Experimental Implementation of Generalized Transitionless Quantum Driving
Chang-Kang Hu, Jin-Ming Cui, Alan C. Santos, Yun-Feng Huang, Marcelo S. Sarandy, Chuan-Feng Li, Guang-Can Guo
EExperimental Implementation of Generalized Transitionless Quantum Driving
Chang-Kang Hu,
1, 2, ∗ Jin-Ming Cui,
1, 2, ∗ Alan C. Santos, † Yun-Feng Huang,
1, 2, ‡ Marcelo S. Sarandy, § Chuan-Feng Li,
1, 2, ¶ and Guang-Can Guo
1, 2 CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People’s Republic of China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, 230026, People’s Republic of China Instituto de F´ısica, Universidade Federal Fluminense,Av. Gal. Milton Tavares de Souza s / n, Gragoat´a, 24210-346 Niter´oi, Rio de Janeiro, Brazil It is known that high intensity fields are usually required to implement shortcuts to adiabaticity via Transi-tionless Quantum Driving (TQD). Here, we show that this requirement can be relaxed by exploiting the gaugefreedom of generalized TQD, which is expressed in terms of an arbitrary phase when mimicking the adia-batic evolution. We experimentally investigate the performance of generalized TQD in comparison with bothtraditional TQD and adiabatic dynamics. By using a Yb + trapped ion hyperfine qubit, we implement aLandau-Zener adiabatic Hamiltonian and its (traditional and generalized) TQD counterparts. We show that thegeneralized theory provides energy-optimal Hamiltonians for TQD, with no additional fields required. In addi-tion, the optimal TQD Hamiltonian for the Landau-Zener model is investigated under dephasing. Even usingless intense fields, optimal TQD exhibits fidelities that are more robust against a decohering environment, withperformance superior than that provided by the adiabatic dynamics. I. INTRODUCTION
Transitionless Quantum Driving (TQD) [1–3] is a usefultechnique to mimic adiabatic quantum tasks at finite time. Ithas been applied for speeding up adiabaticity in several ap-plications, such as quantum gate Hamiltonians [4–6], heat en-gines in quantum thermodynamics [7], quantum informationprocessing [8–10], among others (e.g., Refs. [11–19]). To per-form TQD we need to design a counter-diabatic
Hamiltonian H CD ( t ), given by H CD ( t ) = i (cid:80) n ( | ˙ n t (cid:105) (cid:104) n t | + (cid:104) ˙ n t | n t (cid:105) | n t (cid:105) (cid:104) n t | ),where {| n t (cid:105)} denotes the set of instantaneous eigenstates ofthe original adiabatic Hamiltonian H ( t ) and the dot symboldenotes time derivative. The Hamiltonian H CD ( t ) enables usto cancel any transition between two di ff erent eigenstates of H ( t ). It should be added to the original Hamiltonian H ( t ) toyield the shortcut to adiabaticity Hamiltonian, which is givenby H SA ( t ) = H ( t ) + H CD ( t ).In general, the Hamiltonian H SA ( t ) allows us to exactlymimic an adiabatic dynamics at arbitrary finite time, so that,by starting at a given initial eigenstate | k (cid:105) of H (0), theevolved state is given by | ψ ( t ) (cid:105) = e i (cid:82) t θ ad k ( ξ ) d ξ | k t (cid:105) , where θ ad ( t ) = − E n ( t ) + i (cid:104) n t | ˙ n t (cid:105) is the adiabatic phase [20]. However, thereare several applications where we do not need to exactlymimic an adiabatic phase θ ad ( t ), as long as the system iskept in an eigenstate of the Hamiltonian H ( t ) [4–19, 21–23]. Thus, it has been proposed a generalized approach forTQD, where we consider that the phases that accompany thedynamics can be taken as arbitrary [24, 25]. Therefore, di ff er-ently from traditional TQD, where the dynamics is driven by H SA ( t ) = H ( t ) + H CD ( t ), in generalized TQD the dynamics is ∗ These two authors contributed equally to this work † Electronic address: ac [email protected] ff .br ‡ Electronic address: [email protected] § Electronic address: [email protected] ff .br ¶ Electronic address: cfl[email protected] driven by a generalized shortcut to adiabaticity Hamiltonian H GSA ( t ) written as [24–26] H GSA ( t ) = i (cid:88) n (cid:18) | ˙ n t (cid:105)(cid:104) n t | + i θ n ( t ) | n t (cid:105)(cid:104) n t | (cid:19) , (1)where θ n ( t ) are arbitrary real parameters. In this case, theevolved generalized state is written as | ψ ( t ) (cid:105) = e i (cid:82) t θ k ( ξ ) d ξ | k t (cid:105) ,with arbitrary θ k ( t ). It is worth highlighting that (1) showsthat we can mimic an adiabatic task even when our physicalsystem does not allow for a direct implementation of the adi-abatic Hamiltonian. In addition, as it was discussed in [25],we can also take advantage of these generalized phases θ n ( t )to simplify the Hamiltonian H GSA ( t ), possibly even removingin certain situations its dependence on time.Concerning robustness against decoherence, some experi-mental and theoretical studies of TQD in di ff erent experimen-tal architectures have shown promising features, such as innitrogen-vacancy setups [21], trapped ions [18], atoms in cav-ities [9, 22], nuclear magnetic resonance (NMR) [23], and op-tomechanics [19]. These works consider the traditional ap-proach for shortcuts to adiabaticity, where the adiabatic phaseis taken when mimicking the adiabatic dynamics. As theoret-ically found out in Ref. [25], generalized TQD may providemuch better resistance against decoherence depending on thetime window designed to run the quantum process.In this paper, we report the first experimental implementa-tion aiming at verifying the advantage of generalized TQDwith respect to its adiabatic and traditional TQD counter-parts. More specifically, we implement the energetically op-timal version of generalized TQD in a two-level system real-izing the Landau-Zener model. Our physical system is com-posed of a single Yb + ion confined in a Paul trap, wherethe system dynamics is driven by the Landau-Zener Hamilto-nian. It has previously been shown that the traditional TQDmethod requires high intensity fields [4–6, 27, 28]. As weshall see, generalized TQD theory requires much less intensefields while capable of providing better fidelities. In particu-lar, for the dynamics considered here, while traditional TQD a r X i v : . [ qu a n t - ph ] J u l FIG. 1: Experimental setup for implementing the generalized TQD.(a) The energy spectrum of the Yb + ion, where our two-level sys-tem was encoded in the hyperfine energy levels S / | F = , m F = (cid:105) and S / | F = , m F = (cid:105) and the 369.5 nm laser is used for fluores-cence detection. (b) Diagram of the six needles Paul trap used in ourexperiment. The microwave horn sends out microwaves to drive thehyperfine qubit and the numerical aperture NA = π × requires additional fields, the generalized theory allows us tomimic an adiabatic dynamics without additional fields. More-over, the dynamics under dephasing shows the superiority ofthe generalized TQD approach, with higher fidelities with lessexpenditure of energy resources. II. OPTIMAL TQD FOR THE LANDAU-ZENERHAMILTONIAN
We consider a quantum bit (qubit) undergoing an adia-batic dynamics governed by the Landau-Zener Hamiltonian,which reads H ( s ) = − ∆ σ z − Ω R ( s ) σ x , with a detuning ∆ between the microwave frequency and the atom transitionlevel; Ω R ( s ) is the Rabi frequency and s = t /τ is the normal-ized time. The system is initialized in the ground state | (cid:105) of H (0) and adiabatically evolves to | ψ ( s ) (cid:105) = e − i τ (cid:82) s θ ( ξ ) d ξ | E + ( s ) (cid:105) ,where | E + ( s ) (cid:105) = cos[ ϑ ( s ) / | (cid:105) + sin[ ϑ ( s ) / | (cid:105) , with ϑ ( s ) = arctan [ Ω R ( s ) / ∆ ] a