Para-Hydrogen Raser Delivers Para-hydrogen raser …...V Simulation of SABRE NMR spectra 9 VI Multi-mode EHQE based raser 16 VII Limits of precision for J-coupling measured with the

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Para-Hydrogen Raser DeliversSub-Milli-Hertz Resolution in Nuclear

Magnetic ResonanceSupplementary Information

Martin Suefke, Sören Lehmkuhl, Alexander Liebisch,Bernhard Blümich and Stephan Appelt∗

Section Page

I Maser equations and key dynamical parameters 2II Distant dipolar fields 3III Quantum picture of a raser based on two-spin order 5IV Distant dipolar field immunity of the two-spin based raser 8V Simulation of SABRE NMR spectra 9VI Multi-mode EHQE based raser 16VII Limits of precision for J-coupling measured with the two-spin ordered raser 19VIII Temperature dependence of raser and SABRE signals 21References 23

Figure Page

S1 Distant dipolar field of a 1H pyridine raser with single spin-order. 4S2 Raser of a J-coupled I-S spin system based on single- and two-spin order. 6S3 1H and 15N signals from 15N-acetonitrile. 7S4 Comparison between measured and simulated 15N-pyridine 1H spectra. 15S5 Simulated and measured 15N-pyridine 1H four mode raser signal. 18S6 1H raser signal of 15N-pyridine 1H measured at 41.6 kHz and for 1200 s. 20S7 1H SNR from 1 µL pyridine measured at 166 kHz versus conversion

temperature inside p-H2 generator for SABRE and raser signals. 22∗correspondence to: [email protected]

1

Para-hydrogen raser delivers sub-millihertzresolution in nuclear magnetic resonance

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I Maser equations and key dynamical parameters

In the rotating frame the motion of the transverse and longitudinal magnetization MT(t)and Mz(t) of a raser operating at a single frequency can be described by a set of nonlinearcoupled differential equations6

d

dtMz (t) = −

M2T (t)

τrdM0+

1

τp{M0 −Mz(t)} − Mz(t)

T1, (3)

d

dtMT (t) = MT

{Mz(t)

τrdM0− 1

T ∗2

}. (4)

Numerical simulations (see Fig. S1c) show that the characteristic motion of MT(t) andMz(t) are determined by four constants, namely the radiation damping rate 1/τrd, thepumping rate 1/τp, the overall transverse relaxation rate 1/T ∗

2 = 1/T2 + 1/τp and thelongitudinal relaxation rate 1/T1.Assuming Mz(0) = 0 and MT(0) ≈ 0 (quantum fluctuations) at t = 0 and neglectingany back action of the detection coil, equation (3) predicts an initial exponential growthof Mz(t) with the pumping rate 1/τp towards the equilibrium magnetization M0. Notethat MT is always positive but M0 is negative (population inversion) and Mz(t) startsto grow negatively at first. When the negative magnetization exceeds the thresholdMz(t) = M0τrd/T

∗2 the braced term in equation (4) switches from negative to positive

sign, so dMT(t)/dt > 0 and raser oscillation starts. From equation equation (4) followsthat 1/τrd > 1/T ∗

2 (τrd < T ∗2 ) is indispensable for raser activity. An oscillatory motion of

Mz(t) results from equations (3) and (4). This oscillation is damped until dMz/dt = 0

and the equilibrium value M ez = M0τrd/T

∗2 and a constant M e

T = M0(τrdτp

(1 − τrdT ∗2))1/2

is reached. The superscript “e” stands for equilibrium. In the laboratory frame M eT

oscillates around the B0 field with the Larmor frequency ν0 = γB0/2π and the steadystate motion can be described by a rotating cone with a constant M e

z , which points inopposite direction with respect to the direction of B0. Thus M e

z represents a populationinversion or a state with negative spin temperature. The start-up spikes of the initialrelaxation oscillations, omitting T1 relaxation rate, are damped out with a late timedamping rate 1/(2τp) and are separated by a characteristic late time period given by

TL = 4π τp/√

8τp(1/τrd − 1/T2∗) − 1 . (5)

The values for 1/τp and TL can be determined from the late time decay and the separationbetween subsequent raser pulses. 1/T2 can be measured from the free induction decay ofthe corresponding NMR experiment. From Figs. 1b,c we get τp,py = 20 s, T ∗

2,py = 0.7 s,

2






TL,py = 2.6 s for pyridine and τp,ac = 10 s, T ∗2,ac = 0.6 s, TL,ac = 8.8 s for acetonitrile.

The radiation damping time τrd can be estimated from equation (5) given the measuredvalues for TL, T ∗

2 and τp. This results in τrd,py = 0.016 s for pyridine and τrd,ac = 0.18 sfor acetonitrile. The proton polarization PH for the experiments shown in Figs. 1b,ccan be estimated from equation (2) using the known parameters η = 0.25, Q = 300,and the spin number densities ns,py = 7.5 · 1019 cm−3, ns,ac = 3.4 · 1020 cm−3. We obtainPH,py = −3.8·10−3 and PH,ac = −7.3·10−5. So acetonitrile is about 52 times less polarizedcompared to pyridine. To conclude equations (3–5) together with the key parameterscorrectly describe the dynamics of the raser signals presented here pumped to a statecharacterized by a negative spin temperature (or by a single-spin order ∼ αIz, α < 0)and oscillating at one single frequency.

II Distant dipolar fields

Distant dipolar fields are important since they influence the motion, the observed fre-quency as well as the precision of frequency measurements of molecular rasers. Desvaux30

describes various non-linear effects in liquid state NMR systems based on distant dipolarfields. The average dipolar field for 1H-spins is given by Bdip = µ0M0 = 0.5µ0nsγH�PH,where ns is the spin number density and PH is the proton polarization. For the raserexperiment with pyridine ns,py = 7.5 · 1019 cm−3 and PH,py = 3.8 · 10−3 so the cal-culated dipolar field is Bdip = 5.0 nT. This corresponds to a 1H frequency shift of∆νdip = 214mHz. We verified the dipolar shift experimentally for the pyridine rasersignal shown in Fig. 1b, an expansion of which is shown in Fig. S1 a. In particular thefirst raser burst around t = 0 s is accompanied by a sign change in the correspondingMz magnetization (Fig. S1c) and therefore undergoes roughly a frequency shift ∆νmeas.The highly resolved Fourier transformed spectra before and after t = 0 yield a measuredfrequency difference of ∆νmeas = 224mHz (Fig. S1b), which is close to the calculated∆νdip.

