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Amplitude squeezing in a Mach-Zehnder fiber interferometer: Numerical analysis of experiments with microstructure fiber

Marco Fiorentino, Jay E. Sharping, Prem Kumar, and Alberto Porzio

Open Access Open Access

Abstract

We study a Mach-Zehnder nonlinear fiber interferometer for the generation of amplitude-squeezed light. Numerical simulations of experiments with microstructure fiber are performed using linearization of the quantum nonlinear Shrödinger equation. We include in our model the effect of distributed linear losses in the fiber.

©2002 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. Schematic model of the nonlinear fiber Mach-Zehnder interferometer. BS, beamsplitter; SA, spectrum analyzer.

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Fig. 2.
Fig. 2. (a) Schematic model of symmetric split-step Fourier propagation in a fiber. The fiber is divided in a large number of segments of length Δz. The dispersionis calculated at the midplane of each segment (here indicated with a dashed line). (b) Inclusion of loss is obtained by inserting beamsplitters of reflectivity Γ Δz between segments. The beamsplitters couple in modes νj(k) that are assumed to be in vacuum state.

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Fig. 3.
Fig. 3. Plot of QNR versus fiber transmittivity in case of distributed losses (continuous curve) and lumped losses (dashed curve). Propagation distance in the fiber is 4.3 soliton periods, T 1 = 0.1, the strong-pulse arm is injected with a fundamental soliton, and T 2 = 0.035.

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Fig. 4.
Fig. 4. A schematic of our experimental setup. The shaded area highlights the components that form the Mach-Zehnder polarization interferometer. OPO, optical-parametric oscillator; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarizing beamsplitter; M, mirror.

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Fig. 5.
Fig. 5. Plots of QNR corrected for detection losses in the PM-fiber (a) and MF (b) as a function of the energy of the strong pulse expressed in terms of squared soliton-number N 2. The experimental data (diamonds) are compared with numerical simulations (circles). (a) PM fiber, L = 4.3 soliton periods, T 1 = 0.1, T 2 = 0.032; (b) MF, L = 5.9 soliton periods, T 1 = 0.095, T 2 = 0.037, Γ = 0.01. The numerical simulation in (b) is limited to the maximum value of N 2 by the computational resources at our disposal.

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Equations (20)

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∂ U ̂ ∂ z = i U ̂ † U ̂ U ̂ + i 2 ∂ 2 U ̂ ∂ t 2 ,
U ̂ = U ¯ + u ̂ ,
∂ U ¯ ∂ z = i ∣ U ¯ ∣ 2 U ¯ + i 2 ∂ 2 U ¯ ∂ t 2 ,
∂ u ̂ ∂ z = 2 i ∣ U ¯ ∣ 2 u ̂ + i U ¯ 2 u ̂ † + i 2 ∂ 2 u ̂ ∂ t 2 .
u ̂ ( z , t ) ≡ ∑ j = 1 M u ̂ j ( z ) M ,
u ̂ j ( z ) ≡ ∑ m = 1 M ( μ jm ( z ) u ̂ m ( 0 ) + ν jm * ( z ) u ̂ m † ( 0 ) ) ,
∂ μ jm ∂ z = 2 i ∣ U ¯ ∣ 2 μ jm + iU ¯ 2 ν jm − i 2 FFT − 1 [ ω 2 FFT [ μ jm ] ] ,
∂ ν jm ∂ z = 2 i ∣ U ¯ ∣ 2 ν jm + iU ¯ 2 μ jm − i 2 FFT − 1 [ ω 2 FFT [ ν jm ] ] ,
U ¯ ‴ = R 2 U ¯ ′ + T 2 U ¯ ″ e iϕ ,
u ̂ ‴ = R 2 u ̂ ′ + T 2 u ̂ ″ e iϕ .
Φ 0 ∝ ∑ j , m = 1 M 〈 ( U ¯ j ‴ † U ̂ j ‴ − ∣ U ¯ ‴ ∣ 2 ) ( U ̂ m ‴ † U ̂ m ‴ − ∣ U ¯ ‴ ∣ 2 ) 〉
≃ R 2 ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( μ jm ′ e − i θ j + ν jm ′ e i θ j ) ∣ 2
+ T 2 ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( μ jm ″ e − i θ j + ν jm ″ e i θ j ) ∣ 2 ,
QNR = Φ 0 ∣ ( μ ′ , μ ″ , ν ′ , ν ″ ) Φ 0 ∣ ( μ jm ′ = μ jm ″ = δ jm , ν jm ′ = ν jm ″ = 0 ) .
∂ U ¯ ∂ z = ( i ∣ U ¯ ∣ 2 − Γ ) U ¯ + i 2 ∂ 2 U ¯ ∂ t 2 ,
u ̂ j ( z ) ≡ ∑ m = 1 M ( μ jm ( L ) ( z ) u ̂ m ( 0 ) + ν jm ( L ) * ( z ) u ̂ m † ( 0 ) ) + Γ Δ z ∑ k = 1 P ( ∑ m = 1 M ( ξ jm ( k ) v ̂ m ( k ) + η jm ( k ) * v ̂ m ( k ) † ) ) ,
Φ 0 ∝ R 2 ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( μ jm ′ e − i θ j + ν jm ′ e i θ j ) ∣ 2
+ R 2 ( Γ Δ z ) 2 ∑ k = 1 P ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( ξ jm ′ e − i θ j + η jm ′ e i θ j ) ∣ 2
+ T 2 ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( μ jm ″ e − i θ j + ν jm ″ e i θ j ) ∣ 2
+ T 2 ( Γ Δ z ) 2 ∑ k = 1 P ∑ m = 1 M ∣ ∑ j = 1 M ∣ U ¯ j ‴ ∣ ( ξ jm ″ e − i θ j + η jm ″ e i θ j ) ∣ 2 ,
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