time-dependent dimensionless parameterthat satisfies the boundary conditions ϑ (0) = ϑ (1) = ϑ . Traditional TQD mimics the above dynamics throughthe Hamiltonian H SA ( s ), where H CD ( s ) = [ d s ϑ ( s ) / τ ] σ y ,with the function ϑ ( s ) chosen as ϑ ( s ) = ϑ s for obtaininga time-independent counter-diabatic Hamiltonian (see, e.g.,Ref. [25]).However, in this scenario, we still need to implement atime-dependent contribution due to the adiabatic Hamiltonian H ( s ). On the other hand, if we adopt generalized TQD,we can implement the transitionless dynamics through thesimplest time-dependent Hamiltonian given by H OpSA ( s ) = H CD ( s ). As in Ref. [25], the energetically optimal Hamilto-nian above is obtained by choosing the phases in (1) through ageometric contribution, such as θ n ( t ) = i (cid:104) ˙ n t | n t (cid:105) , which implieshere in θ n ( t ) =
0. The H OpSA ( s ) turns out to be implementedby a flat π -pulse, whose robustness has been investigated forfast population transfer in Ref. [29]. III. EXPERIMENTAL SETUP
We now demonstrate the experimental implementation ofgeneralized TQD, with a single Yb + ion trapped in a sixneedles Paul trap, which is shown in Fig. 1(b). We encodea qubit into two hyperfine energy levels of the S / groundstate, which is denoted by | (cid:105) ≡ S / | F = , m F = (cid:105) and | (cid:105) ≡ S / | F = , m F = (cid:105) , as shown in Fig. 1(a). Applyinga 6.40 G static magnetic field, the clock transition frequencybetween | (cid:105) and | (cid:105) is ω h f = π × ω and ∆ , for im-plementing the target Hamiltonian. In addition, in order tomimic the two level system interacting with an environment,we introduced a Gaussian noise frequency modulation of the2 π × | (cid:105) state with 99.9%e ffi ciency. After the qubit operation with microwave se-quence, a florescence detection method is used to measure thepopulation of the the | (cid:105) state [30]. The florescence of thetrapped ion is collected by an optical lens with a numericalaperture of NA = ffi ciency. Within 300 µ s detection time, the target statepreparation and measurement fidelity is measured as 99.4%[31]. A. Microwave fields and energy resources
In our experiment we used both resonant and o ff -resonantmicrowaves fields to implement adiabatic and TQD dynam-ics. The adiabatic dynamics was performed by using a non-resonant microwave, where the time-dependent e ff ective Rabifrequency is given by Ω e ff ( s ) = [ Ω ( s ) + ∆ ] / . To drivethe system by using the traditional TQD, we use an indepen-dent microwave to simulate the counter-diabatic term H CD ( s ).This additional field is a resonant microwave ( ∆ = Ω R-CD ( s ) = d s ϑ ( s ) / τ .On the other hand, di ff erent from traditional TQD Hamilto-nian, the optimal dynamics driven by H GSA ( s ) could be im-plemented using a single resonant microwave with Rabi fre-quency Ω R-CD ( s ), where we have turned-o ff the non-resonantfield used for simulate H ( s ).In order to quantify the energy resources employed in thequantum evolution, we study the intensity of the fields usedto perform the adiabatic dynamics as well as traditional andoptimal TQD. The field intensity is associated with the Rabifrequency through the relation I ( s ) = ΓΩ ( s ), where Γ is aconstant that depends on the microwave amplifier. Consider-ing the whole evolution time, we can define the average in- DCDCRF RF DC DC AdiabaticmicrowaveCounter-adiabaticmicrowave
FIG. 2: The calculated relative field intensity I ( τ ) for traditionalTQD I SA ( τ ) (magenta continuum line) and optimal TQD I OpSA ( τ )(green dashed-dot line) as function of the total evolution time τ ,where the horizontal black dashed line represents I Ad ( τ ). The grayvertical line denotes the boundary time τ B ≈ .
