i

Contents

Introduction 1

1 Open Quantum Systems 5 1.1 Reduced state and its evolution ...... 5 1.1.1 Complete positivity ...... 7 1.2 Lindblad equation ...... 8

2 Decoherence 9 2.1 Gallis-Flemming master equation ...... 10 2.1.1 Short wavelength limit ...... 12 2.1.2 Long wavelength limit ...... 12 2.2 Rotational Decoherence ...... 13

3 Quantum Brownian Motion 18 3.1 The model ...... 18 3.2 The Calderira-Leggett master equation ...... 20 3.2.1 Complete positivity problem ...... 23 3.3 Non-Markovian Quantum Brownian motion ...... 23 3.3.1 The adjoint master equation ...... 24 3.3.2 The Master Equation for the statistical operator ...... 27 3.3.3 Complete Positivity ...... 29 3.3.4 Time evolution of relevant quantities ...... 31 3.3.5 Non-Gaussian initial state ...... 34

4 Gravitational time dilation 37 4.1 Model for universal decoherence ...... 38 4.2 Heat capacity for gravitational decoherence ...... 39 4.3 Competing effects ...... 40 4.3.1 Comparison of the effects ...... 42

5 Collapse Models 45 5.1 Continuous Spontaneous Localization Model ...... 45 5.1.1 Imaginary noise trick ...... 49 5.2 Optomechanical probing collapse models ...... 49 5.3 Gravitational wave detectors bound collapse parameters space ... 53 5.3.1 Interferometric GW detectors: LIGO ...... 56 5.3.2 Space-based experiments: LISA Pathfinder ...... 57 5.3.3 Resonant GW detectors: AURIGA ...... 58 5.4 Ultra-cold cantilever detection of non-thermal excess noise ..... 61 5.5 Hypothetical bounds from torsional motion ...... 67 5.5.1 Experimental feasibility ...... 69 ii

6 Conclusions 71

Appendices

A Quantum Brownian Motion master equation 74 A.1 Explicit form of (t) ...... 74 A.2 Explicit form of the adjoint master equation ...... 74 A.3 Derivation of the master equation for the states ...... 75 A.4 Explicit expression for ⇤dif(t) and E(t) ...... 76

B Gravitational time dilation 78

C Collapse Models 80 C.1 CSL Diffusion coefficients ...... 80 C.2 Effective frequencies and damping constants ...... 81 C.3 Cantilever ...... 82

Bibliography 92 iii

List of Publications

Published works as outcome of the doctoral project

1. A. Vinante, R. Mezzena, P. Falferi, M. Carlesso and A. Bassi. Improved noninterferometric test of collapse models using ultracold cantilevers. Physical Review Letters, 119 110401 (2017). Link to paper: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.110401 Link to ArXiv: https://arxiv.org/abs/1611.09776 The most important contents of this article are reported in Sec. 5.4.

2. M. Carlesso and A. Bassi. Adjoint master equation for quantum brownian motion. Physical Review A, 95 052119 (2017). Link to paper: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.052119 Link to ArXiv: https://arxiv.org/abs/1602.05116 The most important contents of this article are reported in Sec. 3.3.

3. S. McMillen, M. Brunelli, M. Carlesso, A. Bassi, H. Ulbricht, M. G. A. Paris, and M. Paternostro. Quantum-limited estimation of continuous spontaneous localization. Physical Review A, 95 012132 (2017). Link to paper: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012132 Link to ArXiv: https://arxiv.org/abs/1606.00070 iv

4. M. Carlesso, A. Bassi, P. Falferi, and A. Vinante. Experimental bounds on collapse models from gravitational wave detectors. Physical Review D, 94 124036 (2016). Link to paper: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.124036 Link to ArXiv: https://arxiv.org/abs/1606.04581 The most important contents of this article are reported in Sec. 5.3.

5. M. Carlesso and A. Bassi. Decoherence due to gravitational time dilation: Analysis of competing decoher- ence effects. Physics Letters A, 380 (31–32), pp. 2354 – 2358 (2016). Link to paper: http://www.sciencedirect.com/science/article/pii/S0375960116302407 Link to ArXiv: https://arxiv.org/abs/1602.01979 The most important contents of this article are reported in Chap. 4.

Pre-prints

6. M. Carlesso, M. Paternostro, H. Ulbricht, A. Vinante and A. Bassi. Non-interferometric test of the Continuous Spontaneous Localization model based on the torsional motion of a cylinder. ArXiv, 1708.04812 (2017). Link to ArXiv: https://arxiv.org/abs/1708.04812 The most important contents of this article are reported in Sec. 5.5.

7. M. Carlesso, M. Paternostro, H. Ulbricht and A. Bassi. When Cavendish meets Feynman: A quantum torsion balance for testing the quantumness of gravity. ArXiv, 1710.08695 (2017). Link to ArXiv: https://arxiv.org/abs/1710.08695 v

List of attended Schools, Workshops and Conferences

1. September, 2017 Training Workshop at Instituto Superior Tecnico of Lisbon, Portugal Title Lisbon Training Workshop on Quantum Technologies in Space http://www.qtspace.eu/?q=node/131 Organization Dr. R. Kaltenbaek, Dr. E. Murphy, Dr. J. Leitao and Dr. Y. Omar

2. June, 2017 Workshop at University of Milano, Italy Title Fundamental problems of quantum physics http://www.mi.infn.it/~vacchini/workshopBELL17.html Organization Dr. B. Vacchini

3. May, 2017 Workshop at Laboratory Nazionali in Frascati, Italy Title The physics of what happens and the measurement problem https://agenda.infn.it/conferenceDisplay.py?confId=13169 Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. B. Hiesmayr and Dr. K. Pis- cicchia

4. May, 2017 Junior Symposium in Trieste, Italy Title Trieste Junior Quantum Days http://people.sissa.it/~alemiche/junior-tsqd-2017.html Organization Dr. A. Bassi, Dr. F. Benatti and Dr. A. Michelangeli

5. March, 2017 Conference and Working Group Meeting in Valletta, Malta Title QTSpace meets in Malta http://www.qtspace.eu/?q=node/112 Organization Dr. M. Paternostro, Dr. A. Bassi, Dr. S. Gröblacher, Dr. H. Ul- bricht, Dr. R. Kaltenbaek and Dr. C. Marquardt

6. November, 2016 Autumn School at LMU in Munich, Germany Title Mathematical Foundations of Physics https://light-and-matter.github.io/autumn-school Organization Dr. D.-A. Dercket and Dr. S. Petrat vi

7. May, 2016 Workshop in Pontremoli, Italy Title Quantum control of levitated optomechanics https://quantumlevitation.wordpress.com Organization Dr. A. Serafini, Dr. M. Genoni and Dr. J. Millen

8. September, 2015 International Workshop at Laboratory Nazionali in Frascati, Italy Title Is quantum theory exact? The endeavor of the theory beyond standard quantum mechanics - Second edition http:www.lnf.infn.it/conference/FQT2015 Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. S. Donadi and Dr. K. Pisci- cchia

9. March, 2015 International Conference at Ettore Majorana Foundation in Erice, Italy Title Fundamental Problems in Quantum Physics http:www.agenda.infn.it/conferenceDisplay.py?confId=9095 Organization Dr. A. Bassi and Dr. C.O. Curceanu

10. February, 2015 51 Winter School of Theoretical Physics in Ladek Zdroj, Poland Title Irreversible dynamics: nonlinear, nonlocal and non-Markovian mani- festations http:www.ift.uni.wroc.pl/~karp51 Organization Institut of Theoretical Physics in Wroclaw, Poland 1

Introduction

When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time [. . . ] the whole development [from the mass of experimental facts to its disclosure] seems to me to provide a fresh illustration of the long-since proved saying of Goethe’s that man errs as long as he strivesa. And the whole strenuous intellectual work of an industrious research worker would appear [. . . ] in vain and hopeless, if he were not occasionally through some striking facts to find that he had, at the end of all his criss-cross journeys, at last accomplished at least one step which was conclusively nearer the truth.

aJohann Wolfgang von Goethe, Faust, 1808. Max Karl Ernst Ludwig Planck Nobel Lecture, June 2, 1920 [1]

The question: “How does a chicken move in the atmosphere?” would be typically answered by a physicist: “To start, let us approximate the problem by considering a spherical chicken in vacuum. . . ”. This is for sure a strong and rough approximation, however it can be a good starting point for solving the problem and in certain cases it is more than enough to properly describe the motion of the system of in- terest.

