t)={E}_{1}(t)\rangle \langle {E}_{1}(t)\). We show the experimental results for three different situations, where we have \(\omega \gg {\omega }_{0}\), \(\omega \ll {\omega }_{0}\) and \(\omega \approx {\omega }_{0}\). When we have \(\omega {\omega }_{0}\gg {\omega }_{{\rm{T}}}\), the condition should provide us \({C}_{n}\ll 1\). However, looking at Fig. 3(a) we can see that such result is not obtained in case \(\omega \gg {\omega }_{0}\). Therefore, all the ACs provided by Eq. (1) are not necessary, once we have adiabaticity even in case where the ACs are not obeyed. On the other hand, in the neartoresonance situation we have no adiabaticity (once the fidelity is much smaller than 1), in contrast with Fig. 3(a), where we get \({C}_{n}\lesssim {10}^{2}\). Therefore, the ACs are not sufficient for studying the adiabatic behavior of our system. In conclusion, rather differently from the system considered in ref.^{27}, all the ACs provided by Eq. (1) are not applicable to the dynamics governed by the Hamiltonian in Eq. (2). These results imply that a direct application of the ACs yields neither sufficient nor necessary.
At this point, providing a new condition for adiabaticity could be a natural path to follow. Nevertheless, in order to investigate the applicability of ACs, we will implement, similarly as in classical mechanics, a transformation to a noninertial frame in Schrödinger equation. By considering frame representation in quantum mechanics, Eq. (3) can be taken as Schrödinger equation in an inertial frame^{31}. To introduce a noninertial frame, we can perform a rotation using the unitary timedependent operator \({\mathscr{O}}(t)={e}^{i\omega t{\sigma }_{z}}\). In this frame, the dynamics is given by
where \({H}_{{\mathscr{O}}}(t)={\mathscr{O}}(t)H(t){{\mathscr{O}}}^{\dagger }(t)+i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\) and \({\rho }_{{\mathscr{O}}}(t)={\mathscr{O}}(t)\rho (t){{\mathscr{O}}}^{\dagger }(t)\). The contribution \(i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\) in \({H}_{{\mathscr{O}}}(t)\) can be interpreted as a “fictitious potential”^{31}. This procedure is a common strategy, e.g., in nuclear magnetic resonance, where we use the noninertial frame to describe the system dynamics^{32,33}. By computing the noninertial Hamiltonian \({H}_{{\mathscr{O}}}(t)\) we find \({H}_{{\mathscr{O}}}(t)=({\omega }_{0}\omega ){\sigma }_{z}\mathrm{/2}+\,\sin (\omega t)\,\tan \,\theta {\overrightarrow{\omega }}_{xy}(t)\cdot {\overrightarrow{\sigma }}_{xy}\), with \({\overrightarrow{\omega }}_{xy}(t)={\omega }_{0}[\,\cos (\omega t)\hat{x}\,\sin (\omega t)\hat{y}\mathrm{]/2}\) and \({\overrightarrow{\sigma }}_{xy}={\sigma }_{x}\hat{x}+{\sigma }_{y}\hat{y}\). Now, if we compute the conditions C_{n} considering the set of eigenstate and energies of the new Hamiltonian H′(t) we obtain the curves shown in Fig. 3(b). Thus, considering the results in Figs 2 and 3(b), it is possible to conclude that the coefficients C_{n} computed in the noninertial frame allow us to successfully describe the adiabaticity of the inertial frame. This is in contrast with previous results, which indicated that the ACs may be problematic as we consider oscillating or rotating fields in resonant conditions^{19,20}. In particular, notice that even the traditional AC, when analyzed in this noninertial frame, becomes sufficient and necessary for the adiabatic behavior of the singlequbit oscillating Hamiltonian in Eq. (2). Moreover, it is worth highlighting that the conditions C_{1}, C_{2}, and C_{3} vanish for a farfrom resonance situation, while condition C_{4} presents an asymptotic value of \(2\,{\tan }^{2}\,\theta \sim {10}^{3}\) in the regime \(\omega \gg {\omega }_{0}\) (see Methods section for a detailed discussion).
