1. Towards a unified framework for improving quantum-limited detector sensitivity
Haixing Miao
University of Birmingham
List of contributors:
Rana Adhikari, Dominic Branford, Yanbei Chen, Animesh Datta, Stefan Danilishin, Matthew Evans, Andreas Freise, Farid Khalili, Yiqiu Ma, Denis Martynov, Nicholas Smith, Belinda Pang, Chunnong Zhao
ET Symposium 2017

2. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
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3. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
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4. Quantum noise Advanced LIGO design sensitivity curve:
Quantum noise is one of the limiting noise sources for current and future advanced GW detectors, including ET. This plot show the noise curves for Advanced LIGO and quantum noise is highlighted as red. It is important from the bucket and high frequencies.
Similar situations for ET and other advanced detectors
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5. Quantum noise Phase phase amplitude amplitude ET Symposium 2017 4
Quantum noise arises from quantum nature of light and its interaction with test mass. The field has random amplitude and phase, which can be described by a two-dimensional Gaussian distribution with the size related to the planck constant hbar. Because we are doing a differential measurement, the source of fluctuation comes from the dark port that performs the measurement. In this talk, the input and output are both referred to the dark port.
Phase
phase
amplitude
amplitude
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6. Standard Quantum Limit:
Standard Quantum Limit (SQL)
GW signal
Input field
(fluctuation)
Test mass
Output field (noise + signal)
Quantum fluctuation in phase quadrature
Shot noise
In amplitude quadrature
Radiation pressure noise
As you heard from Stefan and Kentaro’s talk, the quantum fluctuation in the phase gives rise to the shot noise that is dominated at high frequencies, and ampitude fluctuation leads to the radiation pressure noise. The trade-off between these two when varying the optical power defines the standard quantum limit.
Standard Quantum Limit:
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7. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
Standard quantum limit can be surpassed by many different approaches.
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8. Output field (noise + signal)
Frequency-dependent squeezing
GW signal
Input field
(fluctuation)
Input
filter
Test mass
Output field (noise + signal)
The first one is to inject frequency-dependent squeezed state of light at the dark port. It is realized by filtering squeezed light through an optical cavity which creates the proper rotating of the quadratures such that we have amplitude squeezing at low-frequency and phase squeezing at high frequency, which reduces the quantum noise over the entire frequency band.
squeezer
Phase
amplitude
[Kimble, et al. PRD 65, ]
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9. Output field (noise + signal)
Frequency-dependent readout
GW signal
Input field
(fluctuation)
Test mass
Output
filter
Output field (noise + signal)
The frequency-dependent readout also uses filter cavity at the output instead of the input, which allows to measure proper quadrature at different frequency. Its effect is to cancel the radiation pressure at low frequencies.
[Kimble, et al. PRD 65, ]
Coherent cancelation of radiation pressure noise by measuring proper
quadrature at different frequencies.
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10. Output field (noise + signal)
Optical spring effect
GW signal
Input field
(fluctuation)
Test mass
Output field (noise + signal)
As you heard from Kenatro, one can use the so-called optical spring effect to overcome the standard quantum limit. One interpretation is modification of the test mass response via a coherent feedback. Another equvilanent interpretation is the internal ponderomotive squeezing which correlates the amplitude and phase fluctuation via the radiation pressure on the test mass. Indeed, the outgoing field at the optical spring frequency is highly squeezed.
Signal recycling
(Kentaro’s talk)
[Buonanno & Chen, PRD 65, ]
Equivalent interpretation: internal (ponderomotive) squeezing
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11. Output field (noise + signal)
Speed meter
GW signal
Input field
(fluctuation)
Test mass
Output field (noise + signal)
Intra
filter
If an optical cavity is introduced inside the signal recycling cavity, we will obtain a speed meter when the signal-recycling cavity is tuned. This produces the same noise curve as the Sagnac speed meter which was presented by Stefan just before mine. When the signal-recycling cavity is detuned, the test mass dynamics is also modified in a frequency-dependent way as if the test mass has a lower mass. This is called the negative inertia effect.
