The use of weak-value metrology in the gravitational-wave detector

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(1)The use of weak-value metrology in the gravitational-wave detector. Mei-Ting Lin. A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science. Recommended for Acceptance by the Department of Physics Adviser: Feng-Li Lin. July 2019.

(2) Abstract Weak-value metrology refers to quantum measurement with a weak measurement process and post-selection. The outcome, called weak values, can be amplified beyond the eigenvalues of the observable; however, there is some debate on the usefulness of weak-value metrology in increasing the sensitivity of a gravitational wave detector. In this thesis, we investigated the sensitivity limit with regard to quantum shot noise and radiation pressure noise. For this purpose, we formulated an input-output relation with a model via weak-value metrology, which allowed us to understand the optical processes under a condition wherein the weak value is applied intuitively. However, we found that the sensitivity of the modified model was not improved as the modified model destroyed the symmetry of the interferometer, which contributed to additional noise. Despite signal amplification, increasing sensitivity to detect more gravitational wave events is more vital.. Keywords: Gravitational Wave Detector, Quantum Noise, Standard Quantum Limit, Weak Values. ii.

(3) Acknowledgements I would like to express my gratitude to the following people for their unceasing support over the past few years. I am profoundly grateful to my advisor, Prof. Feng-Li Lin, who introduced me to the Taipei Gravitational Wave Group (TGWG) and suggested this exciting project. I am also thankful for his guidance and support. Through him, I met Prof. Yiqiu Ma, with whom I discussed my research and during our discussion, Prof. Ma guided me with his insights and also generously shared the program code he had developed in the past, which I sincerely appreciate. I am also grateful to the co-organizers of TGWG, Prof. Guo-Qing Liu and Prof. Chian-Shu Chen, for their continued support and assistance in all respects. I also want to thank my groupmates in TGWG, Han-Shiang Kuo, Guo-Zhang Huang, Wei Ren Hsu, Chia-Wei Yang, Yun-Jing Huang, and Safdar Imam, and especially my colleagues, Jhao-Hong Peng, Yu-Kuang Chu, Han-Ting Chen, and DengRuei Tan. We have studied and discussed together and have lived a fulfilling life. They are my partners who always share satisfactions and hardships. Above all, I would like to thank my friends and my family for their unwavering support. I also express my deepest gratitude to my boyfriend, Po-Huang Chiu, for his patience and tolerance in the past. Thank you for staying with me and supporting me in my study and research. Finally, I sincerely thank my dear mother and my deceased father, for their significant role in my life and their numerous sacrifices.. iii.

(4) This thesis is dedicated to the memory of my father, Jin-Song Lin. You are gone, but your belief in me has made this journey possible.. iv.

(5) Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ii. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii. 1. Introduction. 1. 1.1. Introduction to Gravitational Waves . . . . . . . . . . . . . . . . . .. 3. 1.2. Introduction to Gravitational-Wave Detection . . . . . . . . . . . . .. 5. 1.2.1. The basic idea of interferometer . . . . . . . . . . . . . . . . .. 5. 1.2.2. Fabry–Perot arm cavities . . . . . . . . . . . . . . . . . . . .. 8. 1.2.3. Sideband . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. Introduction to Weak Measurements and Weak Values . . . . . . . .. 9. 1.3. 1.4 2. 1.3.1. Difference between strong and weak measurements . . . . . .. 11. 1.3.2. Weak-value amplification. . . . . . . . . . . . . . . . . . . . .. 15. Gravitational Wave Detector via Weak Measurement . . . . . . . . .. 16. Mathematical Description of Conventional Gravitational Wave Interferometers 18 2.1. Quantization of the Dynamics . . . . . . . . . . . . . . . . . . . . . .. 18. 2.2. Quantum States of the Optical Field . . . . . . . . . . . . . . . . . .. 21. 2.2.1. Fock or number state. . . . . . . . . . . . . . . . . . . . . . .. 21. 2.2.2. Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. v.

(6) 2.2.3 2.3. 2.4. 2.5 3. 4. Squeezed state . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. Basic Dynamic Processes of the Optical Field . . . . . . . . . . . . .. 26. 2.3.1. Light propagation in the interferometer . . . . . . . . . . . . .. 26. 2.3.2. Dynamics of the test mass. . . . . . . . . . . . . . . . . . . .. 29. Input–Output Relation of Basic Optomechanical System Model . . .. 30. 2.4.1. A single end mirror . . . . . . . . . . . . . . . . . . . . . . . .. 30. 2.4.2. Input–output relation . . . . . . . . . . . . . . . . . . . . . .. 32. Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . .. 39. Gravitational-Wave Detector Using Weak Values. 41. 3.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 3.2. Weak-Value Metrology in Gravitational-wave Detector . . . . . . . .. 42. 3.3. Input–Output Relation Using Weak Values . . . . . . . . . . . . . . .. 46. 3.4. Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 3.5. Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. Conclusion. 53. Bibliography. 55. vi.

(7) List of Figures 1.1. Strain caused by two polarizations of gravitational waves. . . . . . . .. 4. 1.2. Basic idea of gravitational-wave detector. . . . . . . . . . . . . . . . .. 6. 1.3. Basic idea of gravitational-wave detector.. The gravitational wave. passes through the interferometer, which makes the test mass shift and produces a slight difference between the two arms. . . . . . . . . 1.4. A Michelson interferometer with additional ITMX and ITMY (input test mass) to form arm cavities. . . . . . . . . . . . . . . . . . . . . .. 1.5. 7. 8. The spectrum of the optical field reflects from the moving mirror. The center peak represents the carrier light with frequency ω0 , and the two smaller peaks represent the signal sideband with frequency ω0 ± Ω, defined by the mirror motion spectrum x(Ω) . . . . . . . . . . . . . .. 1.6. The distribution of measuring device’s pointer in strong measurement (left) and weak measurement (right). . . . . . . . . . . . . . . . . . .. 2.1. 13. Phasor diagram for a vacuum state and a coherence state |α⟩ with complex amplitude α in Fig 2.1 (a) and (b), respectively. . . . . . . .. 2.2. 9. 24. ˆ2 Phasor diagram for (a) a squeezed state which squeezing in the X ˆ1 quadrature and (b) as a squeezed state which squeezing in the X quadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. 25.

(8) 2.3. ′ Schematic of the optical field Eˆ which is the optical field Eˆ propa-. gates the distance L from the reference position. The dashed line is represented the reference position. . . . . . . . . . . . . . . . . . . . .. 26. 2.4. Transmission and reflection of light by a mirror. . . . . . . . . . . . .. 27. 2.5. Transmission and reflection of light in a Fabry-Perot cavity. . . . . .. 28. 2.6. Schematic of optical field modulation by reflectiing mirror motion. The mirror provides motion x(t) which introduced by the action of external force G(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 2.7. Schematic of interaction between test mass and optical field. . . . . .. 31. 2.8. Schematic of conventional gravitational-wave detection and corresponding notation of field quadratures. . . . . . . . . . . . . . . . . .. 2.9. 33. The quantum noise in the gravitational-waves detector. The yellow curve represents the total quantum noise spectral density. The blue line is the shot noise, and the green line is radiation-pressure noise. The red line illustrates the standard quantum limit hSQL .. . . . . . .. 38. 2.10 Sensitivity curve of the conventional gravitational-waves detector. We set L = 4000 m, λ = 1064 ∗ 10−9 and m = 40 kg. The green curve is laser power I0 = 400kW , the blue curve is laser power I0 = 800kW and the yellow is laser power I0 = 1600kW . Red lines indicate that total quantum noise forms the boundary of sensitivity under different laser power, which is called the standard quantum limits (SQL). . . . 3.1. 40. Setup of Michelson gravitational-wave detector inspired by weak-value metrology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 3.2. Schematic of interferometer optical-field quadratures. . . . . . . . . .. 46. 3.3. Sensitivity curve of modifing gravitational-wave detector which is tuned by different phase. . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4 Weak value amplification in modify gravitational-wave detector. . . . 51 viii. 50.

(9) 3.5. The brown line presents the σ when detecting the signal in frequency 5Hz, and the blue and yellow line illustrate the σ while detecting the signal in frequency 10Hz and 100Hz, respectively. The amplification of weak value is shown as in red line. . . . . . . . . . . . . . . . . . .. 52.

(10) The use of weak-value metrology in the gravitational-wave detector Mei-Ting Lin August 22, 2019.

