2783:
relaxation mechanisms also exist. The chemical shift anisotropy (CSA) relaxation mechanism arises whenever the electronic environment around the nucleus is non spherical, the magnitude of the electronic shielding of the nucleus will then be dependent on the molecular orientation relative to the (fixed) external magnetic field. The spin rotation (SR) relaxation mechanism arises from an interaction between the nuclear spin and a coupling to the overall molecular rotational angular momentum. Nuclei with spin I ā„ 1 will have not only a nuclear dipole but a quadrupole. The nuclear quadrupole has an interaction with the electric field gradient at the nucleus which is again orientation dependent as with the other mechanisms described above, leading to the so-called quadrupolar relaxation mechanism.
196:. This interaction may be through the electrical or magnetic fields generated by other nuclei, electrons, or molecules. Spontaneous emission of energy is a radiative process involving the release of a photon and typified by phenomena such as fluorescence and phosphorescence. As stated by Abragam, the probability per unit time of the nuclear spin-1/2 transition from the + into the - state through spontaneous emission of a photon is a negligible phenomenon. Rather, the return to equilibrium is a much slower thermal process induced by the fluctuating local magnetic fields due to molecular or electron (free radical) rotational motions that return the excess energy in the form of heat to the surroundings.
6004:
5994:
1277:
43:(RF) pulses at the resonant frequency, created orthogonal to the field. The RF pulses cause the population of spin-states to be perturbed from their thermal equilibrium value. The generated transverse magnetization can then induce a signal in an RF coil that can be detected and amplified by an RF receiver. The return of the longitudinal component of the magnetization to its equilibrium value is termed
1003:
191:
is the sample temperature. At room temperature, the number of spins in the lower energy level, Nā, slightly outnumbers the number in the upper level, N+. The energy gap between the spin-up and spin-down states in NMR is minute by atomic emission standards at magnetic fields conventionally used in MRI
2782:
between the magnetic moment of a nucleus and the magnetic moment of another nucleus or other entity (electron, atom, ion, molecule). This interaction depends on the distance between the pair of dipoles (spins) but also on their orientation relative to the external magnetic field. Several other
1374:
However, decoherence because of magnetic field inhomogeneity is not a true "relaxation" process; it is not random, but dependent on the location of the molecule in the magnet. For molecules that aren't moving, the deviation from ideal relaxation is consistent over time, and the signal can be
1362:
In an idealized system, all nuclei in a given chemical environment, in a magnetic field, precess with the same frequency. However, in real systems, there are minor differences in chemical environment which can lead to a distribution of resonance frequencies around the ideal. Over time, this
2817:
that exists at the nuclear site due to surrounding charges. Thermal motion of a nucleus can result in fluctuating electrostatic interaction energies. These fluctuations produce transitions between the nuclear spin states in a similar manner to the magnetic dipole-dipole interaction.
2234:
2079:
1937:
1521:
4612:
1005:
1009:
1008:
1004:
1010:
2777:
Relaxation of nuclear spins requires a microscopic mechanism for a nucleus to change orientation with respect to the applied magnetic field and/or interchange energy with the surroundings (called the lattice). The most common mechanism is the
3481:
1007:
4348:
4215:
873:
674:
1310:
time, as shown in the animation below. The size of the echo is recorded for different spacings of the two applied pulses. This reveals the decoherence which is not refocused by the 180Ā° pulse. In simple cases, an
2838:
proposed the so-called
Bloembergen-Purcell-Pound theory (BPP theory) to explain the relaxation constant of a pure substance in correspondence with its state, taking into account the effect of tumbling motion of
174:
2085:
3945:
1943:
1801:
284:
range from milliseconds to several seconds, depending on the size of the molecule, the viscosity of the solution, the temperature of the sample, and the possible presence of paramagnetic species (e.g.,
509:
3722:
3290:
4233:
times leads to internuclear distances r. One of the examples is accurate determinations of the metal ā hydride (M-H) bond lengths in solutions by measurements of H selective and non-selective T
1233:
4975:. Isobe T, Matsumura A, Anno I, Kawamura H, Muraishi H, Umeda T, Nose T. "Effect of J coupling and T2 Relaxation in Assessing of Methyl Lactate Signal using PRESS Sequence MR Spectroscopy."
5068:
Gusev, Dmitry G.; Nietlispach, Daniel; Vymenits, Alexey B.; Bakhmutov, Vladimir I.; Berke, Heinz (1993). "Synthesis and NMR T1 relaxation study of rhenium and manganese hydride complexes".
1396:
762:
4476:
3550:
3096:
1726:
39:) frequency of the nuclei. At thermal equilibrium, nuclear spins precess randomly about the direction of the applied field. They become abruptly phase coherent when they are hit by
366:
4417:
2889:
1616:
4468:
3321:
1006:
1133:
566:
3604:
3379:
2916:
3638:
3508:
2257:
3352:
1670:
1643:
1340:
1262:
frequency of different spins. As a result, the initial phase coherence of the nuclear spins is lost, until eventually the phases are disordered and there is no net
1070:
1043:
4652:
4647:
2790:, time-dependent interaction energies cause transitions between the nuclear spin states which result in nuclear spin relaxation. The application of time-dependent
3388:
2843:
on the local magnetic field disturbance. The theory agrees well with experiments on pure substances, but not for complicated environments such as the human body.
2802:
of the fluctuating magnetic dipole interactions. The form of the spectral density functions depend on the physical system, but a simple approximation called the
4781:
Jarek, Russell L.; Flesher, Robert J.; Shin, Seung Koo (1997). "Kinetics of
Internal Rotation of N,N-Dimethylacetamide: A Spin-Saturation Transfer Experiment".
5033:
Gusev, D. G.; Vymenits, A. B.; Bakhmutov, V. I. (1991). "Short spin-lattice relaxation times of hydride ligands. Proton-metal dipole-dipole interactions".
216:
The decay of RF-induced NMR spin polarization is characterized in terms of two separate processes, each with their own time constants. One process, called
4657:
4243:
192:
and NMR spectroscopy. Energy emission in NMR must be induced through a direct interaction of a nucleus with its external environment rather than by
3952:
5193:
2388:
Following is a table of the approximate values of the two relaxation time constants for hydrogen nuclear spins in nonpathological human tissues.
1258:
of the transverse nuclear spin magnetization. Random fluctuations of the local magnetic field lead to random variations in the instantaneous NMR
972:) signals, the technique spin saturation transfer (SST) provides information on chemical exchange reactions. The method is widely applicable to
4705:
Relaxation Times and Basic Pulse
Sequences in MR Imaging. in: Magnetic Resonance in Medicine. A Critical Introduction. 12th edition. pp. 65-92
771:
5155:
2846:
This theory makes the assumption that the autocorrelation function of the microscopic fluctuations causing the relaxation is proportional to
575:
237:
is the time constant for the physical processes responsible for the relaxation of the components of the nuclear spin magnetization vector
2229:{\displaystyle {\frac {\partial M_{z}(t)}{\partial t}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{z}-{\frac {M_{z}(t)-M_{0}}{T_{1}}}}
5885:
105:
5786:
5447:
3729:
2074:{\displaystyle {\frac {\partial M_{y}(t)}{\partial t}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{y}-{\frac {M_{y}(t)}{T_{2}}}}
1932:{\displaystyle {\frac {\partial M_{x}(t)}{\partial t}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{x}-{\frac {M_{x}(t)}{T_{2}}}}
5348:
4959:
Kreis R, Ernst T, and Ross BD "Absolute
Quantification of Water and Metabolites in the Human Brain. II. Metabolite Concentrations"
939:. Small amounts of paramagnetic substances in a sample speed up relaxation very much. By degassing, and thereby removing dissolved
35:) is created by a homogeneous magnetic field. This field makes the magnetic dipole moments of the sample precess at the resonance (
4767:
5673:
4642:
2550:
Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human
24:
374:
4713:
3668:
3102:
5997:
5183:
4998:
Bloembergen, N.; Purcell, E. M.; Pound, R. V. (1948-04-01). "Relaxation
Effects in Nuclear Magnetic Resonance Absorption".
2786:
Molecular reorientation or tumbling can then modulate these orientation-dependent spin interaction energies. According to
6033:
6028:
5955:
5531:
2779:
1516:{\displaystyle {\frac {1}{T_{2}^{*}}}={\frac {1}{T_{2}}}+{\frac {1}{T_{inhom}}}={\frac {1}{T_{2}}}+\gamma \Delta B_{0}}
1145:
6007:
5895:
5148:
4897:
2748:. Under these circumstances the relaxation of the magnetization is similar to laboratory frame relaxation in a field
5823:
5481:
4607:{\displaystyle C=10^{7}\left({\frac {{\gamma }_{H}^{2}{\gamma }_{M}^{2}\hbar ^{2}I_{M}(I_{M}+1)}{15}}\right)^{1/6}}
704:
5818:
5546:
5526:
2554:
2680:
The discussion above describes relaxation of nuclear magnetization in the presence of a constant magnetic field
5813:
4738:
3517:
2936:
5325:
1675:
1363:
distribution can lead to a dispersion of the tight distribution of magnetic spin vectors, and loss of signal (
935:) are generally strongly dependent on the NMR frequency and so vary considerably with magnetic field strength
5846:
5668:
5571:
5232:
5851:
5719:
5310:
5141:
4754:
Hoult, D.I.; Bahkar, B. (1998). "NMR Signal
Reception: Virtual Photons and Coherent Spontaneous Emission".
