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The results of experiments in quantum mechanics can be predicted correctly either by assigning a forward-evolving state to the system based on the preparation outcome or by assigning a state that evolves backwards in time based on the measurement outcome. The latter picture admits some retrocausality without allowing messages to be sent at a faster speed than that of light. This retrocausality allows some standard quantum paradoxes to be examined from a different viewpoint. It also allows closed causal cycles to be examined in the context of laboratory experiments. For a particular experiment, we find agreement with the principle that inconsistent causal loops have zero probability of occurring, that is, only self-consistent loops can occur.

We show explicitly how the causal arrow of time that follows from quantum mechanics has already been inserted at a deeper level by the choice of normalisation conditions. This prohibits information being sent backwards in time but does not determine a time direction for state propagation.

We use the impossibility of sending information faster than light to show that it is impossible to distinguish between various types of laser cavity field state, such as a Fock state of uncertain energy and any superposition of Fock states of unknown free evolution time, by means of the physically measurable properties of the laser beam. Superpositions can include coherent states, squeezed states and cat states, indeed any pure intracavity state will produce the characteristic properties of the beam such as intrinsic phase coherence.

We provide an expression for the state of a single field mode, conditioned only on the outcome of an imperfect photon-counting measurement. This suggests that we can view such a measurement as projection onto one of a non-orthogonal set of mixed states. We briefly discuss some of the implications of this result for state preparation.

The applicability is examined of the measured-phase operator describing a simple homodyne measurement scheme to states of the quantum optical field which have the minimum possible phase fluctuations. Such states necessarily have huge intensity fluctuations, so it is expected that this operator should be inapplicable because intensity fluctuations are deliberately ignored in its derivation. However, the surprising result found is that this operator can still provide a reasonably good approximation to the phase properties of fields even in this extreme limit.

Ban [Opt. Commun. 143 (1977) 225] demonstrated an equivalence between a lossless beam splitter with one of the output modes in the vacuum state and a nondegenerate parametric amplifier with a vacuum input. We show that this equivalence can be extended to apply to all states.

We show that the physically absurd state obtained by Vorontsov and Rembovsky [Phys. Lett. A 254 (1999) 7] results from an unphysical choice of approximate phase measurement and not from the phase formalism itself as claimed. Thus their assertion that it is impossible to perform even an approximate measurement of phase is unfounded.

We find the states of light which have minimum phase variance both for a given maximum energy state component and for a given mean energy. When these states contain sufficiently many photon number state components, the number state coefficients approximate sinusoidal and Airy functions respectively. The phase of these new states of light is much more sharply defined than is the phase of a coherent state with the same mean energy.

A question recently raised concerning the possibility of quantitatively determining the timescale of wavefunction collapse is discussed in the context of existing experiments on single atoms.

Established depth pulse sequences, consisting of consecutive phase-cycled pulses, are compared to the same sequences in which the simple phase-cycled pulses are replaced by new broadband composite phase-cycled pulses introduced by A. J. Shaka and R. Freeman ( J. Magn. Reson. 64 , 145 (1985)) to extend the range of usefulness of these methods off resonance. It is found that the composite pulses extend the range by up to about a factor of three, and that their use is warranted if available pulse power does not cover the required spectral width for simple depth pulses.

There are 24 possible polarization transfer pulse sequences having two or three precessional periods in which the effects of heteronuclear coupling and chemical shift are separated by refocusing pulses. Six of the most useful of these pulse sequences are described in detail using the Heisenberg vector approach. The vector description enables a clear analysis of the outcome of each sequence and of the modifications which are necessary to optimize each sequence. Each sequence is analyzed for the production of artifacts and a simple general method for the phase correction of the two-dimensional spectra is described.

Proton spin-lattice ( T 1 ) and spin-spin ( T 2 ) relaxation times have been measured for CH 3 protons in a series of paramagnetic transition-metal acetylacetonate complexes and the results interpreted in terms of current relaxation theory, τ r , the correlation time for molecular reorientation, was estimated from the 13 C T 1 in the analogous diamagnetic Co(III) and Pd(II) complexes. Using this approach and treating in detail the effects of fast CH 3 group internal motion good agreement is obtained between theory and experiment. In all cases electron-nuclear dipolar coupling dominates T 1 whereas the hyperfine contribution can be important for T 2 .

