Here we study the optimum efficiency of the excitation of maximum quantum (MaxQ) coherence using analytical and numerical methods based on optimal control theory. The theoretical limit of the achievable MaxQ amplitude and the minimum time to achieve this limit are explored for a set of model systems consisting of up to five coupled spins. In addition to arbitrary pulse shapes, two simple pulse sequence families of practical interest are considered in the optimizations. Compared to conventional approaches, substantial gains were found both in terms of the achieved MaxQ amplitude and in pulse sequence durations. For a model system, theoretically predicted gains of a factor of three compared to the conventional pulse sequence were experimentally demonstrated. Motivated by the numerical results, also two novel analytical transfer schemes were found: Compared to conventional approaches based on non-selective pulses and delays, double-quantum coherence in two-spin systems can be created twice as fast using isotropic mixing and hard spin-selective pulses. Also it is proved that in a chain of three weakly coupled spins with the same coupling constants, triple-quantum coherence can be created in a time-optimal fashion using so-called geodesic pulses.
REFERENCES
1.
M. H.
Levitt
and R. R.
Ernst
, “Spin-pattern recognition in high-resolution proton NMR spectroscopy
,” Chem. Phys. Lett.
100
, 119
–123
(1983
).2.
D. P.
Weitekamp
, J. R.
Garbow
, J. B.
Murdoch
, and A.
Pines
, “High-resolution NMR spectra in inhomogeneous magnetic fields: Application of total spin coherence transfer echoes
,” J. Am. Chem. Soc.
103
, 3578
–3579
(1981
).3.
R. R.
Ernst
, G.
Bodenhausen
, and A.
Wokaun
, Principles of Nuclear Magnetic Resonance in One and Two Dimensions
(Oxford Science Publications, Clarendon Press
, Oxford
, 1987
).4.
W. P.
Aue
, E.
Bartholdi
, and R. R.
Ernst
, “Two-dimensional spectroscopy. Application to nuclear magnetic resonance
,” J. Chem. Phys.
64
, 2229
–2246
(1976
).5.
G.
Bodenhausen
, “Multiple-quantum NMR
,” Prog. Nucl. Magn. Reson. Spectrosc.
14
, 137
–173
(1980
).6.
L.
Braunschweiler
, G.
Bodenhausen
, and R. R.
Ernst
, “Analysis of networks of coupled spins by multiple quantum NMR
,” Mol. Phys.
48
, 535
–560
(1983
).7.
M.
Munowitz
and A.
Pines
, “Multiple-quantum nuclear magnetic resonance spectroscopy
,” Science
233
, 525
–531
(1986
).8.
G. N. M.
Reddy
and S.
Caldarelli
, “Demixing of severely overlapping NMR spectra through multiple-quantum NMR
,” Anal. Chem.
82
, 3266
–3269
(2010
).9.
G. N. M.
Reddy
and S.
Caldarelli
, “Maximum quantum (MaxQ) NMR for the speciation of mixtures of phenolic molecules
,” Chem. Commun.
47
, 4297
–4299
(2011
).10.
G. N. M.
Reddy
and S.
Caldarelli
, “Identification and quantification of EPA 16 priority polycyclic aromatic hydrocarbon pollutants by maximum-quantum NMR
,” Analyst
137
, 741
–746
(2012
).11.
G. N. M.
Reddy
and S.
Caldarelli
, “Improved excitation uniformity in multiple-quantum NMR experiments of mixtures
,” Magn. Reson. Chem.
51
, 240
–244
(2013
).12.
M. H.
Levitt
and R. R.
Ernst
, “Multiple-quantum excitation and spin topology filtration in high-resolution NMR
,” J. Chem. Phys.
83
, 3297
–3310
(1985
).13.
U.
Piantini
, O. W.
Sørensen
, and R. R.
Ernst
, “Multiple quantum filters for elucidating NMR coupling networks
,” J. Am. Chem. Soc.
104
, 6800
–6801
(1982
).14.
S.
Saxena
and J. H.
Freed
, “Double quantum two-dimensional Fourier transform electron spin resonance: Distance measurements
,” Chem. Phys. Lett.
251
, 102
–110
(1996
).15.
S.
Saxena
and J. H.
Freed
, “Theory of double quantum two-dimensional electron spin resonance with application to distance measurements
,” J. Chem. Phys.