223:, is responsible for the loss of resonance intensity following pulse excitation. The other process, called
5875:
5631:
5491:
5265:
5220:
5164:
4677:
2919:
2769:. Relaxation in the rotating frame is useful because it provides information on slow motions of nuclei.
892:
is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of
333:
303:
277:
limits the pulse repetition rate and affects the overall time an NMR spectrum can be acquired. Values of
1238:
i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant
5747:
5619:
5516:
5392:
5320:
5227:
4637:
4356:
914:
relaxation involves an interaction with the surroundings is the origin of the alternative description,
20:
4708:. Offprint to download: TRTF - The Round Table Foundation / EMRF - European Magnetic Resonance Forum.
3560:
of such species of nuclei, and r the distance between the two nuclei carrying magnetic dipole moment.
2849:
1578:
1273:
relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation.
5556:
5521:
5417:
5360:
2377:
4422:
884:
relaxation involves redistributing the populations of the nuclear spin states in order to reach the
5729:
5702:
5678:
5641:
5432:
5365:
5300:
5285:
5255:
5178:
5128:
4682:
4221:
which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T
2799:
997:
1367:). In fact, for most magnetic resonance experiments, this "relaxation" dominates. This results in
5880:
5614:
5506:
5476:
5275:
3553:
3299:
2814:
2809:
Another relaxation mechanism is the electrostatic interaction between a nucleus with an electric
5112:
1111:
529:
5950:
5714:
5707:
5454:
4818:"Principles, Techniques, and Applications of T2*-based MR Imaging and its Special Applications"
3582:
3357:
2894:
1788:
885:
92:
3613:
3486:
5870:
5496:
5422:
5387:
2623:
2242:
4816:
Chavhan, Govind B; Babyn, Paul S; Thomas, Bejoy; Shroff, Manohar M; Haacke, E. Mark (2009).
3476:{\displaystyle K={\frac {3\mu _{0}^{2}}{160\pi ^{2}}}{\frac {\hbar ^{2}\gamma ^{4}}{r^{6}}}}
5660:
5409:
5260:
4667:
3330:
2831:
2827:
1648:
1621:
1364:
1357:
1318:
1254:
relaxation is a complex phenomenon, but at its most fundamental level, it corresponds to a
1048:
1021:
889:
193:
55:
8:
5979:
5932:
5762:
5536:
5290:
5270:
5205:
5200:
3511:
2791:
2461:
679:
i.e. the magnetization recovers to 63% of its equilibrium value after one time constant
5695:
5541:
5343:
5280:
4842:
4817:
3557:
1527:
973:
691:
5959:
5604:
5511:
5468:
5399:
5315:
5295:
5250:
5210:
5188:
5085:
5050:
5015:
4893:
4847:
4798:
4734:
4709:
2787:
2363:
1312:
1259:
28:
5117:
1553:
relaxation time and is typically milliseconds for water samples in imaging magnets.
976:. This magnetization transfer technique provides rates, provided that they exceed 1/
5576:
5353:
5077:
5042:
5007:
4919:
4837:
4829:
4790:
4763:
4662:
3655:(Simplified. BPP theory uses angular frequency indeed). We can then estimate using
3324:
2795:
2724:
36:
5752:
5690:
5380:
5375:
3382:
2600:
1737:
40:
2762:
is called the spin-lattice relaxation time in the rotating frame and is denoted
5861:
5838:
5805:
5609:
5486:
2422:
1014:
Visual representation of the spin of a proton under a constant magnetic field B
316:
5133:
4343:{\displaystyle r(M-H)=C\left({\frac {(1.4k+4.47)T_{1min}}{\nu }}\right)^{1/6}}
183:
where ĪE is the energy level difference between the two populations of spins,
6022:
5683:
5501:
5427:
5089:
5054:
5019:
4802:
2369:
1139:
magnetization at time zero will decay to zero (i.e. equilibrium) as follows:
32:
4988:
A. Abragam "Principles of
Nuclear Magnetism" (Oxford University Press, 1961)
4210:{\displaystyle T_{2}=\left({\frac {1.02\times 10^{10}}{2}}\left\right)^{-1}}
230:, characterizes the width or broadness of resonances. Stated more formally,
5903:
5828:
5011:
4923:
4851:
3568:
2835:
2794:
in quantum mechanics shows that the relaxation rates (and times) depend on
2755:. The decay constant for the recovery of the magnetization component along
2496:
2393:
2695:, is the relaxation of nuclear magnetization in the presence of the field
77:
nuclei (such as H), the polarization due to spins oriented with the field
5913:
5767:
5305:
4833:
4672:
2650:
2479:
2408:
1792:
1255:
5081:
5046:
4910:
Traficante, Daniel D. (1991). "Relaxation. Can T2, be longer than T1?".
4865:
868:{\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-2e^{-t/T_{1}}\right)}
5974:
5908:
5772:
4237:
times in variable-temperature relaxation experiments via the equation:
3607:
2810:
2558:
669:{\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-e^{-t/T_{1}}\right)}
5106:
4794:
5757:
4768:
10.1002/(SICI)1099-0534(1997)9:5<277::AID-CMR1>3.0.CO;2-W
3572:
2562:
1376:
1368:
1300:
2383:
1290:
values are generally much less dependent on field strength, B, than
1276:
903:
allow nuclear spins to exchange energy with their surroundings, the
5942:
5918:
5777:
5742:
4729:
Abragam, A. (1961). "VII Thermal
Relaxation in Liquids and Gases".
3644:
2840:
2596:
2483:
169:{\displaystyle {\frac {N_{+}}{N_{-}}}=e^{-{\frac {\Delta E}{kT}}}}
5969:
5586:
5067:
3651:) at 1.5 tesla precess at a Larmor frequency of approximately 64
2523:
2380:, which is an important tool in determining molecular structure.
2309:(t)) is the magnetic flux density experienced by the nuclei. The
1742:
Bloch equations are used to calculate the nuclear magnetization
1545:* is influenced by magnetic field gradient irregularities. The T
5946:
5442:
3940:{\displaystyle T_{1}=\left(1.02\times 10^{10}\left\right)^{-1}}
3648:
3640:
3327:
in correspondence with the strength of the main magnetic field
2536:
2373:
1382:
The corresponding transverse relaxation time constant is thus T
940:
4703:
907:, allowing the spin populations to equilibrate. The fact that
5215:
5122:
2551:
2510:
2465:
2436:
5964:
5437:
5370:
1575:
In NMR systems, the following relation holds absolute true
1280:
Spin echo pulse sequence and magnetization decay animation.
957:
of liquid samples easily go up to an order of ten seconds.
888:. By definition, this is not energy conserving. Moreover,
5581:
5566:
3652:
2334:
1534:
the difference in strength of the locally varying field.
504:{\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }-e^{-t/T_{1}}}
50:
while the loss of phase-coherence of the spins is termed
4815:
4653:
Nuclear magnetic resonance spectroscopy of nucleic acids
4648:
Nuclear magnetic resonance spectroscopy of carbohydrates
2669:
2368:
Solomon equations are used to calculate the transfer of
4997:
2337:
and helps with the spatial decoding of the NMR signal.
1345:
5032:
3717:{\displaystyle \omega _{0}\tau _{c}=3.2\times 10^{-5}}
3483:
is defined for spin-1/2 nuclei and is a constant with
3285:{\displaystyle {\frac {1}{T_{2}}}={\frac {K}{2}}\left}
2259:
is the cross-product, Ī³ is the gyromagnetic ratio and
4892:, 2nd edition, John Wiley & Sons, New York 2008,
4479:
4425:
4359:
4246:
3955:
3732:
3671:
3616:
3585:
3520:
3489:
3391:
3360:
3333:
3302:
3105:
2939:
2897:
2852:
2245:
2088:
1946:
1804:
1678:
1651:
1624:
1581:
1399:
1321:
1148:
1114:
1051:
1024:
965:
Especially for molecules exhibiting slowly relaxing (
774:
707:
578:
532:
377:
336:
108:
5129:
Estimation of T1 and T2 relaxation parameters in MRI
5408:
4890:
Spin
Dynamics: Basics of Nuclear Magnetic Resonance
4658:
Nuclear magnetic resonance spectroscopy of proteins
308:The longitudinal (or spin-lattice) relaxation time
4606:
4462:
4411:
4342:
4209:
3939:
3716:
3632:
3598:
3544:
3502:
3475:
3373:
3346:
3315:
3284:
3090:
2910:
2883:
2251:
2228:
2073:
1931:
1720:
1664:
1637:
1610:
1515:
1334:
1227:
1127:
1064:
1037:
867:
756:
668:
560:
503:
360:
168:
4229:. As follows from the BPP theory, measuring the T
2798:functions that are the Fourier transforms of the
2384:Common relaxation time constants in human tissues
1228:{\displaystyle M_{xy}(t)=M_{xy}(0)e^{-t/T_{2}}\,}
84:relative to the spins oriented against the field
16:Decay of nuclear spin polarization in MRI and NMR
6020:
4780:
4621:are measured in Ć
, MHz and s, respectively, and
2741:is typically much smaller than the magnitude of
5163:
1556:
701:values, the initial magnetization is inverted,
5113:Relaxation in high-resolution NMR spectroscopy
2702:together with a time-dependent magnetic field
1773:) as a function of time when relaxation times
1549:* relaxation time is always shorter than the T
1076:The transverse (or spin-spin) relaxation time
259:relaxation affects the coherent components of
5149:
2354:relaxation are those in the Bloch equations.
