International Journal of Magnetics and Electromagnetism
Volume 6, Issue 1
Weak Value Amplification of Spin-Angular Momentum Expectation Values using the HS1 Radio-Frequency Pulse
Dennis J Sorce*
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Dennis J Sorce*
Retired, Center for Magnetic Resonance Research, Minneapolis, USA
Dennis J Sorce, Retired, Center for Magnetic Resonance Research, Minneapolis, MN 55455, USA, Tel: 410-628-2461.
Accepted: December 07, 2020 | Published Online: December 09, 2020
Citation: Sorce DJ (2020) Weak Value Amplification of Spin-Angular Momentum Expectation Values using the HS1 Radio-Frequency Pulse. Int J Magnetics Electromagnetism 6:031
Copyright: © 2020 Sorce DJ. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
We have used Weak Values, as Computed from Weak Measurements of Aharonov, et al.  to Achieve High Incidents of Amplification of Expectation Values for Spin-Angular Momentum Observables for Spin-1/2. The Radio-Frequency used was the HS1 Adiabatic Pulse. When the same Procedure was Applied to the Sin-Cos Pulse, Little or No Amplification was Found.
In 1988 Aharonov, et al.  published their theory paper on the Weak Measurement Formalism. Since then there has been applications of the Formalism to achieve Amplification of Physical Observables in various Fields such as Quantum Optics [2,3]. In this Note we demonstrate via numerical simulations the achievement of Amplifications of the X Spin Angular Momentum for Spin ½ nuclei as high as 150 times the standard Computed Expectation Value. This was achieved with the use of the HS1 Adiabatic Pulse  in the simulations. Such possible behavior is known for the Weak Value computed from the Weak Measurement Formalism [1-3]. In outline, we basically solved the Time-Dependent Schroedinger equation numerically using the simple time dependent RF HS1 Dependence. We then simply used the State Functions or Wave Function computed to calculate numerically the Weak Value Expectation Value for x, y and z Spin Angular Momentum using the Pauli 2 by 2 representation . All Weak Values computed were normalized appropriately .
High Amplifications were also found for y Spin Angular Momentum Weak Value. Little or no Amplification was found for the z Spin Angular Momentum Weak Value.
Orthodox Expectation Values for Spin ½ Angular Momentum is usually computed using the Matrix Representation as:
for α = x, y, z
i, j = 1, 2
Note that the time variable is the same in the Bra as in the Ket (If we use the Dirac Nomenclature ).
In our Application we computed the Wave Function components using the 2 by 2 Matrix Representation of the Radio-Frequency Hamiltonian as:
Using the Pauli Matrix Representation for Spin ½ Nuclei as:
We then use Eq(5) in the Time-Dependent Schroedinger Eq as :
Using Eqs (5,7) in Eq (6) one obtains the following system of First Order Derivative Time-Dependent Equations:
Here, of course, we use Frequency Units for the RF Hamiltonian Elements.
And is a scaling factor
We Numerically Solved the system of equations using the NDSolve Function in the Mathematica Platform vs. 12.1 .
We found, pronounced Amplification effects when we used the HS1 Adiabatic Pulse .
We used the Numerical Solutions to Eqs (8a,b) in the Weak Value Computed as:
Here, of course, we used the Numerical Solutions to the SEq (Eq(6) in Eq(10)).
In Figure 1 we see the Time-Dependence of the x Spin-Angular Momentum for Eq (10).
With tf = ti = t This is a Standard Result which can be easily verified as the solution for the X Magnetization of the Time-Dependent Bloch Equations  using the HS1 Radio-Frequency Pulse Definitions as in Eqs (9a,b).
In Figure 2 we see a time profile for the y Magnetization which is of course Proportional to the y Spin-Angular Momentum .
In Figure 3 we see the Time Dependence of the x Spin-Angular Momentum for Eq (10) with
and One can see the pronounced Amplification for t in the neighborhood of . In Figure 4 we see a profile for the y Spin-Angular Momentum where once again there are Amplification effects in the neighborhood of .
One finds, if one sets and and plots the Weak Value x Dependence versus α that an interesting Amplification Profile can be obtained as demonstrated in Figure 5. Figure 6 shows a similar profile for the y Spin-Angular Momentum. Little or no Amplification was found for the z spin under the conditions used.
When simulations identical to those presented above for the Sin-Cos Pulse  were performed little or no Amplification was found.
The results presented show quite dramatic effects for Amplification of the Expectation Value for the x and y Spin-Angular Momentum, which strongly suggests that similar results May be found experimentally for the HS1 RF Pulse for the x and y Magnetizations. The Formalism Applied is quite Straightforward and Simple to implement. The results, in hindsight are as expected Given the Theoretical Formalism of the Weak Measurement and the results obtained in other Applied Areas. [1-3,6,7,12,13]. The results obtained may have utility in Bio-Medical NMR Applications [4,11].