Article pubs.acs.org/JPCB

Coherence Transfer by Passage Pulses in Electron Paramagnetic Resonance Spectroscopy Gunnar Jeschke,* Stephan Pribitzer, and Andrin Doll ETH Zurich, Lab. Phys. Chem., Vladimir-Prelog-Weg 2, 8093 Zurich, Switzerland S Supporting Information *

ABSTRACT: Linear passage pulses provide a simple approach to ultra-wideband electron paramagnetic resonance (EPR) spectroscopy. We show by numerical simulations that the efficiency of inversion of polarization or coherence order on a single transition by idealized passage pulses is an exponential function of critical adiabaticity during passage, which allows for defining an effective flip angle for fast passage. This result is confirmed by experiments on E′ centers in Herasil glass. Deviations from the exponential law arise due to relaxation and a distribution of the adiabaticity parameter that comes from inhomogeneity of the irradiation field. Such inhomogeneity effects as well as edge effects in finite sweep bands cause a distribution of dynamic phase shifts, which can be partially refocused in echo experiments. In multilevel systems, passage of several transitions leads to generation of coherence on formally forbidden transitions that can also be described by the concept of an effective flip angle. On the one hand, such transfer to coherence on forbidden transitions is a significant magnetization loss mechanism for dipole−dipole coupled electron spin pairs at distances below about 2 nm. On the other hand, it can potentially be harnessed for electron spin echo envelope modulation (ESEEM) experiments, where matching of the irradiation field strength to the nuclear Zeeman frequency leads to efficient generation of nuclear coherence and efficient back transfer to electron coherence on allowed transitions at high adiabaticity.

1. INTRODUCTION Pulsed electron paramagnetic resonance (EPR) spectroscopy1 with monochromatic rectangular pulses suffers from low excitation bandwidths that cannot fully cover the spectral width of most organic radicals, in particular of nitroxide spin labels, and fall short by more than an order of magnitude of covering the spectral width of typical transition metal and rare earth ion complexes. This insufficient spectral coverage reduces sensitivity. Moreover, coherence and polarization transfers are restricted to the subset of transitions that are situated within the excitation band. This poses limits to correlation spectroscopy as well as to the magnitude of nuclear frequencies that can be detected in electron spin echo envelope modulation (ESEEM) experiments. Electron electron double resonance (ELDOR) schemes with excitation at two microwave frequencies can alleviate the latter problem.2,3 The recent advent of arbitrary waveform generators (AWGs) that can cover the full width of typical EPR spectra or at least a sizable fraction thereof has encouraged the development of experimental schemes with amplitude- and phase-modulated pulses that excite different transitions at different times.4−10 From earlier work in NMR spectroscopy, it is known that such schemes lead to dynamic phase shifts11 and, in coupled spin systems, to interference effects.12 The relevance of dynamic phase shifts in EPR has been recognized in the context of pulsed ELDOR experiments.13 Interference effects can be detrimental by introducing magnetization loss to unwanted coherence transfer pathways, © 2015 American Chemical Society

but they can also be advantageous, as was demonstrated in EPR by prepolarization of the central transitions of S = 5/2 Mn(II)14 and S = 7/2 Gd(III)10 ions by field and frequency sweeps, respectively. Wideband EPR excitation is hitherto realized by replacing monochromatic rectangular pulses in established pulse sequences by either optimal-control pulses4,7 or passage pulses,5,6,10 in some cases8,9 considering certain pulse length requirements to achieve echo refocusing without phase dispersion.15 To date, this replacement strategy does not consider signal losses from dynamic phase shifts, although compensation schemes are known in NMR,16 and it does not generally consider the complications and opportunities that arise from interference effects. Optimal control theory can take into account the complications of coupled spin dynamics17 and cooperative effects within sequences consisting of several pulses.18 However, for broad distributions of coupling parameters and differences of Larmor frequencies, as they are commonplace in pulse EPR, it is not necessarily clear how the optimization problem should be posed. Spin dynamics during passage pulses is easier to grasp, and although such pulses are not necessarily time optimal solutions, they often perform quite Special Issue: Wolfgang Lubitz Festschrift Received: March 27, 2015 Revised: May 4, 2015 Published: May 5, 2015 13570

DOI: 10.1021/acs.jpcb.5b02964 J. Phys. Chem. B 2015, 119, 13570−13582

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The Journal of Physical Chemistry B

Figure 1. Inversion of polarization (a, b) and coherence order (c, d) in a two-level system by idealized passage pulses (numerical simulations for tp = 1.6 μs and B = 1.6 GHz). (a) Polarization inversion as a function of critical adiabaticity Qcrit (black solid line) and fit by a function 1 − exp(−cQcrit) giving c = 0.9976π/2 (red dashed line). (b) Semilogarithmic plot of the data shown in part a. (c) Inversion of coherence order +1 (red) to coherence order −1 (blue) during an idealized passage pulse with Qcrit = 10. (d) Semilogarithmic plot for the simulated coherence transfer coefficient T as a function of Qcrit (solid black line) and plot of the empirical expression −πQcrit/2 (red dashed line).

well.19 Moreover, they are amenable to simple compensation for the resonator response function by adapting the instantaneous sweep rate to the enhancement of the microwave field at the instantaneous frequency.8 Further development of ultra-wideband EPR experiments with such pulses requires a quantitative theory of coherence transfers in multilevel systems. Here we set out to provide the basics of such a theory. The paper is organized as follows. In section 2, we reconsider the effects of fast nonadiabatic passage on a two-level system. We first show by numerical simulations for the ideal case, where passage is much faster than transverse relaxation, that a simple expression exists for the flip angle as a function of the critical adiabaticity Qcrit during passage. Then, we demonstrate that, to a rather good approximation, dynamic phase shift is proportional to Qcrit and consider the dependence of this phase shift on the positioning of the transition frequency within the band covered by the passage pulse. The theoretical results of this section are compared to experiments on narrow-line samples of γ-irradiated Herasil that feature E′ centers in SiO2. We find that dynamic phase shifts can be compensated and discuss how such compensation can be achieved in a stimulated echo experiment. By comparing data for samples with different relaxation behavior and different spatial extension, we elucidate effects of transverse relaxation and microwave field inhomogeneity. In section 3, we consider longitudinal and transverse interference effects that occur on consecutive selective passage of two transitions that share a common level. We provide general expressions for the redistribution of populations as well as for transfer of coherence to the third transition of the threelevel system that is not directly excited and may be a forbidden

transition. Furthermore, we consider the limit where the microwave field amplitude is much larger than the difference of the resonance frequencies of the two transitions that are passed. In section 4, we turn to four-level systems that are of particular interest in applications of pulsed EPR spectroscopy. For two weakly coupled electron spins, we find a simple expression for the amplitude of zero- and double-quantum coherence as a function of spin−spin coupling, the difference of the two Larmor frequencies, and sweep rate. By a model computation with parameters typical for a pair of nitroxide spin labels at Qband frequencies (34 GHz), we demonstrate that significant loss of magnetization to zero- and double-quantum coherence may ensue in nanometer-range distance measurements. For a system consisting of an electron spin S = 1/2 and a nuclear spin I = 1/2 with anisotropic hyperfine coupling, we show that passage leads to asymmetric coherence generation on the two forbidden transitions. Finally, we discuss a possible sensitivity enhanced ESEEM experiment based on passage pulses with their irradiation field strength matched to the nuclear Zeeman frequency.