3






Fig. S1: Distant dipolar field of a 1H pyridine raser with single spin-order.a Time expansion of Fig. 1b around first raser burst at t = 0 s showing a starting pyridineraser. b Fourier transformed spectra before (red) and after (blue) the maximum ofthe first raser burst. The difference between the two maxima is given by |∆νmeas| =224mHz, which corresponds to a distant dipolar field of Bdip = −5 nT. The negativeoffset frequency axis comes from the choice of the reference frequency at 166.66 kHz.c Simulation of MT and Mz versus time during the first maser burst based on equations(3) and (4). Simulation parameters: τp = 20 s, T1 = 2.7 s, T ∗

2 = 0.7 s and τrd = 0.016 s.Mz changes sign shortly after MT reaches its maximum. The red and blue bars correspondto the time intervals highlighted in a, which were in turn used to determine |∆νmeas|.

4






III Quantum picture of a raser based on two-spin order

The vector cone representation of raser operation (Fig. 2c) is an intuitive way to explainthe main differences between a raser pumped into a state characterized by a negativespin temperature (population inversion) and by a two-spin state. Fig. S2 shows the cor-responding energy level diagram of a hetero-nuclear J-coupled two-spin system I − S.The four energy levels are indicated by Ei with i = 1..4 . In the case of a populationinversion (Fig. S2a) the highest energy state E4 is the most populated, and the lowerlevel’s population descends in order. Consequently all four possible dipole transitionswith frequencies νI ± J/2, νS ± J/2 have an individual population inversion that can bestimulated to emit radiation (raser). In contrast the two-spin ordered state (Fig. S2b)can be characterized by an increased population of the levels E2 and E3 while E3, E4

are depopulated (quadrupolar distribution). Now only the lower two transitions at fre-quencies νI − J/2, νS − J/2 can be raser active while the upper transitions at νI + J/2,νS + J/2 are raser inactive, since the corresponding energy levels have no populationinversion. All four lines can be observed in a standard NMR experiment after pulseexcitation as either I- or S-spin spectrum with anti-symmetric structure (Fig. S3 a-d).Inversion of a state with population inversion or negative spin temperature (Fig. S2a)inhibits raser activity, while inversion of the two-spin state (Fig. S2b) shifts the raseractive modes from νI,S − J/2, to νI,S + J/2. This fundamental difference distinguishesthe two-spin ordered raser from all other known rasers and masers. With regard to theSABRE mechanism this means that raser action is independent of the individual signsof the quantum mechanical coefficients characterizing the SABRE polarization transfer.

5






Fig. S2: Raser of a J-coupled I-S spin system based on single- and two-spin order.a,b, Energy level diagram (left) and schematic spectrum of raser active modes (right) ofa hetero-nuclear J-coupled two-spin 1/2 system I-S. The energy levels are given by

E1/h = −νI/2− νS/2 + J/4

E2/h = −νI/2 + νS/2− J/4

E3/h = νI/2− νS/2− J/4

E4/h = νI/2 + νS/2 + J/4

where νI = γIB0/(2π) and νS = γSB0/(2π). Circles indicate the population of each level.The transition frequencies of each raser active modes are indicated by arrows. For thecase of a population inversion (α(Iz + Sz), α < 0) both I spin and S-spin transitions areraser active. If the initial state is a two-spin ordered state (β(Iz. Sz), β < 0) only oneraser transition for either I- or S-spin can be observed.

6






a

c

b

d

x10

1

2

2

1

Simulation

Simulation

Measurement

41.650 41.655 41.660 41.665 41.670

Ampl

itude

(a.u

.)

Frequency (kHz)0.0 0.2 0.4 0.6 0.8 1.0 1.2

-0.1

0.0

0.1

-0.1

0.0

0.1

Ampl

itude

(V)

Time (s)

Ampl

itude

(V) Measurement

Simulation

Time (s)

90x°

Pulse

41.658 41.659 41.660 41.661 41.662

Ampl

itude

(a.u

.)

Frequency (kHz)

-1.8 -1.7 -1.6 -1.5

Fig. S3: 1H and 15N signals from 15N-acetonitrile.a, 1H raser signal of 15N-acetonitrile 1© measured at B0 = 0.977mT (41.6 kHz) at t < 0 s.After 90◦ pulse excitation at t = 0 a long FID appears 2©. b, Fourier spectrum of panela, 1© (top), of 2© (middle). The simulated 1H spectrum (bottom) results from pure two-spin order. Note the raser line which appears only on the left side of the spectrum. c,Measured 15N (top) and simulated 15N FID (bottom) of 15N-acetonitrile after 90◦ pulseexcitation (B0 = 9.65mT, 41.6 kHz). d, Measured (top) and simulated (bottom) 15Nspectrum from panel c.

7






IV Distant dipolar field immunity of the two-spin based raser

A further important difference exists between a two-spin ordered raser and a raser basedon population inversion. The raser based on two spin order, which applies to our 15Nlabelled substrates at low field, is nearly immune against distant dipolar fields (DDF).This can be graphically illustrated by two antiparallel magnetizations (M+

z and M−z )

which cancel out all DDF effects30. A slight imperfection arises from the reduction inthe Mz magnitude due to the raser action (see Fig. 2c 1© where |M e-

z | � |M e+z |). For the

case of 15N labelled pyridine raser the magnitude of the 1H−15N two-spin order is at leastone order of magnitude smaller than the dipolar magnetization (single-spin order) of purepyridine. Since we measured a maximum dipolar shift effect of ≈ 224mHz for the pyridineraser (section II) we can estimate the constant dipolar shift for the two-spin based 15N-pyridine raser, taking into account the non-perfect cancelation M e-

z and M e+z , to be in

the order of a few mHz. This is small compared to the maximum frequency fluctuationsof 400mHz as reported in Fig. 4a. Therefore the observed frequency variations are causedmainly by ambient magnetic field fluctuations or by the temporal instability of the B0

