Time-optimal polarization transfer from an electron spin to a nuclear spin
Haidong Yuan, Robert Zeier, Nikolas Pomplun, Steffen J. Glaser, Navin Khaneja
TTime-optimal polarization transfer from an electron spin to a nuclear spin
Haidong Yuan, ∗ Robert Zeier, † Nikolas Pomplun,
2, 3, ‡ Steffen J. Glaser, § and Navin Khaneja Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong, Shatin, Hong Kong Department Chemie, Technische Universit¨at M¨unchen, Lichtenbergstrasse 4, 85747 Garching, Germany Bruker BioSpin GmbH, Silberstreifen 4, 76287 Rheinstetten, Germany Department of Electrical Engineering, IIT Bombay, Powai, Mumbai 400 076, India (Dated: September 7, 2015)Polarization transfers from an electron spin to a nuclear spin are essential for various physi-cal tasks, such as dynamic nuclear polarization in nuclear magnetic resonance and quantum statetransformations on hybrid electron-nuclear spin systems. We present time-optimal schemes forelectron-nuclear polarization transfers which improve on conventional approaches and will havewide applications.
PACS numbers: 03.67.Ac, 33.25.+k, 33.35+r, 02.30.Yy
I. INTRODUCTION
As the gyromagnetic ratio of an electron is two to threeorders of magnitude larger than the one of a nucleus, elec-tron spins are much easier polarized than nuclear spins.This offers a way to improve the polarization of nuclearspins by transferring polarization from electron spins tonuclear spins; much higher nuclear spin polarization canbe achieved as compared to a direct polarization. Thisidea has been widely used in various physical settings,for example dynamic nuclear polarization (DNP) [1–6]employs this idea to dramatically improve the sensitivityof nuclear magnetic resonance (NMR) [7, 8]. It is alsofrequently used on various hybrid electron-nuclear spinsystems, such as organic single crystals [9], endohedralfullerenes [10–12], phosphorous donors in silicon crystals[13], and nitrogen-vacancy centers in diamond [14–17].For example, in the case of nitrogen-vacancy centers indiamond, efficient polarization transfers are used to ini-tialize the quantum state of nuclear spins for quantuminformation processing.Efficient polarization transfers are practically achievedby properly engineered pulse sequences whose design isstudied in the field of quantum control [18–23]. In re-cent years, significant progress has been made in quan-tum control for both numerical [24–37] and analytical[38–41] methods. Extensive knowledge has been gainedon optimal pulse sequences for two- and three-level sys-tems [42–56], two uncoupled spins [57, 58], and two cou-pled spins [59–65]. Further advances have been made onhow to optimally control multiple coupled spins [66–91].These methods have been successfully applied in NMR[92, 93] to designing broad-band [94–96] and decouplingpulse sequences [97–102]. They have also been utilized ∗ [email protected] † [email protected] ‡ [email protected] § steff[email protected] in magnetic resonance imaging [25, 103–105] and electronparamagnetic resonance [106].In this article, we consider time-optimal pulse se-quences for polarization transfers from an electron spinto a nuclear spin. Relaxation and decoherence are inpractice inevitable and result in a loss of signal. Buttheir effect can be mitigated by short pulse sequenceswhich allow for highly sensitive experiments. We ana-lyze and explain how the form of time-optimal sequencesdepends on the direction of the polarization by studyingtime-optimal transfers for different directions.Recent analytical [107] and numerical [108, 109] studiesfocused on low-field single-crystal experiments, where thenuclear Larmor frequency and pseudo-secular hyperfineinteraction (see Sect. 3.5 of Ref. [1]) are comparable inmagnitude. As in [110–113], we focus here on the casesof secular hyperfine coupling (see Sect. 3.5 of Ref. [1]).These assumptions are satisfied in liquid-state and high-field solid-state DNP.We analyze two particular cases of polarization trans-fers and determine the corresponding time-optimal se-quences. In Section II, we consider the transfer from thestate S z of the electron spin to the state I z of the nuclearspin. The second time-optimal transfer from S z to I x is presented in Section III. And most interestingly, thecorresponding optimal transfer time is shorter by 78 . S z to I z , which high-lights that the transfer efficiency depends crucially on thetarget state of the nuclear spin. We discuss our results inSection IV, and the possibility of a non-sinusoidal carrierwave form is entertained in Section V. We conclude inSection VI, and certain details are relegated to Appen-dices A and B. II. TRANSFER FROM S z TO I z In this section, we study the polarization transfer fromthe initial state S z