2. PASSAGE OF A SINGLE TRANSITION 2.1. Equivalence of Critical Adiabaticity to a Flip Angle. Passage through a single transition with frequency ω0 is most easily pictured in a frequency-modulated frame where an irradiation field with constant amplitude ω 1 is time independent.20 In the following, we consider linear sweeps with rate k = dω/dt, where ω is the instantaneous frequency of the irradiation field. We anticipate that the results also apply to the class of passage pulses that compensate variable amplitude 13571

DOI: 10.1021/acs.jpcb.5b02964 J. Phys. Chem. B 2015, 119, 13570−13582

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The Journal of Physical Chemistry B

Figure 2. Inversion of the echo signal of E′ centers in a 3 mm long 1 mm diameter (black solid line, temperature 100 K) and an extended (blue solid line, ambient temperature) sample of γ-irradiated Herasil, prediction by eq 5 (green dotted line). (a) Original data. (b) Semilogarithmic plot. Liouville-space simulations with the experimentally determined transverse relaxation time of T2 = 74 μs for the small sample (red dotted line) and the best-fitting T2 = 0.5 μs for the extended sample (blue dotted line) are also shown. Simulations further assumed the experimentally determined longitudinal relaxation time T1 = 220 μs.

end, we define an idealized passage pulse of duration tp as a constant-amplitude linear sweep from frequency −B/2 + ωc to B/2 + ωc, where ωc is the center frequency, B the bandwidth, and k = B/tp. To avoid the influence from the rising and falling edges, we smooth these edges by sine quarterwaves15 and consider only resonance offsets Ω = ω0 − ωc with respect to the center of the sweep that are well separated from the edges, i.e., Ω + B/2 ≫ ω1 and B/2 − Ω ≫ ω1. Furthermore, we assume that the relaxation-related adiabaticity condition in eq 2 is fulfilled, ω1 ≪ kT2. Violation of the latter condition will be discussed separately. Note that for a given sweep rate such violation is stronger at larger ω1 and thus at larger Qcrit. For a given bandwidth B and critical adiabaticity Qcrit, we can eliminate ω1 from the relaxation condition to give tp ≪ T22/ (BQcrit). We first consider inversion of polarization by performing the simulation with an initial density operator σ0 = Sz and detecting the expectation value ⟨Sz⟩. Inversion efficiency I = (1 − ⟨Sz⟩)/2 is plotted in Figure 1a as a function of Qcrit. We find that the data is nicely approximated by the expression

by nonlinear sweep rate to maintain constant adiabaticity on passage. For convenience, we further define the instantaneous resonance offset Δω = ω − ω0. During a hypothetical sweep with irradiation phase ϕp from Δω = −∞ to Δ ω = ∞, starting with equilibrium magnetization M0 along the quantization axis z, the magnetization vector will follow the effective field Beff =

ℏ (ω1 sin ϕpex + ω1 cos ϕpey + Δωez) gμ B

(1)

if the adiabaticity condition kT2 k ≫1≫ 2 ω1 ω1

(2)

is fulfilled. In eq 1, ℏ is Planck’s quantum of action over 2π, g the electron g value, μB the Bohr magneton, and ex, ey, and ez are unit vectors along the x, y, and z directions, respectively. In general, the approach to this ideal situation can be quantified by an adiabaticity factor20 Q (t ) =

ω12 + Δω 2 dθ /dt

(3)

I = 1 − exp( −πQ crit /2)

where θ is the angle between the effective field and the z axis. For a linear sweep with constant irradiation amplitude, the adiabaticity factor is minimal at the time of passage, where it assumes the value20 Q crit = ω12 /k

(5)

as is also apparent from the semilogarithmic plot in Figure 1b where the deviations at Qcrit > 5 could be traced back to numerical errors. Except for the specific dependence of these deviations on Qcrit, very similar data were obtained by numerically integrating the Bloch equations without relaxation in the frequency-modulated frame (data not shown). Inversion of coherence order can be considered in the same way by assuming an initial density operator σ0 = S+ and defining a coherence transfer coefficient T = |⟨S−⟩|. Formulating the problem with the complex pseudo expectation value ⟨S−⟩ and considering the magnitude of this value obviates, for the moment, the discussion of phase evolution. As is seen in Figure 1c, coherence order inversion is quantitative at large critical adiabaticity. The transfer coefficient T follows the same expression as polarization inversion efficiency I, as is seen in Figure 1d. This is an exact coincidence, as is apparent from even the numerical errors being the same. For both polarization and coherence order inversion, the expression in eq 5 applies irrespective of the choice of bandwidth B and pulse length tp as long as the conditions for idealized passage pulses are fulfilled.

(4)

According to eq 2, adiabatic passage requires that this critical adiabaticity Qcrit is much larger than unity. We have shown that pulses with offset-independent adiabaticity21,22 can compensate for the amplitude modulation of the irradiation field imposed by the resonator, both for inversion of polarization5 and for generation of coherence.8 This suggests critical adiabaticity Qcrit to be the appropriate parameter for deriving expressions for coherence transfer. Unfortunately, derivation of such expressions in closed form fails, as the Bloch equations in the frequency-modulated frame still comprise a system of coupled differential equations with explicitly time-dependent terms on the right-hand side, a problem that cannot be solved by iterative frame transformations.23,24 Therefore, we follow an approach of fitting guessed expressions to results of numerical simulations. To that 13572

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Figure 3. Dynamic phase shifts in a two-pulse chirp echo experiment (numerical simulations for tp = 800 ns). (a) Dependence of the linear phase deviation coefficient sQ on resonance offset Ω with respect to the center of the upward sweep for bandwidths B = 1.6 GHz (solid black line) and B = 0.8 GHz (dotted blue line) and for the latter data horizontally scaled by a factor of 2 and vertically scaled by a factor of (2)1/2 (dashed red line). (b) Dependence of coherence transfer efficiency T (dotted blue line), refocusing efficiency R (dashed red line), and total echo amplitude (solid black line) on critical adiabaticity Qcrit (B = 1.6 GHz).

high concentration of E′ centers in this sample (data not shown). However, a Liouville space density operator simulation with T2 = 3.5 μs does not fit the data (not shown) and even with the best-fitting T2 = 0.5 μs (blue dotted line in Figure 2b) the experimental curve is not reproduced very well. We tentatively assign these deviations to observation of E′ centers in the fringe microwave field of the resonator, where B1 and thus Qcrit is much lower. This conjecture is supported by measurements on a medium-size, high-concentration sample from a 1 mm diameter rod that provided data which could be reasonably well fitted by T2 = 1 μs (see the Supporting Information). 2.2. Dynamic Phase Shifts. When taking into account only the equivalent flip angle, we would predict maximal chirp echo intensity at the maximal attainable Qcrit for the refocusing pulse, corresponding to maximal irradiation power. However, both numerical simulations and experiments8,15 have shown that there exists an optimum irradiation amplitude ω1 for such a pulse. The reason can be traced back to dynamic Bloch−Siegert shifts11 that depend on Qcrit and Ω. To quantify their influence on the echo amplitude, we consider the refocusing condition for the chirp echo.15 After a π/2 idealized passage pulse with pulse length tπ/2 = 2tp, coherence phase ϕ exhibits a parabolic dependence on resonance offset Ω, as coherence of spin packets with different resonance offsets is generated at times proportional to Ω. The refocusing pulse needs to convert this phase dependence to a linear dependence ϕ = ϕ0 − τΩ, so that free evolution under the static spin Hamiltonian H = ΩSz equalizes the phase of all spin packets at time τ after the pulse to the echo phase ϕ0. Disregarding dynamic Bloch−Siegert shifts, this is achieved if the refocusing pulse has half the length of the π/2 pulse, i.e., tπ = tp.15 The dynamic phase shift is thus most conveniently studied by applying to an initial density operator σ0 = −Sz a chirp pulse of length 2tp with Qcrit = 2 ln 2/π and considering ϕ as a function of Ω and of the critical adiabaticity Qcrit of a second, refocusing pulse with length tp. To separate the dynamic phase shift contribution to ϕo from the contributions due to incomplete coherence inversion, we can reduce the density matrix σ1 after the first pulse to σ̃1 = S+Tr{S−σ1 } and determine the phase of ⟨S−⟩ after the second pulse. At sufficiently large Qcrit, this reduction is not required and the phase can be determined from the expectation values ⟨Sx⟩ and ⟨Sy⟩. We consider the