(11) Chapter 1 Introduction Recently, interest in gravitational waves has increased significantly. Gravitational waves are primarily detected using laser interferometers at various facilities across the globe. For instance, the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO)[1] [2] in the United States, Advanced Virgo in Italy [3], and Kamioka Gravitational-Wave Detector (KAGRA) [4] [5] in Japan. In early 2016, when LIGO announced that a gravitational-wave signal was detected in 2015 [6], it immediately became one of the most significant scientific breakthroughs of this century. The gravitational-wave detected by the LIGO on September 14, 2015, which is referred to as GW150914 [6], was the first direct gravitational-wave detected by humans. The existence of gravitational waves validates the last unproven theoretical prediction of general relativity. For GW150914, the two LIGO detectors in Louisiana and Washington States (U.S.) independently recorded the same signal at 0.007 seconds, which is exactly the time required for the gravitational wave to travel from Hanford to Livingston at the speed of light. GW150914 was identified as a combination of two black holes, the mass of 36 M⊙ and 29 M⊙ , respectively, and two black holes merge into one black hole of the mass 62 M⊙ . This event occurs at a distance of ≈ 410 M pc, 1.3 × 109 light-years. This activity is exciting for scientists all over the world. 1.

(12) Gravitational waves are the most crucial prediction of general relativity, and gravitational-wave detection is also one of the essential frontiers in contemporary physics. In particular, gravitational waves are of considerable significance to the study of the origin and evolution of the universe. Unlike methods of conventional astronomy, which employ electromagnetic radiation (i.e., infrared, visible light, ultraviolet light, X-rays, and gamma rays) for detection, gravitational waves explore the universe in entirely new ways and using new concepts. Gravitational-wave detection has become a new window in the human observational universe. Additionally, due to the joint operation of VIRGO, the positioning range of the gravitational-wave source has been made smaller, and the electromagnetic-wave signal for finding the gravitational-wave source has become achievable [7]. In the case of GW170817 [8], it can be said that it is an inspiring and significant discovery of gravitational waves. Astrophysics in the 21st century has officially entered the Multi-Messenger era. However, gravitational waves are difficult to detect. When gravitational waves arrive the Earth, their amplitude with strain is close to 10−21 ; thus, it requires a very sensitive detector and must overcome various noise sources that may overwhelm the target signal. Without extremely precise measurement technology, it is impossible to detect such an insignificant change; therefore, a gravitational-wave detector must be highly sensitive and employ a wideband detector. In order to develop gravitationalwave astronomy, the sensitivity of the detector must be improved. Quantum noise is intrinsic and should be overcome for gravitational-wave detection. Quantum noise covers all of the detection frequency of second-generation of LIGO which comes from the vacuum fluctuation of the electromagnetic field and it is consist of shot noise and radiation pressure noise. The noise caused by the phase fluctuation of the electromagnetic field is called shot noise and is the primary source at high-frequency in LIGO. The amplitude fluctuation of the electromagnetic field is the source of the radiation pressure noise, which dominates at low frequency. The 2.

(13) radiation pressure noise causes the variation of radiation pressure force to the mirror, which leads to the insensitivity of the mirror position. They are collectively called quantum noise. LIGO physicists have studied the use of a frequency-dependent squeezed input to suppress quantum noise[25]. It is impossible to reduce the uncertainty of phase and amplitude of the electromagnetic field at the same time. However, it can be overcome by sacrificing the one quadrature of the electromagnetic field and reduce the uncertainty of the other quadrature. The amplitude and phase quadrature satisfy the minimum of the uncertainty principle. In this thesis, we tried to improve the sensitivity of gravitational-wave detectors by exploring the use of different quantum measurements. Weak measurements and their weak-value amplifications were proposed by Aharonov et al. in 1988 [9]. In particular, weak-value amplification is considered to be one of the techniques that can accurately measure small parameters in a variety of experiments in the following decades. One of the most outstanding experiments is that the beam deflection measurement. This experiment uses the weak value metrology to detect the small deflections which are made with the piezo mirror to 400 frad [10]. The motivation of this thesis is to study the feasibility of using weak-value metrology to improve gravitational-wave detection.. 1.1. Introduction to Gravitational Waves. Albert Einstein first predicted gravitational waves in 1916, which is the result of general relativity. In this theory, mass (or energy) distorts the curvature of spacetime, which propagates through the universe at the speed of light, which we call. 3.

(14) gravitational waves. When gravitational waves propagate in flat space, in a weak field approximation, we can represent it as. gµν = ηµν + hµν , hµν << 1,. (1.1). where ηµν is the metric tensor of Minkowski space and hµν is the metric perturbation from Minkowski space caused by gravitational-wave. We choose the transverse traceless gage for hµν so that the linear Einstein equation becomes a wave equation:. (▽2 −. 1 ∂2 )hµν = 0, c2 ∂t2. (1.2). The solution to Eq. (1.2) is a plane wave with two polarizations denoted by ”+” and ”×”: ˆ + (t − z ) + bh ˆ × (t − z ). hµν = ah c c. (1.3). The strains caused by ”+”(plus) and ”×”(cross) polarizations of gravitational waves are shown in Fig. 1.1. Figure 1.1: Strain caused by two polarizations of gravitational waves.. 4.

(15) 1.2. Introduction to Gravitational-Wave Detection. 1.2.1 The basic idea of interferometer The gravitational-wave detector is a type of the Michelson interferometer, which has a Fabry–Perot cavity in each arm and other functional components. In principle, observing gravitational waves is done by measuring the change in distance between test masses at the ends of each arm and the beam splitter. In this case, to isolate seismic noise and reduce the thermal noise, the test masses are isolated and freely suspended which can be regarded as a pendulum. A monochromatic frequency-stabilized laser incident beam is divided by a beam splitter (BS) that allows half of the light to transmit and the other to reflect. The light after splitting travels into two arms, with distance LX and LY , and then are entirely reflected to BS by the end mirror that is referred to as an end test mass. We represent ETMX and ETMY as the end test mass at x and y arm, respectively. Then, the light in each arm is recombined at the BS and interfere with each other. Finally, the light is transmitted to a photodetector and the port where is set the laser source (Fig. 1.2). Note that we assume that all optics are lossless, the BS is a perfect 50/50 split, and the end mirrors reflect all light. Assuming that the length of two arms of the interferometer is L, the angular frequency of the light is ω0 , and the round trip length of the light traveling back and forth is 2L, the resulting phase shift is given by L ϕ(t) = ω0 tr = 2ωo . c The phase shift is a constant, and its value is proportional to L.. 5. (1.4).

(16) Figure 1.2: Basic idea of gravitational-wave detector.. Initially, the arm length of the interferometer is adjusted so that the two beams demonstrate completely destructive interference. There is no light coming into the photodetector. Because the two arms are equal in length, the phase shift produced by a single round trip in one arm is equivalent to the phase shift caused by the other arm. It means that while the lengths of the two arms are the same, the port where the laser source is set has constructive interference. Moreover, the port where the photodetector is placed has destructive interference. To simplify, the ports which are set laser source and photodetector are usually referred to as bright port and dark port, respectively. In sum, the symmetric laser interferometer gravitational-wave detector output signal is zero. When the gravitational wave arrives, one arm initially stretches, and the other arm shortens due to its polarization characteristics. Then, half a cycle later, the stretched and shortened arms are switched. Thus, the two coherent lights have different paths, which eliminates the initial destructive interference. Therefore, the amount of the photons propagate to the photon detector; thereby, it outputs a signal. The change. 6.

(17) in the relative arm length ∆L is used as the detector output signal to measure the gravitational-wave strain h.. ∆L = ∆Lx − ∆Ly = (L + δL) − (L − δL) = 2δL = hL,. h=. ∆L 2δL = , L L. (1.5). (1.6). where L is the detector arm length, and ∆L is the change in the relative arm length. Note that ∆L/c is also a relative phase difference according to Eq. (1.4). From Eq. (1.6), we can obtain the gravitational-wave strain by measuring the phase difference shown in Fig. 1.3. For typical gravitational-strain amplitude (h ≈ 10−21 ), the typical laser-wave vector k0 = ω0 /c ≈ 3 × 106 m−1 with a 4-km interferometer arm. Thus the phase shift is ∆ϕ ≈ 10−11 , which is too small to measure until we improve the sensitivity of the measuring device.. Figure 1.3: Basic idea of gravitational-wave detector. The gravitational wave passes through the interferometer, which makes the test mass shift and produces a slight difference between the two arms.. 7.

(18) 1.2.2 Fabry–Perot arm cavities. Figure 1.4: A Michelson interferometer with additional ITMX and ITMY (input test mass) to form arm cavities.. From Eq. (1.6), we know that a gravitational-wave signal is related to the ratio between δL and L, where L is the distance between the BS and the end mirror. Therefore, we can increase the sensitivity of the detector directly by lengthening the arm. However, constructing extremely long arms is expensive. Thus, rather than increasing the physical distance between the BS and the end mirror, we increase the length of the optical path. Here, the concept is that additional mirrors can be inserted into each arm between the BS and the end mirror., which is the so-called Fabry–Perot cavity. Fig. 1.4 shows the configuration of a Michelson interferometer with Fabry–Perot cavity. We label the additional mirrors as input test masses (ITMX and ITMY) and the original mirrors as end test masses (ETMX and ETMY). The distances between 8.

