Abstract
The application of adiabatic protocols in quantum technologies is severely limited by environmental sources of noise and decoherence. Shortcuts to adiabaticity by counterdiabatic driving constitute a powerful alternative that speed up timeevolution while mimicking adiabatic dynamics. Here we report the experimental implementation of counterdiabatic driving in a continuous variable system, a shortcut to the adiabatic transport of a trapped ion in phase space. The resulting dynamics is equivalent to a ‘fastmotion video’ of the adiabatic trajectory. The robustness of this protocol is shown to surpass that of competing schemes based on classical local controls and Fourier optimization methods. Our results demonstrate that shortcuts to adiabaticity provide a robust speedup of quantum protocols of wide applicability in quantum technologies.
Introduction
Adiabatic processes play an essential role in many aspects of quantum technology^{1,2}. Quantum adiabatic simulation exploits adiabatic dynamics to track ground states of complex Hamiltonians facilitating the study of quantum manybody phenomena^{3,4}. Schemes for scalable iontrap quantum computer resort to the adiabatic transfer of ions between different trap zones^{5,6}. Adiabatic dynamics plays as well a key role in holonomic quantum computation^{7}, and the design of the geometric phase gate^{8} with its inherent robustness. Adiabatic protocols are also essential in quantum thermodynamics whether studying quantum fluctuations^{9} or the optimization of quantum thermal machines^{10,11,12,13}. These applications are however limited by the requirement of slow driving that conflicts with the feebleness of quantum coherence when the system of interest is embedded in an environment.
According to the adiabatic theorem, a system prepared in a nondegenerate eigenstate will remain in the instantaneous eigenstate during its time evolution under the requirement of slow driving. By contrast, the breakdown of adiabatic dynamics under fast driving couples different energy modes and induces diabatic transitions. Diabatic excitations can however be tailored using shortcuts to adiabaticity (STA) to mimic adiabatic dynamics. Among the available techniques to engineer STA^{14}, counterdiabatic driving (CD), relies on the use of an auxiliary control to explicitly suppress transitions between different energy eigenstates and enforce parallel transport^{15,16}.
The transport can be realized by applying a time dependent force f(t) to a harmonic oscillator of mass m and frequency ω, which is described by
If we increase the force from zero to f(t) slowly, we can transport the ion over a distance q(t)=−f(t)/mω^{2}. The excitations during the nonadiabatic transport can be seen in the instantaneous frame through the positionshift transformation , where we denote ħ≡1 throughout the manuscript. In the instantaneous frame, the timedependent potential minimum is located at x=0 and the state is governed by the Hamiltonian , where a global phase term has been ignored. The first two terms describe the harmonic motion around the potential minimum. The last term is nonlocal in real space and induces diabatic transitions, vanishing only in the adiabatic limit. The CD suppresses these nonadiabatic transition without slowing down the dynamics by adding the auxiliary term^{14,17}
Because is invariant under the positionshift transformation, diabatic transitions are completely suppressed in the instantaneous reference frame under arbitrarily fast transport.
Here we experimentally realize the CD protocol for the nonadiabatic control of a single ^{171}Yb^{+} ion (refs 18, 19) as it is transported in phase space. We use a pair of Raman beams to apply the force on the ion and achieve a precise and flexible control of the quantum evolution that allows us to unveil the superior performance of STA based on CD over alternative schemes. Our experiment provides a faithful realization of various STA protocols and is therefore complementary to previous studies on ion transport with time dependent electric fields^{20,21,22}.
Results
Physical model and quantum control
In the interaction picture with respect to the harmonic oscillation, the force induced by the lasers as configured in Fig. 1 is described by
where , f(t)=Ω(t)Δk/2, Ω(t) is proportional to the intensity of both Raman beams, Δk is the projection of the wavevectors difference of the Raman beams on the motional axis of the ion and ϕ is the phase difference between those two laser beams. Both laser beams are red detuned from the transition between the ground state (^{2}S_{1/2}) and the excited state (^{2}P_{1/2}). Due to the large detuning Δ≈2π × 14 THz, the excited state is adiabatically eliminated. The effective trap frequency ω=2π × 20 kHz in the interaction frame comes from the difference between the beatnote frequency of the laser beams δ and the real trap frequency ν=2π × 3.1 MHz. The effective mass is given by (M_{Yb}: mass of ^{171}Yb^{+}). When the phase ϕ=0, the Hamiltonian (3) describes a dragged harmonic oscillator, with the dragging term . We can implement the CD term , where , by setting ϕ=−π/2 (Supplementary Note 1).
