Algorithmic Cooling of Nuclear Spins Using Long-Lived Singlet Order Bogdan A

Algorithmic Cooling of Nuclear Spins using Long-Lived Singlet Order Bogdan A. Rodin,1, 2 Christian Bengs,3 Alexey S. Kiryutin,1, 2 Kirill F. Sheberstov,4 Lynda J. Brown,3 Richard C. D. Brown,3 Alexandra V. Yurkovskaya,1, 2 Konstantin L. Ivanov,1, 2 and Malcolm H. Levitt3 1)International Tomography Center SB RAS, Novosibirsk, Russia 2)Novosibirsk State University, Novosibirsk, Russia 3)Department of Chemistry, Southampton University, Southampton SO17 1BJ, UK 4)Johannes Gutenberg-Universität,Helmholtz Institute Mainz, Mainz 55099, Germany (Dated: 1 April 2020) Algorithmic cooling methods manipulate an open quantum system in order to lower its temperature below that of the environment. We achieve significant cooling of an ensemble of nuclear spin-pair systems by exploiting the long-lived nuclear singlet state, which is an antisymmetric quantum superposition of the ”up” and ”down” Zeeman states. The effect is demonstrated by nuclear magnetic resonance (NMR) experiments on a molecular system containing a coupled pair of near-equivalent 13C nuclei. The populations of the system are subjected to a repeating sequence of cyclic permutations separated by relaxation intervals. The long-lived nuclear singlet order is pumped well beyond the unitary limit. The pumped singlet order is converted into nuclear magnetization which is enhanced by 21% relative to its thermal equilibrium value.

I. INTRODUCTION is the steady-state Overhauser effect, in which radiofrequency irradiation which is resonant with one spin species A quantum system which is in contact with a thermal may lead to an increase in the magnetization of a second 4,5 reservoir eventually comes into equilibrium, so that the spin species . However, the Overhauser effect does not populations of the quantum states are described by the lead to decrease in entropy for the spin system as a whole, Boltzmann distribution at the reservoir temperature. In and also relies on the dominance of the dipole-dipole re- the case of homonuclear spin systems at ordinary tem- laxation mechanism. More recently, a more general set of peratures in a high magnetic field B0 along the z-axis techniques called algorithmic cooling have been demon- of the laboratory axis system, the high-temperature (or strated. These methods systematically build up the order weak order) approximation applies1–3, so that the ther- of an open quantum system through a deterministic se- 6–22 mal equilibrium state of the spin ensemble may be de- quence of events . Algorithmic cooling methods have 15–22 scribed by a density operator of the form been applied with success to nuclear spin systems . These methods utilise the ongoing contact of the nuclear −1 1 0 spin system with the thermal environment during the ρeq ' N { − ~ω Iz/kBT } (1) H experimental procedure. The nuclear spin system is an where NH is the dimension of Hilbert space (given by open quantum system which constantly exchanges energy the number of independent quantum states of each spin and entropy with the thermal environment, also during a system), ω0 = −γB0 is the nuclear Larmor frequency, radiofrequency pulse sequence. The algorithmic cooling γ is the magnetogyric ratio, and Iz is the spin angular of nuclear spins emerged from the field of NMR quan- 20,23,24 momentum operator along the field. For an ensemble tum computation , and treats the individual nu- of N-spin-1/2 systems, the dimension of Hilbert space is clear spins as independent qubits which may be addressed N NH = 2 . individually by frequency-selective radiofrequency fields. The second term in equation 1 represents the nuclear One example of algorithmic cooling has been demon- spin order generated by thermal equilibration in a strong strated on a spin system containing two different nuclear magnetic field. In the vast majority of nuclear magnetic spin species, one of which relaxes rapidly (and is there- resonance (NMR) experiments, this thermal spin order fore strongly coupled to the thermal environment) while is manipulated by resonant radiofrequency pulses, and the second species relaxes slowly (and therefore has a is eventually converted into transverse magnetization, much weaker environmental coupling)15–19,21. These spin which precesses in the magnetic field, and induces an species have been called “refrigerant” and “computation” electrical signal by Faraday induction. It is a prominent qubits9. The magnetization of the slowly relaxing species paradigm in NMR spectroscopy that the initial spin order is used to temporarily store the initial spin order, while established by thermal equilibration cannot be increased the rapidly relaxing “refrigerant” species acquires a new by the application of resonant radiofrequency pulses, but batch of spin order through contact with the thermal en- only transformed into different spin order modes, with vironment. The total entropy of the nuclear spin system the total quantity of spin order being either conserved, is thereby decreased below its initial value, leading to an or decreased by irreversible decoherence effects. increase in the total nuclear spin magnetization. How- There are, however, some situations in which resonant ever, this particular scheme is restricted to spin systems radiofrequency irradiation does increase the amount of with two species having very different relaxation proper- nuclear spin order, at least locally. A seminal example ties, and with sufficiently well-resolved spectra that each 2 species may be addressed individually by controlling the Nevertheless, we demonstrate in this article an algo- frequencies of the radiofrequency pulses. These are very rithmic cooling procedure which does work for strongly restrictive conditions. coupled homonuclear spin-1/2 pairs. The algorithmic In this paper we demonstrate the algorithmic cooling of cooling procedure exploits the singlet state |1i, which an ensemble of spin-1/2 pair systems which exhibit near- is an antisymmetric entangled state of the two spins-1/2. magnetic-equivalence25–28, meaning that a small differ- The singlet order of the system, defined as the mean pop- ence in chemical environments induces a difference in ulation difference between the singlet state |1i and triplet Larmor frequencies which is much smaller than the scalar states {|2i , |3i , |4i}, is protected against DD relaxation spin-spin coupling. This appears to be an unpromising and may have a much longer lifetime than population dif- substrate for algorithmic cooling for the following rea- ferences within the triplet manifold25,29–48. Techniques sons: (i) the individual spins-1/2 (qubits) are tightly exist for coherent interconversion of singlet and triplet coupled, do not display resolved spectral resonances, and spin order, even in the near-equivalence limit25,35. Spin cannot be addressed selectively using resonant irradia- order established by the initial contact with the ther- tion; (ii) two of the four energy eigenstates are entan- mal molecular environment is stored temporarily as sin- gled superpositions of the ”up-down” and ”down-up” glet order. Thermal contact of the spin system with the qubit states; (iii) the dominant relaxation mechanisms, environment establishes a new batch of nuclear spin or- which couple the quantum system to the environment, der during a subsequent interval, so that the total en- are the dipole-dipole (DD) coupling and chemical shift tropy of the system is lower than after the initial ther- anisotropy (CSA) mechanisms, which affect both spins mal equilibration step. Careful manipulations allow this equally. Unlike previous algorithmic cooling demonstra- enhanced spin order to be converted into enhanced nu- tions, no distinction may be made between fast-relaxing clear magnetization. The result is that by the end of the ”refrigerant” and slowly relaxing ”computation” qubits9. pulse sequence, the nuclear magnetization exceeds that Despite these obstacles, we succeeded in a demonstration established initially by thermal equilibration, breaking of algorithmic cooling on a near-equivalent spin-pair sys- the standard paradigm of most NMR experiments. This tem, by exploiting the extended lifetime of the singlet method also breaks the paradigm of most algorithmic spin order. cooling methods, which rely on the existence of individually accessible qubits.

II. THEORY A. A thought experiment Consider an ensemble of spin systems, each consisting of two coupled spins-1/2 in a molecular environment such The basic principle is illustrated by the “thought ex- that they exhibit near magnetic equivalence25–28, mean- periment” in figure 1. Assume that the spin-lattice relaxing that a small difference in chemical environments in- ation time constant T1 is much shorter than the singlet- duces a difference in Larmor frequencies which is smaller triplet relaxation time constant TS, so that the popula- than the scalar spin-spin coupling. Under these con- tions of the triplet manifold {|2i , |3i , |4i} come rapidly ditions, the four Hamiltonian eigenstates are approxi- into thermal equilibrium with the molecular environ- mately equal to the singlet and triplet states of the spin- ment, while the population of the singlet state |1i remains 1/2 pair, numbered as follows: isolated. Figure 1(a) depicts the thermal equilibrium position of the spin ensemble, adopted when the spin sys- −1/2 |1i = |S0i = 2 (|αβi − |βαi) (2) tem is left in contact with the lattice for a sufficiently

|2i = |T+1i = |ααi long time. The filled balls in figure 1(a) represent positive deviations from a uniform population distribution, |3i = |T i = 2−1/2(|αβi + |βαi) 0 while the empty circles indicate negative population de- |4i = |T−1i = |ββi viations. The excess population in the lower triplet state |2i, and depleted population in the upper triplet state |4i, In many cases, spin-1/2 pairs of this type relax predom- indicate a finite magnetization along the field direction. inantly through the motional modulation of the dipole- An arbitrary population difference of +6 units is shown dipole coupling of the two spins (the DD relaxation mech- between the two extreme triplet states. This number is anism). Since the DD coupling is mutual, the DD re- chosen for arithmetic convenience (see below). laxation mechanism affects both spins equally, investing The algorithmic cooling procedure starts by an ex- both spins with an identical relaxation time. Hence there change of populations between states |1i and |2i, cor- seems to be no possibility of achieving algorithmic cool- responding to the operations: ing. Furthermore, in the case of near magnetic equiv- iφ alence, the individual spins do not have resolved spec- π12 |1i = |2i e tra and cannot be regarded as independent qubits which 0 (3) π |2i = |1i eiφ may be manipulated selectively by radiofrequency fields. 12 Algorithmic cooling therefore appears to be doubly im- where φ and φ0 are arbitrary phase factors. In practice possible for this system. the population swap operation π12 may be implemented 3