In terms of the Heisenberg vector model, heteronuclei undergo a correlated motion via scalar coupling when both have their spins simultaneously in the transverse plane of the doubly rotating reference frame. Several pulse sequences are described in terms of 1 H/ 13 C systems. A theoretical analysis of the correlated motions is given for methine (CH), methylene (CH 2 ), and methyl (CH 3 ) groups for the situation when the 'H magnetization vectors are initially placed in the transverse plane of the doubly rotating frame and are allowed to precess freely for (2 J ) −1 sec, where J is the single-bond 1 H- 13 C coupling constant, prior to the carbon polarization transfer vectors being placed in the transverse plane. The analysis is confirmed experimentally using two-dimensional NMR and the usefulness of the sequences for two-dimensional NMR is assessed. The family of sequences is completed with a theoretical study of sequences which employ polarization transfer in addition to the correlated motion. Experimental verification for the one-dimensional use of these sequences, the exclusive polarization transfer sequence (EPT), in editing 13 C spectra is given. Reverse EPT, the inverse of EPT, is also described and is shown to be useful for editing 1 H spectra. While it is pointed out that we expect there to be a correspondence between correlated motion and Schroedinger picture multiple-quantum coherence, the details of this correspondence are not explored.

Demonstration du transfert de polarisation du deuterium D vers 13 C dans des systemes 13 CD et 13 CD 3 en utilisant les sequences UPT. Accord avec la theorie

Polarization transfer enhancement of the NMR resonance from a nucleus with spin quantum number l (l = 1 2 , 1, 3 2 , …) from n scalar coupled protons using the pulse sequence (90 x H)-τ-(180 x H)(180 x I)-τ-(90 y H)(90 x I)-Δ-(acquire with or without H decoupling) is studied theoretically. It is shown that the enhancement of the components of the I multiplet varies with τ as Σ m I m I sin (4 πJm I τ ) where J is the scalar coupling constant and m I takes the values l, l - l, …, -l . Values of τ opt which result in the maximum enhancement are calculated, as well as the size of these maximum enhancements. Values of Δ opt which yield the maximum decoupled signal are also computed, along with the maximum decoupled enhancements. The theory is confirmed experimentally for l = 3 2 and l = 3 by studying proton polarization transfer spectra of the 10 B and 11 B resonances in NaBH 4 .

A sequence comprising two 13 C pulses and two proton pulses is justified theoretically and experimentally as providing a new form of 13 C- 1 H chemical shift correlation spectroscopy. The sequence is superior to polarization transfer sequences as it contains fewer pulses and distinguishes between carbon groups with different numbers of bonded protons.

Recently, it has been shown that the DEPT sequence may be used to provide separate methyl, methylene, and methine subspectra and preliminary results have indicated it is quite an accurate sequence for this purpose. Here, the errors involved in using DEPT over the normal range of aliphatic and aromatic 1J13C-1H, values are assessed by analyzing DEPT with variable time delay periods using the Heisenberg quantum picture approach. The limitations of the sequence when used across the whole range of organic compounds is discussed. An experimental approach to setting up and using the sequence is described. The possibility of improving the sequence with additional phase-alternation schemes or composite pulses, for example, is explored. The editing of a 13C spectrum is completed by generation of a spectrum containing unprotonated (quaternary) carbons only. The theoretical basis of the pulse sequence used for this purpose is described and errors are assessed.

The “distortionless enhancement by polarization transfer sequence” (DEPT) is a heteronuclear multipulse sequence which provides for polarization transfer from n spin - 1 2 nuclei to a spin - 1 2 heteronucleus subsequent to a period of simultaneous free precession (correlated motion) of both types of nuclei in the transverse plane of the doubly rotating reference frame. A new multipulse sequence is described here which involves polarization transfer in the reverse direction, that is, from one spin - 1 2 heteronucleus to n spin - 1 2 heteronuclei again subsequent to a period of correlated motion. This inverse DEPT sequence has similar advantages to the DEPT sequence, over INEPT and related s sequences, in that it contains fewer pulses and the acquired signal is relatively insensitive to missetting the length of the free precession periods. The sequence is proved both theoretically and experimentally.

We report the observation of transient optical nutations in the D 2 line of sodium vapour subject to an intense monochromatic laser field. The transient signal was obtained using a coherent optical transient technique which involved the sudden application of an electric field pulse to the vapour, a technique used previously only for molecular transitions. The results obtained are in reasonable agreement with theoretical predictions of the transient behaviour of atoms subject to intense optical excitation, using different longitudinal and transverse relaxation times.