Taken together, the two simulations completely describe transformation of any magnetization vector by an idealized, i.e., relaxation-free, passage pulse, except for phase shifts. Thus, we can define an equivalent flip angle β by β = arccos[2 exp( −πQ crit /2) − 1]

(6)

Maximum coherence generation from initial polarization is achieved at β = π/2, i.e., at Qcrit = 2 ln(2)/π. For large Qcrit, the equivalent flip angle asymptotically approaches β = π, which is an upper bound. For an experimental test, we measured inversion of an echo signal of E′ centers in a 3 mm long piece of a 1 mm diameter γirradiated Herasil glass rod at a temperature of 100 K with a spectral width of only about 10 MHz at X-band frequencies (black lines in Figure 2). Such a narrow-line sample mimics, to a good approximation, an individual spin packet within an inhomogeneously broadened line. By fixing all parameters of the microwave setup and varying the magnetic field, we can thus study the dependence of amplitude and phase on offset Ω with respect to the center of the sweep. The behavior of systems with inhomogeneously broadened lines can be predicted by integrating these dependencies over the whole line shape or sweep range, whatever is narrower. As the sample occupies only the center of the 3 mm split-ring resonator, we expected that edge effects would not affect the results for a B = 0.8 GHz sweep with tp = 800 ns and that the inversion curve is not significantly affected by B1 inhomogeneity, which would lead to a distribution in Qcrit. Indeed, at Qcrit < 3, the data agrees very nicely with the prediction by eq 5 (green dotted line). For larger Qcrit, deviations become apparent in the semilogarithmic plot (Figure 2b). These deviations are reproduced within experimental uncertainty by a Liouville-space density operator simulation that considers transverse relaxation with the experimentally determined relaxation time T2 = 74 μs (red dotted line). For a larger sample that extends vertically beyond the resonator, we find a slower rise with Qcrit (blue solid lines) than predicted by eq 5 (green dashed lines). Full inversion is not attained even at large Qcrit. This sample, which was measured at ambient temperature, has a shorter phase memory time of Tm = 3.5 μs. By observing the primary echo decay with rectangular monochromatic pulses and varying the flip angle of the refocusing pulse, the relaxation enhancement could be traced back to instantaneous diffusion, which results from a 13573

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Figure 4. Chirp echo experiments on E′ centers in γ-irradiated Herasil with B = 0.8 GHz and tp = 800 ns. (a) Experimental (black) and simulated (red) dependence of the linear fit coefficient sq, corrected for ϕ0, on resonance offset Ω from the center of the sweep. The blue line shows the corresponding data for an ABSTRUSE16 experiment. (b) Dependence of the linear fit coefficient sq on Ω for a stimulated echo with pulse length ratios 2:1:1 (black) and −1:−2:1 (blue), where a positive sign denotes an upward sweep and a negative sign a downward sweep. (c) Integral echo amplitude as a function of Qcrit for a primary chirp echo (black) and an ABSTRUSE echo (blue) on a 3 mm long 1 mm diameter Herasil sample measured at 100 K. (d) Same experiment as in part c but with a Herasil sample that extends vertically beyond the resonator (measured at ambient temperature).

The linear increase of Δϕ causes a monotonous decrease of refocusing efficiency R with increasing Qcrit. Combined with the monotonous increase in transfer efficiency T, this explains the maximum in echo amplitude (Figure 3b). At which Qcrit this maximum is experimentally observed depends on the spectral line shape function within the band of the passage pulse. In general, the maximum will be attained at larger Qcrit if the spectrum is narrower with respect to the bandwidth B. Inhomogeneity of the irradiation field causes additional signal loss, as it leads to a distribution in Qcrit values and thus in ϕ0. The resulting destructive interference of coherence of spins subjected to different irradiation field strengths ω1 is stronger the higher the mean Qcrit is. Hence, in inhomogeneous B1 fields, one needs to work at lower Qcrit and thus accept lower transfer efficiencies T. In our experimental tests, we first checked that we observe the sign inversion of ϕ0 and Δϕ with sweep reversal (Figure S1a, Supporting Information). In the following, we discuss phase differences between upward and downward sweeps, as this leads to partial compensation of an additional phase dependence on frequency that stems from hardware issues. We find that the dependence of dynamic phase shift on Qcrit in the center of the sweep and off resonance has a linear component that agrees reasonably well with theoretical expectations (Figure S2a and b, Supporting Information). Furthermore, the magnitude and offset dependence of the linear coefficient sQ for the off-resonance contribution (black line in Figure 4a) are also in reasonable agreement with the simulation for idealized

deviation Δϕ from the required linear dependence on Ω as well as the variation of ϕ0 with Qcrit. At a given Qcrit, phase correction of the echo gets rid of ϕ0 and the normalized echo amplitude is described by V = TR with the refocusing efficiency B /2

R=

∫−B/2 cos(Δϕ) dΩ

(7)

where by construction Δϕ = 0 at Ω = 0. Refocusing time τ is determined by a linear fit in a small range around Ω = 0 using Δϕ = ϕ + τΩ. Our simulations reveal a linear dependence Δϕ = sQQcrit at sufficiently large Qcrit (see Figure S4, Supporting Information). As a function of the resonance offset Ω, the linear coefficient sQ increases strongly toward the edges of the band (Figure 3) and sQ(Ω) is close to, but not exactly, an even function. When the bandwidth of the sweep is decreased, phase deviations become prominent already at smaller offsets from the center of the sweep. The function sQ(Ω) scales roughly, but not exactly, with bandwidth along the Ω axis and with the square root of bandwidth along the sQ axis. This scaling behavior was verified by further simulations (data not shown). The sign of sQ changes if the same sweep is performed downward from B/2 to −B/2. Furthermore, to a good approximation, ϕ0 also depends linearly on Qcrit (see Figure S2b, Supporting Information). We cannot exclude the possibility that the small deviations from linearity for sQ and ϕ0 are due to numerical errors. Like sQ, ϕ0 changes sign if the sweep direction of the refocusing pulse is reversed. 13574

