The fictitious spin-

operators were introduced by Vega and Pines to conveniently describe the interaction of a spin

*I*=1 nucleus with a Radio Frequency (RF) field [

10]. In this formalism, the basis consists of nine operators given by

From these definitions the various orders of multiple quantum coherence are as follows

In the following, we express the density operator by the above fictitious spin-

operators and assume an ensemble of spin

*I*=1 nuclei in a large, static magnetic field. In this situation the spin system is subject to the secular part of the first order quadrupolar interaction as well as a possible resonance offset. In the rotating frame the Hamiltonian is

with

where Δ is the resonance offset,

*e*^{2}*qQ/h* is the quadrupolar coupling constant,

*η* is the asymmetry parameter and

*θ* and

are the usual Euler angles with respect to the azimuthal axis [

11]. We will use the notation

*ω*_{Q}_{,0} to distinguish with

*ω*_{Q}which is half the observed splitting of a single crystal doublet when *θ*=0, or equal to the distance between the peaks in a Pake pattern when *η* = 0. Neglecting relaxation processes the time evolution of the density matrix is

with

where *H* is time independent. In what follows, the initial density matrix *ρ*(0) is taken as *ρ*(0)=*I*_{z}. The observable signal is determined by computing

where the operator for the receiver,

*R*_{±y}, is defined as

*R*_{±} ±

*I*_{y,1} *iI*_{x,1}.

Before highlighting the effects of various errors on the alignment echo three-pulse sequence, we will show that the phase cycling scheme given in (from reference [

3]) eliminates double quantum coherence under ideal experimental conditions. In ,

*ϕ*_{i} and

*ϕ*_{r} represent the phases of the

*i*^{th} pulse and receiver, respectively. In the discussion that follows,

*ρ*^{n}(

*t*) represents the density operator for the

*n*^{th} step of the phase cycle given in . For ideal experimental conditions we assume the experiment is performed on-resonance, thus Δ = 0 in

Eq.(3). The propagator for an RF pulse with flip angle

*β* and phase

*ϕ* according to

Eq.(7) is

| **Table 1**A four-step phase cycling scheme for suppressing double quantum coherence in quadrupolar alignment echo spectroscopy [3]. The phases *ϕ*_{1}, *ϕ*_{2}, *ϕ*_{3} and *ϕ*_{r} refer to the three pulses and receiver phase in the pulse sequence shown (more ...) |

where

*β* =

*π*/2 for the first pulse and

*β* =

*π*/4 for the second and last pulses. The factor of 2 in the spin rotation propagator comes from the definition of the fictitious spin-

operators. For the computations that follow we assume

for maximum conversion to quadrupolar order and will expand on this assumption in the proceeding section. By using

Eq.(3),

(6),

(7) and

(9) the various states of the spin system at

*t*=

*τ* +

*T*_{−} for each step of the four-step phase cycling are

In the above expression, *Q* = *D* = 1. The symbols *Q* and *D* have been inserted as coefficients to label the quadrupolar ordered term and double quantum coherence terms respectively. These labels show the presence and magnitude of different quantum coherence created after the second delay *T*, and will be used to track their evolution into detectable signals in the remaining steps of the pulse sequence. It is important to note that the pulse sequence even in the absence of artifacts creates double quantum coherence. Following the last pulse, the signals at 2*τ* + *T* for each step of the phase cycle are

In the above expressions the superscripts denote each step of the phase cycle in . The signal after phase cycling is thus

The result of this computation demonstrates that the four-step phase cycling scheme given in cancels double quantum coherence entirely and that the detected signal results only from the quadrupolar ordered state. In the four subsections below we describe the effect of various pulse errors as well as off-resonance effects. It will be shown that these errors may introduce extra terms, such as single quantum coherence and Zeeman order in the intermittent states. Without proper phase cycling, some of these terms may be transformed into an observable signal. However, with the four-step phase cycling scheme given in , to a good approximation, all terms other than quadrupolar order are suppressed in the detected signal. For the case of a flip error, finite pulse width effect and pulse transient we assume an on-resonance condition with Δ = 0 in

Eq.(3).