757:{\displaystyle M_{z}(0)=-M_{z,\mathrm {eq} }}
323:component of the nuclear spin magnetization,
54:relaxation, which is manifest as an observed
2376:system. They can be employed to explain the
2340:The equation listed above in the section on
1618:. In most situations (but not in principle)
1303:decay experiment can be used to measure the
4957:Chemicals of brain relaxation time at 1.5T.
4753:
1083:is the decay constant for the component of
960:
248:(which is conventionally designated as the
5156:
5142:
5125:Software for the analysis of NMR dynamics
4903:
4841:
3545:{\displaystyle \hbar ={\frac {h}{2\pi }}}
3381:is the correlation time of the molecular
3091:{\displaystyle {\frac {1}{T_{1}}}=K\left}
2772:
2333:(t), is time dependent. It is present in
2317:is typically composed of two terms: one,
1224:
330:, towards its thermal equilibrium value,
241:parallel to the external magnetic field,
4952:
4950:
4733:. Oxford University Press. p. 264.
1721:{\displaystyle 2T_{1}>T_{2}>T_{1}}
1275:
1001:
5674:Continuous-variable quantum information
4728:
4722:
4643:Nuclear magnetic resonance spectroscopy
2313:component of the magnetic flux density
25:nuclear magnetic resonance spectroscopy
6021:
3579:is 1.02Ć10 s and the correlation time
2716:rotates in the plane perpendicular to
2324:, is constant in time, the other one,
1386:, which is usually much smaller than T
1315:is measured which is described by the
5137:
4947:
4701:
694:experiment, commonly used to measure
4695:
2357:
270:. In conventional NMR spectroscopy,
3512:magnetic permeability of free space
361:{\displaystyle M_{z,\mathrm {eq} }}
13:
2780:magnetic dipole-dipole interaction
2689:relaxation in the laboratory frame
2670:Relaxation in the rotating frame,
2116:
2092:
1974:
1950:
1832:
1808:
1791:equations that were introduced by
1731:
1500:
1352:* and magnetic field inhomogeneity
899:relaxation. However, a variety of
812:
809:
748:
745:
616:
613:
442:
439:
415:
412:
352:
349:
147:
14:
6045:
5100:
4412:{\displaystyle k=(f-1)/(0.5-f/3)}
2930:for magnetic dipolar relaxation:
2922:. From this theory, one can get T
1787:are present. Bloch equations are
6003:
6002:
5993:
5992:
4973:Lactate relaxation time at 1.5 T
2884:{\displaystyle e^{-t/\tau _{c}}}
2693:relaxation in the rotating frame
2152:
2135:
2010:
1993:
1868:
1851:
1611:{\displaystyle T_{2}\leq 2T_{1}}
1390:. The relation between them is:
886:thermal equilibrium distribution
5061:
5026:
4991:
4982:
4966:
4909:
4731:Principles of Nuclear Magnetism
2803:
2657:(doublet: 1.27 & 1.39 ppm)
2555:magnetic resonance spectroscopy
2372:as a result of relaxation in a
187:is the Boltzmann constant, and
4930:
4912:Concepts in Magnetic Resonance
4882:
4858:
4809:
4774:
4756:Concepts in Magnetic Resonance
4747:
4577:
4558:
4463:{\displaystyle f=T_{1s}/T_{1}}
4406:
4386:
4378:
4366:
4294:
4279:
4262:
4250:
4176:
4153:
3906:
3883:
3826:
3803:
2197:
2191:
2166:
2162:
2156:
2145:
2139:
2131:
2111:
2105:
2055:
2049:
2024:
2020:
2014:
2003:
1997:
1989:
1969:
1963:
1913:
1907:
1882:
1878:
1872:
1861:
1855:
1847:
1827:
1821:
1728:are rare, but not impossible.
1193:
1187:
1168:
1162:
791:
785:
724:
718:
595:
589:
549:
543:
470:
467:
461:
424:
394:
388:
1:
5669:Adiabatic quantum computation
5118:Field-cycling NMR relaxometry
4961:Journal of Magnetic Resonance
4783:Journal of Chemical Education
4688:
3571:without the contamination of
2821:
2691:. Another technique, called
2577:Relative resonance frequency
764:, and so the recovery follows
27:(NMR), an observable nuclear
5720:Topological quantum computer
4870:Questions and Answers in MRI
199:
7:
5998:Quantum information science
5165:Quantum information science
4963:, Series B 102 (1993): 9-19
4866:"T2* vs T2 relaxation time"
4631:
3316:{\displaystyle \omega _{0}}
2574:Signals of chemical groups
10:
6050:
6034:Magnetic resonance imaging
6029:Nuclear magnetic resonance
5393:quantum gate teleportation
4638:Nuclear magnetic resonance
2361:
2335:magnetic resonance imaging
1735:
1375:recovered by performing a
1355:
1128:{\displaystyle M_{\perp }}
995:
568:and the recovery is simply
561:{\displaystyle M_{z}(0)=0}
301:
21:magnetic resonance imaging
5988:
5931:
5894:
5860:
5837:
5804:
5795:
5728:
5657:
5595:
5555:
5522:Quantum Fourier transform
5467:
5418:Post-quantum cryptography
5361:Entanglement distillation
5334:
5243:
5171:
3599:{\displaystyle \tau _{c}}
3374:{\displaystyle \tau _{c}}
2911:{\displaystyle \tau _{c}}
2569:At a main field of 1.5 T
2450:Whole blood (oxygenated)
2378:nuclear Overhauser effect
522:has been tilted into the
6008:Quantum mechanics topics
5703:Quantum machine learning
5679:One-way quantum computer
5532:Quantum phase estimation
5433:Quantum key distribution
5366:Monogamy of entanglement
4702:Rinck, Peter A. (2022).
4617:where r, frequency and T
3633:{\displaystyle 10^{-12}}
3563:Taking for example the H
3503:{\displaystyle \mu _{0}}
2800:autocorrelation function
2634:white matter: 1220-1410
2610:white matter: 1050ā1360
1135:. For instance, initial
961:Spin saturation transfer
319:for the recovery of the
5615:Randomized benchmarking
5477:Amplitude amplification
4678:Spinālattice relaxation
3554:reduced Planck constant
2815:electric field gradient
2632:gray matter: 1170ā1370,
2608:gray matter: 1150ā1340,
2392:At a main field of 1.5
2252:{\displaystyle \times }
1266:magnetization. Because
1018:. Visualization of the
921:Note that the rates of
916:spin-lattice relaxation
304:Spinālattice relaxation
5715:Quantum Turing machine
5708:quantum neural network
5455:Quantum secret sharing
5012:10.1103/PhysRev.73.679
4924:10.1002/cmr.1820030305
4608:
4464:
4413:
4344:
4211:
3941:
3718:
3634:
3600:
3546:
3504:
3477:
3375:
3348:
3317:
3286:
3092:
2912:
2885:
2773:Microscopic mechanisms
2639:white matter: 436-519
2615:white matter: 194-218
2253:
2230:
2075:
1933:
1722:
1666:
1639:
1612:
1517:
1336:
1281:
1229:
1129:
1073:
1066:
1039:
869:
758:
670:
562:
505:
362:
170:
93:Boltzmann distribution
5787:Entanglement-assisted
5748:quantum convolutional
5423:Quantum coin flipping
5388:Quantum teleportation
5349:entanglement-assisted
5179:DiVincenzo's criteria
4609:
4465:
4414:
4345:
4212:
3942:
3719:
3635:
3601:
3547:
3505:
3478:
3376:
3349:
3347:{\displaystyle B_{0}}
3318:
3287:
3093:
2913:
2886:
2727:of the nuclei in the
2637:gray matter: 388ā426,
2613:gray matter: 198ā207,
2601:Phosphocreatine (PCr)
2254:
2231:
2076:
1934:
1723:
1672:. The cases in which
1667:
1665:{\displaystyle T_{2}}
1640:
1638:{\displaystyle T_{1}}
1613:
1518:
1356:Further information:
1337:
1335:{\displaystyle T_{2}}
1279:
1230:
1130:
1067:
1065:{\displaystyle T_{2}}
1040:
1038:{\displaystyle T_{1}}
1013:
986:
901:relaxation mechanisms
870:
759:
671:
563:
506:
363:
292:
171:
5598:processor benchmarks
5527:Quantum optimization
5410:Quantum cryptography
5221:physical vs. logical
4979:(2005) v25. 2:68-74.