current source. Thus the self-compensating two-spin ordered raser is suitable as a reliablemagnetic field sensor. When measuring J-coupling constants using frequency differencesin the spectra external magnetic fields as well as slight internal DDF fluctuations in themHz regime cancel out, leading to a precision far below the milli-Hertz regime. Finallythe absence of DDF explains why we could not observe non-linear effects associated toDDF, such as dipolar echoes, spectral clustering, spin turbulence30–34, in our 15N-pyridineraser experiments.In summary the immunity against DDF, the absence of non-linear effects and the factthat the bound complex is NMR and raser silent (see section V) distinguishes the raserbased on two-spin order from rasers and masers based on population inversion. Thereforethe hetero-nuclear two-spin ordered raser opens the door for high-precision measurementsof molecular coupling parameters, magnetic fields and rotational motions. This is notonly important for material sciences and sensor-technology but also for fundamentalphysics25,27,28, for example for the two nuclei 3He−129Xe Zeeman raser8,25, which isplagued by distant dipolar field effects.

8






V Simulation of SABRE NMR spectra

Valuable information about the spin state and the possible maser active frequencies for15N labeled pyridine and acetonitrile can be gained from measurements and simulationsof the corresponding 1H and 15N NMR spectra. For 15N-pyridine we define the set ofone 15N spin at position 1 (S-spin) and five proton spin-vector operators (I-spins) atpositions 2–6 as {S, I2, I3, I4, I5, I6} (Fig. 3c). Correspondingly for 15N-acetonitrile wehave {S, I2, I3, I4} (Fig. 2a). The Hamilton operator for 15N-pyridine can be expressedas

Hpy = ωI

6∑j=2

Ijz + ωSSz + 2π

{J12S · (I2 + I6) + J13S · (I3 + I5)

+ J14S · I4 + J23(I2 · I3 + I5 · I6) + J25(I2 · I5 + I3 · I6)

+ J34I4 · (I3 + I5) + J24I4 · (I2 + I6) + J35I3 · I5 + J26I2 · I6}

(6)

and for 15N-acetonitrile it is given by

Hac = ωI

4∑i=2

Iiz + ωSSz + 2πJNHS · (I2 + I3 + I4)

+ 2πJHH (I2 · I3 + I2 · I4 + I3 · I4) (7)

The Larmor frequency of the 1H (15N) spin in equations (6,7) is given by ωI = γHB0

(ωS = γ15NB0). For pyridine the influence of the chemical shift of the two ortho (I2,I6, δo = 8.43 ppm), meta-protons (I3, I5, δm = 7.34 ppm) and the para proton (I4,δp = 7.75 ppm) can be neglected, since their frequency differences at B0 ≈ 1mT are verysmall (< 0.04Hz).For 15N-pyridine at 1 mT the difference in frequency between the ortho, meta and paraprotons is dominated by the difference in the corresponding hetero-nuclear J-couplingconstants, which is in the same order of magnitude compared to the homo-nuclear cou-pling constants. Thus the condition for strong coupling is fulfilled. Therefore a calcula-tion of the frequency difference between possible J-coupled spin states needs to take intoaccount all off-diagonal elements in the scalar products in equations (6,7). We found thecalculated and measured frequency differences in the strong coupling regime are very closeto the J-coupling constants measured in high field. The 1H−15N J-coupling constants of15N-pyridine measured at high field are20: {J12, J13, J14} = {−10.06,−1.56,±0.18} Hz,and the homo-nuclear 1H−1H J-coupling constants are: {J23, J24, J25, J26, J35, J34} =

{4.97, 1.81, 0.90,−0.16, 1.38, 7.83} Hz. For 15N-acetonitrile we find JNH = 1.72Hz.

9






If acetonitrile or pyridine without 15N is attached to the Ir-complex the singlet state ofthe bound para-hydrogen molecule is converted, depending on the field, into observablesingle spin order ∼ −Ijz and/or into multiple spin order by the SABRE mechanism14,15.For example the 1H single spin order of pyridine at B0 = 6.6mT at t = 0 can beexpressed as ρpy(0) = o(I2z+ I6z)+m(I3z+ I5z)+pI4z. The three parameters o = −0.072,m = −0.038 and p = −0.081 represent the amount of spin polarization of the ortho, metaand para protons14,29. This initial state implies that at 6.6mT all protons of pyridineare characterized by a population inversion and are raser active provided the thresholdcondition given by equation (1) is fulfilled.For 15N labelled pyridine and acetonitrile the initial density matrices ρpy(0) and ρac(0)

at fields B0 below a few mT have a different structure. Especially two- and higher spin-order can be generated in the SABRE complex. Let us propose a mechanism whichcreates mainly hetero-nuclear I − S two-spin order at the target molecule. In a simplecase two molecules I − S and I∗ − S∗ are bound to the equatorial plane in the iridium-(Ir) complex, where I, I∗ represent two 1H-spins and S, S∗ represent the 15N spins onboth molecules. On the other side of the equatorial plane of the Ir-complex the twoprotons from para-hydrogen are bound to the Ir. These two protons with their corre-sponding denominators IP and IP∗ are initially in a singlet state

∣∣SP,P∗⟩. The topology

of this six spin system is IP, IP∗ , S, S∗, I, I∗. If both S (15N) spins are either up or down(mS = mS∗ = ±1/2) an efficient transfer of singlet spin-order from para-hydrogen tothe intermolecular pair of protons I-I∗of both molecules exists at a specified low mag-netic field (0.01mT < B0 < 4mT). This so called level anti-crossing point (LAC)35-37,whose effect can be expressed as

∣∣SP,P∗,TI,I∗

±⟩↔

∣∣TP,P∗

± , SI,I∗⟩ occurs at the crossingpoint ±(νP − νI) = JP,P∗ − JI,I∗ ± 0.5 (JP,S + JP,S∗ − JI,S − JI,S∗). The states

∣∣SI,I∗⟩,∣∣SP,P∗⟩are the singlet states and

∣∣TI,I∗±

⟩,∣∣TP,P∗

±⟩

denote the triplet states of the I − I∗

and IP − IP∗ pairs. The constants JP,P∗ , JI,I∗ , JP,S, JP,S∗ , JI,S and JI,S∗ represent theJ-coupling constants between pairs IP − IP∗ , I − I∗, IP − S, IP − S∗, I − S and I − S∗.After the LAC process an intermolecular singlet state