Validation mechanism for ACs and framedependent adiabaticity
We now establish a general validation mechanism for ACs connecting inertial and noninertial frames, with special focus on cases under resonant conditions. This approach is applicable beyond the singlequbit system previously considered, holding for more general multiparticle quantum systems. First, it is important to highlight that rotated frames have been originally considered by Marzlin and Sanders in ref.^{17}. More specifically, they show an example of inapplicability of the adiabatic approximation for highly oscillating Hamiltonians governing a twolevel system. Rotated frames were then applied in order to analytically solve the quantum dynamics. Here, the results of ref.^{17} are generalized, with the role of noninertial frames explicitly formalized and extended to discrete quantum systems of arbitrary dimensions.
Let us consider a Hamiltonian H(t) in an inertial reference frame and its noninertial counterpart \({H}_{{\mathscr{O}}}(t)\), where the change of reference frame is provided by a generic timedependent unitary \({\mathscr{O}}(t)\). The Hamiltonians H(t) and \({H}_{{\mathscr{O}}}(t)\) obey eigenvalue equations given by \(H(t){E}_{n}(t)\rangle ={E}_{n}(t){E}_{n}(t)\rangle \) and \({H}_{{\mathscr{O}}}(t){E}_{n}^{{\mathscr{O}}}(t)\rangle ={E}_{n}^{{\mathscr{O}}}(t){E}_{n}^{{\mathscr{O}}}(t)\rangle \), with \([H(t),H(t^{\prime} )]\ne 0\) and \([{H}_{{\mathscr{O}}}(t),{H}_{{\mathscr{O}}}(t^{\prime} )]\ne 0\), in general. The adiabatic dynamics in the inertial frame, which is governed by H(t), can be defined through its corresponding evolution operator \(U(t,{t}_{0})=\)\({\sum }_{n}\,{e}^{i{\int }_{{t}_{0}}^{t}{\theta }_{n}(\xi )d\xi }{E}_{n}(t)\rangle \langle {E}_{n}({t}_{0})\), where \({\theta }_{n}(t)=\,{E}_{n}(t)/\hslash +i\langle {E}_{n}(t){\dot{E}}_{n}(t)\rangle \) is the adiabatic phase, composed by its dynamic and geometric contributions, respectively. Then, we can connect the adiabatic evolution in the inertial and noninertial frames through the theorem below.
Theorem 1
Consider a Hamiltonian H(t) and its noninertial counterpart \({H}_{{\mathscr{O}}}(t)={\mathscr{O}}(t)H(t){{\mathscr{O}}}^{\dagger }(t)+i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\), with \({\mathscr{O}}(t)\) an arbitrary unitary transformation. The eigenstates of H(t) and \({H}_{{\mathscr{O}}}(t)\) are denoted by \({E}_{k}(t)\rangle \) and \({E}_{m}^{{\mathscr{O}}}(t)\rangle \), respectively. Then, if a quantum system S is prepared at time \(t={t}_{0}\) in a particular eigenstate \({E}_{k}({t}_{0})\rangle \) of \(H({t}_{0})\), then the adiabatic evolution of S in the inertial frame, governed by H(t), implies in the adiabatic evolution of S in the noninertial frame, governed by \({H}_{{\mathscr{O}}}(t)\), if and only if
where \(t\in [{t}_{0},\tau ]\), with \(\tau \) denoting the total time of evolution. Conversely, if the adiabatic dynamics in the noninertial frame starts in \({E}_{m}^{{\mathscr{O}}}({t}_{0})\rangle \), then the dynamics in the inertial frame is also adiabatic if and only if Eq. (5) is satisfied.