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time, the time of passage of an upward sweep is f tp,l, with fraction f being independent of l. For linear sweeps, we have f = (Ω + B/2)/B. The remaining time during the pulse after passage is (1 − f)tp,l. For a downward sweep with θ̃ = 1 − θ, the time before passage is (1 − f)tp,l and the time after passage is f tp,l. Refocusing requires that the f-dependent phase gains during all pulses sum to the same constant for any value of f. Hence, in the refocusing condition, all terms can be dropped that do not depend on f. This allows for indicating sweep direction just by a sign factor sl that is +1 for an upward sweep and −1 for a downward sweep. The general refocusing condition thus boils down to

passage pulses. The deviation is possibly caused by residual variation in Qcrit(Ω) of the compensated pulses (see the Supporting Information). As expected, the dependence simulated including relaxation with T2 = 3.5 μs and T1 = 220 μs virtually coincides with the one simulated excluding relaxation (data not shown). We conclude that relaxation effects influence the transfer efficiency T but not the refocusing efficiency R. As already discussed in the context of NMR spectroscopy,16 the dynamic phase shift changes sign on passage, but at the same time, the already accumulated phase difference is inverted. For this reason, the phase shift does not cancel even in the case of Ω = 0. In the same work, the adjustable broadband sech/ tanh-rotation uniform selective excitation (ABSTRUSE) scheme was introduced, which compensates for the dynamic phase shift by a second refocusing. In this scheme, both the π/2 pulse and the first refocusing pulse have duration 2tp, thus causing an inverted parabolic phase dependence on Ω plus the dynamic phase shift. During the second refocusing pulse with duration tp, the dynamic phase shift is compensated. Indeed, the ABSTRUSE sequence leads to nice compensation of the dynamic phase shift in our experiments (blue line in Figure 4a). Comparison of the echo integral as a function of Qcrit between a primary chirp echo experiment (black line in Figure 4c) and an ABSTRUSE experiment (blue line) for the 3 mm long 1 mm diameter sample shows the expected behavior in that the ABSTRUSE signal shows a slower rise with Qcrit, as it depends on the product of the transfer efficiencies of two chirp pulses, and a later and much slower decay at large Qcrit due to compensation of the dynamic phase shift. At optimum Qcrit, the ABSTRUSE experiment provides a somewhat larger echo signal than the primary chirp-echo experiment. However, this relation inverts for the extended sample where effects of B 1 inhomogeneity and relaxation become important (Figure 4d). In this case, the maximum amplitude of the primary chirp echo is attained at lower Qcrit where the inhomogeneity effects shown in Figure S2c and d (Supporting Information) are less prominent than at larger Qcrit. This leads to a reduction in the maximum echo amplitude due to a lower refocusing efficiency. For the ABSTRUSE sequence, B1 inhomogeneity causes a contribution to phase dispersion that cannot be refocused and thus reduces transfer efficiency at large Qcrit. Furthermore, the transfer efficiency is reduced by transverse relaxation (see Figure 2a). In this situation, the signal loss that is caused in ABSTRUSE by lengthening the pulse sequence and introducing an additional pulse overcompensates the gain from refocusing of the dynamic phase shift. 2.3. Chirp Echo Refocusing with Multiple Pulses. If more than one pulse is involved in refocusing, compensation of dynamic phase shifts does not require a lengthening of the sequence or introduction of additional pulses. The simplest experiment of this type is the stimulated echo experiment. Before we consider this experiment, we discuss general chirp echo refocusing conditions under neglect of dynamic phase shifts. In general, refocusing of resonance offsets in an experiment with several free evolution periods of durations ti occurs if ∑i oi ti = 0, where the oi are the coherence orders in the individual evolution periods. For refocusing by a sequence of idealized passage pulses with indices l, phase gains during free evolution before passage with coherence order o(−) and l must compensate in the after passage with coherence order o(+) l sum over all pulses. If all pulses have the same bandwidth B and the same sweep function ω(θ), where θ = t/tp,l is a normalized

∑ slol(−)ft p,l − slol(+)ft p,l = const l

(8)

which is fulfilled for all f with 0 ≤ f ≤ 1 if and only if

∑ slt p,l(ol(−) − ol(+)) = 0 l

(9)

If coherence transfers are caused exclusively by passage pulses, we further have o(+) = o(−) l l+1 . If a solution in terms of integer multiples of the shortest pulse length is sought, eq 9 becomes a linear Diophantine problem. For a stimulated echo with only upward swept pulses, we (+) (−) (+) (−) (+) have o(−) 1 = 0, o1 = o2 = 1, o2 = o3 = 0, and o3 = −1, so that eq 9 reduces to −tp,1 + tp,2 + tp,3 = 0. This equation has several solutions for the relative pulse lengths. We can thus further require that free evolution between the second and third passage has the same duration for all fractional resonance offsets f, which requires −ftp,2 + f tp,3 = const. This condition can be generally fulfilled only for tp,2 = tp,3, and resubstitution into the refocusing condition provides tp,1 = 2tp,2. Hence, for all pulses being swept upward, the 2:1:1 pulse length ratio is a unique solution if we also require offset-independent mixing time between second and third passage. This scheme does not lead to compensation of the dynamic phase shift, as is apparent from the black line in Figure 4b. By symmetry, the same pulse length ratios apply if all pulses are swept downward. Dynamic phase shift is not compensated in this case either. If we allow for a mix of sweep directions, the refocusing condition is given by −s1tp,1 + s2tp,2 + s3tp,3 = 0 and the constant mixing time condition is given by −s2tp,2 + s3tp,3 = 0. With pulse durations being generally positive, the latter condition can be fulfilled only if the last two pulses have the same sweep direction and further the same length. By substituting this into the refocusing condition, we find that the first pulse must also have the same sweep direction and twice the length of the second and third pulse. In other words, the 2:1:1 solution is still unique if we require constant mixing time. Compensation of dynamic phase shift is impossible with this requirement and only three pulses. However, the constant mixing time condition can be relaxed for experiments with mixing times much longer than the duration of the passage pulses. One easily ascertains that the two possible mixed-direction refocusing conditions are tp,1 − tp,2 + tp,3 = 0 and tp,1 + tp,2 − tp,3 = 0. Here we put the additional requirement that total duration of all passage pulses is the smallest possible multiple of the duration of the shortest pulse. This provides the schemes −1:−2:1 and 1:−1:2, where the negative sign indicates a downward sweep. Again symmetric solutions are obtained by reverting the sweep direction of all three pulses. The −1:−2:1 scheme is a stimulated echo equivalent of the ABSTRUSE scheme and does indeed lead to compensation of the dynamic phase shift (blue line in Figure 13575

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Figure 5. Consecutive idealized passage of the two allowed transitions (1 ↔ 2) and (2 ↔ 3) in an S = 1 system (density operator simulation). (a) Time evolution of the expectation values |⟨S+kl⟩| of pseudo-observables corresponding to the amplitudes of coherence on the allowed transitions (1 ↔ 2) (blue) and (2 ↔ 3) (red) and the forbidden transition (1 ↔ 3) (black). The electron Zeeman frequency coincides with the center of a 1.6 μs long sweep over a bandwidth of 1.6 GHz with Qcrit = 2 ln 2/π. The splitting between the two allowed transitions is 800 MHz. (b) Spinor transfer of coherence on allowed transition (1 ↔ 2) (blue) to forbidden transition (1 ↔ 3) (black) by passage of transition (2 → 3). Red dotted lines correspond to empirical expressions exp(−πQcrit/4) for |S+12| and (1 − exp2(−πQcrit/4))1/2 for |S+13|.

4b). In the stimulated echo, all pulses are π/2 pulses and thus have small Qcrit. Therefore, dynamic phase shift effects on refocusing efficiency R are rather mild and the improvement by compensation is only moderate. However, our approach for finding refocusing conditions that allow for compensation can be easily extended to any multipulse echo sequence, including sequences that contain π pulses. The cases of the primary and stimulated echoes have also been considered for resonance offset distributions that are caused by pulse field gradients in magnetic resonance imaging.25 In this application, pulse durations in eq 9 have to be multiplied with the gradient amplitudes.

