Macroscopic Quantum Mechanics and Black Hole Physics

The third part of this thesis (Chapters 8 – 11) studies the physics of black holes in relation to the detection of gravitational waves. In Part III (Chapters 8 – 11), we study the physics of gravitational wave sources, in particular black hole perturbation theory.

Physics of advanced gravitational wave detectors

An overview of laser interferometer gravitational-wave detectors

Laser light enters and usually returns to the “bright gate” of the Michelson interferometer. Gravitational gradient noise (dark green curve in Figure 1.1), which arises from differential fluctuations of the Newtonian gravitational field at the locations of the test masses.

Figure 1.1: (Color online.) Square root of strain noise spectrum for several of Advanced LIGO’s major noise sources

Brownian thermal noise of mirrors with multilayer coatings (Chapter 2)

Motivation and significance
Summary of main results
My specific contributions

Based on the formulas we obtained for the cross-spectral densities, about 70% of the Brownian noise of the Advanced LIGO coating is due to fluctuations of the coating-substrate interface, because the Young's modulus of the more lossy type of coating material (Ta2O5) was assumed to be much higher than the Young's substrate modulus. This noise turns out to be inversely proportional to the Young's modulus of the substrate material, meaning that it can be reduced by using harder substrate materials.

Macroscopic Quantum Mechanics (MQM)

Background

The input-output formalism of linear quantum measurement theory 8

Open quantum dynamics of single-photon optomechanical devices (Chapter 3) 10

Theory of non-Markovian quantum measurement processes (Chapters 4 and 5) 13

Probing mechanical oscillators near their zero points (Chapter 6)

Motivation and signficance

Macroscopic quantum mechanics in a classical spacetime (Chapter 7)

Motivations and significance

The second term in Eq. 1.18) is the Hamiltonian for the cavity optical mode (the one closest to single photon excitation). In this case, the non-Markovianity of the measurement process does not come from the interaction between

Black hole perturbation theory

Background
First-order perturbative Hamiltonian equations of motion for a point particle

Quasinormal mode spectrum of Kerr black holes and its geometric interpreta-

Quasinormal mode bifurcation for near extreme Kerr black holes (Chapter 10) 27

An analytical approximation for the scalar Green function in Kerr spacetime

With the correspondence we also make other observations about features of the QNM spectrum of Kerr black holes that have simple geometric interpretations. DMs are associated with peaks of the potential barrier; in the eikonal limit they exist when µ ≡m/(l.

Figure 2.1: Drawing of a mirror coated with multiple dielectric layers. Shown here are the various fluctuations that contribute to coating noise, i.e., fluctuations in the amplitude and phase of the returning light caused by fluctuations in the geometry [i

Components of the coating thermal noise

Complex reflectivity
Thermal Phase and Amplitude Noise
Fluctuations δφ j and δr p
Mode selection for phase noise
Conversion of amplitude noise into displacement

Real and imaginary parts of. encode the amplitude/intensity and phase fluctuations of the reflected light at position ~x on the mirror surface. Because we measure the position of the mirror through the additional phase shift gained by the light after it is reflected, through the relation ∆φ= 2k0∆x, Eq. 2.11) shows that ξ(~x) is the displacement noise due to the phase fluctuations of the reflected light imposed by the cladding.

Thermal noise assuming no light penetration into the coating

The Fluctuation-Dissipation Theorem
Mechanical energy dissipations in elastic media
Thermal noise of a mirror coated with one thin layer
Discussions on the correlation structure of thermal noise

It is straightforward to apply Eq. 2.46) to calculate the thermal noise component due to fluctuation of the position of the coating air interface — the weighted average [cf. From such dependence on coating and beam geometries, we infer that (i) each point on the coating–air interface oscillates independently along the z direction, and (ii) materials at different z's in the coating as well. contribute independently to pavement thermal noise.

Cross spectra of thermal noise components

Coating-thickness fluctuations
Fluctuations of Coating-Substrate Interface and their correlations with coating
The anatomy of coating thermal noise
Full formula for thermal noise

To investigate the relationship between the height of the coating-substrate interface, zs(~x) and the thickness of each coating layer, δlj(~x), we apply an identical pair of pressuresf1(~x) =F0w1(~x) on opposite sides of layer 1, and force fs(x, y) =F0ws(~x) onto the coating-substrate interface (along the -z direction), as shown in Fig. This means the thermal bulk stress in a layer drives simultaneously the thickness fluctuation of this layer and a fluctuation of the coating-substrate interface.

Figure 2.2: Illustrations of forces applied onto various interfaces within the coating

Effect of light penetration into the coating

Optics of multi-layer coatings

For completeness of the paper, we briefly consider how the light penetration coefficient ∂logρ/∂φj can be calculated. Thus, assuming the input and output field amplitudes on the top surface of a multilayer in bev1 anddev2, and writing them inside the substrate in bes1ands2, we have.