The proof is provided in Method section. Notice that Theorem 1 establishes that, if Eq. (5) is satisfied, then a nonadiabatic behavior in the noninertial frame ensures a nonadiabatic behavior in the original frame and viceversa, provided that the evolution starts in a single eigenstate of the initial Hamiltonian. Then, we can apply this result to general timedependent resonant Hamiltonians, since we have no restriction on the Hamiltonian in Theorem 1. A typical scenario exhibiting resonance phenomena appears when a physical system is coupled to both a static high intensity field \({\overrightarrow{B}}_{0}\) and a timedependent transverse field \({\overrightarrow{B}}_{{\rm{T}}}(t)\), where \({\overrightarrow{B}}_{{\rm{T}}}(t)\ll {\overrightarrow{B}}_{0}\). Here, we will consider that the transverse field \({\overrightarrow{B}}_{{\rm{T}}}(t)\) is associated to a single rotating or oscillating field with frequency \(\omega \). In this context, we can write a general multiqubit Hamiltonian as
where the contributions \(\hslash {\omega }_{0}{H}_{0}\) and \(\hslash {\omega }_{{\rm{T}}}{H}_{{\rm{T}}}(\omega ,t)\) depend on the fields \({\overrightarrow{B}}_{0}\) and \({\overrightarrow{B}}_{{\rm{T}}}(t)\), respectively. Since \({\overrightarrow{B}}_{0}\perp {\overrightarrow{B}}_{{\rm{T}}}(t)\), we observe that \([{H}_{{\rm{T}}}(\omega ,t),{H}_{0}\mathrm{]\ }\ne \mathrm{\ 0}\). In the case \({\overrightarrow{B}}_{{\rm{T}}}(t)\ll {\overrightarrow{B}}_{0}\), the eigenstates \({E}_{n}(t)\rangle \) of the Hamiltonian \(H(\omega ,t)\) can be written as \({E}_{n}(t)\rangle \approx {E}_{n}^{0}\rangle \), where \({E}_{n}^{0}\rangle \) is a stationary eigenstate of the Hamiltonian \(\hslash {\omega }_{0}{H}_{0}\).
If we have a farfromresonance situation, we can approximate the dynamics obtained from \(H(\omega ,t)\) as that one driven by \(\hslash {\omega }_{0}{H}_{0}\). However, in a neartoresonance field configuration the most convenient way to study the system dynamics is by adopting a change of reference frame. A general approach to frame change is obtained by the choice \({\mathscr{O}}(\omega ,t)={e}^{i\omega {H}_{0}t}\). Then, from Eq. (6), we can show that
where \({H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)={\mathscr{O}}(t){H}_{{\rm{T}}}(t){{\mathscr{O}}}^{\dagger }(t)\). It is worth mentioning that \([{H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t),{H}_{0}\mathrm{]\ }\ne \mathrm{\ 0}\), once \([{H}_{{\rm{T}}}(\omega ,t),{H}_{0}\mathrm{]\ }\ne \mathrm{\ 0}\). In addition, since \({H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)\) is constrained to \({H}_{{\rm{T}}}(t)\) through a unitary transformation, \({H}_{{\rm{T}}}(\omega ,t)={H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)\). Therefore, due to the quantity \({\omega }_{0}\omega \) in the first term of \({H}_{{\mathscr{O}}}(\omega ,t)\), the contribution of \({H}_{{\rm{T}}}(\omega ,t)\) cannot be ignored in this new frame.
As shown in Method section, by considering the generic Hamiltonian in Eq. (6), we obtain that Eq. (5) in Theorem 1 is automatically satisfied if the quantum system is in a farfrom resonance configuration \(\omega {\omega }_{0}\gg {\omega }_{{\rm{T}}}\), so that the adiabatic dynamics in the inertial frame can be always predicted from the adiabaticity analysis in the noninertial frame. For this reason, the curves in Fig. 3(b) can correctly describe the adiabatic behavior exhibited in Fig. 2, yielding \({C}_{n}\ll 1\) for \(\omega \gg {\omega }_{0}\) and \(\omega \ll {\omega }_{0}\). On the other hand, at resonance (or neartoresonance) configuration \(\omega {\omega }_{0}\ll {\omega }_{{\rm{T}}}\), Eq. (5) in Theorem 1 reduces to the rather simple condition \(\langle {E}_{m}^{{\mathscr{O}}}(t){E}_{k}^{0}\rangle =\langle {E}_{m}^{{\mathscr{O}}}({t}_{0}){E}_{k}^{0}\rangle \). Hence, provided a generic Hamiltonian given by Eq. (6) at resonance (or neartoresonance) situation, if the corresponding Hamiltonian in the noninertial frame has timedependent eigenstates obeying \(\langle {E}_{m}^{{\mathscr{O}}}(t){E}_{k}^{0}\rangle ={\rm{constant}}\), \(\forall t\), m, for a particular initial state \({E}_{k}^{0}\rangle \), then a nonadiabatic evolution in the noninertial frame implies in nonadiabatic evolution in the inertial frame. This is exactly the case for the Hamiltonian in Eq. (2), with the violation of adiabaticity at resonance illustrated in Fig. 3(b) for all the ACs considered.