Quantum mechanics is an example of a theory exhibiting a broad collection of theoretical results in complete agreement with experimental evidence: from the black body radiation [2–4] to the double slit experiment [5, 6], from the photo- electric effect [7–9] to the hydrogen atom, from interference fringes in a matter- wave interferometry experiment [10, 11] to Bose-Einstein condensates [12, 13] and many more. In some situations, the unitary dynamics of a quantum isolated sys- tem is not sufficient to well describe the system. One situation is of particular importance due to its ubiquity and unavoidability. Every realistic (quantum) sys- tem interacts with the surrounding environment and consequently is changed by it. In such a case, phenomena like dissipation, diffusion or decoherence emerge and may become important for the system dynamics. External influences on a quantum system must be considered explicitly to get a better description of Na- ture. This is the purpose of the theory of open quantum systems.

In this thesis two different research lines are considered: decoherence and col- lapse models. Albeit conceptually they are far from each other, they both belong to the framework of open quantum systems. Indeed, they refer to systems inter- acting with an external entity: an environment for decoherence models, a noise for collapse models. Although the external influence has a different origin, they 2 can be described by similar dynamical equations and, in order to confirm or fal- sify one of these models, similar experimental tests can be performed.

Decoherence models describe the suppression of the interference fringes of a su- perposition due to the interaction with the surrounding environment, and they also govern other mechanisms, like diffusion and dissipation. The theory has made important contributions in other fields, like chemistry [14, 15], condensed matter [16, 17] and biophysics [18–21], to name a few, which are typically resolved via numerical analysis [22, 23]. By introducing the environment, the complex- ity of the problem grows with the level of detail one gives to the model [24–28]. Consequently, a careful balance between the reliability of the model and its math- ematical idealization becomes a fundamental ingredient to approach the (analyt- ical) resolution of the problem. The seminal works of Caldeira and Leggett [29] and of Joos and Zeh [30] are milestones in this field. They model the system- environment interaction in a very simple way, still being able to capture the most important properties and features of the open system dynamics, cf. Chaps. 2 and 3.

When trying to solve exactly an open quantum system problem, one usually faces several difficulties: the most intriguing example is given by the appearance of non-Markovian features in the system dynamics. A Markovian dynamics is ruled by equations of motion that do not depend on the past of the system: it is a mem- oryless dynamics [24]. If, instead, the dynamics depends on the past, and thus it has a memory, the evolution is said to be non-Markovian. In some situations this memory is responsible for crucial changes in the behaviour of the system: the long-living quantum coherences in a light-harvesting systems are an important non-Markovian effect in quantum-biology [31]. In this thesis, two examples will be discussed explicitly, respectively as an example of a Markovian [cf. Chap. 4] and non-Markovian dynamics [cf. Chaps. 3].

The second line of research is focused on collapse models and their experimen- tal tests. These models unify the two dynamical principles of the quantum me- chanics (the linear and deterministic Schrödinger evolution with the non-linear and stochastic wave-packet reduction) in an unique description. By adding non- linear and stochastic terms to the standard Schrödinger equation, they describe the spontaneous collapse of the wavefunction. With this modification, they re- cover both the quantum and the classical dynamics in the microscopic and in the macroscopic limit respectively, thus answering to the quantum-to-classical tran- sition debate.

Among the broad collection of collapse models [32–39], we focus on a particular collapse model, called Continuous Spontaneous Localization (CSL) model. This model is characterised by a coupling rate CSL between the system and the noise

field allegedly responsible for the collapse, and a typical correlation length rC for 3

the latter. As the CSL model is phenomenological, the values of CSL and rC must be eventually determined by experiments. By now there is a large literature on the subject. Such experiments are important because any test of collapse mod- els is a test of the quantum superposition principle. In this respect, experiments can be grouped in two classes: interferometric tests and non-interferometric ones. The first class includes those experiments, which directly create and detect quan- tum superpositions of the center of mass of massive systems. Examples of this type are molecular interferometry [40–43] and entanglement experiment with di- amonds [44, 45]. Actually, the strongest bounds on the CSL parameters come from the second class of non-interferometric experiments, which are sensitive to small position displacements and detect CSL-induced diffusion in position [46–48]. Among them, measurements of spontaneous X-ray emission gives the 6 strongest bound on CSL for rC < 10 m[49], while force noise measurements on nanomechanical cantilevers [50, 51] and on gravitational wave detectors give the 6 strongest bound for rC > 10 m[52, 53]. Albeit several tests were performed during the last decade, up to date the CSL parameter space still exhibits a vast unexplored region.

Outline

This thesis is organized as follows.

In Chap. 1 we present the basic ingredients of the theory of open quantum sys- tems. Starting from standard quantum mechanics, we introduce the concept of reduced state of the system and derive its evolution. Its dynamics, constructed starting from the global system-environment evolution, needs to satisfy several constrains in order to be a well-defined dynamical map. We discuss these con- strains, with particular attention to complete positivity. Eventually, we introduce the Lindblad structure for the generator of the dynamical map, which naturally satisfies all the above constraints.

In Chap. 2 we introduce the quantum effect of decoherence, which is one of the most important features of an open quantum system. After discussing its main properties, we report the derivation of the Joos and Zeh master equation [30] as an example of dynamical equation dysplaying decoherence effects on a system prepared in a superposition of two different positions in space. In a similar way as the Joos-Zeh master equations has been constructed, we propose the deriva- tion of the master equation describing the decoherence effects on a system whose state is in a superposition of angular configurations. This work becomes of inter- est when we consider systems whose angular configuration can change, and thus one can prepare the system in a superposition of angular configurations. Differ- ences and similarities between the two models are underlined. 4

In Chap. 3 we describe the quantum Brownian motion, which can be safely con- sidered as the most known and used example of an open quantum system. Caldeira and Leggett derived the master equation describing such a dynamics in the mem- oryless limit [29]. This dynamical equation is the dissipative extension of the Joos-Zeh master equation1. Such an equation leads to decoherence and dissipa- tion, and it brings asymptotically the system to the thermal state. However it does not preserve the (complete-)positivity of the dynamics. This drawback can be avoided by considering the exact solution to the problem [54, 55]. In literature one finds different solutions which is very useful for Gaussian initial states. We present an alternative approach to the exact master equation, based on the use of the Heisenberg picture. Beside recovering the already known results, we show how one can benefit from this approach when one is interested in the dynamical evolution of non-Gaussian initial states.