052 ms betweenthe regions ¯ I OpSA ( τ ) > ¯ I ad ( τ ) (left hand side) and ¯ I OpSA ( τ ) < ¯ I ad ( τ )(right hand side). Inset: schematic representation of the fields usedto implement H ( s ), H SA ( s ) and H OpSA ( s ). We set ϑ ( s ) = π s / ∆ = π × Ω R ( s ) = ∆ tan( π s /
3) is used in our experiment. tensity field as ¯ I ( τ ) = (1 /τ ) (cid:82) τ I ( t ) dt = (cid:82) I ( s ) ds . So, for theadiabatic dynamics we have ¯ I Ad ( τ ) = Γ (cid:82) Ω ( s ) ds , for theoptimal TQD we get ¯ I OpSA ( τ ) = Γ (cid:82) [ d s ϑ ( s ) / τ ] ds , andfor the traditional TQD we find ¯ I SA ( τ ) = ¯ I Ad ( τ ) + ¯ I OpSA ( τ ),since the traditional TQD field is composed by both the adi-abatic and the optimal TQD contributions. For convenience,we disregard the constant Γ by taking relative field intensitiesexpressed in unities of the adiabatic intensity ¯ I Ad . Then, wedefine I SA ( τ ) = ¯ I SA ( τ ) / ¯ I Ad and I OpSA ( τ ) = ¯ I OpSA ( τ ) / ¯ I Ad , andadopt the normalization I Ad ( τ ) =
1. These field intensities areplotted in Fig 2, with an schematic representation of each dy-namics indicated in the inset. As we can see from Fig. 2, wecan define a value τ B for the total evolution time for which theintensity fields for implementing optimal TQD becomes lessintense than the adiabatic intensity. By computing τ B we get τ B ≈ .
052 ms and we represent this boundary using a verticalline in Fig. 2. Notice that, after τ B , the shortcut to adiabaticitydefined by the optimal TQD can be implemented by spendingless energy resources, as measured by the field intensity, thanthe adiabatic approach.An alternative approach to compute the energy cost is toestimate the energy scale through the time-average Hamilto-nian norm [4–6, 25, 32]. In this direction, let us consider aquantum dynamics driven by a time-dependent Hamiltonian,with energy cost defined as Σ ( τ ) = (cid:82) τ (cid:112) Tr { H ( t ) } dt . For thespecific case of the Landau-Zener model, this method takesinto account not only the external field intensities but also thedetuning ∆ . Indeed, ∆ exerts influence over the energy gapspectrum, which may justify its accounting in some scenar-ios, such as adiabatic quantum computation. From this defi-nition, we can show that Σ SA ( τ ) ≥ Σ Ad ( τ ), for every τ . More-over, there is again a boundary value τ B, Σ for which we get FIG. 3: Fidelity for unitary dynamics (continuum lines) and non-unitary one (dashed lines) under dephasing. The symbols and linesrepresent experimental data and theoretical results, respectively. Weset the Hamiltonians parameter as in Fig. 2, while the decoheringrate was kept as γ = . τ B ≈ .
052 ms asdiscussed in Fig. 1, while the second vertical lines is associated with τ B, Σ ≈ .
033 ms. Σ Op-SA ( τ ) < Σ Ad ( τ ) for τ > τ B, Σ , as originally predicted inRef. [25]. Here, we obtain τ B, Σ ≈ .
033 ms. Thus, we can seethat an analysis from energy scale of the Hamiltonian providesa (qualitatively) equivalent result to the intensity ields analy-sis. In both cases, it is shown that, by specifically designing asuitable time window, shortcuts to adiabaticity can be imple-mented in such a way to accelerate physical processes whilesaving energy resources.
IV. ROBUSTNESS AGAINST DECOHERENCE
Owing to the long coherence time of the Yb + trappedion hyperfine qubit, the decoherence e ff ects for the timescaleadopted in the experimental setup is negligible. Therefore, tosimulate the interaction of the Landau-Zener system with anenvironment, we introduce a Gaussian noise capable of imple-menting a dephasing channel where the system will be drivenby a Lindblad equation given by ˙ ρ ( t ) = − i (cid:2) H ( t ) , ρ ( t ) (cid:3) + γ (cid:2) σ z ρ ( t ) σ z − ρ ( t ) (cid:3) , where γ is the dephasing rate. To quan-tify the robustness of the protocols, we use the fidelity F ( τ ) = (cid:112) (cid:104) ψ ( s ) | ρ ( s ) | ψ ( s ) (cid:105) , which is related to the Bures length [33],with ρ ( s ) being solution of Lindblad equation and | ψ ( s ) (cid:105) = | E + ( s ) (cid:105) .We start by disregarding decoherence and implementingeach protocol for several choices of τ ∈ [10 − ms ,
10 ms].The results are shown in Fig. 3. Since the maximum to-tal evolution time is τ =
10 ms (i.e., 20 times shorter thanthe coherence time of the qubit), the dynamics can be con-sidered as approximately unitary. The continuum lines inFig. 3 show the fidelities F ( τ ) under a unitary dynamics foreach protocol and the corresponding experimental data points.The black curve shows that the adiabatic behavior is achievedfor long total evolution time τ (cid:29) τ ad ≈ .