Revisiting the problem of the spin1/2 particle in a rotating magnetic field
We now apply our general treatment to the NMR Hamiltonian discussed by Du et al.^{27}. The dynamics describes a single spin1/2 particle coupled to a static field \({\overrightarrow{B}}_{{\rm{0}}}={B}_{{\rm{0}}}\hat{z}\) and a transverse radiofrequency field \({\overrightarrow{B}}_{{\rm{rf}}}(t)={B}_{{\rm{rf}}}[\cos (\omega )\hat{x}+\,\sin (\omega )\hat{y}]\), with Hamiltonian given by
where \({\omega }_{0}\gg {\omega }_{{\rm{rf}}}\). The system is prepared in an eigenstate of σ_{z} and the frequencies are chosen such that the standard AC is satisfied^{27}. In this scenario, the violations and agreements about ACs for this system have widely been discussed in literature^{34,35,36,37}. Here we analyze this Hamiltonian from a different point of view. By writing the system dynamics in the noninertial frame through \({\mathscr{O}}(t)={e}^{\frac{i}{\hslash }\frac{\omega }{2}t{\sigma }_{z}}\), we obtain \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}=({\omega }_{0}\omega ){\sigma }_{z}\mathrm{/2}+({\omega }_{{\rm{rf}}}\mathrm{/2)}{\sigma }_{x}\). Since this Hamiltonian is timeindependent, the dynamics under \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\) is trivially adiabatic, with all ACs in Eq. (1) satisfied. Therefore, the is no direct visualization of the resonant point. However, Theorem 1 cannot be directly applied here near to resonance because the initial state in this case is not an individual eigenstate of \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\), since \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\) is approximately proportional to σ_{x}. We can circumvent this problem by taking advantage of the timeindependence of \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\). More specifically, we start from the evolution operator \({U}_{{\mathscr{O}}}(t,{t}_{0})={e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}(t{t}_{0})}\) in the noninertial frame and investigate under which conditions we may obtain an adiabatic dynamics in the inertial frame. This can be suitably addressed by Theorem 2 below.
Theorem 2.
Consider a Hamiltonian H(t) and its noninertial counterpart \({H}_{{\mathscr{O}}}={\mathscr{O}}(t)H(t){{\mathscr{O}}}^{\dagger }(t)+i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\), with \({\mathscr{O}}(t)\) an arbitrary unitary transformation and \({H}_{{\mathscr{O}}}\) a constant Hamiltonian. The eigenstates of H(t) and \({H}_{{\mathscr{O}}}\) are denoted by \({E}_{k}(t)\rangle \) and \({E}_{m}^{{\mathscr{O}}}\rangle \), respectively. Then, if a quantum system S is prepared at time \(t={t}_{0}\) in a particular eigenstate \({E}_{n}({t}_{0})\rangle \) of \(H({t}_{0})\), then the adiabatic evolution of S in the inertial frame, governed by H(t), occurs if and only if
where \(t\in [{t}_{0},\tau ]\), with \(\tau \) denoting the total time of evolution, and \({U}_{{\mathscr{O}}}(t,{t}_{0})={{\mathscr{O}}}^{\dagger }(t){e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}(t{t}_{0})}{\mathscr{O}}({t}_{0})\).
The proof is provided in Method section. Notice that Theorem 2 can be applied to any timedependent Hamiltonian H(t) associated with a constant noninertial counterpart \({H}_{{\mathscr{O}}}\). The experimental results in ref.^{27} can be validated by Theorem 2, since the Hamiltonian in Eq. (8) satisfies Eq. (9) in a farfrom resonance situation and violates it at resonance. In fact, the initial state \(\psi \mathrm{(0)}\rangle \) can be approximately written as \(\psi \mathrm{(0)}\rangle ={E}_{n}\mathrm{(0)}\rangle \approx n\rangle \) [with \({\sigma }_{z}n\rangle ={(\mathrm{1)}}^{(n+\mathrm{1)}}n\rangle \)]. Thus, Eq. (9) provides the condition \(\langle k{e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}t}n\rangle =\langle kn\rangle ={\delta }_{kn}\), \(\forall k\) and \(\forall t\in [0,\tau ]\). In a farfromresonance situation, we have \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\approx \frac{{\omega }_{0}\omega }{2}{\sigma }_{z}\), and we conclude that \(\langle k{e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}t}n\rangle \approx {\delta }_{kn}\). This shows that the dynamics in the inertial frame is (approximately) adiabatic far from resonance. On the other hand, near to resonance, we get \({H}_{{\mathscr{O}}}^{{\rm{nmr}}}\approx \frac{{\omega }_{{\rm{rf}}}}{2}{\sigma }_{x}\), where we can immediately conclude that \(\langle k{e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}t}n\rangle {\delta }_{kn}\) is not valid for any \(t\in [0,\tau ]\).