(19) the input and end test masses are expressed as LX and LY , respectively. As a result, we increase the distances LX and LY by introducing Fabry–Perot cavities.. 1.2.3 Sideband The input monochromatic laser field Ein with frequency ω0 is reflected on the end mirror with displacement motion x(t), which caused by the effect of external force G (i.e., gravitational waves). The phase of the reflected optical field is modulated by the mechanical motion of the mirror, allowing the output field’s spectrum of Eout to contain two sideband fields with frequencies of ω0 ± Ω, where ω0 is the carrier frequency of the input laser and Ω is the frequency of the gravitational waves as shown in Fig. 1.5.. Figure 1.5: The spectrum of the optical field reflects from the moving mirror. The center peak represents the carrier light with frequency ω0 , and the two smaller peaks represent the signal sideband with frequency ω0 ± Ω, defined by the mirror motion spectrum x(Ω). 1.3. Introduction to Weak Measurements and Weak Values. Weak-value was first defined by Aharonov, Albert, and Vaidman (AAV) in 1988 [9]. From 1987 until today, in the debate on the fundamentals of quantum mechanics, 9.

(20) weak value constructed an innovative and controversial theory. In this section, we introduce weak measurements and their corresponding weak values as follows. Aˆ is assumed to be an observable of the system to be measured, and |ψi ⟩ and |ψf ⟩ denote the initial and final states of the system, respectively. The weak value is then defined by. Aw :=. ⟨ψf | Aˆ |ψi ⟩ , ⟨ψf |ψi ⟩. (1.7). The key to enhancing the observable value is to select a final state that is approximately orthogonal to the initial state of the given system. The most surprising prediction based on this is that it can go far beyond the range of observable eigenvalues of the system’s state unexpectedly. Note that if |ψi ⟩ = |ψf ⟩, then the weak value is equivalent to the expected value. A general weak-value experiment involves the following steps, assuming that we want to measure a quantum system and that an observable of the given system is Aˆ (as mentioned above): 1. Pre-selection: Prepare an initial quantum state of the system. 2. Weak measurement: Standard measurement with weakening of the interaction. 3. Post-selection: Select the desired system state and project out the other state. Note that measurements in pre- and post-selected ensembles are referred to as preselection and postselection measurements. The outcomes of weak measurements are weak values can define as a new quantum variable. The weak-value mainly depends on how the initial and final states are oriented. The results lead to the idea of using weak-value amplification to improve the detection or to measure of small effects. In recent years, an increasing number of weak-value amplification experiments have been performed, and the theoretical and experimental results are mutually consistent [10].. 10.

(21) 1.3.1 Difference between strong and weak measurements Since the system is inevitably disturbed during the measurement process, weak measurement is a type of quantum measurement that uses a weak coupling to measure information in a quantum system and reduces interference with the system. As weak values are proposed on standard quantum measurement foundation, we introduce the standard measurement process first. Therefore, we will discuss the concept of weak measurement we mentioned before. In quantum measurement theory, all physical properties are based on observable that are manipulated by the Hermitian operator of the quantum system. The result of measurement of an observable Aˆ must be the eigenvalues of the given system. Considering the standard measurement procedure, it means that the observed system couple with the measuring device. The interaction Hamiltonian of the quantum system, whose observable Aˆ is the measured observable acting on the system, can be written as ˆ = g Aˆ ⊗ pˆ H. (1.8). where pˆ is the momentum variable related to the measuring device with conjugate ˆ is the variable qˆ and g is a normalized function of the coupling strength. The H Hermitian operator, which determines the dynamics of the system in time. The initial state of the system can be represented as a superposition of the eigenstates of the observable Aˆ |ψi ⟩ =. ∑. an |an ⟩ ,. (1.9). n. where an is the eigenvalue corresponding to |an ⟩. Moreover, the initial state of measuring device can be written as Eq. (1.10), which is initially in a Gaussian distribution centered at zero with variance ∆p and has welldefined initial and final values for pˆ. We read the difference between pf and pi from. 11.

(22) ˆ the measuring device, which corresponds to the observable A. ∫ |ϕi ⟩ =. ∫. dqϕ(q) |q⟩ in q-representation (1.10) ˜ |p⟩ in p-representation, dpϕ(p). =. ˜ where ϕ(q) and ϕ(p) are the Gaussian wave functions in q- and p-representation for the detector, respectively, and ignore the normalization factor. −. ϕ(q) = ⟨q|Φi ⟩ = e. q2 4(∆q)2. p2. ˜ = ⟨p|Φi ⟩ = e− 4(∆p)2 . and ϕ(p). (1.11). Before the interaction, the overall state of the system and the measuring device can be expressed a separable state |Ψ(0)⟩ := |ψi ⟩ ⊗ |ϕi ⟩ ∫ 2 ∑ − q = an |an ⟩ ⊗ dqe 4(∆q)2 |q⟩ .. (1.12). n. The system and measuring device are then coupled together via an interaction Hamiltonian to measure the observable Aˆ which is given by Eq. (1.8). We assume that the system is coupled to the measuring device in a short time, t0 . For simplicity, we use g(t) = δ(t − t0 ) as the Dirac function, where t0 is the time at which the interaction happens. The Eq. (1.12) evolve into the correlated state by the interaction Hamiltonian operator ˆ iHt. |Ψ(t)⟩ := e− ℏ [|ψi ⟩ ⊗ |ϕi ⟩] ∫ 2 ∑ i ˆ − p = an |an ⟩ dqe− ℏ Aˆp e 4(∆p)2 |p⟩ n. =. ∑. ∫ an |an ⟩. dpe. n. =. ∑. an |an ⟩ |ϕan ⟩ ,. n. 12. −. (p−an )2 4(∆p)2. |p⟩. (1.13).

(23) where |ϕan ⟩ represents the measuring device state which is shifted by quantity an after measurement procedure. This interaction results in a perfect correlation between the device and the system. From this indirect measurement of interaction, we can indicate the physical properties by reading out the pointer. Thus, the pointer state is correlated to the observable of the system. However, the wave function will collapse into some eigenstate while we measure quantum states. As the initial quantum state collapses into the other state, the information related to the initial state will be lost due to the observation. Thereby, the basic idea of using weak measurement in the weak-value metrology is that interaction (or perturbation) between the measuring device and the observed system is sufficiently weak that the wave function does not collapse right away. If the variance of the distribution of measuring device’s pointer is much smaller than the difference between the shift of pointer( as shown in Fig1.6), we can obtain the information of the system. But it also means that the interference to the system is very intense. In the opposite case, the variance of the pointer distribution is much larger than the shift of the pointer, as the coupling between the system and the measuring device is weak. It is difficult to obtain information from this type of measurement, but after post-selection, with the final state of the system to read out observable. In sum, weak values can also be considered as a conditional probability.. Figure 1.6: The distribution of measuring device’s pointer in strong measurement (left) and weak measurement (right).. 13.

(24) Because the interaction between the system and the device is weak, we up to the first order of gt |Ψ(t)⟩ = e−. ˆ iHt ℏ. |ψi ⟩ ⊗ |ϕi ⟩. ˆ ig pÂt ) |ψi ⟩ |ϕi ⟩ ℏ igt ˆ = |ψi ⟩ |ϕi ⟩ − A |ψi ⟩ pˆ |ϕi ⟩ . ℏ. ≈ (I −. (1.14). We then project out the evolved state |Ψ(t)⟩ into the post-selected state |ψf ⟩, leaving only the state of the measuring device as the pointer state to be readout. After renormalizing the Eq (1.14), we get the final pointer state |Φf ⟩ =. ⟨ψf |Ψ(t)⟩ ⟨ψf |ψi ⟩. = |ϕi ⟩ −. igt ⟨ψf | Aˆ |ψi ⟩ pˆ |ϕi ⟩ ℏ ⟨ψf |ψi ⟩. (1.15). ≈ e−iˆpAw /ℏ |ϕi ⟩ = |ϕAw ⟩ , where AAV [9] is defined by the Eq. (1.7). Aw :=. ⟨Ψf | Aˆ |Ψi ⟩ , ⟨Ψf |Ψi ⟩. (1.16). ˆ and |ϕAw ⟩ represents the pointer is shifted by which is called the “weak value of A” Aw after measurement procedure. We rewrite the weak value as Aw = ReAw + iImAw and the shift in pointer momentum qˆ can be represented as δ qˆ = ⟨Φf | qˆ |Φf ⟩ − ⟨Φi | qˆ |Φi ⟩ =−. igt ReAw ⟨Φi | (ˆ q pˆ − pˆqˆ) |Φi ⟩ ℏ. gt q pˆ + pˆqˆ) |Φi ⟩ + ImAw ⟨ψf |ψi ⟩ ⟨Φi | (ˆ ℏ = −gtReAw . 14. (1.17).

