Quantum noise of white light cavity using double-gain medium Yiqiu Ma, Haixing Miao, Chunong Zhao and Yanbei Chen For more details: DCC-P1400162 LVC meeting 2014 Stanford
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
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
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
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
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
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
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
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
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, 022002 (2001). LVC meeting 2014 Stanford 7
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
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
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
White-light-cavity idea 11 Introducing a medium: Negative-dispersion medium LVC meeting 2014 Stanford 11
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
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
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
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
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
Double gain medium Three-level atomic system with double pumps: 17 Three-level atomic system with double pumps: LVC meeting 2014 Stanford 17
Double gain medium Three-level atomic system with double pumps: 17 Three-level atomic system with double pumps: LVC meeting 2014 Stanford 17
Double-gain medium Some math: pumping power Line-width 18 Some math: pumping power Line-width Phase cancellation: LVC meeting 2014 Stanford 18
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
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
Stability condition: Nyquist theorem 21 Open-loop transfer function: Gain and phase margin: positive feedback Nyquist plot: Stable Unstable LVC meeting 2014 Stanford 21
Stability condition: Nyquist theorem 22 Open-loop transfer function: Fixing other parameters while changing Im[ H ] Re[ H ] LVC meeting 2014 Stanford 22
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
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
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
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
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