shortcuts to dynamic polarizationshortcuts to dynamic polarization tamiro villazon, 1pieter w....
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Shortcuts to Dynamic Polarization
Tamiro Villazon,1 Pieter W. Claeys,2, โ Anatoli Polkovnikov,1 and Anushya Chandran11Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA
2TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK
Dynamic polarization protocols aim to hyperpolarize a spin bath by transferring spin polarization from a well-controlled qubit such as a quantum dot or a color defect. Building on techniques from shortcuts to adiabaticity, wedesign fast and efficient dynamic polarization protocols in central spin models that apply to dipolarly interactingsystems. The protocols maximize the transfer of polarization via bright states at a nearby integrable point, exploitthe integrability-breaking terms to reduce the statistical weight on dark states that do not transfer polarization, andrealize experimentally accessible local counterdiabatic driving through Floquet-engineering. A master equationtreatment suggests that the protocol duration scales linearly with the number of bath spins with a pre-factor thatcan be orders of magnitude smaller than that of unassisted protocols. This work opens new pathways to cool spinbaths and extend qubit coherence times for applications in quantum information processing and metrology.
I. INTRODUCTION
A prevalent goal in several fields of physics and chemistryis to efficiently polarize an ensemble of spin particles. Innuclear magnetic resonance spectroscopy (NMR) and mag-netic resonance imaging (MRI), polarizing nuclear spins en-hances sensitivity and resolution [1โ4]. In applications toquantum information processing, hyperpolarization schemescan be used to initialize large-scale quantum simulators [5] orto extend qubit coherence times by cooling the surroundingspin bath [6, 7]. Where costly or difficult to polarize the spinensemble directly, dynamic polarization protocols have beendeveloped to repeatedly transfer polarization from readily po-larized control spins [1, 8โ14]. In simple experimental setups,a spin bath is polarized by controlling a single qubit, suchas a nitrogen vacancy (NV) center in diamond [15โ17] or aquantum dot [18โ20], whose polarization can be repeatedlyreset, effectively generating a zero temperature reservoir forthe bath [21]. A key goal of this article is to introduce a fastand efficient scheme for dynamic polarization in central spinmodels.
Polarization transfer relies on the spin-flip interactions be-tween a control spin and the spin ensemble to be polarized.The Hamiltonian can be schematically represented as
๐ป = ฮฉ(๐ก) ๐๐ง + ๐ปspin-flip , (1)
consisting of an electromagnetic field ฮฉ(๐ก) acting on the con-trol spin along the z-direction and spin-flip interactions be-tween control spin and spin bath. Given an initially polarizedcontrol spin, ฮฉ(๐ก) can be tuned to transfer polarization [22โ24].Specifically, dynamic polarization protocols can be separatedinto two classes: (i) sudden protocols in which the control fieldฮฉ(๐ก) is quenched to resonance with the spin-flip interactionsto induce polarization transfer and (ii) adiabatic protocols inwhich polarization transfer is induced by slowly driving ฮฉ(๐ก)across resonances [25]. Adiabatic protocols offer an advantageover sudden protocols as they do not require precise resonancetuning and pulse timing. They also can cover a broader range of
bath spin resonances, enabling robust transfer in the presenceof field and interaction inhomogeneities [26โ28]. Their maindisadvantage is the requirement of slow speeds, which can beinefficient or unfeasible in experiments limited by spin diffu-sion in the bath and decoherence of the control spins [29, 30].
Apart from the limitations on control speeds, the achievablepolarization is also limited by the presence of dark states, mak-ing it seldom possible to completely polarize the bath evenat slow speeds [21, 31โ34]. Dark states are many-body qubit-bath eigenstates in which the qubit is effectively decoupledfrom the bath. Since such states have a fixed control spin po-larization and cannot be depopulated through changes in thequbit control field, any initial nonzero population of dark stateswill limit hyperpolarization. Experiments in different materialsystems have found maximum saturation at about 60% fullpolarization [35โ37].
Several schemes have been proposed to enhance hyperpo-larization by depopulating dark states effectively [20], for ex-ample by modulating the electron wavefunction of the qubit inquantum dots [21, 31] or by alternating resonant drives whichreduce quantum correlations in the bath [38]. While studies sofar mainly focused spin systems where the central spin interactswith its environment through fully isotropic (XXX) interac-tions, arising in e.g. quantum dots in semiconductors, weconsider a model where the interactions are anisotropic (XX),as in resonant dipolar spin systems [17, 19, 23, 24, 38โ40].
Overcoming the requirement of slow speeds in adiabaticprotocols is the aim of the field of shortcuts to adiabaticity[41, 42]. Shortcut methods such as counterdiabatic driving(CD) suppress diabatic transitions between the eigenstates of adriven Hamiltonian ๐ป (๐ก) by evolving the system with a Hamil-tonian ๐ป๐ถ๐ท (๐ก) containing additional counter terms [43โ49].CD preserves the systemโs adiabatic path through state spaceeven during ultra-fast protocols. CD protocols typically requireengineering operators which are highly complex and many-body, making them difficult to implement in practice [49].Recent progress has focused on reducing the complexity ofCD Hamiltonians, for example by mapping them to simplerunitary equivalents [50โ53], or by approximating them with lo-cal (few-body) operators [54โ56]. The required local operatorscan be realized through e.g. Floquet-engineering techniques,using high-frequency oscillations to realize the CD Hamilto-
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nian as an effective high-frequency Hamiltonian using onlycontrols present in the original adiabatic protocol [55, 57โ60].Local counterdiabatic driving has recently been realized ex-perimentally in synthetic tight-binding lattices [61], in IBMโssuperconducting quantum computer [62], and in a liquid-stateNMR system for a nonintegrable spin chain [63]. Such meth-ods have also gained attention in the context of quantum ther-modynamics, where (approximate) CD can be used to speed upunderlying adiabatic processes and increase the performanceof quantum engines [58, 64โ71].
We develop a dynamic polarization scheme which imple-ments approximate counterdiabatic driving (CD) to quicklyand efficiently polarize a spin bath using a tunable qubit whilesimultaneously depopulating dark states. In the absence ofinhomogeneous bath fields, the model Hamiltoniani is inte-grable [34]. We first exploit the integrability of this modelto design a CD protocol explicitly targeting all polarization-transferring bright (i.e., not dark) states. Within all protocolsthe bright states arise in pairs acting as independent two-levelLandau-Zener systems, for which CD protocols can be straight-forwardly designed. While the exact CD protocol targets allbright states and gives rise to a highly involved control Hamilto-nian, we show how the CD protocol can be well approximatedusing local (few-body) operators and experimentally imple-mented using Floquet engineering (FE). In the presence ofinhomogeneous bath fields the system is no longer integrable.However, the proposed protocols still lead to a remarkableincrease in transfer efficiency. Furthermore, the local counter-diabatic (LCD) protocol dynamically couples dark states tobright states, such that dark states can be depopulated. Notonly are such LCD protocols much easier to implement thanthe exact CD ones, we find that they outperform CD protocolsand lead to a complete hyperpolarization of the spin bath.
The FE protocols also lead to natural quantum speed limits:there exists a lower bound for the protocol durations belowwhich the FE protocol can no longer accurately mimic theLCD protocol. The emergence of speed limits is ubiquitousin shortcut protocols and control theory [42, 49, 52, 71โ78].Interestingly, our work now suggests that speed limits are alsointrinsic in approximate local counterdiabatic protocols.
This paper is organized as follows. In Section II, we presentthe qubit-bath model system and the hyperpolarization schemeused throughout this work. In Section III, we construct anddetail the CD and LCD protocols and compare their efficiencywith unassisted (UA) protocols which do not use shortcut meth-ods. In Section IV, we show how our shortcut protocols canbe applied to fully polarize a spin bath. A master equationfor the hyperpolarization is introduced in Section V, which isused to analyze the protocols at large system sizes and showthat all protocol durations scale linearly with the number ofbath spins. In Section VI, we show how to realize LCD withFE and discuss the emergence of a quantum speed limit. Weconclude in Section VII with a discussion of our results in abroader context.
II. MODEL AND HYPERPOLARIZATION SCHEME
A. Hamiltonian
We focus on a concrete central spin model describing a qubitinteracting with ๐ฟ โ 1 spin-1/2 bath spins. The Hamiltonian isgiven by
๐ป (๐ก) = ฮฉ๐ (๐ก) ๐๐ง0+๐ฟโ1โ๏ธ๐=1
ฮฉ๐ต, ๐ ๐๐ง๐+ 1
2
๐ฟโ1โ๏ธ๐=1๐ ๐
(๐+0๐
โ๐ +๐โ0 ๐
+๐
), (2)
where ฮฉ๐ (๐ก) is the magnetic field strength on the qubit, ฮฉ๐ต, ๐is the magnetic field strength on the ๐ th bath spin, and ๐ ๐ is thecoupling strength between the qubit and the ๐ th bath spin, with๐ = 1, 2, . . . , ๐ฟ โ1. Eq. (2) describes several physical setups inrotating frames, such as color defects or quantum dots coupledto ensembles of nuclear spins via dipolar interactions [17, 19,23, 24, 38]. Spin conserving (โflip-flopโ) transitions dominatethe dipolar interaction provided ๐ ๐ ๏ฟฝ ฮฉ๐ + ฮฉ๐ต, with thelatter set by the amplitudes of the continuous driving fields; astandard derivation is given in Appendix A. The top panel ofFig. 1 shows a schematic of the model.
Experimentally, the bath field and qubit-bath couplings arespatially inhomogeneous. For simplicity, we model these in-homogeneities as uncorrelated disorder: we draw each ฮฉ๐ต, ๐independently from a uniform distribution
ฮฉ๐ต, ๐ โ [ฮฉ๐ต โ ๐พ๐ง ,ฮฉ๐ต + ๐พ๐ง], (3)
where ฮฉ๐ต sets the mean value and ๐พ๐ง sets the z-disorderstrength. We also draw each ๐ ๐ independently from a uniformdistribution
๐ ๐ โ [๐ โ ๐พ๐ฅ๐ฅ , ๐ + ๐พ๐ฅ๐ฅ], (4)
where ๐ sets the mean value and ๐พ๐ฅ๐ฅ sets the xx-disorderstrength. In this work, we probe the weak coupling and disor-der regime given by ๐พ๐ฅ๐ฅ , ๐พ๐ง < ๐ ๏ฟฝ ฮฉ๐ต.
Since ๐ป conserves total magnetization [๐ป, ๐] = 0, where
๐ โก๐ฟโ1โ๏ธ๐=0
๐๐ง๐, (5)
its eigenspectrum splits into ๐ฟ + 1 polarization sectors (seeleft of lower panel in Fig 1). Each sector can alternativelybe specified by the number ๐ = ๐ + ๐ฟ/2 of spin flips abovethe fully-polarized state |โใ โ |โโ . . . โใ. The aim of hyperpo-larization is then to find protocols that systematically reduce๐, where a fully polarized state corresponds to ๐ = 0 or๐ = โ๐ฟ/2.
B. Spectrum
The eigenstates of ๐ป capture essential features common inapplications of dynamic polarization: bright states which allowresonant polarization transfer when ฮฉ๐ (๐ก) is varied, and darkstates which limit transfer. In the ๐พ๐ง = 0 limit, the model is
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FIG. 1. Model schematic, spectrum, and hyperpolarizationscheme. (Top panel) Schematic of the central spin model in Eq. (2).(Bottom panel) On the left, we illustrate the polarization sectors andspectrum for a system with an even number ๐ฟ of spins and smalldisorder strengths. On the right, we illustrate the spectrum in twopolarization sectors with ๐ < 0, ฮmax sets the maximal width of theresonance region, where together with an outline of the transfer-resethyperpolarization scheme.
integrable and the structure of its eigenstates is known [34].While our proposed protocols are not restricted to integrablemodels, the known eigenstate structure at the integrable pointallows for a quantitative understanding of the general coolingprotocols. We briefly review these eigenstates and their basicproperties in the integrable limit, and subsequently extend thediscussion to ๐พ๐ง > 0.