Levels of light penetration in Advanced LIGO ETM Coatings

These plots show that for both structures light penetration is limited in the first 10 layers. Note that in order to focus on the effect of light penetration, we have shown only the first 10 layers.

Figure 2.6: Light penetration into the first 10 layers of a 38-layer coating (left panel for conventional coating and right panel for Advanced LIGO coating)

Thermal noise contributions from different layers

We should also expect that the effect of photoelasticity (dashed lines) will be small, and that the effect of backscattering (giving rise to Tjξc and Tjξs, blue and purple dashed lines) will be even smaller. Dotted lines of each type, calculated without introducing the backscatter terms, are superimposed on the solid lines; the effect is noticeable during the first few coats.

Dependence of thermal noise on material parameters

Dependence on ratios between loss angles
Dependence on Young’s moduli and Poisson’s ratios
Dependence on photoelastic coefficients
Optimization of coating structure

When we change the ratio between the loss angles, there are moderate changes in the thermal noise. Hz (the thermal noise for the target φB/φS is given in bold, and the numbers in bold must be the smallest in its column); thermal noise spectra of a 38-layer λ/4 stack assuming a target φB/φS are also listed for comparison.

Figure 2.9: (Color online.) Variations in total noise when φ B /φ S is varied: (solid) total noise, (dotted) total bulk noise, (dashed) total shear noise

Measurements of loss angles

Bending modes of a thin rectangular plate

The first torsional eigenmodes of such a shell can be used to measure the shear loss angle of the pavement. However, in the case where the Poisson's ratio σc of the coating vanishes, the thickness fluctuation depends on the loss angle of Young's modulus.

Torsional modes of a coated hollow cylinder

For a cylinder shell, according to Donnell scale theory, the natural frequency of the nth torsional mode is given by [36]. By measuring both the thin plate and the cylinder shell, we can obtain φBandφS of the coating.

Conclusions

After the completion of this manuscript, we noted that the optimization of the pavement structure for the case assuming φB = φS (and β = 0) has been carried out by Kondratiev, Gurkovsky and Gorodetsky [17]. In table 2.7 we give the structure of the coating jointly optimized for dichroic operation and thermal noise.

φ B /φ S 0.2 – 5 1 ±37% Sec. 2.6.1, Figs. 2.8, 2.9.

A Single cavity with one movable mirror

The Hamiltonian
Structure of the Hilbert space
Initial, final States and photodetection
Evolution of the photon-mirror quantum state

Free evolution
Junction condition
Coupled evolution
Full evolution

Here |ψ1(x, t)im,−∞< x <+∞, is a set of vectors, parametrized byx, in the Hilbert space of the mechanical oscillator, while |ψ2(x, t)ii is a single vector in the Hilbert space of the mechanical oscillator. Equation (3.29) corresponds to the free evolution of the outgoing photon and the mechanical oscillator.

Figure 3.3: (Color online.) Three regions of the t-x plane and the free evolutions of |ψ 1 i

Single-photon interferometer: Visibility

The configuration
The role of the beamsplitter and a decomposition of field degrees of freedom 104
The final state
Examples

If we idealize the arrival time of the incoming photon (at the front mirror) to be t = 0, and ignore the macroscopic distance between the front mirrors, the beam splitter and the photodetector, then we are interested in inp(t) att ≥0. At any time, if ψA is proportional to ψB (differs by a phase), the state of the moving mirror does not change, and therefore we have perfect visibility.

Figure 3.4: We illustrate the fields entering and exiting each of the four ports of the interferometer.

Conditional quantum-state preparation

The configuration

The chance of the photon being detected at such late times is exponentially small – as indicated by the left panels of figure. Even if there is a finite chance that the photon will come out through the western arm or the bright gate, as soon as we detect a photon at the moment of the dark gate we know that it must come out of arm A and that it must also have a certain has spent time in cavity A.

Preparation of a single displaced-Fock state

This leads to the interesting effect that in the asymptotic limit τ → +∞ the conditional state will be independent of τ. This dependence (3.89) on β comes from two sources, which can be better understood if we go to a phase-space reference frame centered at the equilibrium position of the oscillator when the photon is inside the cavity.

Figure 3.7: A sketch of the phase-space trajectory of the mechanical oscillator. The Wigner function of the initial state |0i is represented by the shaded disk, the dot marked with β on the real axis is the new equilibrium position of the oscillator when t

Preparation of an arbitrary state

To provide a concrete measure of the capability of our state preparation scheme, we chose to calculate the minimum success probabilities of creating all the states in the mechanical oscillator's Hilbert subspaces spanned by the lowest displaced Fock states, e.g. H1≡Sp {|˜0i,|˜1i},H2≡Sp{|˜0i,|˜1i,|˜2i}, etc. The rapid transition between the two extremes indicates that when trying to prepare states in Hj, the difficulty lies either in|˜0i, or in |˜ji, and only for a small region of β can the two difficulties competing with each other - while none of the intermediate states add to the difficulty of state preparation.