4834:10.1148/rg.295095034
4683:Spināspin relaxation
4668:Relaxation (physics)
4477:
4423:
4357:
4244:
3953:
3730:
3669:
3614:
3583:
3518:
3487:
3389:
3358:
3331:
3300:
3103:
2937:
2895:
2850:
2832:Edward Mills Purcell
2828:Nicolaas Bloembergen
2620:N-Acetyl group (NA),
2243:
2086:
1944:
1802:
1676:
1649:
1622:
1579:
1397:
1365:free induction decay
1358:T2*-weighted imaging
1319:
1146:
1112:
1049:
1022:
998:Spināspin relaxation
928:relaxation (i.e., 1/
890:spontaneous emission
772:
705:
576:
530:
375:
334:
194:spontaneous emission
106:
56:free induction decay
5311:Quantum speed limit
5206:Quantum programming
5201:Quantum information
5082:10.1021/ic00067a013
5070:Inorganic Chemistry
5047:10.1021/ic00016a003
5035:Inorganic Chemistry
4888:Malcolm H. Levitt:
4537:
4520:
3606:is on the order of
3418:
3273:
3258:
3210:
3195:
3079:
3064:
3016:
3001:
2792:perturbation theory
2734:. The magnitude of
2570:
2462:Cerebrospinal fluid
2396:
1564:always longer than
1526:where Ī³ represents
1419:
974:fluxional molecules
514:In specific cases:
5960:Forest/Rigetti QCS
5696:quantum logic gate
5482:BernsteināVazirani
5469:Quantum algorithms
5344:Classical capacity
5228:Quantum processors
5211:Quantum simulation
4940:, Physical Review
4628:is the spin of M.
4604:
4521:
4504:
4460:
4409:
4340:
4207:
3937:
3714:
3630:
3596:
3558:gyromagnetic ratio
3542:
3500:
3473:
3404:
3371:
3344:
3313:
3282:
3259:
3244:
3196:
3181:
3088:
3065:
3050:
3002:
2987:
2908:
2881:
2568:
2391:
2249:
2226:
2071:
1929:
1718:
1662:
1635:
1608:
1528:gyromagnetic ratio
1513:
1405:
1332:
1282:
1225:
1125:
1074:
1062:
1035:
865:
754:
692:inversion recovery
666:
558:
501:
358:
166:
6016:
6015:
5927:
5926:
5824:Linear optical QC
5605:Quantum supremacy
5559:complexity theory
5512:Quantum annealing
5463:
5462:
5400:Superdense coding
5189:Quantum computing
5107:The Basics of NMR
5076:(15): 3270ā3276.
5041:(16): 3116ā3118.
4938:Nuclear Induction
4795:10.1021/ed074p978
4715:978-3-7460-9518-9
4584:
4320:
4186:
4106:
3998:
3916:
3836:
3540:
3471:
3435:
3275:
3212:
3134:
3121:
3081:
3018:
2955:
2788:quantum mechanics
2687:. This is called
2667:
2666:
2624:N-acetylaspartate
2548:
2547:
2464:(similar to pure
2364:Solomon equations
2358:Solomon equations
2224:
2123:
2069:
1981:
1927:
1839:
1492:
1472:
1440:
1420:
1313:exponential decay
1087:perpendicular to
1072:relaxation times.
1011:
263:perpendicular to
162:
131:
29:spin polarization
6041:
6006:
6005:
5996:
5995:
5802:
5801:
5732:error correction
5661:computing models
5627:Relaxation times
5517:Quantum counting
5406:
5405:
5354:quantum capacity
5301:No-teleportation
5286:No-communication
5158:
5151:
5144:
5135:
5134:
5094:
5093:
5065:
5059:
5058:
5030:
5024:
5023:
4995:
4989:
4986:
4980:
4970:
4964:
4954:
4945:
4944:, 460ā473 (1946)
4934:
4928:
4927:
4907:
4901:
4900:, Section 11.9.2
4886:
4880:
4879:
4877:
4876:
4862:
4856:
4855:
4845:
4828:(5): 1433ā1449.
4813:
4807:
4806:
4778:
4772:
4771:
4751:
4745:
4744:
4726:
4720:
4719:
4699:
4663:Protein dynamics
4613:
4611:
4610:
4605:
4603:
4602:
4598:
4589:
4585:
4580:
4570:
4569:
4557:
4556:
4547:
4546:
4536:
4531:
4526:
4519:
4514:
4509:
4502:
4495:
4494:
4469:
4467:
4466:
4461:
4459:
4458:
4449:
4444:
4443:
4418:
4416:
4415:
4410:
4402:
4385:
4349:
4347:
4346:
4341:
4339:
4338:
4334:
4325:
4321:
4316:
4315:
4314:
4277:
4216:
4214:
4213:
4208:
4206:
4205:
4197:
4193:
4192:
4188:
4187:
4185:
4184:
4183:
4174:
4173:
4139:
4138:
4137:
4112:
4107:
4105:
4104:
4103:
4098:
4094:
4093:
4092:
4061:
4060:
4059:
4034:
4029:
4028:
3999:
3994:
3993:
3992:
3976:
3965:
3964:
3946:
3944:
3943:
3938:
3936:
3935:
3927:
3923:
3922:
3918:
3917:
3915:
3914:
3913:
3904:
3903:
3869:
3868:
3867:
3842:
3837:
3835:
3834:
3833:
3824:
3823:
3795:
3794:
3793:
3774:
3767:
3766:
3742:
3741:
3723:
3721:
3720:
3715:
3713:
3712:
3691:
3690:
3681:
3680:
3639:
3637:
3636:
3631:
3629:
3628:
3605:
3603:
3602:
3597:
3595:
3594:
3551:
3549:
3548:
3543:
3541:
3539:
3528:
3509:
3507:
3506:
3501:
3499:
3498:
3482:
3480:
3479:
3474:
3472:
3470:
3469:
3460:
3459:
3458:
3449:
3448:
3438:
3436:
3434:
3433:
3432:
3419:
3417:
3412:
3399:
3380:
3378:
3377:
3372:
3370:
3369:
3353:
3351:
3350:
3345:
3343:
3342:
3325:Larmor frequency
3322:
3320:
3319:
3314:
3312:
3311:
3291:
3289:
3288:
3283:
3281:
3277:
3276:
3274:
3272:
3267:
3257:
3252:
3233:
3232:
3231:
3218:
3213:
3211:
3209:
3204:
3194:
3189:
3173:
3172:
3171:
3158:
3153:
3152:
3135:
3127:
3122:
3120:
3119:
3107:
3097:
3095:
3094:
3089:
3087:
3083:
3082:
3080:
3078:
3073:
3063:
3058:
3039:
3038:
3037:
3024:
3019:
3017:
3015:
3010:
3000:
2995:
2979:
2978:
2969:
2956:
2954:
2953:
2941:
2920:correlation time
2917:
2915:
2914:
2909:
2907:
2906:
2890:
2888:
2887:
2882:
2880:
2879:
2878:
2877:
2868:
2806:is widely used.
2796:spectral density
2725:Larmor frequency
2571:
2567:
2397:
2390:
2258:
2256:
2255:
2250:
2235:
2233:
2232:
2227:
2225:
2223:
2222:
2213:
2212:
2211:
2190:
2189:
2179:
2174:
2173:
2155:
2138:
2124:
2122:
2114:
2104:
2103:
2090:
2080:
2078:
2077:
2072:
2070:
2068:
2067:
2058:
2048:
2047:
2037:
2032:
2031:
2013:
1996:
1982:
1980:
1972:
1962:
1961:
1948:
1938:
1936:
1935:
1930:
1928:
1926:
1925:
1916:
1906:
1905:
1895:
1890:
1889:
1871:
1854:
1840:
1838:
1830:
1820:
1819:
1806:
1789:phenomenological
1727:
1725:
1724:
1719:
1717:
1716:
1704:
1703:
1691:
1690:
1671:
1669:
1668:
1663:
1661:
1660:
1645:is greater than
1644:
1642:
1641:
1636:
1634:
1633:
1617:
1615:
1614:
1609:
1607:
1606:
1591:
1590:
1522:
1520:
1519:
1514:
1512:
1511:
1493:
1491:
1490:
1478:
1473:
1471:
1470:
1446:
1441:
1439:
1438:
1426:
1421:
1418:
1413:
1401:
1341:
1339:
1338:
1333:
1331:
1330:
1234:
1232:
1231:
1226:
1223:
1222:
1221:
1220:
1211:
1186:
1185:
1161:
1160:
1134:
1132:
1131:
1126:
1124:
1123:
1071:
1069:
1068:
1063:
1061:
1060:
1044:
1042:
1041:
1036:
1034:
1033:
1012:
874:
872:
871:
866:
864:
860:
859:
858:
857:
856:
847:
817:
816:
815:
784:
783:
763:
761:
760:
755:
753:
752:
751:
717:
716:
675:
673:
672:
667:
665:
661:
660:
659:
658:
657:
648:
621:
620:
619:
588:
587:
567:
565:
564:
559:
542:
541:
510:
508:
507:
502:
500:
499:
498:
497:
488:
460:
459:
447:
446:
445:
420:
419:
418:
387:
386:
367:
365:
364:
359:
357:
356:
355:
289:or metal ions).
