Spin echo decays

The two pulse spin echo decay in Equation 3.10 is based on the assumption that the lifetime of the excited state is much longer than the coherence time. In this case, the T1

process which tends to degrade an individual Bloch vector is considered to occur long after the ensemble loses coherence. This assumption holds true for the hyperfine transitions in Eu3+:Y2SiO5, the system investigated in this thesis. The coherence time at cryogenic

temperature of a diluted sample is at the order of 10 ms, which is much shorter than the lifetime of 23 days [17].

For hyperfine transitions in Eu3+:Y2SiO5, the dephasing of Eu3+ Bloch vectors are

due to the fluctuating magnetic field caused by the random spin flips of host Y3+ spins. In order to work out the decay function of the echo amplitude in Equation3.10, this can be treated by considering the problem to be about the echo decay of the spins of interest , orAspins, in the presence of a random magnetic field due to an array of spin-1/2B spins. In this model, the concentration of theAspins is assumed to be sufficiently low thatA-A interactions can be ignored. Such a problem of the echo decay of the A system in a bath ofB spins was studied by Mims et al. [85,85,97]. The rest of this section provides a brief review of these studies.

I start the discussion from Equation3.10. For a spin echo, the rephasing for different subgroups will be the same. The echo amplitude decay shape is then given by the phase factor of theAspins due to the phase accumulation over the total evolution timeτ[85,85]. The effect of the B spin bath can be understood by considering how it affects a single subgroup. For the j-th subgroup, the phase accumulation is written

ϕj(τ) =

Z τ 0

s(t)∆ωj(t)dt. (3.12)

The phase accumulation is due to the dynamic frequency shift caused by the magnetic dipolar interaction of the A spin with the array ofB spins. The dynamic frequency shift

§3.1 Decoherence of two-level spin ensemble systems 47

∆ωj(t) can, therefore, be written ∆ωj(t) =γAX

k

[mk(t)−mk(0)](1−3 cos2θk)/|rk|3, (3.13)

where γA is the gyromagnetic ratio of theA spin1,rk is the distance between the A spin and the k-th B spin and θk is the angle between rk and the z axis of the Bloch vector.

The sum is taken oven all B spins in the lattice. mk(t) is the time-varying z component

of the k-thB-spin magnetic moment, i.e.,

mk(t) =γBSzk(t), (3.14)

where γB is the gyromagnetic ratio andSzk is the nuclear spin ofB spins. AsB ions are

a spin-1/2 system,

|Sk_z|= 1/2. (3.15)

The effect of the dynamics of the B spin bath on an A spin is due to the various B spin lattice configurations and the time-varing magnetic momentmk(t)for eachB spin. From

Equation 3.10 and Equation 3.12, the echo amplitude of the ensemble A spin for a given total evolution timeτ is then written [85]

E(τ) = ** expi Z τ 0 s(t)γA ( X k mk(t)−mk(0) (3.16) ×(1−3 cos2θk)/r_k3 ) dt + Av(1) + Av(2) ,

where Av(1) is the average of different subgroups of A spins while Av(2) is the average over the possible configurations ofB spins surrounding an Aspin.

An exact solution of Equation 3.16 is not possible and statistical models describing the reconfiguration of theBspin bath are needed to find suitable averages. In all statistical models, the decay rate of theAspin is directly related to two parameters, one is theB spin bath reconfiguration rate R, which is equivalent to the inverse of the population lifetime, R = 1/τ1 of the bath. The other is the magnitude of the frequency shift caused by the

reconfiguration of the B spin bath as defined in Equation 3.13.

Herzog et al demonstrated it is reasonable to assume that the average over all bath configurations results in a Gaussian lineshape independent of time [97]. They also assumed that allBspins are the same and therefore can be averaged over using the same probability distribution. When the spin B bath configuration rate is fast compared to the total evolution time τ of spin A, namely Rτ 1, a large number of randomly selected B spins reverse their orientation over the time τ. These randomly selected elements of the Gaussian distribution also forms a Gaussian distribution. Then, an average of Equation

3.16 results in a single exponential decay of the two pulse spin echo amplitude

E(τ)∝exp −∆ω 2 Aτ 2R , Rτ1, (3.17) 1

I explain the concept by considering that A spins are isotropic to simplify the explanation, though typically, the rare earth spins in our system are not isotropic.

where ∆ωA is the averaged frequency shift of an A spin due to the reconfiguration of B spin bath and R, as defined in the last paragraph, is the B spin configuration rate. Equation 3.17can be reformulated as

E(τ)∝E0exp(−τ/T2), (3.18)

where

T2 = 2R/∆ω2A (3.19)

If the configuration rate of B spins is slow compared to the evolution time τ, such thatRτ1, then over the period ofτonly a small number ofB spins flip. Though, over a long time scale, the average over all bath configurations is assumed to be a Gaussian lineshape, it is not possible to argue that a small number of randomly selected elements of a Gaussian distribution also form a Gaussian distribution [98]. A different method was used to average Equation 3.16by Klauder and Anderson [99] in this limit, where the B-spin flips were considered to result in a “diffusion” of the distribution of the local field perturbation. In a time period ∆t 1/R, there is a small number of randomly selected B spins flips, which is equivalent to the insertion of nBR∆t additional spins, each with magnitude of γB, at random sites in the lattice. Here nB is the total number of the B spins in the bath. An average over the A spins that independently experience such a B configuration results in a Lorentzian distribution of the A transition frequencies [98]. That is, an A spin subgroup with an initial transition frequency ωa will broaden into a distribution [100]

K(ω−ωa,∆t) =

(2R∆ω_1/2∆t)/π (ω−ωa)2+ (2R∆ω1/2∆t)2

(3.20) where ∆t=t−t0 and ∆ω1/2 is the full width at half maximum of theA spin frequency

deviation defined by the distribution of all possible bath configurations and ∆t. The concept of ∆ω1/2 is equivalent to ∆ωA in Equation3.17, but were labelled differently in the corresponding papers. Klauder and Anderson [99] show that this results in the echo decay function:

E(τ)∝exp−(τ/TM)2, Rτ1 (3.21) where

TM = 1.41(R∆ω1/2)−1/2. (3.22)

In this equation, TM, the phase memory time, is used rather than the coherence time T2, because the decay rate is time dependent. TM is defined as the evolution time which brings about a 1/eattenuation of the echo amplitude.

In between these two extremes, when Rτ ≈1, there is no simple statistical approxi- mation [85] and consequently no decay functions have been derived.

The two distinctly different spin echo decays, Equation 3.18 and Equation3.21, are derived depending on the relative time scale of the echo andB spin bath reconfigurations. In the coherence measurement, as we are only concerned with the delay for the echo am- plitude drops to 1/eof the maximum value, the evolution timeτ is generally chosen to be comparable to the inverse of the perturbation magnitude caused by the bath reconfigura- tion. Thus, the limiting conditions for the two different echo envelopes can be reformulated in terms of the static local field broadening ∆ω1/2. The conditionRτ1 is equivalent to

§3.1 Decoherence of two-level spin ensemble systems 49

formulas can then be summarized as : the echo decay has an exponential format when the B spin bath reconfigures at a rate fast compared to the frequency perturbation it causes on theAspins and the echo decay has a non-exponential format when the reconfiguration rate is slow compared to the frequency perturbation.