Conclusion
We have introduced a framework to validate ACs in generic discrete multiparticle Hamitonians, which is rather convenient to analyze quantum systems at resonance. This is based on the analysis of ACs in a suitably designed noninertial reference frame. In particular, we have both theoretically and experimentally shown that several relevant ACs [provided by Eq. (1)], which include the traditional AC, are sufficient and necessary to describe the adiabatic behavior of a qubit in an oscillating field given by Eq. (2). In this case, sufficiency and necessity are fundamentally obtained through the noninertial frame map, with all the conditions failing to point out the adiabatic behavior in the original reference frame. The experimental realization has been performed through a single trapped Ytterbium ion, with excellent agreement with the theoretical results. More generally, the validation of ACs has been expanded to arbitrary Hamiltonians through Theorems 1 and 2, with detailed conditions provided for a large class of Hamiltonians in the form of Eq. (6). Therefore, instead of looking for new approaches for defining ACs, we have introduced a mechanism based on “fictitious potentials” (associated with noninertial frames) to reveal a correct indication of ACs, both at resonance and offresonant situations. In addition, as a further example, we discuss how the validation mechanism through noninertial frames can be useful to describe the results presented in ref.^{27}, where the adiabatic dynamics of a single spin1/2 in NMR had been previously investigated. More general settings, such as decoherence effects, are left for future research.
Methods
Proof of theorem 1
Let us consider two Hamiltonians, an inertial frame Hamiltonian H(t) and its noninertial counterpart \({H}_{{\mathscr{O}}}(t)\), which are related by a timedependent unitary \({\mathscr{O}}(t)\). The dynamics associated with Hamiltonians H(t) and \({H}_{{\mathscr{O}}}(t)\) are given by
where \({H}_{{\mathscr{O}}}(t)={\mathscr{O}}(t)H(t){{\mathscr{O}}}^{\dagger }(t)+i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\) and \({\rho }_{{\mathscr{O}}}(t)={\mathscr{O}}(t)\rho (t){{\mathscr{O}}}^{\dagger }(t)\). Then, the connection between the evolved states \(\psi (t)\rangle \) and \({\psi }_{{\mathscr{O}}}(t)\rangle \) in inertial and noninertial frames, respectively, is given by \({\psi }_{{\mathscr{O}}}(t)\rangle ={\mathscr{O}}(t)\psi (t)\rangle \), \(\forall t\in [{t}_{0},\tau ]\). By considering the initial state in inertial frame given by a single eigenstate of H(t), namely \(\psi ({t}_{0})\rangle ={E}_{k}({t}_{0})\rangle \), the adiabatic dynamics in this frame is written as
where \({\theta }_{k}(t)=\,{E}_{k}(t)/\hslash +i\langle {E}_{k}(t)(d/dt){E}_{k}(t)\rangle \) is the adiabatic phase composed by the dynamical and geometrical phase, respectively. On the other hand, an adiabatic behavior is obtained in noninertial frame if and only if
Therefore, we can write
Thus, Eq. (14) establishes a necessary and sufficient condition to obtain an adiabatic evolution in the noninertial frame, assuming an adiabatic evolution in the original frame. To conclude our proof, let us consider the converse case, where the system starts in a eigenstate of \({E}_{m}^{{\mathscr{O}}}({t}_{0})\rangle \) in noninertial frame. If the dynamics is adiabatic we write
where \({\theta }_{m}^{{\mathscr{O}}}(t)\) is the adiabatic phase collected in this frame. The dynamics will be adiabatic in the inertial frame if and only if
Therefore, by using the same procedure as before, we get the condition
which is equivalent to Eq. (14). This ends the proof of Theorem 1.