to the final state I z [114]. We assumea secular hyperfine coupling (see Sect. 3.5 of Ref. [1]). In a r X i v : . [ qu a n t - ph ] S e p the lab frame, the resulting Hamiltonian is given by H = ω S S z + ω I I z + 2 πAS z I z + 2 π ˜ u x ( t ) S x + 2 π ˜ v x ( t ) I x , (1)where ω S and ω I denote the respective Larmor frequen-cies of the electron and the nuclear spin, A representsthe strength of the secular hyperfine coupling, ˜ u x ( t ) and˜ v x ( t ) are the amplitudes of the control fields. Here, S j =( σ j ⊗ σ ) / I k = ( σ ⊗ σ k ) / j, k ∈ { x, y, z } , where σ := ( ) denotes the identity matrix and the Pauli ma-trices are σ x := ( ), σ y := (cid:0) − ii (cid:1) , and σ z := (cid:0) − (cid:1) .For typical NMR settings, only a single radio-frequencycoil is used which can be assumed to be oriented alongthe x axis of the lab frame. Hence, only a single control˜ v x ( t ) appears for the nuclear spin in the lab frame Hamil-tonian of Eq. (1). We assume in this work that the carrierwave form for the nuclear spin has a sinusoidal shape, i.e.˜ v x ( t ) = v ( t ) cos[ ω rf I t + φ ( t )] with amplitude v ( t ) ≤ v max and phase φ . Here, ω rf I is the carrier frequency of theradio-frequency irradiation and 2 v max denotes the max-imal control amplitude (in the lab frame). This choiceof ˜ v x ( t ) is motivated by the properties (e.g., bandwidthlimitations) of the usually available wave form genera-tors and amplifiers. More general carrier wave forms arediscussed in Section V.By switching to the rotating frame of ω S S z + ω rf I I z corresponding to the carrier frequencies ω S and ω rf I = ω I − ω off I and applying the rotating wave approximation,we get an effective Hamiltonian H rot = + ω off I I z + 2 πAS z I z + H mwrot + H rfrot , where (2) H mwrot := 2 πu x ( t ) S x + 2 πu y ( t ) S y ,H rfrot := 2 πv x ( t ) I x + 2 πv y ( t ) I y . One can obtain any desired offset term ω off I I z in the driftterm of H rot in Eq. (2) by suitably choosing the carrierfrequency ω rf I [115]. For simplicity, ω off I is set to zero inthe following. The microwave-frequency control pulses onthe electron spin and the radio-frequency control pulseson the nuclear spin are given by H mwrot and H rfrot , respec-tively. The control amplitudes u x ( t ), u y ( t ), v x ( t ), and v y ( t ) satisfy the bounds (cid:113) u x ( t ) + u y ( t ) ≤ u max and (cid:113) v x ( t ) + v y ( t ) ≤ v max , where u max and v max denote the maximal available am-plitudes of the control fields in the rotating frame fora given experiment. This is a result of the rotatingwave approximation, which reduces the maximal con-trol amplitude of 2 v max in lab frame to v max in the ro-tating frame [8]. In the following, we will assume that u max (cid:29) A (cid:29) v max and neglect the time needed to ap-ply operations that can be generated by the hyperfinecoupling and the controls on the electron spin [116].Time-optimal transformations are essentially only lim-ited by the weak controls on the nuclear spin. The opti-mal strategy to achieve a desired transfer can be inferred (cid:1) I /(2 (cid:2) )+ A /2 (cid:1) I /(2 (cid:2) ) (cid:1) I /(2 (cid:2) )- A /2(a) 2 S ③ I ③ (cid:3) (cid:4) (cid:1) I /(2 (cid:2) )+ A /2 (cid:1) I /(2 (cid:2) ) (cid:1) I /(2 (cid:2) )- A /2(b) I ③ (cid:3) (cid:4) FIG. 1. Schematic depiction of the absorption profiles for(a) 2 S z I z and (b) I z . x yzβ x yzα FIG. 2. In the polarization transfer from 2 S z I z to I z , the β component of the nuclear spin doublet is rotated by anangle of π around the y axis and the α component is leftinvariant (both visualized in the interaction frame). Notethat the electron spin is in the state | β (cid:105) on the left-hand sideand in the state | α (cid:105) on the right-hand side. from the structure of cosets with respect to the fast op-erations [59, 66, 110]. Here, the fast operations are givenby the hyperfine coupling and the strong controls on theelectron spin. The transformation U which transfers S z to I z = U S z U − will be suitably decomposed into a prod-uct U = U U . The unitary U transfers the initial state S z to the intermediate state 2 S z I z = U S z U − , and itcan be generated using only fast operations. In addition,the unitary U transfers the intermediate state 2 S z I z tothe final state I z = U (2 S z I z ) U − , and one has to usethe weak controls on the nuclear spin in order to gener-ate U . Below, we will provide a time-optimal scheme toproduce U . This results also in a time-optimal schemefor U as any faster scheme for U would also imply a fasterone for U = U U − .The polarization transfer from S z to I z can be decom-posed into the following steps [117]: S z π S y −−−→ S x πS z I z −−−−→ S y I z π S x −−−→ S z I z πS β I y −−−−→ I z , (3)where we denote S α := ( ) ⊗ σ and S β := ( ) ⊗ σ , then πS β I y = − πS z I y + πI y /