NMR26,27 and EPR10 spectroscopy for polarization enhancement of the central transitions of half-integer spins S > 1/2. Note that for an (mS ↔ mS + 1) transition of a spin S ω1 =

S(S + 1) − mS (mS + 1) ·gμB B1/ℏ}

(11)

i.e., Qcrit is proportional to S(S + 1) − mS(mS + 1). Hence, for spins S > 1, inversion efficiency differs on the transitions that are consecutively passed. 3.2. Coherence Transfer. Passage with moderate adiabaticity generates coherence on all passed transitions. At the same time, it leads to some polarization change, usually polarization enhancement, on transitions that have not yet been passed but share a level with a transition that has already been passed. Therefore, more coherence is generated on transitions that are passed later in the sweep.12 Using eq 6, we find for the coherence amplitude c(+) lm after passage of transition (l ↔ m)

3. CONSECUTIVE PASSAGE OF TWO TRANSITIONS WITH A COMMON LEVEL 3.1. Polarization Transfer. In a three-level system with transitions (1 ↔ 2) and (2 ↔ 3), passage of the first transition changes polarization on the second transition. With level populations p1(0) and p2(0) of transition (1 ↔ 2) before passage, eq 5 provides for the populations after passage

(−) (+) clm = 2Δplm exp(−πQ crit)(exp(πQ crit /2) − 1)

(12)

where Δp(−) lm = pl − pm is the polarization on this transition before passage. Coherence phase exhibits a constant shift with respect to the phase of the passage pulse. Hence, phase cycling [(+ϕ) − (−ϕ)] of the passage pulse will keep the coherence and cancel the remaining polarization. Note that eq 12 defines coherence amplitudes by considering each transition as a twolevel system. With this definition, an ideal nonselective π/2 pulse generates coherence with amplitude (2)1/2 on both allowed transitions of an S = 1 system. Coherence amplitude enhancement is illustrated in Figure 5 for a pulse with a duration of 1.6 μs, a bandwidth of 1.6 GHz, and Qcrit = 2 ln 2/π corresponding to maximum coherence generation on passage of each individual transition. We have assumed an S = 1 system where both allowed transitions have the same transition amplitude and thus the same Qcrit, an electron Zeeman frequency coinciding with the center of the sweep, and a zero-field splitting of the two transitions of 800 MHz. Passage of transition (1 ↔ 2) enhances polarization on transition (2 ↔ 3) by 50%, resulting in coherence amplitude c(+) 23 = 1.5 after passage of transition (2 ↔ 3). This amplitude is slightly larger than the one obtained with an ideal nonselective pulse, whereas the coherence amplitude c(+) 12 immediately after passage of the first transition is only 1, significantly smaller than

p1 = p1 (0) + [1 − exp( −πQ crit /2)][p2 (0) − p1 (0)] p2 = p2 (0) + [1 − exp( −πQ crit /2)][p1 (0) − p2 (0)] (10)

For large critical adiabaticity, populations are exchanged and the polarization on transition (2 ↔ 3) becomes Δp23 = p1(0) − p3(0). If the three levels belong to a triplet state S = 1 and the high-field and high-temperature approximations apply, thermal equilibrium populations are given by p1(0) = 1/3 + ϵ, p2(0) = 1/3, and p3(0) = 1/3 − ϵ, where ϵ ≈ ℏω0/(kB T) with electron Zeeman angular frequency ω0, Boltzmann constant kB, and temperature T. Polarization ϵ is the same on both transitions in thermal equilibrium. After adiabatic passage of the first transition, polarization on transition (2 ↔ 3) is twice as large. For a cascade of ideally selective passages of transitions in a system with S > 1 and 2S + 1 levels, eq 10 can be consecutively applied for the transitions that are passed. If again the high-field and high-temperature approximations apply and Qcrit is very large, passage of the n transitions (1 ↔ 2) to (n ↔ n + 1) leads to an n + 1-fold enhancement of polarization on the transition (n + 1 ↔ n + 2). Such schemes have been applied in 13576

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Figure 6. Consecutive idealized passage with bandwidth B and Qcrit = 2 ln(2)/π of the four allowed transitions of a system of two coupled electron spins S1 = 1/2 and S2 = 1/2 with a difference of resonance frequency offsets ΔΩ and a dipole−dipole coupling d/(2π) = 100 MHz (density operator simulation). Coherence amplitudes |⟨S+12⟩| (blue solid lines), |⟨S+34⟩| (blue dotted lines), |⟨S+13⟩| (green solid lines), |⟨S+24⟩| (green dotted lines), |⟨S+23⟩| (red solid lines), and |⟨S+14⟩| (red dotted lines) are plotted. (a) Coherence amplitudes during a passage pulse of duration tp = 794 ns with B/(2π) = 1.6 GHz, ΔΩ/(2π) = 0.5 GHz, and d/(2π) = 100 MHz. (b) Coherence amplitudes during a passage pulse of duration tp = 810 ns with B/(2π) = 1.6 GHz, ΔΩ/(2π) = 0.5 GHz, and d/(2π) = 100 MHz. (c) Coherence amplitudes after a passage pulse of variable length tp with B/(2π) = 1.6 GHz, ΔΩ/(2π) = 0.5 GHz, and d/(2π) = 10 MHz. The black dotted line corresponds to sin[dtpΔΩ/(2B)]. (d) Coherence amplitudes as a function of the spin−spin distance after a passage pulse with length tp = 200 ns, bandwidth B/(2π) = 500 MHz, and resonance offsets of the two spins corresponding to nitroxide spin pairs with uncorrelated mutual orientation at Q-band frequencies near 34 GHz.

data of the numerical simulation is nicely fitted by the empirical expressions

the one for an ideal nonselective pulse. Related phenomena have been considered for adiabatic passage of half-integer nuclear spins and explained in terms of level anticrossings in the rotating frame.28 Passage of transition (2 ↔ 3) further reduces c12 by coherence transfer to the forbidden transition (1 ↔ 3). For an ideally selective pulse acting on transition (2 ↔ 3), the dependence on flip angle and on phase of the initial coherence on transition (1 ↔ 2) is as follows

+ + |⟨S12 ⟩|(Q crit) = exp( −πQ crit /4)|⟨S12 ⟩|(0) + |⟨S13 ⟩|(Q crit) =

(14)

where the second equation follows from the first one by considering that the sum of squares of the two coherence amplitudes must be constant. According to eqs 12 and 14, a pulse with Qcrit = 2 ln 2/π that passes both transitions of an S = 1 system creates coherence with the same amplitude (2)1/2/2 on the allowed transition (1 ↔ 2) and the forbidden transition (1 ↔ 3). The maximum coherence amplitude attainable on the forbidden transition is only slightly larger, |⟨S+13⟩| = 4/(3(3)1/2) and is attained at Qcrit = ln(9)/π, corresponding to a flip angle of β = arccos(−1/3) ≈ 109.5°. The maximum coherence amplitude on the allowed transition (1 ↔ 2) is the same but attained at Qcrit = 2 ln(3/2)/π, corresponding to β = arccos(1/ 3) ≈ 70.5°. The analytical expression for maximum coherence amplitude on the allowed transition (2 ↔ 3) is lengthy. We find max(|⟨S+13⟩|) ≈ 1.57 at Qcrit ≈ 0.65. Hence, a pulse with Qcrit = 2 ln(2)/π comes close to generating maximum coherence on all three transitions. Taking into account eqs 4 and 11, it is straightforward to extend such considerations to coherence generation on any transition in an S > 1/2 system in situations

βSx(23)

Sx(12) ⎯⎯⎯⎯⎯→ cos(β /2)Sx(12) − sin(β /2)S(13) y βSx(23)