In Chap. 4 an example of memoryless system-environment interaction discussed. We analyze a recently proposed source of decoherence, based on the gravitational time dilation [56]. We show that modifications to the proposed model are needed in the low temperature regime, which is the most favourable one to detect such a decoherence source. We performed a detailed analysis by comparing the “grav- itational” decoherence to the more standard decoherence sources, like the colli- sions with the surrounding residual gas in the vacuum chamber and the emis- sion, absorption and scattering of thermal radiation. Eventually, we show that the proposed source of decoherence is orders of magnitude off to be detected with present technology.

In Chap. 5 we introduce the Continuous Spontaneous Localization (CSL) model. We show how optomechanical systems provide a particularly promising experi- mental setup to infer bounds on the CSL parameters CSL and rC. We report the analyzis of three examples we considered during the doctorate project. The first is related to the gravitational wave detectors LIGO, LISA Pathfinder and AURIGA [52]. These experiments set strong bounds on the collapse parameters, and, for the first time, enclose the still unexplored parameter space in a finite region. The second one reports an improved cantilever experiment where a non-thermal ex- cess noise of unknown origin is measured [51]. In principle such a noise is com- patible with the predictions on CSL given by Adler [57]. The last example is an experimental proposal we recently presented in [58], which is based on the tor- sional motion of a system affected by the CSL noise. The proposed experiment will eventually probe the unexplored CSL parameter space, and confirm, or fal- sify, the hypothesis that the excess noise measured in [51] is due to CSL.

In Chap. 6 we draw the conclusions of the thesis.

1Since the work by Joos and Zeh [30] appeared two years later than the one by Caldeira and Leggett, it is more appropriate to say that the Joos-Zeh master equation is the restriction of the Caldeira-Leggett master equation to the regime where dissipative effects can be neglected. Notice that the physical motivations for the two models are very different, cf. Chap. 2 and Chap. 3. 5

Chapter 1 Open Quantum Systems

The usual approach to open quantum systems consists in considering the system plus the environment as an isolated system, which evolves under the usual uni- tary quantum dynamics. In the most general case, the degrees of freedom (d.o.f.) of the system plus environment are infinite and it is prohibitive to follow them all in time. However, since we are interested in the evolution of the system only, we focus on it, and we take into account the influence of the environment by av- eraging over its d.o.f. In this case we speak of an open quantum system

In the following we introduce the general properties and features of an open quantum system. For an extended description of the theory of open quantum systems we refer to [24, 26–28].

1.1 Reduced state and its evolution

The initial state ⇢ˆSE at time t =0of the global system, composed by the system of interest S and its environment E, is usually considered as uncorrelated and it evolves according to the unitary evolution ˆ : Ut ⇢ˆ =ˆ⇢ ⇢ˆ , (1.1a) SE S ⌦ E ⇢ˆ (t)= ˆ ⇢ˆ ˆ†. (1.1b) SE Ut SEUt where ⇢ˆS and ⇢ˆE are the system and environmental states respectively. After some time t, the interaction between the system and its environment correlates the two, and ⇢ˆSE(t) cannot be written in the form as in Eq. (1.1a). To extract the system properties from ⇢ˆSE(t) one needs to average over the d.o.f. of the environment

(E) ⇢ˆS(t)=Tr ⇢ˆSE(t) , (1.2) ⇥ ⇤ where ⇢ˆS(t) is called reduced state of the system S, and it is obtained by taking the partial trace (Tr (E) ) over the d.o.f. of E. This operation is performed by (E)· choosing a basis of the Hilbert space associated to E and applying the {| ⇥i i⇤}i following map ⇢ˆ (t)= (E) ⇢ˆ (t) (E) . (1.3) S h i | SE | i i i X Chapter 1. Open Quantum Systems 6

This definition does not depend on the choice of the basis. The state ⇢ˆS(t) obtained in this way preserves all the properties of a quantum state: it is an hermitian, (S) linear and positive operator, whose trace is equal to one (Tr ⇢ˆS(t) =1). The dynamical map determining its time evolution is given by: ⇥ ⇤

(E) ˆ :ˆ⇢ ⇢ˆ (t)=Tr ˆ ⇢ˆ ˆ† , (1.4) t S 7! S Ut SEUt ⇥ ⇤ which is different from the one describing the unitary dynamics ˆ of the global Ut system, because the partial trace operation breaks the unitarity of the dynam- ics. This map is given by the combination of two operations: the unitary evolu- tion provided by Eq. (1.1b), and the trace over the d.o.f. of the environment as described in Eq. (1.2). The construction of the reduced dynamical map can be represented by the following scheme1:

unitary evolution ˆ ˆ ⇢ˆSE ⇢ˆSE(t)= t⇢ˆSE t† !ˆt U U U Tr(E) Tr(E) (1.5) ? ˆ ? t (E) ˆ ˆ ⇢?ˆS ⇢ˆS(t)=Tr? t⇢ˆSE t† . y !reduced evolution y U U ⇥ ⇤ We can also express the reduced dynamical map as:

t ˆ t[ ]= exp ds Ls [ ], (1.6) · T · ✓Z0 ◆ where is the time-ordering operator and Ls is the generator of the dynamics. Ls T describes the most important dynamical equation in the theory of the open quan- tum system: the quantum master equation

dˆ⇢S(t) = Lt[ˆ⇢S(t)]. (1.7) dt One can easily verify the relation between Eq. (1.6) and Eq. (1.7). In fact, by con- sidering the time derivative of Eq. (1.6) applied to ⇢ˆS, one finds:

t d ˆ ˆ t[ˆ⇢S]=Lt exp ds Ls [ˆ⇢S]=Lt t[ˆ⇢S], (1.8) dt T ✓Z0 ◆ which corresponds to Eq. (1.7).

All dynamical maps ˆ t realized following the scheme in Eq. (1.5) satisfy by con- struction some important features. The first is the linearity of the dynamical map:

ˆ t[ˆ⇢1 +ˆ⇢2]=ˆ t[ˆ⇢1]+ˆ t[ˆ⇢2], (1.9)

1 If the initial state ⇢ˆSE does not present the structure described in Eq. (1.1a), the operation (B) defined in Eq. (1.4) with ⇢ˆS =Tr ⇢ˆSE in general is not a (dynamical) map. For an exaustive description we refer to [59] and references therein, where the first attempts in constructing a ⇥ ⇤ reduced dynamical map starting from a correlated initial state are reported. Chapter 1. Open Quantum Systems 7 which is fundamental to comply with the superposition principle. The second property of ˆ t is its continuity: ˆ ˆ lim t+⌧ [ˆ⇢S] t[ˆ⇢S]=0, (1.10) ⌧ 0 ! which naturally follows from the continuity of ˆ . The third property is related Ut to the probability interpretation we give to ⇢ˆS: its diagonal terms describe the probabilities of finding the system in a particular state. Thus it follows that, to maintain consistently this interpretation, ˆ t must preserve the positivity, the her- miticity and the trace of ⇢ˆS:

ˆ [ˆ⇢ ] 0, t S ˆ † ˆ t [ˆ⇢S] = t [ˆ⇢S] , (1.11) ⇣ (S) ⌘ Tr ⇢ˆS =1.

A dynamical map owning these features⇥ ⇤ can be considered as well constructed. However, since in general it is difficult to derive exactly a dynamical map start- ing from the global unitary dynamics, approximations are often needed. This implies that the above features are not granted, and one has to require them ex- plicitly. While the conditions on the hermiticity and on the trace can be simply verified, the conservation of the positivity is difficult to characterize. This is not however a problem since, as we will see, there are physical motivations that re- quire a stronger condition on the dynamical map: ˆ t must be completely positive.