107
, 1317
–1340
(1997
).16.
P. P.
Borbat
and J. H.
Freed
, “Multiple-quantum ESR and distance measurements
,” Chem. Phys. lett.
313
, 145
–154
(1999
).17.
A.
Schweiger
and G.
Jeschke
, Principles of Pulse Electron Paramagnetic Resonance
(Oxford University Press
, New York
, 2001
).18.
O. W.
Sørensen
, “Polarization transfer experiments in high-resolution NMR spectroscopy
,” Prog. Nucl. Magn. Reson. Spectrosc.
21
, 503
–569
(1989
).19.
N.
Nielsen
and O.
Sørensen
, “Conditional bounds on polarization transfer
,” J. Magn. Reson., Ser. A
114
, 24
–31
(1995
).20.
N. C.
Nielsen
, T.
Schulte-Herbrüggen
, and O. W.
Sørensen
, “Bounds on spin dynamics tightened by permutation symmetry—Application
,” Mol. Phys.
85
, 1205
–1216
(1995
).21.
J.
Stoustrup
, O.
Schedletzky
, S. J.
Glaser
, C.
Griesinger
, N. C.
Nielsen
, and O. W.
Sørensen
, “A generalized bound on quantum dynamics: Efficiency of unitary transformations between non-Hermitian states
,” Phys. Rev. Lett.
74
, 2921
–2924
(1995
).22.
S. J.
Glaser
, T.
Schulte-Herbrüggen
, M.
Sieveking
, O.
Schedletzky
, N. C.
Nielsen
, O. W.
Sørensen
, and C.
Griesinger
, “Unitary control in quantum ensembles, maximizing signal intensity in coherent spectroscopy
,” Science
280
, 421
–424
(1998
).23.
T. S.
Untidt
, S. J.
Glaser
, C.
Griesinger
, and N. C.
Nielsen
, “Unitary bounds and controllability of quantum evolution in NMR spectroscopy
,” Mol. Phys.
96
, 1739
–1744
(1999
).24.
M. H.
Levitt
, “Symmetry constraints on spin dynamics: Application to hyperpolarized NMR
,” J. Magn. Reson.
262
, 91
–99
(2016
).25.
N.
Khaneja
, F.
Kramer
, and S. J.
Glaser
, “Optimal experiments for maximizing coherence transfer between coupled spins
,” J. Magn. Reson.
173
, 116
–124
(2005
).26.
T. O.
Reiss
, N.
Khaneja
, and S. J.
Glaser
, “Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates
,” J. Magn. Reson.
165
, 95
–101
(2003
).27.
N.
Khaneja
, S. J.
Glaser
, and R.
Brockett
, “Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer
,” Phys. Rev. A
65
, 032301
(2002
).28.
H.
Yuan
and N.
Khaneja
, “Time optimal control of coupled qubits under nonstationary interactions
,” Phys. Rev. A
72
, 040301
(2005
).29.
N.
Khaneja
, T.
Reiss
, C.
Kehlet
, T.
Schulte-Herbrüggen
, and S. J.
Glaser
, “Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms
,” J. Magn. Reson.
172
, 296
–305
(2005
).30.
Z.
Tošner
, T.
Vosegaard
, C. T.
Kehlet
, N.
Khaneja
, S. J.
Glaser
, and N. C.
Nielsen
, “Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON
,” J. Magn. Reson.
197
, 120
–134
(2009
).31.
S.
Machnes
, U.
Sander
, S. J.
Glaser
, P.
de Fouquières
, A.
Gruslys
, S.
Schirmer
, and T.
Schulte-Herbrüggen
, “Comparing, optimising and benchmarking quantum control algorithms in a unifying programming framework
,” Phys. Rev. A
84
, 022305
(2011
).32.
P.
de Fouquieres
, S. G.
Schirmer
, S. J.
Glaser
, and I.
Kuprov
, “Second order gradient ascent pulse engineering
,” J. Magn. Reson.
212
, 412
–417
(2011
).33.
S. J.
Glaser
, U.
Boscain
, T.
Calarco
, C. P.
Koch
, W.
Köckenberger
, R.
Kosloff
, I.
Kuprov
, B.
Luy
, S.
Schirmer
, T.
Schulte-Herbrüggen
, D.
Sugny
, and K.
Wilhelm
, “Training Schrödinger’s cat: Quantum optimal control
,” Eur. Phys. J. D
69
, 279
(2015
).34.
