# Fileset

[LY17664.pdf](https://mdr.nims.go.jp/filesets/d019faf3-6153-44d8-a20f-dacd3184ecbc/download)

## Creator

[Kento Sasaki](https://orcid.org/0000-0002-5880-2116), [Yuki Nakamura](https://orcid.org/0000-0002-9038-468X), [Tokuyuki Teraji](https://orcid.org/0000-0002-7731-0547), [Takashi Oka](https://orcid.org/0000-0003-1746-5368), [Kensuke Kobayashi](https://orcid.org/0000-0001-7072-5945)

## Rights

©2025 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Demonstration of geometric diabatic control of quantum states](https://mdr.nims.go.jp/datasets/f2953298-b20f-4484-bfcb-ee1772a27862)

## Fulltext

main_recent.dviDemonstration of geometric diabatic control of quantum statesKento Sasaki∗ and Yuki NakamuraDepartment of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, JapanTokuyuki TerajiNational Institute for Materials Science, Tsukuba, Ibaraki 305-0044, JapanTakashi OkaThe Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, JapanKensuke Kobayashi†Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, JapanInstitute for Physics of Intelligence, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan andTrans-scale Quantum Science Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan(Dated: April 25, 2023)Geometric effects can play a pivotal role in streamlining quantum manipulation. We demonstrate a geometricdiabatic control, that is, perfect tunneling between spin states in a diamond by a quadratic sweep of a drivingfield. The field sweep speed for the perfect tunneling is determined by the geometric amplitude factor andcan be tuned arbitrarily. Our results are obtained by testing a quadratic version of Berry’s twisted Landau-Zenermodel. This geometric tuning is robust over a wide parameter range. Our work provides a new basis for quantumcontrol in various systems, including condensed matter physics, quantum computation, and nuclear magneticresonance.Tunneling is an exotic yet ubiquitous quantum phe-nomenon. To control quantum states, a common strategyknown as adiabatic control avoids it by moving a large en-ergy barrier slowly. Another ubiquitous feature of quantumphysics is geometric effects [1]. A well-known example is thegeometric phase [2] that a particle acquires during an adia-batic motion. However, geometric effects are not restricted byadiabaticity. Even during diabatic tunneling events, geomet-ric effects take place and lead to grave consequences in thedynamics.The simplest system that demonstrates the marriage oftunneling and geometric effects is the twisted Landau-Zener(TLZ) model introduced by M. V. Berry, which describes aparticle in two quantum states driven by an external field [3].In the original untwisted Landau-Zener (LZ) model [4–7],when two energy levels change in time, quantum tunnelingacross an energy gap ∆ occurs depending on the speed ofthe change [Fig. 1(a)] [8, 9]. The tunneling probability Pdepends on the sweep speed F [Fig. 1(c)]; P = 0 in theadiabatic limit (F → 0), while P is unity in the diabaticlimit (|F | → ∞). Such speed-dependent tunneling has beendemonstrated in various systems [10–16]. In the TLZ model,the driving field has a “twist” and the adiabatic to diabatictransition is geometrically modulated [17, 18]. Recently, theimportance of the geometric effects has been recognized notonly in equilibrium [19] but also in nonequilibrium [20–23].The TLZ model, which possesses a new nonequilibrium tun-ing knob on top of the LZ model, should be widely applied tomaterials engineering [24, 25] and quantum controls [26–29].Despite a few experiments [30–33], the opportunity of utiliz-ing such geometric tuning for quantum control has long beenoverlooked, and their robustness remains unexplored.Here, using an electron spin in a diamond, we realize andtest an ideal