1. Bright States (๐พ๐ง = 0)
Bright eigenstates can be written as
|B๐ผ (๐)ใ = ๐๐ผโ (๐) |โใ โ |B๐ผโ ใ + ๐
๐ผโ (๐) |โใ โ |B
๐ผโ ใ , (6)
where
๐(๐ก) โก ฮฉ๐ (๐ก) โฮฉ๐ต (7)
measures the detuning between the qubit and bath fields. Onresonance, ฮฉ๐ = ฮฉ๐ต and ๐ = 0. The index ๐ผ distinguishesbetween the different bright states. Crucially, the bath states|B๐ผโ,โใ do not depend on ๐.
As such, when varying ๐ the bright states only couple inpairs (๐ผ = ยฑ๐), behaving as independent two-level Landau-Zener systems. The Hamiltonian in each such two-dimensionalsubspace can be written as
๐ป๐ผ (๐) = ๐๐๐ง๐ผ + ฮ๐ผ๐๐ฅ๐ผ +ฮฉ๐ต ๐, (8)
where ฮ๐ผ sets the energy splitting (gap) of the pair at resonance[34] and we have introduced generalized spin operators ๐๐ฅ,๐ฆ,๐ง๐ผacting on the two-dimensional space spanned by |B๐ผ+ ใ = |โใ โ|B๐ผโ ใ and |B
๐ผโ ใ = |โใ โ |B๐ผโ ใ.
๐๐ฅ๐ผ =12
(|B๐ผโ ใ ใB๐ผ+ | + |B๐ผ+ ใ ใB๐ผโ |
),
๐๐ฆ๐ผ =
๐
2(|B๐ผโ ใ ใB๐ผ+ | โ |B๐ผ+ ใ ใB๐ผโ |
),
๐๐ง๐ผ =12
(|B๐ผ+ ใ ใB๐ผ+ | โ |B๐ผโ ใ ใB๐ผโ |
).
๐๐ง๐ผ corresponds to ๐๐ง0 projected on a bright pair subspace. The
apparent simplicity of the problem in this subspace hides thecomplexity of the qubit-bath interactions present in the orig-inal spin basis, where the bath states |B๐ผโ,โใ and the gap ฮ๐ผare obtained by solving a set of nonlinear Bethe equations[34]. Within each magnetization sector ๐ = ๐ โ ๐ฟ/2, we la-bel bright state pairs by ๐ผ = |๐ |, where ๐ โ {1, 2, . . . , ๐๐ต}and
๐๐ต =
(๐ฟ โ 1๐ โ 1
), (9)
is the number of pairs in the sector for ๐ < 0 1.The Hamiltonian (8) returns the bright state energies
๐ธ ๐ผB (๐) = ฮฉ๐ต ๐ ยฑ12
โ๏ธ๐2 + ฮ2๐ผ . (10)
We refer to the set of bright states with positive๐ธ ๐ผB โฮฉ๐ต๐ > 0 as the top bright band, and those with neg-ative ๐ธ ๐ผB โฮฉ๐ต๐ < 0 as the bottom bright band (see red bandsin bottom panel of Fig. 1).
In bright states, polarization can be transferred between thequbit and the bath. At resonance (๐ = 0) the bright state pairsare fully hybridized with ๐ยฑ๐โ = ยฑ๐
ยฑ๐โ . Initializing the system
in a fully polarized central spin state and then quenching toresonance, as is done in sudden protocols, transfers polarizationon the timescale ฮโ1๐ผ . Adiabatic protocols induce a qubit-bathpolarization transfer in bright states by slowly varying ๐(๐ก)resonance. As ๐ โ ยฑโ, bright states approach a product formand the initial eigenstate |โใ โ |B๐ผโ ใ is adiabatically connectedto |โใ โ |B๐ผโ ใ and vice versa.
Each Landau-Zener problem is fully characterized by itsgap. The distribution of bright pair gaps at resonance ฮ๐ผ isshown in Fig. 2 for various polarization sectors. As shown in
1 Since the aim of the proposed protocols is to reduce magnetization, wefocus on ๐ < 0.
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Appendix B, the gap distribution can be obtained analyticallyat zero disorder (๐พ๐ฅ๐ฅ = 0) in the thermodynamic limit wherewe take ๐ฟ โ โ, holding ๐ โก |๐ |/๐ฟ fixed. Since these gapsset the necessary time scales for adiabatic protocols, we brieflydetail some relevant gap scales. The typical gap scale is givenby
ฮtyp โกโ๏ธโ๏ธ
๐2๐โผโ๐ฟ โ 1 ๐, (11)
shown as a gold vertical dashed line in Fig. 2, whereas themaximal gap, also setting the maximal width of the resonanceregion, is given by
ฮmax โผ ๐ฟ ๐, (12)
scaling extensively in ๐ฟ. The smallest bright gap in the homo-geneous model can be found as
ฮmin = ๐โ๏ธ
2(๐ + 1) โผ ๐โ
2๐ ๐ฟ. (13)
At sufficiently small gaps ฮ & ฮmin, the distribution of brightpair gaps is given by
๐(ฮ) โ ฮ(1 + 2๐1 โ 2๐
)โ(ฮ/ฮmin)2; ฮ โฅ ฮmin. (14)
This distribution is shown in Fig. 2 as a dashed black curve.We find good qualitative agreement between the analyticalcurve at ๐พ๐ฅ๐ฅ = 0 and our numerical results for small but finitedisorder ๐พ๐ฅ๐ฅ = 0.05 in the magnetization sector ๐ = โ1 withthe largest Hilbert space dimension.
Fig. 2 also shows the numerically obtained distribution ofbright pair gaps in the ๐ = โ4 and ๐ = โ7 sectors. Thedistribution in the ๐ = โ4 sector exhibits three broad peakswhich are centered around the three bright pair gap energiesin the ๐พ๐ฅ๐ฅ = 0 limit (Appendix B). As the width of each peakis proportional to ๐พ๐ฅ๐ฅ๐ฟ while the bright pair gaps at ๐พ๐ฅ๐ฅ = 0are order one, we expect the three-peak structure to be washedout at larger ๐ฟ, and the distribution to be captured by Eq. (14)instead. In the ๐ = โ7 sector, we expect a single pair ofbright states with pair gap โ ฮtyp at small ๐พ๐ฅ๐ฅ , as confirmedby Fig. 2. We note that Eq. (14) only applies to sectors withfinite magnetization density at large ๐ฟ.
The main difference comes from the non-zero density ofgaps for ฮ < ฮmin. However, as will be shown in followingsections, this non-zero density does not qualitatively influenceour protocols.
In sum, the bright bands consist of an ensemble of indepen-dent Landau-Zener systems with a non-trivial distribution ofgaps. For each bright pair, an adiabatic passage of ๐ across res-onance prevents excitations across its gap and flips polarizationof the qubit, transferring polarization to the bath.
2. Dark States (๐พ๐ง = 0)
Dark states have the following product form with the centralqubit fully polarized:
|D๐ผใ = |โใ โ |D๐ผโ ใ or |D๐ผใ = |โใ โ |D๐ผโ ใ , (15)
FIG. 2. Distribution of bright pair gaps at resonance. Histogramof number n(ฮ๐ผ) of bright state pairs with gap ฮ๐ผ. Data is shown fora typical disorder realization in multiple polarization sectors. The goldvertical dashed line denotes the typical gap ฮtyp. The black dashedcurve denotes the distribution of gaps from Eq. (14). Parameters:๐ฟ = 16, ๐ = 0, ๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0, and 60 bins.
where the index ๐ผ distinguishes between the different darkstates. The bath states |D๐ผโ,โใ depend implicitly on {๐ ๐ }, butcrucially not on ๐, and are obtained by solving a set of โdarkโBethe equations [34].
In a given polarization sector ๐ = ๐ โ ๐ฟ/2, there are
๐๐ท =
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ(๐ฟ โ 1๐ ) โ (๐ฟ โ 1๐ โ 1)๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ (16)dark states. Dark states with central qubit polarized along +๐งonly exist in sectors ๐ > 0, while dark states with central spinpolarization along โ๐ง only exist in sectors ๐ < 0, and no darkstates exist in the sector ๐ = 0. Dark state are eigenstates of ๐๐ง0and are annihilated by the interaction part of the Hamiltonian[34], such that the energies given by
๐ธ ๐ผD (๐) = ฮฉ๐ต ๐ + sgn[๐]๐
2, (17)
change linearly with the qubit field detuning ๐. Their wavefunctions however do not change with ๐, preventing polariza-tion transfer to the bath.
3. Bright & Dark States (๐พ๐ง > 0)
In the presence of z-disorder (๐พ๐ง > 0), the system is notintegrable. However, the same qualitative picture for the eigen-states holds: on adiabatically changing the detuning ๐ andcomparing the polarization of the central spin far away fromresonance (๐ = ยฑโ), there exists a subset of โbright statesโ inwhich the polarization is changed and a subset of โdark statesโfor which the polarization is unchanged.
Since the central spin is polarized far away from resonance,a counting argument can be used to determine the number ofbright and dark states. Consider a sector with magnetization
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๐ < 0 and dimension ๐๐ =( ๐ฟ๐
): there are ๐โ =
(๐ฟโ1๐
)states
in which the qubit is fully polarized along the โ๐ง-direction and๐โ =
( ๐ฟโ1๐โ1
)states in which the qubit is fully polarized along
the +๐ง-direction. Far from resonance, the energies are given byฮฉ๐ต๐ โ ๐/2 and ฮฉ๐ต + ๐/2 respectively. Comparing the totalnumber of states in the top and bottom band far away fromresonance, there must be ๐๐ท = ๐โ โ ๐โ dark states in whichthe polarization of the qubit does not flip for an adiabaticpassage across resonance (see also Fig. 1). The remaining๐๐ โ ๐๐ท = 2 ๐โ states are bright states in which the spin of thequbit flips during such an adiabatic process, consistent withEqs. (9) and (16). This simple counting argument only usesconservation of total z-magnetization and produces the samequalitative eigenstate band structure as one sthe Bethe ansatzin the integrable limit (๐พ๐ง = 0) [34].
There are, however, important differences between the in-tegrable (๐พ๐ง = 0) and non-integrable (๐พ๐ง > 0) models in theresonance regime. When ๐พ๐ง > 0, the simple product state struc-ture of dark states and the Landau-Zener structure of brightstates is no longer exact: the non-integrable eigenstates aremixtures of the unperturbed states and exhibit ergodic behavior(Appendix E).
While adiabatic protocols transferring polarization in theinhomogeneous model are qualitatively similar to those inthe homogeneous model, and bright and dark states can gen-erally be defined by their central spin polarization far awayfrom resonance, non-adiabatic effects can enhance polariza-tion transfer in the inhomogeneous model. Finite z-disorderis useful for the purposes of dynamic polarization: in a non-adiabatic protocol dark states can be excited to bright statessince |ใD๐ผ |๐๐ง0 |B
๐ผโฒใ| > 0. Dark states in the inhomogeneousmodel can be depopulated during a non-adiabatic passageacross resonance, such that the limit on hyperpolarization canbe overcome by preferentially inducing transitions from darkstates to bright states.
C. Hyperpolarization Scheme
We now discuss the basic hyperpolarization scheme as illus-trated in Fig. 1.