∣∣SI,I∗⟩ is formed between the twomolecules I− S and I∗ − S∗. In a further step the symmetry of

∣∣SI,I∗⟩ is broken by a CHactivation mechanism in the N-heterocyclic carbene Ir-complex, where the methyl groupof the Imes ligand coordinates to the Ir-para-hydrogen binding sites39,40. This breaks thechemical and magnetic equivalence between the S−I and S∗−I∗ target molecules, leadingto JI,S �= JI∗,S∗ , JI,S∗ �= JI∗,S and to a small chemical shift difference δν between spins Iand I∗. As a result the I − I∗ singlet-order is broken up into different forms of hetero-and homo-nuclear spin-order terms, which can be approximated by two complementary

10






three spin density matrices ρI,I∗,S and ρI∗,I,S∗ as shown in38:

ρI,I*,S = Iz. I∗z +

1

2

[1

1 + (y + x)2+

1

1 + (y − x)2

]ZQx

+1

2

[y + x

1 + (y + x)2+

y − x

1 + (y − x)2

](Iz − I∗z) . Sz

+

[1

1 + (y + x)2− 1

1 + (y − x)2

]ZQx. Sz

+1

4

[y + x

1 + (y + x)2− y − x

1 + (y − x)2

](Iz − I∗z) . (8)

The lower dot between two operators, for example Iz. Sz, denotes the direct operatorproduct between Iz and Sz. The first term Iz. I

∗z in equation (8) represents homo-nuclear

I−I∗ two-spin order. ZQx = Ix. I∗x+Iy. I

∗y denotes homo-nuclear zero-quantum spin-order.

The terms Iz. Sz and I∗z . Sz represent hetero-nuclear two-spin order between spins I − Sand I∗−S. The term ZQx. Sz corresponds to three-spin order and Iz−I∗z represents homo-nuclear anti-Zeeman order. The complementary density matrix ρI∗,I,S∗ is introduced heresince the Imes ligand on the Ir-complex is symmetric and its methyl group coordinatesto other possible para-hydrogen binding sides of Ir. Therefore ρI∗,I,S∗ is obtained fromequation (8) by replacing Iz by I∗z and Sz by S∗

z . Each spin-order has an amplitude whichis proportional to the coefficients represented by the square brackets in equation (8).The quantities y and x are defined as y =

(JI,S − JI∗,S∗ + JI∗,S − JI,S∗

)/(2 JI,I∗

)and

x = δν/JI,I∗ where JI,S, JI∗,S∗ , JI∗,S and JI,S∗ are the hetero-nuclear I−S, I∗ − S∗, I∗ − Sand I − S∗ J-coupling constants and JI,I∗ is the homo-nuclear coupling constant betweenI and I∗. At low field (< 1mT) δν is small (< 0.04Hz) compared to JI,I∗ ≈ 0.2Hz− 1Hzso x � y. Thus the last two terms in equation (8) (ZQx. Sz and Iz− I∗z) can be neglected.Similar arguments apply for ρI∗,I,S∗ , especially the operator part in the third term ofequation (8) becomes (I∗z − Iz) . S

∗z In summary in the bound complex only the first three

terms Iz. I∗z , ZQx and (Iz − I∗z) . Sz and their complementary terms in ρI∗,I,S∗ are different

from zero, but they are NMR and raser silent. After dissociation of the Ir-complex allthe correlations between spins I − I∗, S − S∗, I − S∗ and I∗ − S are lost, which meansthat only the two-spin order terms Iz. Sz and I∗z . S

∗z survive. At this point it is important

to realize that the reduction of the six spin system IP, IP∗ , S, S∗, I, I∗, which undergoesa symmetry break, into two three-spin systems ρI,I∗,S and ρI∗,I,S∗ (see equation (8)) isa simplification which qualitatively describes the dominant spin order terms which areproduced in the bound complex. Evaluation of the simpler four-spin system I,I∗,S,S∗ leadsto complex analytical expressions with less insight and with additional spin-order termsup to 4th order. Nevertheless during symmetry break in I,I∗,S,S∗, meaning JI,S �= JI∗,S∗

11






and JI,S∗ �= JI∗,S, the main effect on the singlet I−I∗ seen from the I-site is equivalent to asingle spin K forming a three spin system I, I∗,K where JI,K �= JI∗,K, JI,K = JI,S−JI,S∗ andJI∗,K = JI∗,S∗−JI∗,S holds. The same arguments apply to the I∗-site in the I∗, I,K∗ three-spin system. This leads to the simplified model represented by equation (8). Expandingthis model to the case of 15N-acetonitrile with three magnetically equivalent protonscoupled to 15N leads to the following ansatz for the density matrix ρac(0) after dissociation

ρac (0) = c2 Sz. (I2z + I3z + I4z) + e4 Sz. I2z. I3z. I4z . (9)

ρac is composed of a sum of three equivalent two-spin order terms Sz. Ijz, j = 2..4 and a fourspin order term Sz. I2z. I3z. I4z which might be generated in the full quantum mechanicaltreatment of the (I3S, I∗3S

∗)–system. The coefficients c2 and e4 describe the magnitudeof the corresponding spin order. For 15N-pyridine at B0 ≈ 1mT we have x � yo, ym, yp,where o, m and p stand for ortho, meta and para protons and yo, ym, yp, might havedifferent values (three different but overlapping level crossing terms). Therefore thedensity matrix can be expressed as

ρpy (0) = γ2,o Sz. (I2z + I6z) + γ2,m Sz. (I3z + I5z) + γ2,p Sz. I4z

+ higher spin order terms . (10)

The coefficients γ2,o, γ2,m and γ2,p describe the magnitude of the relevant 1H−15N two-spin order terms. Higher spin-order terms (four and six-spin order) are not given ex-plicitly, since they have not been observed experimentally. A simulation shows that onthe I-channel the four and six-spin order terms are not observable, but on the S-channelthey could be detected, as shown later for the case of 15N-acetonitrile (see Fig. S3).After 90◦y-pulse excitation on 1H (15N) all Ijz (Sz) operators in equations (9,10) becomeIjx (Sx). We denote the resulting density matrices after 90◦ pulse excitation as ρk(π/2)

where k = ac or py. Evolution of ρk(π/2) under the influence of the Hamiltonian givenby equations (6,7) and introducing a damping term ∼ exp(−t/T2) in order to obtainfinite line-widths leads to:

ρk (t) = exp (−iHkt) ρk (π/2) exp (iHkt) exp(−t/T2) . (11)

The observed 1H (15N) signal decay (FID) can be calculated according to⟨Idetk

⟩= Tr

{ρk (t) Idet

k}

, (12)

where Idetk is the 15N (1H) detection operator Sx (

∑Ijx). The theory represented by

equations (6–12) is the base for computer simulations where the evolution of the FID

12






is calculated successively by dividing the time t into small enough discrete steps. Thesimulated FID can be Fourier transformed to obtain the NMR spectrum.The model described above leads to the conclusion that for free 15N-acetonitrile and 15N-pyridine molecules the NMR spectrum is dominated by observable two-spin order termswhile for bound molecules the 1H and 15N dipolar signal contributions are practicallyzero. This is because of the terms (Iz − I∗z) . Sz and Iz. I

∗z in equation (8), which have zero

NMR signal contributions at low field. SABRE experiments with 15N labelled pyridine atearth’s field and subsequent observation in high-field confirm a 1H−15N two-spin order onthe 15N NMR signal of the free pyridine and no 15N-NMR signal for the bound pyridinespecies35,36.Finally at ultra-low fields (B0 < 1 µT) another efficient pathway for polarization transferexists from para-hydrogen to the 15N spins (SABRE-SHEATH41,42), which leads to single-spin order of 15N with a high degree of polarization P 15N ≈ 0.1. SABRE-SHEATH canbe explained as a LAC process between three spins IP, IP∗ and S, where the crossingcondition at ultra-low field is given by νI − νS = JP,P∗ .To prove whether the model described above, especially the two- and four-spin ordergiven by equations (9,10), is an adequate description for the spin-physics involved in lowfield, several SABRE and raser experiments with 15N labelled acetonitrile and pyridinehave been performed at 1mT field.Fig. S3 a shows the measured 1H FID from SABRE polarized 15N-acetonitrile after 90◦

pulse excitation measured at B0 = 0.977mT (νH = 41.6 kHz). Before the pulse theprotons of 15N-acetonitrile are in a state just slightly above the raser threshold, as canbe seen by the small oscillations in Fig. S3a and by the corresponding Fourier spectrumshowing a single line (Fig. S3b, top). The FID after the pulse starts at zero amplitudeand has a long lasting beat pattern, and the associated transverse decay time (≈ 5 s) islonger compared to the measured T2 relaxation time (≈ 0.7 s). This can be explainedby the anti-phase character of the two co-rotating magnetization components, whichreduce their dipolar interaction with respect to each other. The spectrum of this longliving FID (Fig. S3b, center) shows two lines with equal magnitudes but opposite sign,which are separated by J 15N−H = 1.7Hz. This measured antisymmetric 1H spectrumis in agreement with the simulation (Fig. S3b, bottom), in which pure two spin order≈ Sz. (I2z + I3z + I4z) with a corresponding antisymmetric spectrum is assumed. Wecould also measure the corresponding 15N FID of 15N-acetonitrile at B0 = 9.65mT(ν 15N = 41.6 kHz) after 90◦ pulse excitation in a single scan (Fig. S3c), indicating thatthe SABRE process at low field also polarizes the 15N nuclei significantly. The FID whichstarts at zero and the observed beat pattern cannot be explained by pure single- or two-

13






spin order. The corresponding Fourier transformed spectrum (Fig. S3 d, top) shows anantisymmetric quartet with line intensity ratio of 1 : 2 : (−2) : (−1). The correspondingsimulated 15N spectrum (Fig. S3 d, bottom) is in agreement with the measured spectrumif ρac(0) from equation (9) is assumed with c2 = 1 · 10−4 and e4 = −2 · 10−4. For15N-acetonitrile the 1H and the 15N-spectra at low field suggest that besides two-spinorder there is a significant amount of four-spin order (both result in an anti-symmetricspectrum) and other single- and three-spin order terms (corresponding to a symmetricspectrum) are completely absent.To compare the theory above with the measured 1H SABRE spectrum of 15N-pyridine andfor the purpose of spectral assignment of the observed NMR lines to possible raser activemodes with their associated spin states, different simulations were performed. Fig. S4a-jshows the simulated results for different initial conditions and spin configurations. Themeasured anti-symmetric 1H SABRE spectrum (grey) is plotted in each panel in orderto compare with the simulated amplitudes and line positions. In Fig. S4 a-e (blue) puretwo-spin order I2z. Sz between ortho 1H pyridine (I2-spin) and the 15N-spin (S-spin) isassumed. The measured I-spin (here I2) and some possible spin states of the other spins(S, I3 − I6) are indicated in the insets. The resulting line position for one specific spinstate is marked by an arrow. In Fig. S4a the initial state is I2z. Sz. After 90◦y

1H pulseexcitation the two-spin state is I2x. Sz and the corresponding simulated spectrum is anantisymmetric doublet separated by J12 = −9.8Hz. In Fig. S4b one additional meta-proton I3 is included, resulting in a strongly coupled three spin S − I2 − I3 system, andthe corresponding simulated I2-spectrum shows an additional splitting into two doublets(each splitted by J23 = 5.05Hz). Due to strong coupling effects four additional lines(inter-combination lines) are visible close to the center frequency (41666 Hz). The resultof the simulations for the strongly coupled four ( S−I2−I3−I4 ), five ( S−I2−I3−I4−I5 )and six-spin ( S−I2−I3−I4−I5−I6 ) system including the para (I4), meta (I5) and ortho(I6) spins are shown in Figs. S4c-e. One specific line position in Fig. S4 e, where spin I3= up, I4 = up, I5 = down and I6 = up (see inset), is exactly where the first raser line(41663.06 Hz) is observed (Fig. 3c, line 1). The third raser line (41663.32 Hz) results froma flip of spin I6 to the down position, the expected frequency shift of J26 = −0.26Hzbeing equal to the observed frequency difference between raser line 1 and 3. Fig. S4fshows the simulated spectrum (orange) for the case that both ortho-protons are in a twospin-state ρ(0) = (I2z + I6z) . Sz and spin I2, I6 are detected after a 90◦ pulse excitationresulting in ρ(π/2) = (I2x + I6x) . Sz. The simulated spectrum reflects basically all linepositions of the measured SABRE spectrum but not the line intensities.