(25) Since for a Gaussian measuring device iℏ 2 −iℏ ⟨Φi | pˆqˆ |Φi ⟩ = . 2 ⟨Φi | qˆpˆ |Φi ⟩ =. (1.18). The pointer shift in weak measurement and post-selection is proportional to the real part of the weak value. In summary, there is a low strength g, where the observed system is least interfered by the interaction. The pointer of measuring device shift but the shape of the probability distribution does not change significantly. However, the observable can still be measured after post-selection, and the quantity of observable is proportional to the pointer shift.. 1.3.2 Weak-value amplification The most interesting thing is that weak value is not limited by the minimum and maximum values of observable A. From the definition of the weak value in Eq. (1.7), we can see that the weak value depends on the orientation of the initial state and the final state. If the initial state and the final state are close to orthogonal, the denominator in Eq. (1.7) approaches zero so that the weak value will enhance. We can use weak measurements for the amplification process if we wisely choose the state after the post-selection. In other words, if |ψf ⟩ is almost orthogonal to |ψi ⟩, we will get the amplification we want so that we can use it to amplify the signal. However, we need to consider a trade-off between the process of amplification and post-selection since there are costs in the Weak-value amplification. For example, when |ψf ⟩ and |ψi ⟩ are almost orthogonal, and the output signal strength is quite small while we only choose the post-selection state and drop out others, whereas, the 15.

(26) amplification factor of the weak value is high. The probability of weak value with post-selection is P = | ⟨ψf |ψi ⟩ |2 .. (1.19). The weak value has been proven to be useful in the amplification of detecting small parameters in quantum metrology. In the research of an beam deflection measurement [10], weak measurements are used to amplify small momentum disturbances in the Sagnac interferometer.. 1.4. Gravitational Wave Detector via Weak Measurement. In particular, weak-value metrology is considered to be one of the methods for measuring precise small parameters in various experimental environments. One of the accurate measurements is gravitational-wave detection. This paper is based on the accuracy measurement research of quantum theory and discusses quantum noise in the gravitational-wave detector. In recent years, Nishizawa[15][16], Nakamura, and Fujimoto [14] have published discussions on the topic of the gravitational-wave detector inspired d by weak-valuevalue metrology. As a brief introduction, in the paper[16], the primary purpose is to investigate the weak value in the interferometer in the Heisenberg picture and take into account quantum shot noise and radiation pressure noise. In the Heisenberg picture, the operators corresponding to the observable evolve, while the state of quantum systems is time-dependent. It is crucial to notice that we use the observable value in dynamics, such as position, momentum. Since the Heisenberg picture focuses on the evolution of observable over time; the dynamics can be easily observed. However, the conventional gravitational-wave detector uses continuous monochromatic light for the input source [25], while the research in [14] uses the pulse as 16.

(27) an input source. Thus Nakaruma and Fujimoto reconfirmed the extension of the input-output relation for the Michelson interferometer with coherent light sources to compare with the conventional gravitational-wave detector. In general, the input-output relations in the gravitational-wave detector are derived through the two-photon formalism[21][22]. Nevertheless, the input-output relations for the gravitational-wave detector in these two references are derived without the two-photon formulation [16][14]. In this thesis, to compare with the standard input-output relation, and to investigate whether the weak-value metrology can overcome quantum limit and amplify the signal. We try to derive input-output relation using two-photon formalism and a linear optical path algorithm to consider the quantum noise and the weak-value metrology in a more intuitive way.. 17.

(28) Chapter 2 Mathematical Description of Conventional Gravitational Wave Interferometers To analyze quantum noise, we will discuss the quantization of the optical field in Section 2.1 and the typical quantum states usually used in the gravitational-wave interferometers in Section 2.2. After understanding the dynamics of the optical system in Section 2.3, we then also derive the input-output relation of the gravitational-wave interferometer and plot the sensitivity curve in Section 2.4.. 2.1. Quantization of the Dynamics. Here, we briefly introduce the quantization of an optical field. The input optical field is represented as { } (+) (−) ˆ ˆ ˆ Ein (x, y, z; t) = u(x, y, z) Ein (t) + Ein (t) ,. (2.1). (+). where u(x, y, z) is the spatial mode, Ein is the positive frequency component of the optical field that contains all terms that oscillate as e−iωt for ω > 0 , and Ein is the (−). negative frequency component. 18.

(29) We can write the positive frequency part of the optical field as a standard expression:. ∫ (+) Eîn (t). = 0. ∞. dω 2π. √. 2πℏω a ˆω e−iωt , Ac. (2.2). where A is the effective cross-section area of the light beam, a ˆω and a ˆ†ω are the annihilation and creation operators of a single photon in the frequency ω of the field. The commutation relations are accordingly satisfied: [ˆ aω , a ˆω′ ] = 0, [ˆ aω , a ˆ†ω′ ] = 2πδ(ω − ω).. (2.3). Note that we ignore the spatial mode u(x, y, z) because it does not contribute to quantum noise spectral density. Similarly, the negative-frequency part of the optical field is (−) (+)† Eîn (t) = Eîn √ ∫ ∞ dω 2πℏω † iωt = a ˆ e . 2π Ac ω 0. (2.4). Eqs. (2.2) and (2.4) satisfy the following equation: ∫ (1/A). dxdy|u(x, y, z)|2 = 1.. (2.5). In the gravitational-wave interferometer, the input monochrome laser has a carrier frequency of ω0 , while the gravitational-wave signal Ω (ranging from 10 Hz to 104 Hz) will produce ω0 modulation, forming pairs of sidebands of frequency ω = ω0 ± Ω; therefore, we introduce the sideband operators to analyze quantum noise by defining the upper and lower sideband operators as. a ˆ+ ≡ a ˆω0 +Ω ,. a ˆ− ≡ a ˆω0 −Ω .. 19. (2.6).

(30) Then, the Eq. (2.2) can be rewritten as √ (+) Eîn (t). =. 2πℏω −iω0 t e Ac. ∫. ∞. 0. dΩ (ˆ a+ e−iΩt + a ˆ− e+iΩt ). 2π. (2.7). We also invoke two-photon formalism, which was developed in 1985 by Caves and Schumaker and is widely used in gravitational-wave detectors [21][22]. We can define the field quadratures as. a ˆ1 =. a ˆ+ + a ˆ† √ −, 2. a ˆ2 =. a ˆ+ − a ˆ† √ −, 2i. (2.8). where a ˆ1 and a ˆ2 correspond to amplitude and phase quadrature operators, respectively. This operators, which is referred to as a two-photon quadrature operator, satisfy the following commutation relation in the frequency domain:. [ˆ a1 , a ˆ1′ ] = [ˆ ˆ†1′ ] = [ˆ a†1 , a ˆ†1′ ] = [ˆ a1 , a ˆ2′ ] = [ˆ ˆ†2′ ] = 0, a1 , a a†1 , a. (2.9). ′. where the operator a ˆ1′ and a ˆ2′ denote frequency ω0 + Ω . Therefore, we can represent the optical field operator as √ Eîn (t) =. ∫ ∞ dΩ 4πℏω0 [cos(ω0 t) (ˆ a1 e−iΩt + a†1 e+iΩt ) Ac 2π 0 ∫ ∞ dΩ + sin(ω0 t) (ˆ a2 e−iΩt + a†2 e+iΩt )]. 2π 0. (2.10). Also, we can simplify Eq. (2.10) as √ Eîn (t) = where. 4πℏω0 [ˆ a1 cos(ω0 t) + a ˆ2 sin(ω0 t)], Ac. ∫ a ˆ1,2 ≡ 0. ∞. dΩ (ˆ a1,2 e−iΩt + a†1,2 e+iΩt ). 2π. 20. (2.11). (2.12).

(31) Similarly, the output field can be expressed in the same way as for the input field with the annihilation operator now denoted as ˆb: √ Eôut (t) =. 4πℏω0 ˆ [b1 cos(ω0 t) + ˆb2 sin(ω0 t)]. Ac. (2.13). These are known as two-photon formalism.. 2.2. Quantum States of the Optical Field. In this section, we introduce three quantized optical states that are important in the gravitational-wave detector.. 2.2.1 Fock or number state The definition of fock state is 1 ˆ |n⟩ = ℏω(ˆ H a† a ˆ + ) |n⟩ . 2. (2.14). In general, the operators annihilate and create photons, √ a ˆ |n⟩ = n |n − 1⟩ , √ a ˆ† |n⟩ = n + 1 |n + 1⟩ .. (2.15a) (2.15b). If we repeat this process, we move the energy ladder down until we get. a ˆk |0⟩ = 0.. (2.16). The state |0⟩ is referred to as the vacuum state. Note that even for vacuum state |0⟩, there are non-zero fluctuations that originate from random fluctuations in the electric. 21.