To polarize the spin bath, we apply a cyclical scheme. Ineach cycle, we (i) reset the polarization of the qubit to |โใ atlarge detuning, and (ii) we transfer polarization from the qubitto the bath by sweeping the central field detuning ๐(๐ก) acrossresonance over a timescale ๐๐ . The reset step is a routine ex-perimental step in quantum computing platforms; for example,in a NV set-up, the qubit can be reset using a rapid opticalpulse [59, 79]. The ramp varies ๐(๐ก) from an initial value ๐๐to a final value ๐ ๐ = โ๐๐ , such that the cycle starts and endsfar from resonance ๐0 โก |๐๐ | = |๐ ๐ | ๏ฟฝ ฮmax, where the qubitis completely polarized in every eigenstate. From one cycle tothe next, the direction of the ramp is reversed (after each reset)in a forward-backward fashion.
During a single reset and sweep cycle probability is trans-
ferred in every magnetization sector2 (๐) from states with upqubit polarization |โใ in sector ๐ to states with down qubitpolarization |โใ in the magnetization sector (๐ โ 1) (as de-picted in Fig. 1). The effects on a single bright state can bereadily understood: suppose the system is initially in a brighteigenstate |โใ โ |B๐ผโ ใ, factorizable far away from resonanceand with fixed magnetization ๐ . Then the total magnetizationof the bath state is necessarily ๐ + 1/2. After an ideal adi-abatic transfer across resonance, this bright state is given by|โใ โ |B๐ผโ ใ, again far away from resonance. From conservationof magnetization, the bath state now has total magnetization๐ โ 1/2. Following the reset step of the central spin, this stateis reset to |โใ โ |B๐ผโ ใ, which is no longer an eigenstate butrather a superposition of eigenstates. Crucially, these statesall have magnetization ๐ โ 1: the total bath magnetizationhas been reduced. Dark states of the form |โใ โ |D๐ผโ ใ are leftinvariant by these steps. After several cycles, dark state popula-tions build up and ultimately saturate the bath spin polarizationwell above its fully polarized value.
The success of the protocol depends on the suppressionof diabatic excitations. However, transitions between brightstates within their own band are irrelevant for the purposes ofpolarization transfer, and thus we only require that transitionsbe suppressed between the bands. Specifically, we mimic aslow smooth ramp ๐(๐ก) with ramp time ๐๐ ๏ฟฝ ๐0, where
๐0 = 2๐0/ฮ2min (18)
sets the timescale for the onset of diabatic transitions betweeneigenstate bands (Appendix C). While such a protocol maystill be too slow in practical applications, here it serves only asa starting point which guarantees efficient transfer.
III. POLARIZATION TRANSFER PROTOCOLS
We detail how to speed up adiabatic ramps with the assis-tance of CD and LCD protocols in a single sweep. Such (L)CDprotocols can be exactly analyzed in the integrable limit. Wefurther compare our CD protocols to unassisted (UA) protocolswhich, unlike CD, attempt to polarize the bath without engi-neering additional controls. A full cooling protocol consistingof repeated sweeps will be analyzed in Section IV.
We simulate sweeps ๐(๐ก) across resonance by numericallysolving the time-dependent Schroฬdinger equation 3 in a specificpolarization sector and measure efficiency. The system isinitialized at ๐๐ = โ๐0 ๏ฟฝ โฮtyp in a mixed state:
๐(๐ก = 0) = |โใ ใโ| โ ๐๐ต, (19)
2 The reset and transfer steps have an effect on all polarization sectors simul-taneously.
3 The specific ramp function used in this work is a smooth polynomial๐(๐ก) = ๐0 (12 (๐ก/๐๐ )5 โ 30 (๐ก/๐๐ )4 + 20 (๐ก/๐๐ )3 โ 1) , which monoton-ically increases from ๐(0) = โ๐0 to ๐(๐๐ ) = ๐0 and has vanishing firstand second derivatives at the protocol boundaries. The minimal order ofa polynomial in ๐ก/๐๐ that satisfies these constraints is five. However, anyform ๐(๐ก) with sufficiently smooth boundary conditions can be used [49].
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where the bath is in an infinite-temperature state ๐๐ต. Thischoice of a spatially uncorrelated and unpolarized bath stateis motivated by experimental conditions. We also expect anycoherences in the initial bath state to be lost during the repeatedcycling of the qubit. This gives an initial probability ๐๐ตโ(๐ก =0) of starting in the top bright band and ๐๐ทโ(๐ก = 0) of startingin the dark band, with
๐๐ตโ,โ(๐ก) =โ๏ธ๐ผ
Tr[๐(๐ก) Pโ,โ |B๐ผใใB๐ผ |Pโ,โ], (20)
๐๐ทโ,โ(๐ก) =โ๏ธ๐ผ
Tr[๐(๐ก) Pโ,โ |D๐ผใใD๐ผ |Pโ,โ], (21)
in which Pโ โก |โใ ใโ| โ ๐ผ is the projection operator to thesubspace with down qubit polarization and similarly Pโ โก|โใ ใโ| โ ๐ผ. At the end of the ramp (๐ ๐ = +๐0), the state ๐(๐ก =๐๐ ) has a probability ๐๐ตโ(๐ก = ๐๐ ) of being in the top brightband, ๐๐ทโ(๐ก = ๐๐ ) of being in the dark band, and ๐๐ตโ(๐๐ ) ofhaving transitioned to the bottom bright band. For protocolefficiency, we use two measures: (i) the transfer efficiency,
๐๐ โก ๐๐ตโ(๐๐ )/๐๐ตโ(0), (22)
which measures how effectively the qubit polarization in brightstates is flipped during a single sweep, and (ii) the kick effi-ciency,
๐๐พ โก 1 โ ๐๐ทโ(๐๐ )/๐๐ทโ(0), (23)
which measures how effectively dark states are depopulated(or โkickedโ) into the bright manifold. Throughout this section,we continually refer to Fig. 3, which plots these efficienciesover a range of ramp times ๐๐ for numerically simulated UAand CD protocols. Note that we average over ๐๐ realizationsof disorder in ฮฉ๐ต, ๐ and ๐ ๐ , which we denote by an overline as๐๐ or ๐๐พ .
A. Unassisted Driving (UA)
We first discuss unassisted (UA) protocols, corresponding toa sweep of ๐ over a finite time. Adiabatic protocols correspondto infinite ramp times ๐๐ โ โ, where all bright state polariza-tion is transferred across a single sweep: ๐๐ = 1 while ๐๐พ = 0.At finite ramp times diabatic effects become important andgenerally ๐๐ < 1 and ๐๐พ > 0. In the fast limit (๐๐ โ 0) thesystem does not have time to respond to the drive, completelypreventing polarization transfer and dark state depletion suchthat ๐๐ , ๐๐พ โ 0.
In a system with a homogeneous bath field (๐พ๐ง = 0), theoperator ๐๐ง0 only couples bright state pairs
4, and within each๐ sector all excitations induced by a finite ramp speed ยค๐ > 0occur only between bright state pairs [34]. Each bright statepair can be treated as an independent two-level Landau-Zenerproblem following Eq. (8), for which the known transition
4 Note that dark states at ๐พ๐ง = 0 are eigenstates of ๐๐ง0 , so they cannot coupleto bright states on changing the qubit z-field in time.
FIG. 3. Efficiency vs. ramp time. Disorder-averaged transferefficiency (top) and kick efficiency (bottom) of UA, CD, and LCDprotocols in systems with ๐พ๐ง = 0.00 (crosses) and ๐พ๐ง = 0.05 (boxes).Dashed lines show the analytic prediction for the UA transfer effi-ciency (26), the analytic predictions for LCD transfer (41) and kickefficiencies (43) at large ramp velocities. Parameters: ๐๐ = 150,๐ฟ = 10, ๐ = โ1, ฮฉ๐ต = 10, ๐0 = 5, ๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, and๐0 โ 1000.
probability for a ramp ๐(๐ก) across a resonant gap ฮ๐ผ is givenby [80, 81]
๐trans [ฮ๐ผ] = exp(โ ๐
2ฮ2๐ผยค๐
). (24)
Averaging this transition probability over the gap distribution(14) returns an approximate transfer efficiency for a givenmagnetization sector
๐๐ = 1 โ
โซ โฮmin
๐trans [ฮ] ๐(ฮ) ๐ฮโซ โฮmin
๐(ฮ) ๐ฮ, (25)
which can be evaluated to return
๐๐ = 1 โยค๐๐๐
1 + ยค๐๐๐exp
(โ๐๐๐
2๐ฟ
ยค๐
), (26)
in which ๐ = ๐/๐ฟ is the magnetization density, ยค๐ โ ๐0/๐๐ ,and
๐๐ =1
๐๐๐2๐ฟln
(1 + 2๐1 โ 2๐
). (27)
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7
The transfer efficiency in Eq. (26) is plotted in Fig. 3 as adashed black curve and shows excellent agreement with theUA calculations for ๐พ๐ง = 0 and ๐พ๐ง = 0.05.
As shown in Fig. 3, the distinction between a system with ahomogeneous bath field (crosses) and an inhomogeneous bathfield (squares) has little impact on the UA transfer efficiency.The transfer efficiency in UA dynamics varies drastically with๐๐ . When ๐๐ is sufficiently large (๐๐ ๏ฟฝ ๐0; cf. Eq. (18)),transitions between eigenstate bands become suppressed andthe system becomes effectively adiabatic for the purposes ofpolarization transfer: the qubit flips in bright state bands, butnot in dark bands. Fig. 3 shows the tendency of simulated UAprotocols toward unit transfer efficiency.
The difference between homogeneous and inhomogeneoussystems is important when considering the kick efficiency. Ina system with inhomogeneous bath fields (๐พ๐ง > 0), ๐๐ง0 couplesbright and dark eigenstates. As such, inhomogeneous fieldslead to a nonzero kick efficiency because diabatic transitionscan depopulate dark states, whereas homogeneous fields leadto a zero kick efficiency at all ramp rates.
The convergence ๐๐ โ 1 (shown) occurs much faster thanthe convergence ๐๐พ โ 0+ (not shown). The former is deter-mined by the gap between bright state pairs, which remainfinite throughout the ramp at numerically accessible systemsizes, whereas the latter is determined by the dark-bright gaps,which tend to close away from resonance. This leads to dark-bright transitions at large yet finite ๐๐ . (ฮ๐ธ)โ2, where ฮ๐ธ ison the order of the level spacing (Appendix C). Any attempt todrive the system faster (๐๐ . ๐0) leads to diabatic excitationsbetween eigenstates. When ๐พ๐ง = 0, speeding up UA protocolsdecreases the transfer efficiency due to transitions betweenbright bands, but again does not deplete dark states. At finitedisorder strength ๐พ๐ง = 0.05, UA protocols suffer a similar lossof transfer efficiency, but gain the ability to kick dark states,with a peak kick efficiency at intermediate speeds ๐๐ โผ ๐0.
B. Exact Counterdiabatic Driving (CD)
CD protocols suppress transitions between the eigenstatesof an instantaneous Hamiltonian by evolving the system withan assisted Hamiltonian that exactly cancels all diabatic ex-citations [49]. The inclusion of counterdiabatic terms in ahyperpolarization protocol can hence be used to increase thetransfer efficiency.
Within each two-dimensional Landay-Zener subspace (8),the system remains in an instantaneous eigenstate of ๐ป๐ผ (๐(๐ก))at all times when evolved with a time-dependent Hamiltonian[49]
๐ปCD,๐ผ (๐ก) = ๐ป๐ผ (๐(๐ก)) โ ยค๐(๐ก)ฮ๐ผ
๐(๐ก)2 + ฮ2๐ผ๐๐ฆ๐ผ, (28)
where the auxiliary (counterdiabatic) term โ ๐๐ฆ๐ผ exactly can-cels diabatic transitions between the bright states for arbitraryramp speeds provided ยค๐ = 0 at the beginning and end of theramp.