1.1.1 Complete positivity

ˆ 1 ˆ A map t is Completely Positive (CP) if (ˆn t) is a positive map n N, where ⌦ 8 2 1ˆn identifies the identity map acting on a n dimensional Hilbert space. The physical motivation for the request of a CP dynamical map can be simply understood with the following example. Consider the Universe as composed by two systems of interest immersed in a common environment: the usual system

S and an ancilla A. The corresponding states are ⇢ˆS and ⇢ˆA, respectively. Sup- pose the environment acts independently on the two systems, with the dynami- cal maps ˆ t and ⇤ˆ t respectively. Then, the dynamical map describing the effect of the environment on the two systems is constructed as

(⇤ˆ ˆ ):ˆ⇢ ⇢ˆ (t), (1.12) t ⌦ t AS 7! AS where ⇢ˆ =ˆ⇢ ⇢ˆ , ⇢ˆ (t)=ˆ⇢ (t) ⇢ˆ (t) with ⇢ˆ (t)=⇤ˆ [ˆ⇢ ] and ⇢ˆ (t)=ˆ [ˆ⇢ ]. AS A ⌦ S AS A ⌦ S A t A S t S Then, we require the positivity of the final total state ⇢ˆ (t) 0, i.e. the total map AS (⇤ˆ ˆ ) must be a positive map. This must hold for any form of the map ⇤ˆ , t ⌦ t t even when ⇤ˆ t = 1ˆn, and for any dimension n of the Hilbert space associated to the system A. These are the requirements for a CP dynamical map. Chapter 1. Open Quantum Systems 8

1.2 Lindblad equation

A simple and important example of the Eq. (1.7) is given by the so called Lind- blad equation [60, 61]. Here we report its expression for a N dimensional Hilbert space:

N 2 1 i ˆ i ˆ ˆ ˆ 1 ˆ ˆ L [ˆ⇢S(t)] = HS, ⇢ˆS(t) H,⇢ˆS(t) + Kab La⇢ˆS(t)L† L†La, ⇢ˆS(t) , ~ ~ b 2 b h i h i a,b=1  n o X (1.13) where the first term describes the free coherent evolution with respect to the sys- ˆ tem Hamiltonian HS, as it is isolated. The last two terms result from the influence of the environment: Hˆ modifies the coherent evolution, and is called Lamb shift. The last term in Eq. (1.13) is called dissipator and it is distinguished by its characteristic structure. Here, Lˆa are called Lindblad operators and Kab is the Kossakowski matrix. Effects like dissipation, diffusion and decoherence are due to this latter term, which breaks the unitarity of the dynamical map. More- over, Eq. (1.13) represents the most general time independent generator for a trace preserving completely positive dynamical map [24]. Thus, a generator having a time independent structure different from, or that cannot be rewritten in terms of, Eq. (1.13), is not a generator of a good dynamical map. In particular, given Eq. (1.13), the complete positivity requirement is satisfied if and only if the Kos- sakowski matrix is positive K 0. ab

In general, one can consider extensions of the generator defined in Eq. (1.13) to cases with time dependent coefficients, i.e. Hˆ (t) and Kab(t). If Kab(t) remains a positive matrix for any time t, then the corresponding dynamical map satisfies all the requirements needed. It is worth noticing that in the time dependent case, the requirement of a positive Kossakowski matrix can be relaxed without compro- mising the structure of the corresponding dynamical map ˆ t. In fact, there exist cases where, for some finite time intervals, Kab(t) < 0 and still the dynamical map is CP. We will come back to this in Sec. 3.3.3, where an example is explicitly studied. 9

Chapter 2 Decoherence

One of the most interesting features of an open quantum system is decoherence. This is an intrinsically quantum feature, appearing due to the interaction with the surrounding environment. It cannot be completely avoided, one can only try to screen its action and soften its effect on the system. Consider a system S prepared in the superposition = 1 ( L + R ), where | i p2 | i | i L and R identify the states centered on two different positions, e.g. on the left | i | i and on the right of the origin, respectively. We consider these positions suffi- ciently apart to make sure that the two states can be considered as orthonormal, i.e. L R =0. The total state, system S plus environment E, at time t =0reads h | i = 1 L + R , (2.1) | SEi p2 | i | i ⌦| Ei where is the state of one environmental particle. After some time t, the inter- | Ei action between the system and the environment correlates S with E. Suppose the total state becomes

(t) = 1 ( L L + R R ), (2.2) | SE i p2 | i⌦| Ei | i⌦| E i where L is the state the environmental particle takes if the system is in L , and | Ei | i R is the state of the particle if the system is in R . Let us now take the reduced | E i | i states of S associated to the states in Eq. (2.1) and Eq. (2.2), and represent them on the system basis L , R . The corresponding density matrices are {| i | i}

R L 1 11 1 1 E E ⇢S = and ⇢S(t)= h | i . (2.3) 2 11 2 L R 1 ✓ ◆ ✓h E| E i ◆ As we can see, the populations (probabilities of finding the system on the left or on the right), which are the diagonal elements of the density matrix, do not change during the evolution. Conversely, the off diagonal terms, called coher- ences and which represent the possibility of measuring interference among the different terms of the superposition, are modified. Since, in general, L and R | Ei | E i are not equal, we have L R 1. Consequently, the coherences are reduced. |h E| E i| If we consider N environmental particles instead of one only, the coherences be- come: L R L R L R L R ... L R . (2.4) h E| E i!h 1 | 1 ih 2 | 2 ih 3 | 3 i h N | N i As time passes, more and more environmental particles interact with the system and consequently the coherences are suppressed in time: this effect is called de- coherence. Chapter 2. Decoherence 10

It is important to underline that decoherence is a fully quantum mechanism. It is due to a typically quantum feature which has no classical counterpart: the en- tanglement. In fact, when the system and the bath particle interact, their states entangle and thus decoherence occurs [cf. Eq. (2.2)].

In the next sections we will discuss qualitatively and quantitatively the decoher- ence effects, in particular we will focus on two of the most common decoherence sources: the scattering with thermal background radiation [30, 62] and the colli- sions with the residual gas in the vacuum chamber [29, 55]. Other decoherence sources act and are described in a similar way. In Chap. 4 we will analyze a re- cently proposed source of decoherence induced by gravity and we will compare it with other common decoherence sources, which action is derived in the next Section.

2.1 Gallis-Flemming master equation

In typical experiments, decoherence effects cannot be fully avoided. The residual gas in a ultra-high vacuum chamber, the thermal radiation of the chamber itself, and even the 3 K cosmic background radiation represent decoherence sources on the system dynamics. In their seminal paper [30] Joos and Zeh set the basis for the description of such effects. Their master equation can be obtained from the long wavelength limit of the Gallis and Flemming master equation [63], which is here derived.

