# Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems

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## Abstract

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## 1. Introduction

## 2. Weak Value Amplification Effect

**Figure 1.**A quantum system in an initial state ${\rho}_{S}$ is subjected to a measurement of the observable ${A}_{S}$. The measurement is described by a set of effects ${E}_{r}$. In a single instance, the measurement produces an outcome r, a random variable that distributes according to $P\left(r\right)\equiv \mathrm{Tr}\left\{{E}_{r}{\rho}_{S}\right\}$. Then, the system is subjected to the action of a filter, a second quantum measurement with two outcomes: “yes” and “no”. The idea is to consider $r|yes$, i.e., the results of the first measurement conditioning on the successful operation of the filter, a random variable that distributes according to (10).

## 3. Amplification of Photons in a Kerr Medium

**Figure 2.**Description of the experiment performed by HFDSS [22,23]. The upper interferometer is associated with the system (a single photon), while the lower interferometer is associated with the measurement device. Both the system and the measurement device have a two-mode structure. Modes 1 and c interact in a Kerr medium. Detectors ${D}_{3}$ and ${D}_{4}$ are used to post-select a final system state, while detectors ${D}_{e}$ and ${D}_{f}$ are used to read the phase generated on the classical beam. Note: this figure was completely redrawn by the authors of this article, taking Figure 1 of [23] and Figure 1 of [22] as references.

## 4. Amplification of Photons in Opto-Mechanical Systems

## 5. Amplification in Spin-Mechanical Systems

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Mach-Zehnder interferometer with an opto-mechanical cavity in arm 1 and a conventional Fabry-Pérot cavity in arm 2. Single photons are sent across the first beam splitter. The position of the moving mirror, ${x}_{0}(b+{b}^{dagger}),$ should be observed only in those cases when the photons are detected at ${D}_{2}$. In this scenario, the average displacement of the mirror is amplified. Note: this figure was completely redrawn by the authors of this article, taking Figure 1 of [34] as reference.

**Figure 4.**Schematic description of an experiment performed in [43,44], in which the axial displacement of a trapped ion is amplified. (

**a**) The vibrational degree of freedom (measurement device) starts cooling down to its ground state ${|\psi \left(z\right)\rangle}_{M}$, while the internal electronic state (measured system) begins in the ${\sigma}_{z}$-eigenstate ${|1\rangle}_{S}$. (

**b**) An interaction Hamiltonian ${H}_{SM}$ couples the system spin (along the x-direction) with the momentum of the meter by means of a bichromatic laser, producing the entangled (system-meter) state ${|\psi \left(t\right)\rangle}_{SM}$. Please note that the effective coupling constant is $gt$. Therefore, control of the interaction time allowed the exploration of the full weak-to-strong transition. (

**c**) After applying a qubit rotation along the y-direction, ${R}_{y}\left(2\theta \right)={e}^{-i\theta {\sigma}_{y}}$, postselection of the internal state ${|0\rangle}_{S}$ is made by no observing fluorescence. In this case, the conditional (un-normalized) meter state is ${|\tilde{\psi}\rangle}_{M}$. (

**d**) Whenever postselection has been successful, the probability density of the meter’s position is reconstructed using the method described in [45]. This figure was completely redrawn by the autors, taking Figure 1 of [43] and Figure 1 of [44] as references.

**Table 1.**Weak measurement protocol for the nonlinear interaction between: (a) a single photon and a classical beam (optical Kerr medium), (b) a single photon and a mechanical oscillator (OM interaction), and (c) the spin component of a spin 1/2 particle and a mechanical oscillator (SM interaction). In all cases, the measurement of ${A}_{S}$ is described by the unitary operator $U=\mathrm{exp}\left\{-ig{A}_{S}{O}_{M}\right\}$. In cases a and b, the system variable being amplified via weak values is a photonic number operator. In case c, the amplified variable is ${\sigma}_{z}$, which in turn enlarges the number of phonons (${b}^{\u2020}b$) in the measurement device. In the last line, regarding the initial probe (measurement device) states, $|\alpha \rangle $ denotes a coherent state.

Optical Kerr Medium | OM Interaction | SM Interaction | |
---|---|---|---|

${A}_{S}$ | ${n}_{1}\otimes {\mathbb{1}}_{2}$ | ${a}_{1}^{\u2020}{a}_{1}\otimes {\mathbb{1}}_{2}$ | ${\sigma}_{z}$ |

${O}_{M}$ | ${n}_{c}\otimes {\mathbb{1}}_{d}$ | $\phi \left(t\right){b}^{\u2020}+{\phi}^{*}\left(t\right)b$ | $\eta {b}^{\u2020}+{\eta}^{*}b$ |

${R}_{M}$ | $c{d}^{\u2020}+{c}^{\u2020}d$ | ${x}_{0}({b}^{\u2020}+b)$ | ${b}^{\u2020}b$ |

Initial system state | $\frac{{|1\rangle}_{1}{|0\rangle}_{2}-{|1\rangle}_{1}{|0\rangle}_{2}}{\sqrt{2}}$ | $\frac{{|1\rangle}_{1}{|0\rangle}_{2}-{|1\rangle}_{1}{|0\rangle}_{2}}{\sqrt{2}}$ | $\frac{|0\rangle +|1\rangle}{\sqrt{2}}$ |

Final system state | $\frac{(1+\delta ){|1\rangle}_{1}{|0\rangle}_{2}+(1-\delta ){|0\rangle}_{1}{|1\rangle}_{2}}{\sqrt{2}}$ | $\frac{(1+\delta ){|1\rangle}_{1}{|0\rangle}_{2}+(1-\delta ){|0\rangle}_{1}{|1\rangle}_{2}}{\sqrt{2}}$ | $\frac{cos(\theta /2)|0\rangle +sin(\theta /2){e}^{i\varphi}|1\rangle}{\sqrt{2}}$ |

Initial probe state $\left({|\psi \rangle}_{M}\right)$ | ${|\alpha \rangle}_{c}{|i{e}^{i\theta}\alpha \rangle}_{d}$ | ground state | ${|\alpha \rangle}_{M}$ |

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**MDPI and ACS Style**

Carrasco, S.; Orszag, M.
Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems. *Photonics* **2024**, *11*, 291.
https://doi.org/10.3390/photonics11040291

**AMA Style**

Carrasco S, Orszag M.
Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems. *Photonics*. 2024; 11(4):291.
https://doi.org/10.3390/photonics11040291

**Chicago/Turabian Style**

Carrasco, Sergio, and Miguel Orszag.
2024. "Weak Value Amplification of Photons in Optical Nonlinear Medium, Opto-Mechanical, and Spin-Mechanical Systems" *Photonics* 11, no. 4: 291.
https://doi.org/10.3390/photonics11040291