021 ms, with τ ad computed from τ ad = max s ∈ [0 , |(cid:104) E − ( s ) | d s H ( s ) | E + ( s ) (cid:105) / g ( s ) | ,where g ( s ) = E + ( s ) − E − ( s ) is the gap between fundamental | E − ( s ) (cid:105) and excited | E + ( s ) (cid:105) energy levels [34]. Notice that, forfast evolutions ( τ < .
021 ms), both optimal and traditionalTQD provide a high fidelity protocol, while adiabatic dynam-ics fails. Naturally, this high performance is accompanied bycostly fields used for implementing the TQD protocols, as pre-viously shown in Fig. 2. On the other hand, it is importantto highlight that for total evolution time τ > τ B , where theoptimal TQD field is smaller than both adiabatic and tradi-tional TQD ones, the high performance of optimal TQD iskept. By looking now at the e ff ect of the dephasing channel,we can see that optimal TQD exhibits the highest robustnessfor every τ , with its advantage increasing as τ increases. Forfast evolution times, this high performance is again associ-ated with costly fields in comparison with the adiabatic fields.However, for large evolution times, while the traditional TQDfidelity converges to adiabatic fidelity, the fidelity behavior ofoptimal TQD is much better than both adiabatic dynamics andtraditional TQD, with less energy resources spent in the pro-cess. This is a remarkable result, since we can achieve bothbetter fidelities with less energy fields involved, where we get F OpSA > F Ad even when ¯ I OpSA (cid:28) ¯ I Ad . V. CONCLUSION
In this paper we have experimentally investigated the per-formance of shortcuts to adiabaticity by exploiting the gaugefreedom as we fix the phase accompanying the evolution dy-namics. In particular, we have focused on the optimal TQD,comparing it with its associated adiabatic dynamics and tradi-tional TQD counterparts. Our main results are: i) The optimalversion of generalized TQD has been shown to be a useful pro-tocol for obtaining the optimal shortcut to adiabaticity. Whileadiabatic and traditional TQD require time-dependent quan-tum control, optimal TQD can be experimentally realized byusing time-independent fields. In addition, the necessity ofauxiliary fields in traditional TQD is not a requirement forimplementing TQD via its optimal version. ii) Optimal TQDis an energetically optimal protocol of shortcut to adiabatic-ity. By considering the average intensity fields as a measure of energy cost for implementing the protocols discussed here,we were able to show that the optimal version of generalizedTQD may be energetically less demanding. This result is keptalso for alternatives definitions of energy cost, e.g. takinginto account the detuning contribution. iii) By simulating anenvironment-system coupling associated with the dephasingchannel, we have shown that optimal TQD can be more robustthan the adiabatic dynamics and the traditional TQD while atthe same time spending less energy resources for a finite timerange. In addition, we were able to to mimic the adiabatic be-havior through time-independent Hamiltonian, showing thatthe optimal theory works. Once the Landau-Zener Hamilto-nian can be implemented in others physical systems, e.g. nu-clear magnetic resonance [35] and two-level systems drivenby a chirped field [1], it is reasonable to think that the resultsobtained here can be also realized in other experimental ar-chitectures. Moreover, inverse engineering protocols are aninteresting candidate to implement fast and robust elementaryquantum gates for quantum computing [4–6, 36]. Thus, an ex-perimental investigation of these protocols can be a promisingdirection as a future research, as can the robustness of optimalTQD against others classes of errors (as it has been done fortraditional TQD in Ref. [37]).