Let us consider the following situation. A system S of mass M, described by a localized eigenstate x of the position, collides with a particle of the surrounding | i gas, whose state is and mass is m. The scattering (or collision) between the | i two particles can be schematically represented by

scattering x Sˆ ( x ) , (2.5) | i⌦| i ! | i⌦| i where Sˆ is the unitary scattering operator describing the collision process. In order to simplify the problem, we consider the recoil-less limit, i.e. the infinite mass limit1 (M m). In this limit, we can safely assume that the scattering operator heavily affects only the gas particle state and not that of S. Then Eq. (2.5) can be approximated by

scattering ˆ x x Sx , (2.6) | i⌦| i ! | i⌦ | i ˆ where the scattering operator Sx depends on the position x of the system. In a realistic situation, the system is described, in place of x , by a wavepacket : | i 1The scattering process for finite mass has been widely discussed, see for example [64–67]. Chapter 2. Decoherence 11

' = dx '(x) x . Consequently, also the scattering process changes to | i | i R scattering ˆ ' dx '(x) x Sx . (2.7) | i⌦| i ! | i⌦ | i Z In terms of the statistical operator of the reduced system, the scattering process can be described by

⇢ˆ = dx dx0 '(x)'⇤(x0) x x0 , S | ih | R R scattering (2.8) ? ⇢ˆS(t)= dx dx0 '(x?)'⇤(x0) x x0 ⌘(x, x0). y | ih | ˆ ˆ ˆ where ⌘(x, x0)= S† SxR . SinceR Sx is unitary, it does not change the popula- h | x0 | i tions of the system: ⌘(x, x0)=1. Instead, for x = x0 the environmental state 6 | i is changed according to the position x where the scattering occur, and we have

ˆ ˆ S† Sx = x x < 1, (2.9) |h | x0 | i| |h 0 | i| i.e., the coherences are reduced by the scattering event. ⌘(x, x0) quantifies the re- duction of coherences, and thus it is called decoherence term.

By assuming that the environment is in the thermal equilibrium, whose state2 is

⇢ˆB, we find that ⌘(x, x0) evolves according to

⌘(x, x0)=1 t⇤(x, x0), (2.10) where we defined [68]

(q qnˆ ) (x x ) 2 i 0 · 0 ⇤(x, x0)=n dq µ (q)v(q) dnˆ 0 f(q,qnˆ 0) 1 e ~ . (2.11) gas B | | Z Z ⇣ ⌘

Here ngas is the number density of the gas, µB(q) is the momentum thermal dis- tribution of the scattered particles, f(q,qnˆ 0) is the scattering amplitude of the process with an incoming (outgoing) scattered particle momentum q (qnˆ 0). The phase in parenthesis is due to the translational invariance of the scattering pro- cess. Indeed, because of it the scattering operator can be translated to the origin: qˆ x qˆ x ˆ i · ˆ i · Sx = e ~ S0e ~ . The velocity v(q) take different expressions depending on the scattering particle involved: for a massive bath particle we have v(q)=q/mgas, while v(q)=c the light. Substituting Eq. (2.10) in Eq. (2.8) we can simply express the time evolution of the state:

d⇢S(x, x0; t) = ⇤(x, x0)⇢ (x, x0; t), (2.12) dt S

2 We will denote the environmental state with ⇢ˆB only if it is a thermal state. Commonly, an environment in thermal equilibrium is referred to as a bath. This clarifies the change of notation (E B) with respect to the one used in Chap. 1. ! Chapter 2. Decoherence 12

where ⇢ (x, x0; t)= x ⇢ˆ (t) x0 . This is the Gallis and Flemming master equation S h | S | i [63].

It is interesting to investigate the explicit expression of Eq. (2.11) in two limiting cases. These are related to the ratio between the average wavelength of the en- vironmental particles , which can be evaluated by using the de Broglie formula =2⇡~/ q [68], and the coherent separation x = x x0 . h i | |

2.1.1 Short wavelength limit

Let us first consider the so-called short wavelength limit, i.e. . In such a ⌧ x limit the phase appearing in Eq. (2.11) oscillates very rapidly and gives no contri- bution to the integral. Thus it can be safely neglected, and the expression becomes independent from the coherent separation x. Assuming that µB(q) depends only on the modulus of q, we can evaluate the angular part of the integral as follows:

dnˆ dnˆ 0 2 f(q,qnˆ 0) = (q), (2.13) 4⇡ | | TOT Z with TOT(q) denoting the total cross section of momentum q, averaged over the directions. Consequently, Eq. (2.11) becomes position independent:

2 ⇤(x, x0) =4⇡n dqq µ (q)v(q) (q). (2.14) ' TOT gas B TOT Z The corresponding master equation in the position representation takes the fol- lowing form:

d⇢S(x, x0; t) = ⇢ (x, x0; t), (2.15) dt TOT S where TOT is referred as the total scattering rate. The state evolution is simply given by TOTt ⇢S(x, x0; t)=⇢S(x, x0;0)e , (2.16) showing that the coherences of the system are exponentially suppressed in time, independently from the coherent separation x.

2.1.2 Long wavelength limit, Joos and Zeh master equation

In the opposite limit, when , the phase in Eq. (2.11) is small, allowing x to Taylor expand the expression to the second order in q. By also assuming that

µB(q)=µB(q), Eq. (2.11) reduces to

ngas 2 4 ⇤(x, x0)= (x x0) dqµ (q)v(q)q (q), (2.17) 2 B eff ~ Z Chapter 2. Decoherence 13 where 2⇡ (q)= d(cos ✓) f(q, cos ✓) 2(1 cos ✓), (2.18) eff 3 | | Z is the effective cross section of the process, with ✓ denoting the angle between the incoming and the outgoing scattered vectors.

The master equation corresponding to the localization rate in Eq. (2.17) reads

dˆ⇢ (t) ⌘ S = [ˆx, [ˆx, ⇢ˆ (t)]] , (2.19) dt 2 S where the diffusion constant takes the following expression

ngas 4 ⌘ = 2 dqµB(q)v(q)q eff(q). (2.20) ~ Z Typically one refers to Eq. (2.19) as the Joos and Zeh master equation [24, 30]. We stress that in this case the evolution of the state depends explicitly also on the coherent separation = x x0 : x | | 2 ⌘xt ⇢S(x, x0; t)=⇢S(x, x0;0)e . (2.21)

In a similar way as in the short wavelength limit, the coherences are exponentially suppressed in time and the populations are not affected by the process.

2.2 Rotational Decoherence

We consider a situation similar to the one described in Sec. 2.1: a system in super- position interacting with the surrounding enviromnent, but now the system is in a superposition of angular configurations. This can be the case of interest for a system showing an anisotropy under rotations, e.g. a system more elongated in one direction. By exploiting the derivation of Eq. (2.12), in this section we derive the master equation describing rotational decoherence [69].

In a similar way to what was done starting from Eq. (2.8), we define the initial state of the system as

⇢ˆ = d⌦ d⌦0 '(⌦)'⇤(⌦0) ⌦ ⌦0 , (2.22) S | ih | Z Z where ⌦ is the state representing the system prepared in the angular configu- | i ration ⌦. Such a configuration can be prepared by starting from some reference configuration 0 (chosen, for example, in a way that the anisotropy of the system | i Chapter 2. Decoherence 14 is aligned along the x axis) and applying a rotation defined by the three Euler3 angles [71, 72]. Starting from the configuration ⌦, a scattering process can be described as follows

scattering ˆ ⌦ ⌦ S⌦ , (2.23) | i⌦| i ! | i⌦ | i where the recoil-less limit is considered [cf. Eq. (2.6)] and thus the scattering oper- ˆ ˆ ator S⌦ acts on the environmental state only. S⌦ can be related to the standard ˆ | i scattering operator S0 acting in the ⌦ = 0 configuration through a rotation from ˆ ˆ ˆ ˆ the configuration 0 to ⌦ : S⌦ = R(⌦)S0R†(⌦). | i | i After the scattering process, the reduced state of the system is given by