S(12) ⎯⎯⎯⎯⎯→ cos(β /2)S(12) + sin(β /2)Sx(13) y y

+ 1 − exp2 ( −πQ crit /4) |⟨S12 ⟩|(0)

(13)

Note that the transfer occurs independently of phase of the initial coherence and that it has spinor behavior: A pulse with flip angle 2π inverts the sign of the coherence, and the flip angle period is 4π. This has interesting consequences if the transfer is caused by a passage pulse, as is seen in Figure 5. Adiabatic passage leads to complete coherence transfer, confirming the intuitive picture that such passage corresponds to a π pulse. Furthermore, exponential loss of the original coherence with increasing Qcrit occurs with only half the rate of inversion of a two-level transition. If we denote coherence amplitudes with absolute expectation values of pseudo observables |⟨S+lm⟩|, the 13577

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the remaining two transitions of spin S2 are passed, no other coherence exists in the system (Figure 6a and b). What happens on passage of the transitions of spin S2 depends on the relative phase of coherence on the transitions of spin S1. If the phase difference is zero, coherence on the S1 transitions is transferred to coherence on the forbidden zero- and doublequantum transitions during passage of the first allowed S2 transition and is then quantitatively transferred back to coherence on the allowed S1 transitions during passage of the second allowed S2 transition (Figure 6a). If, on the contrary, S1 spin coherence is in antiphase, passage of the second S2 transition completes quantitative transfer to zero- and doublequantum coherence (Figure 6b). Phase difference between the two S1 transitions accumulates during the sweep due to the difference of their resonance frequencies, which is the dipole−dipole splitting d. The evolution time t between passage of the center of the S1 doublet and the center of the S2 doublet is given by t = tpΔΩ/B. Neglecting the pseudosecular term in eq 15, the amplitude of the antiphase coherence at passage of the center of the S2 doublet and thus the amplitude of zero- and double-quantum coherence after complete passage should thus be given by sin[dtpΔΩ/(2B)]. This prediction is confirmed by simulations with ΔΩ/(2π) = 500 MHz and d/(2π) = 10 MHz even when including the pseudosecular term (Figure 6c). As seen in Figure 6a−c, coherence generation on the two S2 transitions is slightly asymmetric. The transition that is passed first acquires slightly less than unit coherence, whereas the one that is passed last acquires slightly more than unit coherence. This effect vanishes if the pseudosecular term in the spin Hamiltonian is omitted (data not shown). Zero- or double-quantum coherence generation during a passage pulse by this mechanism cannot easily be utilized for measuring distance distributions, as in general ΔΩ is broadly distributed and this distribution is not known. Since such coherence generation on forbidden transitions reduces coherence amplitude on the allowed transitions, it is an unwanted effect in most experimental contexts. Figure 6d presents simulations that were performed for estimating such coherence loss under conditions that are typical for Q-band DEER experiments on pairs of nitroxide labels.29 We assumed a length tp = 200 ns for a passage pulse with a bandwidth B/(2π) = 500 MHz and Qcrit = 2 ln 2/π that covers the whole nitroxide spectrum and is realistic with a high-power Q-band EPR spectrometer for an oversized sample. Coherence amplitudes for all six transitions in the system after the passage pulse were computed for dipole−dipole couplings in the equatorial plane of the coupling tensor at distances between 1.5 and 8 nm corresponding to the typical range of DEER measurements. Because spins S1 and S2 are assumed to have the same resonance frequency distribution corresponding to a simulated nitroxide EPR spectrum at 34 GHz, the problem is symmetric with respect to exchange of the two spins. Asymmetry is observed between the two single-quantum transitions of the same spin as well as between the single- and double-quantum transition. A significant reduction of sensitivity by more than 10% due to the generation of zero- and double-quantum coherence is predicted at distances below 2 nm. At distances above 5 nm, loss in coherence on the single-quantum transitions is negligible. This indicates that losses due to intermolecular dipole−dipole interactions should be negligible in the typical concentration range between 5 and 200 μM, where distances to the next neighbor spin are typically much

where several, but not necessarily all, of the 2S + 1 transitions are passed consecutively. In spin systems with a more complicated topology of the transitions, coherence transfer according to eq 14 occurs independently for all coherences that share one level with the transition that is passed. Note however that in such systems coherence on the same transition may be generated at various times during the sweep by transfer from different transitions. This generally results in different phase and thus in interference effects. Such a situation will be discussed in section 4.1. In the transfer described by eq 14, coherence on transition (1 ↔ 3) experiences a phase shift Δϕc = Δϕ0 + ϕ with respect to initial coherence on transition (1 ↔ 2), where ϕ is the phase of the passage pulse and Δϕ0 is constant. Considering also previous generation of coherence on transition (1 ↔ 2) during the same pulse, the phase of double-quantum coherence varies with 2ϕ, whereas the phase of single-quantum coherence varies with ϕ. A phase cycle [+(+x) − (+y) + (−x) − (−y)] thus keeps double-quantum coherence and cancels polarization and single-quantum coherence. This is the same phase cycling as for a double-quantum coherence generator based on rectangular monochromatic pulses. In general, selection of coherence order change by phase cycling of a passage pulse follows the same rules as for monochromatic pulses. If the amplitude ω1 of the microwave irradiation becomes comparable to the splitting between the two transitions, the approximation of consecutive selective passage breaks down. In our simulations, we noticed first visible effects for splittings smaller than about 5ω1. For degenerate transitions, we find that the approach to the adiabatic limit is only half as fast as that for passage of a two-level system, I = 1 − exp(−πQcrit/4). To achieve the same equivalent flip angle, ω1 needs to be increased by a factor of 4. Adiabatic passage of two degenerate transitions that share a common level exchanges populations of the two nonshared levels. The π/2 pulse equivalent passage with Qcrit = 4 ln 2/π generates coherence with amplitude (2)1/2 on both allowed transitions and no coherence on the forbidden transition. In the limit of degenerate transitions, a passage pulse thus resembles an ideally nonselective pulse.

4. FOUR-LEVEL SYSTEMS 4.1. Two Coupled Electron Spins. Measurement of distance distributions in the nanometer range is one of the major applications of pulsed EPR spectroscopy. Optimization of passage pulses for such measurements requires understanding of coherence transfer by passage pulses in a system of two dipole−dipole coupled electron spins S1 = 1/2 and S2 = 1/ 2 with the Hamiltonian H = Ω1S1, z + Ω 2S2, z + dS1, zS2, z − (d /2)(S1, xS2, x + S1, yS2, y) (15)

where d is the dipole−dipole coupling and the last term on the right-hand side of eq 15 is negligible for |d| ≪ |ΔΩ| with ΔΩ = Ω2 − Ω1. In the following, we relate Ω1 and Ω2 to the center of the sweep and assume, without loss of generality, Ω1 < Ω2. We consider the case typical in double electron electron resonance (DEER, also termed PELDOR) experiments where d < ΔΩ. The sweep covers only the four allowed single-quantum transitions. In this situation, the two allowed transitions of spin S1, which do not share a common level, are passed first and are excited independently of each other. By π/2-pulse equivalent passage (Qcrit = 2 ln 2/π), coherence with unit amplitude is generated on both of these transitions and, before 13578