TLZ model with a quadratic twist [24] that mani-fests perfect tunneling and nonreciprocity over a wide range ofgap and twist parameters. We measure the tunneling probabil-ities with high precision and obtain an average of 95.5 % un-der condition where perfect tunneling occurs. The conditionof perfect tunneling can be smoothly tuned by adjusting thecurvature of the quadratic sweep. These geometrical effectsare robust beyond the framework of the existing theory [24].This geometric diabatic control is ubiquitous and can be ap-plied to various quantum systems.As a geometric diabatic control, we aim to realize perfecttunneling (P = 1) and change the state at the same time. TheHamiltonian for the TLZ model in the natural units is definedas [24],Ĥ = b · σ̂ = mσ̂x + vqσ̂y +12κ‖v2q2σ̂z , (1)where σ̂j (j = x, y, and z) is the Pauli operator, b =(bx, by, bz) ≡ (m, νq, 1/2k‖ν2q2) is a driving field. Wechange the parameter q in time as q = −F (t − T/2) be-tween time t = 0 and t = T with a dimensionless sweepspeed F . This is a quadratic version of the original TLZmodel [3], and ∆ = 2m is the gap, 2v(> 0) is the energyslope. Figure 1(b) depicts the initial and final fields as a solidred arrow (t = 0) and a dotted red arrow (t = T ), respec-tively. The bz component, which depends quadratically ontime, induces a “twist” of the field. This twist appears inthe trajectory of the field [the solid red line in Fig. 1(b)], andits strength is determined by the geodesic curvature κ‖. Sit-uations in which the spin and driving field are always keptparallel or antiparallel are adiabatic; situations that deviatefrom this are diabatic. The diabatic geometric effect is cap-2F01PF01P|F|F  0 (spin flip)P = 0P = 1xyxyF FPTP ~ 1F = FPT (spin flip)precession at minimaPy y(a) t = 0t = Tmgap minima (t = T/2) LZTLZcurvature||x yz(b)(c) (d)(e) (g)TLZ(|| > 0)LZ(|| = 0)(f)(h)FPTTimeEnergyPPgap | >| > | >| >| >| >| >| >LZ TLZdiabaticadiabatic00z zx xFIG. 1. Comparison of the Landau-Zener (LZ) transition and thetwisted Landau-Zener (TLZ) transition. (a) Tunneling at level anti-crossing. (b) Sweeping of the driving field. The fields at t = 0and t = T for the LZ/TLZ model are indicated by solid and dashedblue/red arrows, respectively. (c,d) Predicted LZ transition probabil-ity (c) and TLZ transition probability (d) as a function of the speedF . (e,f) Dynamics of the field (black arrow) and spin (blue arrow)in the LZ model in the adiabatic (e) and diabatic (f) limits [Also see(b,c)]. (g,h) Dynamics of the field (black arrow) and spin (red arrow)in the TLZ model at F = FPT (g) and F = −FPT (h) [Also see(b,d)]. The solid black line indicates the field amplitude in the xyplane in (e,f) and in the yz plane in (g,h). The origin of each arrowcorresponds to the field amplitude at each instant.tured by the geometric amplitude factor [3] (also known asthe quantum geometric potential [17] or shift vector [25])R12(q) = −A11(q)+A22(q)+∂qargA12(q), where the Berryconnection is defined by Anl(q) = 〈n(q)| i∂q |l(q)〉 usingthe instantaneous eigenstate |n(q)〉 satisfying Ĥ(q) |n(q)〉 =En(q) |n(q)〉. The tunneling probability P from |1〉 to |2〉 isgiven by [24],P ≈ exp[−π4v|F |(∆+FR12(0)2)2], (2)where R12(0) = vκ‖ holds in the present model. Equa-tion (2), referred to as “TLZ formula” in this work, is de-rived using a twisting coordinate transformation [24], and itrecovers the LZ formula when κ‖ = 0 [Fig. 1(c)]. We stress(b)(a)fdetfRmw0 TInit. Read.tTransition{ { (c)0.0 0.5 1.00.91.0P||95.5±1.3%F = FPT0.00.51.0Exp.TLZ Eq.(2)LZ0.00.51.0PExp.LZ-1.0 -0.5 0.0 0.5 1.0F0.00.51.0Exp.TLZ Eq.(2)LZ||LZ||TLZ||TLZFIG. 2. Demonstration of the TLZ transition at m = 0.5 MHz. (a)Measurement sequence. Laser and microwave pulses are used forinitialization and readout of the NV center. (b) Dependence of tun-neling probability P on speed F . The square and dots indicate ex-perimental result, solid black lines indicate the TLZ formula [Eq.