Signal enhancement of magneto-optical Kerr effect measurements by weak value amplification

Abstract

Magneto-optic effects are widely applied in the analysis of magnetic thin films. Specifically, due to its local probing characteristic, sensitivity, and experimental simplicity, the surface magneto-optic Kerr effect has been shown to be very useful in the investigation of the magnetization structure of materials. In this work, we propose a weak measurement-based protocol to enhance the pointer deflection, namely the polarization rotation of an incident light beam upon reflection from a magnetic surface, in some orders of magnitude, without loss in precision. It could provide a new insight into magnetic research.

Appendix: Post-selecting a frequency qubit

Our protocol is based on the post-selection of an arbitrary frequency qubit (filtering process). In doing so, after the weak interaction between the frequency and polarization degrees of freedom, if a prism is used to spatially separate the photons in the states $\left| {\omega _{1}} \right\rangle$ and $\left| {\omega _{2}} \right\rangle$, and a phase shift $\varphi$ is applied in the path $\left| {2} \right\rangle$, we obtain the entangled state

$$\begin{aligned} \left| {\varPhi } \right\rangle = \frac{1}{\sqrt{2}}(\left| {\omega _{1}} \right\rangle \left| {1} \right\rangle + e^{i\varphi }\left| {\omega _{2}} \right\rangle \left| {2} \right\rangle ). \end{aligned}$$

(21)

Now, if these two paths are recombined in a symmetric beam splitter (See Fig. 2), the transformations $\left| {1} \right\rangle \rightarrow \frac{1}{\sqrt{2}} \left( \left| {1^{'}} \right\rangle + i\left| {2^{'}} \right\rangle \right)$ and $\left| {2} \right\rangle \rightarrow \frac{1}{\sqrt{2}} (i\left| {1^{'}} \right\rangle + \left| {2^{'}} \right\rangle )$ take place, where $\left| {1^{'}} \right\rangle$ and $\left| {2^{'}} \right\rangle$ represent the two output modes of the beam splitter [32]. After this stage, we obtain the state

$$\begin{aligned} \left| {\varPhi ^{'}} \right\rangle&= \frac{1}{2} \left( \left| {\omega _{1}} \right\rangle + e^{i(\varphi +\pi /2)}\left| {\omega _{2}} \right\rangle \right) \left| {1^{'}} \right\rangle \\&\quad + \frac{i}{2} \left( \left| {\omega _{1}} \right\rangle - e^{i(\varphi +\pi /2)}\left| {\omega _{2}} \right\rangle \right) \left| {2^{'}} \right\rangle . \end{aligned}$$

(22)

Then, if, after the weak interaction, we apply a phase shift in the path $\left| {2} \right\rangle$ given by

$$\begin{aligned} \varphi = \hbox {arccos} \left[ \frac{\sin \alpha - \cos \alpha }{ \sin \alpha + \cos \alpha } \right] - \frac{\pi }{2}, \end{aligned}$$

(23)

and select only the photons emerging from the output mode $\left| {1^{'}} \right\rangle$ of the beam splitter, the post-selected state, $\left| {\psi _{f}} \right\rangle = (1/\sqrt{2}) [(\cos \alpha + \sin \alpha )\left| {\omega _{1}} \right\rangle - (\cos \alpha - \sin \alpha )\left| {\omega _{2}} \right\rangle ]$, necessary for obtaining the amplification factor of Eq. (18), is accomplished.

Fig. 2

A symmetric beam splitter, with input modes $\left| {1} \right\rangle$ and $\left| {2} \right\rangle$ and output modes $| {1^{'}} \rangle$ and $| {2^{'}} \rangle$, operates according to the relation $\hat{B} = | {1^{'}} \rangle \left\langle {1} \right| + | {2^{'}} \rangle \left\langle {2} \right| + i(| {1^{'}} \rangle \left\langle {2} \right| + | {2^{'}} \rangle \left\langle {1} \right| )$. This device acts as a fundamental ingredient in the post-selection (filtering) process of a frequency qubit