(Stefan’s talk for Sagnac speed meter)
[Purdue & Chen, PRD 66, ]
Signal-recycling detuned
Negative inertia
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12. Output field (noise + signal)
White-light-cavity schemes
GW signal
Input field
(fluctuation)
Test mass
Output field (noise + signal)
Intra
filter
In addition to using passive filter cavities which do not have external energy input, one can also use active filters, e.g., atomic gain medium or unstable filter based upon radiation-pressured mediated optomechanical interaction. This is the idea of white light cavity, which is to enhance the signal response over a broader frequency range than the original detector.
[Wicht et al., Opt. Comm. 134, 431]
[Miao et al., PRL 115, ]
Improving shot-noise-limited sensitivity by increasing bandwidth.
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13. Output field (noise + signal)
General cases
GW signal
Input field
(fluctuation)
Input
filter
Test mass
Output
filter
Output field (noise + signal)
Intra
filter
Each filter can be a cascade of several passive/active filter cavities.
Hundreds and thousands of possible combinations.
Each filter
In principle, one can combine all these approaches together and use more complicated filters. The problem is that there are hundreds and thousands of possible combinations, and it is not possible to exhaust all the possibility. How can we combine them in a systematic way is the key question.
Question:
How do we combine different approaches/filters in a systematic way?
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14. Future study An old slide What is the next? Freq.-dep. squeezing Dual32
What is the next?
Freq.-dep.
squeezing
Dual
Carrier
Negative
inertia
Long
SR cavity
Variational
readout
Khalili
cavity
Sloshing cavity
Optical
bar
This is an old slide of mine presented in 2012 at the workshop in Hawaii when we were working on optimization of interferometer configurations. It was a kind of daunting task for me to complete the list of configurations. We were quite confused at that time and did not know what to do next.
Local readout
Sagnac
Optical
lever
Intra-cavity
filtering
GWADW 2012 Waikoloa Hawaii

15. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
The breakthrough comes from recent understanding of the fundamental quantum limit or energetic quantum limit, which is a sensitivity limit more stringent than the standard quantum limit.
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16. Fundamental (Energetic) Quantum Limit
A sensitivity limit that is more stringent than the SQL
In terms of quantum noise spectrum [1, 2]:
optical power spectrum
Intuitive understanding:
Number-phase uncertainty relation:
Time-energy uncertainty relation:
In terms of quantum noise spectrum, it says that the minimum quantum-noise level, or the maximum sensitivity, is bounded by the amount of optical power fluctuation inside the arm. You can also rewrite it either in terms of the spectrum for photon number or the optical energy. For intuitive understanding, we can look at number-phase uncertainty relation. Since we are measure the phase shift of light induced by GW, the error is bounded by the number fluctuation. Or equivalently, if we view GW detection as a timing measurement, the time resolution is ultimately bounded by the energy fluctuation that we have. That is why it is also called the energetic quantum limit.
Just having this lower bound is not enough, 0 is also a adequate lower bound but not so useful. It can provide useful guideline only if we know how to achieve the lower bound, namely, the inequality takes the equal sign.
How to achieve the minimum?
[1] V. Braginsky, M. Gorodetsky, F. Khalili, and K. Thorne, Energetic quantum limit in large-scale interferometers, AIP Conference Proceedings 523, 180 (2000).
[2] M. Tsang, H. Wiseman, and C. Caves, Fundamental Quantum Limit to Waveform Estimation, Phys. Rev. Lett. 106, (2011).
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17. General conditions for achieving FQL
The minimum quantum-noise spectrum:
Conditions for achieving it at different frequencies [1]:
1. Frequency-dependent readout (in general)
Not needed if focusing on a small frequency range.