(2)],and vertical dotted black lines indicate F = FPT. The blue dashedlines at the top and bottom panels indicate the LZ formula [TLZ withR12(0) = 0]. (c) Tunneling probability at F = FPT in the range ofκ‖ = 0 µs to κ‖ = 1 µs. The error bars indicate 65 % confidenceintervals estimated from the shot noise of PL measurement.that TLZ formula is approximate in contrast to the LZ formulawhich is asymptotically exact. Figure 1(d) shows the behaviorof the transition described by the TLZ formula when κ‖ > 0.P is nonreciprocal to the sign reversal of the speed F corre-sponding to the field sweep direction [25]. In Eq. (2), the gap∆ in the LZ model is effectively shifted to ∆ + FR12(0)2 bythe geometric amplitude factor [17, 24]. In particular, whenthe speed is,FPT = −2∆/R12(0) (3)the effective gap closes, and the tunneling probability satu-rates P ≈ 1. We call this behavior “perfect tunneling” [24],and the speed at which P is maximized is referred to as the“PT condition”. In contrast to the LZ case, the quantum statechanges during the diabatic transition from the initial state|1(q = FT/2)〉 to the final state |2(q = −FT/2)〉, and thusallows us to realize geometric diabatic control of the quantumstates. Our main purpose is to extensively test the behaviorspredicted by the TLZ formula [Eq. (2)].We realize the TLZ transition with an electron spin of asingle nitrogen-vacancy (NV) center in a diamond [12, 13,34]. We use the NV center’s mS = 0 and mS = −1 states as a3two-level system and manipulate it with microwave pulses. Ina suitable rotating frame [see Supplementary Materials (SM)],the Hamiltonian is expressed as (Ŝi: S=1/2 spin operators),Ĥr = fR[cos(φmw)Ŝx − sin(φmw)Ŝy]+d(fdett)dtŜz, (4)where fR is the Rabi frequency corresponding to the mi-crowave field amplitude, φmw is the microwave phase, andfdet is the detuning between the resonance frequency andthe microwave frequency. We generate a microwave pulsesatisfying fR =√b2x + b2y , φmw = − atan(by/bx), andfdet =∫ t0bz(t′)dt′, so that Eq. (4) reproduces the driving fieldb in the TLZ Hamiltonian [Eq. (1)]. This conversion to theS=1/2 system in MKS units corresponds to making the fol-lowing changes to each parameter; m → πm, v → πv, andκ‖ → κ‖/π (see SM). We adjust the sweep durationT consid-ering the coherence time and available microwave parameterranges. Figure 2(a) shows the measurement sequence. We usegreen laser pulses and photoluminescence (PL) measurementsfor spin initialization and readout. We prepare the initial andfinal states using rectangular microwave pulses after and be-fore the laser pulse to match the instantaneous field directionwith the projection direction. The obtained PL intensity isprecisely converted to a tunneling probability using referencePL intensities of mS = 0 state and mS = −1 state [35].We show our experimental results obtained when the gapparameter is fixed as m = 0.5 MHz. Without loss of gener-ality, we investigate the probability P [Eq. (2)] by selectingthe energy slope v to (10 MHz)2 and adjusting only the di-mensionless speed F . First, we set κ‖ = 0 µs to address theconventional LZ model. The blue dots in the middle panel ofFig. 