CD is realized for the full system if the system is evolvedwith the time-dependent CD Hamiltonian
๐ป๐ถ๐ท (๐ก) = ๐ป (๐(๐ก)) + ยค๐(๐ก) A๐ (๐(๐ก)), (29)
where the CD term A๐, also known as the adiabatic gaugepotential, follows as
A๐ (๐) = โโ๏ธ๐ผ
ฮ๐ผ๐2 + ฮ2๐ผ
๐๐ฆ๐ผ . (30)
The summation index ๐ผ runs over all bright pairs in all magne-tization sectors. The effect of the counterdiabatic term can beunderstood in the limit ยค๐ โ โ, where the Hamiltonian reducesto ยค๐A๐ and the evolution operator for a single sweep can bewritten as
exp(๐
โซ โโโ
A๐ ๐๐)=
โ๐ผ
exp(โ ๐ ๐ ๐๐ฆ๐ผ
). (31)
The gauge potential generates a rotation around the y-axis thatexchanges |B๐ผ+ ใ โ |B๐ผโ ใ when ๐ is swept across resonance,exactly as happens in the adiabatic protocol.
Alternatively, the gauge potential can be written in closedform as (see Appendix F)
A๐ = โ๐
4(๐ป โฮฉ๐ต๐)โ2 [๐ป, ๐๐ง0] . (32)
The first (inverse) term in the product is to be interpreted in thesense of a pseudo-inverse, and the second (commutator) termin the product is given by:
[๐ป, ๐๐ง0] = ๐โ๏ธ๐
๐ ๐ (๐๐ฅ0 ๐๐ฆ
๐โ ๐๐ฆ0 ๐
๐ฅ๐ ). (33)
The gauge potential is a complex many-body operator, dif-ficult to compute in theory and even harder to implement inpractice [49]. Only in certain special cases, for example when๐๐๐ป is an integrable perturbation of an integrable model ๐ป,is this operator sufficiently local [49, 82, 83]. Fortunately,๐๐๐ป = ๐
๐ง0 is an integrable perturbation of ๐ป in the ๐พ๐ง = 0 limit
of our present model, and the pair structure of the bright statecould be used to immediately write down the adiabatic gaugepotential. A similar two-level structure for the gauge potentialalso arises in integrable free-fermionic systems [82].
In a system with an inhomogeneous bath field (๐พ๐ง > 0), wecan no longer express the adiabatic gauge potential explicitly.Nevertheless, as CD mimics an adiabatic protocol, the transferefficiency will be maximal.
Fig. 3 showcases the effect of exact CD in a transfer proto-col across resonance for systems with ๐พ๐ง = 0 and ๐พ๐ง = 0.05(crosses and squares respectively). In both cases, the completesuppression of bright state transitions yields a maximally ef-ficient transfer protocol ๐๐ = 1, systematically improving onthe UA protocol, while the complete suppression of dark statetransitions results in zero kick efficiency ๐๐พ = 0.
C. Local Counterdiabatic Driving (LCD)
In practice, it is hard to realize exact CD, and we mustresort to approximation schemes. In this section, we follow the
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8
method devised in Ref. [55] to develop a local approximationALCD to A๐. We refer to assisted driving (see Eq. (29)) withALCD as local counterdiabatic driving (LCD).
As proposed in Ref. [55] and detailed in Appendix G, aformal expansion for the adiabatic gauge potential can be foundin terms of nested commutators:
A๐ = ๐๐โ๏ธ๐=1๐ผ ๐ [๐ป, [๐ป, . . . [๐ป๏ธธ ๏ธท๏ธท ๏ธธ
2 ๐โ1
, ๐๐๐ป]]] . (34)
For ๐ โ โ Eq. (34) reproduces the exact gauge potential. Alocal approximation for A๐ is obtained by truncating the com-mutator expansion of Eq. (34) to a desired order ๐, and using avariational minimization scheme [54] to set the coefficients ๐ผ ๐for ๐ = 1, . . . , ๐.
We focus on the leading order term because it (i) is simpleenough to be implemented by Floquet driving on the qubit field(see Section VI) and (ii) is already remarkably effective forpolarization transfer. One can always refine the approximationto CD by adding higher-order commutators in Eq. (34); therapid convergence of higher-order LCD to CD is shown inAppendix G.
To leading order (๐ = 1), we obtain:
ALCD (๐) = ๐ ๐ผ1 (๐) [๐ป, ๐๐ง0], ๐ผ1 (๐) = โ1
๐2 + ฮ2typ, (35)
This scheme leads to a coefficient ๐ผ1 (๐) depending on a singleenergy scale which coincides exactly with the typical gap ฮtyp,in contrast with exact CD, where the prefactor depends eitherexplicitly (28) or implicitly (32) on all bright state gaps ฮ๐ผ. Insum,
๐ปLCD (๐ก) = ๐ป (๐(๐ก)) + ๐ ยค๐(๐ก) ๐ผ1 (๐(๐ก)) [๐ป, ๐๐ง0]
= ๐ป (๐(๐ก)) +ยค๐(๐ก)
๐(๐ก)2 + ฮ2typ
โ๏ธ๐
๐ ๐ (๐๐ฅ0 ๐๐ฆ
๐โ ๐๐ฆ0 ๐
๐ฅ๐ ).
(36)
Fig. 3 shows the resulting LCD curves as unmarked solidred curves (๐พ๐ง = 0.0), and circle-marked red curves (๐พ๐ง =0.05). LCD is approximate, ๐๐ < 1 (see top panel of Fig. 3).Nevertheless, LCDโs transfer efficiency is high (๐๐ & 0.75)over the whole range of ramp times ๐๐ , even as ๐๐ โ 0 whereUA becomes completely transfer inefficient.
A finer comparison between ALCD and A๐ can be made for๐พ๐ง = 0 by expressing the gauge potential in the Landau-Zenerpicture (8),
ALCD (๐) = โโ๏ธ๐ผ
(ฮ๐ผฮtyp
)ฮtyp
๐2 + ฮ2typ๐๐ฆ๐ผ . (37)
Rather than targeting individual gaps as in Eq. (30), the LCDprotocols effectively target a single typical energy splittingscale to suppress diabatic transitions between the bright bands.In contrast with Eq. (30), the Lorentzian prefactor of ๐๐ฆ๐ผ hasa fixed width ฮtyp, which does not vary with bright state gap,and a modulated amplitude ฮ๐ผ/ฮtyp.
This discrepancy introduces polarization transfer errors inLCD at intermediate and fast ramps (๐๐ . ๐0). Comparing withEq. (31), in the limit ยค๐ โ โ the LCD protocol again generatesa rotation within bright state pairs:
exp(๐
โซ โโโ
A๐ฟ๐ถ๐ท ๐๐)=
โ๐ผ
exp(โ ๐ ๐ ฮ๐ผ
ฮtyp๐๐ผ๐ฆ
). (38)
LCD strongly suppresses transitions between those bright pairswith a gap ฮ๐ผ โ ฮtyp, but otherwise yield only partial suppres-sion.
From Eq. (38) we can define a mismatch error between CDand LCD for each bright pair with gap ฮ๐ผ as
E[ฮ๐ผ] = cos2(๐
2ฮ๐ผฮtyp
). (39)
Averaging over the gap distribution (14), the transfer efficiencyat large ramp rates is
๐๐ = 1 โ
โซ โฮmin
E[ฮ] ๐(ฮ) ๐ฮโซ โฮmin
๐(ฮ) ๐ฮ, (40)
The saddle-point approximation returns
๐๐ =
โซ โ1 ๐๐ก ๐ก sin
2(๐2โ
2๐๐ก) (
1โ2๐1+2๐
) ๐ก2โซ โ1 ๐๐ก ๐ก
(1โ2๐1+2๐
) ๐ก2 . (41)This expression agrees with the LCD transfer efficiency inFig. 3 (dashed red line) and will be discussed in more detail inthe following section.
Fig. 3 (bottom panel) also shows the LCD kick efficiencyover several ๐๐ orders. In the ๐พ๐ง = 0 limit, LCD has no effecton dark states, just like UA and CD, again leading to a zerokick efficiency (crosses in bottom panel of Fig. 3). When๐พ๐ง > 0, LCD protocols do not prevent dark-bright transitionsand exhibit non-zero kick efficiency. Since the bright-dark gapis smaller than the typical bright band gap ฮtyp, especially farfrom resonance where the bright-dark gap tends to close, LCDallows dark-bright transitions as in UA driving.
The difference in gap scales gives LCD both the advantagesof CD for efficient transfer, and the advantages of diabatic UAfor depopulating dark states. For slow ramps ๐๐ > ๐0, LCD andUA have similar transfer efficiencies as the diabatic transitionprobabilities are small. In faster ramps (๐๐ . ๐0), bright bandtransitions are suppressed by LCD but not UA. Meanwhile,LCD saturates to a maximum kick efficiency for ๐๐ < ๐0, incontrast with UA protocols which peak around ๐๐ โผ ๐0 and thenlose kick efficiency as ๐๐ โ 0. The distinction between LCDand UA protocols in this fast limit will be further quantified inEq. (43), following Appendix E, where it is argued that in thelimit ๐๐ โ 0 the kick efficiency is proportional to the transferefficiency.
Finally, note that this work focuses on the weak xx-disorderlimit ๐พ๐ฅ๐ฅ < ๐ where there is a finite gap between bright bandsat numerically accessible system sizes. However, a finite brightpair gap is not necessary to design efficient LCD protocols that
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9
need only target a typical gap between the bright bands. InAppendix D we show that LCD maintains high transfer andkick efficiencies in the presence of strong xx-disorder evenin the presence of small gap as long as the bulk of the brightspectrum still has a gap โผ ฮtyp.
D. Protocol Efficiency and Polarization Sector
We compare the efficiencies of CD, LCD, and UA protocolsfor a single sweep across different polarization sectors ๐.Since all protocols systematically reduce polarization, it iscrucial to understand how the transfer and kick efficienciesdepend on the polarization sector. The results are summarizedin Fig. 4 for a fast ramp (๐๐ = 0.05 ๐0) in a system with aninhomogeneous bath field (๐พ๐ง = 0.05) for multiple systemsizes ๐ฟ.
FIG. 4. Efficiency vs. polarization sector. The vertical axes showtransfer efficiency (top) and kick efficiency (bottom) for fast LCD(colored) and UA (grey) sweeps across resonance in a system withan inhomogeneous bath field. The horizontal axis shows the density๐/๐ฟ = ๐/๐ฟ + 1/2 of spin flips above the fully polarized state in๐ < 0 sectors. We plot theoretical predictions to ๐๐ based on thethermodynamic limit calculations at zero disorder (cf. Eqs. (26) and(41)), and to ๐๐พ based on ๐๐ (cf. Eq. (43)). For reference, wealso plot corresponding efficiencies for CD protocols. Parameters:๐๐ = 150, ฮฉ๐ต = 10, ๐0 = 5, ๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, and ๐๐ = 500/๐ฟ.
As expected, CD always produces unit transfer efficiencyand zero kick efficiency. Moreover, LCD outperforms UA
by both efficiency measures in every polarization sector. Thetop panel shows the transfer efficiency ๐๐ , plotted against๐/๐ฟ = ๐/๐ฟ โ 1/2.For both LCD and UA protocols, the trans-fer efficiency decreases with ๐/๐ฟ because the minimal gapand the number of bright state pairs ๐๐ต =
( ๐ฟโ1๐โ1
)increases with
๐/๐ฟ, in turn increasing the likelihood of diabatic transitionsbetween bright pairs. For LCD, the transfer efficiency is max-imal (๐๐ = 1) in the sector with ๐ = 1 because there is onlyone bright state pair with gap ฮtyp = ฮLCD to target.
The bottom panel shows the kick efficiency ๐๐พ , plottedagainst ๐/๐ฟ. For both LCD and UA, the kick efficiency in-creases with polarization. This increase can be understoodby comparing the number ๐๐ท of dark states to the number ofbright pairs ๐๐ต within each sector. In the sector with ๐ = 1,there are (๐ฟ โ 2) dark states compared to a single pair ofbright states, which severely limits the pool of bright states thatdark states can transition to. As we probe increasingly larger๐ , ๐๐ต eventually surpasses ๐๐ท such that ๐๐ต/๐๐ท โ O(๐ฟ)as ๐/๐ฟ โ 0.5. The number of available bright states thatdark states can transition to increases and thus enhances kickefficiency.