Application of theorem 1 to a timedependent hamiltonian for a single oscillating field
Let us consider a generic system under action of a single timedependent oscillating field with characteristic frequency \(\omega \), whose Hamiltonian reads
where we consider the transverse term \(\hslash {\omega }_{{\rm{T}}}{H}_{{\rm{T}}}(\omega ,t)\) as a perturbation, so that \({\omega }_{0}{H}_{0}\gg {\omega }_{{\rm{T}}}{H}_{{\rm{T}}}(\omega ,t)\), \(\forall t\in [0,\tau ]\). In this case, the eigenstates \({E}_{n}(t)\rangle \) and energies \({E}_{n}(t)\) of \(H(\omega ,t)\) can be obtained as perturbation of eigenstates \({E}_{n}^{0}\rangle \) and energies \({E}_{n}^{0}\) of \(\hslash {\omega }_{0}{H}_{0}\) as (up to a normalization coefficient)
On the other hand, in the noninertial frame, we have \({H}_{{\mathscr{O}}}(t)={\mathscr{O}}(t)H(t){{\mathscr{O}}}^{\dagger }(t)+i\hslash \dot{{\mathscr{O}}}(t){{\mathscr{O}}}^{\dagger }(t)\), which yields
where \({H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)={\mathscr{O}}(t){H}_{{\rm{T}}}(\omega ,t){{\mathscr{O}}}^{\dagger }(t)\). Now, we separately consider two specific cases:

Farfrom resonance situation \({\omega }_{0}\omega \gg {\omega }_{{\rm{T}}}\): In this case, the term \(\hslash {\omega }_{{\rm{T}}}{H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)\) in Eq. (21) works as a perturbation. Therefore the set of eigenvectors of \({H}_{{\mathscr{O}}}(\omega ,t)\) reads
$${E}_{n}^{{\mathscr{O}}}(t)\rangle ={E}_{n}^{0}\rangle +{\mathscr{O}}(\hslash {\omega }_{{\rm{T}}}{H}_{{\rm{T}}}(\omega ,t)),$$(22)where we have used that the energy gaps \({\tilde{E}}_{n}^{0}{\tilde{E}}_{k}^{0}\) of the Hamiltonian \(\hslash ({\omega }_{0}\omega ){H}_{0}\) are identical to energy gaps \({E}_{n}^{0}{E}_{k}^{0}\) of \(\hslash {\omega }_{0}{H}_{0}\) and \(\hslash {\omega }_{{\rm{T}}}{H}_{{\mathscr{O}},{\rm{T}}}(\omega ,t)=\hslash {\omega }_{{\rm{T}}}{H}_{{\rm{T}}}(\omega ,t)\). Thus, from Eqs (19) and (22) we conclude, for any eigenstate \({E}_{k}(t)\rangle \), that
$$\langle {E}_{m}^{{\mathscr{O}}}(t){\mathscr{O}}(t){E}_{k}(t)\rangle \approx {e}^{i\frac{\omega }{{\omega }_{0}}\frac{{E}_{k}^{0}}{\hslash }t}{\delta }_{mk},$$(23)so that we get \(\langle {E}_{m}^{{\mathscr{O}}}(t){\mathscr{O}}(t){E}_{k}(t)\rangle ={\rm{constant}}\), \(\forall m\), \(\forall t\in [{t}_{0},\tau ]\).

Resonance situation \({\omega }_{0}\omega \ll {\omega }_{{\rm{T}}}\): Now, we have a more subtle situation. Firstly, we can use Eqs (19) and (20) to write
so that
Now, it is possible to see that if \(\langle {E}_{m}^{{\mathscr{O}}}(t){E}_{k}^{0}\rangle =\langle {E}_{m}^{{\mathscr{O}}}({t}_{0}){E}_{k}^{0}\rangle \), \(\forall t\in [{t}_{0},\tau ]\), then we obtain \(\langle {E}_{m}^{{\mathscr{O}}}(t){\mathscr{O}}(t){E}_{k}(t)\rangle ={\rm{constant}}\).
Proof of theorem 2
Let us consider a timeindependent Hamiltonian \({H}_{{\mathscr{O}}}\) in the noninertial frame, so that its evolution operator can be written as \({U}_{{\mathscr{O}}}(t,{t}_{0})={e}^{\frac{i}{\hslash }{H}_{{\mathscr{O}}}(t{t}_{0})}\). Thus, we can write the dynamics in noninertial frame as
Moreover, assuming adiabatic dynamics in the inertial frame, we get
By using the relationship between inertial and noninertial frames as \(\psi (t)\rangle ={{\mathscr{O}}}^{\dagger }(t){\psi }_{{\mathscr{O}}}(t)\rangle \), we can write
where we have used the Eq. (27). Now, by taking \({\psi }_{{\mathscr{O}}}({t}_{0})\rangle ={\mathscr{O}}({t}_{0})\psi ({t}_{0})\rangle \), we obtain
Thus, by inserting the initial state \(\psi ({t}_{0})\rangle ={E}_{n}({t}_{0})\rangle \) in Eq. (30), we get
This concludes the proof of Theorem 2.