A five-pulse sequence is studied theoretically, with experimental support, which can be used for the controllable transfer of nuclear magnetization from a system S ( l 2 ) n of the n nuclei of spin l 2 , to a different system I ( l 1 ) m . of m nuclei of spin l 1 , when the scalar coupling between the two systems can be characterized by a single constant. It is found that the polarization transfer signal is governed by a remarkably simple formula of universal validity for all values of l 1 , l 2 , n , and m , which is applicable for all initial conditions. Consequently, the sequence is called a universal polarization transfer (UPT) sequence. It is easily seen that three previous sequences, EPT, DEPT, and inverse-DEPT, which have been shown to be of considerable value for editing spectra, are merely special cases of the UPT sequence. Like these sequences, the UPT sequence should also prove to be a valuable tool for spectral editing, but with a much improved versatility.

The theory, with supporting experimental evidence, of possible two-dimensional extensions to the DEPT polarization transfer sequence of NMR is given. Some of the extensions give results similar to the corresponding two-dimensional INEPT sequences, but others exhibit the correlated motion, or multiple-quantum transition, nature of the DEPT sequence. The DEPT two-dimensional sequences have the advantages that they contain a minimal number of pulses, thus being less susceptible to artifact production, and that the mathematical expressions for the signals contain the θ dependence as a separable factor, where θ is the variable DEPT pulse angle. This latter property enables the generation of accurate edited two-dimensional spectra by suitable settings of θ just as for one-dimensional DEPT spectra.

The conventional hyperfine energy states associated with the D transitions of atomic sodium are transformed to a new basis where the response to coherent light is more readily interpreted. Relaxation processes are not considered. For each of the transitions, it is found that for excitation by plane-polarised light the atomic dynamics can be considered in terms of an effective Rabi precession similar to that for a simple two-state system. For circularly polarised light, the states involved in the transitions can be expressed in terms of isolated subsystems of two, three and four states only. This represents a considerable simplification over the multiple-state systems in the conventional basis.

The simple Heisenberg vector model, though useful for numerous nmr pulse sequences, was found inadequate to describe the DEPT sequence, which was subsequently analyzed by an unwieldy Schroedinger picture approach. Because this latter approach is not amenable to treating either deviations from, or applications of, the ideal DEPT sequence which arise in practical cases, a Heisenberg picture analysis is presented which is more mathematically formal than the simple vector model. In terms of efficiency, flexibility, and generality this represents a vast improvement on the previous Schroedinger picture approach which was restricted to treating CH, CH2, and CH3 groups individually for specific initial conditions. The new approach is applicable to a general ISn group, where the spin of I is one-half but that of S is arbitrary, for a variety of initial conditions. More importantly it is efficient enough to form the basis of analyses of the more complicated DEPT-type sequences which will be used for practical applications.

Transfer of polarization from deuterium to carbon using the universal polarization transfer (UPT) sequence, in conjunction with gated proton decoupling, enables the editing of a 13 C spectrum of deuterated carbon groups into separate CD, CDH, CDH 2 , CD 2 H, CD 2 , and CD 3 subspectra. The method has some advantage over rival techniques.

We study theoretically, and confirm experimentally for specific cases, the forms of the coupled spectra arising from polarization transfer from and to systems of multiple nuclei of higher spins by use of the UPT and INEPT pulse sequences. We find that we cannot obtain a simple separable universal formula for the polarization transfer signal for the INEPT sequence, as we have done for both decoupled and coupled signals obtained by the UPT sequence. Remarkably, however, we find that, for the more common cases where the polarization is transferred from multiple spin-one and spin-half nuclei, the universal polarization transfer formula that we obtained for the decoupled UPT signals can be used to calculate directly both decoupled and coupled INEPT signals.

A recently observed triple-peaked intensity profile in the transmission of a Fabry-Perot etalon containing sodium vapour has hitherto been interpreted in terms of a three-state lambda configuration model atom. Here we show that for π polarized laser excitation the simplest appropriate sodium atom model is a four-state system. This also predicts a triple-peaked profile, which differs from the three-state model prediction mainly by the absence of non-absorptive resonance.

Simple vector models have been considered as useful heuristic descriptions of the behavior of magnetization vectors of nuclei under the influence of heteronuclear scalar coupling during some straightforward multipulse sequences. However, it is generally held in the literature that the rigorous quantum-mechanical evolution should be treated by a density matrix approach, particularly where neither of the nuclei involved is in a pure z -spin eigenstate. In this paper, based on several selected pulsed NMR experiments, we develop and extend the simple vector model to deal with more complicated multipulse situations. We show that the vector model not only provides a good simple physical insight into these selected problems but is also a rigorous description of the quantum-mechanical evolution. As such, it will prove a viable and useful simple alternative to the density matrix approach for the understanding and design of new specific pulse sequences.