Fig. 4 also shows the analytic predictions (black curves) forthe transfer efficiency from Eq. (26) and (41), consistent withthe collapse of the curves. For LCD in fast ramps (41), the onlydependence of ๐๐ is on ๐ = ๐/๐ฟ, consistent with the collapseof ๐๐ curves at different system sizes as a function of spinflip density (๐/๐ฟ โผ ๐/๐ฟ + 1/2). The transfer efficiency inEq. (26) depends on both ๐ and ๐2๐ฟ/ ยค๐. To achieve a collapseof curves at different system sizes, one must also scale ยค๐ โผ ๐ฟto eliminate the residual ๐ฟ dependence, yielding a transfer effi-ciency which depends only on ๐/๐ฟ. In practice, the collapsecan be achieved by scaling up ๐0 โผ ๐ฟ at fixed ramp-time ๐๐ orscaling down ๐๐ โผ 1/๐ฟ at fixed ramp range; in our simulations,we have implemented the latter. Both predictions show excel-lent agreement with simulation results in most magnetizationsectors, except near ๐ โผ O(1) where finite size effects aresignificant. Such finite-size effects also lead to the deviation ofthe numerically observed bright pair gap distribution in Fig. 2from the analytical one at large negative values of ๐ .
Remarkably, there is a simple approximate relation between๐๐ and ๐๐พ for LCD and UA protocols in the presence ofz-disorder at moderate-to-fast ramps ๐ . ๐0. Along withEqs. (41) and (26), this relation provides an analytical predic-tion for the kick efficiency, such that the protocol efficiencycan be fully characterized analytically. Assume that the proba-bility weight that is not successfully transferred to states withup qubit polarization ergodically mixes between the availabledark and bright states with spin down. Then,
๐๐ทโ(๐๐ ) โ ๐๐ทโ(0)๐๐ต (1 โ ๐๐ ) + ๐๐ท
๐๐ต + ๐๐ท, (42)
which implies a kick efficiency
๐๐พ = 1 โ๐๐ทโ(๐๐ )๐๐ทโ(0)
โ ๐๐ต๐๐ต + ๐๐ท
๐๐ =๐
(๐ฟ โ ๐) ๐๐ . (43)
The black curves in Fig. 4 show ๐๐พ computed using Eqs. (43),(41) and (26). These analytic curves are in good agreement
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10
with the numerical data for both LCD and UA. Note that thederivation of Eq. (43) assumes equal mixing between dark andbright states: closer to integrable points, this assumption breaksdown, and an analytic relation between the transfer and kickefficiency is no longer possible.
The resulting protocols hence depend on the interplay ofdifferent effects across magnetization sectors: increasing ๐,the total number of bright states capable of transfering polar-ization increases, the average transfer efficiency decreases, andthe kick efficiency increases.
IV. HYPERPOLARIZING THE BATH
FIG. 5. Polarizing the spin bath. Expectation value of the bathpolarization per spin ใใ๐๐ง
๐ตใใ along a sequence of ๐c = 100 back and
forth cycles of the detuning ๐ across resonance. Top and bottom panelsshow a typical realization for systems with zero and finite z-disorder,respectively. The system is initialized in an infinite temperature state.The forward-backward transfer flow between resets is illustrated bythe gold arrows. Grey dotted lines denote the polarization of the fullypolarized state. Parameters: ๐ฟ = 4, ฮฉ๐ต = 10, ๐0 = 5, ๐ = 0.1,๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0 (top), ๐พ๐ง = 0.05 (bottom), and ๐๐ /๐0 โ 0.05.
We now turn to the performance of the various protocolsover multiple reset-transfer cycles. In particular, we show howthe ability of LCD to kick dark states enables complete bathspin polarization.
Figs. 5 and 6 illustrate the progressive polarization of thebath over multiple (๐๐ = 100) reset-transfer cycles in UA, CD,
FIG. 6. Spin bath polarization vs. cycle. Average bath polarizationper spin vs. cycle number with the same setup as in Fig. 5. Parameters:๐๐ = 1, ๐ฟ = 4, ฮฉ๐ต = 10, ๐0 = 5, ๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0 (top),๐พ๐ง = 0.05 (bottom), and ๐๐ /๐0 โ 0.05.
and LCD protocols. We focus on fast sweeps ๐๐/๐0 โ 0.05,where the effects of LCD and UA are significantly differen-tiated. For this simple demonstration, we consider a qubitcoupled to 3 bath spins; however, the observed qualitativebehavior generalizes to larger baths (see Section V). In bothfigures, we measure the expectation value of the average bathspin polarization per spin:
ใใ๐๐งBใใ =1
๐ฟ โ 1๐ฟโ1โ๏ธ๐=1
ใ๐๐ง๐ใ, (44)
where ใ๐๐ง๐ใ โก Tr[๐(๐ก)๐๐ง
๐] is the expectation value of ๐๐ง
๐in the
density matrix ๐(๐ก) of the system at time ๐ก.In Fig. 5, the bath polarization per spin is shown as a func-
tion of the detuning ๐(๐ก) across resonance. After each transfersweep, the qubit polarization is reset and the direction of theramp reversed; the resulting forward-backward motion is de-picted by the gold arrows. Fig. 6 shows the corresponding bathpolarization per spin after every cycle.
In a typical realization of a system with a homogeneousbath field (๐พ๐ง = 0), CD protocols at first quickly reduce thebath polarization due to their maximal transfer efficiency, butsoon slow down and saturate as dark states become populated.The saturation point lies well above the fully polarized state(see grey dotted line ใใ๐๐งBใใ = โ0.5). In contrast, UA protocolsare relatively inefficient and much slower to reach saturation,requiring many more sweeps. LCD protocols perform onlyslightly worse than CD and much better than UA; they eventu-ally also saturate above the fully polarized state.
In a typical realization of a system with an inhomogeneousbath field (๐พ๐ง = 0.05), CD protocols behave the same as in thehomogeneous limit, quickly polarizing the bath to a saturationpoint. LCD protocols no longer saturate and can polarize thebath close to the fully polarized state due to their non-zero kickefficiency. Since the hyperpolarization scheme progressivelypopulates smaller ๐ sectors, and the kick efficiency decreaseswith decreasing ๐ (see Fig. 4), the polarization rate per cycledecreases as we ใใ๐๐ง
๐ตใใ โ โ1/2. UA protocols, like LCD, are
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11
able to fully polarize the bath, but their smaller kick efficiencyrequires many more sweeps.
V. SCALING TO LARGE BATHS
In this section, we explore how the number of cycles neededto hyperpolarize the bath scales with system size ๐ฟ. So farwe focused on relatively small system sizes ๐ฟ . 10 to designand test our protocols with accessible exact dynamic simu-lations. To circumvent the resource cost of simulating exactdynamics with larger system sizes, we introduce a scalablemaster equation of the hyperpolarization process in terms ofprobability flow equations which should be accurate at largetransfer speeds.
The state of the system after ๐ transfer-reset polarizationcycles is given by:
๐(๐) =[ยฎ๐๐ตยฎ๐๐ท
]=
๐๐ต,โ[0]๐๐ต,โ[1]
...
๐๐ต,โ[๐ฟ]๐๐ท,โ[0]๐๐ท,โ[1]
...
๐๐ท,โ[๐ฟ]
(45)
where ๐๐ต,โ[๐] is the probability of finding the system in abright state with down qubit polarization in the sector with ๐spin flips above the fully polarized state, and ๐๐ท,โ[๐] is theprobability of finding the system in a dark state with downqubit polarization in the same sector. We do not track the prob-abilities of bright and dark states with up qubit polarization, asthey are always converted to states with down qubit polariza-tion after reset. Moreover, as there exist no dark states withdown qubit polarization for ๐ โฅ 0, ๐๐ท,โ[๐] = 0 for ๐ โฅ ๐ฟ/2.Finally, we assume that the bath is fully characterized by theprobabilities in Eq. (45) and ignore any correlations in thedensity matrix between individual dark and bright states sincethe bath generally decoheres between different polarizationcycles. This assumption is justified a posteriori by compar-ing the efficiencies predicted by the master equation to exactsimulations.
The dynamics of the system is obtained by applying a trans-fer matrix ๐ :
๐(๐ + 1) = ๐ ๐(๐), (46)
where the transfer matrix can be schematically written as
๐ =
[๐๐ต๐ต ๐๐ต๐ท
๐๐ท๐ต ๐๐ท๐ท
]. (47)
The transfer efficiency ๐๐ sets the probability that the qubitpolarization is flipped in bright states during a sweep acrossresonance. On the other hand, the kick efficiency ๐๐พ sets the
FIG. 7. Effective model. Schematic representing the action of thetransfer matrix through the efficiency functions ๐ and reset rates ๐in bright (red) and dark (black) manifolds in two neighboring po-larization sectors ๐ = ๐ + 1 and ๐ = ๐ during a single polarization(transfer+reset) cycle.
probability that dark states are kicked into bright states. Weassume that dark states with down qubit polarization are onlykicked into bright states with down qubit polarization. Whenthe qubit polarization is reset after each sweep, bright stateswith qubit state |โใ transition to either bright states (with |โใ)or dark states (with |โใ) in a lower magnetization sector, withrelative probability ๐๐ต and ๐๐ท , respectively. Therefore, thenon-zero matrix elements of the transfer matrix are given by:
๐๐ต๐ต [๐, ๐] = 1 โ ๐๐ [๐] (48)๐๐ต๐ต [๐ โ 1, ๐] = ๐๐ต [๐] ๐๐ [๐] (49)๐๐ท๐ต [๐ โ 1, ๐] = ๐๐ท [๐] ๐๐ [๐] (50)๐๐ท๐ท [๐, ๐] = 1 โ ๐๐พ [๐] (51)๐๐ต๐ท [๐, ๐] = ๐๐พ [๐] (52)
for every sector index ๐ = 0, . . . , ๐ฟ. Fig. 7 illustrates the trans-fer and reset rates for a single cycle. Note ๐๐ต [๐] + ๐๐ท [๐] = 1,so only one reset rate needs to be specified.
As shown in Section III, the different protocols CD, UA, andLCD have different efficiency functions ๐๐ [๐] and ๐๐พ [๐]. Here,we consider moderate-to-fast ramp speeds (๐๐ . ๐0) whereLCD and UA have distinct effects. Since we are interestedin obtaining the scaling of the full protocol, we linearize theprevious expressions and model the kick efficiency for LCDand UA as
๐๐พ [๐] = ๐0๐
๐ฟ; (๐ โค ๐ฟ/2), (53)
and model the corresponding transfer efficiency ๐๐ [๐] (๐ โค๐ฟ/2) using Eq. (43).
The reset rates ๐๐ท [๐] and ๐๐ต [๐] depend on the probabilitydistribution of the state within each bright/dark band and detailsof the structure of eigenstates. Furthermore, these rates candrastically change from one disorder realization to another andare hence difficult to predict. To get a reasonable estimate forour master equation, we take ๐๐ท [๐+1] and ๐๐ต [๐+1] proportionalto to the number of accessible dark and bright states in the ๐๐กโ
sector, respectively. For ๐ โฅ 0, all the weight is transferred tobright states,
๐๐ต [๐ + 1] = 1, ๐ โฅ 0, (54)
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since dark states have qubit spin up and are not accessibleduring reset. For ๐ < 0, dark states have qubit spin down andbecome accessible, such that
๐๐ต [๐ + 1] =๐๐ต [๐]
๐๐ต [๐] + ๐๐ท [๐], ๐ < 0, (55)
with ๐๐ท and ๐๐ต given by Eqs. (9) and (16), and we refer to thisreset rate as well-mixed. Although we cannot generally satisfythe equiprobable condition within every bright band, we expectto approximately have well-mixed reset rates on average inlarge disordered systems.
We test the master equation in Fig. 8, which compares UA,CD, and LCD dynamics with Eq. (46) to the correspondingexact dynamics for different system sizes (๐ฟ = 4, 8). The figureplots the average bath polarization per spin over many cyclesfor a single disorder realization. Our master equation agreeswell with the results from exact dynamics for all protocols,justifying the assumptions in Eq. (45). The exact dynamicsis computed at small ramp times ๐๐ โ 0.05 ๐0. To properlycapture protocol efficiencies at this ramp speed, we set ๐0 โ 1.0for LCD and ๐0 โ 0.4 for UA in accordance with the results inFig. 4.
FIG. 8. Spin bath polarization vs. cycle. Average bath spinpolarization after every transfer-reset cycle over many cycles. Theleft and right panels show simulation results for systems of size ๐ฟ = 4and ๐ฟ = 8, respectively. Colored markers indicate numerical resultsusing our scalable master equation. Solid colored lines correspond toexact dynamics simulations. Parameters: ๐๐ = 1, ฮฉ๐ต = 10, ๐0 = 5,๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0.05, and ๐๐ = 500/๐ฟ.
The master equation allows acces to much larger systemsizes compared to exact diagonalization. Fig. 9 shows thenumber of cycles ๐๐ required to reach 99% of the polarizationof the fully polarized state against system size ๐ฟ. The opaqueand faint curves denote master equations capturing ๐๐ = 0.01 ๐0and ๐๐ = 0.05 ๐0, respectively.
We find that the number of polarization cycles needed tofully polarize the bath in both LCD and UA protocols scaleslinearly with system size ๐ฟ. The main difference between LCDand UA is in the prefactor, depending on protocol duration, andwhich can be orders of magnitude larger in the UA protocolcompared to the LCD protocol at sufficiently fast ramps. Inslower ramps ๐๐ > ๐0 (not shown), LCD and UA have similar
prefactor, but the prefactor for UA however increases as ๐๐is decreased. Our results are consistent with the expectationthat as ๐๐ โ 0, UA takes progressively more cycles to fullypolarize the bath. Thus, moderate-to-fast LCD is not only time-efficient but also optimizes the number of cycles required toreach the fully polarized state.
We conclude this section with a couple of remarks. (i)The master equation is applicable at sufficiently fast ramptimes ๐๐ < ๐0 โผ ๐0 ฮโ2min, where ฮmin โผ
โ๐ฟ ๐ in the lowest
polarization sectors. To ensure this condition holds as ๐ฟ โ โ,we scale ๐0 โผ ๐ฟ. Otherwise at fixed ๐0 and sufficiently large๐ฟ โผ ๐0/(๐๐ ๐2), the master equation would need to be refinedto properly account for more complicated speed dependenciesin the transfer and kick efficiencies. (ii) Similarly, our masterequation is based on efficiency measurements at sufficientlylarge z-disorder, where ๐พ๐ง & ฮ2min/๐0. In the lowest energysectors, this requires ๐พ๐ง & ๐ฟ ๐2/๐0. Again, the ๐ฟ dependencecan be cancelled by scaling ๐0 โผ ๐ฟ.
VI. FLOQUET ENGINEERING (FE) OF LCD
The physical implementation of the LCD protocol requiresrealizing a non-trivial operator [๐ป, ๐๐ง0]. We show how it ispossible to obtain this LCD Hamiltonian as an effective high-frequency Hamiltonian through Floquet engineering.
Floquet engineering focuses on the design and physical ef-fects of periodic drives [84]. A periodically driven systemexhibits dynamics which can be described stroboscopicallyusing an effective slow/static Floquet Hamiltonian ๐ป๐น . Fre-quently, a control is periodically modulated at a frequencyscale ๐ larger than any other dynamical frequency in the sys-tem, and ๐ป๐น can be (Magnus) expanded in powers of ๐โ1 [84].In addition to capturing high-frequency physics, the Magnus
FIG. 9. Number of cycles to 99% polarization vs. system size.Master equation simulation results are shown for UA (dashed black)and LCD (solid red) protocol. Opaque curves show results for ๐0 =1.0 set for LCD and ๐0 = 0.1 set for UA, which model ramps with๐๐ = 0.01 ๐0. At this ramp speed, we find ๐๐ โ 4๐ฟ for LCD and๐๐ โ 40๐ฟ for UA. Faint curves show results for ๐0 = 1.0 set for LCDand ๐0 = 0.4 set for UA, which model ramps with ๐๐ = 0.05 ๐0. Atthis ramp speed, we find ๐๐ โ 4๐ฟ for LCD and ๐๐ โ 10๐ฟ for UA.
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expansion has a commutator structure closely related to thestructure of the gauge potential in Eq. (34), and can be used torealize local counterdiabatic driving at every order [55].
A. Two-level system
In order to give some intuition for the many-body Floquetprotocol, we illustrate the general ideas on the two-level sys-tem of Eqs. (8) and (28). Temporarily dropping the brightstate label and making the time-dependence implicit, the CDHamiltonian to be realized can be written as
๐ปCD = ๐๐๐ง + ฮ๐๐ฅ + ยค๐๐ผ1ฮ๐๐ฆ . (56)
In an experimental set-up only ๐ is an accessible control pa-rameter, whereas ฮ is constant and the ๐๐ฆ term is absent (asit corresponds to a complex many-body operator acting onthe bright pair states). In order to realize ๐ปCD as an effectiveHamiltonian, we consider a LZ Hamiltonian and add high-frequency oscillations modulated by a slowly-varying ampli-tude. Specifically, we consider a time-dependent Hamiltonianof the form
๐ปFE (๐ก) = ๐พ(๐ก)๐๐ง + ฮ๐๐ฅ+
[๐ฝ(๐ก)๐ sin(๐๐ก) + ยค๐ฝ(๐ก) (1 โ cos(๐๐ก))
]๐๐ง , (57)
with ๐ฝ(๐ก) and ๐พ(๐ก) slowly-varying functions to be determined.The choice of this time-dependence is motivated by the re-
sulting effective Hamiltonian: in the limit of a large driving fre-quency ๐, the stroboscopic dynamics for this time-dependentHamiltonian is generated by the Floquet Hamiltonian (derivedin Appendix H)
๐ปF = ๐พ๐๐ง + ๐ฝ0 (๐ฝ)ฮ[cos(๐ฝ) ๐๐ฅ โ sin(๐ฝ) ๐๐ฆ
], (58)
where the slow time-dependence has been made implicit and๐ฝ0 is a Bessel function of the first kind.
The effective Hamiltonian is of the form (56), containing a๐๐ฆ term not present in the instantaneous Hamiltonian. How-ever, in the CD Hamiltonian the prefactor of ๐๐ฅ is constantand the prefactor of ๐๐ฆ is time dependent. Since the (slow)time dependence of these terms in the Floquet Hamiltonian isdetermined by the same factor ๐ฝ(๐ก), it is not possible to directlyrealize the CD Hamiltonian in this way. Rather, we can realizea Hamiltonian proportional to the CD Hamiltonian.
Demanding ๐ป๐น = ๐บ (๐ก)๐ป๐ถ๐ท , the prefactor for ๐๐ฅ immedi-ately returns the time-dependent prefactor of the full Hamilto-nian as
๐บ (๐ก) = ๐ฝ0 (๐ฝ(๐ก)) cos(๐ฝ(๐ก)). (59)
Time evolution follows the time-dependent Schroฬdinger equa-tion
๐๐๐ก |๐(๐ก)ใ = ๐บ (๐ก)๐ปCD |๐(๐ก)ใ . (60)
Defining a โrescaled timeโ ๐ (๐ก) such that ๐๐ = ๐บ (๐ก)๐๐ก , Eq. (60)can be used to realize counterdiabatic control in the rescaled
time provided ๐๐๐ |๐(๐ก (๐ ))ใ = ๐ปCD (๐ (๐ก)) |๐(๐ (๐ก))ใ. The coun-terdiabatic term is obtained by setting
tan(๐ฝ(๐ก)) = โ๐ผ1 (๐ (๐ก)) ยค๐(๐ (๐ก)), (61)
determining ๐ฝ(๐ก) as function of ๐ผ1 (๐ก), leaving
๐พ(๐ก) = ๐บ (๐ก)๐(๐ (๐ก)), (62)
to finally return ๐ป๐น = ๐บ (๐ก)๐ปCD (๐ (๐ก)). Note that the experi-mental time necessarily runs in the positive direction, requiring๐บ (๐ก) > 0 and ๐ฝ โ [โ๐/2, ๐/2].
B. FE protocol
The ideas in Section VI A can be immediately extended tothe many-body Hamiltonian and LCD of Eq. (36). Given atarget LCD ramp with ๐(๐ก) = ฮฉ๐ (๐ก) โฮฉ๐ต, we drive the systemwith the Floquet engineered (FE) Hamiltonian:
๐ปFE = ๐ป (ฮ(๐ก)), (63)
with a modified field detuning ฮ(๐ก) = ฮฉ๐ (๐ก) โฮฉ๐ต given by
ฮ(๐ก) = ๐ฝ0 (๐ฝ(๐ก)) cos(๐ฝ(๐ก)) ๐(๐ (๐ก))+ ๐ฝ(๐ก) ๐ sin(๐ ๐ก) + ยค๐ฝ(๐ก) (1 โ cos(๐ ๐ก)). (64)
Following Eq. (61), we set
๐ฝ(๐ก) โก arctan(โ ๐๐(๐ (๐ก))
๐๐ ๐ผ1 (๐ (๐ก))
), (65)
and the rescaled time ๐ = ๐ (๐ก) satisfying ๐๐ = ๐บ (๐ก)๐๐ก is definedas
๐ =โซ ๐ก
0๐ฝ0 (๐ฝ(๐ก โฒ)) cos(๐ฝ(๐ก โฒ)) ๐๐ก โฒ > 0. (66)
This FE Hamiltonian is designed precisely so that the leadingorder approximation to its Floquet Hamiltonian ๐ป๐น in thehigh-frequency limit is the LCD Hamiltonian in the rescaledtime:
๐ปF = ๐ป (๐(๐ ))+๐๐๐(๐ )๐๐
๐ผ1 (๐ ) [๐ป (๐(๐ )), ๐๐๐ป]+O(
1๐
). (67)
More specifically, the effective Floquet Hamiltonian is foundas (see Appendix H)
๏ฟฝฬ๏ฟฝF =๐บ (๐ก)[๐(๐ (๐ก)) ๐๐ง0 +
โ๏ธ๐
๐ฟฮฉโฒ๐ ๐๐ง๐
+โ๏ธ๐
๐ ๐ (๐๐ฅ0 ๐๐ฅ๐ + ๐
๐ฆ
0 ๐๐ฆ
๐) โ tan(๐ฝ(๐ก)) ๐ [๐ป, ๐๐ง0]
], (68)
where ๐ฟฮฉโฒ๐โก (ฮฉ๐ต, ๐ โฮฉ๐ต)/๐บ (๐ก) is the renormalized z-
disorder.The FE protocol is stroboscopically equivalent to LCD with
ALCD in Eq. (35). Moreover, in a smooth ramp ๐ with ยค๐๐ =ยค๐ ๐ = 0, ๐ป๐น๐ธ and ๐ป๐น yield the exact same initial and final
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states, which guarantees that FE and LCD produce the samepolarization transfer during our hyperpolarization scheme.