K.
Kobzar
, T. E.
Skinner
, N.
Khaneja
, S. J.
Glaser
, and B.
Luy
, “Exploring the limits of broadband excitation and inversion pulses
,” J. Magn. Reson.
170
, 236
–243
(2004
).35.
K.
Kobzar
, T. E.
Skinner
, N.
Khaneja
, S. J.
Glaser
, and B.
Luy
, “Exploring the limits of broadband excitation and inversion: II. RF-power optimized pulses
,” J. Magn. Reson.
194
, 58
–66
(2008
).36.
K.
Kobzar
, S.
Ehni
, T. E.
Skinner
, S. J.
Glaser
, and B.
Luy
, “Exploring the limits of broadband 90 and 180 universal rotation pulses
,” J. Magn. Reson.
225
, 142
–160
(2012
).37.
M.
Nimbalkar
, B.
Luy
, T. E.
Skinner
, J. L.
Neves
, N. I.
Gershenzon
, K.
Kobzar
, W.
Bermel
, and S. J.
Glaser
, “The fantastic four: A plug ‘n’ play set of optimal control pulses for enhancing NMR spectroscopy
,” J. Magn. Reson.
228
, 16
–31
(2013
).38.
J. L.
Neves
, B.
Heitmann
, T. O.
Reiss
, H. H. R.
Schor
, N.
Khaneja
, and S. J.
Glaser
, “Exploring the limits of polarization transfer efficiency in homonuclear three spin systems
,” J. Magn. Reson.
181
, 126
–134
(2006
).39.
N.
Pomplun
, B.
Heitmann
, N.
Khaneja
, and S. J.
Glaser
, “Optimization of electron-nuclear polarization transfer
,” Appl. Magn. Reson.
34
, 331
–346
(2008
).40.
N.
Pomplun
and S. J.
Glaser
, “Exploring the limits of electron-nuclear polarization transfer efficiency in three spin systems
,” Phys. Chem. Chem. Phys.
12
, 5791
–5798
(2010
).41.
M.
Nimbalkar
, R.
Zeier
, J. L.
Neves
, S. B.
Elavarasi
, H.
Yuan
, N.
Khaneja
, K.
Dorai
, and S. J.
Glaser
, “Multiple-spin coherence transfer in linear Ising spin chains and beyond: Numerically optimized pulses and experiments
,” Phys. Rev. A
85
, 012325
(2012
).42.
S.
Ehni
and B.
Luy
, “BEBEtr and BUBI: J-compensated concurrent shaped pulses for 1H–13C experiments
,” J. Magn. Reson.
232
, 7
–17
(2013
).43.
T.
Vosegaard
, C.
Kehlet
, N.
Khaneja
, S. J.
Glaser
, and N. C.
Nielsen
, “Improved excitation schemes for multiple-quantum magic-angle spinning for quadrupolar nuclei designed using optimal control theory
,” J. Am. Chem. Soc.
127
, 13768
–13769
(2005
).44.
T.
Schulte-Herbrüggen
, A.
Spörl
, N.
Khaneja
, and S. J.
Glaser
, “Optimal control-cased efficient synthesis of building blocks of quantum algorithms seen in perspective from network complexity towards time complexity
,” Phys. Rev. A
72
, 042331-1
–042331-7
(2005
).45.
N. I.
Gershenzon
, K.
Kobzar
, B.
Luy
, S. J.
Glaser
, and T. E.
Skinner
, “Optimal control design of excitation pulses that accommodate relaxation
,” J. Magn. Reson.
188
, 330
–336
(2007
).46.
T.
Schulte-Herbrüggen
, A.
Spörl
, N.
Khaneja
, and S. J.
Glaser
, “Optimal control for generating quantum gates in open dissipative systems
,” Phys. Rev. B
44
, 154013
(2011
).47.
T. E.
Skinner
, N. I.
Gershenzon
, M.
Nimbalkar
, and S. J.
Glaser
, “Optimal control design of band-selective excitation pulses that accommodate relaxation and RF inhomogeneity
,” J. Magn. Reson.
217
, 53
–60
(2012
).48.
S. S.
Köcher
, T.
Heydenreich
, and S. J.
Glaser
, “Visualization and analysis of modulated pulses in magnetic resonance by joint time-frequency representations
,” J. Magn. Reson.