2(b) show the experimental result. The lower the speed(F → 0), the lower the transition probability P , and the be-havior is symmetric between positive and negative speeds. Itagrees well with the LZ formula (solid black line), and provesthat our system reproduces the LZ model with high accuracy(for more detail, see SM).We then address the TLZ transition when κ‖ = +0.2 µsshown in the top panel of Fig. 2(b). The experimental result(red dots) is asymmetric in F → −F and becomes higher forF < 0 than for F > 0. P reaches maxima in the vicinity ofthe predicted PT condition (F = FPT) indicated by the ver-tical dashed line. Specifically, as shown in Fig. 2(c), we findP = 95.5 ± 1.3 %, on average, in a range of κ‖ = 0.0 µsto κ‖ = 1.0 µs. The bottom panel of Fig. 2(b) shows the re-sults when κ‖ = −0.2 µs. Compared to the κ‖ = +0.2 µscase [the top panel of Fig. 2], it shows totally inverted behav-ior to the speed F . These behaviors are qualitatively differentfrom the LZ transition (dashed blue line), and well reproducedby the TLZ formula without any adjustable parameters (solidblack line). These are our central results, proving that the tun-neling probability is successfully modulated by the geodesiccurvature κ‖ of the driving field, resulting in perfect tunnelingand nonreciprocity. The fact that perfect tunneling, which hasonly been possible in the extreme fast speed limits of the LZmodel, is achieved even at finite speeds is essentially new inthe long history of the LZ physics.Here, we give an intuitive picture of the perfect tunnel-ing phenomenon. Figure 1(g) shows the driving field (blackarrow) and spin (red arrow) dynamics at F = FPT. Thequadratic sweep produces adiabatic dynamics in the initialstage (t ∼ 0), and diabatic dynamics near the gap minima(t ∼ T/2). Near the gap minima, the x component of the driv-ing field b, i.e. the gap itself (bx = m), causes spin precessionand rotates the spin around the x-axis. When the PT conditionis fulfilled, this rotation of the spin is synchronized with thecounterclockwise twist of the field (also around the x-axis),and the transition to the excited state is achieved smoothly.Thus a spin flipping is realized [Fig. 1(g)]. When the sweepdirection is reversed (F = −FPT), as shown in Fig. 1(h), theclockwise field twist cannot synchronize with the spin preces-sion. This geometric motion near the gap minima increasesthe effective gap ∆+ FR122 and prevents tunneling. More gen-erally, the observed nonreciprocity is analogous to the well-known selective absorption of circularly polarized light, butin the non-perturbative regime.As described above, the spin flips during the perfect tun-neling. In terms of quantum control, spin-flip can also beachieved differently using the Rabi oscillation and the adia-batic control (or its shortcut[36]). The driving field and spinare orthogonal, parallel, and antiparallel in the Rabi oscilla-tion, the adiabatic control, and the TLZ model, respectively.This difference in the restriction of the driving field to thespin direction makes difference in control speed, robustness,and implementability. Our geometric diabatic control is an ef-fective means of increasing the versatility of quantum control(see SM).Next, we study the validity of the TLZ formula [Eq. (2)]when the twist becomes stronger; the higher-order terms ig-nored in the derivation of the TLZ formula increase, andthe precession is no longer perfectly synchronized with thequadratic twist. We investigate the tunneling probability ob-tained at m = 0.5 MHz for a curvature range from κ‖ = 0 µsto κ‖ = 3 µs. The left middle panel of Fig. 3(a) shows the ex-perimental result, representing a clear nonreciprocal behaviorto the speed F . As κ‖ increases, the PT condition approacheszero. A similar trend is observed in the TLZ formula shownin the left top panel of Fig. 3(a), indicating that this character-istic is consistent with FPT = − 2∆R12(0). This result proves thatthe speed of the quantum control is tunable by the geodesiccurvature κ‖ of the driving field.For a more quantitative comparison, we show a cross-section at κ‖ = 1.4 µs in the left panel of Fig. 3(b) [white linein the middle panel of Fig. 3(a)]. The experimental result (reddot) exhibits P ∼ 1 near FPT = −0.045 in good agreementwith the TLZ formula (solid black line). On the other hand, in(negatively) large speeds F < FPT, P decreases almost expo-nentially in the TLZ formula [24], whereas the change is grad-ual in the experimental result. This deviation becomes promi-nent as the gap parameterm and/or the curvatureκ‖ are larger.The right panels of Figs. 3(a,b) show the corresponding data4(b)(a) m = 0.5 MHz m = 2.0 MHz-1 0 10.00.51.0PTLZ Eq.