We remark that ๐ป๐น equals ๐ปLCD only in the absence of z-disorder (๐พ๐ง = 0). At finite z-disorder (๐พ๐ง > 0), the two differdue to the renormalization ๐ฟฮฉโฒ
๐of the bath fields. Away from
this point, the renormalization tends to enhance z-disordersince ๐บ (๐ก) โ [0, 1]. No significant quantitative differences inperformance were found between FE (in lab time) and LCD(in rescaled time) with renormalized disorder, as shown next.
FIG. 10. Floquet engineered ramp. Top panel shows the FE rampฮ(๐ก) (solid colored curves) as a function of time ๐ก for ramp times๐๐ = 0.025, 0.5, 1.0. The target ramp ๐(๐ก) (dashed black curve) isshown for reference. The bottom panel shows the effect of FE rampson the mean qubit polarization ใ๐๐ง0 (๐ก)ใ over the course of the ramp at๐๐ = 0.25, 0.5. The corresponding LCD curves are shown to coincidewith the FE curves. Curves for UA and CD at ๐๐ = 0.05 ๐0 areshown for reference. Parameters: ๐ฟ = 8, ฮฉ๐ต = 10, ฮ0 = ๐0 = 5,๐ = 0.1, ๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0.05, ๐0 โ 1000, and ๐ = 100 (Note: Fordisplay, we have graphically reduced ๐ by a factor 10 to decreasecurve density).
The upper panel of Fig. 10 showcases the FE ramp ฮ(๐ก)in Eq. (64) for various ramp times ๐๐ . The vertical axis is re-scaled by the magnitude of the initial/final detunings ฮ0 โก ๐0,which are designed to coincide with the target ramp at the rampendpoints. The target ramp ๐(๐ก) is shown for reference (dashedblack curve). Near the adiabatic breakdown time ๐๐/๐0 = 1,the FE ramp ฮ(๐ก) has a base profile (averaging out the oscil-lations) similar to ๐(๐ก), with small oscillation amplitudes thatget slightly more pronounced in the middle of the ramp aroundresonance. For progressively faster ramps ๐๐/๐0 = 0.05, 0.025,
the FE ramp ฮ(๐ก) shows more pronounced deviations from๐(๐ก). First observe that the base profile of FE changes, keepingthe system near resonance for a progressively larger amountof time. Moreover, the amplitude of the high-frequency oscil-lations around resonance progressively increases due to ๐ฝ(๐ก)in ฮ(๐ก). Physically, these properties ensure the qubit and bathinteract strongly and long enough to effect polarization transferin accordance with LCD.
The lower panel of Fig. 10 serves two purposes: (i) to showthe effect of FE on the mean qubit z-polarization ใ๐๐ง0ใ overthe course of a sweep, and (ii) to highlight the equivalenceof FE and LCD protocols. The qubit is initialized with spindown ใ๐๐ง0ใ = โ0.5 in a mixed state. Over the course of theramp ฮ(๐ก), FE (solid colored curves) transfers a large fractionof the qubit polarization to the bath in perfect agreement withLCD (dashed white lines). For reference we also show an UAprotocol at ๐๐/๐0 = 0.05 (dashed black curve); as expected it ismuch less efficient compared to FE/LCD and CD at this rampspeed.
In sum, we can systematically realize LCD with FE, wherethe LCD protocol can be implemented indirectly in experi-ments by driving the local qubit field ฮฉ๐ (๐ก) periodically athigh-frequencies ๐ ๏ฟฝ ฮฉ๐,ฮฉ๐ต, ๐โ1๐ . Importantly, the FE pro-tocol requires no controls which are not already present in ๐ปin Eq. (2), similar in spirit to Ref. [59]. It can be achieved bysetting a fixed global field ฮฉ๐ต and dynamically varying ฮฉ๐ (๐ก),without modifying the qubit-bath interactions. This result dif-fers from other schemes which require controlling interactionsto realize LCD with Floquet engineering [55, 57, 58].
C. Quantum Speed Limit
The distinction between the lab time ๐ก and the rescaled time๐ gives rise to a quantum speed limit. Namely, there exists acritical ramp time ๐๐ = ๐๐๐ฟ > 0 in the lab frame for whichthe protocol duration ๐๐ in rescaled time becomes zero and๐ฝ โ ๐/2. Given sufficiently large driving frequencies, it isalways possible to realize LCD using FE if the LCD ramp timeis larger than this critical ramp time. However, at shorter ramptimes, the proposed protocol would lead to negative protocoldurations in stretched times, and the FE protocol can no longerrealize LCD. The speed limit can derived (see Appendix I) byinverting Eq. (66):
๐๐๐ฟ = lim๐ โ0
โซ ๐ 0
[๐บ (๐ก (๐ โฒ)]โ1๐๐ โฒ โผ ฮโ1typ. (69)
The timescale ๐๐๐ฟ is set by the typical bright pair gap ฮtyp,which is on the order of the time needed to transfer polarizationfrom the qubit to the bath while sitting at resonance. In fact,the profile ๐บ (๐ก)๐(๐ (๐ก)) of the FE protocol in Eq. (63) reducesto a sudden quench protocol to resonance as ๐ โ ๐๐๐ฟ .
Eqs. (64) and (69) provide an additional connection be-tween sudden and adiabatic polarization protocols: Floquet-engineering implements the adiabatic polarization protocol ina transformed frame, which resembles a sudden protocol inthe lab frame when the speed limit is approached. For rampsfaster than the speed limit ๐๐/๐๐๐ฟ < 1, FE can be extended
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FIG. 11. Power vs. protocol time. Disorder-averaged transfer poweragainst the scaled ramp time ๐๐ /๐0 for UA, CD, and FE protocols. Thevertical axis is scaled by (1/๐๐๐ฟ) (grey dash-dotted line). The verticalblue solid line marks the disorder-averaged speed limit timescale ๐๐๐ฟ .Parameters: ๐๐ = 50, ๐ฟ = 8, ฮฉ๐ต = 10, ฮ0 = ๐0 = 5, ๐ = 0.1,๐พ๐ฅ๐ฅ = 0.05, ๐พ๐ง = 0.05, ๐0 โ 1000, and ๐ = 100.
by taking ฮ(๐ก) = ๐2 ๐ sin(๐ ๐ก); then FE effectively oscillatesaround resonance for a shorter time than ๐๐๐ฟ and is no longeras effective as LCD.
This speed limit is quantified in Fig. 11, showing the transferpower vs. ramp time for several transfer protocols acrossresonance. The transfer power is defined as
ฮ๐๐ง0๐๐
=ใ๐๐ง0 (๐ ๐ )ใ โ ใ๐
๐ง0 (๐๐)ใ
๐๐, (70)
measuring the rate at which polarization is extracted from thequbit per unit ramp time. In the absence of tunable qubit-bathinteractions, the maximum possible polarization transfer is aunit of polarization on the timescale ๐๐๐ฟ , shown in the plot asa grey dash-dotted line. We find that UA protocols operate farbelow this rate over the whole range of ramp times. On theother hand, FE protocols significantly enhance power, peakingin the vicinity of the speed limit. At protocol durations ๐ > ๐๐๐ฟFE nears the efficiency of CD protocols which, as expected,transfers polarization more effectively than UA and FE perunit time. At protocol durations ๐ < ๐๐๐ฟ , the FE power liessignificantly below the CD power.
The presence of this speed limit suggests a broader physicallimitation: Any ramped protocol which does not directly tunesystem-bath couplings or add extra controls cannot transferpolarization at a faster rate than a sudden resonant exchange.
VII. CONCLUSION
In this work, we apply the tools of shortcuts to adiabaticityto a class of hyperpolarization protocols. In a single cycle ofeach such protocol, the qubit is reset along the โ๐ง direction byan external pulse, after which the ๐ง-field of the qubit is sweptacross a resonance region. Polarization is transferred from thequbit to the spin bath during the sweep.
We introduce local counterdiabatic driving (LCD) proto-cols that mimic an adiabatic protocol. The LCD protocols
simultaneously tackle two problems: (i) the small sweep ratesnecessary for the adiabatic transfer of polarization to the bath,and (ii) the limits on hyperpolarization imposed by dark states.The LCD protocols tackle (i) by efficiently suppressing dia-batic transitions between bright bands. They tackle (ii) bydepleting dark states in the presence of inhomogeneous bathfields (i.e., when the system is non-integrable). In this way,LCD protocols outperform both unassisted protocols and exactcounterdiabatic protocols since the former does not suppresstransitions between bright bands and the latter suppresses tran-sitions from dark to bright bands.
Using exact numerics and a master equation, we show thatthe LCD protocols outperform the unassisted ones by variousmetrics (efficiency, power, and number of cycles). Addition-ally, the LCD can be experimentally implemented through ahigh-frequency Floquet drive on the qubit. These engineeredprotocols have a natural quantum speed limit; once the sweeprate exceeds this limit, the LCD protocols cannot be realizedthrough Floquet drives.
The LCD may be used to speed up hyper-polarization inseveral experimental systems with dipolarly interacting spins.Indeed, Eq. (2) models the (rotating-frame) Hamiltonian of ashallow nitrogen-vacancy (NV) defect coupled to surface elec-tronic spins in high-purity diamond [79, 85, 86], as well as the(rotating-frame) Hamiltonian of a NV defect coupled to bulkC-13 nuclei [7, 12] or the nuclei of external molecules in solu-tion [17]. A promising avenue for future work is to compare theperformance of LCD protocols to sudden protocols that satisfythe Hartmann-Hahn condition [87] in these systems. Hyper-polarization of powdered diamond using NV-centers [12, 14]is also an important goal for magnetic resonance imaging. Itwould be interesting to develop LCD protocols that account forthe random orientation of the NV center axes (the ๐ง-directionof the central spin in Eq. (2)) in these systems. An additionaldirection for future work is to use optimal control theory todesign possibly more efficient protocols and test the validityof the speed limit. However, such a numerical optimizationproblem in the many-body setting is expected to be highlycomplex. In contrast, the LCD approach, while not guaranteedto be optimal, is readily extended to more complex systemswith an arbitrary number of spins and arbitrary interactions.
Theoretically, our work raises questions about the precise in-terplay between integrability and hyperpolarizability in centralspin models. The model in Eq. (2) with ๐พ๐ง = 0 is (i) integrable,and (ii) has exact dark states for any choice of ๐ ๐ [34]. How-ever, the closely related XXX model with ๐พ๐ง = 0 and isotropicqubit-bath interactions
โ๐ ๐ ๐ ยฎ๐0 ยท ยฎ๐ ๐ is integrable without ex-
hibiting dark states [88, 89]. The XXX model describes thehyperfine interactions of the electronic spin of a quantum dotwith surrounding nuclei [90, 91]. Previous work [31, 92] sug-gests that the spin bath can be efficiently polarized in the XXXmodel despite its integrability. A natural direction for futurework is to quantify the general role of integrability in the po-larization process. Another possible direction is to examinethe role of interactions in the spin bath. The accompanyingdiffusive spin transport is expected to aid in the polarization ofdistant bath spins with negligible ๐ ๐ .
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ACKNOWLEDGEMENTS.
The authors thank E. Boyers, M. Pandey, C.R. Laumann,and A. Sushkov for insightful discussions. The authors ac-knowledge support from the Sloan Foundation through a SloanResearch Fellowship (A.C.), from the Belgian American Edu-cational Foundation (BAEF) through the Francqui FoundationFellowship (P.W.C.), and from the BU CMT Visitor Program(P.W.C.). Numerics were performed on the BU Shared Com-puting Cluster with the support of the BU Research Comput-ing Services. This work was supported by EPSRC Grant No.EP/P034616/1 (P.W.C.), NSF DMR-1813499 (T.V. and A.P.),NSF DMR-1752759 and AFOSR FA9550-20-1-0235 (T.V. andA.C.), and AFOSR FA9550-16- 1-0334 (A.P.).