(32) field. Furthermore, the vacuum fluctuations that enter the dark port can limit the sensitivity of the gravitational-wave detector.. 2.2.2 Coherent state For any complex number α, we first define the unitary displacement operator as † ∗ ˆ D(α) ≡ eαâ −α aˆ .. (2.17). Then the coherent state |α⟩ is defined as ˆ |α⟩ ≡ D(α) |0⟩ .. (2.18). The coherent state |α⟩ is considered as a displacement vacuum state. The coherent state |α⟩ can also be defined as an eigenstate of annihilation operator a with an eigenvalue α, i.e., a ˆ |α⟩ = α |α⟩ .. (2.19). Next, we can discuss the uncertainty of the coherent state. We define the quadrature operators, which are analogues of position and momentum operators, are given as follows, respectively, √ ˆ= Q. ℏ † (ˆ a +a ˆ), 2ω. Pˆ =. 22. √ 2ℏωi(ˆ a† − a ˆ).. (2.20).

(33) We can derive the variance of an operator O from the relation (∆O)2 = ⟨O2 ⟩ − ⟨O⟩2 . Thus, we have the variances of position and momentum operators in a coherent state. 2. ℏ , 2ω ℏω = . 2. ˆ (∆Q) coh = 2. (∆Pˆ )coh. (2.21a) (2.21b). We derive the minimize Heisenberg uncertainty relation. ℏ ˆ ˆ (∆Q) . coh (∆P )coh = 2. (2.22). In order to introduce the dimensionless operators, we redefine the equivalent quadrature operators as ˆ 1 = 1 (ˆ X a† + a ˆ), 2 ˆ 2 = i (ˆ X a† − a ˆ). 2. (2.23a) (2.23b). The corresponding uncertainty principle of coherence state is. ˆ 1 ) (∆X ˆ 2 ) = 1. (∆X coh coh. (2.24). ˆ 1 and X ˆ 2 quadrature phase, and The coherence state has equal uncertainty in the X ˆ 1 and X ˆ 2 as it can be represented by error circle in a phasor diagram in the axes X shown in Fig 2.1.. 23.

(34) Figure 2.1: Phasor diagram for a vacuum state and a coherence state |α⟩ with complex amplitude α in Fig 2.1 (a) and (b), respectively.. 2.2.3 Squeezed state Coherent states are specific states with minimum and equal uncertainty in two quadrature phases as Eq (2.24). We seek to determine whether other states can be minimal uncertainty. ˆ1, X ˆ 2 ). In Fig. 2.1, the coherent state is represented as a circle in phase space (X ˆ 1 ) (∆X ˆ2) = There are other series of minimum uncertainty states to satisfy (∆X coh coh 1. In the case of quadrature squeezing which satisfies the Heisenberg uncertainty principle if we write. ˆ 1 < 1, ∆X ˆ 2 > 1 and ∆X ˆ 2 ∆X ˆ 2 ≥ 1. ∆X. (2.25). In another case, we write. ˆ 1 > 1, ∆X ˆ 2 < 1 and ∆X ˆ 2 ∆X ˆ 2 ≥ 1. ∆X. 24. (2.26).

(35) The states which satisfy the conditions in Eqs. (2.25) and (2.26) will have less uncertainty in one of the quadratures than for a coherent state while the fluctuations in another quadrature will be enhanced, as shown in Fig. 2.2.. ˆ 2 quadrature Figure 2.2: Phasor diagram for (a) a squeezed state which squeezing in the X ˆ and (b) as a squeezed state which squeezing in the X1 quadrature.. Thus, graphically, the squeezed states are ellipses. We define a unitary squeeze operator as 1 ˆ S(ξ) ≡ exp[ (ξ ∗ a â ˆ − ξˆ a† a ˆ† )], 2. (2.27). ξ = reiϕ ,. (2.28). where. here r is represented as the squeeze parameter and 0 ≤ r < ∞. The phase ϕ is 0 ≤ ϕ ≤ 2π. A more general squeezed state is obtained by the squeeze operator act on the coherent state, ˆ D(α) ˆ |α, ϵ⟩ = S(ϵ) |0⟩ . Note that we obtain a coherent state when ξ = 0. 25. (2.29).

(36) Although we sacrificed the uncertainty of a quadrature, we also added another precision, and we can use the squeezing on the variables we want to measure.. 2.3. Basic Dynamic Processes of the Optical Field. In this section, we further study the dynamic processes of the optical field. Thus, the relation of the optical field we encounter in gravitational-wave interferometers will become very simple.. 2.3.1 Light propagation in the interferometer. ˆ ′ which is the optical field E ˆ propagates the Figure 2.3: Schematic of the optical field E distance L from the reference position. The dashed line is represented the reference position.. To understand how to calculate quantum noise in the gravitational-wave interferometer, we need to understand the situations that light is reflected from optical elements such as mirrors and beamsplitter. To simplify the problem, we first consider where the optical components are fixed. The effects of the mirror motion will be considered in the next subsection 2.3.2. Now we consider the most straightforward case, which is a free propagation distance of L, as shown in Fig 2.3. The new field Eˆ ′ (t) can be written as ˆ − τ ), Eˆ ′ (t) = E(t where τ ≡ L/c is retarded time. 26. (2.30).

(37) Next, we will discuss the continuity condition in the interferometer. The mirror can be approximated as geometrically thin; thus, the output field E1 is linearly related to the input field E2 . We can express the continuity condition on the mirror (Fig. 2.4) as Eˆ2 (t) = rEˆ1 (t) + tEˆ4 (t),. (2.31). Eˆ3 (t) = t′ Eˆ1 (t) + r′ Eˆ4 (t).. Figure 2.4: Transmission and reflection of light by a mirror.. While the mirror is lossless, we can infer that the transfer matrix M of the mirror is a unitary operator (M−1 M = M† M = 1). Thus, the operator can be represented in . matrix form as. . r t  M= . ′ ′ t r. (2.32). Energy conservation can bring about the relationship between reflectivity and transmittance which yields |r| = |r′ |, and |t| = |t′ |,. (2.33a). |r|2 + |t|2 = 1,. (2.33b). r∗ t′ + r′ t∗ = 0, and r∗ t + r′ t∗ = 0.. (2.33c). We can rewrite the transfer matrix M with reflectivity R and transmissivity T as R = |r|2 , T = |t|2 . 27. (2.34).

(38) Thus, we can obtain the transfer matrix of the mirror as .  √ √ T − R M= √ √ . T R. (2.35). We derive the continuity condition on the mirror as √ √ Eˆ2 (t) = − REˆ1 (t) + T Eˆ4 (t), √ √ Eˆ3 (t) = T Eˆ1 (t) + REˆ4 (t).. (2.36). Also, the continuity condintion is such that for the cavity configuration (Fig. 2.5): √ √ Eˆ2 (t) = − REˆ1 (t) + T Eˆ4 (t), √ √ Eˆ3 (t) = T Eˆ1 (t) + REˆ4 (t), Eˆ4 (t) = Eˆ6 (t − τ ), Eˆ5 (t) = Eˆ3 (t − τ ), Eˆ5 (t) = Eˆ6 (t).. Figure 2.5: Transmission and reflection of light in a Fabry-Perot cavity.. 28. (2.37).

(39) 2.3.2 Dynamics of the test mass Assuming the end mirrors are test masses, the equation of motion for the end-mirror displacement xˆ becomes. xˆ˙ (t) = pˆ(t)/m,. (2.38a). ¨ ˆ = Fˆrp + m Lh(t), pˆ(t) = Fˆrp + G(t) 2. (2.38b). where xˆ and pˆ are the position and momentum operators of the mirror, which have the commutator relation [ˆ x, pˆ] = iℏ. In Eq. (2.38b), the first term is the radiation– pressure force, which is due to the quantum fluctuation of the optical amplitude and ˆ provides radiation–pressure noise. The second term G(t) is the external force (i.e., the gravitational waves tidal force) on the end mirror. Combining the equations, we can get m ¨ mx¨ˆ(t) = Fˆrp + Lh(t). 2. (2.39). The radiation-pressure force is given by: [. 2 I0 A Fˆrp = 2 |Eîn (t − τ )| = 2 1+ 4π c. √. ]. ℏω0 a ˆ1 (t − τ ) , I0. (2.40). where I0 is laser power, c is the speed of light and a ˆ1 is amplitude quadrature of the carrier field. However, we are only interested in the perturbed part, which represents radiation– pressure noise, while the trivial DC component can be safely ignored. Thus, the equation of motion for the test mass in Eq. 2.39 can be written as √ m¨ x=2. ℏω0 I0 1 ¨ mLh(t). a ˆ (t − τ ) + 1 c2 2. 29. (2.41).

