Researching | Weak-value amplification for the optical signature of topological phase transitions

Journals >Photonics Research >Volume 8 >Issue 12 >Page B47 > Article

Photonics Research
Vol. 8, Issue 12, B47 (2020)

Weak-value amplification for the optical signature of topological phase transitions

Weijie Wu^1、2, Shizhen Chen¹, Wenhao Xu¹, Zhenxing Liu³, Runnan Lou², Lihua Shen², Hailu Luo^1、4、*, Shuangchun Wen¹, and Xiaobo Yin^{2、3、5、*}

Author Affiliations

¹Laboratory for Spin Photonics, School of Physics and Electronics, Hunan University, Changsha 410082, China

²Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309, USA

³Materials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA

⁴e-mail: hailuluo@hnu.edu.cn

⁵e-mail: xiaobo.yin@colorado.edu

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DOI: 10.1364/PRJ.401531 Cite this Article Set citation alerts

Weijie Wu, Shizhen Chen, Wenhao Xu, Zhenxing Liu, Runnan Lou, Lihua Shen, Hailu Luo, Shuangchun Wen, Xiaobo Yin. Weak-value amplification for the optical signature of topological phase transitions[J]. Photonics Research, 2020, 8(12): B47 Copy Citation Text

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Abstract

We show that weak measurements can be used to measure the tiny signature of topological phase transitions. The signature is an in-plane photonic spin Hall effect, which can be described as a consequence of a Berry phase. It is also parallel to the propagation direction of a light beam. The imaginary part of the weak value can be used to analyze ultrasmall longitudinal phase shifts in different topological phases. These optical signatures are related to the Chern number and bandgaps; we also use a preselection and postselection technique on the spin state to enhance the original signature. The weak amplification technique offers a potential way to determine the spin and valley properties of charge carriers, Chern numbers, and topological phases by direct optical measurement.

H_{η} = ℏ ν_{F} (η k_{x} τ_{x} + k_{y} τ_{y}) + λ_{SO} σ_{z} η τ_{z} - e ℓ E_{z} τ_{z} - Λ η τ_{z} .

(1)

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m_{s}^{η} = η s λ_{SO} - e ℓ E_{z} - η Λ .

(2)

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\frac{{\tilde{σ}}_{x x}^{η s}}{σ_{0} / 2 π} = \frac{4 μ^{2} - {| m_{s}^{η} |}^{2}}{2 ℏ μ Ω} Θ (2 μ - | m_{s}^{η} |) + (1 - \frac{{| m_{s}^{η} |}^{2}}{ℏ^{2} Ω^{2}}) \times \arctan (\frac{ℏ Ω}{M}) + \frac{{| m_{s}^{η} |}^{2}}{ℏ Ω M}, \frac{{\tilde{σ}}_{x y}^{η s}}{σ_{0} / 2 π} = \frac{2 η m_{s}^{η}}{ℏ Ω} \arctan (\frac{ℏ Ω}{M}) .

(3)

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r_{h h} = \frac{α_{-} β_{+} + σ_{h v} σ_{v h}}{α_{+} β_{+} + σ_{h v} σ_{v h}}, r_{v v} = \frac{α_{+} β_{-} - σ_{h v} σ_{v h}}{α_{+} β_{+} + σ_{h v} σ_{v h}}, r_{h v} = - \frac{2}{Z_{i}} \frac{σ_{h v}}{α_{+} β_{+} + σ_{h v} σ_{v h}}, r_{v h} = \frac{2}{Z_{i}} \frac{σ_{v h}}{α_{+} β_{+} + σ_{h v} σ_{v h}} .

(4)

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(\begin{matrix} r_{h h} & r_{h v} \\ r_{v h} & r_{v v} \end{matrix}) .

(5)

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| H (k_{i}) ⟩ \to (r_{h h} - \frac{k_{r x}}{k_{0}} \frac{\partial r_{h h}}{\partial θ_{i}}) | H (k_{r}) ⟩ + (r_{v h} - \frac{k_{r x}}{k_{0}} \frac{\partial r_{v h}}{\partial θ_{i}}) | V (k_{r}) ⟩,

(6)

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| V (k_{i}) ⟩ \to (r_{h v} - \frac{k_{r x}}{k_{0}} \frac{\partial r_{h v}}{\partial θ_{i}}) | H (k_{r}) ⟩ + (r_{v v} - \frac{k_{r x}}{k_{0}} \frac{\partial r_{v v}}{\partial θ_{i}}) | V (k_{r}) ⟩ .