2. Balanced response to upper and lower signal sidebands
Otherwise a factor of two degradation in the maximum sensitivity.
3. No optical loss.
Effect of loss is under study.
This question is answered by our recent work in which we derive the general condition for achieving the fundamental limit. Firstly, because this limit is frequency dependent, it requires frequency-dependent readout. If we are only interested in a small frequency range, this can be relaxed. Secondly, the response to upper and lower signal sidebands needs to be balanced. As I will show in a minute, a factor of two degradation to the sensitivity will happen if this is not the case as we loss half of the signal power. Finally, it assumes a lossless interferometer. We are still studying the effect of loss.
[1] H. Miao, R. Adhikari, Y. Ma, and Y. Chen, arXiv: (2016)
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18. Example of tuned signal recycling
Freq. dep. readout
Just give you a visual impression of this limit, here I will use dual-recycled Michelson as an example. When the signal-recycling is tuned, the fundamental quantum limit is simply the shot-noise limited sensitivity which is achieved by using the frequency-dependent readout.
FQL is equal to that
of the optimal frequency dependent readout in this case.
FQL is simply the shot-noise limited sensitivity
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19. Example of detuned signal recycling
Freq. dep. readout
When the signal-recycling cavity is detuned, the situation is more complicated, but nevertheless, the FQL overlaps with that of frequency-dependent readout pretty well apart from the detuning frequency where the issue of unbalanced signal response appears. You can see the factor or 2 or square root 2 in term of amplitude spectrum.
FQL is not exactly equal to that of optimal frequency dependent readout but within a factor of 2.
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20. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
Because the fundamental quantum limit only depends on one quantities, namely, the power fluctuation inside the arm. We can now put all different approaches in a unified framework.
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21. Output field (noise + signal)
A unified framework (picture)
GW signal
Input field
(fluctuation)
Input
filter
Test mass
Output
filter
Output field (noise + signal)
Intra
filter
Input filtering (squeezing):
Equivalent to increasing the optical power
The input filtering (squeezing) is equivalent to increasing the optical power which enhances the power fluctuation. The intra-cavity filters allow us to shape the frequency dependence of power fluctuation. The output filtering is needed to achieve the fundamental quantum limit at different frequencies.
Intra-cavity filtering:
Shaping the frequency dependence of power fluctuation
Output filtering (frequency-dependent readout):
Reaching the FQL at difference frequencies
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22. An example by applying this framework
SRM
With this idea, one can already create some rather futuristic configurations which cannot be found by using optimization. You can see that we can lower the noise spectrum before the quantum noise become irrelevant any more. This is just an example to highlight the principle, and not to take it too seriously at the moment. We stopped playing with this framework because we do not yet have a full understanding of the role of optical loss, which is crucial.
One passive filter to shape
the internal pondermotive
squeezing. Two unstable
filters to shape the phase.
Ideal lossless case
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23. Outline Quantum noise and Standard Quantum Limit (SQL)
Different approaches to surpassing SQL
Fundamental (Energetic) Quantum Limit
A unified framework
Remaining issues for completing the framework
Optical loss is one of the remaining issues in order to complete the framework.
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24. Output field (noise + signal)
Remaining issues
Optical loss:
GW signal
Input field
(fluctuation)
Input
filter
Test mass
Output
filter
Output field (noise + signal)
Intra
filter
Optical losses are presented in these filters.
Multiple-wavelength (carrier) interferometry:
Because all these filters will inevitably have some losses which degrades the quantum coherence. Another issue has to with multiple-wavelength or carrier interfemoetry which incorportates multiple carriers insight one interferometer. Right now, we have some preliminary results on this but not yet ready to present.
[D. Branford, Y. Ma et al. (in preparation)]
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25. Takeaway messages Different approaches in one “No” quantum limit
Ultimate limit: optical loss
These are the takeaway message that best summarizes this talk. The last one may be obvious but not so a year ago just from my personal viewpoint.
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