Figure 3.9: Minimum success probability for states in Hilbert spaces H 1,2,...7 (solid curves with markers), together with success probability for producing single displaced Fock states, P 0,1,2,...,7

Practical considerations

By combining the above two conditions, we get the following relation λ. 3.102) where λ is the optical wavelength of the photon, F is the finesse of the cavity. Finally, we require the ability to generate a single photon with an arbitrary wave function with a duration comparable to the mechanical oscillation frequency of the photon.

Conclusions

On the theoretical side of the study, the usual approach is to investigate the decoherence of the quantum system using either the stochastic non-Markovian Schr¨odinger equation (also called the state diffusion equation) or equivalently the master equation [1-3] . Di'osi tried to give a quantum trajectory interpretation of the non-Markovian stochastic Schr¨odinger equation (SSE), i.e., claiming that it describes a pure state evolution under a continuous measurement of the system operator.

Quantum-classical interface: the measurement device

To be an adequate quantum probe, its output must be accurately measured by the detector (that is, a von Neumann projective measurement), and there is no further quantum feedback from the detector on the probe. This serves as the Heisenberg cut [8], which is essential for an unambiguous interpretation of the measurement result.

Markovian measurement of a cavity mode

Before the interaction, we assume that the probe field and the cavity mode are separable, that is, ˆρ(t) = ˆρa(t)⊗ρˆo(t), and the input probe field is in vacuum state ˆρo(t) =|0ih0 |4. If we ignore the fact that the measurement result and mean of dW are equal to zero, we simply get the Markovian master equation.

Non-Markovian quantum measurement: the first scenario

Cavity QED: atom-cavity interaction

Instead of evolving the cavity mode and the atom all together, as in the usual approach, we only need to consider a finite dimensional density matrix for the atom. The bottom panel is the convergence of the accumulated numerical difference between the SSE and SME simulation results, given a different number of cavity mode grid points.

Optomechanical device

Similar to the previous case, after tracking the cavity state and the continuous optical field, the reduced non-Markovian SME is given by . In the large cavity, the bandwidth limit γωm,∆, andG, ˆO is simply given by:. and we again recover the Markovian SME:. 4.30).

Generalization to bath with many degrees of freedom

Here ˆHs is the free part of the system Hamiltonian; ˆL is an arbitrary operator of the system; gk. In other words, we are effectively coupled to a dynamical quantity of the system formed by the bath - a quantum filter.

Non-Markovian quantum measurement: The second scenario

To derive the conditional quantum state of the oscillator, we follow the general formalism outlined in Ref. This formula gives the conditional Wigner function of the oscillator as a function of the initial non-Gaussian state of the input field.

Phonon number measurement

The only addition [ζ∗Γˆ−ζΓˆ†,B] is the c-number, since ˆˆ B is a linear function of the optical field operator; so we can directly use the result obtained in the previous example with a Gaussian input field. In conclusion, we have derived the conditional Wigner oscillator function in both cases of correlated Gaussian and more general non-Gaussian input probe fields.

Nonlinear measurement with initially entangled detector state

Therefore we can write (t) in terms of a classical random process y(t)dt=−√. with dW2 = dt Wiener addition. 4.77) and (4.79), we can obtain a differential equation for the normalized density matrix, and up to the first order of dt, we obtain.

Non-Markovian quantum measurement: General case

Conclusion

The wave function and operator in Schr¨odinger and the interaction picture are related to. The corresponding stochastic Schr¨odinger equation in the coherence basis of the cavity mode reads d|ψ(α∗)i=− i.

Figure 5.1: (Color online.) A schematic showing how the reduced dynamics of the plant emerges from the full dynamics of the plant-bath system by tracing over the bath state at each step.

Plant – bath dynamics

5.3 we show the change of the track spacing and also the coincidence with and without introducing a local unit operation on the atom ˆU = ˆσz⊗Iˆc at t= 1 (ˆIc is the identity operator for the cavity state). By controlling g(ω), the time evolution of the track distance can either decrease monotonically or fluctuate.

Figure 5.2: (Color online.) A schematic showing the atom-cavity system. The cavity mode is coupled to the external continuum field (a Markovian bath), and they together form an effective non-Markovian bath for the atom

Conclusion

Needless to say, a continuous observation of the zero-point fluctuation of a macroscopic mechanical oscillator requires excellent displacement sensitivity. 24], this contribution is equal to the zero point fluctuation of the oscillator for one tuning of the readout beam and exactly opposite for the other tuning.