(40) The reflected light is modulated by mechanical motion so that the spectrum of the output field Eôut out contains two sidebands with information about the external force.. Figure 2.6: Schematic of optical field modulation by reflectiing mirror motion. The mirror provides motion x(t) which introduced by the action of external force G(t).. 2.4. Input–Output Relation of Basic Optomechanical System Model. In this section, we discuss the basis of the gravitational-wave detection model, which is an optomechanical system by coupling the propagating light field and moving mirrors driven by gravitational waves. We assume that the incident laser beam is reflected off a mirror with displacement xˆ(t) that is much less than the wavelength of the light.. 2.4.1 A single end mirror An ideal single-mode laser with a center frequency ω0 can be regarded as quantum fluctuations superimposed on top of the classic carrier field, and we represent it in two-photon formalism as Eq. (2.11) √ Eîn (t) =. 4πℏω0 Ac. {√ (. } I0 +a ˆ1 ) cos ω0 t + a ˆ2 sin ω0 t , ℏω0 30. (2.42).

(41) Figure 2.7: Schematic of interaction between test mass and optical field.. where I0 is laser power. a ˆ1 and a ˆ2 are amplitude and phase modulations of the carrier field. In addition, the output optical field is given as follows. √ Eôut (t) =. 4πℏω0 Ac. {√ (. } I0 + ˆb1 ) cos ω0 t + ˆb2 sin ω0 t , ℏω0. (2.43). where ˆb1 and ˆb2 are the output quadratures. In order to derive the input–output relation of the gravitational-wave detector, we must consider the relation between the input and output optical fields, Eîn and Eôut . The input optical field is reflected off end mirrors, which have tiny displacements xˆ driven by gravitational waves. xˆ(t − τ ) Eôut (t) = Eîn (t − 2τ − 2 ), c. (2.44). where delay time can be defined as τ ≡ L/c, and xˆ is the motion of the test mass, which is determined by external force and the mirror response function. We employ Taylor expansion on Eq. (2.44) in a series of ω0 xˆ/c because the displacement of the test mass is small. For simplicity, we assume that ω0 L/c = nπ ,. 31.

(42) where n is the integer. { √ ( ) ( I 4πℏω xˆ ) xˆ 0 0 ˆ Eout (t) = +a ˆ1 (t − 2τ − 2 ) cos ω0 (t − 2 ) Ac ℏω0 c c ( )} xˆ +a ˆ2 (t − 2τ − 2ˆ xc) sin ω0 (t − 2 ) . c √. (2.45). Therefore, the first order of Eq. (2.45): √ Eôut (t) ≈. { 4πℏω0 a ˆ1 (t − 2τ ) cos (ω0 (t − 2τ )) Ac [ ] } √ I0 ω0 + a ˆ2 (t − 2τ ) + 2 xˆ(t − τ ) sin(ω0 (t − 2τ )) . ℏω0 c. (2.46). We obtain the input–output relation in the form of amplitude and phase quadratures below.. ˆb1 (t) =ˆ a1 (t − 2τ ),. √. ˆb2 (t) =ˆ a2 (t − 2τ ) + 2. (2.47a) I0 ω0 xˆ(t − τ ). ℏω0 c. (2.47b). 2.4.2 Input–output relation While gravitational waves pass by, the test-mass (end mirror) can move and provide relative motion, thus we can read out the signal at the output port (Fig. 2.7). From Eq. (2.47), we derived the equation of motion for the end mirror in one arm. Next we considered the motion of two end mirrors (x and y arms) represented in the frequency domain. Because the tidal effect of the gravitational waves on the x and y arms are opposite, thus we have. 32.

(43) Figure 2.8: Schematic of conventional gravitational-wave detection and corresponding notation of field quadratures.. m 2ˆ Ω Xx (Ω) = Frp + G(Ω) 2 √ ℏω0 I0 1m 2 =2 e ˆ + LΩ h(Ω), 1 c2 22 m 2ˆ Ω Xy (Ω) = Frp − G(Ω) 2 √ ℏω0 I0 ˆ 1 m 2 =2 f1 − LΩ h(Ω). c2 22 Note the sign difference caused by the strain h(Ω). Here. m 2. (2.48a). (2.48b). is the reduce mass. of the end mirror and the BS, which have equal mass m, and Ω is the frequency of ˆ x (Ω) and X ˆ y (Ω) represent the small mirror displacement in x gravitational waves. X and y arm respectively, which caused by gravitational waves. Following the notation illustrated in Fig. 2.7, î represents the quadrature of the input laser, and a ˆ, ˆb describe the quadrature of the input and output field at the dark ′ ˆ dˆ′ are the quadratures of the optical field at the BS in each port, respectively. cˆ, cˆ , d, ′ ′ arm; eˆ, eˆ , fˆ, fˆ denote the quadratures of the optical field at the end mirrors in each. 33.

(44) arm. From Eq. (2.47), we can write down the input-output relation for x arm. ′. eˆ1 (Ω) = eˆ1 (Ω), ′. (2.49a). √. ˆ x (Ω) I0 2ω0 X ℏω0 c √ 8I0 ω0 I0 ω0 L2 e ˆ (Ω) + h(Ω). = eˆ2 (Ω) + 1 mΩ2 c2 ℏc2. eˆ2 (Ω) = eˆ2 (Ω) +. (2.49b). Similarly, we can write the input-output relation for y arm. ′ fˆ1 (Ω) = fˆ1 (Ω),. (2.50a). √. ˆxΩ I0 2ω0 X , ℏω0 c √ 2 8I ω 0 0 ˆ1 (Ω) − I0 ω0 L h(Ω). f = fˆ2 (Ω) + mΩ2 c2 ℏc2. ′ fˆ2 (Ω) = fˆ2 (Ω) +. (2.50b). From Eqs. (2.48), (2.49), and (2.50), we obtain the coupling relation between the optical field and the moving mirror (driven by the gravitational-wave). For a more explicit representation, we represent these equations in matrix form:   eˆ1 (Ω)  1  ′ = 8I0 ω0 eˆ2 (Ω) mΩ2 c2    ′ ˆ f1 (Ω)  1 =  ′ 8I0 ω0 ˆ f2 (Ω) mΩ2 c2 . ′.    0 eˆ1 (Ω)  0    + √  h(Ω), I0 ω0 L2 1 eˆ2 (Ω) ℏc2     ˆ 0 f1 (Ω)  0   − √   h(Ω). I0 ω0 L2 ˆ 1 f2 (Ω) ℏc2 . (2.51a). (2.51b). This relation is an essential part of the gravitational-wave detector. We can detect the optical field at the output port and read the mirror displacement, which is driven by gravitational waves.. 34.

(45) Next, we consider how the optical field propagates along the x and y arms which we consider to be as free space. The delay time as photons propagate from the BS to the end mirrors is given by τ = L/c, where L is the length of each arm. .        ′ ′ ′ ˆ dˆ (Ω) eˆ1 (Ω) cˆ (Ω) f1 (Ω) −iΩτ  1 −iΩτ  1  ′ =e  ,  ′ =e ,  ′  ′ fˆ2 (Ω) eˆ2 (Ω) dˆ2 (Ω) cˆ2 (Ω)         ˆ ˆ cˆ (Ω) d (Ω) eˆ1 (Ω) f1 (Ω) iΩτ  1 iΩτ  1 .  ,  =e   =e  ˆ ˆ cˆ2 (Ω) d2 (Ω) eˆ2 (Ω) f2 (Ω) ′. (2.52). ˆ Using Eq. (2.52), replacing eˆ(Ω) and fˆ(Ω) of Eq. (2.51) with cˆ(Ω) and d(Ω), respectively yields.       1 0 cˆ1 (Ω) 0  cˆ1 (Ω) 2iΩτ  iΩτ  =e   ′   + e √  h(Ω), 8I0 ω0 I0 ω0 L2 cˆ2 (Ω) 1 c ˆ (Ω) 2 mΩ2 c2 ℏc2 . ′. (2.53). .       ˆ 1 0 dˆ1 (Ω) 0  d1 (Ω) 2iΩτ  iΩτ   ′ =e   − e √   h(Ω). 8I0 ω0 I0 ω0 L2 ˆ ˆ d2 (Ω) d2 (Ω) 1 mΩ2 c2 ℏc2 ′. (2.54). Finally, we consider the junction conditions at the BS to derive the input-output relation of the gravitational wave detector. In Eq. (2.36), we know the junction condition on the mirror and the reflectivity and transmissivity are equal to. 35. 1 2. in the.

