1. 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

2. 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

3. 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

4. Quantum noise Input Vacuum noise Output noise + signal1
Input
Vacuum noise
Output noise
+ signal
Input vacuum noise
Test mass
(mechanical)
Output
(noise+signal)
Purely
optical
LVC meeting 2014 Stanford 1

5. 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

6. 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

7. Quantum noise Input vacuum noise Test mass (mechanical) Output4
Input vacuum noise
Test mass
(mechanical)
Output
(noise+signal)
Shot noise
Purely optical
LVC meeting 2014 Stanford 4

8. Quantum noise Input vacuum noise Test mass (mechanical) Output5
Input vacuum noise
Test mass
(mechanical)
Output
(noise+signal)
Shot noise
Purely optical
LVC meeting 2014 Stanford 5

9. Quantum noise Input vacuum noise Test mass (mechanical) Output6
Input vacuum noise
Test mass
(mechanical)
Output
(noise+signal)
Purely
optical
LVC meeting 2014 Stanford 6

10. Cancelling radiation pressure noise7
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

11. Outline Background Quantum noise8
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

12. Mizuno theorem Shot-noise-only sensitivity Given different9
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

13. 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

14. White-light-cavity idea11
Introducing a medium:
Negative-dispersion medium
LVC meeting 2014 Stanford 11

15. 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

16. 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

17. Stable linear response: KK relation14
Rule of thumb for Kramers-Kronig (KK) relation (global):
Real part / Magnitude
Imaginary part / Phase
LVC meeting 2014 Stanford 14

18. Stable linear response: KK relation15
Rule of thumb for Kramers-Kronig (KK) relation (global):
Lossless: Transmissivity around = 1
Gain medium with amplified “wing”
LVC meeting 2014 Stanford 15

19. 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

20. Double gain medium Three-level atomic system with double pumps:17
Three-level atomic system with double pumps:
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21. Double gain medium Three-level atomic system with double pumps:17
Three-level atomic system with double pumps:
LVC meeting 2014 Stanford 17

22. Double-gain medium Some math: pumping power Line-width18
Some math:
pumping power
Line-width
Phase cancellation:
LVC meeting 2014 Stanford 18

23. Outline Background Quantum noise19
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

24. 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

25. Stability condition: Nyquist theorem21
Open-loop transfer function:
Gain and phase margin: positive feedback
Nyquist plot:
Stable
Unstable
LVC meeting 2014 Stanford 21

26. Stability condition: Nyquist theorem22
Open-loop transfer function:
Fixing other parameters
while changing
Im[ H ]
Re[ H ]
LVC meeting 2014 Stanford 22

27. Stability condition: Nyquist theorem23
Open-loop transfer function:
Fixing other parameters
while changing
Im[ H ]
Sweeping through 1
when becomes larger
Re[ H ]
LVC meeting 2014 Stanford 23

28. Stability condition: Nyquist theorem24
Open-loop transfer function:
Purple: Stable
Red: Unstable (Lasing)
with SR
White: Unstable by the
medium itself
LVC meeting 2014 Stanford 24

29. Sensitivity gain No additional noise White regime: gain > 125
No additional noise
White regime:
gain > 1
Also in the unstable regime
LVC meeting 2014 Stanford 25

30. Sensitivity gain With additional noise White regime: gain > 126
With additional noise
White regime:
gain > 1
Also in the unstable regime
LVC meeting 2014 Stanford 26

31. The end Background Quantum noise27
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