⇢ˆ (t)= d⌦ d⌦0 '(⌦)'⇤(⌦0) ⌦ ⌦0 ⌘(⌦, ⌦0), (2.24) S | ih | Z Z where (B) ˆ ˆ ⌘(⌦, ⌦0)=Tr ⇢ˆB † ⌦ . (2.25) S⌦0 S For the sake of simplicity, let us suppose that⇥ the system⇤ is in a superposition of angular configurations obtained only from rotations around the z axis. Then, the state of the system can be identified by ↵ = Rˆ (↵) 0 , with Rˆ (↵)=exp( i Lˆ ↵) | i z | i z ~ z where Lˆz is the angular momentum operator with respect to the z axis. The natu- ral basis for the computation of ⌘(⌦, ⌦0) is given by the energy-angular momen- tum representation

+ l 1 ˆ ˆ ⌘(↵,↵0)= dEµB(E) E,l,m S† S↵ E,l,m , (2.26) h | ↵0 | i l=0 m= l Z X X where µB(E) is the energy distribution of the environmental states, l and m are ˆ ˆ respectively the eigenvalues of the total angular momentum and Lz. Writing S↵ in

3These three angles describe three consecutive rotations of the system, as depicted in the fol- lowing scheme [70]:

x0 x1 x1 x3 arround z0 arround x1 arround z2 y0 y1 y2 y3 0 1 !angle ↵ 0 1 !angle 0 1 !angle 0 1 z0 z0 z2 z2 @ A @ A @ A @ A Chapter 2. Decoherence 15

ˆ 1 ˆ ˆ terms of S0 = ˆ +iT,where T is well known T-matrix from the standard quantum- mechanical scattering theory [73], we obtain

i(m m )(↵ ↵) ⌘(↵,↵0)=1 dEµ(E) 1 e 0 0 B · Xl,m lX0,m0 Z ⇣ ⌘ (2.27) 2 dE0 E,l,m Tˆ † E0,l0,m0 , · |h | | i| Z ˆ ˆ ˆ ˆ where we used the relation for the T-matrix: i(T† T)=T†T, and introduced a completeness: 1ˆ = dE0 E0,l0,m0 E0,l0,m0 . We proceed by evaluating l0,m0 | ih | the matrix elements of the T-matrix, which are defined in terms of to the scattering R P amplitude f(p, p0) of the process: i p Tˆ p0 = (E E0)f(p, p0), (2.28) h | | i 2⇡~mgas where p and p0 are respectively the incoming and outgoing momentum of the environmental particle of mass mgas, and (E E0) accounts for the energy con- 2 servation of the gas particle in the recoil-less limit (E = p /2mgas). Due to the modulus square in Eq. (2.27), we have

2 t (p p0) 2 ˆ 0 0 p T p = 3 f(p, p ) , (2.29) |h | | i| (2⇡~) mgas p | | where t is the interaction time and we handled the squared energy delta function in the usual way: 2 t mgas (E E0)= lim (p p0). (2.30) t + ! 1 2⇡~ p Now, ⌘(↵,↵0) can be rewritten in terms of Eq. (2.28) by using the following repre- sentation of the energy-angular momentum states on the momentum basis:

~3/2 p E,l,m = (E E0)Y (pˆ), (2.31) h | i pMp lm where Ylm(pˆ) are the spherical harmonics with quantum numbers l and m and with an angular dependence defined with respect to the direction pˆ. By using this representation and applying Eq. (2.30), we can express Eq. (2.27) in terms of the scattering amplitude functions f(p, p0):

⌘(↵,↵0)=1 t⇤(↵,↵0), (2.32) Chapter 2. Decoherence 16 with 2m gas i(m m0)(↵0 ↵) ⇤(↵,↵0)= dEEµB(E) 1 e (2⇡~)3 · Z Xlm Xl0m0 ⇣ ⌘ dpˆ dpˆ0 dpˆ00 dpˆ000Y ⇤ (pˆ)Ylm(pˆ0)Yl m (pˆ00)Y ⇤ (pˆ000)f(p, p0)f ⇤(p00, p000). · l0m0 0 0 lm Z Z Z Z (2.33) The angular integrations in Eq. (2.33) can be simply evaluated since the spheri- cal harmonics form a complete set of functions and the following completeness relation holds l 1 (✓ ✓0) Y ⇤ (pˆ0)Y (pˆ)= ( 0), (2.34) lm lm sin✓ l=0 m= l X X where the two direction pˆ and pˆ0 are parametrized by the angles (✓,) and (✓0,0), respectively [74]. By exploiting Eq. (2.34), Eq. (2.33), multiplied by the number density ngas of particles, becomes

p 2 ⇤(↵,↵0)=n dp µ (p) dnˆ 0 f(p,pnˆ 0) (1 R(p,pnˆ 0,!)) , (2.35) gas B m | | Z gas Z where f ⇤(p!,pnˆ !0 ) R(p, n0,!)= , (2.36) f ⇤(p,pnˆ 0) where p is the vector p rotated by an angle ! = ↵ ↵0 around the z axis. Conse- ! quently the master equation reads [69]

d⇢S(↵,↵0; t) = ⇤(↵,↵0)⇢ (↵,↵0; t), (2.37) dt S where ⇢ (↵,↵0; t)= ↵ ⇢ˆ (t) ↵0 . S h | S | i

It is interesting to notice that the expression for ⇤(↵,↵0) in Eq. (2.35) has the same structure of Eq. (2.11), with R(p, n0,!) replaced by

f ⇤(qx x0 ,qnˆ x0 x ) R(q, nˆ 0, x x0)= 0 . (2.38) f ⇤(q,qnˆ 0)

Here qx x is the vector q translated in space by x x0. This result can be under- 0 stood once we consider the expression for the scattering amplitude generated by the potential V (r), under the Born approximation [72]:

(q qnˆ ) r ~mgas i 0 · f(q,qnˆ 0)= dr e ~ V (r). (2.39) 2⇡ Z Chapter 2. Decoherence 17

qˆ x qˆ x ˆ i · ˆ i · Implementing the translation in space: Sx = e ~ S0e ~ , as it was done to obtain Eq. (2.11), we have

(q qnˆ ) (r+x x ) ~mgas i 0 · 0 ~ f ⇤(qx x0 ,qnˆ x0 x )= dr e V (r), (2.40) 0 2⇡ Z and by considering the ratio between the two scattering amplitudes, i.e. Eq. (2.39) and Eq. (2.40), one obtains the known result:

f ⇤(q ,qnˆ 0 ) (q qnˆ 0) (x x0) x x0 x x0 i · = e ~ , (2.41) f ⇤(q,qnˆ 0) which is the exponential factor in Eq. (2.11).

As recent theoretical [58, 71, 72, 75–83] and experimental [84–92] works can tes- tify, there is a growing interest in systems exploiting rotational or torsional de- grees of freedom of a quantum system. Thus, the master equation (2.37) becomes particularly useful in the description of decoherence on these systems. 18

Chapter 3 Quantum Brownian Motion

Brownian motion is considered the paradigm of an open system, both in the clas- sical and in the quantum case. Originally [93], it was observed as the motion of pollen grains suspended in a viscous liquid, for which different classical models were proposed [94, 95]. In the classical case, two different but related, ways to model the liquid have been developed. The first one is the so-called collisional model where the environment is represented by free particles in thermal equilibrium, interacting with the sys- tem through instantaneous collisions. This is the model considered by Einstein [94] and Langevin [95], and its description is given by the Langevin equation [95]

1 1 x¨(t)+2mx˙(t)+ M @xV (x)= M F (t), (3.1) where the evolution of the position x of the system of mass M in a external poten- tial V (x) is damped by a friction term proportional to the velocity (Stokes term), where m is a damping rate. The stochastic force F (t) describes the noise induced by the environment, and is assumed to be gaussian, i.e. fully described by its av- erage F (t) and two-time correlation function F (t)F (s) , where denotes a h i h i h·i statistical average. For the Brownian motion they read

F (t) =0, h i (3.2) F (t)F (s) =4M k T(t s), h i m B where kB is the Boltzmann constant, T is the temperature of the environment.