A family of consecutive phase-cycled pulses, known as depth pulse sequences, has previously been proposed for spatial localization of the sensitive volume in in vivo NMR spectroscopy by utilizing the inhomogeneity of rf fields. More recently, composite pulses have also been suggested for this purpose. Comparison of the two rival techniques reveals that the present composite pulses show poor behavior off resonance relative to depth pulses and have no other significant advantages over depth pulses. As yet unknown 90° composite pulses might have an advantage if no phase cycling was required for their implementation.

Recently introduced unitary and hermitian phase operators are used to examine the phase properties of squeezed states of light with particular reference to the squeezed vacuum. The results differ markedly from previous calculations involving the Susskind and Glogower operators. The new formalism allows the construction of a phase probability density which, on a polar diagram, is a circle for the vacuum state, becomes elliptical with gentle squeezing and collapses to a line through the origin for full squeezing. This probability density together with the calculation of expectation values of various trigonometrical functions of phase show how squeezing impresses phase information onto the vacuum.

A form of J cross-polarization in liquids can be readily performed with a standard cross-coil probe; the result is that the signal intensity is only moderately sensitive to the Hartmann-Hahn condition and off-set in the carrier frequencies of the two rf fields. The cross-polarization process is a quasi-stochastic process with the enhancement of 13C in CH and CH2 groups being γH/2γC. For a CH3 group the enhancement is 5γH/8γC. There is a redistribution of the normal line intensities in the CH, CH2, and CH3 multiplets on cross-polarization which can be altered by the addition of a (π/2)[H] pulse at the end of the pulse sequence. Theoretically predicted results for coupled spectra are confirmed experimentally. From the theory, a vector model of this J cross-polarization process is developed, and its immediate consequences are checked by some test pulse sequences.

Inhomogeneous noise decoupling may be used in short bursts to leave the nuclear spins of the irradiated nuclei randomized in space. It is shown that the n protons on each individual 13C atom experience the same decoupling field and thus nutate through the same net angle even though this field and angle vary randomly across the sample due to inhomogeneity in the rf field. Sequences employing decoupling bursts may be used to cancel 13C signals arising from methine (CH) and methyl (CH3) groups, leaving signals from uaternary carbons at full intensity and signals from methylene (CH2) groups at a precise reduced intensity in the coupled or decoupled spectrum. The mechanisms of these sequences are described.

Two known spin randomization techniques, which might be used for separating a methylene (CH 2 )/quatemary (q) 13 C subspectrum into separate CH 2 and q subspectra, are compared. Both techniques utilise the heteronuclear scalar coupling interaction between the carbon and directly bonded proton spins. However, the first technique involves randomization of the 13 C spins in the transverse plane of the rotating reference frame as a result of off resonance 1 H decoupling, whereas the second technique involves randomization of the 1 H spins in three dimensions by a powerful noise-modulated pulse. The latter method is superior in that it is much less sensitive to the nonequivalence of methylen protons.

We study the general conditions under which a three-level atom can exhibit population trapping and non-absorption resonance while interacting with two laser beams which have simultaneous phase and amplitude variations. We find there are three separate conditions necessary for the effect and, further, when a particular correlation exists between phase and amplitude variations, the atomic time-displacement operator without damping can be expressed exactly as a finite series, allowing the dynamics of the atom in two-photon resonance with the phase and amplitude modulated light to be easily determined.

In order to justify the mathematical postulate of a distribution of correlation times, the concept of time-dependent correlation times is introduced. The time dependence arises because molecules may change shape during nuclear relaxation. The model may be used to explain qualitatively the field dependence of the temperature dependence of 13 C T 1 values in isopropyl alcohol.

A correspondence is derived between the extended INEPT sequence for polarization transfer from n equivalent protons to a spin-half nucleus I and the DEPT sequence. This allows a calculation of the optimum variable pulse angle and enhancement value for the coupled and decoupled DEPT spectra of the group IHn, and enables us to show that the coupled DEPT spectra are distortion-free for all values of n. These results are checked experimentally for specific values of n. The correspondence relies on a trigonometrical operator relation which does not hold for spin I greater than a half, making the DEPT and INEPT sequences quite distinct for higher spins.