(2)Exp .Sim.-1 0 10123TLZ Eq.(2)0 1-1 0 10123Exp.-1 0 1F0123Sim.-1 0 1-1 0 1TLZ Eq.(2)-1 0 1Exp.-1 0 1FSim.|| (|| (|| (|| = 1.4 FIG. 3. The gap parameter and curvature dependence of the TLZtransition probability. (a) The left (right) panels denote the resultsat m = 0.5 MHz (m = 2.0 MHz). The solid black (dashed green)line indicates the PT condition in the TLZ formula (simulation). (b)Tunneling probability at κ‖ = 1.4 µs [white line in the middle panelof (a)]. The black arrow in the right panel indicates the PT condition.sets obtained with a larger gap parameter (m = 2.0 MHz).The PT condition in the experiment (red dot) shifts to the leftfrom what the TLZ formula (solid black line) predicts [blackarrow on the right panel of Fig. 3(b)]. The maximumP is thenslightly suppressed from unity.We obtain exact solutions by numerical simulations (seeSM) to discuss this deviation. The simulation results arethe bottom panels of Fig. 3(a) and the dashed green lines inFig. 3(b). They reproduce the experimental results satisfacto-rily over the entire speed range. The solid black and dashedgreen lines in Fig. 3(a) show the perfect tunneling conditionsobtained by the TLZ formula and the simulation, respectively.The results show that the larger gap parameter m and cur-vature κ‖ become, the exact PT condition shifts toward the(negatively) large speed side. Our precise measurements re-veal that the higher-order terms are essential for quantitativelyunderstanding the TLZ transition.As shown above, we find that nonreciprocity and high tun-neling probability at finite speed always persist even when theTLZ formula is invalid. Thus, we conclude that these geo-metric effects are robust. Introducing a field twist can be aGapless (m = 0.0 MHz)(a)(b)-1.0 -0.5 0.0 0.5 1.00.00.51.0PExp.LZ Eq.(2)Sim.-1.0 -0.5 0.0 0.5 1.0F0.00.51.0PExp.TLZ Eq.(2)Sim.||||LZTLZgaplesstEnergyFIG. 4. Sweep speed dependence of transition probability of the gap-less (m = 0.0 MHz) system. (a) The LZ transition (κ‖ = 0). Theinset is a schematic of the energy change. (b) The TLZ transition(κ‖ = 2.5 µs).ubiquitous method of adjusting tunneling probabilities at arbi-trary speeds, making the present TLZ model a new frameworkfor quantum control at various energy scales. When appliedto quantum materials, such control induces nontrivial proper-ties such as the nonreciprocity of DC current and photocurrent[24, 25].In the case of an infinitesimal gap (m = 0.0 MHz), theTLZ formula predicts a counter-intuitive behavior, i.e., tun-neling is suppressed as we increase the speed. Since this isrelevant to the study of laser field driven dynamics in Diracand Weyl semimetals [24], we study this situation in detail.The energy change is shown in the inset of Fig. 4 (a), whichmimics the situation where electrons in the valence band ac-celerated by the electric field are excited through the Dirac(Weyl) point into the conduction band. Here, the LZ modeland the TLZ model correspond to the case where the drivingfields are DC and AC electric fields, respectively. We examinethe LZ model and observe that it yields P ∼ 1, as shown inFig. 