π12 τ π12

a b c d 4

1 3

FIG. 1. Cartoon showing how the magnetization of a near-equivalent spin-1/2 pair ensemble may be enhanced beyond the thermal equilibrium limit by exploiting the long-lived singlet state. The energy levels (horizontal lines) are labelled as in equation 2. Filled balls represent positive deviations in population from the state of complete disorder; Empty circles represent negative deviations in population from a state of complete disorder. The operator π12 exchanges populations between states |1i and |2i. The waiting interval τ, where T1 τ TS , allows thermal equilibrium to be established in the triplet manifold through spin-lattice relaxation, leaving the singlet population almost undisturbed. (a) The initial state of thermal equilibrium, with a population difference of ∆ = 6 units between the lower and upper triplet states; (b) The populations of states |1i and |2i are swapped. This leads a mean population deviation of −3/3 = −1 for the three triplet states; (c) A waiting interval τ allows thermal equilibration of the triplet manifold, re-establishing the population difference of ∆ = +6 between the lower and upper triplet states, superposed on the mean triplet population deviation of −1 established in (b). (a) Swapping the populations of |1i and |2i leads to an enhanced population difference of ∆ = +7 between the lower and upper triplet states. This corresponds to a magnetization enhanced by a factor 7/6 relative to thermal equilibrium. with high fidelity by a sequence of applied radiofrequency A more formal analysis of the thought experiment in fields, as described later. The consequence of swapping figure 1 is provided by considering the transformations of the populations of |1i and |2i is shown in figure 1(b). a vector of state populations, defined as follows: Note that the effect of the population swap is to de-   plete slightly the overall population of the triplet man- p1 p ifold {|2i , |3i , |4i}. The mean population of the triplet p =  2 (4) manifold in figure 1(b) is −1 unit, (relative to a uni- p3 form population distribution), as opposed to figure 1(a), p4 where the mean deviation in triplet populations is zero. where P p = 1. The thermal equilibrium state at the At the same, the relatively large population deviation of r r beginning of the sequence may be written, in the high- +3 units is “stored” in the long-lived singlet state |1i. temperature limit, as The next step is to leave the spin system undisturbed in the magnetic field for a time long compared to T1, but  1  short compared to T . This allows the triplet manifold 1 1 + S p =   (5) to come back into thermal equilibrium with the environ- eq 4  1  ment, while the singlet population is left approximately 1 − undisturbed. The result is the population distribution shown in figure 1(c). This is the same as the initial pop- where ulation distribution shown in figure 1(a), but superposed 0 on the mean population depletion of −1 for the triplet = ~γB /kBT (6) manifold. The population difference between the lower showing that the lower triplet state is slightly enhanced and upper triplet states is restored to +6 units. in population relative to the upper triplet state. If the population swap π12 is again executed, the re- The Zeeman and singlet orders of the spin ensemble sulting population distribution is as shown in figure 1(d). are given in terms of the populations as follows: Notably, the population difference between the lower and upper triplet states is now +7 units, i.e. a factor of 7/6 hZi = p2 − p4 (7) larger than its thermal equilibrium value. This is the same Zeeman polarization as would be exhibited by a 1 system with a spin temperature equal to 6/7 of the envi- hSi = p − (p + p + p ) (8) 1 3 2 3 4 ronmental temperature. Although the effect is not large, this simple thought experiment establishes the principle These quantities may be derived from the population vec- of algorithmic cooling using long-lived spin states. tor p at any time through hZi = Z · p, and hSi = S · p, 4 where to first order in . This corresponds to the final Zeeman  0  order: 1 7 7 Z =   (9) hZi = Z · p = = hZi (16)  0  12 6 eq −1 Hence, the protocol in figure 1 enhances the Zeeman and order by a factor of 7/6, relative to thermal equilibrium.  1  − 1  S =  3  (10) B. Cooling cycles  1  − 3  1 − 3 1. Pumping of singlet order The Zeeman order hZi is proportional to the population difference between the lower and upper triplet states, A greater cooling effect is established by applying a while the singlet order hSi is given by the difference be- repeated sequence of cooling cycles to the spin system. tween the singlet population and the mean triplet popu- Several possible cooling cycles exist: the most effective lation. one we have found may be written as follows (in left-to- In thermal equilibrium, the Zeeman order is given by right chronological order): 1 C = (τ − π − τ − π ) (17) hZi = Z · p = /2 = γB0/k T, (11) 124 142 eq eq 2~ B The symbol τ denotes a relaxation delay, while π de- using the thermal equilibrium populations in equation 5. ijk notes a unitary transformation with the following prop- Both sets of spins have the same spin temperature. There erties: is no nuclear singlet order in thermal equilibrium: iϕ πijk |ii = |ji e hSieq = S · peq = 0. (12) iϕ0 πijk |ji = |ki e (18) iϕ00 The first step in figure 1 may be represented by an πijk |ki = |ii e application of the following permutation matrix to the population vector: where {ϕ, ϕ0, ϕ00} are arbitrary phase factors. The cyclic permutations of three spin state populations may be rep- 0 1 0 0 resented by the following matrices: 1 0 0 0 π =   (13) 12 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 π =   124 0 0 1 0 The second step in figure 1 represents an idealised re- 0 1 0 0 laxation process, which brings the three triplet popu- (19) lations back into thermal equilibrium with the environ- 0 1 0 0 0 0 0 1 ment, while leaving the singlet population unchanged. π =   We call this manipulation a triplet thermal reset, and 142 0 0 1 0 represent it by the following matrix: 1 0 0 0   1 0 0 0 The cooling cycle in equation 17 involves alternating 0 1 (1 + ) 1 (1 + ) 1 (1 + ) cyclic permutation operations on the states {|1i, |2i, |4i} reset  3 3 3  Θ =   (14) in the opposite sense, separated by relaxation delays τ, T 0 1 1 1   3 3 3  which in the idealized case implement triplet thermal re- 1 1 1 0 3 (1 − ) 3 (1 − ) 3 (1 − ) set operations, as described by the matrix in equation 14. The effect of the cycle C on the spin-state populations, In practice, the triplet thermal reset may be achieved in the case of idealized triplet thermal reset operations, approximately by leaving an appropriate relaxation in- may be expressed by the matrix terval, in the case that TS T1. Matrix-vector multiplications lead to the end result of C = π · Θreset · π · Θreset the operations in figure 1: 142 T 124 T   3 + 2 3 + 3 2 + 3 2 + 3 2 + 3 1 3 − 3 2 − 2 − 2 − reset 1 3 + 3 =   (20) p = π12 · Θ · π12 · peq '   (15)  3 2 + 2 + 2 +  T 12  3 −  9 3 − 4 0 3 − 3 3 − 3 3 − 3 5 to first order in . 3. Zeeman order enhancement If the cycle C is applied repeatedly to the system, a steady-state population vector is eventually established. The entropy reduction of the spin system may be This steady-state population vector pss is defined by the cashed in as an enhancement of the Zeeman magneti- condition zation by applying an additional π12 permutation operation. This leads to the following population vector: C · pss = pss (21)  2 −  1 1 2 + 3 This implies that pss lies in the nullspace of C− , where p = π · p '   (28) 1 is the unit matrix. Linear algebra analysis leads to the fin 12 ss 8  2 +  following vector of steady-state populations: 2 − 3 which exhibits the following Zeeman order: 2 + 3 1 2 − 3 3 pss '   (22) hZi = Z · p ' = hZi (29) 8  2 +  fin fin 4 2 eq 2 − 3 Hence, repeated cooling cycles C followed by a π12 permutation operation lead to an enhancement of the Zee- to first order in . man order with respect to thermal equilibrium. A com- The singlet order in the steady state is given by parison with equation 11 indicates that the enhancement 1 factor is 3/2 in the ideal case. In the language of spin 50 hSiss = S · pss ' (23) thermodynamics an equivalent amount of Zeeman or- 2 der would be generated if the spin temperature were re- while the Zeeman order in the steady state is given by duced by a factor of 2/3. However, it should be noted that the population distribution in equation 28 does not 1 1 conform to a Boltzmann distribution under a Zeeman hZi = Z · p ' = hZi (24) ss ss 4 2 eq Hamiltonian and hence cannot strictly be described by a spin temperature. Hence the cooling cycles do not immediately enhance the Note that the enhancement in Zeeman order applies to Zeeman order. both sets of spins at the same time, unlike the nuclear 2. Entropy Overhauser effect, which enhances the Zeeman order of one set of spins while destroying the order of a different The cooling cycles reduce the overall entropy of the set. spin system. A useful quantity is the von Neumann en- It is shown in appendix D that the cycle C implements tropy49, defined as follows: a “greedy” algorithmic cooling protocol, as defined in reference8. S = −Tr(ρ ln ρ) (25)

For small values of spin polarization, the population vec- C. Liouville space analysis tor in equation 4 corresponds to the following von Neu- mann entropy: An alternative insight into the algorithmic cooling protocol is achieved by using a geometric interpretation in 1 2 51,52 Seq ' ln 4 − (26) the Liouville space of orthogonal spin operators . In 4 the following discussion, superoperators are denoted by hats (ˆ), while operators go bare-headed and are recog- The first term is the entropy of a completely disordered nized by context. four-level system. The last term represents the reduction Express the density operator as a sum of orthogonal in entropy achieved by thermal contact with the finite- operator terms: temperature environment. In the steady state established by repeated pumping NL X cycles, the von Neumann entropy is given by applying ρ = ρqQq (30) equation 25 to the population vector in equation 22. This q=1 leads to the following expression: 2 where the dimension of Liouville space is NL = NH , ρq are coefficients and the operators {Q } are orthonormal: 5 2 q Sss ' ln 4 − (27) 8 0 Qq Qq = δqq0 (31) In ideal circumstances, the reduction in entropy achieved The Liouville bracket is defined as follows52 by the cooling cycles is therefore a factor of 5/2 more than 0 † 0 that achieved by thermal equilibration. Qq Qq = Tr QqQq (32) 6