A 13 C spin-echo sequence with a single variable θ proton pulse coincident with the carbon refocusing pulse has been proposed which provides complete editing of a 13 C spectrum into separate quaternary, methine, methylene, and methyl subspectra. This sequence, named SEMUT, has been found to be as accurate as the DEPT sequence with regard to errors in the subspectra resulting from divergence of the single-bond 13 C 1 H coupling constant from the value assumed in setting the DEPT and SEMUT delay periods. In this paper, however, it is found that the similarity in the signal dependence of methine and methyl groups as a function of the θ pulse, and the additional complexity of the linear combinations of SEMUT spectra required for editing, leads to higher noise levels in the final subspectra for the SEMUT method as compared to the DEPT/q-only method. It is also shown that the basic SEMUT method is more sensitive to inhomogeneity and missetting of the θ pulse. Such inaccuracies, combined with the more complicated linear combinations, will make basic SEMUT more difficult to use. The problem of sensitivity to θ pulse missetting and inhomogeneity can be alleviated by a modification to the basic SEMUT sequence which unfortunately also removes its advantage as a minimal pulse number sequence not requiring phase control, but which gives very good insensitivity to moderate inhomogeneity. It is also found that other recently proposed editing methods have little, if any, more to offer than DEPT.

It is known that cross-correlation effects can induce nonexponential proton magnetization decay in some isolated methyl groups on large molecules in liquids. Here it is shown theoretically and confirmed experimentally that these cross correlations can also give rise to a more instantly recognizable effect, a negative methyl 13C signal in an otherwise accurate methine carbon subspectrum obtained from an editing polarization transfer sequence. Although this technique offers a new method for detecting directly and studying cross-correlation spectral densities, and thus the underlying molecular dynamics, this paper is concerned mainly with its potential for the rapid identification of isolated methyl groups in edited spectra of medium to large molecules.

The DEPT sequence (I) is readily extended to two-dimensional (2D) heteronuclear NMR applications, a number of which we describe briefly. In more detail, we show that it can be extended to ‘H-13C chemical shift correlation spectroscopy, by including appropriate incremented delay periods, but without additional pulses. The method has several valuable advantages over the normal form (2) of ‘H-13C chemical shift correlation spectroscopy. First, provided phase-sensitive 2D software is available, separate CH3, CH2, and CH 2D subspectra, i.e., 2D spectrum editing, may be achieved. Second, phase errors are fully refocused using the DEPT sequence and thus also the 2D DEPT sequence. This enables phase correction in both dimensions, thus improving lineshapes by eliminating the need to do a magnitude calculation as is normally done. Third, the normal form of heteronuclear chemical shift correlation spectroscopy may also be improved by the addition of refocusing pulses so that phase errors are refocused. The resulting 7 or 8 pulse sequences (sequences [C] and [B] of Ref. (3)) may be considered to be 2D extensions of the INEPT (4) sequence. Thus it is also possible to achieve 2D spectrum editing with these methods in the same way as one-dimensional editing using INEPT (3). But the 2D DEPT sequence described here contains fewer pulses and is also more accurate for spectrum editing, just as DEPT is more accurate than INEPT for one-dimensional editing (1).

Modification of the INEPT sequence enables application of the sequence to two-dimensional NMR and provides a simplified form of two-dimensional 13C J spectroscopy. The problem of phase correcting the final spectrum has been solved and the source of artifact signals has been identified and the problem alleviated. The uses and advantages of this form of two-dimensional spectroscopy are briefly described.

The absorptive and dispersive effects of three-level atoms moving in the optical standing wave field of a Fabry-Perot etalon are studied. When the frequency of modulation experienced by the moving atom is of the splitting of the lower two levels, an intensity-dependent non-absorption resonance can occur. Associated dispersive effects can account for a recently-observed triple-peaked intensity profile.

In studying the amplification of squeezed light by a linear atomic light amplifier, various authors have found that the output light will not be squeezed for intensity gains greater than a limiting value of 2. In this paper we examine the general conditions under which this limiting value is applicable and find that for the atomic light amplifier it is associated with the vanishing of the initial average coherence of the atoms. A multi-stage atomic light amplifier is discussed, in which the atoms are in specially prepared initial states, which allows in principle a squeezed output for any value of the gain.

We use the phase states to analyze phase evolution in optical amplifiers and attenuators. The resulting diffusion of the phase probability distribution is compared with that obtained in conventional Fokker-Planck treatments.