14






ρ(0) = I2z⊗Sz

a

j

Ampl

itude

(a.u

.)

Frequency (kHz)

S I2

ρ(0) = -0.07 (I2z+I6z) -0.04 (I3z+I5z) -0.08 I4z

meas.

d

S I2

I3I4

I5

ρ(0) = I2z⊗Sz

meas.

2nd raser line

ρ(0) =(I2z+I6z)⊗Sz

f

S I2

I3I4

I5

I6meas. meas.

ρ(0) =(I3z+I5z)⊗Sz

g

S I2

I3I4

I5

I6

meas. meas.

ρ(0) = I4z⊗Szh

S I2

I3I4

I5

I6

meas.

i ρ(0) = 10-4(I2z+I6z)⊗Sz +5⋅10-5 (I3z+I5z)⊗Sz +5⋅10-5 I4z⊗Sz

S I2

I3I4

I5

I6 meas.

meas.meas.

meas.

meas.

all protons measured(I2-I6)

bρ(0) = I2z⊗Sz

S I2

I3meas.

c

S I2

I3I4

ρ(0) = I2z⊗Sz

meas.

1st raser line

e

S I2

I3I4

I5

I6

ρ(0) = I 2z⊗Sz

meas.

41.660 41.665 41.670 41.67541.65541.660 41.665 41.670 41.67541.655

( )

(3rd raser line)

4th raser line

Fig. S4: Comparison between measured and simulated 15N-pyridine 1H spectra.1H (I-) spectrum measured at 41.6 kHz from 15N-pyridine (grey) and simulated 1H spec-tra (colour) based on a linear combinations of pure 1H−15N two-spin order. Insetsshow initial density matrix ρ(0) (upper left) and spin states including measured Ij-spins (j = 2..6, see the chemical structure insets). Simulation parameters: (J∗

12, J∗13, J

∗14)

= (−9.8,−1.56, 0.18) Hz, (J∗23, J

∗34, J

∗24, J

∗25, J

∗35, J

∗26) = (5.05, 7.83, 1.78, 0.95, 1.38,−0.26)

Hz, center frequency νI = 41 666Hz, T2 = 1.5 s. Arrows in (a–d) indicate the frequencyfor a specific spin configuration shown in the inset. Arrows in (e–g) correspond to thefrequencies for the four raser transitions observed.

15






The largest deviations in line intensities appear close to the center frequency at 41 666Hzand at the wings (41 657Hz and 41 675Hz). This indicates that the total initial state isnot purely ρ(0) = (I2z + I6z) . Sz but other two- or higher-spin order contribution from themeta- and para- hydrogens have to be taken into account. Another specific line positionassociated with the spin state I3 = up, I4 = down, I5 = up, as indicated in Fig. S4 f,corresponds exactly to the second raser active line (line 2 in Fig. 3c) at 41 662.32Hz. Thesimulated spectrum in Fig. S4g results from both meta-protons being in a two spin-stateρ(0) = (I3z + I5z) . Sz and spin I3 and I5 are detected (green). The fourth raser activeline (line 4 in Fig. 3c) at 41 663.91Hz with large negative amplitude and with S = up,I2 = I6 = down and I4 = up can be identified. Fig. S4h is the result with the para-protonbeing in a two spin-state ρ(0) = I4z. Sz while spin I4 is detected (pink).In summary the analysis of the position, amplitudes and sign of the lines at the region ofraser activity (≈ 41 662Hz−41 664Hz) in Figs. S4f,g,h lead to the conclusion that in theseexperiments only ortho- and meta-protons are raser active in 15N-pyridine. In additionall four raser active lines in Fig. 3c are identified with respect to their frequency positionand their corresponding spin state. Figure S4 i shows the complete simulated spectrum(red) assuming a superposition of ortho-, meta- and para-protons in a two spin state(see equation (10)). The state ρ(0) = γ2,o Sz. (I2z + I6z) + γ2,m Sz. (I3z + I5z) + γ2,p Sz. I4z

with (γ2,o, γ2,m, γ2,p) = 10−4 · (1, 0.5, 0.5) fits best to the measured SABRE spectrum.The small deviations can be explained by the assumption that in the simulations alllinewidths are assumed to be equal (T2 = 1.5 s). This is not the case in the measuredSABRE spectrum, where different lines have different widths. Especially two specificraser lines (at 41 663Hz and 41 663.26Hz) show large narrow peaks, while the mirroredlines on the other side of the spectrum at 41 668.74Hz and 41 669Hz are broader.Finally Fig. S4j shows the simulated spectrum (brown) if all protons of the 15N-pyridineare prepared in a single-spin order ρ(0) = o (I2z + I6z) + m (I3z + I5z) + pI4z with o =

−0.07, m = −0.04 and p = −0.08. The spectrum is symmetric and not compatible withany observed SABRE spectrum.

VI Multi-mode EHQE based raser

While a simulation of the amplitudes of the SABRE NMR lines is possible, a model todescribe the four observed raser amplitudes at different flow rates is not possible yet. Thisis due to the complexity of the multi-mode raser operation which in our case involvesup to four non-linear coupled differential equations. A qualitative explanation for theincreasing number of raser modes with increasing p-H2 flow rate is possible (Fig. 3b,c).Each mode has its own raser threshold that is crossed in sequence as their polarizationincreases with rising flow rate. At higher flow rate there is a better mixing of p-H2 gas

16






bubbles with the methanol solution, resulting in a higher p-H2 concentration at the siteof the dissolved catalyst. This increases the degree of polarization on the target molecule.It is expected that at very high degrees of polarizations using an improved exper-imental setup all raser modes can be observed. This means that the whole NMRSABRE spectrum can be recovered via its antisymmetric properties with a vastly im-proved frequency resolution from the raser modes. From the difference in frequencybetween the four lines ∆νi,j with i,j = 1..4 1H homo-nuclear J-coupling values as wellas linear combinations of J-coupling constants can be deduced. The most narrow lineshown in Fig. 4b, corresponds to the two mode raser and the frequency difference is∆ν1,2 = J∗