249
, 63
(2014
).49.
F.
Kramer
, M. V.
Deshmukh
, H.
Kessler
, and S. J.
Glaser
, “Residual dipolar coupling constants: An elementary derivation of key equations
,” Concepts Magn. Reson., Part A
21A
, 10
–21
(2004
).50.
L.
Braunschweiler
and R. R.
Ernst
, “Coherence transfer by isotropic mixing: Application to proton correlation spectroscopy
,” J. Magn. Reson.
53
, 521
–528
(1983
).51.
A. J.
Shaka
, C. J.
Lee
, and A.
Pines
, “Iterative schemes for bilinear operators; application to spin decoupling
,” J. Magn. Reson.
77
, 274
–293
(1988
).52.
S. J.
Glaser
and J. J.
Quant
, “Homonuclear and heteronuclear Hartmann-Hahn transfer in isotropic liquids
,” in Advances in Magnetic and Optical Resonance
, edited byW. S.
Warren
(Academic Press
, 1996
), Vol. 19
, pp. 59
–252
.53.
S. J.
Glaser
, “Coupling topology dependence of polarization-transfer efficiency in TOCSY and TACSY experiments
,” J. Magn. Reson., Ser. A
104
, 283
–301
(1993
).54.
T.
Untidt
, T.
Schulte-Herbrüggen
, B.
Luy
, S. J.
Glaser
, C.
Griesinger
, O. W.
Srensen
, and N. C.
Nielsen
, “Design of NMR pulse experiments with optimum sensitivity: Coherence-order-selective transfer in I2 S and I3 S spin systems
,” Mol. Phys.
95
, 787
–796
(1998
).55.
See supplementary material at http://dx.doi.org/10.1063/1.4945781 for more details on the approaches of pulse sequence optimization, data of optimized pulse sequences, a product operator analysis of simple sequences consisting of two 90° pulses separated by a delay, a more detailed analysis of time-optimal generation of double- and triple-quantum coherence in a system consisting of two and three coupled spins, details on determination of flip angles from spectrograms and information about the fidelity of frequency-selective spectrogram-based pulse sequences.
56.
R.
Freeman
, K. A.
McLauchlan
, J. I.
Musher
, and K. G. R.
Pachler
, “The relative signs of geminal and vicinal proton spin coupling constants
,” Mol. Phys.
5
, 321
(1962
).57.
A.
Garon
, R.
Zeier
, and S. J.
Glaser
, “Visualizing operators of coupled spin systems
,” Phys. Rev. A
91
, 042122
(2015
).58.
59.
V.
Vijayan
, J.-P.
Demers
, J.
Biernat
, E.
Mandelkow
, S.
Becker
, and A.
Lange
, “Low-power solid-state NMR experiments for resonance assignment under fast magic-angle spinning
,” ChemPhysChem
10
, 2205
–2208
(2009
).60.
S.
Hediger
, B. H.
Meier
, N. D.
Kurur
, G.
Bodenhausen
, and R. R.
Ernst
, “NMR cross polarization by adiabatic passage through the Hartmann-Hahn condition (APHH)
,” Chem. Phys. Lett.
223
, 283
–288
(1994
).61.
G. A.
Morris
and R.
Freeman
, “Selective excitation in Fourier transform nuclear magnetic resonance
,” J. Magn. Reson.
29
, 433
–462
(1978
).62.
K.
Dorai
, Arvind
, and A.
Kumar
, “Implementing quantum-logic operations, pseudopure states, and the Deutsch-Jozsa algorithm using noncommuting selective pulses in NMR
,” Phys. Rev. A
61
, 042306
(2000
).63.
N.
Linden
, B.
Hervé
, R. J.
Carbajoc
, and R.
Freeman
, “Pulse sequences for NMR quantum computers: How to manipulate nuclear spins while freezing the motion of coupled neighbours
,” Chem. Phys. Lett.
305
, 28
–34
(1999
).64.
N.
Khaneja
, R.
Brockett
, and S. J.
Glaser
, “Time optimal control in spin systems
,” Phys. Rev. A
63
, 032308
(2001
).65.
N.
Chandrakumar
and K.
Nagayama
, “A new experimental method for heternuclear correlation spectroscopy with cross-polarization
,” Chem. Phys. Lett.
133
, 288
–292
(1987
).© 2016 Author(s).
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