2. As shown in Eq. (3), thepolarization transfer from S z to 2 S z I z is accomplishedusing an INEPT-type transfer [8, 118]: First, we apply ahard π/ y direc-tion (i.e. S y ). Then, we let the hyperfine coupling evolvefor the duration of 1 / (2 A ) units of time. Another hard π/ x directioncompletes the transfer to 2 S z I z . All of these steps takenegligible time, since they are either local operations onthe electron spin or operations which can be generatedby the coupling. In conclusion, we can completely focuson the last step in Eq. (3) where we need to generate thepropagator U βy ( θ ) = exp( − iθS β I y ) = exp[ − i ( − θS z I y + θ I y )] (4)for θ = π . The operator in the exponent of U βy ( θ ) inEq. (4) is a single-transition operator [1, 8]. In partic-ular, the operator U βy ( π ) = exp( − iπS β I y ) describes atransition-selective π rotation around the y axis in thesubspace spanned by the basis states | βα (cid:105) and | ββ (cid:105) ,where the subspace corresponds to the β component ofthe nuclear spin doublet at frequency ω I / (2 π ) + A/ | α (cid:105) and | β (cid:105) are eigenstates of S z and I z , e.g., S z | α (cid:105) = | α (cid:105) / S z | β (cid:105) = −| β (cid:105) / U βy ( π ). The set of all unitarieswhich transfer 2 S z I z to I z are discussed in Appendix A 1,where we also show by extending the results in the cur-rent section that choosing a different element from thisset of unitaries does not lead to a shorter transfer time.To determine the optimal transfer, we switch to the in-teracting frame of 2 πAS z I z by applying the transforma-tion exp( i πAS z I z t ) H rot exp( − i πAS z I z t ). The Hamil-tonian of Eq. (2) changes to [120] H int = + 2 πu x ( t )[cos( πAt ) S x − sin( πAt )2 S y I z ]+ 2 πu y ( t )[cos( πAt ) S y + sin( πAt )2 S x I z ]+ 2 πv x ( t )[cos( πAt ) I x − sin( πAt )2 S z I y ]+ 2 πv y ( t )[cos( πAt ) I y + sin( πAt )2 S z I x ] (5)which can be also written as H int = + 2 πu x ( t )[cos( πAt ) S x − sin( πAt )2 S y I z ]+ 2 πu y ( t )[cos( πAt ) S y + sin( πAt )2 S x I z ]+ 2 π [ v x ( t ) cos( πAt ) I x + v y ( t ) sin( πAt )2 S z I x ] − π [ v x ( t ) sin( πAt ) − v y ( t ) cos( πAt )] S α I y + 2 π [ v x ( t ) sin( πAt ) + v y ( t ) cos( πAt )] S β I y , (6)where S α I y = S z I y + I y / S β I y = − S z I y + I y / U βy ( π ) in minimumtime, which corresponds to maximizing the coefficient2 π [ v x ( t ) sin( πAt ) + v y ( t ) cos( πAt )] in front of S β I y . TheCauchy-Schwarz inequality implies[ v x ( t ) sin( πAt ) + v y ( t ) cos( πAt )] ≤ [ v x ( t ) + v y ( t )][sin ( πAt ) + cos( πAt ) ] ≤ v , where the second inequality is a consequence of the con-straints on the amplitude of the control fields. The max-imal value of 2 π [ v x ( t ) sin( πAt ) + v y ( t ) cos( πAt )] is de-noted by 2 πv max and it can be achieved by choosingthe controls u x ( t ) = u y ( t ) = 0, v x ( t ) = v max sin( πAt ), v y ( t ) = v max cos( πAt ). To understand that this choicegenerates the desired operator, one can substitute thecontrols in the Hamiltonian with the chosen values andobtains H int = + 2 πv max sin( πAt ) cos( πAt ) I x + 2 πv max cos( πAt ) sin( πAt )2 S z I x + 2 πv max cos(2 πAt ) S α I y + 2 πv max S β I y . Since A (cid:29) v max , average Hamiltonian theory implies thatthe first three terms average out to zero; and one is leftwith the desired Hamiltonian 2 πv max S β I y .The minimum time to generate U βy ( π ) is then fixed bythe relation 2 πv max T min = π , and one obtains T min = 1 / (2 v max ) . The presented time-optimal control corresponds to aradio-frequency irradiation at frequency ω I / (2 π ) + A/ T min , which results in a transition-selectiveinversion of the β line of the nuclear spin doublet. Thisbelongs to the class of controls presented in Ref. [110]and is also closely related to selective population inver-sion (SPI) experiments [121–124]. -40-2002040 (a) ❆(cid:0)✁✂✄☎✆✝✞✟✠✡ H z ) u ① (b) u ② -2 ✲☛ ☛ ☛✶ ☛☞ ❆(cid:0)✁✂✄☎✆✝✞✟✠✡ H z ) ❚✌✍✎ ✏ (cid:1) s) v ① ☛✶ ☛☞ ❚✌✍✎ ✏ (cid:1) s) v ②✾✑✒ ✓✔✑✒ ✾✑✒ ✓✔✑✒ FIG. 3. Numerically optimized pulses for the polarizationtransfer from S z to I z . The coupling strength is 10 MHz andthe bounds on the micro-wave and radio-frequency amplitudesare u max = 1 MHz and v max = 20 kHz, respectively. Themaximal transfer efficiency is reached after 25 µs . The insetsshow magnified parts of the controls v x ( t ) and v y ( t ) in orderto illustrate their form. We can also compute the maximal transfer efficiency η max ( T ) for a given time T . The operator U βy ( θ ) transfersthe state 2 S z I z to the state U βy ( θ )(2 S z I z )[ U βy ( θ )] † =cos ( θ )2 S z I z + cos( θ ) sin( θ )2 S z I x − cos( θ ) sin( θ ) I x + sin ( θ ) I z . (7)For θ = 2 πv max T , we get the maximal transfer efficiency η max ( T ) = sin ( θ ) = sin ( πv max T ) (8)for the transfer to I z . Note that η max ( T min ) = 1.We compare our analytic results with numerical op-timizations for achieving the transfer from S z to I z asshown in Fig. 3, cf. [111–113]. For these optimizations,the hyperfine coupling constant is chosen as A = 10 MHzand the maximal allowed radiation amplitudes are set to u max = 1 MHz and v max = 20 kHz [125]. In Fig. 3, thetransfer is completed after 25 µs = 1 / (2 v max ) units oftime which agrees with the analytically computed time.Moreover, the form of the numerically optimized controlscompares nicely with the analytic results: the values of u x ( t ) and u y ( t ) are most of the time small (except for thebeginning), and v x ( t ) and v y ( t ) have a sinusoidal formwith the maximal allowed amplitude. III. TRANSFER FROM S z TO I x OR I y We analyze now how to time-optimally transfer po-larization from the state S z to I x (and similarly for thetransfer to I y ). The considered transfer consists of thefollowing steps: S z π S y −−−→ S x πS z I z −−−−→ S y I z π S x −−−→ S z I z π S α I y − π S β I y −−−−−−−−−−→ I x , where π S α I y − π S β I y = πS z I y . As in Sec. II, wecan focus on generating the final propagator ˜ U =exp( − iπS z I y ) in the product ˜ U = ˜ U U . The transfer I x = ˜ U S z ˜ U − is decomposed into a fast transfer to theintermediate state 2 S z I z = U S z U − and a slow transferto final state I x = ˜ U (2 S z I z ) ˜ U − . Building on the re-sults in this section, we prove in Appendix A 2 that onecannot reduce the transfer time by substituting ˜ U witha different unitary V satisfying I x = V (2 S z I z ) V − .Previously, the propagator exp( − iπS z I y ) in the finalstep has been achieved [110] by applying a transition-selective radio-frequency − π/ y direc-tion at the β transition with frequency ω I / (2 π ) + A/ π/ y direction at the α transition with frequency ω I / (2 π ) − A/
2, see Fig. 4. In the rotating frame ofEq. (2), this irradiation scheme on the nuclear spin cor-responds to a radio-frequency Hamiltonian of the form H rfrot = − π v max πAt ) I x − π v max πAt ) I y x yzβ x yzα FIG. 4. In the polarization transfer from 2 S z I z to I x the β component of the nuclear spin doublet is rotated by − π/ y axis and the α component is rotated by π/ y axis (both visualized in the interaction frame). + 2 π v max − πAt ) I x + 2 π v max − πAt ) I y = − πv max sin( πAt ) I y . Note that in this scheme the α and β transitions canonly be irradiated with a radio-frequency amplitude of v max / v max for the overall irradia-tion at the nuclear spin. Hence, the duration for thesimultaneous ± π/ y direction at the α and β transitions is equal to 1 / (2 v max ). This conven-tional transfer is optimal if one considers only pulses atthe frequencies ω I / (2 π ) ± A/ v max , shorter pulses can be obtained by irradiat-ing at the frequencies ω I / (2 π ) ± A/ additional well selected frequen-cies. Our approach is quite effective although it mightseem counterintuitive at first.In the interaction frame of 2 πAS z I z , the Hamiltonianis again given by Eq. (5). In order to generate the op-erator exp( − iπS z I y ) in minimum time, we maximize thecoefficient − πv x ( t ) sin( πAt ) of 2 S z I y . Note that − πv x ( t ) sin( πAt ) ≤|− πv x ( t ) sin( πAt ) | ≤ πv max | sin( πAt ) | , where the second equality is implied by the constraint | v x ( t ) | ≤ (cid:113) v x ( t ) + v y ( t ) ≤ v max on the control ampli-tudes. Therefore, the maximal value 2 πv max | sin( πAt ) | for − πv x ( t ) sin( πAt ) can be attained by choosing thecontrols u x ( t ) = u y ( t ) = v y ( t ) = 0 and v x ( t ) = − sgn[sin( πAt )] v max . This means that v x ( t ) is a squarewave such that v x ( t ) = v max when sin( πAt ) < v x ( t ) = − v max when sin( πAt ) >
0. In this case, theradio-frequency Hamiltonian in the rotating frame isgiven by H rfrot = − πv max sgn[sin( πAt )] I y . (9) -40-2002040 (a) ❆(cid:0)✁✂✄☎✆✝✞✟✠✡ H z ) u ① (b) u ② -2 ✲☛ ☛ ☛✶ ☛☞ ❆(cid:0)✁✂✄☎✆✝✞✟✠✡ H z ) ❚✌✍✎ ✏ (cid:1) s) v ① ☛✶ ☛☞ ❚✌✍✎ ✏ (cid:1) s) v ②✾✑✒ ✓✔✑✒ FIG. 5. Numerically optimized pulses for the polarizationtransfer from S z to I x . The coupling strength is 10 MHz andthe bounds on the micro-wave and radio-frequency amplitudesare given by u max = 1 MHz and v max = 20 kHz, respectively.The maximal transfer efficiency is reached after 20 µs . Theinset shows a magnified part of the control v x ( t ) in order toillustrate its form. We obtain the minimum time T min = π/ (8 v max )for generating exp( − iπS z I y ) in the interaction frame.The duration of the transfer is reduced to 78 .