Appendix A: Derivation of central spin Hamiltonian
Consider a driven qubit-bath spin system in a magnetic field๐ต along the z-direction described by the Hamiltonian:
๐ป0 = ๐ป๐ + ๐ป๐ต + ๐ป๐ท + ๐ป๐๐ต + ๐ป๐ต๐ต, (A1)
Here ๐ป๐ = ๐พ๐ ๐ต ๐๐ง0 is the Zeeman energy of the central qubitwith gyromagnetic ratio ๐พ๐, and ๐ป๐ต = ๐พ๐ต ๐ต
โ๐ฟโ1๐=1 ๐
๐ง๐
is theZeeman energy of the bath spins with gyromagnetic ratio ๐พ๐ต.The driving term is given by
๐ป๐ท = 2ฮฉ๐ cos(๐๐๐ก) ๐๐ฅ0 + 2 cos(๐๐ต๐ก)๐ฟโ1โ๏ธ๐=1
ฮฉ๐ต, ๐ ๐๐ฅ๐ (A2)
where ฮฉ๐ is a Rabi amplitude on the central spin, ฮฉ๐ต, ๐ =ฮฉ๐ต + ๐ฟฮฉ ๐ is a generally inhomogeneous Rabi amplitude on thebath, and ๐๐, ๐๐ต are respectively the corresponding drivingfrequencies. The qubit-bath coupling is given by a dipoleinteraction term:
๐ป๐๐ต =โ๏ธ๐
๐พ๐๐พ๐ต
๐3๐
[ยฎ๐0 ยท ยฎ๐๐ โ 3 ( ยฎ๐0 ยท ๐๐) ( ยฎ๐๐ ยท ๐๐)
](A3)
where ยฎ๐๐ is the vector between the central qubit and the ๐thbath spin. The interaction term ๐ป๐ต๐ต between bath spin pairsis generally also dipolar. In this work, we assume bath-bathinteractions are small compared to the qubit-bath couplings,which is realized in experiments with sufficiently low bathspin density or with bath spins that satisfy ๐พ๐ต ๏ฟฝ ๐พ๐. TheHamiltonian ๐ป0 in Eq. (A1) describes single qubit systems,such as NV centers in diamond or quantum dots, interactingwith an ensemble of spins (e.g. spins on the surface of di-amond) and driven by continuous irradiation fields (such asradio waves) [5, 16, 17, 19, 23, 24, 38].
In a doubly rotated frame defined by the unitary transforma-tion
๐ = exp[โ ๐
(๐๐ ๐
๐ง0 + ๐๐ต
๐ฟโ1โ๏ธ๐=1
๐๐ง๐
)๐ก
], (A4)
๐ป0 can be simplified by matching the driving frequencies tothe Zeeman energies (๐๐ = ๐พ๐๐ต, ๐๐ต = ๐พ๐ต๐ต), and applying
a rotating wave approximation to eliminate rapidly rotatingnon-secular terms which average to zero on the timescale ofthe dynamics [24]. Relabeling our axes (๐ฅ, ๐ง) โ (๐ง,โ๐ฅ), thedominant time-averaged motion is described by
๐ปrot (๐ก) = ฮฉ๐๐๐ง0 +๐ฟโ1โ๏ธ๐=1
ฮฉ๐ต, ๐๐๐ง0 +
๐ฟโ1โ๏ธ๐=1
2 ๐ ๐ ๐๐ฅ0 ๐๐ฅ๐ (A5)
where
๐ ๐ โก๐พ๐๐พ๐ต
2 ๐3๐
[1 โ 3 cos2 (๐๐)], (A6)
and ๐๐ is the angle between ยฎ๐ต and ยฎ๐๐ in the frame of ๐ป0.We note our rotating wave approximation requires |๐ ๐/๐ต | ๏ฟฝ๐พ๐ , ๐พ๐ต , |๐พ๐ ยฑ ๐พ๐ต | , which is readily satisfied in NV centers orquantum dot experiments [16, 18].
The interaction term
๐๐ฅ0 ๐๐ฅ๐ =
14(๐+0๐
โ๐ + ๐โ0 ๐
+๐ + ๐+0๐
+๐ + ๐โ0 ๐
โ๐
)(A7)
describes zero quantum (flip-flop) transitions in its first twoterms, and double quantum (flip-flip/flop-flop) transitions in-teractions in its last two terms. Zero quantum transitions dom-inate when ๐ ๐ ๏ฟฝ ฮฉ๐ + ฮฉ๐ต, ๐ [24], yielding the Hamiltonianpresented in the main text:
๐ป (๐ก) = ฮฉ๐ ๐๐ง0 +๐ฟโ1โ๏ธ๐=1
ฮฉ๐ต, ๐ ๐๐ง๐+ 1
2
๐ฟโ1โ๏ธ๐=1๐ ๐
(๐+0๐
โ๐ + ๐โ0 ๐
+๐
). (A8)
Appendix B: Distribution of energy gaps in the homogeneouslimit
In this section, we compute the approximate distribution ofbright pair gaps ฮ๐ผ (Eq.(14)) in a system with a homogeneousbath field (๐พ๐ง = 0).
In the homogeneous limit (๐พ๐ฅ๐ฅ = 0), the central spin modelreduces to a two-body Hamiltonian
๐ป = ๐๐๐ง0 +๐
2(๐+0๐
โ + ๐โ0 ๐+) , (B1)
with ๐ยฑ =โ๐ฟโ1๐=0 ๐
ยฑ๐. The spectrum of this Hamiltonian can be
obtained in the collective bath spin basis, as the Hamiltonianonly couples the states: |โใ โ |๐ , ๐ใ , |โใ โ |๐ , ๐ + 1ใ. Here ๐ isthe total spin quantum number of the bath, and ๐ is the totalz-projection of the bath state, leading to ๐ = ๐ + 1/2. We takeโ๐ < ๐ < ๐ , since the states |โใ โ |๐ , ๐ ใ and |โใ โ |๐ ,โ๐ ใ aredark eigenstates of the Hamiltonian.
The energy in each two-dimensional subspace is fully deter-mined by the quantum numbers ๐ and ๐, leading to energiesยฑฮ๐ ,๐/2 at resonance given by
ฮ๐ ,๐ = ๐โ๏ธ(๐ โ ๐) (๐ + ๐ + 1). (B2)
The number of gaps equal to ฮ๐ ,๐ is fully determined by thenumber of ways the (๐ฟโ1) spin-1/2 bath spins can be coupled
-
17
to a collective spin ๐ with spin projection ๐. Furthermore,since ๐ < ๐ and ๐ is fixed by specifying ๐, this also leadsto a minimal gap within each polarization sector ๐ , given byฮ๐ = ๐
โ2๐ + 1, obtained by setting ๐ = ๐ + 1 = ๐ + 1/2.
Increasing ๐ resulting in an increasing ฮ๐ ,๐, whereas smallervalues of ๐ are not allowed within this polarization sector.
Given (๐ฟ โ 1) bath spins, the number of spin-๐ representa-tions is given by Catalanโs triangle as
๐๐ (๐ฟ) = ๐ถ ((๐ฟ โ 1)/2 + ๐ , (๐ฟ โ 1)/2 โ ๐ ) (B3)
=(๐ฟ โ 1)!(2๐ + 1)
(๐ฟ/2 โ 1/2 โ ๐ )!(๐ฟ/2 + 1/2 + ๐ )! , (B4)
which can be approximated for large ๐ฟ as
๐๐ (๐ฟ) โ๐๐ฟ ๐๐ (1/2โ๐ /๐ฟ)
โ2๐
2๐ + 1๐ฟ/2 + ๐ + 1
โ๏ธ๐ฟ
(๐ฟ/2 โ ๐ ) (๐ฟ/2 + ๐ ) ,
(B5)with ๐๐ (๐) = โ๐ ln(๐) โ (1 โ ๐) ln(1 โ ๐). The total magneti-zation ๐ fixes ๐ such that the energy gap only depends on ๐ ,and we can introduce a gap density as
๐(ฮ) = ๐๐ (ฮ) (๐ฟ)๐๐
๐ฮ, (B6)
with ๐ (ฮ) =โ๏ธ(ฮ/๐)2 + ๐2 โ 1/2 and
๐๐
๐ฮ=
ฮ/๐2โ๏ธ(ฮ/๐)2 + ๐2
. (B7)
As mentioned before, every fixed magnetization sector has aminimal gap, which we write as ฮ๐ โก ๐
โ2๐ + 1, such that
๐(ฮ < ฮ๐) = 0. Due to the presence of the exponentialterm, in the limit of large ๐ฟ all integrals over ๐(ฮ)๐ฮ willbe dominated by the boundary terms where ฮ โ ฮ๐. Ap-proximating the exponential factor at ๐ /๐ฟ = |๐ |/๐ฟ โก ๏ฟฝฬ๏ฟฝ forฮ๐ < ฮ ๏ฟฝ ๐๐ , we find
๐(ฮ) โ ๐พ (๏ฟฝฬ๏ฟฝ) ๐๐ฟ ๐๐ (1/2โ๏ฟฝฬ๏ฟฝ)โ
2๐๐ฟฮ
ฮ2๐
(1 โ 2๏ฟฝฬ๏ฟฝ1 + 2๏ฟฝฬ๏ฟฝ
) ฮ2ฮ2๐, (B8)
with
๐พ (๏ฟฝฬ๏ฟฝ) = 4๏ฟฝฬ๏ฟฝ1/2 + ๏ฟฝฬ๏ฟฝ
โ๏ธ1
1/4 โ ๏ฟฝฬ๏ฟฝ2. (B9)
Appendix C: Diabatic transitions in the Landau-Zener problem
The Landau-Zener (LZ) problem, described in Eq. (8) ofthe main text, consists of a two-level system with gap ฮ๐ฟ๐ =โ๐2 + ฮ2, where ๐ is a control field and ฮ is the minimum
gap [80, 81].When the control field is varied at a speed ยค๐ โผ ๐0/๐๐ , we
can estimate the speed scale below which the system remainsadiabatic. Adiabaticity occurs when the rate of change of thegap ฮ is smaller than the dynamical timescale over the whole
range of the control field ๐. In particular, this condition holdsif it is satisfied near resonance (|๐ | โผ ฮ) where the gap issmallest. ( ยคฮ๐ฟ๐
ฮ๐ฟ๐๏ฟฝ ฮ๐ฟ๐
) ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐=ฮ
=โ ยค๐ ๏ฟฝ ฮ2. (C1)
Therefore we satisfy the adiabatic condition when the ramptimescale ๐๐ ๏ฟฝ ๐0/ฮ2. In faster ramps moreover, the scale๐0 โผ ๐0/ฮ2 sets the scale for the onset of diabatic transitions.
As discussed in the main text for our central spin model, theLZ problem directly captures the interactions between brightbands. Nevertheless, the LZ problem can also help understandtransitions between dark and bright bands, as we discuss next.
In system with ๐พ๐ง = 0, the gap ฮ๐ท๐ต between a dark statewith energy ๐ธ๐ท and a neighboring bright state with energy ๐ธ๐ตis given by
ฮ๐ท๐ต = |๐ธ๐ต โ ๐ธ๐ท | =12
โ๏ธ๐2 + ฮ2min ยฑ
12๐, (C2)
since ๐ธ๐ท = ยฑ๐/2 and ๐ธ๐ต = ยฑ 12โ๏ธ๐2 + ฮ2min and where
ฮmin โก min๐ผฮ๐ผ is the minimum bright-bright gap at reso-nance. At resonance, the gap is ฮ๐ท๐ต = ฮmin/2, comparableto the minimum bright-bright gap. In contrast with bright-bright gaps, dark-bright gaps are smallest furthest away fromresonance ๐ = ยฑ๐0:
minฮ๐ท๐ต โ14
(ฮmin๐0
)2๐0. (C3)
In systems with ๐พ๐ง = 0, the presence of a small bright-dark gapdoes not imply fast ramps yield diabatic transitions becausethe driving operator ๐๐ง0 does not couple bright a