A two-beam experiment that measures zero-point mechanical oscillation

Experimental setup and results
Interpretation in terms of quantum measurement

The mechanical oscillator near ground state
The quantum-measurement process
Asymmetry between spectra

Detailed theoretical analysis
Connection with the scattering picture

The observed spectrum of the readout laser is asymmetric with respect to the detuning ∆≡ωr−ωlr. This allows us to relate the output spectrum of the amplitude quadrature to the emission spectrum.

Figure 6.1: Figure illustrating the observed spectra of the readout laser in the positive-detuning case (left) and the negative-detuning case (right).

General linear measurements of the zero-point fluctuation

The nature of zero-point mechanical fluctuation

The replenishment of the position-momentum Bracket Poisson by environmental variables, in classical mechanics, can also be seen as a consequence of the conservation of phase space volume, following Liouville's theorem. Therefore, in the steady state, the zero-point oscillation of the mechanical oscillator can be seen as imposed by the environment due to the linearity of the dynamics.

Measuring the zero-point fluctuation

Therefore, the zero-point fluctuation of the mechanical oscillator in the steady state can be seen as being imposed by the environment due to the linearity of the dynamics. for all values often where we have defined. In particular, the non-vanishing commutator [ˆxq(t),xˆq(t0)] underlying the existence of the zero-point fluctuation is canceled in a simple way by the non-vanishing commutator between the sense noise and the back-action noise [cf.

Measuring an external classical force in presence of zero-point fluctuation

The first term is the standard quantum limit for force sensitivity with mechanical probes, which arises from the zero-point fluctuation due to mechanical quantization also limits the sensitivity. However, the limit imposed by zero-point fluctuations cannot be exceeded - although it can be mitigated by lowering κm.

Figure 6.3: (Color online.) Figure illustrating that total quantum limitation S F Qtot (red) for force sensitivity and contribution from zero-point fluctuation S F zp (blue)

Conclusion

Although the above relative comparison between the SQL and zero-point fluctuation indicates that the later plays a more important role near the oscillator's resonant frequency, it is simple to see from Eq. 6.71) and (6.72) that on an absolute scale: (i) for a given oscillator at ground state, SFSQL(ω) is lower near mechanical resonance, while SFzp(ω) is independent of frequency, and (ii) at any frequency, lowering κm, while fixing it and keeping the oscillator in ground state, always results in lower noise, as illustrated in Fig. This additional noise does disappear when the oscillator's bandwidth approaches zero, i.e. when the oscillator becomes ideal. is supported by the Russian Foundation for Basic Research grant No. was supported by the DARPA/MTO ORCHID program through a grant from AFOSR, and the Kavli Nanoscience Institute at Caltech.

SN theory for macroscopic objects

Discussion and conclusion

Review of Moncrief’s Hamiltonian approach

First-order perturbation of a static space-time in 3+1 form

Degrees of freedom

Schwarzschild perturbations

Odd parity (l ≥ 2)

Even parity (l ≥ 2)

Monopole and dipole perturbations

Odd parity (l ≥ 2)

Even parity (l ≥ 2)

Regularization of test particle equation of motion

General Discussion

Odd parity

An algebraic gauge
Fixing Lorenz gauge

Even parity

Conclusions and discussions

Overview of quasinormal modes and their geometric interpretation

Methods and results of this article

Organization of the paper

WKB approximation for the quasinormal-mode spectrum of Kerr black holes

The Teukolsky equations

The angular eigenvalue problem

Real part of A lm for a real-valued ω
Complex A lm for a complex ω

The radial eigenvalue problem

Computing ω R
Computing ω I

Accuracy of the WKB approximation

Geometric optics in the Kerr spacetime

Geometric optics: general theory

Null geodesics in the Kerr spacetime

Correspondence with quasinormal modes

Leading order: conserved quantities of rays and the real parts of
Next-to-leading order: radial amplitude corrections and the imagi-
Next-to-leading order: angular amplitude corrections and the imag-

Features of the spectra of Kerr black holes

Spherical photon orbits and extremal Kerr black holes

A mode’s orbital and precessional frequencies

Degenerate quasinormal modes and closed spherical photon orbits

Slowly spinning black holes
Generic black holes

Conclusions and discussion

Matched expansions

WKB analysis

Phase boundary

Bifurcation

Numerical investigation

Bound state formulation of the radial Teukolsky equation

Late time tails of NEK excitations

Conclusion

Spectral decomposition

QNMs in the eikonal limit

The radial wavefunction

The angular wavefunction

Matched expansions

Wavefunction near the peak of the scattering potential

WKB wavefunctions away from the peak of the scattering potential

Matching solutions

The Green fucntion

Summation over of all QNM contributions and the singular structure

The Green function

Conclusion and future work