5% of thelength of the conventional pulse sequence. By transform-ing the operator back to the rotating frame, we obtain theoperator exp( − iφS z I z ) exp( − iπS z I y ) exp( iφS z I z ) where φ = 2 πAT min denotes the phase accumulated during thetime T min = π/ (8 v max ). The effect of this superfluousphase φ can be reversed using the hyperfine coupling2 πAS z I z which takes only a negligible time period asthe coupling strength A is much larger than the controlstrength v max of the nuclear spin. Thus, the minimumtime in the rotating frame is also given by π/ (8 v max ).Similarly as in Eq. (7), we compute the maximal transferefficiency η max ( T ) = sin(4 v max T ) (10)that can be reached for the polarization transfer from S z to I x in a specified time T .Transferring the state from S z to I y is similar. We set u x ( t ) = u y ( t ) = v x ( t ) = 0 and maximize the coefficient − πv y ( t ) sin( πAt ) of − S z I x in Eq. (5) by setting v y ( t ) = − sgn[sin( πAt )] v max . The minimum time for this case isalso given by π/ (8 v max ).A numerically optimized pulse sequence for transfer-ring polarization from S z to I x is shown in Fig. 5, cf. [111–113]. The maximal transfer efficiency is reached af-ter 20 µs which is consistent with the analytical result of π/ (8 v max ) ≈ . µs . IV. DISCUSSION
We see that the minimum time for transferring S z to I x or I y is shorter by a factor of π/ S z to I z . This factorcan be explained by a closer examination of the pulsesequences. The radio-frequency sequence for the trans-fer from S z to I z shows a sine-cosine wave modulation ofmaximal amplitude for the v x - and v y -components (seeFig. 3). However, the radio-frequency sequence for thetransfer from S z to I x consists of a square wave of max-imal amplitude for the v x -component of the control (seeFig. 5). The higher effective amplitude at the two fre-quencies ω I / (2 π ) ± A/ f square ( t ) = sgn[sin( πAt )] = 4 π (cid:88) n odd , n ≥ n sin( nAt ) . This is illustrated in Fig. 6 where the first sine wavefunction has an amplitude which is larger by a factorof 4 /π when compared to the amplitude of the squarewave. Therefore, the square wave contains implicitly asine wave with an higher effective amplitude. This im-plies that the duration of the simultaneous ± π/ α and β components of the nuclear spin dou-blet (see Fig. 4) is shorter by a factor of π/ -4/ (cid:1) (cid:1) -1.5-1-0.5 0 0.5 1 1.5 0 (cid:1) /2 (cid:1) (cid:1) /2 2 (cid:1) A m p li t ude Time
FIG. 6. Decomposition of a square wave into sine waves: alarge number of harmonics sum to an approximate squarewave. The first harmonic has an amplitude of 4 /π while thesquare wave has an amplitude of 1. ❋(cid:0)✁✂✄(cid:0)☎✆ (a) zz (analytic)zx (analytic)zzzx 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ❋(cid:0)✁✂✄(cid:0)☎✆ (b) zz (analytic)zzzz (5.0 MHz)zz (4.8 MHz)zz (4.6 MHz) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 5 10 15 20 25 30 35 40 45 50 55 ❋(cid:0)✁✂✄(cid:0)☎✆ Time ( (cid:1) s)(c) zx (analytic)zxzx (5.0 MHz)zx (4.8 MHz)zx (4.6 MHz)
FIG. 7. The maximal transfer efficiency (i.e. fidelity) is shownfor the transfers from S z to I z and I x in (a). The couplingstrength is 10 MHz, and the control strengths are u max =20 kHz and v max = 1 MHz. In (b), data points for the transferfrom S z to I z are shown also for the bandwidth-limited caseswith bounds of 5 . . . S z to I x are shown in (c). The square-modulated transfer sequence is optimal butneeds infinite bandwidth. We also studied numericallyhow the maximal transfer efficiency varies as a functionof time and bandwidth limitations. The results are shownin Fig. 7, cf. [111–113]. In the case of infinite bandwidth,the results are consistent with the analytical results. Thetransfer functions sin ( πv max t ) and sin(4 v max t ) for therespective transfers from S z to I z and I x have been ob-tained in Eqs. (8) and (10).We compare our results to the time-optimal synthe-sis of unitary transformations in [110]. Motivated byenergy considerations, only irradiations at the two res-onance frequencies ω I / (2 π ) ± A/ F r equen cy ( M H z ) 'amp_1u.txt' matrix-40-30-20-10010203040 (a) 'amp_2u.txt' matrix(b) F r equen cy ( M H z ) Time ( (cid:1) s)'amp_1v.txt' matrix-40-30-20-10010203040 0 5 10 15 20 25(c) Time ( (cid:1) s)'amp_2v.txt' matrix0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1(d)