Counterdiabatic transport
In the experiment, after Doppler and motional sideband cooling, is prepared with 0.02±0.02 average phonon number. Because we cannot measure the phonon distribution in the interaction picture directly, the STA performance is probed with the quench echo method^{23} in which the ion is first transported adiabatically and then brought back to the initial location using the STA protocol. During the first adiabatic process, we linearly increase the force f(t) from 0 to f_{max}=Ω_{max}Δk/2 within one period of the harmonic motion T_{0}=2π/ω=50 μs, where Ω_{max}=2π × 378 kHz corresponds to the maximum value allowed by the laser. This linear ramp has been well studied in experiments^{9,22,24}, and can be regarded as perfectly adiabatic. Following it, the force is linearly reduced from f_{max} to 0 within a duration of sT_{0}, where s is defined as the shortcut ratio. The backward dynamics is assisted by turning on the laser to implement the CD term according to equation (2). The relation between the strength of the CD term and the shortcut ratio is given by h(t)≡h_{max}/(2πs). Finally, we apply blue sideband transitions to measure the phonon distribution^{18}. The timedependent laser intensity profiles (waveforms) during the forward and backward transport stages are shown in Fig. 2. We vary s from 0.95 to 0.15 with a step of 0.1 and obtain the final average phonon number 0.016±0.018, which confirms that the CD protocol does not excite the motion after the transport for any duration (Supplementary Notes 2 and 3).
We also measure phonon excitations in the instantaneous basis during the transport in order to certify that the dynamics is following the adiabatic ground state. During the forward linear ramp and the backward CD transport, we stop at different instants and add another CD transport with s=0.15 to adiabatically change back to the lab frame. As shown in Fig. 3a, we do not observe any significant excitation during the transport, which confirms that the CD is speeding up the adiabatic trajectory associated with as in a ‘fast motion video’. We also measure the excitation in the lab framework (Supplementary Note 4).
Furthermore, the CD is shown to be robust against the trap frequency drift error. We design STA’s waveforms with the nominal trap frequency ω=2π × 20 kHz. In the first linear adiabatic ramp, we keep the trap frequency ω, but change the effective trap frequency to ω′ during the STA transport. Then we measure the final average phonon excitation as a function of ω′/ω. The result in Fig. 3b shows that the CD is extremely robust against the drift of the trap frequency. This feature can be qualitatively explained by the results shown in Fig. 3a, where almost no excitation appears during the CD driving. Since the higher excited states are more fragile to errors, the protocol with the smaller excitations during the transport is surely more robust. The higher robustness also results from the lower amplitude of the required control field. In the experiment, for the shortcut ratio s=0.4, the CD protocol uses three times less intensity than the other protocols, which naturally reduces the amount of noise proportionally.
Unitarily equivalent transport
The CD stands out among STA protocols for its robustness and the adiabatic following during the whole evolution. Yet, the realization of the auxiliary control is hardly feasible with classical electrical fields. Many efforts have been devoted to identify alternative controls requiring only local potentials^{14,17,25,26,27}. To this end, we resort to controls related to CD via its unitary equivalence (UE)^{14,17,25,28}. The exact solution to timedependent Schrödinger equation with Hamiltonian is given by the adiabatic approximation to the dynamics generated by . Under a momentumshift transformation , the time evolving state becomes , which is governed by the Hamiltonian,
where a global phase term has been gauged away. The auxiliary control in the driving Hamiltonian can be realized with a local potential. As long as , the state reproduces exactly the desired target state on completion of the STA protocol. This suggests a route to design the UE transport waveform. The boundary conditions f(0)=0 and f(t_{f})=f_{max} define the transport problem. Vanishing firstorder derivatives guarantee that . Considering that generally we do not suddenly turn on or off control fields, we further impose secondorder boundary conditions . These constraints are satisfied by a polynomial waveform f(t)=10(t/t_{f})^{2}−15(t/t_{f})^{3}+6(t/t_{f})^{4} (refs 17, 29). In the experiment, we apply the UE transport in the backward process, as shown in Fig. 2.
For the UE transport, we measure the final average phonon number 0.026±0.019 for various shortcut ratios s from 1 to 0.4 with a step of 0.1. As shown in Fig. 3a, we also examine the process of the UE transport in the instantaneous basis and in the lab framework (Supplementary Note 4). We observe large excitations in the process, which shows the UE protocol does not follow the adiabatic evolution, but succeeds in preparing the adiabatic target state at the final stage. As shown in Fig. 3b, the robustness against the drift of the trap frequency is below that of the CD transport. Note that the f(t) used is not the only solution. Simulation results (Supplementary Note 5) show that the waveform will be more sensitive to the trap frequency error, when higher order boundary conditions are considered. The firstorder polynomial waveform can also be used to mimnimize the DC Stark shift during the transport with the electric fields^{30}.
