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Mid-range adiabatic wireless energy transfer via a mediator coil Andon A. Rangelov1,2 and Nikolay V. Vitanov1 1Department of Physics, Sofia University, 5 James Bourchier blvd, 1164 Sofia, Bulgaria 2Corresponding author: [email protected]fia.bg (Dated: January 24, 2012) A techniquefor efficient mid-range wireless energy transfer between two coils viaa mediator coil is proposed. By varying the coil frequencies three resonances are created: emitter-mediator (EM), mediator-receiver (MR) and emitter-receiver (ER). If the frequency sweeps are adiabatic and such 2 that the ER resonance precedes the MR resonance, the energy flows sequentially along the chain 1 emitter-mediator-receiver. If the MR resonance precedes the ER resonance, then the energy flows 0 directly from the emitter to the receiver via the ER resonance; then the losses from the mediator 2 are suppressed. This techniqueis robust to noise, resonant constraints and external interferences. n PACSnumbers: a J 2 Since the pioneering workof Tesla the searchfor wire- G G r G 2 w less power transfer is a hot topic for its vast potential m ] to substitute electrical cables and batteries. Until very G w r h recentlytraditionalmagneticinduction,whereintwocon- e w p ductivecoilsinducteachother,hasbeenusedtotransfer m receiver - energywirelesslyoververyshortdistancesonly. Thecoils w m e donotmakedirectelectricalcontactwitheachotherbut o they must be very close because the efficiency of power mediator t emitter a transferdrops by ordersofmagnitude when the distance s. between the coils becomes larger than their sizes. FIG. 1: (Color online) Three coils with intrinsic frequencies ic Recently, some novel ideas for medium-range wireless ωe,ωm andωr. Thelargesizeofthemediatorleadstostrong s energy transfer have emerged [1, 2]. Specially designed couplingswiththeotherscoils,butalsotolargerlossrateΓm y magneticresonatorsachievestrongcouplingbetweenthe than theloss rates Γe and Γr of theemitter and receiver. h coils that enable efficient energy transfer over distances p much larger than the size of the coils [1, 2]. This tech- [ nique,whichdemandsthesamefrequenciesoftheemitter againstvariationsoftheparameters,suchastheintrinsic 1 andreceivercoils,canbemademorerobusttovariations frequencies of the coils and the couplings between them. v ofthe coilparametersbysweepingthe emitter frequency WeconsiderthreeparallelhelixcoilsasshowninFig.1: 2 through resonance with the receiver frequency (or vice twosmallcoils—anemitterandareceiver—andalarge 9 versa) [3] in a fashion reminiscent of adiabatic passage mediator,whichinduces strongcouplings. We follow the 5 4 through a level crossing in quantum physics [4]. description of the coupled mode theory as presented by Hamam et al. [5]. The energy transfer, in the strong- . A recent theoretical paper [5] proposed a setup of two 1 coupling regime, is described by a set of three coupled identical coils (emitter and receiver) strongly coupled to 0 differential equations (in matrix form) [5], 2 an intermediate coil (mediator) of different properties 1 butwiththe sameintrinsic frequency. The energytrans- dA(t) : fer occurs due to adiabatic following of an instantaneous i =H(t)A(t), (1a) v dt i (“dark”)eigenstateofthethree-coilsystem,anideasim- X ilar to the quantum three-state technique of stimulated where A=[a (t),a (t),a (t)]T and e m r r Raman adiabatic passage (STIRAP) [6–8]. In this tech- a niqueboththeemitterandthereceivershouldberotated ω −iΓ κ 0 e e em insynchroinordertoengineertime-dependentcouplings H= κem ωm−iΓm κmr . (1b) that reduce the energy stored at the mediator [5]. It is, 0 κ ω −iΓ −iΓ mr r r w however,crucialtomaintainanexactresonancebetween   the emitter andreceiverfrequencies [5]; a frequency mis- Here a (t), a (t) and a (t) are defined so that the en- e m r match, e.g., due to differences the coils or random noise ergies contained in the emitter, the mediator and the (whichmaybecausedbyexternalobjectsnearthecoils), receiver are, respectively, |a (t)|2, |a (t)|2 and |a (t)|2, e m r reduces the transfer efficiency significantly. while Γ , Γ and Γ are the corresponding intrinsic loss e m r Inthispaper,weproposeadifferentapproachtowire- rates(duetoabsorptionandradiation),andω (t), ω (t) e m lessenergytransferbetweentwocoilsviaamediatorcoil and ω (t) are the corresponding intrinsic frequencies. r by using the ideas of adiabatic population transfer in a The extraction of work from the receiver is described three-statequantumsystemwithcrossingenergies[9–11]. by the term