FIG. 8. (Color online) Normalized amplitudes of the short-time Fourier transform for different controls: (a) u -controls ofFig. 3, (b) u -controls of Fig. 5, (c) v -controls of Fig. 3, and (d) v -controls of Fig. 5. In (c), (essentially) only the characteristicfrequency of A/ ± A/ nite bandwidth. It provides in general superior results,but its benefit depends on the available bandwidth.The irradiations at the different frequencies can beclearly observed in Fig. 8 where the normalized ampli-tudes of the short-time Fourier transform [126] are plot-ted for the relevant cases (using the method and imple-mentation of [127]). The important difference betweenthe controls of Fig. 3 for the polarization transfer from S z to I z and the controls of Fig. 5 for the transfer from S z to I x manifest itself in the short-time Fourier trans-forms of the v parts of the controls which are visible insubfigures (c) and (d) of Fig. 8. In subfigure (c), onenotices the characteristic frequency of A/ ± A/
2. This agreeswith the square-wave form of the control v x in Fig. 5.In general context, our results can also interpreted as aconnection between energy and time optimizations. Theenergy optimization leads to a sinusoidal solution whilethe time optimization leads to a square wave (see Fig. 5).This phenomenon has been also observed for the simul-taneous inversion of two uncoupled spins [57, 58] wherethe minimum energy solution was related to the first har-monic in the Fourier expansion of the time-optimal solu-tion. The same reasoning applies here to the solutions ofSection III. V. NON-SINUSOIDAL CARRIER WAVEFORMS
We discuss now the possibility and opportunities ofnon-sinusoidal carrier wave forms for the electron andnuclear spin. Here, we focus on the nuclear spin as themaximal amplitude v max limits the minimum polariza-tion transfer times in the rotating frame. But similararguments might also be used for the micro-wave carrierwave form in applications where the minimum durationof an experiment is limited by u max . As explained inSection II, we have considered so far a sinusoidal carrierwave form ˜ v x ( t ) = v ( t ) cos[ ω rf I t + φ ( t )], which is motivatedby the limited bandwidth of typical radio-frequency waveform generators and amplifiers. Equation (9) states thetime-optimal radio-frequency Hamiltonian for transfer-ring polarization from the state S z to I x in the rotat-ing frame. In the lab frame, the corresponding radio-frequency Hamiltonian is given by H rf = 4 πv max cos( ω rf I t − π/
2) sgn[sin( πAt )] I x . (11)However, it is conceivable (e.g., for applications at lowmagnetic fields or for nuclei with small gyromagnetic ra-tios) that the resonance frequency and the correspond-ing carrier frequency ω rf I of the controls are sufficientlysmall such that non-sinusoidal wave forms (containinghigher harmonics of the carrier frequency ω rf I ) can be cre-ated and amplified. One can therefore envision a radio-frequency Hamiltonian for the ideal case of infinite band-width as given by˜ H rf = 4 πv max sgn[cos( ω rf I t − π/ πAt )] I x . (12)Assuming the same maximal radio-frequency 2 v max (inthe lab frame) and switching from the Hamiltonian H rf in Eq. (11) to the Hamiltonian ˜ H rf in Eq. (12), the radio-frequency amplitude of the carrier frequency is implicitlyincreased in the rotating frame by another factor of 4 /π (similarly as in discussed in Section IV). Consequently,the polarization transfer from the state S z to I x would beachievable using only ( π/ ≈ .
7% of the conventionaltransfer time.
VI. CONCLUSION
We have presented time-optimal polarization transfersfrom an electron spin to a nuclear spin for the case ofsecular hyperfine couplings. In particular, we have ana-lyzed the transfers from the electron-spin state S z to thenuclear-spin states I z and I x . For the transfer to I x , wecould improve on the duration of on-resonance sinusoidalsolutions by applying a control which has the form of asquare wave. Our results also highlight differences be-tween optimizations for minimum energy and minimumtime. We have also discussed how these differences arerelated to bandwidth limitations. ACKNOWLEDGMENTS
H.Y. acknowledges the financial support from the
Re-search Grants Council (RGC) of Hong Kong (Grant538213). R.Z. and S.J.G. acknowledge support from the
Deutsche Forschungsgemeinschaft (DFG) through grantsGL 203/7-1 and GL 203/7-2.