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Figure 7. Coherence generation and transfer in a system consisting of an electron spin S = 1/2 and a nuclear spin I = 1/2 with nuclear Zeeman frequency of νI = −51.66 MHz, secular hyperfine coupling A/(2π) = 6.43 MHz, and pseudosecular hyperfine coupling B/(2π) = 3.37 MHz. (a) Coherence amplitudes after a passage pulse of duration tp = 240 ns with B/(2π) = 0.5 GHz starting from thermal equilibrium. Electron coherence on allowed transitions is depicted by green lines [(1 ↔ 3) solid, (2 ↔ 4) dotted], coherence on forbidden electron−nuclear transitions by red lines [(2 ↔ 3) solid, (1 ↔ 4) dotted], and nuclear coherence by blue lines [(1 ↔ 2) solid, (3 ↔ 4) dotted]. The vertical black dotted line denotes Qmin = 2 ln 2/π and the horizontal black dotted line the amplitude of coherence on the two forbidden transitions after an ideal π/2 pulse. A zoom into the range of small transfer amplitudes is provided in Figure S6a (Supporting Information). (b) Coherence transfer by matched (ω1 = ωI) passage pulses as a function of pulse length tp and critical adiabaticity Qcrit. The red dotted line depicts the efficiency T of the transfer from electron coherence on allowed transitions to nuclear coherence and the green dotted line efficiency of the back transfer of nuclear coherence to electron coherence on allowed transitions. The black line is the product of these two functions, corresponding to modulation depth in an ESEEM experiment.

longer than 10 nm. Note that in multispin systems, such as spin-labeled protein homooligomers, magnetization is lost to several zero- and double-quantum transitions. In such cases, significant sensitivity loss may occur at distances longer than 2 nm. 4.2. Electron Spin S = 1/2 Coupled to a Nuclear Spin I = 1/2. Another important application of pulsed EPR is the measurement of the transition frequencies of hyperfine coupled nuclei by electron spin echo envelope modulation (ESEEM) and hyperfine sublevel correlation (HYSCORE) experiments. Such experiments rely on the excitation of coherence on formally forbidden electron−nuclear zero- and double-quantum transitions or on nuclear transitions by applying solely microwave pulses. Typically, only a small fraction of total magnetization is converted to such coherence and converted back to allowed electron coherence for detection or is detected directly on forbidden transitions. In experiments with monochromatic rectangular pulses, generation and backtransfer of coherence on forbidden transitions30 and nuclear transitions31 can be enhanced by matching the pulse amplitude ω1 to the nuclear Zeeman frequency ωI. It is of some interest whether similar enhancement can be achieved with passage pulses. We consider this problem in a rotating frame for the electron spin S = 1/2 and in the laboratory frame for the nuclear spin I = 1/2. The spin Hamiltonian is then given by H = ΩSz + ωI Iz + ASzIz + BSzIx

The dependence of coherence amplitude for all six transitions on critical adiabaticity Qcrit is shown in Figure 7a for a passage pulse that starts from only electron polarization −Sz, which to a good approximation corresponds to thermal equilibrium. At Qcrit = 2 ln(2)/π (vertical black dotted line), corresponding to a nominal π/2 passage pulse, the coherence amplitude for the two allowed electron spin transitions (green) is almost unity, the coherence amplitude for the two forbidden transitions (red) is almost equal and similar to the one obtained with an ideal π/2 pulse (horizontal black dotted line), and almost no nuclear coherence (blue) is created (see also Figure S6a, Supporting Information). At Qcrit = 2 ln(2)/π, coherence amplitudes do not significantly depend on pulse length for tp > 64 ns (data not shown). Shorter pulses are incompatible with our edge smoothing, but as in the range tested the outcome is basically the same as for an ideal π/2 pulse, no other phenomena are expected for even shorter pulses. At larger Qcrit, the effective flip angle for both forbidden transitions increases by the same factor, but an asymmetry arises with respect to coherence generated on the two transitions: for the forbidden transition (2 ↔ 3) that is passed first (red solid line), coherence amplitude decreases, whereas it increases for the other forbidden transition (1 ↔ 4) (red dotted line). This can be understood by considering that coherence on the firstpassed forbidden transition is transferred to coherence on the last-passed forbidden transition at the time when the two allowed transitions are inverted. Note however that this explanation assumes consecutive idealized passage, a condition that is violated by the strong microwave fields at large Qcrit. Furthermore, significant nuclear coherence is generated, although its maximum amplitude of 0.042, attained at Qcrit = 1.9 for our parameter set, remains much smaller than the maximum coherence amplitude of 0.438 for the last passed forbidden transition (attained at Qcrit = 5.75). By applying a second passage pulse with equivalent flip angle π/2, a large part of the coherence on the forbidden transition (1 ↔ 4) can be transferred to coherence on both nuclear

(16)

where for illustration we assume a nuclear Zeeman frequency νI = ωI/(2π) = −51.65 MHz, corresponding to a proton coupled to an electron spin with g = ge = 2.002319 at a Q-band frequency of 34 GHz, a secular part of the hyperfine coupling A/(2π) = 6.43 MHz, and a pseudosecular part of the hyperfine coupling of B/(2π) = 3.37 MHz. Furthermore, we assume that a band of 500 MHz width is passed during a pulse with a duration of 240 ns. 13579

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that the first pulse with low Qcrit needs to be twice as long as the second and third pulses with large Qcrit in order to achieve refocusing of electron spin resonance offsets. Nevertheless, a pulse length of 128 ns used in recent chirp echo experiments8 already provides significant modulation depth enhancement and is probably well below the relaxation limit. Note that transfer of nuclear to electron coherence leads to sensitivity loss if the π pulse in a HYSCORE experiment is replaced by a matched pulse with large adiabaticity. For the intended exchange of nuclear coherence between transitions (1 ↔ 2) and (3 ↔ 4), matching should be avoided and Qcrit should be only moderate. For our parameter set, we find >99% nuclear coherence exchange at ω1 = 0.6ωI with a pulse length of 400 ns.

transitions (Figure S6b, Supporting Information). In the adiabatic limit, a second passage pulse transfers coherence on the forbidden transition (1 ↔ 4) to coherence on the other forbidden transition (2 ↔ 3). A sequence of two passage pulses, the first one having a large Qcrit and the second one having Qcrit = 2 ln(2)/π, thus creates much more nuclear coherence than can be created by single passage at any Qcrit. This is reminiscent of one of the nuclear coherence generators suggested for matched three-pulse ESEEM that consists of a high-turning-angle (HTA) pulse, an interpulse delay, and a π/2 pulse, with the amplitude of the HTA pulse fulfilling the condition ω1 = ωI.31 In both cases, nuclear coherence is generated efficiently by enhancing the formally forbidden transfer from polarization to coherence on a forbidden electron−nuclear transition in a first step and effecting the allowed transfer from electron−nuclear to nuclear coherence in a second step. In matched ESEEM, the reversed scheme was also shown to work, where first coherence on allowed electron transitions is created by a π/2 pulse and then the forbidden transfer from electron to nuclear coherence is enhanced by a matched HTA pulse.31 Furthermore, it was demonstrated that the nuclear coherence can be back transferred directly to electron coherence on allowed transitions by another HTA pulse. As the allowed electron coherence is detectable with high efficiency, such a scheme can strongly enhance sensitivity of three-pulse ESEEM and HYSCORE experiments. To see whether this principle translates to wideband excitation with passage pulses, we focus on the sequence consisting of a first passage pulse with Qcrit = 2 ln(2)/π for generating electron coherence on allowed transitions, a second passage pulse with larger Qcrit for the transfer to nuclear coherence, and a third passage pulse with larger Qcrit for back transfer of the nuclear coherence to electron coherence on allowed transitions. In a first set of simulations, we attempted to iteratively find the Qcrit and pulse duration tp that maximize formally forbidden coherence transfers. We found that transfer efficiency increases monotonously with Qcrit and that optimal pulse duration at given Qcrit is proportional to B/ωI2. Furthermore, with increasing Qcrit, the optimal ω1 appears to approach the nuclear Zeeman frequency ωI, corresponding to the same level of anticrossing in the rotating frame that underlies matched ESEEM.30,31 This suggested to study transfer while maintaining the matching ω1 = ωI, which implies that pulse duration is proportional to Qcrit. Neglecting relaxation, transfer of almost half of the electron coherence on allowed transitions to nuclear coherence is observed at large Qcrit (red dotted line in Figure 7b). Note that transfer of half the coherence would correspond to a transfer efficiency of T = (2)1/2/2 ≈ 0.707. The transfer is symmetric: With the same transfer efficiency, nuclear coherence is transferred back to electron coherence on allowed transitions (green solid line). The product of the two transfer efficiencies can exceed 0.45, which compares to a modulation depth factor1 of k = 0.0043 in a conventional three-pulse ESEEM experiment for this set of spin Hamiltonian parameters. A chirp ESEEM experiment consisting of a first pulse with Qcrit = 2 ln 2/π, a second, matched pulse with large Qcrit, a variable delay for nuclear coherence evolution, and a third matched pulse with large Qcrit and pulse lengths tp,1 = 2tp,2 = 2tp,3 can thus be expected to simultaneously increase excitation bandwidth and modulation depth. As a further advantage, no blind spots occur. In practice, relaxation during the pulses will limit transfer efficiency. With respect to this limitation, it is disadvantageous