Γ . The coupling coefficient between the w This technique promises to be both efficient and robust emitterandthemediatorisκ andtheonebetweenthe em 2 mtdhiaeedtmoiaratiotsrrimxanuHdc(htt)hladeregrreeivrceetihfvraeonrmitshtheκemearms.situTtmehrpetaninoudnlltthehleaetmrteehcneetisvmeiern-; encies w r e3 e2 A e2w r e3 w mB w mw r e3 w e C hence the direct emitter-receiver coupling is negligible Frequ w m e1 e e2 comparedtothecouplingsκem andκmr andisneglected. w e w e 1 e1 Theefficiencyoftheenergytransferisdescribedbythe 1.0 tehffiecwieonrckyecxoterffiacctieedntfroηm[1t–h3e],rewcheiicvherifsorthteheratitmioebinettwereveanl Energies 00..68 |ae|2 |ar|2 D |ae|2 |ar|2 E |ae|2 |ar|2 F ftoi−r tthdeivsiadmedebtiymteheinttoetravlael,nergy(absorbedandradiated) ormalized 00..24 |am|2 |am|2 |am|2 N 0 Γ E (t) SwchhEerqröeudaEintηkig(o=ten)r=Γe(eq1ERu)tetai(tt|iia)sok+n(tiΓdf)o|em2rndEtatmicw((takthl)rr=e+teoe-(s,Γtmartht,+eer)Γq.wtuia)mEnert(-udtm)e,pseynsdte(en2mt) Normalized Energies 00001.....246800 |a|ea|2m|2 |ar|2hG |ae||2am|2 |ar|2hH |ae||a2m|2 |ar|2hI -50 -25 0 25 50 -50 -25 0 25 50 -50 -25 0 25 50 which is studied in great detail [7, 8]; the vector A(t) and the driving matrix H(t) correspond to the quan- Time (units of a -1) Time (units of a -1) Time (units of a -1) tum state vector and the Hamiltonian, respectively. By FIG.2: (Coloronline)Wirelessenergytransfer. Leftcolumn: definition, in the adiabatic limit the system stays in sequentialtransfer(δ=7α);middlecolumn: bow-tietransfer an eigenvector of H(t). We first assume that the loss (δ=0);rightcolumn: directtransfer(δ=−7α). Topframes: rates Γe, Γm, Γr and Γw are all zero; then the quantity Diagonalelements(solidlines)andeigenvalues(dashedlines) |A(t)|2 =|a (t)|2+|a (t)|2+|a (t)|2isconserved,likethe of H of Eq. (1). Middle frames: The energies in the coils e m r totalpopulationin a coherentlydrivenquantum system. calculatednumericallyfromEq.(1)withnolosses,Γe =Γm = The key of our proposal is the assumption that the Γr = Γw = 0 for κem = κmr = 3.5α. Bottom frames: The frequencyofthemediatorcoilisfixed,whilethefrequen- energies in the coils and the efficiency coefficient η for Γe = cies of the emitter and receiver coils change in time in Γr =0.001α, Γm =Γw =0.05α. opposite directions, ω (t)=ω +δ+α2t, (3a) column and δ <0 in the right column. In the beginning e m ω =const, (3b) andtheendeacheigenfrequencyεn(t)coincideswithone m ofthe intrinsicfrequenciesωk(t)ofthe coils,while inbe- ω (t)=ω +δ−α2t, (3c) r m tweeneachεn(t)isasuperpositionofωe(t),ωmandωr(t). Intheadiabaticlimit,thesystemfollowsaneigenstateof whereδisasuitablychosenstaticfrequencyoffset,which H(t)andtheeigenfrequencyofthesystematanyinstant controls the energy flow. For the sake of generality, we of time is equal to the (time-dependent) eigenfrequency take hereafter α>0 as the unit of frequency and 1/α as in which it is initially, i.e. ε (t). However, the com- 1 the unit of time. We assume thatthe fixed mediator fre- position of ε (t) is different for different δ because the quencyω ismuchlargerthanthe termsδ±α2tinω (t) 1 m e topology of the crossings differ. and ω (t); therefore the couplings κ and κ can be r em mr For δ > 0 (left column of frames), the frequency reso- assumed constant for the sake of simplicity. Taking the nances occur in the intuitive manner: first the emitter- variations of the couplings due to the frequency sweeps mediator resonance and then the mediator-receiverreso- into account does not change the results significantly. nance. The eigenfrequency ε (t) is initially nearly equal For δ 6= 0, the intrinsic frequencies of the three coils 1 to ω , at intermediate times to ω and in the end to cross each other at three different instants of times, e m ω . Consequently,the energyflows sequentially from the therebycreatingatrianglecrossingpattern[9–11],shown r emittertothemediatorandthentothereceivercoils;this in Fig. 2 (top frames). For δ = 0, the three frequencies sequentialflowisdemonstratedinframeD.Thistwo-step crossatthesameinstantoftime,therebycreatingabow- scheme extends the single-step adiabatic wireless energy tiecrossing(frameB)[12]. Thesecrossingpatternsallow transfer proposed earlier [3]. ustodesignrecipesforefficientadiabaticwirelessenergy transfer, in analogy to adiabatic passage techniques in For δ = 0 (middle column of frames) — the bow-tie quantum physics [7–11]. crossing — all resonances occur at the same time. The The proposed technique is illustrated in Fig. 2. The compositionofε1(t)is similarasforδ >0,thedifference topframesshowtheevolutionoftheintrinsicfrequencies