Appendix A: Decomposition of unitaries1. Unitaries which transfer S z I z to I z All unitaries in SU(4) can be be decomposed as K AK with a slow evolution A := exp[ − i ( αS α I y + βS β I y )] andfast unitaries K and K which can be generated by con-trols on the electron spin and the secular hyperfine cou-pling (cf. [59, 66, 110, 128]); recall that S α = ( ) ⊗ σ and S β = ( ) ⊗ σ . The unitaries that transfer 2 S z I z to I z can be determined as solutions to the matrix equation I z = K AK (2 S z I z ) K † A † K † . The fast unitaries K and K can be parameterized using canonical coordinates ofthe second kind (see Section 2.8 of [129], Section 2.10 of[130], or Chapter III, Section 4.3 of [131]), i.e. K := e − ia S x e − ia S y e − ia S z × e − ia S x I z e − ia S y I z e − ia S z I z e − ia I z , (A1a) K := e − ib S x e − ib S y e − ib S x I z e − ib S y I z × e − ib S z e − ib S z I z e − ib I z . (A1b)The surjectivity of the representations in Eq. (A1) is veri-fied in Appendix B. As the unitary K commutes with I z and parts of K commute with 2 S z I z , the matrix equa-tion simplifies to I z = Ae − ib S x e − ib S y e − ib S x I z e − ib S y I z (2 S z I z ) × e ib S y I z e ib S x I z e ib S y e ib S x A † . With the help of the computer algebra system Maple[132], one can verify that either α = 2 πz and β = π +2 πz or α = π +2 πz and β = 2 πz with z , z ∈ Z holds.In Sec. II of the main text, we focused on the first caseassuming that α = 0 and β = π (i.e. z = z = 0), allother cases are similar.
2. Unitaries which transfer S z I z to I x Similarly as in Appendix A 1, the unitaries can bedecomposed into a product K AK of fast unitaries K , K and a slow evolution A = exp[ − i ( αS α I y + βS β I y )]. In particular, all unitaries which transfer2 S z I z to I x have to satisfy the matrix equation I x = K AK (2 S z I z ) K † A † K † . By observing trivial commuta-tors, the matrix equation simplifies to I x = e − ia S x I z e − ia S y I z e − ia S z I z e − ia I z A × e − ib S x e − ib S y e − ib S x I z e − ib S y I z (2 S z I z ) × e ib S y I z e ib S x I z e ib S y e ib S x A † × e ia I z e ia S z I z e ia S y I z e ia S x I z . With the help of the computer algebra system Maple[132], one can infer that β = α − π + 2 πz holds for z ∈ Z .In Sec. III of the main text, we consider the case of A =exp[ − πiS z I y ] which corresponds to α = π/ β = − π/ z = 0. This choice is actually optimal: It followsfrom Eq. (6) in the main text that α = (cid:90) T − π [ v x ( t ) sin( πAt ) − v y ( t ) cos( πAt )] dt,β = (cid:90) T π [ v x ( t ) sin( πAt ) + v y ( t ) cos( πAt )] dt holds for any given time T . Consequently, β − α = (cid:82) T πv x ( t ) sin( πAt ) dt . One applies the condition β = α − π + 2 πz and obtains (cid:82) T πv x ( t ) sin( πAt ) dt = − π +2 πz . This implies that | (cid:90) T πv x ( t ) sin( πAt ) dt | ≥ | − π + 2 πz | ≥ π. On the other hand, one has | (cid:82) T πv x ( t ) sin( πAt ) dt | ≤ (cid:82) T πv max | sin( πAt ) | dt = 8 v max T (see Sec. III). In orderto satisfy the condition β = α − π + 2 πz , the time T hasto fulfill the inequality 8 v max T ≥ π . One gets a lowerbound T min ≥ π/ (8 v max ) on the minimum time T min . Insummary, the scheme presented in Sec. III of the maintext is optimal as it saturates the lower bound. Appendix B: Verification of the surjectivity of therepresentations in Eq. (A1)
In order to verify the surjectivity of K in Eq. (A1a)it is sufficient to verify the surjectivity of the product˜ K = ˜ K ( a , a , a , a , a , a ) := e − ia S x e − ia S y e − ia S z e − ia S x I z e − ia S y I z e − ia S z I z which consists of the first six elements of K as the sev-enth element commutes with all the other ones. First weshow that there exists a (cid:48) , a (cid:48) , and a (cid:48) such that e − ia S x e − ia S y e − ia S z e − ia S x I z e − ia S y I z e − ia S z I z = e − ia (cid:48) S x I z e − ia (cid:48) S y I z e − ia (cid:48) S z I z e − ia S x e − ia S y e − ia S z , (B1)which can be