2.1. Flip error

For the case of a pulse flip error, where the flip angle of any given pulse deviates from a perfect rotation, we study the possible leakage of various multiple quanta in the detected signal. The propagator for the RF pulse with a flip error is modeled as

In the above expression *δ*_{β} is the flip error for a pulse with flip angle *β*. For the *π*/4 pulses the error *δ*_{45} is taken to be half of that of the *π*/2 pulse, *δ*_{90}. The phase of each pulse in the cycle, *ϕ*, is again listed in . Using this model and the procedure described above, the various states of the spin system at time *τ* + *T*_{−} for each step of the phase cycling are

where

In the above expressions *Q* denotes quadrupolar order, *D* denotes double quantum coherence, *Z* denotes Zeeman order, and *S*_{1}, *S*_{2}, *S*_{3}, *S*_{4} all denote single quantum coherence. Following the third pulse the signals detected at *t*=*T* + 2*τ* for each step of the phase cycling are

and the signal after phase cycling is

From the above expressions it is clear that Zeeman order (labeled by

*Z*) which appears at time

*t*=

*τ* +

*T*_{−} in

Eq.(14a) and

(14b) does not contribute to the phase cycled signal in

Eq. (17); the first two steps of the phase cycling are able to cancel single quantum coherence (labeled

*S*_{2} and

*S*_{4}). Together with the last two steps of the phase cycling scheme, double quantum coherence (labeled

*D*) is also canceled. The only term which is created and detected is the quadrupolar ordered term (labeled

*Q*). Thus, the computation shows that even in the presence of a pulse flip error there is no leakage of quantum coherence other than quadrupolar order after phase cycling. The final detected signal is real and is amplitude modulated by cos

^{3}(

*δ*_{90}). Note that when the error

*δ*_{90} = 0 we find that the results agree with the situation of no error, given in

Eq. (12). The signal intensity as a function of the

*π*/2 flip angle error,

*δ*_{90}, in the range of 0 ~ 10° is plotted in . The figure shows that the relative reduction of the signal intensity is less then 5% when the flip error is as large as 10°. The pulse flip error may be experimentally measured by a flip-flip sequence [

12].

2.2. Finite pulse width effects

In the situation of a rigid solid, the quadrupolar coupling constant for the deuteron can be as large as 200kHz and therefore the evolution under the quadrupolar interaction during the RF pulses cannot be ignored. The propagator for an RF pulse of width *t*_{p} thus includes the quadrupolar interaction,

In the following we assume that *ω*_{Q}t_{p} is a small quantity. This assumption is justified under the conditions where the *π*/2 pulse width *t*_{p} is on the order of 1 to 3 *μ*s, and the quadrupolar frequency *ω*_{Q}/2*π* is less than 20 kHz, so that *ω*_{Q}t_{p} ≤ 0.12*π* < *π*/2. We have performed a Taylor expansion of the solution to second order of this small quantity to simplify the complex algebraic expressions and give insight into the dynamics of the spin system in the presence of this error. Following the evolution under the sequence of pulses and delays shown in , one obtains the following density matrices before the last pulse for each step of the phase cycle

where

Again, the symbols *Q* and *D*_{1} indicate the magnitude of quadrupolar order and double quantum coherence respectively, *S*_{1}, *S*_{2}, *S*_{3} and *S*_{4} denote single quantum coherence and *Z*_{1} denotes Zeeman order. In addition, the term *D*_{2} is a second double quantum coherence term created due to finite pulse width effects.