5. EXPERIMENT AND SIMULATIONS 5.1. Experimental Section. All experiments were performed on a home-built AWG spectrometer operating at X-band frequencies8 with a fully overcoupled Bruker MS3 splitring resonator (see also the Supporting Information). Chirp pulses had their sweep rate adapted to the experimental resonator profile5 and their edges smoothened by sine quarter waves with 50 ns length. For the inversion I of the echo signal reported in Figure 2, the pulse sequence (β)chirp−(π/2)−(π) was used, with delays between starting pulse flanks of 3 μs and pulse lengths of 16 and 32 ns for the π/2 and π pulse, respectively. A [+(+x) − (-x)] phase cycle on the π/2 pulse combined with a [+(+x) − (+y) + (−x) − (−y)] phase cycle on the π pulse was used. For the offset dependence of sQ, as shown in Figure 4a, the resonance offset Ω was varied by stepping the static magnetic field. At each field position, chirp echoes with refocusing pulses having a variety of critical adiabaticities Qcrit were recorded and the dynamic phase was extracted as described in the Supporting Information. In order to vary Qcrit for two-pulse chirp echoes the amplitude of the second pulse was changed. For the threepulse sequences, the second and third pulses’ amplitudes were varied concurrently while maintaining identical critical adiabaticity for each of these two pulses. The two-pulse chirp echo data were obtained with a delay of 3 μs between starting pulse flanks and a [+(+x) − (−x)] phase cycle on the first pulse. The three-pulse refocused chirp echo data (ABSTRUSE) were obtained with a delay of 6 μs between starting pulse flanks of the first and last pulse, while the second pulse immediately followed the first pulse. A four-step phase cycle, [+(+x) − (−x)], on the first and the third pulse was used to record the stimulated echo. To reject stimulated echoes in the ABSTRUSE experiment, a phase cycle [+(+x, +x) − (−x, +x) + (+x, −x) − (−x, −x)] was applied to the first two pulses. For the 2:1:1 scheme and the −1:−2:1 scheme, the delays between starting pulse flanks of the first and second pulses were 3.6 and 3.2 μs, respectively. The third pulse immediately followed the second pulse. To compare refocused echo data to primary echo data (Figure 4c and d), the timing of the primary echo was adjusted to equalize the total coherent evolution time. Furthermore, the four-step phase cycle of the refocused echo was here also used for the primary echo. The small Herasil sample with a 1 mm diameter was recorded at 100 K using a repetition time of 40 ms. The extended sample with 2 mm diameter was recorded at ambient temperature using a repetition time of 5 ms. 13580

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5.2. Simulations. All simulations were performed and analyzed with home-written Matlab scripts based on the Matlab library SPIDYAN for simulating experiments with arbitrary waveform excitation with density operator formalism in Hilbert space (without relaxation) or Liouville space (with relaxation). A detailed description of SPIDYAN will be given elsewhere. Briefly, during pulses, the density matrix is propagated stepwise with the time resolution of the AWG, which was 12 GS/s (83.33 ps) as in our experiments, except for the simulations shown in Figures 1, 2, and 7 where it was 200 GS/s. Hilbertspace or Liouville-space propagators for these time steps are precomputed for all digital output levels of the AWG and tabulated. Usually 1024 levels were used corresponding to the 10-bit vertical resolution of our AWG; the simulations shown in Figures 1, 2, and 7 were performed with 14-bit vertical resolution. Evolution was computed in a rotating frame in which the center of the frequency sweeps was at 1 GHz and the single-channel output mode of the simulated AWG was used, corresponding to double sideband irradiation, except for simulation of dynamic phase shifts in Figure 3 and Figure 4a where the complex excitation feature of SPIDYAN was used to suppress the modulation sideband below the local oscillator frequency. All simulated pulses were linear frequency sweeps with constant amplitude, except for smoothing of the leading and trailing edges by sine quarter waves with 20 ns length.

AUTHOR INFORMATION

Corresponding Author

*Phone: +41 (0)44 6325702. Fax: +41 (0)44 6331448. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Shimon Vega for a helpful remark on the importance of relaxation, Takuya Segawa for helpful discussions, and an anonymous reviewer for suggesting the term “critical adiabaticity”. For the experimental and simulation part of this work, funding from the ETH research grant ETHIIRA-23 11-2 and from the DFG priority programme SPP 1601 (grant JE 246/5-1), respectively, is acknowledged.

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6. CONCLUSION The extent of magnetization inversion after linear fast passage of a single transition is an exponential function of critical adiabaticity during passage. This applies to inversion of coherence order as well as of polarization, which allows for defining an equivalent flip angle for such a fast passage pulse. The concept can be extended to transfer of coherence from a transition that shares one level with the passed transition to the transition that shares the other level. Limitations to such an idealized description of passage pulses arise from relaxation and from dynamic phase shifts. The dynamic phase shifts are particularly large near the edges of the passed frequency band and can be largely refocused. For echo experiments consisting of several passage pulses, a general Diophantine equation determines the pulse length ratios required for refocusing of resonance offsets. In multilevel systems, passage of several transitions may result in unavoidable excitation of coherence on forbidden transitions, which may adversely affect sensitivity of experiments that rely on only allowed coherence transfer or may be harnessed for sensitivity enhancement in experiments that rely on formally forbidden transfers. The previously introduced concept of matched pulses can be extended to passage pulses for enhancing generation of electron coherence on forbidden transitions or of nuclear coherence by excitation of only electron spin transitions.

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S Supporting Information *

Experimental and data analysis procedures, description of the AWG spectrometer, and supplementary figure on nuclear coherence generation. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b02964. 13581

DOI: 10.1021/acs.jpcb.5b02964 J. Phys. Chem. B 2015, 119, 13570−13582

Article

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NOTE ADDED IN PROOF For inversion of polarization, eq 5 agrees with the LandauZener formula describing a diabatic transition at a level anticrossing.32 It is the essence of Sections 2.1 and 3.2 that the same or similar expressions apply more widely to coherence transfers.

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DOI: 10.1021/acs.jpcb.5b02964 J. Phys. Chem. B 2015, 119, 13570−13582