4 (a). This is a straightforward phenomenon caused bythe complete reversal of the field in the y axis. We then exam-ine the TLZ transition at κ‖ = 2.5 µs as in Fig. 4(b). The hightunneling probability near the adiabatic limit F ∼ 0 is con-sistent with FPT = − 4πmvκ‖= 0 (for m = 0). This behavior,where the probability decreases with increasing sweep speed,is opposite to the LZ transition at a finite gap [Fig. 2(b)]. Thiscounter-intuitive result is caused by the monocyclic nature ofthe quadratic twist, where the initial and final fields point inthe same direction. It is qualitatively reproduced by the TLZformula (solid black line) and is perfectly reproduced in thesimulation (dashed green line).We experimentally confirmed the nonadiabatic geometriceffects of nonreciprocity and perfect tunneling in the quadraticTLZ model over a wide range of parameters. Specifically, we5showed that we could utilize the geometric effects to controlthe quantum state dynamically. Geometric diabatic controlcan be applied to control systems of various energy scales,from nuclear spins to quantum materials. An important chal-lenge to improving this method is to find a way to enhancethe tunneling probability and bring it even closer to 100 %.We think this is possible by engineering the shape of the fieldtwist to cancel the higher-order terms ignored in the derivationof the TLZ formula.We thank K. M. Itoh (Keio University) for letting us to usethe confocal microscope system, and MEXT-NanotechnologyPlatform Program “Microstructure Analysis Platform” fortechnical support. This work was supported by JSPSGrants-in-Aid for Scientific Research (Nos. JP22K03524,JP19H00656, JP19H05826, JP20H02187, and JP20H05661),JST CREST (JPMJCR19T3 and JPMJCR1773), MEXTQ-LEAP (JPMXS0118068379), JST Moonshot R&D (JP-MJMS2062), and MIC R&D for construction of a globalquantum cryptography network (JPMI00316).∗ kento.sasaki@phys.s.u-tokyo.ac.jp† kensuke@phys.s.u-tokyo.ac.jp[1] F. Wilczek and A. Shapere, Geometric Phases in Physics(WORLD SCIENTIFIC, 1989).[2] M. V. Berry and M. Wilkinson, Diabolical points in the spectraof triangles, Proc. Math. Phys. Eng. Sci. 392, 15 (1984).[3] M. V. Berry, Geometric amplitude factors in adiabatic quantumtransitions, Proc. Math. Phys. Eng. Sci. 430, 405 (1990).[4] L. D. Landau, Zur theorie der energiebertragung ii, Phys. Z.Sow. 2, 46 (1932).[5] C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc.Lond. A 137, 696 (1932).[6] E. C. G. Stückelberg, Theorie der unelastichen stsse zwischenatomen, Helv. Phys. Acta 5, 369 (1932).[7] E. Majorana, Atomi orientati in campo magnetico variabile,Nuovo Cimento. 9, 43 (1932).[8] S. N. Shevchenko, S. Ashhab, and F. Nori, Landau–Zener–Stückelberg interferometry, Phys. Rep. 492, 1 (2010).[9] O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, Nonadiabaticlandau–zener–stückelberg–majorana transitions, dynamics, andinterference, Physics Reports 995, 1 (2023).[10] W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, HanburyBrown and Twiss-type experiment with electrons, Science 284,299 (1999).[11] J. R. Petta, H. Lu, and A. C. Gossard, A coherent beam splitterfor electronic spin states, Science 327, 669 (2010).[12] P. Huang, J. Zhou, F. Fang, X. Kong, X. Xu, C. Ju, and J. Du,Landau-Zener-Stückelberg interferometry of a single electronicspin in a noisy environment, Phys. Rev. X 1, 011003 (2011).[13] J. Zhou, P. Huang, Q. Zhang, Z. Wang, T. Tan, X. Xu, F. Shi,X. Rong, S. Ashhab, and J. Du, Observation of time-domainRabi oscillations in the Landau-Zener regime with a single elec-tronic spin, Phys. Rev. Lett. 112, 010503 (2014).[14] S. Gasparinetti, P. Solinas, and J. P. Pekola, Geometric Landau-Zener interferometry, Phys. Rev. Lett. 107, 207002 (2011).[15] J. Zhang, J. Zhang, X. Zhang, and K. Kim, Realization of geo-metric Landau-Zener-Stückelberg interferometry, Phys. Rev. A89, 013608 (2014).[16] L. Wang, T. Tu, B. Gong, C. Zhou, and G.