The second model instead considers a particle S immersed in an environment of independent harmonic oscillators in thermal equilibrium. This is called the harmonic bath model [96, 97] and its quantum version is the main subject of this Chapter.

3.1 The model

The model consists of a particle S of mass M, with position xˆ and momentum pˆ, harmonically trapped at frequency !0 and interacting with a thermal bath of independent harmonic oscillators, with positions Rˆk, momenta Pˆk, mass mk and ˆ ˆ ˆ ˆ ˆ frequencies !k. The total Hamiltonian HT of system plus bath is HT = HS +HB +HI, Chapter 3. Quantum Brownian Motion 19 where

2 ˆ2 ˆ pˆ 1 2 2 ˆ Pk 1 2 ˆ2 ˆ ˆ HS = + M!0xˆ , HB = + mk!kRk, HI =ˆx CkRk, (3.3) 2M 2 2mk 2 Xk Xk are respectively system, bath and interaction Hamiltonians. The total initial state is assumed to be uncorrelated

⇢ˆ =ˆ⇢ ⇢ˆ , (3.4) T S ⌦ B where ⇢ˆB is the state of the environment. A common assumption is to consider the environment in thermal equilibrium, and its state described by a Gibbs state with respect to the free Hamiltonian of the bath:

Hˆ e B ⇢ˆ = , (3.5) B (B) Hˆ Tr e B ⇥ ⇤ where =1/(kBT ) is the inverse temperature. The characterization of the set of 1 coupling constants Ck is provided by the spectral density which is defined as

2 Ck J(!)= (! !k). (3.6) 2mk!k Xk In terms of the latter, we can define the two-time correlation function of the bath ˆ ˆ operator B = k CkRk:

CP(t s)=Tr(B) Bˆ(t)Bˆ(s)ˆ⇢ = 1 D (t s) i D(t s), (3.7) B 2 1 2 ˆ ˆ iHBt/~ iHBt/⇥~ ⇤ where Bˆ(t)=e Beˆ . D1(t) and D(t) are the noise and the dissipative kernels, describing respectively the noisy action of the environment, related to the temperature of the latter, and the corresponding dissipative effect on the system. They read:

+ 1 D1(t)=2~ d!J(!)coth(~!/2) cos(!t), (3.8a) 0 Z + 1 D(t)=2~ d!J(!)sin(!t). (3.8b) Z0 Such kernels are related through the Fluctuation-Dissipation theorem [98–102]:

+ + 1 ~! 1 dt cos(!t)D (t)=coth dt sin(!t)D(t), (3.9) 1 2 Z1 ✓ ◆ Z1 1J(!) describes the distribution in frequency of the environmental harmonic oscillators, weighted by the corresponding coupling constant. This expression does not imply a limit where the frequencies !k form a continuous spectrum. Chapter 3. Quantum Brownian Motion 20 which provides the following time symmetries

C( t)=C⇤(t)=C(t i~). (3.10) This relation, known as Kubo-Martin-Schwinger (KMS) condition, allows for a proper definition of the thermodynamical limit of the environmental state. In- deed, one can easily see that in the (thermodynamical) infinite bath particle limit the Gibbs formula in Eq. (3.5) becomes meaningless [103, 104] and one needs an alternative way to describe the bath state. Typically, instead of considering the whole state (that becomes meaningless in the limit), one considers only some of its properties, that are expected to remain stable in the limit. The KMS condition [cf. Eq. (3.10)] survives the limit and it provides the basis to construct a general- ization of Eq. (3.5) valid also in the thermodynamical limit [104]. Consequently, the spectral density J(!) must be chosen in a way that the two kernels exist and that Eq. (3.9) holds: these are two important as well as trivial conditions one must respect to obtain a well defined description of the environmental state in the ther- modynamical limit.

To make an explicit comparison between the classical and quantum description, we write down the quantum Langevin equation derived form Eq. (3.3):

1 t i 1 xˆ¨(t) dsD(t s)ˆx(s) V (ˆx), pˆ = Bˆ(t). (3.11) M M M ~ Z0 ~ ⇥ ⇤ Although Eq. (3.1) describes the evolution of a classical system, while Eq. (3.11) is the dynamical equation for a quantum one, the two equations have much in common. The potential V appears in both Langevin equations in the usual way, and the stochastic force F (t) in Eq. (3.1) is here replaced by the bath operator Bˆ(t). The classical dissipative term proportional to the velocity of the system is now replaced by an integral term, containing the position operator at all previous times. In this sense, Eq. (3.11) contains a memory of the history of the system, which is weighted by the dissipative kernel D(t). In the memoryless limit, we have D(t s) @ (t s) and we recover / s ¨ ˙ i 1 ˆ xˆ(t)+2mxˆ(t) V (ˆx), pˆ = B(t), (3.12) ~M M ⇥ ⇤ which is the quantum analog of Eq. (3.1).

3.2 The Calderira-Leggett master equation

The first master equation describing the model introduced in Sec. 3.1 was derived by Caldeira and Leggett [29] by using three approximations. The Born approxi- mation assumes a weak coupling between the system and the bath, guaranteering that the total state can be expressed as factorized at any time t and that the bath Chapter 3. Quantum Brownian Motion 21 state remains unperturbed: ⇢ˆ (t) ⇢ˆ (t) ⇢ˆ . The Markov approximation instead T ⇡ S ⌦ B neglects all memory effects in the dynamics, and the corresponding master equa- tion becomes time local. In order to implement the latter approximation a spectral density J(!) ! must be considered. This implies that the dissipation kernel has / a memoryless structure D(t s) @ (t s) [cf. Eq. (3.12)]. However, a drawback / s of this choice is that the noise kernel D1(t) defined in Eq. (3.8) diverges, and the KMS condition in Eq. (3.10) does not hold anymore. One can cure this divergence by introducing the (third) high temperature approximation for which we obtain D (t) (t). Under these approximations, one obtains the Caldeira-Leggett (CL) 1 / master equation in the Lindblad form with constant coefficients:

dˆ⇢S(t) i ˆ im 2Mm = [HS, ⇢ˆS(t)] [ˆx, p,ˆ ⇢ˆS(t) ] [ˆx, [ˆx, ⇢ˆS(t)]] . (3.13) dt ~ ~ { } ~2 The first term describes the coherent evolution due to the system Hamiltonian, the second governs the dissipation, while the last term determines the tempera- ture dependent diffusive action of the bath and it is the one responsible for deco- herence. In this sense, the CL master equation can be considered as the dissipative extension of the Joos and Zeh master equation (2.19).