(46) BS. Thus we can write the following equations using the notation depicted in Fig. (2.7): 1 cˆ(Ω) = √ {î(Ω) − a ˆ(Ω)}, 2 1 ˆ d(Ω) = √ {î(Ω) + a ˆ(Ω)}, 2 ˆb(Ω) = √1 (dˆ′ {Ω) − cˆ′ (Ω)}. 2. (2.55). From Eqs. (2.53), (2.54) and (2.55), the input–output relation of the gravitationalwave detector may be represented directly in matrix form:.        ˆ 1 0  a ˆ1 (Ω) 0 b1 (Ω)  2iΩτ  iΩτ  =e     − e √  h(Ω) 8I0 ω0 2I0 ω0 L2 ˆb2 (Ω) 1 a ˆ2 (Ω) mΩ2 c2 ℏc2      0  h(Ω) ˆ1 (Ω)  1 0  a iΩτ  , = e2iΩτ    − e √  2κ hSQL −κ 1 a ˆ2 (Ω). (2.56). where we denoted 8I0 ω0 , κ= mc2 Ω2. √ hSQL =. 8ℏ . mΩ2 L2. (2.57). This is the input–output relation for gravitational-wave detection [17] [20]. We can read the gravitational-wave signal from the output phase-quadrature ˆb2 . However, ˆb2 contains both signal and noise components. ⟨ ⟩ ˆb2 = ˆb2 (Ω) + ∆ˆb2 (Ω) √ 2κ ˆ iΩτ b2 h(Ω) + ∆ˆb2 . =e hSQL. 36. (2.58).

(47) Here, the first term. ⟨ ⟩ ˆb2 (Ω) is the expectation value of the output, which is. proportional to the gravitational wave signal. The second term ∆ˆb2 is the noise component due to quantum fluctuation, which contains the following two parts. ∆ˆb2 = e2iΩτ a ˆ (Ω) | {z2 }. −. shot noise. e2iΩτ κˆ a (Ω) {z 1 } |. .. (2.59). radiation−pressure noise. The first term, shot noise, is arisen from the phase-quadrature fluctuation of the optical field. The phase fluctuations of light cause counting noise of photon at the photodetector. The second term is radiation–pressure noise, which originated from the amplitude-quadrature fluctuation of the optical field. The back-action noise causes the inaccuracy in the measurement of the position of the mirror, it will be detected at the dark port. After normalizing by the signal, the spectrum of shot noise can be written as below √ Ssh (Ω) = 1/(. 2κ. hSQL. )2 =. ℏc2 . 2I0 ω0 L2. (2.60). The shot noise decreases with the laser power I0 increases. This reason is that the √ ¯ number of photons N follows the Poisson statistics and has a deviation of ∆N = N ¯ represents the average number of photons detected. This means that the , where N relative phase difference between the arms of the interferometer, based on the number of photons detected at the output port, will be affected by the relative uncertainty of the. ∆N ¯ N. =. √1 . ¯ N. Furthermore, the spectrum of radiation–pressure noise can be obtained as √ 2. Srp (Ω) = κ /(. 2κ. hSQL. )2 =. 32ℏI0 ω0 . m2 c2 L2 Ω4. (2.61). Note that the radiation pressure noise increases as the laser power I0 increases and can be reduced by increasing the mass m of the end mirror. Since the radia37.

(48) tion pressure noise is raised by the power fluctuations interacting with mirrors, it is proportional to the laser power and inversely proportional to the m2 Ω4 . From Eqs (2.60) and (2.61), we can understand that the radiation–pressure noise is significant at low frequencies while the shot noise is significant at high frequencies which are illustrated in Fig 2.8. Besides, we can obtain the noise spectral density form Eq. (2.56): (0, 1)MMT (0, 1)T (0, 1)DDT (0, 1)T h2SQL 1 = ( + κ), 2 κ. S h (Ω) =. where. . . . (2.62). . 0   1 0  M=  , D = √ . κ 1 2κ/hSQL. (2.63). The quantum noise spectral density of the standard detector is shown in Fig. 2.8.. Figure 2.9: The quantum noise in the gravitational-waves detector. The yellow curve represents the total quantum noise spectral density. The blue line is the shot noise, and the green line is radiation-pressure noise. The red line illustrates the standard quantum limit hSQL .. 38.

(49) The quantum noise is consist of shot-noise and radiation-pressure noise. As we can see in Fig 2.8, the shot noise dominates at high frequency, while the radiation-pressure noise dominates at low frequency.. 2.5. Standard Quantum Limit. In Fig 2.9, If the power of laser grows the shot-noise decrease, but the radiationpressure-noise increase. When the power of laser decrease, the radiation-pressurenoise decrease, but the shot-noise increase. No matter how we change the power laser, the sensitivity can not suppress the Standard quantum Limit. Moreover, in Eq. (2.62) we notice that since κ is always large than zero. We write 12 ( κ1 + κ) as the value y 1 1 y =: ( + κ). 2 κ. (2.64). We can derive that y ≥ 1, so that κ is greater or equal to zero. It means that the noise spectral density is always large than h2SQL , which is independent of laser power, as shown in Eq. (2.65). Thereby, the standard quantum Limit (SQL) is defined by the trajectory of the lower limit of the total spectrum.. S h (Ω) ≥ h2SQL =. 8ℏ . mΩ2 L2. (2.65). The standard quantum limit can also be regarded as the basic quantum properties of light. The light does a series of measurements on the difference of end mirror displacement x. Since the end mirror positions do not commute to each other at dif′. ferent times [x(t), x(t )] = 0. While the measurement is too accurate, the uncertainty of measurement increases at the next measuring time. In other words, shot noise and radiation pressure noise produce the SQL of the interferometer and SQL is the limit of detector sensitivity we need to consider. 39.

(50) However, the squeezed state in the interferometer may benefit non-classical features of light. As we mentioned in Section 2.2.1, even there is no light, vacuum fluctuations entering from dark ports can produce quantum noise in the interferometer. By squeezing the vacuum state into the dark port, the quantum noise will be reduced while it can reduce the variance in one quadrature, decreasing the corresponding noise at the cost of increasing variation of the orthogonal quadrature. Furthermore, frequency-dependent squeezing will allow quantum noise to be reduced at all frequencies.[25]. Figure 2.10: Sensitivity curve of the conventional gravitational-waves detector. We set L = 4000 m, λ = 1064 ∗ 10−9 and m = 40 kg. The green curve is laser power I0 = 400kW , the blue curve is laser power I0 = 800kW and the yellow is laser power I0 = 1600kW . Red lines indicate that total quantum noise forms the boundary of sensitivity under different laser power, which is called the standard quantum limits (SQL).. Since the standard quantum limit arises from the quantum properties in the measurement process, it is deserving of investigating from the perspective of quantum measurement, i.e., weak-value metrology. In the next chapter, we will investigate whether the weak value metrology will overcome the standard quantum limit or not.. 40.

(51) Chapter 3 Gravitational-Wave Detector Using Weak Values In this chapter, we will introduce a model of gravitational-wave detection based on weak-value metrology [15][16][14]. To compare the performance between the conventional gravitational-wave detector and the gravitational-wave detector, which is inspired by weak-value metrology. We need to investigate the quantum optical system and derive the sensitivity curve of the detector to know whether or not it improves the sensitivity.. 3.1. Setup. Consider the scheme of the Michelson gravitational-wave detector inspired by weakvalue metrology shown in Fig. 3.1. The optical beam from the laser beam enters the Michelson interferometer, and the laser beam separates into two beams by the BS after it reaches the 50/50 BS. Then, the two beams propagate in x and y arms separately and are reflected at the end mirrors which are sensing small mirror displacement l attributed to gravitational waves.. 41.

(52) Figure 3.1: Setup of Michelson gravitational-wave detector inspired by weak-value metrology.. In the weak-value metrology, the difference with the conventional gravitationalwave detector is that we introduce the phase rotators in each arm, which cause the relative phase offset θ inspired by the original setup of weak value metrology [9] and the idea of interferometer with weak-value [16][14][10]. One optical beam in the x arm is phase-shifted by −θ/2, and the other beam in the y arm is phase-shifted by θ/2. After propagating by phase rotators, the two beams recombine at the BS and interfere with each other. At the dark port set the photodetector, we can measure the small displacements of the end mirrors.. 3.2. Weak-Value Metrology in Gravitational-wave Detector. In the discussion of section 1.3, we have discussed the scheme of weak-value metrology. The process of weak value consists of pre-selection, weak measurement, and post42.

(53) selection. We know that to write down the measurement results under the weak-value scheme, we must describe the observable, quantum state of the observed system and measuring device. As the introduction of gravitational-wave detection in Section 1.2, the goal of the configuration is to measure the small mirror displacement l by the interferometer. In an ideal symmetry interferometer, all light exit into the bright port, because it creates destructive interference at the dark port, and the photodetector can detect no photons. We can read out the number of photons to identify whether the interferometer has gravitational waves passing through. Since the displacements of the end mirrors caused by gravitational waves break the symmetry of the interferometer, allowing a portion of the light to the dark port after passing through the BS. Therefore, it can also understand that the mirror displacement l is couple with which-path the photons propagate. In this setup, we can ˆ The definition of the system operator Aˆ is written as write the observable as 2lA.. Aˆ := |y⟩ ⟨y| − |x⟩ ⟨x| .. (3.1). The system operator contains information about which arm the photon propagates from, and has the eigenvalue ±1; it is called the which-path operator. The state of the system is represented by the path information of the beam, which is written as |x⟩ and |y⟩ to represent photons propagating along the x and y arms, respectively. Then, according to the configuration in Fig. 3.1, the phase rotator provides phase offset, which can be adjusted to tune the pre-selection system state by the relative phase shift by ±θ/2. Thus, the pre-selection state of the system can be denoted by 1 |ψi ⟩ = √ (eiθ/2 |y⟩ + e−iθ/2 |x⟩). 2 43. (3.2).