175:
173:
172:
167:
165:
164:
163:
161:
153:
145:
132:
130:
129:
120:
119:
110:
91:is given by the
76:
74:
73:
70:
67:
6049:
6048:
6044:
6043:
6042:
6040:
6039:
6038:
6019:
6018:
6017:
6012:
5984:
5934:
5923:
5896:Superconducting
5890:
5856:
5847:Neutral atom QC
5839:Ultracold atoms
5833:
5798:implementations
5797:
5791:
5731:
5724:
5691:Quantum circuit
5659:
5653:
5647:
5637:
5597:
5591:
5558:
5551:
5507:Hidden subgroup
5459:
5448:other protocols
5404:
5381:quantum network
5376:Quantum channel
5336:
5330:
5276:No-broadcasting
5266:GottesmanāKnill
5239:
5167:
5162:
5103:
5098:
5097:
5066:
5062:
5031:
5027:
5000:Physical Review
4996:
4992:
4987:
4983:
4971:
4967:
4955:
4948:
4935:
4931:
4908:
4904:
4887:
4883:
4874:
4872:
4864:
4863:
4859:
4814:
4810:
4779:
4775:
4752:
4748:
4741:
4727:
4723:
4716:
4700:
4696:
4691:
4634:
4626:
4620:
4594:
4590:
4565:
4561:
4552:
4548:
4542:
4538:
4532:
4527:
4522:
4515:
4510:
4505:
4503:
4501:
4497:
4496:
4490:
4486:
4478:
4475:
4474:
4454:
4450:
4445:
4436:
4432:
4424:
4421:
4420:
4398:
4381:
4358:
4355:
4354:
4330:
4326:
4301:
4297:
4278:
4276:
4272:
4271:
4245:
4242:
4241:
4236:
4232:
4228:
4224:
4198:
4179:
4175:
4166:
4162:
4140:
4130:
4126:
4113:
4111:
4099:
4085:
4081:
4074:
4070:
4069:
4062:
4052:
4048:
4035:
4033:
4021:
4017:
4004:
4000:
3988:
3984:
3977:
3975:
3974:
3970:
3969:
3960:
3956:
3954:
3951:
3950:
3928:
3909:
3905:
3896:
3892:
3870:
3860:
3856:
3843:
3841:
3829:
3825:
3816:
3812:
3796:
3786:
3782:
3775:
3773:
3772:
3768:
3762:
3758:
3751:
3747:
3746:
3737:
3733:
3731:
3728:
3727:
3724:(dimensionless)
3705:
3701:
3686:
3682:
3676:
3672:
3670:
3667:
3666:
3661:
3645:hydrogen nuclei
3621:
3617:
3615:
3612:
3611:
3590:
3586:
3584:
3581:
3580:
3575:, the value of
3567:O molecules in
3566:
3532:
3527:
3519:
3516:
3515:
3494:
3490:
3488:
3485:
3484:
3465:
3461:
3454:
3450:
3444:
3440:
3439:
3437:
3428:
3424:
3420:
3413:
3408:
3400:
3398:
3390:
3387:
3386:
3365:
3361:
3359:
3356:
3355:
3338:
3334:
3332:
3329:
3328:
3307:
3303:
3301:
3298:
3297:
3268:
3263:
3253:
3248:
3234:
3227:
3223:
3219:
3217:
3205:
3200:
3190:
3185:
3174:
3167:
3163:
3159:
3157:
3148:
3144:
3140:
3136:
3126:
3115:
3111:
3106:
3104:
3101:
3100:
3074:
3069:
3059:
3054:
3040:
3033:
3029:
3025:
3023:
3011:
3006:
2996:
2991:
2980:
2974:
2970:
2968:
2967:
2963:
2949:
2945:
2940:
2938:
2935:
2934:
2929:
2925:
2902:
2898:
2896:
2893:
2892:
2873:
2869:
2864:
2857:
2853:
2851:
2848:
2847:
2824:
2813:moment and the
2775:
2768:
2761:
2754:
2747:
2740:
2733:
2722:
2715:
2708:
2701:
2686:
2678:
2676:
2660:(To be listed)
2656:
2649:
2647:
2638:
2633:
2621:
2614:
2609:
2590:
2583:
2559:physiologically
2557:(MRS) studies,
2439:(deoxygenated)
2423:Adipose tissues
2416:
2406:
2386:
2366:
2360:
2353:
2346:
2332:
2323:
2308:
2299:
2288:
2275:
2244:
2241:
2240:
2218:
2214:
2207:
2203:
2185:
2181:
2180:
2178:
2169:
2165:
2151:
2134:
2115:
2099:
2095:
2091:
2089:
2087:
2084:
2083:
2063:
2059:
2043:
2039:
2038:
2036:
2027:
2023:
2009:
1992:
1973:
1957:
1953:
1949:
1947:
1945:
1942:
1941:
1921:
1917:
1901:
1897:
1896:
1894:
1885:
1881:
1867:
1850:
1831:
1815:
1811:
1807:
1805:
1803:
1800:
1799:
1786:
1779:
1772:
1763:
1754:
1740:
1738:Bloch equations
1734:
1732:Bloch equations
1712:
1708:
1699:
1695:
1686:
1682:
1677:
1674:
1673:
1656:
1652:
1650:
1647:
1646:
1629:
1625:
1623:
1620:
1619:
1602:
1598:
1586:
1582:
1580:
1577:
1576:
1573:
1570:
1563:
1552:
1548:
1544:
1540:
1533:
1507:
1503:
1486:
1482:
1477:
1454:
1450:
1445:
1434:
1430:
1425:
1414:
1409:
1400:
1398:
1395:
1394:
1389:
1385:
1360:
1354:
1351:
1326:
1322:
1320:
1317:
1316:
1309:
1296:
1289:
1272:
1253:
1244:
1216:
1212:
1207:
1200:
1196:
1178:
1174:
1153:
1149:
1147:
1144:
1143:
1119:
1115:
1113:
1110:
1109:
1107:
1100:
1093:
1082:
1056:
1052:
1050:
1047:
1046:
1029:
1025:
1023:
1020:
1019:
1017:
1002:
1000:
994:
992:
982:
971:
963:
956:
949:
934:
927:
913:
898:
883:
852:
848:
843:
836:
832:
822:
818:
808:
801:
797:
779:
775:
773:
770:
769:
744:
737:
733:
712:
708:
706:
703:
702:
700:
685:
653:
649:
644:
637:
633:
626:
622:
612:
605:
601:
583:
579:
577:
574:
573:
537:
533:
531:
528:
527:
493:
489:
484:
477:
473:
455:
451:
438:
431:
427:
411:
404:
400:
382:
378:
376:
373:
372:
348:
341:
337:
335:
332:
331:
328:
314:
306:
300:
298:
288:
283:
276:
269:
258:
247:
236:
229:
222:
214:
212:
205:
154:
146:
144:
140:
136:
125:
121:
115:
111:
109:
107:
104:
103:
89:
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71:
68:
65:
64:
62:
17:
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6013:
6011:
6010:
6000:
5989:
5986:
5985:
5983:
5982:
5980:many others...
5977:
5972:
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5939:
5937:
5929:
5928:
5925:
5924:
5922:
5921:
5916:
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5892:
5891:
5889:
5888:
5883:
5878:
5873:
5867:
5865:
5858:
5857:
5855:
5854:
5852:Trapped-ion QC
5849:
5843:
5841:
5835:
5834:
5832:
5831:
5826:
5821:
5816:
5810:
5808:
5806:Quantum optics
5799:
5793:
5792:
5790:
5789:
5784:
5783:
5782:
5775:
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5652:
5651:
5650:
5649:
5645:
5639:
5635:
5624:
5623:
5622:
5612:
5610:Quantum volume
5607:
5601:
5599:
5593:
5592:
5590:
5589:
5584:
5579:
5574:
5569:
5563:
5561:
5553:
5552:
5550:
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5539:
5534:
5529:
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5489:
5487:Boson sampling
5484:
5479:
5473:
5471:
5465:
5464:
5461:
5460:
5458:
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5452:
5451:
5450:
5445:
5440:
5430:
5425:
5420:
5414:
5412:
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5402:
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5396:
5395:
5385:
5384:
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5368:
5363:
5358:
5357:
5356:
5351:
5340:
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5332:
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5329:
5328:
5323:
5321:SolovayāKitaev
5318:
5313:
5308:
5303:
5298:
5293:
5288:
5283:
5278:
5273:
5268:
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5241:
5240:
5238:
5237:
5236:
5235:
5225:
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5197:
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5173:
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5160:
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5146:
5138:
5132:
5131:
5126:
5120:
5115:
5110:
5102:
5101:External links
5099:
5096:
5095:
5060:
5025:
5006:(7): 679ā712.