ρ The thermal equilibrium state is indicated by the blue Z thermal dot in ﬁgure 2. equilibrium τ-(��)-τ-(��) �� (��) ﬁnal thermal equilibrium state ρZ

steady state

ρS

FIG. 2. Bounds on unitary evolution. The projection of the density operator onto the subspace spanned by the singlet FIG. 3. Trajectory of the spin density operator in the sub- order operator (horizontal axis) and Zeeman order operator space spanned by the singlet order (horizontal axis) and Zee- (vertical axis) is represented as a point on the plane with man order (vertical axis), starting from thermal equilibrium coordinates (ρS , ρZ ), defined by equations 34 and 35. The (blue dot). Three repetitions of the cycle C (equation 17) thermal equilibrium density operator is represented by the lead to the steady-state density operator indicated by the red eq point (0, ρZ ) (blue dot). The entropy bound on unitary evo- dot, generating singlet order well beyond the unitary bound eq √ lution corresponds to a circle of radius ρZ = /(2 2) (dashed (right edge of the yellow polygon). Enhanced magnetization line). The Sørensen bound is represented by a non-equilateral is generated by applying a π12 permutation to the steady- hexagon (yellow shape), derived in appendix A. Unitary evo- state density operator. This leads to a final density operator lution, starting from thermal equilibrium, leads to a trajec- indicated by the green dot, clearly exceeding the thermal equi- tory which is bounded by the hexagon (blue line). librium magnetization. The displayed trajectory is simulated for the following parameters: T1 = 7.3 s, TS = 240 s, and assumes ideal permutation operations. All trajectories were Define the first two operators in the orthogonal set as calculated using SpinDynamica53. follows: 2 QS = −√ I1 · I2 3 (33) 1. Unitary Bounds 1 1 QZ = √ Iz = √ (I1z + I2z) 2 2 Coherent evolution of the spin ensemble leads to a unitary transformation preserving the Frobenius norm The projection of a spin density operator ρ onto the P 2 kρk = r,s |hr| ρ |si| . Hence coherent evolution starting {QS,QZ } Liouville subspace may be represented as a eq from the thermal equilibrium position (0, ρZ ) leads to a point with coordinates (ρS, ρZ ), defined as follows: eq point (ρS, ρZ ) within a circle of radius |ρZ |, called the √ entropy bound54 (dashed circle in figure 2). 3 54 ρS = QS ρ = hSi (34) Sørensen showed that the bounds on unitary evo- 2 lution are in general more restrictive than the entropy 1 ρZ = QZ ρ = √ hZi (35) bound. The Sørensen bounds on unitary evolution, start- 2 ing from thermal equilibrium, may be written As indicated above, the coefficients ρS and ρZ are proportional to the singlet order hSi and Zeeman order hZi Q ρ ≤ Λ(Q) · Λ(ρeq) (37) of the spin ensemble, respectively. For example, the where Q is a normalized Hermitian operator, Q Q = 1, thermal equilibrium density operator corresponds to a eq and Λ(Q) is a vector whose elements are the eigenvalues point (0, ρ ), where the equilibrium Zeeman coordinate Z of Q, arranged in ascending order. is given by The regions of Liouville space which are excluded by eq equation 37 may be derived by a graphical construc- ρ = √ (36) 55 Z 2 2 tion . As shown in Appendix A, the bounds on unitary 7 evolution restrict the point (ρS, ρZ ) to the interior of a Figure 3 shows that this point is essentially reached after non-equilateral hexagon, shown in yellow in figure 2. Uni- three repetitions of C, starting from thermal equilibrium. tary propagation of the thermal equilibrium state follows The application of further cooling cycles serve no pur- a trajectory which lies entirely within the yellow hexagon pose. (or touches its boundary). The steady-state point indicated by the red circle lies An important consequence of the Sørensen bound is not only outside the yellow hexagon, but even beyond the impossibility of converting the initial Zeeman order the dashed circle in figure 3. This indicates that three eq ρZ completely into singlet order by a unitary transfor- cooling cycles decrease the entropy of the system beyond mation. The singlet order which is accessible by uni- the Frobenius bound. tary transformation of the thermal equilibrium state is The same steady-state loop is attained by the applica- U,max bounded by |ρS| ≤ ρS where tion of repeated cycles C to any initial state. Note that the steady state loop never corresponds to r 2 an enhancement of Zeeman order beyond the thermal ρU,max = ρeq (38) S 3 Z equilibrium value. The application of repeated cycles C pumps the singlet order beyond the unitary bound, but In singlet NMR experiments25,29–48, the relevant ex- not the Zeeman order. perimental parameter is the “two-way” amplitude for the conversion of Zeeman order (magnetization) into singlet 8 order by unitary transformation, and back again, denoted 1.1 aMSM, and given by 1.0 2 ˆ ˆ ˆ aMSM = (QZ | V |QS)(QS| V |QZ ) = (QZ | V |QS) 6 0.9 (39) 0.8 where Vˆ is the propagation superoperator for the spin evolution. 1 0.7 For any unitary transformation, the two-way conver- 4 ★★

T / τ 0.6 U U,max sion amplitude is bounded by aMSM ≤ aMSM , where the theoretical maximum is given by 0.5 0.4 2 2 ρU,max 2 aU,max = S = (40) 0.3 MSM ρeq 3 Z 0.2 and the subscript may be read as “magnetization-to- 0 0.1 singlet-order-to-magnetization”. Hence only ∼ 66.7% of 0 10 20 30 40 50 the nuclear magnetization may be passed through singlet TS/T1 order and back again. This theoretical limit applies to any coherent manipulation of the spin system, and has FIG. 4. Contour plot of the steady-state singlet order ρss/ρeq been approached closely in practice by using robust adi- S Z (equation 41), as a function of the relaxation time ratio TS /T1 42,43,45,47 abatic transformations . (horizontal axis) and the cooling cycle interval τ (in units of the relaxation time constant T1, vertical axis). The dashed line shows the contour for the unitary bound of p2/3. The 2. Pumping of Singlet Order area to the right and above the dashed line indicates the regime where cooling beyond the unitary bound is possi- Figure 3 shows numerical simulations of the density op- ble. The star indicates the parameters for the experimental erator trajectory under repeated cooling cycles C (equa- demonstration. tion 17, starting from thermal equilibrium (blue dot). The blue arrows show the trajectory of the density oper- The behaviour of the steady state singlet order may ator under three repetitions of C. After an initial phase be investigated in detail with the help of SpinDynam- of rather irregular motion, the evolution settles into a ica software53. Deﬁne the relaxation rate constants as −1 −1 repeated “bow tie” loop. The upwards-pointing arrows R1 = T1 and RS = TS , with T1 and TS being the represent the relaxation during the τ intervals, while the relaxation time constants for Zeeman magnetization and cyclic permutations π124 and π142 provide the diagonal singlet order respectively. In the cases of interest here, lines which form the “wings” of the “bow tie”. singlet order decays much more slowly than longitudinal The red circle in ﬁgure 3 indicates the steady-state magnetization, RS R1, and TS T1. An analytical ss ss position of the spin density operator, which returns ex- expression for the steady-state point (ρS , ρZ ) under the actly to the same position after application of a further C cooling cycle C may be obtained by making suitable ap- ss ss cycle. The steady-state point has coordinates (ρS , ρZ ). proximations and assumptions (see Appendix B). This 8 leads to the following expression for the steady-state singlet order: √ ss eq 2 2 6 ρS = 2 6ρZ eS(1 − e1)(e1 + eS)/D (41)

R1τ/2 RS τ/2 where e1 = e , eS = e , and

6 8 2 D = 5e1 + 2e1 + 3e1eS + 6eS 3 6 2 8 2 2 3 − 2eS − 3e1eS − 6e1eS − 5e1eS (42) The dependence of the steady-state singlet order on the ratio of relaxation times TS/T1 and the cooling cycle intervals τ is shown by the contour plot in figure 4. The cooling effect is most effective for large values of TS/T1 and intervals τ which are long compared to T1 but short 13 relative to T . The limit of equation 41 for ideal condi- FIG. 5. (a) Molecular structure of the compound C2- S I, with the filled circles indicating the positions of the 13C tions (R → 0 and R → ∞) is S 1 labels. The chemical equivalence of the 13C nuclei is deliber- ately broken by the isopropyloxy moiety shown in blue. (b) r 13 13 ss 3 eq 3 U,max C NMR spectrum of a 20 mM solution of C2-I in acetone- lim ρS = ρZ = ρS (43) RS →0 2 2 d6 at a field of B0 = 16.4 T. Expanded views are shown of R1→∞ the small outer peaks of the AB spectrum. (c) Energy level 13 diagram of the C2 pair giving the numbering of the basis In the limiting case, singlet order is enhanced by a factor states. The assignments of the observable NMR transitions of 3/2 relative to that achievable by unitary transforma- are given in (b). tion of a thermal equilibrium state (equation 38).