Asymptotic behavior of the adiabaticity parameters C _{n}
Due to the large analytical expressions for the eigenstates and eigenvalues of the Hamiltonian, it is suitable to analyze the quantities C_{1}, C_{2} and C_{3} from a numerical perspective. By applying a such a treatment, we can show that in the regime where \(\omega \ll {\omega }_{0}\) and \(\omega \gg {\omega }_{0}\), we have \({C}_{n}\to 0\) for \(n=\{1,2,3\}\). However, the same numerical treatment shows that C_{4} has a nonvanishing asymptotic behavior. In this case, since the expression for C_{4} just depends on the spectrum of H(t), an analytical study about the asymptotic behavior can be considered. Therefore, let us write
By computing the quantities in the above equation for the Hamiltonian in Eq. (2), we get
where we already used that \(\omega =r{\omega }_{0}\) and we adopted the total evolution time as \(\tau =1/{\omega }_{0}\). A first point to be highlighted is that in the limit \(r\to 0\) (\(\omega \ll {\omega }_{0}\)), we have \(\{{C}_{4}^{(1)}(t),{C}_{4}^{(2)}(t)\}\to \{0,0\}\), so that \({C}_{4}\ll 1\). On the other hand, let us look at the regime where \(r\gg 1\) (\(\omega \gg {\omega }_{0}\)). First, notice that, for \(r\gg 1\), we can approximate
since the terms \(\mathrm{[3}+\,\cos \,\mathrm{(2}\theta )2\,\cos \,\mathrm{(2}rt{\omega }_{0})\,{\sin }^{2}\,\theta ]\) in the denominators of Eqs (33) and (34) are at zeroth order in r and, therefore, can be neglected. Now, the maximum value of \({C}_{4}^{\mathrm{(1)}}(t)\) as a function of t is obtained when \(\cos (rt{\omega }_{0})=1\), which happens at \(t=\pi \mathrm{/(2}r{\omega }_{0})\). Therefore, we obtain
Finally, we can now expand the above equations for values of \(r\gg 1\) and we can show that
where we are neglecting terms depending on r^{−n}, with n ≥ 1 in the limit \(r\gg 1\). Thus, we conclude that
In the particular case of the Fig. 3(b), we have \({C}_{4}{}_{r\gg 1}\approx 0.8\cdot {10}^{3}\).
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Acknowledgements
We thank YuanYuan Zhao for valuable discussion. This work was supported by the National Key Research and Development Program of China (No. 2017YFA0304100), National Natural Science Foundation of China (Nos 61327901, 61490711, 11774335, 11734015), Anhui Initiative in Quantum Information Technologies (AHY070000, AHY020100), Anhui Provincial Natural Science Foundation (No. 1608085QA22), Key Research Program of Frontier Sciences, CAS (No. QYZDYSSWSLH003), the National Program for Support of Topnotch Young Professionals, the Fundamental Research Funds for the Central Universities (WK2470000026). A.C.S. is supported by Conselho Nacional de Desenvolvimento Cientfico e Tecnológico (CNPqBrazil). M.S.S. is supported by CNPqBrazil (No. 303070/20161) and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) (No. 203036/2016). A.C.S., F.B. and M.S.S. also acknowledge financial support in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior  Brasil (CAPES) (Finance Code 001) and by the Brazilian National Institute for Science and Technology of Quantum Information (INCTIQ).
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A.C.S., F.B. and M.S.S. developed and performed the theoretical analysis. C.K.H., J.M.C., Y.F.H. and C.F.L. designed the experiment. C.K.H., J.M.C. and Y.F.H. performed the experiment. C.K.H., Y.F.H., A.C.S., F.B. and M.S.S. wrote the manuscript. C.F.L. and G.C.G. supervised the project. All authors discussed and contributed to the analysis of the experimental data.
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Correspondence to JinMing Cui or Alan C. Santos or YunFeng Huang.
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