4990:
4981:
4965:
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4929:
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3468:
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3427:
3423:
3416:
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3407:
3403:
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3293:
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3271:
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3237:
3230:
3226:
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2990:
2986:
2983:
2977:
2973:
2966:
2962:
2959:
2952:
2948:
2944:
2927:
2923:
2918:is called the
2905:
2901:
2876:
2872:
2867:
2863:
2860:
2856:
2823:
2820:
2774:
2771:
2766:
2759:
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2630:
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2617:
2616:
2611:
2606:
2603:
2593:
2592:
2588:
2585:
2581:
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2575:
2563:pathologically
2546:
2545:
2542:
2539:
2533:
2532:
2529:
2526:
2520:
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2516:
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2489:
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2475:
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2432:
2431:
2428:
2425:
2419:
2418:
2414:
2411:
2404:
2401:
2385:
2382:
2362:Main article:
2359:
2356:
2351:
2344:
2328:
2321:
2304:
2297:
2284:
2271:
2248:
2237:
2236:
2221:
2217:
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2206:
2202:
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2196:
2193:
2188:
2184:
2177:
2172:
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2164:
2161:
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2154:
2150:
2147:
2144:
2141:
2137:
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2130:
2127:
2121:
2118:
2113:
2110:
2107:
2102:
2098:
2094:
2081:
2066:
2062:
2057:
2054:
2051:
2046:
2042:
2035:
2030:
2026:
2022:
2019:
2016:
2012:
2008:
2005:
2002:
1999:
1995:
1991:
1988:
1985:
1979:
1976:
1971:
1968:
1965:
1960:
1956:
1952:
1939:
1924:
1920:
1915:
1912:
1909:
1904:
1900:
1893:
1888:
1884:
1880:
1877:
1874:
1870:
1866:
1863:
1860:
1857:
1853:
1849:
1846:
1843:
1837:
1834:
1829:
1826:
1823:
1818:
1814:
1810:
1784:
1777:
1768:
1759:
1750:
1736:Main article:
1733:
1730:
1715:
1711:
1707:
1702:
1698:
1694:
1689:
1685:
1681:
1659:
1655:
1632:
1628:
1605:
1601:
1597:
1594:
1589:
1585:
1572:
1568:
1561:
1555:
1550:
1546:
1542:
1538:
1531:
1524:
1523:
1510:
1506:
1502:
1499:
1496:
1489:
1485:
1481:
1476:
1469:
1466:
1463:
1460:
1457:
1453:
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1437:
1433:
1429:
1424:
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1412:
1408:
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1387:
1383:
1353:
1349:
1344:
1329:
1325:
1307:
1294:
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1270:
1251:
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1236:
1235:
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1206:
1203:
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1195:
1192:
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1184:
1181:
1177:
1173:
1170:
1167:
1164:
1159:
1156:
1152:
1122:
1118:
1105:
1098:
1091:
1080:
1059:
1055:
1032:
1028:
1015:
996:Main article:
993:
990:
985:
980:
969:
962:
959:
954:
947:
932:
925:
911:
896:
881:
876:
875:
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851:
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839:
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831:
828:
825:
821:
814:
811:
807:
804:
800:
796:
793:
790:
787:
782:
778:
766:
765:
750:
747:
743:
740:
736:
732:
729:
726:
723:
720:
715:
711:
698:
683:
677:
676:
664:
656:
652:
647:
643:
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632:
629:
625:
618:
615:
611:
608:
604:
600:
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594:
591:
586:
582:
570:
569:
557:
554:
551:
548:
545:
540:
536:
512:
511:
496:
492:
487:
483:
480:
476:
472:
469:
466:
463:
458:
454:
450:
444:
441:
437:
434:
430:
426:
423:
417:
414:
410:
407:
403:
399:
396:
393:
390:
385:
381:
368:. In general,
354:
351:
347:
344:
340:
326:
317:decay constant
312:
302:Main article:
299:
296:
291:
286:
281:
274:
267:
256:
245:
234:
227:
220:
213:
210:
203:
198:
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176:
160:
157:
152:
149:
143:
139:
135:
128:
124:
118:
114:
87:
80:
41:radiofrequency
15:
9:
6:
4:
3:
2:
6046:
6035:
6032:
6030:
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6026:
6024:
6009:
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5991:
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5981:
5978:
5976:
5973:
5971:
5968:
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5963:
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5905:
5902:
5901:
5899:
5897:
5893:
5887:
5884:
5882:
5879:
5877:
5876:Spin qubit QC
5874:
5872:
5869:
5868:
5866:
5863:
5859:
5853:
5850:
5848:
5845:
5844:
5842:
5840:
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5830:
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5737:
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5727:
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5709:
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5705:
5704:
5701:
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5694:
5693:
5692:
5689:
5685:
5684:cluster state
5682:
5681:
5680:
5677:
5675:
5672:
5670:
5667:
5666:
5664:
5662:
5656:
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5638:
5634:
5630:
5629:
5628:
5625:
5621:
5618:
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5606:
5603:
5602:
5600:
5594:
5588:
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5580:
5578:
5575:
5573:
5570:
5568:
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5564:
5562:
5560:
5554:
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5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5492:DeutschāJozsa
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5474:
5472:
5470:
5466:
5456:
5453:
5449:
5446:
5444:
5441:
5439:
5436:
5435:
5434:
5431:
5429:
5428:Quantum money
5426:
5424:
5421:
5419:
5416:
5415:
5413:
5411:
5407:
5401:
5398:
5394:
5391:
5390:
5389:
5386:
5382:
5379:
5378:
5377:
5374:
5372:
5369:
5367:
5364:
5362:
5359:
5355:
5352:
5350:
5347:
5346:
5345:
5342:
5341:
5339:
5337:communication
5333:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
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5287:
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5282:
5279:
5277:
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5272:
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5264:
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5257:
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5248:
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5242:
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5231:
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5207:
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5202:
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5195:
5192:
5191:
5190:
5187:
5185:
5182:
5180:
5177:
5176:
5174:
5170:
5166:
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5145:
5140:
5139:
5136:
5130:
5127:
5124:
5121:
5119:
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5111:
5108:
5105:
5104:
5091:
5087:
5083:
5079:
5075:
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5064:
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5048:
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5040:
5036:
5029:
5021:
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5009:
5005:
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4985:
4978:
4977:Igaku Butsuri
4974:
4969:
4962:
4958:
4953:
4951:
4943:
4939:
4933:
4925:
4921:
4917:
4913:
4906:
4899:
4898:0-470-51117-6
4895:
4891:
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4871:
4867:
4861:
4853:
4849:
4844:
4839:
4835:
4831:
4827:
4823:
4822:Radiographics