(7)

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| Φ ⟩ = \frac{w_{0}}{\sqrt{2 π}} \exp [- \frac{w_{0}^{2} (k_{i x}^{2} + k_{i y}^{2})}{4}] .

(8)

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| Φ_{r}^{H} ⟩ \approx \frac{r_{h h} - i r_{v h}}{\sqrt{2}} \exp (i s k_{r x} δ_{x}^{H} + k_{r x} Δ_{x}^{H}) | + ⟩ | Φ ⟩ + \frac{r_{h h} + i r_{v h}}{\sqrt{2}} \exp (i s k_{r x} δ_{x}^{H} + k_{r x} Δ_{x}^{H}) | - ⟩ | Φ ⟩,

(9)

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| Φ_{r}^{V} ⟩ \approx \frac{r_{h v} - i r_{v v}}{\sqrt{2}} \exp (i s k_{r x} δ_{x}^{V} + k_{r x} Δ_{x}^{V}) | + ⟩ | Φ ⟩ + \frac{r_{h v} + i r_{v v}}{\sqrt{2}} \exp (i s k_{r x} δ_{x}^{V} + k_{r x} Δ_{x}^{V}) | - ⟩ | Φ ⟩ .

(10)

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δ_{x}^{H} = \frac{r_{h h}}{k_{i} (r_{h h}^{2} + r_{v h}^{2})} \frac{\partial r_{v h}}{\partial θ_{i}} - \frac{r_{v h}}{k_{i} (r_{h h}^{2} + r_{v h}^{2})} \frac{\partial r_{h h}}{\partial θ_{i}},

(11)

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δ_{x}^{V} = \frac{r_{h v}}{k_{i} (r_{v v}^{2} + r_{h v}^{2})} \frac{\partial r_{v v}}{\partial θ_{i}} - \frac{r_{v v}}{k_{i} (r_{v v}^{2} + r_{h v}^{2})} \frac{\partial r_{h v}}{\partial θ_{i}} .

(12)

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⟨ Δ x_{\pm}^{H, V} ⟩ = \frac{⟨ Φ_{r \pm}^{H, V} | i \partial_{k_{r x}} | Φ_{r \pm}^{H, V} ⟩}{⟨ Φ_{r \pm}^{H, V} | Φ_{r \pm}^{H, V} ⟩},

(13)

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⟨ Θ x_{\pm}^{H, V} ⟩ = \frac{1}{k_{i}} \frac{⟨ Φ_{r \pm}^{H, V} | k_{r x} | Φ_{r \pm}^{H, V} ⟩}{⟨ Φ_{r \pm}^{H, V} | Φ_{r \pm}^{H, V} ⟩} .

(14)

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⟨ Δ x_{s}^{H, V} ⟩ = - s Re [δ_{x}^{H, V}],

(15)

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⟨ Θ x_{s}^{H, V} ⟩ = - \frac{1}{z_{R}} s Im [δ_{x}^{H, V}] .

(16)

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| Φ_{f} ⟩ = ⟨ ψ_{f} | \exp (i σ_{z} k_{r x} δ_{r x}) | ψ_{i} ⟩ | Φ ⟩ = ⟨ ψ_{f} | 1 + i σ_{z} k_{r x} δ_{r x} | ψ_{i} ⟩ | Φ ⟩ \approx ⟨ ψ_{f} | ψ_{i} ⟩ (1 + i k_{r x} δ_{r x} \frac{⟨ ψ_{f} | σ_{z} | ψ_{i} ⟩}{⟨ ψ_{f} | ψ_{i} ⟩}) | Φ ⟩ = ⟨ ψ_{f} | ψ_{i} ⟩ (1 + i k_{r x} A_{w} δ_{r x}) | Φ ⟩ .