A. Sample 3. Enhanced Zeeman Order

13 A suitable molecular system is given by the C2- Although the steady state loop contains points of en- labelled naphthalene derivative shown in Figure 5 and hanced singlet order, the Zeeman order is not enhanced. 13 referred to hereafter as C2-I. This system has been Zeeman order enhancement may be achieved by apply- 13 shown to support long-lived C2 singlet order with a ing a further π12 permutation to the steady-state system. decay time constant TS exceeding 1 hour in low mag- The trajectory of the density operator under the entire netic field38. The experiments described below were per- sequence is shown in figure 3, with the transformation formed at a high magnetic field of 16.4 T on a sample of under the terminal π12 permutation shown by the green 13 C2-I dissolved in acetone-d6. Under these conditions, arrow. The final state is shown by the green dot. Under the relaxation time constants for the 13C pair were de- ideal conditions, the thermal equilibrium magnetization termined to be T1 = 7.36 ± 0.02 s and TS = 214 ± 5 s is enhanced by a factor of 3/2, with respect to thermal respectively, corresponding to a relaxation time ratio of equilibrium. TS/T1 = 29.1 ± 0.7. 13 13 The C NMR spectrum of C2-I in acetone-d6 at final 3 eq lim ρZ = ρZ (44) a field of 16.4 T is shown in Figure 5. The spin sys- RS →0 2 13 13 R1→∞ tem is well-described by a C- C scalar coupling J = 54.141 ± 0.001 Hz and a small chemical shift difference In the language of spin thermodynamics50, this quantity ∆δ = 0.057 ± 0.001 ppm, due to the deliberate chemical 13 of Zeeman order corresponds to the cooling of the nuclear asymmetry of C2-I. The difference in the chemically spin system to a temperature equal to 2/3 of the sample shifted Larmor frequencies for the two 13C sites in the 0 temperature. 16.4 T magnetic field is ω∆ = ω ∆δ = 2π × 10.01 Hz. 13 Since |2πJ| |ω∆|, the C pair comprises a near- equivalent AB spin system giving rise to a four-peak pat- III. MATERIALS AND METHODS tern with two strong inner peaks and two weak outer peaks, as shown in Figure 5. The splitting between the 2 2 Experimental demonstration of this algorithmic cool- strong inner peaks is given by ∼ ω∆/(8π J) ' 0.9 Hz. ing procedure requires (1) a suitable molecular system The outer peaks are very weak but may be enhanced by displaying near magnetic equivalence and a large relax- specialised pulse techniques56. 13 ation time ratio TS/T1, and (2) a robust and high-fidelity The work described here used a solution of C2-I in experimental procedure for implementing the population acetone-d6 sealed into a micro-cell of length 15 mm, in- permutations. serted into a standard 5 mm NMR tube and positioned 9 about 1 cm above the bottom of the tube. This proce- shape of the APSOC pulse is optimized for the given dure avoids convection41 and provides high homogeneity spin system parameters using previously described algo- for both the static field and the radiofrequency field. rithms42–45, and is specified in appendix C. The APSOC pulses are applied with phase φ = 0 but off-resonance by ±35 Hz, relative to the centre of the NMR spectrum, a 57 180 90 taking into account the sign of the Larmor frequency , 30 150 as depicted in Figure 6(a,b). APSOC(-) As explained in refs.42–45, an APSOC pulse induces a high-fidelity population exchange between the singlet π = φ 124 = 0 state |1i and one of the outer triplet states |2i or |4i. However, care is needed, since these states are not defined in the laboratory frame, but in a frame rotated by ±π/2 about the rotating-frame y-axis. The propagator for the 180 90 b 330 210 two varieties of APSOC pulse may be written as follows: APSOC(+) (−) π 14 (−) U ' exp{−i Iy} exp{−iπI }Φ π APSOC 2 y 142 = φ (45) = 0 (+) π 12 (+) U ' exp{+i I } exp{−iπI }Φ APSOC 2 y y rs 62,63 Here Iy is a single-transition operator and the op- c erators Φ(±) induce phase shifts of the individual spin 180 90 180 90 90 180 0 MA 0 MA 180 0 states: ( 4 ) G2 X G G Φ(±) = exp −i φ(±) |ri hr| (46) T = 1 3 r 00 r=1

(±) where φr are uncontrolled phase factors, which are τ τ τ unimportant in the experiments described here, since only the spin state populations are involved.

FIG. 6. Pulse sequence elements used in this work. The AP- SOC (Adiabatic Passage Spin Order Conversion) pulses? are 2. Cyclic Permutation Sequences specified in appendix C and have phase φ = 0, a duration 0.36 s and a maximum amplitude corresponding to a 13C nu- The pumping cycle C (equation 17) requires cyclic per- tation frequency of 181 Hz. APSOC(±) pulses are applied at a mutation operations π124 and π142 for the spin state pop- 57 frequency shifted (in the algebraic sense ) by ±35 Hz relative ulations, defined by the transformations in equation 18. to the centre of the spectrum (red arrows in the right-hand Each cyclic permutation is realised by applying an AP- pane). (a) The permutation element π142 is implemented by SOC pulse immediately followed by a π/2 pulse of ap- applying a APSOC(-) pulse followed by a composite 90◦ pulse propriate phase. of the form 180 90 24,58–61. (b) The permutation element 30 150 For example, consider an APSOC(-) pulse, immedi- π124 is implemented in a similar fashion but with the opposite frequency offset of the APSOC pulse and reversed signs of ately followed by a strong π/2 pulse with phase π/2. This the composite pulse phases. (c) Pulse sequence for the singlet combination has a propagator given through equation 45 order filtration element T00 where MA indicates the magic by: angle, φ ' 54.74◦, and the durations are τ = 18.2 ms. The magnetic field gradient pulses {G1,G2,G3} have strengths U(APSOC(-),(π/2)y) {0.75, 2.25, 0.51} mT cm−1 and durations {4.4, 2.4, 2.0} ms. π = exp{−i I }U (−) 2 y APSOC 14 (−) ' exp{−iπIy} exp{−iπIy }Φ (47) B. Pulse Sequences The APSOC-pulse combination transforms the four spin states as follows: 1. APSOC pulses (−) U(APSOC(-),(π/2)y) |1i = exp{−iφ1 } |2i (−) The experimental demonstrations use APSOC (Adia- U(APSOC(-),(π/2)y) |2i = exp{−iφ2 } |4i batic Passage Spin Order Conversion) pulses42–45. These (−) U(APSOC(-),(π/2)y) |3i = − exp{−iφ3 } |3i are amplitude-modulated radiofrequency pulses of care- (−) fully optimized shape, applied at a frequency slightly U(APSOC(-),(π/2)y) |4i = − exp{−iφ4 } |1i displaced from the centre of the spectral pattern. The (48) 10

This shows that the combination of an APSOC(-) pulse signal amplitude builds up rapidly, reaching a steady- with an appended (π/2)y pulse fulfils the properties of state of around 0.85 times the thermal equilibrium signal the cyclic permutation operator π124 defined in equa- after approximately 6 permutations. Further permutation 18. tions do not increase the observed singlet order. Similarly, an APSOC(+) pulse with an appended The unitary bound on passing thermal equilibrium (π/2)−y pulse has the functionality of the cyclic permu- magnetization through singlet order is given by the fac- tation operator π142. tor 2/3 (equation 40). This two-way magnetization-to- In practice, high fidelity for the π/2 rotations is singlet-order-to-magnetization bound is shown by the red ensured by using composite pulses24,58–61 of the form dashed line in Fig. 8. The results show that the algorith- 180±3090±150, where the notation βφ indicates the pulse mic cooling cycles successfully generated singlet spin or- flip angle β and the phase φ, both in units of degrees der with an amplitude 1.27 times the unitary limit. The (see figure 6(a,b)). These sequences are derived by tak- loss with respect to the ideal enhancement factor of 3/2 ing the second halves of composite π-pulses of the form (equation 43) may be attributed to non-ideality of the 90±150360±3090±150, and are very well-compensated for triplet thermal reset operations, due to the finite ratio of 59 errors in the radio-frequency field amplitude . The TS to T1, and experimental infidelities in the population phases of the composite pulses take into the sign of permutations. the Larmor frequency and the radio-frequency mixing The singlet pumping efficiency depends on the relax- 57,64 scheme . The APSOC(∓)-180±3090±150 sequences ation delays τ, as shown in Figure 9(a). If the delays provide high-fidelity cyclic permutations of the spin-state are too short, there is not enough time to establish fresh populations, well-compensated for radiofrequency field polarization in the triplet manifold before the next per- errors. mutation; if the delays are too long, singlet-triplet transitions allow the loss of singlet order during the delays. In the case considered, the generation of singlet order is IV. RESULTS optimised for delays of duration τ = 28.0 s. The generation of pumped singlet order was verified A. Pumping of Singlet Order by allowing an evolution time for the singlet order to decay, before conversion into transverse magnetization for The build-up of singlet order under repeated pumping detection. As shown in Figure 9(b), the singlet order cycles C was assessed by the protocols shown in figure 7. decays with the characteristically long time constant of A fine-grained study is achieved by varying the number TS = 209 ± 6 s, which greatly exceeds the spin-lattice of permutations nP , instead of the number of cycles nC . relaxation time constant of T1 = 7.36 ± 0.02 s. This be- Since each cycle C contains two cyclic permutations, even haviour verifies that the state generated by the pumping values of nP are accessed by completing nP /2 cycles (fig- protocol does indeed correspond to long-lived nuclear sin- ure 7(a)). Odd values of nP are accessed by appending an glet order. additional delay and permutation element to (nP − 1)/2 cycles (figure 7(b)). The intervals τ are chosen such that T τ T . 1 S B. Enhancement of Magnetization Under these circumstances, each relaxation interval τ provides an acceptable approximation to the triplet ther- reset mal reset ΘT (equation 14), allowing the triplet popu- The low-entropy state achieved by pumping the singlet lations to approach thermal equilibrium with the environ- order may be used to enhance the nuclear magnetization. ment, while the singlet population is left approximately An enhanced NMR signal may be generated from the unchanged. In practice the value τ = 28.0 s was used. steady state by an additional relaxation delay τ 0 followed The pumped singlet order was allowed to evolve for an by an APSOC pulse (figure 7(c)). The optimal value of 0 optional interval τev, followed by a T00 signal filtration τ was found empirically to be 18.0 s, which is smaller element, as specified in figure 6(c). The T00 filter sup- than the optimal pumping interval τ = 28.0 s. presses all signal components that do not pass through Figure 10 shows experimental results demonstrating singlet order37. A final APSOC pulse efficiently converts the enhancement of the nuclear spin magnetization with the singlet order to transverse magnetization, which in- respect to its thermal equilibrium value. Figure 10(b) 13 13 duces an NMR signal in the subsequent interval of free shows a C NMR signal of C2-I, enhanced by 21% precession. The integrated amplitude of the NMR sig- with respect to the thermal equilibrium spectrum in fig- nal is therefore a reliable measure of the singlet order ure 10(a). The same enhancement in nuclear magne- generated by the repeated cycles C. tization would be accomplished by a reduction in spin Fig. 8 shows the singlet-filtered signal amplitude, nor- temperature by a factor of 0.83. This result proves that malized with respect to the thermal equilibrium signal it is possible to enhance the nuclear magnetization by induced by a π/2 pulse, as a function of the number of algorithmic cooling, even on a strongly coupled spin sys- permutations nP , using a relaxation interval τ = 28.0 s, tem which lacks any distinction between fast-relaxing and and zero evolution interval, τev = 0. The singlet-filtered slowly relaxing spins. 11