Fourier optimization transport
Finally, we implement the Fourier optimization scheme as proposed in^{27}. When the applied force f(t) for transport satisfies the conditions f(0)=0 and , the final excitation energy can be expressed as the Fourier transform of the acceleration of the force at the trap frequency. In principle, this method allows us to find a driving f(t) that simultaneously minimizes the final excitation energy for an ensemble of N different trap frequencies. When they are equalized, the final excitation is set by (ω′^{2}−ω^{2})^{N}, which enhances the robustness with N. The cost of the enhanced robustness is the increase of the amplitude of the control field with the order N. In our experiment, we choose N=3 that results in a oscillatory waveform, shown in Fig. 2. The required amplitude of the control field greatly surpasses f_{max} for a small shortcut ratio, thus we only test the scheme for s=1.5. The excitation in the instantaneous base and its robustness are shown in Fig. 3a,b, respectively.
Discussion
We have provided a realization of shortcuts to adiabaticity based on counterdiabatic driving in a continuous variable system. By demonstrating the robust adiabatic following, we have shown that the resulting timeevolution follows a ‘fastmotion video’ of the adiabatic dynamics. This protocol is also known to be the optimal solution of the quantum brachistochrone problem^{31}. We have further realized two competing STA protocols for the transport problem: local UE driving and Fourier optimization methods. In the UE scheme, while the auxiliary control field takes the form of a timedependent linear potential, its amplitude scales as surpassing the value required for counterdiabatic driving, . We note that by further modulating the trap frequency during transport, these shortcuts can still be accelerated within a maximum control field with the ‘rapid scan method’, that has been realized for a two level system^{28}. The total duration can then be reduced to half for the tested UE protocol reported here. As for the Fourier optimization scheme, its robustness is reduced even with respect to the UE scheme for a given amplitude of the control field, but could be increased with the order N and a higher amplitude of the control field (Supplementary Note 6).
In our experiment, we demonstrate that the challenging nonlocal CD term can be generated in the interaction frame. Therefore, our results will be directly influential and beneficial to the other experimental works that require adiabatic evolution in short time and are performed in the interaction picture including quantum thermodynamics, quantum simulation and quantum computation. The transport of a harmonic oscillator can be a test bath for quantum thermodynamics^{9} or as part of a quantum engine^{11,12,13}, for which the CD protocol can be used to boost the performance. For many quantumsimulation experiments, adiabatic evolution is essential to prepare a complex ground state of nontrivial Hamiltonian from a simple Hamiltonian whether or not in the interaction frame. The nontrivial ground state of a bosonic Hamiltonian or spinboson Hamiltonian could be implemented via the CD protocol, overcoming the limitation imposed by the coherence time of the system. The CD protocol can also speed up routines in holonomic quantum computation^{7,8,32,33}, and enable the implementation of topological quantum computation with nonAbelian braiding operations^{34} that need not be adiabatic.
Methods
The dragged harmonic oscillator model
As mentioned in the main text, the Hamiltonian of the dragged harmonic oscillator in the interaction picture about the harmonic motion is equation (3). Here we apply a pair of Raman beams to the ion with a beatnote, which is red detuned to the real trap frequency ν with the nominal trap frequency ω to simulate this Hamiltonian. We can find the detail of the laser ion interaction in the section A of the Supplementary Information. And the interaction Hamiltonian is , which equals the equation (3) when f(t)=Ω(t)Δk/2 and , where the effective mass .
Dynamics in the instantaneous basis
To study the STA dynamics we measure phonon excitations in the instantaneous basis during the transport, and use a short CD protocol to change to the lab frame, where the measurements can be made. To choose the protocol for the frame change, we measure the fidelity of different STA with various shortcut ratios (Supplementary Note 2) and find that the CD transport with the smallest shortcut ratio s=0.15 is optimal. In addition to its robustness against the trap frequency error, its shortest duration protects the motion of the ion from the heating effect.
Data availability
Raw data for any of the results reported in the text are available from the authors on request.
Additional information
How to cite this article: An, S. et al. Shortcuts to adiabaticity by counterdiabatic driving for trappedion displacement in phase space. Nat. Commun. 7, 12999 doi: 10.1038/ncomms12999 (2016).
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Acknowledgements
We thank Mathieu Beau and Bala Sundaram for a critical reading of the manuscript. This work was supported by the National Basic Research Program of China under Grants 2011CBA00300 (2011CBA00301), the National Natural Science Foundation of China 11374178, 11574002. Funding support from UMass Boston (project P20150000029279) is acknowledged.
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S.A. and D.L. performed the experiments and data taking. S.A. Analysed the data. A.d.C. provided the idea for the experiments. K.K. supervised the project. All authors contribute to writing the manuscript.
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Supplementary Information
Supplementary Figures 17, Supplementary Notes 16 and Supplementary References. (PDF 214 kb)
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An, S., Lv, D., del Campo, A. et al. Shortcuts to adiabaticity by counterdiabatic driving for trappedion displacement in phase space. Nat Commun 7, 12999 (2016). https://doi.org/10.1038/ncomms12999
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DOI: https://doi.org/10.1038/ncomms12999
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