24 − J∗25 + J∗

26/2 = 0.696 80Hz measured over 1000 s. In Fig. 4c, the frequencydifferences of the four mode raser are shown. Especially ∆ν1,3 = J∗

26 = −0.2561Hz cor-responds to a spin flip of I6 while I2 is measured (see spin states 1 and 3 in Fig. 3c). Notethe frequency difference ∆ν1,2 = J∗

24−J∗25+J∗

26/2 = 0.7357Hz from Fig. 4c differs slightlyfrom Fig. 4b because of slightly different experimental conditions. A spin flip of I6 leadsto ∆ν2,3 = J∗

24−J∗25−J∗

26/2 = 0.9918Hz as shown in Fig. 3c and Fig. S4 e. The frequencydiffernence ∆ν2,4 in Fig. 4c cannot be interpreted as combination of J-couplings since itarises from the difference between the ortho- and meta-raser modes we have identified inFig. S4 f,g.We remark that the values of J-coupling constants J∗

ik for the simulations are very close(up to 0.2Hz) to the J-coupling constants of 15N-pyridine measured in high field20. Thevariation can be explained by different experimental conditions, such as the tempera-ture, pyridine and catalyst concentration. The experimenters are aware of the effect of‘frequency pulling’ in masers and seek to quantify it in future raser experiments.Another interesting property of the multi-mode raser is the near-chaotic appearance ofits waveform recorded over time. Actually, the signal shown in Fig. 3b bottom (orange)line looks quite chaotic but can be synthesized by a superposition of four sinusoids,each with their own frequency and amplitude. We used Python and MatPlotLib44,45

in the interactive program SimNatPhysMain3.py which is available in another SI file toproduce the data given in Fig. S5. For optimum comparability with the experiment, thefour frequencies and amplitudes can be extracted from the spectrum shown in Fig. 3c.We took advantage of the precisely tabulated values coming straight from the Fouriertransform to determine the eight parameters with exceedingly high precision. Fig. S5ashows a plot of 100 seconds of the sum of the four sinusoids. Fig. S5b shows a timeslice from 15 to 40 seconds of the simulated data, the peak patterns look distinctivelyrepeated yet are each unlike each other. On the scale 18.74 s–28.74 s (Fig. S5c) the signallooks very much like the experimental recording (Fig. S5d). In summary, the superposedsinusoids completely describe the time evolution of the raser signal. Non-linearities30

are not involved in explaining the waveform observed. At very high polarizations andQ-factors, real chaotic behaviour may emerge.

17






Fig. S5: Simulated and measured 15N-pyridine 1H four mode raser signal.a, Raser signal simulated as a linear superposition of sinusoids plotted over 100 s. S(t) =A1 ·sin(2π·t·31.088 64Hz)+A2 ·sin(2π·t·31.821 968Hz)+A3 ·sin(2π·t·32.077 521Hz)+A4 ·sin(2π · t · 32.599 739Hz) with A1 = 0.381134327, A2 = 0.306345952, A3 = 0.180330163,A4 = 0.132189557. b, Enlarged section from a (15 s–40 s) showing repeating patternsthat are modulated by very low frequencies. c, Section from b (18.74 s–28.74 s) showinga signal comparable to the experimental result in d which is the same as main Fig. 3b,bottom line (orange).

18






VII Limits of precision for J-coupling measured with the two-spin ordered raser

The fundamental limit of precision for frequency measurements in a raser follows fromthe Cramér-Rao lower bound (CRLB)43 for the standard deviation of frequency σf :

σf ≥√3

π SNR√

fBW T3/2m

, (13)

where SNR denotes the signal to noise ratio, fBW is the detection bandwidth and Tm is themeasurement time. Equation (13) can be used as an estimate for the lower bound σraser

f

of the 15N-pyridine raser at 1mT field (Fig. S5 d, Fig. S6 a). Given a bandwidth fBW =

100Hz and Tm = 1000 s, SNR = 50 (Fig. S6a) or Tm = 100 s, SNR = 25 (Fig. S5d), weobtain for the lower bound σraser

f = 3.6 · 10−8 Hz (Fig. S6a) and σraserf = 2.2 · 10−6 Hz

(Fig. S5d). Following equation (13) these two limits can theoretically be further loweredby either increasing the measurement time Tm or the SNR. Let us compare two lowerbounds σraser

f for the raser with σNMRf for a corresponding NMR experiment with N =

Tm/∆tm single scans, where each scan has a duration ∆tm. According to [25] σNMRf

is given by replacing Tm by the transverse relaxation time T2 in equation (13). Theimprovement of the SNR for N scans is SNR ·

√N = SNR ·

√Tm/∆tm. If we assume

equal SNR and bandwidth fBW for the NMR signal per scan and the raser signal, theratio is given by:

σraserf

σNMRf

=T3/22

Tm√∆tm

. (14)

For the case of the 15N-pyridine raser T2 = 1 s, ∆tm = 10 s (inverse of SABRE pumpingrate 1/τp) and Tm = 1000 s the ratio becomes σraser

f /σNMRf ≈ 1/(3.3Tm) ≈ 3.3 · 10−4.

Therefore for large enough Tm raser-operation has a frequency precision which is ordersof magnitude higher compared to the NMR experiment with N = Tm/∆tm scans.The precision of our measurements is far away from the bounds given by equation (13).In order to get an idea about the statistical fluctuations and the long term behaviourof a specific J-coupling constant of 15N-pyridine (two mode raser), a 1H raser signal ismeasured over a time of 1200 s. The data measured at a constant p-H2 flow rate of100 cm3 min−1 is shown in Fig. S6 a. A beat pattern consisting of two lines with similaramplitude is observed over the whole measurement time (Fig S6a, inset). The Fouriertransformed spectrum of the entire raser signal shows two groups of broadened lineswith nearly identical shape and amplitude (Fig. S6b). The broadening is caused by themagnetic field fluctuations over a time scale of 1200 s. The observed splitting of ≈ 0.7Hzis caused by simultaneous oscillation of two raser modes with different frequencies.