(17)

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A_{w} = \frac{⟨ ψ_{f} | σ_{z} | ψ_{i} ⟩}{⟨ ψ_{f} | ψ_{i} ⟩},

(18)

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Im [A_{w} δ_{r x}^{H, V}] = Re [A_{w}] Im [δ_{r x}^{H, V}] + Im [A_{w}] Re [δ_{r x}^{H, V}],

(19)

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Re [A_{w} δ_{r x}^{H, V}] = Re [A_{w}] Re [δ_{r x}^{H, V}] - Im [A_{w}] Im [δ_{r x}^{H, V}] .

(20)

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| ψ_{i} ⟩ = \cos (\frac{Θ}{2}) | + ⟩ + e^{- i Φ} \sin (\frac{Θ}{2}) | - ⟩,

(21)

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| ψ_{m} ⟩ = \cos (\frac{Θ}{2} + α) | + ⟩ + e^{- i (Φ + 2 δ)} \sin (\frac{Θ}{2} + α) | - ⟩ .

(22)

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| ψ_{f} ⟩ = \sin (\frac{Θ}{2} + α + Δ_{1}) | + ⟩ - e^{- i (Φ + 2 δ + 2 Δ_{2})} \cos (\frac{Θ}{2} + α + Δ_{1}) | - ⟩,

(23)

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A_{w} = \frac{⟨ ψ_{f} | σ_{z} | ψ_{m} ⟩}{⟨ ψ_{f} | ψ_{m} ⟩} .

(24)

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γ \approx | ⟨ ψ_{f} | ψ_{m} ⟩ |^{2} = \cos^{2} Δ_{2} \sin^{2} Δ_{1} + \sin^{2} Δ_{2} \sin^{2} (Θ + Δ_{1} + 2 α),

(25)

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Re [A_{w}] = \frac{\sin Δ_{1} \sin (2 α + Δ_{1} + Θ)}{| ⟨ ψ_{f} | ψ_{m} ⟩ |^{2}},

(26)

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Im [A_{w}] = \frac{\sin 2 Δ_{2} \sin (2 α + Θ) \sin [2 (α + Δ_{1}) + Θ]}{2 | ⟨ ψ_{f} | ψ_{m} ⟩ |^{2}} .

(27)

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⟨ Δ x_{w}^{H, V} ⟩ = \frac{⟨ Φ_{f} | i \partial_{k_{r x}} | Φ_{f} ⟩}{⟨ Φ_{f} | Φ_{f} ⟩},

(28)

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⟨ Θ x_{w}^{H, V} ⟩ = \frac{1}{k_{i}} \frac{⟨ Φ_{f} | k_{r x} | Φ_{f} ⟩}{⟨ Φ_{f} | Φ_{f} ⟩} .

(29)

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⟨ Δ x_{w}^{H, V} ⟩ = Re [A_{w} δ_{r x}^{H, V}] = Re [δ_{r x}^{H, V}] \frac{\sin (2 α + Δ_{1} + Θ)}{\sin Δ_{1}},

(30)

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⟨ Θ x_{w}^{H, V} ⟩ = \frac{1}{z_{R}} Im [A_{w} δ_{r x}^{H, V}] = Im [δ_{r x}^{H, V}] \frac{\sin (2 α + Δ_{1} + Θ)}{z_{R} \sin Δ_{1}} .

(31)

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| Φ_{f} ⟩ = ⟨ ψ_{f} | \exp (- i τ ω \hat{σ}) M_{r} | ψ_{i} ⟩ \otimes | Φ ⟩ = 0.5 f (ω) (e^{- i τ ω - i ϵ} r_{h h} - e^{+ i τ ω - i ϵ} r_{v h} + e^{- i τ ω + i ϵ} r_{h v} - e^{+ i τ ω + i ϵ} r_{v v}) .

(32)

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\frac{\int ω F (ω) d ω}{\int F (ω) d ω} = ω_{0} + Δ ω, T = \int F (ω) d ω .

(33)

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Δ λ = \frac{λ^{2}}{2 π c} Δ ω .

(34)

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Weijie Wu, Shizhen Chen, Wenhao Xu, Zhenxing Liu, Runnan Lou, Lihua Shen, Hailu Luo, Shuangchun Wen, Xiaobo Yin. Weak-value amplification for the optical signature of topological phase transitions[J]. Photonics Research, 2020, 8(12): B47

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