Upon implementing the four-step phase cycling scheme again given in , the detected signal at *t* = 2*τ* + *T* is given by

The result shows that when finite pulse widths are taken into account the detected signal contains Zeeman order as well as quadrupolar order. However, the Zeeman order contribution to the signal intensity is cubic in *t*_{p} and can be ignored to second order. The quadrupolar order contribution to the signal intensity decreases as the pulse width increases, and is plotted in as a function of the pulse width *t*_{p} for several values of the quadrupolar frequency *ω*_{Q}/2*π* from 2kHz to 20kHz. The figure highlights that the relative reduction of the signal intensity is less than 5% for the largest value of *ω*_{Q}t_{p} we considered *ω*_{Q}t_{p}/2*π*=0.06.

2.3. Anti-symmetric Transients

In this section we model the effects of RF transients. In a previous work [

13], we adopted a model after A. J. Vega [

14], where any RF pulse of our pulse sequence includes a transient effect. The model consists of an orthogonal RF field that is applied to the spin system before and after the main pulse. The propagator for an antisymmetric pulse transient is given by

where the first and last propagators represent a pulse transient that rotate the ensemble by an angle *α*_{t} and have phases orthogonal to that of the main pulse. For example, if the main pulse is about the *x*-axis then the transient is about the + *y*-axis before and - *y*-axis after the pulse. We performed a computation that accounts for pulse transients on all pulses in the spin alignment sequence, assuming that the transient flip angle *α*_{t} is the same for all pulses. Because *α*_{t} will typically be small relative to the main pulse duration we have taken a Taylor expansion to second order in our density matrix calculation. The states of the spin system before the last pulse for each step of the phase cycling are

where

The notation again for the various symbols *Q*, *D*_{1}, *S*_{1}, *S*_{2}, *S*_{3}, *S*_{4} and *Z*_{1}, *D*_{2} are the same as those in the previous section on finite pulse errors. The computation shows the creation of single quantum coherence, double quantum coherence and Zeeman order in addition to the quadrupolar order terms resulting from RF transients. After implementing the four-step phase cycling one obtains the following expression for the signal at *t*=2*τ* + *T*

The result shows that the detected signal in the presence of RF transients arises only from the quadrupolar ordered state. However, the detected signal contains both real and imaginary components whose individual amplitudes depend on the transient flip angle

*α*_{t}. The real part and the magnitude of the complex signal as functions of

*α*_{t} in the range of 0 to 30° are plotted in . When the flip angle of the RF transient is as large as 30° the signal intensity on the real channel reduces to approximately 70%, and the magnitude of the complex signal reduces to approximately 75% of the signal intensity under ideal conditions. Experimentally RF transients may be characterized by a flip-flop sequence by measuring the modulation frequency induced by the transient [

12], [

15]. From average Hamiltonian theory, the first order term of Magnus expansion for our transient model in the flip-flop sequence is

where

*τ*_{c} is cycle time of the multiple pulse sequence and the modulation induced by the RF transient in a stroboscopic measurement is given by [

13]

The relation between *J*_{1} and the transient flip angle *α*_{t} in our model is

2.4 B_{z} offset

Lastly we consider the case when the internal Hamiltonian has a nonzero resonance offset, i.e. Δ ≠ 0 in

Eq.(3). We assume the propagators for the RF pulses are the same as that for the ideal condition given in

Eq.(9). The density matrices at

*t* =

*τ* +

*T*_{−} for the various steps in the phase cycling are

where

With the four-step phase cycling the final detected signal is

The above result shows that in the presence of a resonance offset that the detected signal still only arises from quadrupolar order with no additional quantum coherence leaked into the detected signal. There are two observed effects that the resonance offset error has on the signal; there is a phase and amplitude modulation of the signal that is proportional to Δ/*ω*_{Q}. Because the resonance offset is typically much smaller than the quadrupolar frequency, the effect on the detected signal is negligible. highlights the variation in the amplitude of the signal as a function of the resonance offset for four different values of quadrupolar frequencies, *ω*_{Q}/2*π*, from 2kHz to 20kHz. The results show the reduction in signal in the alignment echo amplitude in the presence of an off-resonance effect as large as 40 Hz is negligible for quadrupolar frequencies *ω*_{Q}/2*π*=20 kHz.