(54) This state represents the phase-weighted superposition of states of photons propagating from the BS via the x arm and y arms to the end mirrors. Besides, the meter in the current setup of the weak-value metrology is the momentum degree of freedom of the propagating photon, and its initial state is prepared as ∫ |ϕ⟩ =. dpϕ(p) |p⟩ ,. (3.3). where p is the photon momentum. Thereby, the pointer variable of the meter is photon’s momentum to measure the phase shift caused by the mirror displacements. In the process of measurement, the Hamiltonian of interaction between the system and the meter at time t is written as. ˆ = g(t)δ(t − t0 )Aˆ ˆp, H. (3.4). ˆy − X ˆ x means the interaction strength and to represents the moment where g := X when the beam is reflected at the end mirror. As with standard quantum measurement, if we want to measure the observable 2lAˆ of the system, we need to couple the observed system with the meter that has the pointer variable pˆ. Therefore, we can represent the state evolution of the entire system–meter as. |Φ⟩ = e−iH |ψi ⟩ |ϕ⟩ = e−ig(t)δ(t−t0 )Aˆp |ψi ⟩ |ϕ⟩ . ˆ. ˆ. (3.5). In the weak-value scheme, we consider the quantum system not only by preselected state (initial state of the system) but also by the post-selected state (final state of the system). In the configuration shown in Fig (3.1), we define the post-. 44.

(55) selected state of the system as 1 |ψf ⟩ = √ (|y⟩ − |x⟩). 2. (3.6). Eq. (3.6) describes the photons propagate into dark port of the Michelson interferometer, which refers to post-selection done automatically by detecting only at the dark port. Consequently, in the post-selection process, we projecting Eq. (3.5) to the post-selected state to derive the final state of the meter. |Φ′ ⟩ = ⟨ψf | e−iH |ψi ⟩ |ϕ⟩ ∫ ˆ = dpϕ(p) |p⟩ ⟨ψf | e−igAp |ψi ⟩ . ˆ. (3.7). We can expand exponential to first order due to weak interaction: ∫. ′. |ϕ ⟩ =. dpϕ(p) |p⟩ [⟨ψf |ψi ⟩ − igp ⟨ψf | Aˆ |ψi ⟩],. (3.8). After rewriting the Eq (3.8), the final state of the meter is given by. ′. ∫. |ϕ ⟩ = ⟨ψf |ψi ⟩. dpϕ(p) |p⟩ exp(−igpAw ),. (3.9). where Aw is the weak-value defined in Eq. (1.7), from which we can derive the weakvalue in this configuration as ⟨ψf | Aˆ |ψi ⟩ ⟨ψf |ψi ⟩ θ = − i cot 2. Aw =. (3.10). ≈ − 2i/θ. This is so called weak value amplification (AAV). . By tuning the phase θ, weak value amplification allows small displacement of mirror to be detected.. 45.

(56) 3.3. Input–Output Relation Using Weak Values. However, there is no more intuitive insight from Eq. (3.10) to know that whether the detector inspired by weak-value metrology can improve the sensitivity. In addition to amplifying the signal, we also have to consider the quantum noise under this setup. To investigate a clearer understanding of signal and quantum noise in the detector, we will follow the process in section 2.4, which we derived the input-output relation for conventional gravitational-wave detection, to calculate the modifying input-output relation based on the configuration in Fig. 3.1.. Figure 3.2: Schematic of interferometer optical-field quadratures.. In Fig. 3.2, the optical fields are represented as quadrature operators along the path in the detector to derive the input-output relation. According to the strategy. 46.

(57) discussed in section 2.4, we start from the motion of two mirrors which driven by gravitational waves with the following results.. .   eˆ1 (Ω)  1  ′ = eˆ2 (Ω) κ    ′ ˆ f1 (Ω)  1 =  ′ ˆ f2 (Ω) κ ′.    0 eˆ1 (Ω)  0    +  √  h(Ω), κ 1 eˆ2 (Ω) hSQL     ˆ 0 f1 (Ω)  0   −  √  h(Ω).  κ ˆ f2 (Ω) 1 hSQL . (3.11a). (3.11b). We obtain the relationship between optical fields reaching and reflecting from the end mirrors in each arm. The optical fields, gravitational-wave strain h(Ω), and radiation–pressure force are coupled in the Eq. (3.11). Next, we consider the optical fields propagating along the x and y arms. In the weak value metrology scheme, we introduce the phase offset ±θ/2 into x and y arm respectively. ˆ Since the optical field cˆ(Ω) and d(Ω) propagate a distance 2L to cˆ′ (Ω) and dˆ′ (Ω), and the gravitational wave strain h(Ω) propagates a distance L. After introducing the phase rotator matrix, we can reorganize Eq. (3.10) to get the equations of the optical field return to BS. .      θ θ cˆ1 (Ω) 2iΩτ cos 2 − sin 2   1 0 cˆ1 (Ω) =e  ′      cˆ2 (Ω) κ 1 cˆ2 (Ω) sin 2θ cos 2θ    θ θ cos 2 − sin 2   0  + eiΩτ    √  h(Ω), κ sin 2θ cos 2θ hSQL ′. 47. (3.12).

(58) .      θ θ ˆ ˆ d1 (Ω) 2iΩτ  cos 2 sin 2   1 0 d1 (Ω)  ′  =e     dˆ2 (Ω) − sin 2θ cos 2θ κ 1 dˆ2 (Ω)    θ θ  cos 2 sin 2   0  − eiΩτ    √  h(Ω), κ − sin 2θ cos 2θ hSQL ′. (3.13). where the rotator operator is due to the phase offset ±θ/2 and τ = L/c is the delay time while photons propagate from the BS to the end mirrors. Next, we consider the junction conditions at the BS (Eq. 2.55), yielding   θ θ 1 2iΩτ cos 2 − κ sin 2 cˆ1 (Ω) √ = e  ′   2 sin 2θ + κ cos 2θ cˆ2 (Ω)  θ θ 1 2iΩτ cos 2 − κ sin 2 −√ e  2 sin 2θ + κ cos 2θ   √ κ θ sin 2  − h + eiΩτ  √SQL  h(Ω), κ θ cos hSQL 2. . . ′. . − sin.   î2 (Ω)   θ − sin 2  a ˆ1 (Ω)   cos 2θ a ˆ2 (Ω).   θ θ ˆ 1 2iΩτ  cos 2 + κ sin 2 d1 (Ω) √ = e  ′   2 dˆ2 (Ω) − sin 2θ + κ cos 2θ  θ θ 1 2iΩτ  cos 2 + κ sin 2 +√ e  2 − sin 2θ + κ cos 2θ   √ κ θ  hSQL sin 2  − eiΩτ  √  h(Ω). κ θ cos 2 hSQL ′. 48. . θ î (Ω)  2  1. cos 2θ.   sin 2θ  î1 (Ω)   î2 (Ω) cos 2θ   θ sin 2  a ˆ1 (Ω)   cos 2θ a ˆ2 (Ω). (3.14). (3.15).

(59) Through Eq. (2.55), Eq. (3.14) and Eq. (3.15), we obtain the input–output relation of gravitational wave detector inspired by the weak-value metrology..         θ θ θ ˆ κ sin 2 sin 2  î1 (Ω) cos 2 0  a ˆ1 (Ω) b1 (Ω) 2iΩτ  2iΩτ   +e  =e     ˆb2 (Ω) î2 (Ω) 0 − sin 2θ κ cos 2θ cos 2θ a ˆ2 (Ω)   θ  0  h(Ω) − cos eiΩτ √  . 2 h SQL 2κ (3.16) When θ = 0 it returns to the input–output relation of a conventional gravitational waves detector as Eq. (2.56). It is worth noting that there is no term for î in the conventional input-output relation. In the conventional input-output relation, the optical field of x arm and y arm combine in the BS, and the î propagates to the dark port is completely offset, so the input-output relation does not have a î term. However, this symmetry is broken by the phase rotator, which introduces a relative phase between the two paths, allowing photons to escape to the dark port.. 3.4. Sensitivity. We draw the sensitivity curve (Fig. 3.3) to evaluate whether the weak-value scheme overcomes the standard quantum limit of the gravitational-wave detector. The noise spectral density is given by T T T (0, 1)M1 MT 1 (0, 1) + (0, 1)M2 M2 (0, 1) (0, 1)DDT (0, 1)T sin2 2θ + κ2 cos2 2θ + cos2 2θ = cos2 2θ 2κ/h2SQL. Sh =. =. κ2 cos2 2θ + 1 h2SQL . 2κ cos2 2θ 49. (3.17).