written as e − ia (cid:48) S x I z e − ia (cid:48) S y I z e − ia (cid:48) S z I z = e − ia S x e − ia S y e − ia S z × ( e − ia S x I z e − ia S y I z e − ia S z I z ) e ia S z e ia S y e ia S x . The effect of the conjugation with exp( − ia S z ) is e − ia S z e − ia S x I z e − ia S y I z e − ia S z I z e ia S z = e − ia S z e − ia S x I z e ia S z × e − ia S z e − ia S y I z e ia S z e − ia S z e − ia S z I z e ia S z = e − ia [cos( a )2 S x I z +sin( a )2 S y I z ] × e − ia [cos( a )2 S y I z − sin( a )2 S x I z ] e − ia S z I z = e − i a (cid:48)(cid:48) S x I z e − ia (cid:48)(cid:48) S y I z e − ia (cid:48)(cid:48) S z I z , where the last step follows from the Euler-angle decom-position. Similar arguments for the conjugations with e − ia S y and e − ia S x demonstrate Eq. (B1). Any ele-ment in the connected Lie group that is infinitesimallygenerated by the elements − iS x , − iS y , − iS z , − i S x I z , − i S y I z , and − i S z I z can be achieved by a finite prod-uct of elements having the form of ˜ K ; this is a con-sequence of Lemma 6.2 in [133]. We apply Eq. (B1)and the Euler-angle decomposition multiple times andobtain ˜ K ( a , a , a , a , a , a ) ˜ K (˜ a , ˜ a , ˜ a , ˜ a , ˜ a , ˜ a ) =˜ K ( c , c , c , c , c , c ) for certain values of c , c , c , c , c , and c . In summary, we have verified the surjectivityof the representations ˜ K and K .Similar as for Eq. (B1), one can verify that e − ia S z e − ia S x I z e − ia S y I z = e − ia (cid:48) S x I z e − ia (cid:48) S y I z e − ia S z holds for some a (cid:48) and a (cid:48) . Consequently, the surjectivityof K implies the surjectivity of K .An alternative second argument for the surjectiv-ity of Eq. (A1a) applies the decomposition K (cid:48) A (cid:48) K (cid:48) for the set K = exp( k ) of all fast operations where K (cid:48) i = exp( k (cid:48) ) and A (cid:48) = exp( a (cid:48) ). This decompo-sition is a consequence of the Cartan decomposition k = k (cid:48) ⊕ p (cid:48) where the corresponding linear subspacesare given by k (cid:48) := span {− iS x , − iS y , − iS z , − iI z } , p (cid:48) :=span {− i S x I z , − i S y I z , − i S z I z } , and the abelian sub-algebra a (cid:48) := span {− i S z I z } ⊆ p (cid:48) [128]. The decomposi-tion K (cid:48) A (cid:48) K (cid:48) implies that the decomposition U (cid:48) = U e iπS z I z = e − id S x e − id S y e − id S z e − id S z I z × e − id S z e − id S x e − id S y e − id I z is a surjective parameterization of the set of all fast op-erations. Therefore, the surjectivity is also verified for U = e − id S x e − id S y e − id S z e − id S z I z × e − id S z e − id S x e − id S y e − id I z e − iπS z I z = e − id S x e − id S y e − i ( d + d ) S z e − i ( d + π/ S z I z × e id S y I z e − id S x I z e − id I z = e − id S x e − id S y e − id (cid:48) S z e − id (cid:48) S x I z × e − id (cid:48) S y I z e − id (cid:48) S z I z e − id I z , where the last equality follows from the Euler-angle de-composition. This completes the second argument forthe surjectivity of Eq. (A1a). [1] A. Schweiger and G. Jeschke, Principles of pulse electronparametric resonance (Oxford University Press, Oxford,2010).[2] H. Brunner, R. H. Fritsch, and K. H. Hausser, Z. Natur-forsch. A , 1456 (1987).[3] V. Weis and R. G. Griffin, Solid State NMR , 66(2006).[4] G. W. Morley, J. van Tol, A. Ardavan, K. Porfyrakis,J. Zhang, and G. A. Briggs, Phys. Rev. Lett. , 220501(2007).[5] T. Maly, G. T. Debelouchina, V. S. Bajaj, K.-N. Hu, C.-G. Joo, M.-L. Mak-Jurkauskas, J. R. Sirigiri, P. C. A.van der Wel, J. Herzfeld, R. J. Temkin, and R. G.Griffin, J. Chem. Phys. , 052211 (2008).[6] C. Griesinger, M. Bennati, H. M. Vieth, C. Luchinat,G. Parigi, P. H¨ofer, F. Engelke, S. J. Glaser, V. Deny-senkov, and T. F. Prisner, Prog. NMR Spectrosc. ,4 (2012).[7] M. H. Levitt, Spin Dynamics: Basics of Nuclear Mag-netic Resonance (Wiley, New York, 2008).[8] R. R. Ernst, G. Bodenhausen, and A. Wokaun,
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