VI. FUNDING INFORMATION
This work was supported by the National Key Research andDevelopment Program of China (No. 2017YFA0304100), Na-tional Natural Science Foundation of China (Nos. 61327901,61490711,11774335, 11734015, 11474268, 11374288,11304305,11404319), Anhui Initiative in Quantum Informa-tion Technologies (AHY070000), Key Research Program ofFrontier Sciences, CAS (No. QYZDY-SSWSLH003), the Na-tional Program for Support of Top-notch Young Professionals(Grant No. BB2470000005), the Fundamental ResearchFunds for the Central Universities (WK2470000026). A.C.S.is supported by CNPq-Brazil. M.S.S. is supported by CNPq-Brazil (No. 303070 / / [1] M. Demirplak and S. A. Rice, J. Phys. Chem. A , 9937(2003).[2] M. Demirplak and S. A. Rice, J. Phys. Chem. B , 6838(2005).[3] M. Berry, J. Phys. A: Math. Theor. , 365303 (2009).[4] A. C. Santos and M. S. Sarandy, Sci. Rep. , 15775 (2015).[5] A. C. Santos, R. D. Silva, and M. S. Sarandy, Phys. Rev. A ,012311 (2016).[6] I. B. Coulamy, A. C. Santos, I. Hen, and M. S. Sarandy, Fron-tiers in ICT , 19 (2016).[7] M. Beau, J. Jaramillo, and A. del Campo, Entropy , 168(2016).[8] M. Herrera, M. S. Sarandy, E. I. Duzzioni, and R. M. Serra, Phys. Rev. A , 022323 (2014).[9] Z. Chen, Y. Chen, Y. Xia, J. Song, and B. Huang, Sci. Rep. ,22202 (2016).[10] X. Chen, I. Lizuain, A. Ruschhaupt, D. Gu´ery-Odelin, and J. G.Muga, Phys. Rev. Lett. , 123003 (2010).[11] D. Stefanatos, Phys. Rev. A , 023811 (2014).[12] M. Lu, Y. Xia, L.-T. Shen, J. Song, and N. B. An, Phys. Rev. A , 012326 (2014).[13] S. De ff ner, New J. Phys. , 012001 (2016).[14] Y.-H. Chen, Y. Xia, Q.-Q. Chen, and J. Song, Phys. Rev. A ,033856 (2014).[15] X.-K. Song, H. Zhang, Q. Ai, J. Qiu, and F.-G. Deng, New J.Phys. , 023001 (2016). [16] A. del Campo, Phys. Rev. Lett. , 100502 (2013).[17] H. Saberi, T. c. v. Opatrn´y, K. Mølmer, and A. del Campo, Phys.Rev. A , 060301 (2014).[18] S. An, D. Lv, A. Del Campo, and K. Kim, Nat. Commun. ,12999 (2016).[19] H. Zhang, X.-K. Song, Q. Ai, M. Zhang, and F.-G. Deng,arXiv:1610.09938 (2016).[20] M. V. Berry, Proc. R. Soc. A , 45 (1984).[21] Z.-T. Liang et al. , Phys. Rev. A , 040305 (2016).[22] S. L. Wu, X. L. Huang, H. Li, and X. X. Yi, Phys. Rev. A ,042104 (2017).[23] J. Vandermause and C. Ramanathan, Phys. Rev. A , 052329(2016).[24] E. Torrontegui et al. , Adv. At. Mol. Opt. Phys , 117 (2013).[25] A. C. Santos and M. S. Sarandy, J. Phys. A: Math. Theor. ,025301 (2018).[26] X. Chen, E. Torrontegui, and J. G. Muga, Phys. Rev. A ,062116 (2011). [27] S. Campbell and S. De ff ner, Phys. Rev. Lett. , 100601(2017).[28] O. Abah and E. Lutz, Europhys. Lett. (EPL) , 40005 (2017).[29] X.-J. Lu et al. , Phys. Rev. A , 033406 (2013).[30] S. Olmschenk et al. , Physical Review A , 052314 (2007).[31] R. Blume-Kohout et al. , Nature Communications , 14485(2017).[32] M. Kieferov´a and N. Wiebe, New J. Phys. , 123034 (2014).[33] D. Bures, Trans. Am. Math. Soc. , 199 (1969).[34] M. S. Sarandy, L.-A. Wu, and D. A. Lidar, Quantum Informa-tion Processing , 331 (2004).[35] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information: 10th Anniversary Edition , 10th ed.(Cambridge University Press, New York, NY, USA, 2011).[36] A. C. Santos, J. Phys. B: At. Mol. Opt. Phys. , 015501 (2018).[37] A. Ruschhaupt, X. Chen, D. Alonso, and J. Muga, New J. Phys.14