APSOC(+) observe τ τ τ π π ev 124 142 T00

nP / 2 b

APSOC(-) observe τ τ τ τ π π π ev 124 142 124 T00

(nP -1) / 2 c

APSOC(+) observe τ τ τ’ π π 124 142

nP / 2

FIG. 7. NMR Protocols. (a) Algorithmic cooling followed by detection of enhanced singlet order for an even number of permutations nP . The implementation of the cyclic permutations and APSOC pulses is shown in Fig. 6. Singlet order is pumped by nC = nP /2 repetitions of the cycle C (equation 17), followed by an optional evolution interval τev, a T00 ﬁlter (ﬁgure 6(c)) and an APSOC(+) pulse for generation of the NMR signal. (b) Protocol for an odd number of permutations nP . The number of cycles is nC = (nP − 1)/2, and an additional delay and permutation is appended; (c) Protocol for detecting enhanced nuclear magnetization for an even number of permutations nP . After pumping the singlet order by nP /2 cycles, the system is allowed to relax for a delay τ 0, followed by an APSOC(+) pulse to generate observable transverse magnetization. In all cases, a singlet-order-destruction (SOD) sequence was applied after signal detection, in order to allow faster repetition of the experiments65 (not shown).

V. CONCLUSIONS several paradigms which have played a prominent role in the theory and implementation of algorithmic cool- Since the achieved enhancement of nuclear magneti- ing. The algorithmic cooling of nuclear spin systems has generally been framed in the language of NMR quantum zation is modest, we do not claim that this method is 23 in serious competition with true hyperpolarization tech- computation , in which a prevailing assumption is that niques such as dynamic nuclear polarization66. Never- the individual nuclear spins may be treated, to a good theless the phenomenon is of interest, since it transcends approximation, as independent qubits which are indepen- 12

0.85 a b

0.80

0.75

0.70 signal amplitude 0.65 -4 -2 0 2 4 -4 -2 0 2 4 0.60 13C frequency offset / Hz 1 2 3 4 5 6 nP 13 13 FIG. 10. C NMR spectra of C2-I. (a) Conventionally ac- FIG. 8. Experimental dependence of the singlet-filtered signal quired spectrum, obtained from the free-induction decay induced by a π/2 pulse applied to a system in thermal equilib- amplitude on the number of permutations nP . Even values rium. (b) Spectrum obtained from an algorithmically cooled of nP (open symbols) and odd values of nP (filled symbols) spin system, obtained by the protocol in Figure 7(c), with were acquired using the protocols in Figure 7(a) and (b) re- 0 spectively. The signal amplitude was normalized against the parameters nP = 2nC = 6, τ = 28 s, τev = 0, τ = 18 s. The signal induced by a π/2 pulse applied to a system in ther- signal intensity is enhanced by a factor of 1.21, with respect mal equilibrium. The intervals between permutations was to thermal equilibrium. τ = 28.0 s. No evolution interval was used, τev = 0. The red dashed line indicates the theoretical bound of 2/3 for two-way magnetization-to-singlet-order conversion under any pulse se- state in the current case); (iii) algorithmic cooling does quence that only exploits coherent interactions (equation 40). not necessarily require qubits with distinct relaxation behaviour; (iv) algorithmic cooling may be used to enhance modes of spin order which do not correspond to spin po- 0.8 a b larization. The last point is salient: The primary target of the 0.6 cooling algorithm in the current work is not the estab- lishment of a low effective spin temperature in the spin ensemble but the enhancement of non-magnetic singlet 0.4 order. This is an anticorrelated state of the spin ensemble which does not correspond to macroscopic spin signal amplitude 0.2 polarization and which cannot be described by a spin temperature. Nevertheless, as shown here, singlet or- 0.0 der may be pumped by an algorithmic procedure to a 0 20 40 60 80 100 0 400 800 1200 1600 level which significantly exceeds that which is accessible τ τ / s ev / s by a unitary transformation of the thermal equilibrium state. We also show that pumped singlet order may be FIG. 9. Experimental dependence of the normalized singlet- converted in a subsequent step to nuclear magnetization filtered signal amplitude on the delays τ and τev. (a) De- that is substantially enhanced with respect to its thermal pendence on the relaxation delays τ in Figure 7(a), where equilibrium value, and which does therefore correspond nP = 6 and τev = 0. The pumped singlet order is max- to a reduced spin temperature. imised for a duration τ = 28.0 s (vertical dashed line). (b) Dependence on the evolution delay τ in Figure 7(a), for This work exploits a very simple system: an ensemble ev of strongly coupled spin-1/2 pairs. It is not currently nP = 6 and τ = 28.0 s. The decay is fitted to a monoexpo- nential decay function with the time constant TS = 209 ± 6 s, known how the phenomenon described here translates to verifying the generation of long-lived singlet order. The red more complex quantum systems, including those which dashed line indicates the theoretical bound of 2/3 for two-way use substrates other than nuclear spins. The fundamen- magnetization-to-singlet-order conversion under any pulse se- tal limits on algorithmic cooling of the type described quence that only exploits coherent interactions (equation 40). here, in which the relevant system eigenstates are not discrete qubit states but entangled superposition states, are not yet known. It is also unknown whether the the- dently addressable by suitable radiofrequency fields15–22. oretical bounds on algorithmic cooling in discrete qubit The current work has shown that (i) algorithmic cooling systems9–13 apply here too. We expect the current re- may be achieved on systems which lack independently port to stimulate interest, and further development, in addressable qubits; (ii) algorithmic cooling may exploit the NMR and quantum information communities, and entangled superpositions of the qubit states (the singlet elsewhere. 13

DATA AVAILABILITY STATEMENT As the angle θ is varied, the eigenvalues in equation A3 cross. Eigenvalue degeneracies occur at the crossing The data that support the ﬁndings of this study are points, which are found at angles θ given by available from the corresponding author upon reasonable request. θ ∈ {0, Θ, π/2, π − Θ, π, π + Θ, 3π/2, 2π − Θ} (A4) where

ACKNOWLEDGMENTS r3 Θ = arctan ' 31.48◦ (A5) 8 M.H.L. thanks S. S. Roy for discussions. This research was supported by EPSRC-UK (grants EP/P009980/1 From the Sørensen bound (equation 37), the projection and EP/P030491/1), the European Research Council of ρ onto Qθ is subject to the following constraint: (grant 786707-FunMagResBeacons), the Russian Foun- dation for Basic Research, grant No. 19-29-10028, Qθ ρ ≤ rB(θ) (A6) the Russian Ministry of Science and Higher Education (project AAAA-A16-116121510087-5), and the Marie where Sklodowska-Curie program of the European Commission, r (θ) = ρeq Λ(Q ) · Λ(Q ) (A7) (grant agreement No 766402). B Z θ 0 Here Λ(Q) represents a vector containing the eigenvalues of the operator Q, arranged in ascending order. Appendix A: Unitary Bounds The set of points (rB(θ), θ), ∀ 0 ≤ θ < 2π, defines a boundary line B, shown by the black curve in figure 11. Any operator Q may be projected onto the Liouville The curve B displays discontinuities at the eigenvalue subspace spanned by the normalized operators {QS,QZ } crossings, which occur for θ values given in equation A4, (equation 33). The projection has Cartesian coordinates with the exception of θ ∈ {0, π}. All points (r, θ) which are outside the curve B are phys- {x, y} = { QS Q , QZ Q } (A1) ically inaccessible by unitary transformation of the thermal equilibrium state. However, not all points within the which may by specified by the notation (r, θ), where curve B are physically accessible. Consider, for example, 2 2 1/2 the polar coordinates are r = {x + y } and θ = the boundary point (rB(Θ), Θ), which is shown by the arctan(x/y). filled orange disk in figure 11. The orange dashed line The thermal equilibrium density operator corresponds passes through this point and the origin. The point (r, θ) eq to a point with polar coordinates (ρZ , 0) (filled blue disk is only physically accessible if its projection onto the or- in figure 11). Unitary evolution of the density operator, ange dashed line is less than, or equal to, r (Θ). This eq B starting from the thermal equilibrium point (ρZ , 0), leads excludes all points (r, θ) in the half-plane shown by the to a new point in the Liouville subspace, with polar coor- orange-shaded region in figure 11. Part of this excluded dinates (r, θ). We now derive the constraints on the phys- region lies within the boundary line B. ically accessible points (r, θ), imposed by the Sørensen Since the curve B has re-entrant projections at the six θ bound (equation 37), by adapting the reasoning in refer- angles {Θ, π/2, π−Θ, π+Θ, 3π/2, 2π−Θ}, the constraints ence55. imposed by these six values of θ delineate the boundaries Define the operator Qθ as a superposition of the nor- of the allowed region of Liouville space. Excluding all malized Zeeman order operator QZ and singlet order op- forbidden regions leaves the central hexagonal region in erator QS with mixing angle θ: figure 12. This corresponds to the yellow hexagon shown in figure 2. The vertices of the hexagon have the fol- eq Qθ = QZ cos θ + QS sin θ (A2) lowing Cartesian coordinates (in units of ρZ ): {0, 1}, {p2/3, 1/2}, {p2/3, −1/2}, {0, −1}, {−p2/3, −1/2}, This corresponds to a point with polar coordinates (1, θ). {−p2/3, 1/2}. The hexagon appears to be regular at The eigenvalues of Q are given by θ first glance but a close inspection shows that it does not n 1 have exact six-fold rotational symmetry. √ sin θ, 2 3 √ 3 − sin θ, Appendix B: Relaxation superoperator and steady-state 2 solutions 1 √ √ (−3 2 cos θ + 3 sin θ), 6 Detailed analysis of the relaxation of the 2-spin-1/2 1 √ √ o (3 2 cos θ + 3 sin θ) (A3) system requires a theoretical model of the relaxation 6 processes. A dipole-dipole relaxation model would be a 14

