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Quantum noise of white light cavity using double-gain medium
Yiqiu Ma, Haixing Miao, Chunong Zhao and Yanbei Chen For more details: DCC-P LVC meeting 2014 Stanford
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Outline Background Quantum noise
Background Quantum noise Radiation-pressure noise and shot noise Mizuno theorem and white-light cavity gain (peak sensitivity) and bandwidth product double-gain medium with negative dispersion Stability condition and sensitivity gain Nyquist theorem Resulting sensitivity gain LVC meeting 2014 Stanford
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Outline Background Quantum noise
Background Quantum noise Radiation-pressure noise and shot noise Mizuno theorem and white-light cavity gain (peak sensitivity) and bandwidth product double-gain medium with negative dispersion Stability condition and sensitivity gain Nyquist theorem Resulting sensitivity gain LVC meeting 2014 Stanford
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Quantum noise Input Vacuum noise Output noise + signal
1 Input Vacuum noise Output noise + signal Input vacuum noise Test mass (mechanical) Output (noise+signal) Purely optical LVC meeting 2014 Stanford 1
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Radiation-pressure noise
Quantum noise 2 Radiation-pressure noise Input vacuum noise Test mass (mechanical) Output (noise+signal) Purely optical LVC meeting 2014 Stanford 2
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Radiation-pressure noise
Quantum noise 3 Radiation-pressure noise Input vacuum noise Test mass (mechanical) Output (noise+signal) Purely optical LVC meeting 2014 Stanford 3
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Quantum noise Input vacuum noise Test mass (mechanical) Output
4 Input vacuum noise Test mass (mechanical) Output (noise+signal) Shot noise Purely optical LVC meeting 2014 Stanford 4
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Quantum noise Input vacuum noise Test mass (mechanical) Output
5 Input vacuum noise Test mass (mechanical) Output (noise+signal) Shot noise Purely optical LVC meeting 2014 Stanford 5
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Quantum noise Input vacuum noise Test mass (mechanical) Output
6 Input vacuum noise Test mass (mechanical) Output (noise+signal) Purely optical LVC meeting 2014 Stanford 6
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Cancelling radiation pressure noise
7 Frequency-dependent readout In principle, this allows for a shot-noise only sensitivity. Reference: H. Kimble, Y. Levin, A. Matsko, K. Thorne, and S. Vyatchanin, PRD 65, (2001). LVC meeting 2014 Stanford 7
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Outline Background Quantum noise
8 Background Quantum noise Radiation-pressure noise and shot noise Mizuno theorem and white-light cavity gain (peak sensitivity) and bandwidth product double-gain medium with negative dispersion Stability condition and sensitivity gain Nyquist theorem Resulting sensitivity gain LVC meeting 2014 Stanford 8
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Mizuno theorem Shot-noise-only sensitivity Given different
9 Shot-noise-only sensitivity Increasing peak-sensitivity Given different signal recycling mirror reflectivity Decreasing detection bandwidth Only depends on power & arm cavity length LVC meeting 2014 Stanford 9
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Mizuno theorem For a single cavity (free space): Resonant condition:
10 For a single cavity (free space): A positive feedback but with a phase delay: Resonant condition: Only valid at one single frequency: Feedback changes sign LVC meeting 2014 Stanford 10
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White-light-cavity idea
11 Introducing a medium: Negative-dispersion medium LVC meeting 2014 Stanford 11
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Negative dispersion For usual mediums at low frequencies:
12 For usual mediums at low frequencies: positive (normal) dispersion Around absorption (attenuation) line: LVC meeting 2014 Stanford 12
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Negative dispersion For usual mediums at low frequencies:
13 For usual mediums at low frequencies: positive (normal) dispersion Around absorption (attenuation) line: LVC meeting 2014 Stanford 13
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Stable linear response: KK relation
14 Rule of thumb for Kramers-Kronig (KK) relation (global): Real part / Magnitude Imaginary part / Phase LVC meeting 2014 Stanford 14
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Stable linear response: KK relation
15 Rule of thumb for Kramers-Kronig (KK) relation (global): Lossless: Transmissivity around = 1 Gain medium with amplified “wing” LVC meeting 2014 Stanford 15
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Double gain medium Three-level atomic system with double pumps:
16 Three-level atomic system with double pumps: Pump fields: Signal field 1. Virtually populated level 3 2. Two frequencies at which there is gain for signal LVC meeting 2014 Stanford 16
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Double gain medium Three-level atomic system with double pumps:
17 Three-level atomic system with double pumps: LVC meeting 2014 Stanford 17
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Double gain medium Three-level atomic system with double pumps:
17 Three-level atomic system with double pumps: LVC meeting 2014 Stanford 17
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Double-gain medium Some math: pumping power Line-width
18 Some math: pumping power Line-width Phase cancellation: LVC meeting 2014 Stanford 18
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Outline Background Quantum noise
19 Background Quantum noise Radiation-pressure noise and shot noise Mizuno theorem and white-light cavity gain (peak sensitivity) and bandwidth product double-gain medium with negative dispersion Stability condition and sensitivity gain Nyquist theorem Resulting sensitivity gain LVC meeting 2014 Stanford 19
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Stability condition Open-loop transfer function:
20 Open-loop transfer function: Close-loop transfer function: Stability in terms of poles: LVC meeting 2014 Stanford 20
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Stability condition: Nyquist theorem
21 Open-loop transfer function: Gain and phase margin: positive feedback Nyquist plot: Stable Unstable LVC meeting 2014 Stanford 21
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Stability condition: Nyquist theorem
22 Open-loop transfer function: Fixing other parameters while changing Im[ H ] Re[ H ] LVC meeting 2014 Stanford 22
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Stability condition: Nyquist theorem
23 Open-loop transfer function: Fixing other parameters while changing Im[ H ] Sweeping through 1 when becomes larger Re[ H ] LVC meeting 2014 Stanford 23
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Stability condition: Nyquist theorem
24 Open-loop transfer function: Purple: Stable Red: Unstable (Lasing) with SR White: Unstable by the medium itself LVC meeting 2014 Stanford 24
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Sensitivity gain No additional noise White regime: gain > 1
25 No additional noise White regime: gain > 1 Also in the unstable regime LVC meeting 2014 Stanford 25
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Sensitivity gain With additional noise White regime: gain > 1
26 With additional noise White regime: gain > 1 Also in the unstable regime LVC meeting 2014 Stanford 26
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The end Background Quantum noise
27 Background Quantum noise Radiation-pressure noise and shot noise Mizuno theorem and white-light cavity gain (peak sensitivity) and bandwidth product double-gain medium with negative dispersion Stability condition and sensitivity gain Nyquist theorem Resulting sensitivity gain Thank you! LVC meeting 2014 Stanford 27
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