4819:
4812:
4804:
4800:
4796:
4792:
4788:
4784:
4777:
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4761:
4757:
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4506:
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4487:
4483:
4480:
4473:
4472:
4455:
4451:
4446:
4440:
4437:
4433:
4429:
4426:
4403:
4399:
4395:
4392:
4389:
4382:
4375:
4372:
4369:
4363:
4360:
4353:
4352:
4335:
4331:
4327:
4322:
4317:
4311:
4308:
4305:
4302:
4298:
4291:
4288:
4285:
4282:
4273:
4268:
4265:
4259:
4256:
4253:
4247:
4240:
4239:
4238:
4202:
4199:
4194:
4189:
4180:
4170:
4167:
4163:
4159:
4156:
4150:
4147:
4144:
4141:
4134:
4131:
4127:
4123:
4120:
4117:
4114:
4108:
4100:
4095:
4089:
4086:
4082:
4078:
4075:
4071:
4066:
4063:
4056:
4053:
4049:
4045:
4042:
4039:
4036:
4030:
4025:
4022:
4018:
4014:
4011:
4008:
4005:
4001:
3995:
3989:
3985:
3981:
3978:
3971:
3966:
3961:
3957:
3949:
3932:
3929:
3924:
3919:
3910:
3900:
3897:
3893:
3889:
3886:
3880:
3877:
3874:
3871:
3864:
3861:
3857:
3853:
3850:
3847:
3844:
3838:
3830:
3820:
3817:
3813:
3809:
3806:
3800:
3797:
3790:
3787:
3783:
3779:
3776:
3769:
3763:
3759:
3755:
3752:
3748:
3743:
3738:
3734:
3726:
3709:
3706:
3702:
3698:
3695:
3692:
3687:
3683:
3677:
3673:
3665:
3664:
3663:
3658:
3654:
3650:
3646:
3642:
3625:
3622:
3618:
3609:
3591:
3587:
3578:
3574:
3570:
3561:
3559:
3555:
3536:
3533:
3529:
3524:
3521:
3513:
3495:
3491:
3466:
3462:
3455:
3451:
3445:
3441:
3429:
3425:
3421:
3414:
3409:
3405:
3401:
3395:
3392:
3384:
3366:
3362:
3339:
3335:
3326:
3308:
3304:
3278:
3269:
3264:
3260:
3254:
3249:
3245:
3241:
3238:
3235:
3228:
3224:
3220:
3214:
3206:
3201:
3197:
3191:
3186:
3182:
3178:
3175:
3168:
3164:
3160:
3154:
3149:
3145:
3141:
3137:
3131:
3128:
3123:
3116:
3112:
3108:
3099:
3084:
3075:
3070:
3066:
3060:
3055:
3051:
3047:
3044:
3041:
3034:
3030:
3026:
3020:
3012:
3007:
3003:
2997:
2992:
2988:
2984:
2981:
2975:
2971:
2964:
2960:
2957:
2950:
2946:
2942:
2933:
2932:
2931:
2921:
2903:
2899:
2874:
2870:
2865:
2861:
2858:
2854:
2844:
2842:
2837:
2833:
2829:
2819:
2816:
2812:
2807:
2805:
2801:
2797:
2793:
2789:
2784:
2781:
2770:
2765:
2758:
2751:
2744:
2737:
2730:
2726:
2719:
2712:
2705:
2698:
2694:
2690:
2683:
2673:
2662:
2659:
2654:
2652:
2643:
2642:
2636:
2631:
2628:
2625:
2619:
2618:
2612:
2607:
2604:
2602:
2598:
2597:Creatine (Cr)
2595:
2594:
2587:Approximate T
2586:
2580:Approximate T
2579:
2576:
2573:
2572:
2566:
2564:
2560:
2556:
2553:
2543:
2540:
2538:
2535:
2534:
2530:
2527:
2525:
2522:
2521:
2517:
2514:
2512:
2509:
2508:
2504:
2501:
2498:
2495:
2494:
2490:
2487:
2485:
2481:
2478:
2477:
2473:
2470:
2467:
2463:
2460:
2459:
2455:
2452:
2449:
2448:
2444:
2441:
2438:
2434:
2433:
2429:
2426:
2424:
2421:
2420:
2413:Approximate T
2412:
2410:
2403:Approximate T
2402:
2399:
2398:
2395:
2389:
2381:
2379:
2375:
2371:
2370:magnetization
2365:
2355:
2350:
2343:
2338:
2336:
2331:
2327:
2320:
2316:
2312:
2307:
2303:
2296:
2292:
2287:
2283:
2279:
2274:
2270:
2266:
2262:
2246:
2219:
2215:
2208:
2204:
2200:
2194:
2186:
2182:
2175:
2170:
2159:
2148:
2142:
2128:
2125:
2119:
2108:
2100:
2096:
2082:
2064:
2060:
2052:
2044:
2040:
2033:
2028:
2017:
2006:
2000:
1986:
1983:
1977:
1966:
1958:
1954:
1940:
1922:
1918:
1910:
1902:
1898:
1891:
1886:
1875:
1864:
1858:
1844:
1841:
1835:
1824:
1816:
1812:
1798:
1797:
1796:
1794:
1790:
1783:
1776:
1771:
1767:
1762:
1758:
1753:
1749:
1745:
1739:
1729:
1713:
1709:
1705:
1700:
1696:
1692:
1687:
1683:
1679:
1657:
1653:
1630:
1626:
1603:
1599:
1595:
1592:
1587:
1583:
1567:
1560:
1554:
1535:
1529:
1508:
1504:
1497:
1494:
1487:
1483:
1479:
1474:
1467:
1464:
1461:
1458:
1455:
1451:
1447:
1442:
1435:
1431:
1427:
1422:
1415:
1410:
1406:
1402:
1393:
1392:
1391:
1380:
1378:
1372:
1370:
1366:
1359:
1348:
1343:
1327:
1323:
1314:
1306:
1302:
1298:
1293:
1286:
1278:
1274:
1269:
1265:
1261:
1257:
1250:
1246:
1241:
1217:
1213:
1208:
1204:
1201:
1197:
1190:
1182:
1179:
1175:
1171:
1165:
1157:
1154:
1150:
1142:
1141:
1140:
1138:
1120:
1116:
1104:
1097:
1094:, designated
1090:
1086:
1079:
1057:
1053:
1030:
1026:
999:
989:
984:
979:
975:
968:
958:
953:
946:
942:
938:
931:
924:
919:
917:
910:
906:
902:
895:
891:
887:
880:
861:
853:
849:
844:
840:
837:
833:
829:
826:
823:
819:
805:
802:
798:
794:
788:
780:
776:
768:
767:
741:
738:
734:
730:
727:
721:
713:
709:
697:
693:
689:
688:
687:
682:
662:
654:
650:
645:
641:
638:
634:
630:
627:
623:
609:
606:
602:
598:
592:
584:
580:
572:
571:
555:
552:
546:
538:
534:
525:
521:
517:
516:
515:
494:
490:
485:
481:
478:
474:
464:
456:
452:
448:
435:
432:
428:
421:
408:
405:
401:
397:
391:
383:
379:
371:
370:
369:
345:
342:
338:
329:
322:
318:
311:
305:
295:
290:
280:
273:
266:
262:
255:
251:
244:
240:
233:
226:
219:
209:
202:
197:
195:
190:
186:
158:
155:
150:
141:
137:
133:
126:
122:
116:
112:
102:
101:
100:
99:
98:
97:
96:
94:
90:
83:
59:
57:
53:
49:
46:
42:
38:
34:
33:magnetization
30:
26:
22:
5904:Charge qubit
5829:KLM protocol
5778:
5642:
5632:
5626:
5326:Purification
5256:EastināKnill
5073:
5069:
5063:
5038:
5034:
5028:
5003:
4999:
4993:
4984:
4976:
4972:
4968:
4960:
4956:
4941:
4937:
4932:
4915:
4911:
4905:
4889:
4884:
4873:. Retrieved
4869:
4860:
4825:
4821:
4811:
4786:
4782:
4776:
4759:
4755:
4749:
4730:
4724:
4704:
4697:
4622:
4616:
4220:
3656:
3576:
3569:liquid phase
3562:
3295:
2845:
2836:Robert Pound
2825:
2808:
2785:
2776:
2763:
2756:
2749:
2742:
2735:
2728:
2717:
2710:
2709:. The field
2703:
2696:
2692:
2688:
2681:
2679:
2671:
2622:mainly from
2549:
2499:of cerebrum
2497:White matter
2471:4200 - 4500
2417:value in ms
2400:Tissue type
2387:
2367:
2348:
2341:
2339:
2329:
2325:
2318:
2314:
2310:
2305:
2301:
2294:
2290:
2285:
2281:
2277:
2272:
2268:
2264:
2260:
2238:
1781:
1774:
1769:
1765:
1760:
1756:
1751:
1747:
1743:
1741:
1574:
1565:
1558:
1536:
1525:
1381:
1379:experiment.
1373:
1361:
1346:
1304:
1299:
1291:
1284:
1283:
1267:
1263:
1248:
1247:
1239:
1237:
1136:
1102:
1095:
1088:
1084:
1077:
1075:
987:
977:
966:
964:
951:
944:
936:
929:
922:
920:
915:
908:
904:
900:
893:
878:
877:
695:
680:
678:
526:plane, then
523:
519:
513:
324:
320:
309:
307:
293:
278:
271:
264:
260:
253:
249:
242:
238:
231:
224:
217:
215:
207:
200:
188:
184:
182:
85:
78:
60:
51:
47:
45:spin-lattice
44:
18:
5935:programming
5914:Phase qubit
5819:Circuit QED
5291:No-deleting
5233:cloud-based
4673:Relaxometry
3608:picoseconds
2591:value (ms)
2584:value (ms)
2480:Gray matter
1793:Felix Bloch
1256:decoherence
6023:Categories
5975:libquantum
5909:Flux qubit
5814:Cavity QED
5763:BaconāShor
5753:stabilizer
5281:No-cloning
4875:2018-08-13
4789:(8): 978.
4740:019852014X
4689:References
3662:= 5Ć10 s:
3510:being the
2822:BPP theory
2811:quadrupole
2804:BPP theory
2474:2100-2300
1260:precession
48:relaxation
23:(MRI) and
5881:NV center
5316:Threshold
5296:No-hiding
5261:Gleason's
5090:0020-1669
5055:0020-1669
5020:0031-899X
4936:F Bloch,
4803:0021-9584
4540:ℏ
4524:γ
4507:γ
4393:−
4373:−
4318:ν
4257:−
4217:= 3.92 s,
4200:−
4168:−
4160:×
4151:⋅
4132:−
4124:×
4118:⋅
4087:−
4079:×
4054:−
4046:×
4040:⋅
4023:−
4015:×
4009:⋅
3982:×
3930:−
3898:−
3890:×
3881:⋅
3862:−
3854:×
3848:⋅
3818:−
3810:×
3788:−
3780:×
3756:×
3707:−
3699:×
3684:τ
3674:ω
3623:−
3588:τ
3573:oxygen-17
3537:π
3522:ℏ
3492:μ
3452:γ
3442:ℏ
3426:π
3406:μ
3363:τ
3305:ω
3261:τ
3246:ω
3225:τ
3198:τ
3183:ω
3165:τ
3146:τ
3067:τ
3052:ω
3031:τ
3004:τ
2989:ω
2972:τ
2900:τ
2871:τ
2859:−
2841:molecules
2826:In 1948,
2407:value in
2247:×
2201:−
2176:−
2149:×
2129:γ
2117:∂
2093:∂
2034:−
2007:×
1987:γ
1975:∂
1951:∂
1892:−
1865:×
1845:γ
1833:∂
1809:∂
1795:in 1946.
1593:≤
1501:Δ
1498:γ
1416:∗
1377:spin echo
1369:dephasing
1301:Hahn echo
1202:−
1121:⊥
838:−
827:−
731:−
639:−
631:−
479:−
449:−
422:−
148:Δ
142:−
127:−
61:For spin-
52:spin-spin
5943:OpenQASM
5919:Transmon
5796:Physical
5596:Quantum
5497:Grover's
5271:Holevo's
5244:Theorems
5194:timeline
5184:NISQ era
4852:19755604
4632:See also
4225:equals T
3947:= 3.92 s
3643:, while
3556:, Ī³ the
3385:motion.