ρZ ρZ

ρ ρ * S * S

FIG. 11. Liouville subspace spanned by the operators FIG. 12. Construction of the allowed region of Liouville {QS ,QZ } (equation 33). The asterisk marks the origin {0, 0}. subspace spanned by the operators {QS ,QZ }. The bound- The thermal equilibrium density operator is represented by ary curve B has inward projections at the six θ angles eq the blue point with polar coordinates (r, θ) = (ρZ , 0). The {Θ, π/2, π − Θ, π + Θ, 3π/2, 2π − Θ}. The points on the curve black curve is the boundary line B, defined by the set of B at these values of θ are shown by the coloured disks. For points (rB(θ), θ), with 0 ≤ θ < 2π and rB given by equa- each of these θ values, the construction in figure 11 identi- tion A7. All points which are outside this curve are physically fies an infinite half-plane in Liouville subspace which is physi- inaccessible by unitary transformation of the thermal equilib- cally inaccessible under unitary transformation of the thermal rium state. The orange point is at coordinates (rB(Θ), Θ), equilibrium density operator (coloured shaded regions). The p where Θ = arctan 3/8. The dashed line shows a line pass- remaining unshaded region in the centre of the figure corre- ing through this point and the origin. The orange shading sponds to the yellow hexagon in figure 2. indicates the half-plane of Liouville space for which the projection of any point onto the dashed line exceeds the value 3 rB(Θ). This region is physically inaccessible, starting from the extreme-narrowing limit by thermal equilibrium. Note that there is a region of Liouville +1 subspace which lies within B but which is nevertheless phys- X 1 Γˆ = 2ω2 τ exp µ × ically inaccessible. ran c 2 µ=−1 ˆ (j) (j)† ˆ (k) (k)† ˆ (j) (k)† D[T1µ ,T1µ ] + D[T1µ ,T1µ ] + κD[T1µ ,T1µ ] (B1) Here the two spins-1/2 are denoted I and I , ω is natural choice; however, pure DD relaxation would lead j k ran the root-mean-square amplitude of the fluctuating ran- to an infinite value of T . A realistic relaxation model S dom fields, expressed in frequency units, and τ is the would need to be relatively complex, including DD, chem- c correlation time of the random fields. The amplitude of ical shift anisotropy, and external random field mecha- the random fields is assumed to be the same on both nisms28,34. For simplicity, a partially-correlated random- spins, and their correlation coefficient is −1 ≤ κ ≤ 1. field mechanism was used in the current work. (j) The symbol Tλ,µ represents a component of the rank- λ irreducible spherical tensor operator for spin I . The The analysis of algorithmic cooling requires the use j ˆ 3 of relaxation superoperators which take into account the Lindbladian dissipator D[A, B] is defined as follows : finite temperature of the environment. Several methods Dˆ[A, B] = A • B − 1 (BA • + • BA) (B2) have been proposed for the construction of such superop- 2 erators3,52,67,68. In this work, we use a Lindbladian for- where the “bullet” symbol indicates the position of the mulation, which is rigorously consistent with open quan- operand, for example: tum systems theory49, avoiding the pitfalls of the other (A • B)Q = AQB (B3) methods3. The relaxation superoperator in equation B1 leads to The thermalized relaxation superoperator is given in the following expressions for the characteristic relaxation 15 rate constants: following polynomial coefficients:

−1 ˆ 2 {c0, c1 . . . c20} = R1 = T1 = − (QZ | Γ |QZ ) = 2ωranτc (B4) −3 3 −1 ˆ 2 {−3.58531 × 10 , 2.91521, −160.639, 6.58521 × 10 , RS = TS = − (QS| Γ |QS) = 4ωran(1 − κ)τc −145.473 × 103, 2.01387 × 106, −18.9512 × 106, The matrix elements given above are conveniently eval- 126.672 × 106, −617.009 × 106, 2.22025 × 109, 53 uated using SpinDynamica software . −5.9162 × 109, 11.5203 × 109, −15.6681 × 109, In the high-temperature and high-ﬁeld approxima- 12.7611 × 109, −1.0036 × 109, −12.8512 × 109, tions, the superoperator in equation B1 leads to the following matrix for the thermalized transition probabilities 18.9193 × 109, −14.8478 × 109, 7.07406 × 109, per unit time, between the four spin eigenstates: −1.93568 × 109, 235.046 × 106} (C2) W θ =  3 1 (−) 1 1 (+)  − 4 RS 4 RS 4 RS 4 RS Appendix D: Interpretation using Quantum Information   Theory  1 (+) (−) (+) 1 (+)   RS −R1 R1 − RS 0   4 4   (−) (−) (+) (+) In this appendix we show that the cooling cycle C im-  1 R R − 1 R −2R + 1 R R − 1 R   4 S 1 4 S 1 4 S 1 4 S    plements a “greedy” optimisation protocol that aims to 1 R(−) 0 R(−) − 1 R(−) −R(+) maximise the total order of the system at each iteration 4 S 1 4 S 1 step8. The total order of the system is quantiﬁed in terms (B5) of its purity, which is closely related to the entropy of the system8,49. The von Neumann entropy (equation 25) (±) 1 (±) 1 may be approximated to lowest order as follows: where R1 = (1 ± 2 )R1 and RS = (1 ± 2 )RS. The θ null space of W contains the vector of thermal equilib- 2 rium populations, given by equation 5. S = −Tr(ρ ln ρ) ≈ 1 − Tr(ρ ). (D1) The evolution of spin populations over the cycle C in The quantity Tr(ρ2) represents the purity (γ) of the sys- equation 17 is given by the propagation matrix tem49

θ θ γ = Tr(ρ2), (D2) C = π142 exp{W τ}π124 exp{W τ} (B6) and has been utilised previously in the study of algorith- and the permutation matrices π124 and π142 are given mic cooling protocols8,11,13. In contrast to the entropy S, in equation 19. The steady-state population vector pss the purity γ is maximised for pure states and minimised of C is found by determining the null space of the ma- for fully mixed states. A pure state is characterised by trix C − 1, followed by normalisation so that the sum a purity of γ = 1, whereas a fully mixed state is charac- of all elements is equal to 1. The full expression for the −1 terised by a purity of γ = NH . steady-state vector is rather lengthy, but leads after some The iterative application of the cooling cycle C leads to manipulation to the steady-state singlet order given by the following recurrence relation for the density matrix equation 41. of the system

ρn = C(ρn−1), ρ0 = ρeq, (D3)

Appendix C: APSOC pulses with the corresponding set of purities 2 2 γn = Tr(ρn), γ0 = Tr(ρeq) = γeq. (D4) The APSOC shape used in this work is well- approximated by a polynomial of the form The equilibrium purity γeq for a weakly polarised spin- 1/2 pair at moderate magnetic fields is given by 20 2 max X i 1 ωAPSOC(t) = ωAPSOC ci(t/TAPSOC) (C1) γeq = + . (D5) i=0 4 8 The connection between a greedy protocol and the The maximum value of the rf field amplitude, expressed cooling strategy C may be established by considering an max as a nutation frequency, is ωAPSOC = 2π × 181 Hz, and arbitrary two-step cooling cycle C˜ the duration of the APSOC pulse is given by TAPSOC = 0.36 s. The shape of the APSOC pulse is specified by the C˜ = (πa − τ − πb − τ). (D6) 16

0 Here we have exchanged the order slightly to simplify the The quantities δj represent the individual displacements discussion. This change of order is inconsequential for 1 from 4 and are constrained by the normalisation condi- the steady-state. The cooling cycle C˜ may be analysed in tion terms of its matrix representation as discussed in section 0 0 0 0 IIB δ1 + δ2 + δ3 − δ4 = 0, (D12)