-C. Guo, Experimen-tal realization of non-adiabatic universal quantum gates usinggeometric Landau-zener-stückelberg interferometry, Sci. Rep.6, 19048 (2016).[17] J. Wu, M. Zhao, J. Chen, and Y. Zhang, Adiabatic condition andquantum geometric potential, Phys. Rev. A 77, 062114 (2008).[18] C. Xu, J. Wu, and C. Wu, Quantized interlevel character inquantum systems, Phys. Rev. A 97, 032124 (2018).[19] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec-tronic properties, Rev. Mod. Phys. 82, 1959 (2010).[20] Q. Ma, A. G. Grushin, and K. S. Burch, Topology and geometryunder the nonlinear electromagnetic spotlight, Nat. Mater. 20,1601 (2021).[21] D. N. Basov, R. D. Averitt, and D. Hsieh, Towards propertieson demand in quantum materials, Nat. Mater. 16, 1077 (2017).[22] J. Orenstein, J. E. Moore, T. Morimoto, D. H. Torchinsky, J. W.Harter, and D. Hsieh, Topology and symmetry of quantum ma-terials via nonlinear optical responses, Annu. Rev. Condens.Matter Phys. 12, 247 (2021).[23] A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W.McIver, and M. A. Sentef, Colloquium: Nonthermal pathwaysto ultrafast control in quantum materials, Rev. Mod. Phys. 93,041002 (2021).[24] S. Takayoshi, J. Wu, and T. Oka, Nonadiabatic nonlinear opticsand quantum geometryApplication to the twisted Schwinger ef-fect, SciPost Phys. 11, 075 (2021).[25] S. Kitamura, N. Nagaosa, and T. Morimoto, NonreciprocalLandau–Zener tunneling, Commun. Phys. 3, 63 (2020).[26] F. Gaitan, Temporal interferometry: A mechanism for control-ling qubit transitions during twisted rapid passage with possibleapplication to quantum computing, Phys. Rev. A 68, 052314(2003).[27] F. Gaitan, Controlling qubit transitions during non-adiabaticrapid passage through quantum interference, J. Mod. Opt. 51,2415 (2004).[28] R. Li, M. Hoover, and F. Gaitan, High-fidelity single-qubit gatesusing non-adiabatic rapid passage, Quantum Inf. Comput. 7,594 (2007).[29] R. Li and F. Gaitan, Robust high-fidelity universal set of quan-tum gates through non-adiabatic rapid passage, J. Mod. Opt. 58,1922 (2011).[30] J. W. Zwanziger, S. P. Rucker, and G. C. Chingas, Measuringthe geometric component of the transition probability in a two-level system, Phys. Rev. A 43, 3232 (1991).[31] D. Bouwmeestert, G. P. Karman, N. H. Dekker, C. A. Schrama,and J. P. Woerdman, Observation of the geometric amplitudefactor in an optical system, J. Mod. Opt. 43, 2087 (1996).[32] D. Bouwmeester, G. P. Karman, C. A. Schrama, and J. P. Wo-erdman, Observation of interference in transitions due to localgeometric phases, Phys. Rev. A 53, 985 (1996).[33] J. W. Zwanziger, U. Werner-Zwanziger, and F. Gaitan, Non-adiabatic rapid passage, Chem. Phys. Lett. 375, 429 (2003).[34] G. Fuchs, G. Burkard, P. Klimov, and D. Awschalom, A quan-tum memory intrinsic to single nitrogen–vacancy centres in di-amond, Nat. Phys. 7, 789 (2011).[35] D. Misonou, K. Sasaki, S. Ishizu, Y. Monnai, K. M. Itoh, andE. Abe, Construction and operation of a tabletop system fornanoscale magnetometry with single nitrogen-vacancy centersin diamond, AIP Adv. 10, 025206 (2020).[36] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui,S. Martı́nez-Garaot, and J. G. Muga, Shortcuts to adiabatic-ity: Concepts, methods, and applications, Rev. Mod. Phys. 91,045001 (2019).[37] V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen,6J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, andJ. Wrachtrup, Dynamic polarization of single nuclear spins byoptical pumping of nitrogen-vacancy color centers in diamondat room temperature, Phys. Rev. Lett. 102, 057403 (2009).[38] A. Dréau, M. Lesik, L. Rondin, P. Spinicelli, O. Arcizet, J.-F.Roch, and V. Jacques, Avoiding power broadening in opticallydetected magnetic resonance of single NV defects for enhanceddc magnetic field sensitivity, Phys. Rev. B 84, 195204 (2011).