19






a

0 200 400 600 800 1000 1200

0.685

0.690

0.695

0.700

Time (s)

b

c

41.665 41.666 41.667

Ampl

itude

(a.u

.)

Frequency (kHz)

0 200 400 600 800 1000 1200

-1

0

1

2

3

Ampl

itude

(V)

Time (s)

190 200 210 220-2

0

2

(Hz

) 1

,2

Fig. S6: 1H raser signal of 15N-pyridine 1H measured at 41.6kHz and for 1200 s.a, The signal amplitude slowly decreases due to evaporation of liquid methanol. Thep-H2 flow rate is 100 cm3 min−1. Inset is a section of 30 s duration showing two modesoscillating simultaneously. b, Fourier transformed spectrum of entire raser signal in a,resulting in two broad lines caused by magnetic field fluctuations. c, Evolution of linesplitting ∆ν1,2 from 0.7004Hz to 0.6858Hz free of magnetic field noise versus time. Thedata was obtained by dividing the 1H raser signal in a into time blocks of 30 s and byapplying a Fourier transformation and a correction of magnetic field fluctuations. Theerror bar of each data point is ≈ 2 · 10−4 Hz. The red line is for guiding the eyes.

20






Based on the observed frequencies and the experimental conditions we conclude thatthese two modes correspond to the lines 1 and 2 in Fig. 3c. From the associated spin-states of 15N-pyridine (Fig. 3c, insets 1,2) the frequency difference of modes 1 and 2 isgiven by ∆ν1,2 = J∗

24 − J∗25 + J∗

26/2.Fig. S6 c shows the evolution of ∆ν1,2 versus time. The data was obtained by dividing the1H raser signal in Fig. S6a into time blocks of 30 s, applying the correction of magneticfield fluctuations and measuring the frequency in the Fourier transformed spectrum. Asthe main trend ∆ν1,2 decreases from 0.7004Hz at t = 0 to 0.6858Hz at t = 1200 s.Sudden fluctuations of ±2mHz on the 100 s time scale are superimposed on the longterm trend. We believe that the slow decrease is due to constant evaporation of thesolvent (d4-methanol) which leads to a continuous increase of the 15N-pyridine and Ircatalyst concentration. The dependence of J-coupling constants on the 15N-pyridineconcentration is known from literature20, but has never been reported for highly dilutedsolutions in the milli-Hz regime. The data in Fig. S6 c implies that the linewidths ∆ν ofthe measured raser signals from Fig. 4b,c (2.6mHz and 13mHz, respectively) for a giventime interval ∆t (800 s and 130 s, respectively) is determined by systematic fluctuationsin concentration of the raser active molecules in solution and not by the inverse of themeasuring time ∆ν = 1/(π∆t). In future a significant improvement in precision whichcomes closer to the bound given by equation (13) is expected by using a better continuousflow apparatus eliminating the concentration dependence effect.

VIII Temperature dependence of raser and SABRE signals

A major obstacle for all technology involving para-hydrogen is the need for cryogenictemperatures (T < 123K) to produce enriched para-hydrogen from the 3:1 ortho-/para-hydrogen mixture at room temperature15. The cryo-coolers (e.g. Gifford-McMahon cyclecooler or pulse tube) working at T ≈ 30K require a lot of input power and thick insula-tion, resulting in noisy and bulky equipment. This is a serious disadvantage for realizing aminiaturized SABRE based raser, even though the sample is kept at room temperature.Therefore starting from 60K we tested the limit up to which conversion temperature(diminishing p-H2 concentration) SABRE signals and associated raser operation are ob-servable in a continuous mode (Fig. S7). We found raser activity for pyridine up to 155K.The SNR ≈ 8 000 for the 1H SABRE signal at 60K corresponds to a proton polarizationPH ≈ 1 · 10−2 and 1H signal is observed up to 270K. Considering the linewidth ∆ ofthe SABRE spectra versus the p-H2 conversion temperature (blue triangles in Fig. S7),we observe a knee at T ≈ 155K. Above this temperature, the linewidth ∆ ≈ 0.4Hz isconstant. Below T = 155K, ∆ rises exponentially with falling conversion temperature.The observed linewidth is given by ∆ ≈ 1

π

(1T2

+ 1τrd

)where 1

τrd= µ0 �

4 η Qγ2H ns |PH| is

21






proportional to the proton polarization PH which is proportional to the molar fractionof xp of p-H2 in the feed gas flow. The molar fraction xp is well approximated by anexponential function in the temperature range from 50K–150K, e.g. xp ∝ exp(−Ea/kT )where Ea is the activation energy for conversion from ortho- to para-H2. Therefore, therising part of ∆ below 155K reflects the ortho-para activation energy of hydrogen. Theposition of the knee (our limit for raser activity) at T = 155K could be raised usingbetter EHQE circuits and para hydrogen delivery to a temperature of T > 200K, wheresmall Peltier coolers can be applied.

50 100 150 200 250100

101

102

103

104

p-H2 conversion temperature (K)

Sign

al-to

-noi

se ra

tio (S

NR

)

SABREraser

0.1

1

10

SABRE linew

idth ∆ (Hz)

Fig. S7: 1H SNR from 1µL pyridine measured at 166kHz versus conversiontemperature inside p-H2 generator for SABRE and raser signals.

SNR of steady state 1H raser signal (circles) and of corresponding SABRE FID after30 s crusher pulse and subsequent 90◦x pulse excitation (squares). The p-H2 flow is35 cm3 min−1 at 4 bar. The linewidth ∆ is determined from the Fourier-transformedspectrum of the SABRE FID (triangles). Solid lines are for guiding the eyes. Withdecreasing temperature raser activity starts at 155K. At this point the radiation damp-ing rate is equal to the 1H linewidth of the SABRE FID (equation (1)). At even lowertemperatures the 1H linewidth increases exponentially, reflecting the energy of activationbetween ortho- and para-hydrogen states.

22






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Documents

Para-Hydrogen Raser Delivers Para-hydrogen raser …...V Simulation of SABRE NMR spectra 9 VI Multi-mode EHQE based raser 16 VII Limits of precision for J-coupling measured with the