# International Journal of Magnetics and Electromagnetism

(ISSN: 2631-5068)

Volume 6, Issue 1

Research Article

#### DOI: 10.35840/2631-5068/6531

# Weak Value Amplification of Spin-Angular Momentum Expectation Values using the HS1 Radio-Frequency Pulse

Dennis J Sorce^{*}

### References

- Y Aharonov, DZ Albert, L Vaidman (1988) How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys Rev Letts 60: 1351-1354.
- J Dressel, M Malik, FM Miatto, AN Jordan, RW Boyd (2014) Colloquium: Understanding weak values: Basics and applications. arXiv:1305.7154 v2 [quant-ph].
- OS Magana-Loaiza, M Mirhusseini, B Rodenberg, RW Boyd (2014) Amplification of angular rotations using weak measurements. arXiv:1312.2981v3 [quant-ph].
- M Garwood, L DeLaBarre (2001) Advances in magnetic resonance: The return of the frequency sweep: Designing adiabatic pulses for contemporary NMR. J Magn Reson 153: 155-177.
- D Dubbers, HJ Stockmann (2013) Quantum-physics: The bottom-up approach. Springer.
- BEY Svensson (2013) Pedagogical review of quantum measurement theory with an emphasis on weak measurement. Quanta 2: 18-49.
- R Flack, BJ Hiley (2014) Weak measurement and its experimental realization. arXiv: 1408.5685v1 [quant-ph].
- A Brodutch, E Cohen (2017) A scheme for performing strong and weak sequential measurements of non-commuting observables. Quantum Stud: Math Found 4: 13-27.
- Wolfram Research, Urbana IL, USA.
- A Abragam (1986) Principles of nuclear magnetism. OxFord.
- DJ Sorce, S Mangia, T Liimatainen, M Garwood (2014) Exchange-induced relaxation in presence of fictitious field. J Magn Reson 245: 12-16.
- AG Kofman, S Ashhab, F Nori (2012) Non-perturbative theory of pre- and post-selected measurements. ArXiv:1109.6315v2 [quant- ph].
- NDH Dass, R Krishna, SS Samantaway (2017) Optimal weak value measurements. arXiv:1702.00347v3[quant-ph].

**Author Details**

Dennis J Sorce^{*}

Retired, Center for Magnetic Resonance Research, Minneapolis, USA

**Corresponding author**

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.

## Abstract

We have used Weak Values, as Computed from Weak Measurements of Aharonov, et al. [1] 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.

## Introduction

In 1988 Aharonov, et al. [1] 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 [4] 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 [5]. All Weak Values computed were normalized appropriately [1].

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.

## Theoretical Formalism

Orthodox Expectation Values for Spin ½ Angular Momentum is usually computed using the Matrix Representation as:

$$({\Psi}_{1}{}^{*}\left[t\right],{\Psi}_{2}{}^{*}\left[t\right])({S}_{i,j}{}^{\alpha})\left[\begin{array}{c}{\Psi}_{1}\left[t\right]\\ {\Psi}_{2}\left[t\right]\end{array}\right]\text{(1)}$$

for *α = x, y, z*

*i, j* = 1, 2

Where ${\Psi}_{i}\left[t\right]\text{}i=1,\text{}2$

Note that the time variable is the same in the Bra as in the Ket (If we use the Dirac Nomenclature [5]).

The Innovation suggested by the Weak Measurement Formalism [1,2,6-8].

$$({\Psi}_{1}{}^{*}[{t}_{f}],{\Psi}_{2}{}^{*}[{t}_{f}])({S}_{i,j}{}^{\alpha})\left[\begin{array}{c}{\Psi}_{1}\left[{t}_{i}\right]\\ {\Psi}_{2}\left[{t}_{i}\right]\end{array}\right]\text{(2)}$$

Where ${t}_{f}\text{}\ge \text{}{t}_{i}.$

In our Application we computed the Wave Function components using the 2 by 2 Matrix Representation of the Radio-Frequency Hamiltonian as:

$${H}^{RF}\left[t\right]={\omega}_{1}\left[t\right]{I}_{x}+\Delta \omega \left[t\right]{I}_{z}\text{(3)}$$