˜ reset reset 0 C = ΘT · πb · ΘT · πa, (D7) The displacement δ4 takes a special role in our discussion 0 and is assumed to be positive δ4 > 0. We additionally where πa represents the matrix representations of the assume that the relationship population permutations πa, and similarly for πb. The 0 2 0 2 0 2 relaxation delay τ is assumed to implement a good ap- (δ2) , (δ3) ≤ (δ4) (D13) proximation to the triplet thermal reset. The cooling cycle C˜ is assumed to act on arbitrary pop- holds. This assumption is motivated by the form of ulation vectors p the thermal equilibrium density operator in equation 5. 0 Within this setup the population p4 is initially displaced   1 p1 the furthest from 4 . The purity is therefore maximised p by choosing πa = π14 as the first permutation. The pop- p =  2 , (D8) p3 ulation vector for the first half of the cycle then takes the p4 form 0 P  p˜1  which fulfil the usual normalisation condition j pj = 1. 0 0 reset 0 p˜ The permutations πa and πb are arbitrary elements of ˜  2  p = ΘT · π14 · p =  0  (D14) the permutation group S . p˜3 4 p˜0. The effect of the first permutation πa may be repre- 4 sented as follows To determine the permutation πb it is necessary to con-   sider the displacements of the updated populationsp ˜0 pπ−1(1) j a 1 p −1 from 4 . The updated populations satisfy the following  πa (2) πa · p =   . (D9) relations p −1  πa (3) p −1 1 1 2 πa (4) (˜p0 − )2 − (˜p0 − )2 = δ0(3 + 4δ0) + (3 + 4δ0)2 2 4 3 4 8 4 4 144 4 The elements pπ−1(j) represent the populations pk that 1 1 a (˜p0 − )2 − (˜p0 − )2 = δ0(3 + 4δ0). get permuted to the j’th position of the population vec- 2 4 4 4 9 4 4 tor. The purity of the system after the permutation πa (D15) 0 followed by a thermal reset is then given by Since > 0 and δ4 > 0 this implies that the probability 0 1 p˜2 is always displaced the furthest from 4 . The purity is reset γ(ΘT · πa · p) = therefore maximised by choosing πa = π12 as the second permutation. The population vector p1 after the first 1 2 2 2 2 (1 + ) + (2p −1 − pπ−1(1)) cycle then reads as 3 3 3 πa (1) a 2 2 2 p1 = Θreset · π · Θreset · π · p0. (D16) + (p −1 − 2pπ−1(1)), (D10) T 12 T 14 9 πa (1) a The populations of p may again be decomposed in sim- where we have made use of the normalisation condition. 1 ilar fashion to equation D11 Firstly, the purity of the system only depends on the population p −1 that gets permuted to the singlet popula-    1  πa (1) 1 δ1 tion before the thermal reset. Secondly, the purity takes 1 1 δ1 p1 =   +  2  . (D17) the simple form of a parabola in p −1 with a minimum 1 πa (1) 4 1  δ3  1 1 (to first order in ) at p −1 = . The purity of the 1 −δ πa (1) 4 4 system is thus maximised by choosing the permutation The displacement δ1 is given by π such that the population p −1 is furthest displaced 4 a πa (1) 1 from 4 . δ0 0 δ1 = 4 (1 − 2). (D18) The 0’th step population vector p may always be ex- 4 9 pressed as follows 0 1 Since 0 < < 1 and δ4 > 0, it follows that δ4 > 0. 0 0 p1 1  δ1  Similarly it is straightforward, but slightly tedious, to 0 0 0 p2 1 1 δ2 show that p =  0 =   +  0  . (D11) p3 4 1  δ3  0 0 1 2 1 2 1 2 p4 1 −δ4 (δ2) , (δ3) ≤ (δ4) . (D19) 17

The population vector p1 thus fulfils the same assump- 7J. M. Fernandez, S. Lloyd, T. Mor, and V. Roychowdhury, “Al- tions that have been used to determine the optimal strat- gorithmic cooling of spins: A practicable method for increasing egy for the 0’th step. Since there is nothing special about polarization,” Int. J. Quantum Inform. 02, 461–477 (2004). 8S. E. Sklarz, D. J. Tannor, and N. Khaneja, “Optimal control of the first step, it follows that for a general step j one has quantum dissipative dynamics: Analytic solution for cooling the three-level $\ensuremath{\Lambda}$ system,” Phys. Rev. A 69, j δ4 > 0 (D20) 053408 (2004). 9L. J. Schulman, T. Mor, and Y. Weinstein, “Physical Limits of and Heat-Bath Algorithmic Cooling,” Phys. Rev. Lett. 94, 120501 (2005). 10 j 2 j 2 j 2 H. Yuan, “Reachable set of open quantum dynamics for a sin- (δ2) , (δ3) ≤ (δ4) . (D21) gle spin in Markovian environment,” Automatica 49, 955–959 (2013). This implies that at each step of the cooling process the 11S. Raeisi and M. Mosca, “Asymptotic Bound for Heat-Bath Al- application of the cycle gorithmic Cooling,” Phys. Rev. Lett. 114, 100404 (2015). 12J. Li, D. Lu, Z. Luo, R. Laflamme, X. Peng, and J. Du, “Approx- reset reset imation of reachable sets for coherently controlled open quantum C˜ = Θ · π12 · Θ · π14 (D22) T T systems: Application to quantum state engineering,” Phys. Rev. A 94, 012312 (2016). results in a local optimisation of the purity. 13N. A. Rodr´ıguez-Brionesand R. Laflamme, “Achievable Polariza- The cycle given by equation D22 is slightly different tion for Heat-Bath Algorithmic Cooling,” Phys. Rev. Lett. 116, to the cycle C given in equation 17. However, in the 170501 (2016). limit of infinitely long singlet and infinitely short triplet 14N. A. Rodr´ıguez-Briones,J. Li, X. Peng, T. Mor, Y. Weinstein, relaxation times the two cycles coincide with each other. and R. Laflamme, “Heat-bath algorithmic cooling with corre- reset lated qubit-environment interactions,” New J. Phys. 19, 113047 Inspection of the triplet thermal reset ΘT reveals that (2017). 15 it is right invariant under elements of S3 J. Baugh, O. Moussa, C. A. Ryan, A. Nayak, and R. Laflamme, “Experimental implementation of heat-bath algorithmic cooling reset reset using solid-state nuclear magnetic resonance,” Nature 438, 470 Θ = Θ · πj, (D23) T T (2005). 16J. M. Fernandez, T. Mor, and Y. Weinstein, “Paramagnetic ma- with πj ∈ S3 acting on the indices {2, 3, 4}. It is therefore terials and practical algorithmic cooling for nmr quantum com- permissible to replace π14 by π24π14 = π124 and π12 by puting,” Int. J. Quantum Inform. 03, 281–285 (2005). 17 π24π12 = π142. With this choice the two cooling cycles O. Moussa, On Heat-Bath Algorithmic Cooling and Its Imple- are identical C = C˜ and the cycle C does indeed follow a mentation in Solid-State NMR, MSc, University of Waterloo, “greedy” protocol. Ontario, Canada (2005). 18C. A. Ryan, O. Moussa, J. Baugh, and R. Laflamme, “Spin Based The situation for finite relaxation times τ may be anal- Heat Engine: Demonstration of Multiple Rounds of Algorithmic ysed in a similar fashion. Here, the permutation decom- Cooling,” Phys. Rev. Letters 100, 140501 (2008). 19 positions π24π14 = π124 and π24π12 = π142 may be given Y. Elias, H. Gilboa, T. Mor, and Y. Weinstein, “Heat-bath cool- a physical interpretation. Application of the permuta- ing of spins in two amino acids,” Chemical Physics Letters 517, tions π and π , respectively, increases the purity of 126–131 (2011). 14 12 20J. A. Jones, “Quantum computing with NMR,” Progress in Nu- the system. The permutation π24, on the other hand, ex- clear Magnetic Resonance Spectroscopy 59, 91–120 (2011). changes triplet populations in order to facilitate quicker 21G. Brassard, Y. Elias, J. M. Fernandez, H. Gilboa, J. A. Jones, thermal equilibration within the triplet manifold. For T. Mor, Y. Weinstein, and L. Xiao, “Experimental heat-bath the idealised triplet thermal reset this permutation is su- cooling of spins,” Eur. Phys. J. Plus 129, 266 (2014). 22D. Khurana and T. S. Mahesh, “Bang-bang optimal control of perfluous as the reset operation is considered to restore large spin systems: Enhancement of 13C–13C singlet-order at triplet thermal equilibrium ”instantaneously”. natural abundance,” Journal of Magnetic Resonance 284, 8–14 (2017). 23D. G. Cory, A. F. Fahmy, and T. F. Havel, “Ensemble Quantum REFERENCES Computing by NMR Spectroscopy,” Proc. Natl. Acad. Sci. USA 94, 1634–1639 (1997). 24L. M. K. Vandersypen and I. L. Chuang, “NMR techniques for quantum control and computation,” Rev. Mod. Phys. 76, 1037– 1 A. Abragam, The Principles of Nuclear Magnetism (Clarendon 1069 (2005). Press, Oxford, 1961). 25M. C. D. Tayler and M. H. Levitt, “Singlet nuclear magnetic 2 R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nu- resonance of nearly-equivalent spins,” Phys. Chem. Chem. Phys. clear Magnetic Resonance in One and Two Dimensions (Claren- 13, 5556–5560 (2011). don Press, Oxford, 1987). 26G. Pileio, J. T. Hill-Cousins, S. Mitchell, I. Kuprov, L. J. Brown, 3 C. Bengs and M. H. Levitt, “A master equation for spin sys- R. C. D. Brown, and M. H. Levitt, “Long-Lived Nuclear Singlet tems far from equilibrium,” Journal of Magnetic Resonance 310, Order in Near-Equivalent 13C Spin Pairs,” J. Am. Chem. Soc. 106645 (2020). 134, 17494–17497 (2012). 4 A. W. Overhauser, “Polarization of Nuclei in Metals,” Physical 27G. Pileio, S. Bowen, C. Laustsen, M. C. D. Tayler, J. T. Hill- Review 92, 411 (1953). Cousins, L. J. Brown, R. C. D. Brown, J. H. Ardenkjaer-Larsen, 5 I. Solomon, “Relaxation Processes in a System of Two Spins,” and M. H. Levitt, “Recycling and Imaging of Nuclear Singlet Phys. Rev. 99, 559–565 (1955). Hyperpolarization,” J. Am. Chem. Soc. 135, 5084–5088 (2013). 6 P. O. Boykin, T. Mor, V. Roychowdhury, F. Vatan, and R. Vrijen, 28G. Stevanato, S. Singha Roy, J. Hill-Cousins, I. Kuprov, L. J. “Algorithmic cooling and scalable NMR quantum computers,” Brown, R. C. D. Brown, G. Pileio, and M. H. Levitt, “Long-lived PNAS 99, 3388–3393 (2002). 18