being the lesser contribution of the mediator frequency of the three coils ωk(t) (k = e,m,r) and the eigenval- ωm at intermediate times. Consequently, the mediator ues εn(t) (n = 1,2,3) (the eigenfrequencies) of H(t) of receives less transient energy, as seen in frame E. Eq. (1). The three columns of frames differ by the offset For δ < 0 (right column of frames), the frequency frequencyδ: δ >0intheleftcolumn,δ =0inthemiddle resonances occur in a counterintuitive manner, with the 3 mediator-receiverresonanceoccurringbeforetheemitter- κ = κ for the mediator-receiver resonance. For the mr mediator resonance. In fact, the system never comes bow-tie crossing, the condition is (for κ =κ ) [12] em mr close to the “nearest-neighbor” emitter-mediator and mediator-receiver resonances but it passes through the κ ln(1/ǫ) direct emitter-receiver resonance (frame C). The eigen- > . (5) α r π frequency ε (t) remains very far from the mediator fre- 1 quencyωm;consequently,verylittleenergyattendstran- For the directemitter-receiverresonance,we canuse the siently the mediator coil, as seen in frame F. This fea- LZSM formula again, by taking into account that the tureisreminiscentoftheenergytransferintheSTIRAP- sweep rate is 2α and the effective emitter-receiver cou- based technique [5]. pling is κ= κ2 +κ2 +δ2/4−δ/2 [11], em mr The coil parameters in Fig. 2 — the intrinsic frequen- p cies, the couplings and the sweep rate — are chosen in κ 2ln(1/ǫ) a such way that in the absence of losses the evolution is > . (6) adiabaticandhencetheenergyistransferredalmostcom- α r π pletely to the receiver coil in the end, as seen in frames In addition to these conditions, efficient energy transfer D, E and F of Fig. 2. However, the temporary transfer requires that the loss rates be small compared to the of energy to the mediator coil makes the energy transfer couplings and the interaction time T, efficiency very different in the presence of losses, as seen inthebottomframesofFig.2. Forδ >0,alargeamount ofenergyvisitsthemediatorwhereissubjectedtostrong Γk ≪1/T,κ. (7) Onecanreadilyverifythatallthese conditionsaresatis- dissipationandmost ofit is lostbefore it has the chance fied for the parameters in Fig. 2. to reach the receiver (frame G). For δ = 0, less energy In conclusion, the proposed technique for adiabatic attends the mediator coil and hence some more energy wirelessenergytransferbetweenanemitterandareceiver reaches the receiver (frame H). For δ <0, only little en- coil via a larger mediator coil allows one to transfer energy visits the mediator coil and a significant amount of ergywithhighefficiencybyvaryingtheintrinsicfrequen- energy is transferred to the receiver (frame I). cies of the coils in the counterintuitive way, in which the We therefore identify the directadiabatic path, occur- mediator-receiver resonance occurs before the emitter- ring for δ < 0, as the optimal path for energy transfer mediatorresonance. Thenonlyasmallamountofenergy from the emitter to the receiver. visits the mediator and hence the losses from the media- Next we turn our attention to the conditions for adi- tor are minimized. The presence of a large mediator coil abatic evolution in the three distinct cases δ > 0, δ = 0 allows one to increase the coil-coil couplings and there- and δ < 0. By using the formula for the transition foretoincreasethedistancebetweentheemitterandthe probability for the Landau-Zener-Stu¨ckelberg-Majorana (LZSM) model [4], p=1−exp(−2πκ2/α2), we find that receivercomparedtothesimple two-coilemitter-receiver setup[3]. ComparedtotheSTIRAP-basedenergytrans- the condition for transition probability larger than 1−ǫ fer technique [5], the present technique does not require at each crossing reads identicalcoilfrequenciesandtime-varyingcouplings,and therefore it may be easier to implement. κ ln(1/ǫ) α >r 2π , (4) This work is supported by the European Commission project FASTQUAST, and the Bulgarian National Sci- where κ = κ for the emitter-mediator resonance and ence Fund grants D002-90/08and DMU-03/103. em [1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Bergmann, Annu.Rev. Phys.Chem. 52, 763 (2001). Fisher, and M. Soljaˇcić, Science317, 83 (2007). [8] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. [2] A. Karalis, J. D. Joannopoulos, and M. Soljaˇcić, Ann. Bergmann, Adv.At.Mol. Opt.Phys. 46, 55 (2001). Phys.323, 34 (2008). [9] B.Broers,L.D.Noordam,andH.B.vanLindenvanden [3] A. A. Rangelov, H. Suchowski, Y. Silberberg, and N. V. Heuvell, Phys. Rev.A 46, 2749 (1992). 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