3383:tumbling
2891:, where
2655:1.33 ppm
2648:group of
2629:2.0 ppm
2605:3.0 ppm
2541:860-900
2484:cerebrum
2427:240-250
1537:Unlike T
1530:, and ĪB
1297:values.
252:-axis).
5933:Quantum
5871:Kane QC
5730:Quantum
5658:Quantum
5587:PostBQP
5557:Quantum
5542:Simon's
5335:Quantum
5172:General
4843:2799958
4419:, with
3649:protons
3514:of the
3323:is the
2723:at the
2651:Lactate
2537:Muscles
2524:Kidneys
2374:dipolar
905:lattice
690:In the
315:is the
75:
63:
58:(FID).
5951:IBM QX
5947:Qiskit
5886:NMR QC
5864:-based
5768:Steane
5739:Codes
5537:Shor's
5443:SARG04
5251:Bell's
5088:
5053:
5018:
4896:
4850:
4840:
4801:
4737:
4712:
3296:where
2926:> T
2834:, and
2626:(NAA)
2531:60-75
2435:Whole
2430:60-80
2239:Where
1342:time.
943:, the
941:oxygen
37:Larmor
5773:Toric
5216:Qubit
5123:relax
5109:, RIT
2663:1040
2552:brain
2511:Liver
2466:water
2453:1350
2442:1350
2437:blood
2267:) = (
1108:, or
5965:Cirq
5956:Quil
5862:Spin
5758:Shor
5438:BB84
5371:LOCC
5086:ISSN
5051:ISSN
5016:ISSN
4894:ISBN
4848:PMID
4799:ISSN
4735:ISBN
4710:ISBN
4292:4.47
3979:1.02
3753:1.02
3552:the
2599:and
2528:650
2515:490
2502:780
2491:100
2488:920
2456:200
2347:and
1780:and
1706:>
1693:>
1045:and
206:and
5779:gnu
5743:CSS
5620:XEB
5582:QMA
5577:QIP
5572:EQP
5567:BQP
5547:VQE
5502:HHL
5306:PBR
5078:doi
5043:doi
5008:doi
4920:doi
4838:PMC
4830:doi
4791:doi
4764:doi
4390:0.5
4283:1.4
4157:3.2
4076:3.2
3887:3.2
3807:3.2
3696:3.2
3653:MHz
3647:H (
3422:160
2644:āCH
2561:or
2544:50
2518:40
2505:90
2482:of
2445:50
2293:),
2280:),
1746:= (
1557:Is
1541:, T
518:If
19:In
6025::
5970:Q#
5084:.
5074:32
5072:.
5049:.
5039:30
5037:.
5014:.
5004:73
5002:.
4949:^
4942:70
4914:.
4868:.
4846:.
4836:.
4826:29
4824:.
4820:.
4797:.
4787:74
4785:.
4758:.
4582:15
4488:10
4164:10
4135:12
4128:10
4083:10
4057:12
4050:10
4026:12
4019:10
3990:10
3986:10
3894:10
3865:12
3858:10
3814:10
3791:12
3784:10
3764:10
3760:10
3703:10
3626:12
3619:10
3610:=
3354:.
2830:,
2767:1Ļ
2675:1Ļ
2565:.
2468:)
2409:ms
2300:+
1764:,
1755:,
1371:.
1264:xy
1245:.
1137:xy
1101:,
1099:xy
983:.
918:.
686:.
524:xy
95::
5958:ā
5949:ā
5945:ā
5646:2
5643:T
5636:1
5633:T
5157:e
5150:t
5143:v
5092:.
5080::
5057:.
5045::
5022:.
5010::
4926:.
4922::
4916:3
4878:.
4854:.
4832::
4805:.
4793::
4770:.
4766::
4760:9
4743:.
4718:.
4625:M
4623:I
4619:1
4600:6
4596:/
4592:1
4587:)
4578:)
4575:1
4572:+
4567:M
4563:I
4559:(
4554:M
4550:I
4544:2
4534:2
4529:M
4517:2
4512:H
4499:(
4492:7
4484:=
4481:C
4456:1
4452:T
4447:/
4441:s
4438:1
4434:T
4430:=
4427:f
4407:)
4404:3
4400:/
4396:f
4387:(
4383:/
4379:)
4376:1
4370:f
4367:(
4364:=
4361:k
4336:6
4332:/
4328:1
4323:)
4312:n
4309:i
4306:m
4303:1
4299:T
4295:)
4289:+
4286:k
4280:(
4274:(
4269:C
4266:=
4263:)
4260:H
4254:M
4251:(
4248:r
4235:1
4231:1
4227:2
4223:1
4203:1
4195:)
4190:]
4181:2
4177:)
4171:5
4154:(
4148:4
4145:+
4142:1
4121:5
4115:2
4109:+
4101:2
4096:)
4090:5
4072:(
4067:+
4064:1
4043:5
4037:5
4031:+
4012:5
4006:3
4002:[
3996:2
3972:(
3967:=
3962:2
3958:T
3933:1
3925:)
3920:]
3911:2
3907:)
3901:5
3884:(
3878:4
3875:+
3872:1
3851:5
3845:4
3839:+
3831:2
3827:)
3821:5
3804:(
3801:+
3798:1
3777:5
3770:[
3749:(
3744:=
3739:1
3735:T
3710:5
3693:=
3688:c
3678:0
3660:c
3657:Ļ
3641:s
3592:c
3577:K
3565:2
3534:2
3530:h
3525:=
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3467:6
3463:r
3456:4
3446:2
3430:2
3415:2
3410:0
3402:3
3396:=
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3292:,
3279:]
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3124:=
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3085:]
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2965:[
2961:K
2958:=
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2947:T
2943:1
2928:2
2924:1
2904:c
2875:c
2866:/
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2315:B
2311:z
2306:z
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2289:(
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2278:t
2276:(
2273:x
2269:B
2265:t
2263:(
2261:B
2220:1
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2209:0
2205:M
2198:)
2195:t
2192:(
2187:z
2183:M
2171:z
2167:)
2163:)
2160:t
2157:(
2153:B
2146:)
2143:t
2140:(
2136:M
2132:(
2126:=
2120:t
2112:)
2109:t
2106:(
2101:z
2097:M
2065:2
2061:T
2056:)
2053:t
2050:(
2045:y
2041:M
2029:y
2025:)
2021:)
2018:t
2015:(
2011:B
2004:)
2001:t
1998:(
1994:M
1990:(
1984:=
1978:t
1970:)
1967:t
1964:(
1959:y
1955:M
1923:2
1919:T
1914:)
1911:t
1908:(
1903:x
1899:M
1887:x
1883:)
1879:)
1876:t
1873:(
1869:B
1862:)
1859:t
1856:(
1852:M
1848:(
1842:=
1836:t
1828:)
1825:t
1822:(
1817:x
1813:M
1785:2
1782:T
1778:1
1775:T
1770:z
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1600:T
1596:2
1588:2
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1571:?
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1551:2
1547:2
1543:2
1539:2
1532:0
1509:0
1505:B
1495:+
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1484:T
1480:1
1475:=
1468:m
1465:o
1462:h
1459:n
1456:i
1452:T
1448:1
1443:+
1436:2
1432:T
1428:1
1423:=
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1407:T
1403:1
1388:2
1384:2
1350:2
1347:T
1328:2
1324:T
1308:2
1305:T
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1271:2
1268:T
1252:2
1249:T
1243:2
1240:T
1218:2
1214:T
1209:/
1205:t
1198:e
1194:)
1191:0
1188:(
1183:y
1180:x
1176:M
1172:=
1169:)
1166:t
1163:(
1158:y
1155:x
1151:M
1117:M
1106:T
1103:M
1096:M
1092:0
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1054:T
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1016:0
991:2
988:T
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978:T
970:1
967:T
955:2
952:T
950:/
948:1
945:T
937:B
933:1
930:T
926:1
923:T
912:1
909:T
897:1
894:T
882:1
879:T
862:)
854:1
850:T
845:/
841:t
834:e
830:2
824:1
820:(
813:q
810:e
806:,
803:z
799:M
795:=
792:)
789:t
786:(
781:z
777:M
749:q
746:e
742:,
739:z
735:M
728:=
725:)
722:0
719:(
714:z
710:M
699:1
696:T
684:1
681:T
663:)
655:1
651:T
646:/
642:t
635:e
628:1
624:(
617:q
614:e
610:,
607:z
603:M
599:=
596:)
593:t
590:(
585:z
581:M
556:0
553:=
550:)
547:0
544:(
539:z
535:M
520:M
495:1
491:T
486:/
482:t
475:e
471:]
468:)
465:0
462:(
457:z
453:M
443:q
440:e
436:,
433:z
429:M
425:[
416:q
413:e
409:,
406:z
402:M
398:=
395:)
392:t
389:(
384:z
380:M
353:q
350:e
346:,
343:z
339:M
327:z
325:M
321:z
313:1
310:T
297:1
294:T
287:2
285:O
282:1
279:T
275:1
272:T
268:0
265:B
261:M
257:2
254:T
250:z
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239:M
235:1
232:T
228:2
225:T
221:1
218:T
211:2
208:T
204:1
201:T
189:T
185:k
159:T
156:k
151:E
138:e
134:=
123:N
117:+
113:N
88:+
86:N
81:ā
79:N
72:2
69:/
66:1
31:(
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