We demonstrate coherence amplification in off-diagonal density matrix elements for a decoherence model for spin ½ particles with applied radio-frequency pulse without relaxation. The method of coherence amplification developed may be of general utility in applications in NMR/MRI.
The manifestation of coherence phenomena is a hallmark of quantum mechanics, differentiating it from classical phenomenon due to the property of superposition in quantum reality due to the linearity of the Schroedinger equation [1-4].
Methods of possible amplification of coherence as quantitated by the absolute value of off-diagonal density matrix elements, may be of use in applications such as quantum computing [5] and MRI tissue contrast [6,7].
In the Letter, we develop a Possible Methodology of Coherence Enhancement. It is easily implemented and may have value in Venues of NMR/MRI [8].
We base our Treatment on the Following Expression often used in Formalism of Coherence [1,9].
dρˆ[t]dt=−I[HˆRF[t],ρˆ[t]]−T[τdph][HˆRF[t],[HˆRF[t],ρˆ[t]]] (1)
Here, ρˆ[t] is the time-dependent Density Matrix. HˆRF[t] is a Time Dependent Radio-Frequency Hamiltonian, defined as:
HˆRF[t]=Ixw1[t]+IzΔw[t] (2)
Iˆαα=x,z are Spin ½ Angular Momentum Operators [8] . The Radio-Frequency terms w1[t] and Δw[t] are respectively, Amplitude and Frequency Offset Variable defined in Appendix I.
T[τdph]=kBThτ2dph (3)
Here, kB is Boltzmann's Constant.
T is Temperature in Kelvins.
h is Planck's Constant Divided by 2 π .
τdph is a Time Constant Characterizing the Decoherence.
For the Specific Cases Treated here, to Explore a Model of Coherence Enhancement Effects, we specialize to the HS1 Adiabatic Pulse as detailed by Garwood, et al. [10] [see Appendix II].
If one substitutes Eq (2) into Eq (1), one obtains after "straightforward but tedious algebra":
dρˆ[t]dt=−I[HˆRF[t],ρˆ[t]]+T[τdph](w12[t]Iˆxρˆ[t]Iˆx+Δw2[t]Iˆzρˆ[t]Iˆz (4)−1ˆ4w2eff[t]ρˆ[t]+w1[t]Δw[t](Iˆxρˆ[t]Iˆz+Iˆzρˆ[t]Iˆx))
Where:
w2eff[t]=w21[t]+Δw2[t] (5)
And:
1ˆ is the 2 by 2 Identity Matrix.
For Clarity and Completeness, we Explicitly Define the Following Terms:
ρˆ[t]=(ρ11[t]ρ21[t]ρ12[t]ρ22[t]) (6)
We adopt the Standard Definition of the Spin-1/2 Cartesian Spin Angular Momentum Operators
As:
Iˆx=12σx,Iˆy=12σy,Iˆz=12σz, (7a)
σˆx=(0110),σˆy=(0I−I0),σˆz=(100−1), (7b)
Where, Eqs (7b) are the Pauli Matrices [9].
If one substitutes Eqs (5,6,7a-b) into Eq (4) one finds after manipulations the following set of four first order differential equations in time for the four matrix elements of the defined Density Matrix. We note that the derived system of equations below were numerically verified versus Eq(9) below:
dρ11[t]dt=−I2w1[t](ρ21[t]−ρ12[t])+T[τdph]14(w21[t](ρ22[t]−ρ11[t])+w1[t]Δw[t](ρ21[t]+ρ12[t])) (8a)
dρ22[t]dt=−dρ11[t]dt (8b)
dρ12[t]dt=−I2(2Δw[t]ρ12[t]+w1[t](ρ22[t]−ρ11[t])+ (8c)T[τdph]14(w21[t](ρ21[t]−ρ12[t])−2Δ2w[t]ρ12[t]+w1[t]Δw[t](ρ11[t]−ρ22[t]))
dρ21[t]dt=I2(2Δw[t]ρ21[t]+w1[t](ρ22[t]−ρ11[t])− (8d)T[τdph]14(w21[t](ρ21[t]−ρ12[t])+2Δ2w[t]ρ12[t]−w1[t]Δw[t](ρ11[t]−ρ22[t]))
We note that for ease of manipulation and coding, one can rewrite Eqs (1) as:
dρˆ[t]dt=−I(HˆRF[t]ρˆ[t]−ρˆ[t]HˆRF[t])+T[τdph]12(2HˆRF[t]ρˆ[t]HˆRF[t]−Hˆ2RF[t]ρˆ[t]−ρˆ[t]Hˆ2RF[t]) (9)
Using a Numerical Platform such as Mathematica [11] one can readily Numerically Solve the four Differential equations Eqs (8 a,b,c,d) to obtain the Time-Dependent Matrix Elements ρij[t];i,j=1,2.
In Figure 1a, Figure 1b, Figure 1c and Figure 1d we see plotted the [t,τdph] dependence in Three-Dimensional Figures of the Four Density Matrix elements which are numerical solutions of Eqs [8a,b,c,d]. We note that the density matrix elements are considered dimensionless [4,6,8,9]. One can readily see there are Maxima at [0.002s,τf1.068] τf=1.010−5s over the domain considered.
In Summary, the Key Results of this effort, is the finding of Maxima for the dependence of Density Matrix element which indicate amplification of the coherences and populations for spin ½ nuclei which are solutions of the system of Eqs [8a,b,c,d] that incorporate Decoherence effects at [t,τdph]. The system of Eqs [8] are to the knowledge of the author unique to the Magnetic Resonance literature.
Such Enhancement may prove to be of utility in Biomedical Applications in NMR/MRI, because the spin angular momenta can be expressed as sums and differences of the spin-1/2 Density Matrix elements which can be shown to exhibit pronounced amplification that are proportional to the Magnetization.