nuclear spin states far from magnetic equivalence,” Phys. Chem. alence regime and field-dependent study of singlet–triplet leak- Chem. Phys. 17, 5913–5922 (2015). age,” Phys. Chem. Chem. Phys. 21, 2595–2600 (2019). 29M. Carravetta, O. G. Johannessen, and M. H. Levitt, “Beyond 47A. S. Kiryutin, M. S. Panov, A. V. Yurkovskaya, K. L. Ivanov, the T1 Limit: Singlet Nuclear Spin States in Low Magnetic and G. Bodenhausen, “Proton Relaxometry of Long-Lived Spin Fields,” Phys. Rev. Lett. 92, 153003 (2004). Order,” ChemPhysChem 20, 766–772 (2019). 30M. Carravetta and M. H. Levitt, “Long-Lived Nuclear Spin 48“Long live the singlet state!” Journal of Magnetic Resonance 306, States in High-Field Solution NMR,” J. Am. Chem. Soc. 126, 69–74 (2019). 6228–6229 (2004). 49H.-P. Breuer and F. Petruccione, The Theory of Open Quantum 31 R. Sarkar, P. R. Vasos, and G. Bodenhausen, “Singlet-State Ex- Systems (Oxford University Press, Oxford [England]; New York, change NMR Spectroscopy for the Study of Very Slow Dynamic 2010) oCLC: 876166237. Processes,” J. Am. Chem. Soc. 129, 328–334 (2007). 50M. Goldman, Spin Temperature and Nuclear Magnetic Reso- 32 G. Pileio, M. Carravetta, E. Hughes, and M. H. Levitt, “The nance in Solids (Clarendon Press, Oxford, 1970). Long-Lived Nuclear Singlet State of 15N-Nitrous Oxide in Solu- 51C. N. Banwell and H. Primas, “On the analysis of high-resolution tion,” J. Am. Chem. Soc. 130, 12582–12583 (2008). nuclear magnetic resonance spectra,” Molecular Physics 6, 225– 33 W. S. Warren, E. Jenista, R. T. Branca, and X. Chen, “Increas- 256 (1963). ing Hyperpolarized Spin Lifetimes Through True Singlet Eigen- 52J. Jeener, “Superoperators in Magnetic Resonance,” in Advances states,” Science 323, 1711–1714 (2009). in Magnetic and Optical Resonance, Vol. 10, edited by J. S. 34 G. Pileio, “Relaxation theory of nuclear singlet states in two spin- Waugh (Academic Press, 1982) pp. 1–51. 1/2 systems,” Progress in Nuclear Magnetic Resonance Spec- 53C. Bengs and M. H. Levitt, “SpinDynamica: Symbolic and nu- troscopy 56, 217–231 (2010). merical magnetic resonance in a Mathematica environment,” 35 G. Pileio, M. Carravetta, and M. H. Levitt, “Storage of nuclear Magnetic Resonance in Chemistry 56, 374–414 (2018). magnetization as long-lived singlet order in low magnetic field,” 54O. W. Sørensen, “A Universal Bound on Spin Dynamics,” J. Proc. Natl. Acad. Sci. U. S. A. 107, 17135–17139 (2010). Magn. Reson. 86, 435–440 (1990). 36 M. H. Levitt, “Singlet Nuclear Magnetic Resonance,” Annu. Rev. 55M. H. Levitt, “Unitary Evolution, Liouville Space and Local Spin Phys. Chem. 63, 89–105 (2012). Thermodynamics,” J. Magn. Reson. 99, 1–17 (1992). 37 M. C. D. Tayler and M. H. Levitt, “Accessing Long-Lived Nu- 56K. F. Sheberstov, A. S. Kiryutin, C. Bengs, J. T. Hill-Cousins, clear Spin Order by Isotope-Induced Symmetry Breaking,” J. L. J. Brown, R. C. D. Brown, G. Pileio, M. H. Levitt, A. V. Am. Chem. Soc. 135, 2120–2123 (2013). Yurkovskaya, and K. L. Ivanov, “Excitation of singlet–triplet co- 38 G. Stevanato, J. T. Hill-Cousins, P. H˚akansson, S. S. Roy, L. J. herences in pairs of nearly-equivalent spins,” Phys. Chem. Chem. Brown, R. C. D. Brown, G. Pileio, and M. H. Levitt, “A Nuclear Phys. 21, 6087–6100 (2019). Singlet Lifetime of More than One Hour in Room-Temperature 57M. H. Levitt, “The Signs of Frequencies and Phases in NMR,” Solution,” Angew. Chem. Int. Ed. 54, 3740–3743 (2015). Journal of Magnetic Resonance 126, 164–182 (1997). 39 Y. Zhang, K. Basu, J. W. Canary, and A. Jerschow, “Singlet 58M. H. Levitt and R. Freeman, “NMR population inversion using lifetime measurements in an all-proton chemically equivalent spin a composite pulse,” J. Magn. Reson. 33, 473–476 (1979). system by hyperpolarization and weak spin lock transfers,” Phys. 59M. H. Levitt, “Symmetrical Composite Pulses for NMR Popula- Chem. Chem. Phys. 17, 24370–24375 (2015). tion Inversion. Part I: Compensation of RF Inhomogeneity,” J. 40 Y. Zhang, X. Duan, P. C. Soon, V. Sychrovský, J. W. Ca- Magn. Reson. 48, 234 (1982). nary, and A. Jerschow, “Limits in Proton Nuclear Singlet-State 60M. H. Levitt, “Composite Pulses,” Prog. NMR Spectrosc. 18, 61 Lifetimes Measured with para-Hydrogen-Induced Polarization,” (1986). ChemPhysChem 17, 2967–2971 (2016). 61S. Wimperis, “Broadband, Narrowband, and Passband Compos- 41 B. Kharkov, X. Duan, J. W. Canary, and A. Jerschow, “Effect ite Pulses for Use in Advanced NMR Experiments,” Journal of of convection and B1 inhomogeneity on singlet relaxation exper- Magnetic Resonance, Series A 109, 221–231 (1994). iments,” Journal of Magnetic Resonance 284, 1–7 (2017). 62A. Wokaun and R. R. Ernst, “Selective excitation and detection 42 A. N. Pravdivtsev, A. S. Kiryutin, A. V. Yurkovskaya, H.-M. in multilevel spin systems: Application of single transition oper- Vieth, and K. L. Ivanov, “Robust conversion of singlet spin order ators,” J. Chem. Phys. 67, 1752–1758 (1977). in coupled spin-1/2 pairs by adiabatically ramped RF-fields,” J. 63S. Vega, “Fictitious spin 1/2 operator formalism for multiple Magn. Reson. 273, 56–64 (2016). quantum NMR,” J. Chem. Phys. 68, 5518–55127 (1978). 43 B. A. Rodin, A. S. Kiryutin, A. V. Yurkovskaya, K. L. Ivanov, 64M. H. Levitt and O. G. Johannessen, “Signs of Frequencies and S. Yamamoto, K. Sato, and T. Takui, “Using optimal control Phases in NMR: The Role of Radiofrequency Mixing,” Journal methods with constraints to generate singlet states in NMR,” of Magnetic Resonance 142, 190–194 (2000). Journal of Magnetic Resonance 291, 14–22 (2018). 65B. A. Rodin, K. F. Sheberstov, A. S. Kiryutin, L. J. Brown, 44 K. F. Sheberstov, H.-M. Vieth, H. Zimmermann, K. L. Ivanov, R. C. D. Brown, M. Sabba, M. H. Levitt, A. V. Yurkovskaya, A. S. Kiryutin, and A. V. Yurkovskaya, “Cis Versus trans- and K. L. Ivanov, “Fast destruction of singlet order in NMR Azobenzene: Precise Determination of NMR Parameters and experiments,” J. Chem. Phys. 151, 234203 (2019). Analysis of Long-Lived States of 15N Spin Pairs,” Appl Magn 66J. H. Ardenkjaer-Larsen, B. Fridlund, A. Gram, G. Hansson, Reson 49, 293–307 (2018). L. Hansson, M. H. Lerche, R. Servin, M. Thaning, and K. Gol- 45 B. A. Rodin, K. F. Sheberstov, A. S. Kiryutin, J. T. Hill-Cousins, man, “Increase in signal-to-noise ratio of > 10,000 times in liquid- L. J. Brown, R. C. D. Brown, B. Jamain, H. Zimmermann, state NMR,” Proc. Natl. Acad. Sci. USA 100, 10158–10163 R. Z. Sagdeev, A. V. Yurkovskaya, and K. L. Ivanov, “Constant- (2003). adiabaticity radiofrequency pulses for generating long-lived sin- 67M. H. Levitt and L. Di Bari, “Steady state in magnetic resonance glet spin states in NMR,” J. Chem. Phys. 150, 064201 (2019). pulse experiments,” Phys. Rev. Lett. 69, 3124–3127 (1992). 46 B. Kharkov, X. Duan, E. S. Tovar, J. W. Canary, and A. Jer- 68T. O. Levante and R. R. Ernst, “Homogeneous versus inhomoge- schow, “Singlet excitation in the intermediate magnetic equiv- neous quantum-mechanical master equations,” Chemical Physics Letters 241, 73–78 (1995).