Using the Pauli Matrix Representation for Spin ½ Nuclei as:

$${I}_{x}=\text{}\frac{1}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\text{}{I}_{y}=\frac{1}{2}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\text{}{I}_{z}=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\text{(4)}$$

$${H}^{RF}\left[t\right]=\frac{1}{2}\left(\begin{array}{cc}\Delta \omega \left[t\right]& {\omega}_{1}\left[t\right]\\ {\omega}_{1}\left[t\right]& -\Delta \omega \left[t\right]\end{array}\right)\text{(5)}$$

We then use Eq(5) in the Time-Dependent Schroedinger Eq as [5]:

$$\frac{d\Psi \left[t\right]}{dt}=\text{}-i{H}^{RF}\left[t\right]\Psi \left[t\right]\text{(6)}$$

Where:

$$\Psi \left[t\right]=\text{}\left(\begin{array}{c}{\Psi}_{1}\left[t\right]\\ {\Psi}_{2}\left[t\right]\end{array}\right)\text{(7)}$$

Using Eqs (5,7) in Eq (6) one obtains the following system of First Order Derivative Time-Dependent Equations:

$$\begin{array}{l}\frac{d{\Psi}_{1}\left[t,\eta \right]}{dt}=\text{}-\frac{in(\Delta \omega \left[t\right]{\Psi}_{1}\left[t,\eta \right])+{\omega}_{1}\left[t\right]{\Psi}_{2}\left[t,\eta \right]}{2}\text{(8a,b)}\\ \frac{d{\Psi}_{2}\left[t,\eta \right]}{dt}=\text{}-\frac{in({\omega}_{1}\left[t\right]{\Psi}_{1}\left[t,\eta \right])-\Delta \omega \left[t\right]{\Psi}_{2}\left[t,\eta \right]}{2}\end{array}$$

Here, of course, we use Frequency Units for the RF Hamiltonian Elements.

And $\eta $ is a scaling factor $0\le \eta \le 1$

We Numerically Solved the system of equations using the NDSolve Function in the Mathematica Platform vs. 12.1 [9].

We found, pronounced Amplification effects when we used the HS1 Adiabatic Pulse [4].

Defined as:

$$\begin{array}{l}{\omega}_{1}\left[t\right]=\text{}{\omega}_{1\mathrm{max}}\mathrm{Sech}[b(2t\text{}/\text{}tp-1)]\text{(9)}\\ \Delta \omega \left[t\right]=\text{}{\Omega}_{p}-amp\Delta \mathrm{Tanh}[b(2t\text{}/\text{}tp-1)]\end{array}$$

We used the Numerical Solutions to Eqs (8a,b) in the Weak Value Computed as:

$$\frac{<\Psi \left[{t}_{f},\alpha \right]{S}^{\delta}\Psi \left[{t}_{i},\beta \right]>}{<\Psi \left[{t}_{f},\alpha \right]|\Psi \left[{t}_{i},\beta \right]>}\text{(10)}$$

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 *t _{f}* =

*t*=

_{i}*t*This is a Standard Result which can be easily verified as the solution for the X Magnetization of the Time-Dependent Bloch Equations [10] 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 [10].

In Figure 3 we see the Time Dependence of the x Spin-Angular Momentum for Eq (10) with

$\alpha =\beta =1$ and ${t}_{i}=0,0\le {t}_{f}\le {t}_{p}$ ${t}_{p}=0.002s$ One can see the pronounced Amplification for t in the neighborhood of ${t}_{p}$. In Figure 4 we see a profile for the y Spin-Angular Momentum where once again there are Amplification effects in the neighborhood of ${t}_{p}$.

One finds, if one sets ${t}_{i}=0s,{t}_{f}=0.0019s$ and $\beta =1$ 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 [11] were performed little or no Amplification was found.

## Discussion

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].