There are several approaches [105–107] one can use to derive the CL master equa- tion in (3.13). The original one [29] is based on the Feynman-Vernon theory [108, 109], which we now briefly describe. We consider the position represen- tation of the total state at time t in terms of the total state at time t =0

x, R ⇢ˆSE(t) y, Q = dx0 dy0 dR0 dQ0 K(x, R,t; x0, R0, 0) h | | i · (3.14) Z Z Z Z K⇤(y, Q,t; y0, Q0, 0) x0, R0 ⇢ˆ y0, Q0 , h | SE| i where R and Q identify the positions of the environmental particles, x and y the position of the system and K(x, R,t; x0, R0, 0) is the position representation of ˆ iHTt/~ e , which can be expressed via a path integral

x R ˆ i iHTt/~ ST[x,R] K(x, R,t; x0, R0, 0) = x, R e x0, R0 = x R e ~ , (3.15) h | | i D D Zx0 ZR0 with S [x, R]= t ds denoting the action of the total system and the total T 0 LT LT Lagrangian. R The reduced state of the system S in the position representation is given by

⇢ (x, y, t)= dR x, R ⇢ˆ (t) y, R , (3.16) S h | SE | i Z Chapter 3. Quantum Brownian Motion 22 which, under the assumption of uncorrelated initial states in Eq. (3.4), can be expressed as

⇢S(x, y, t)= dx0 dy0 J(x, y, t; x0,y0, 0)⇢S(x0,y0, 0). (3.17) Z Z

Here the propagator J(x, y, t; x0,y0, 0) takes into account the influence of the sur- rounding environment:

x y i (SS[x] SS[y]) J(x, y, t; x0,y0, 0) = x ye~ [x, y], (3.18) D D F Zx0 Zy0 where SS[x] is the action of the system S alone and

R Q i (SI[x,R] SI[y,Q]+SB[R] SB[Q]) [x, y]= dR0 dQ0⇢ (R0, Q0, 0) R Q e ~ , F B D D Z Z ZR0 ZQ0 (3.19) is the influence functional. This is defined in terms of the matrix elements of the bath state at time t =0

mk!k 2 2 mk!k exp [(R0 + Q0 )cosh(~!) 2R0 Q0 ] 2⇡~ sinh(~!k) k k k k ⇢B(R0, Q0, 0) = , n 2⇡ sinh ( !k) o k ~ ~ Y (3.20) and of the actions SB and SI derived from the bath and interaction Lagrangians respectively. By differentiating Eq. (3.17) with respect to time we can derive, un- der the approximations already stated, the CL master equation (3.13). Precisely, this is done by substituting the above actions with the following expressions:

t

SS[x]= ds S, 0 L Z t 1 ˙ 2 1 2 2 SB[R]= ds 2 mkRk 2 mk!kRk , (3.21) 0 Z k ⇣ ⌘ t X

SI[x, R]= dsx CkRk, 0 Z Xk where is the Lagrangian of the system. LS

The CL master equation has two limitations. First, it is restricted to the high temperature regime, which cannot be always fulfilled: the latest attempts to reach the ground state [110, 111] is an opto-mechanical example. Second, the master equation is the generator of a dynamical map which is not CP [112, 113], i.e. it does not map all quantum states ⇢ˆS into quantum states. Accordingly, one needs to “reduce” the class of initial states in order to avoid gross miscalculations [54, 114, 115]. This will be discussed in the next subsection. Chapter 3. Quantum Brownian Motion 23

3.2.1 Complete positivity problem

The CP problem in the CL model can be handled by modifying the master equa- tion by introducing suitable correcting terms. Consider the CL master equation written in the form displayed in Eq. (1.13) and here reported:

2 i ˆ i ˆ ˆ ˆ 1 ˆ ˆ L [ˆ⇢S(t)] = HS, ⇢ˆS(t) H,⇢ˆS(t) + Kab La⇢ˆS(t)L† L†La, ⇢ˆS(t) , ~ ~ b 2 b h i h i a,b=1 h n oi X (3.22) As already mentioned, for a master equation with a time independent Lindblad structure, the positivity of the Kossakowski matrix is sufficient to satisfy the re- quirement of completely positivity of the corresponding dynamical semigroup [24]. In the case of the Caldeira-Leggett master equation, the Lamb shift reads ˆ m ˆ ˆ H = 2 (ˆxpˆ +ˆpxˆ), the Lindblad operators are L1 =ˆx and L2 =ˆp and the Kos- sakowski matrix takes the form

2 4Mm/~ im/~ Kab = . (3.23) im/~ 0 ✓ ◆ As one can see, the Kossakowski matrix has a negative determinant and conse- quently the dynamical map is not CP. However, we can modify the Kossakowski matrix in such a way that its determinant is zero, this is the minimally invasive modification [113]. We do so by adding a term to the master equation propor- tional to a double commutator in pˆ dˆ⇢S(t) i ˆ im 2Mm m = [HS, ⇢ˆS(t)] [ˆx, p,ˆ ⇢ˆS(t) ] [ˆx, [ˆx, ⇢ˆS(t)]] p,ˆ [ˆp, ⇢ˆS(t)] . dt ~ ~ { } ~2 8M ⇥ (3.24)⇤ With this modification Eq. (3.13) can be written in the form in Eq. (3.22) with

ˆ 4Mm m L = 2 xˆ + i p.ˆ (3.25) s ~ r4M So there is only one Lindblad operator, and the Kossakowski matrix becomes K =1, and the dynamics satisfies the CP condition. For high temperatures, 0, the term we added by hand is small compared to the others and its action ! is negligible. On the other hand, for low temperatures one obtains different pre- dictions from the one given by the CL master equation. This will be discussed in detail in Sec. 3.3.4.

3.3 Non-Markovian Quantum Brownian motion

The main contributions in overcoming the limitations given by the CL model were given by Haake and Reibold [54] and later by Hu, Paz and Zhang [55], who provided the exact master equation for the particle S given the total Hamiltonian Chapter 3. Quantum Brownian Motion 24

ˆ HT:

dˆ⇢S(t) i ˆ i(t) = [H(t), ⇢ˆS(t)] [ˆx, p,ˆ ⇢ˆS(t) ] dt ~ ~ { } h(t)[ˆx, [ˆx, ⇢ˆ (t)]] f(t)[ˆx, [ˆp, ⇢ˆ (t)]] , (3.26) S S where Hˆ (t) and the coefficients (t), h(t) and f(t) now are time dependent. We refer to this model as to the Quantum Brownian Motion (QBM) model. Contrary to the CL master equation, which is valid only for the specific ohmic spectral density (J(!) !), Eq. (3.26) is valid for arbitrary spectral densities J(!) and / temperatures T . The explicit form of the coefficients, beyond the weak coupling regime, was provided by Haake and Reibold [54] and later by Ford and O’Connell in [116]. The generality of such a solution is outstanding; however, as noticed in [116], solving the time-dependent master equation is in general a formidable problem. In [116] the authors show that the dynamics of the system can be more easily solved by working with the Wigner function of the system and bath at time t and then averaging over the degrees of freedom of the bath. According to their procedure, the reduced Wigner function W at time t can be expressed in terms of that at time t =0as follows

+ + 1 1 W (x, p, t)= dr dqP(x, p; r, q; t)W (r, q, 0), (3.27) Z1 Z1 where P describes the transition probability of Gaussian form [116]. The draw- back of such a procedure is the limited set of initial states ⇢ˆS for which the Wigner function is analytically computable. For Gaussian states this is not a problem; however there exist physically relevant situations where this is not the case [117– 119]. An example is provided by a system initially confined in an infinite square potential [cf. Sec. 3.3.5]. Here we report an alternative approach to the master equation in Eq. (3.26) de- rived in the Heisenberg picture [120]. The approach exploited is valid for a gen- eral environment at arbitrary temperatures, regardless of the strength of the cou- pling and of the form of the initial state. The master equation we derive is exact and equivalent to that in [54, 55], however it can be use also for non-Gaussian states.

3.3.1 The adjoint master equation

We start by deriving the adjoint master equation for QBM. This is the dynamical equation describing the time evolution of a generic operator Oˆ of the system S, once the average over the bath is taken. Chapter 3. Quantum Brownian Motion 25

For reasons that will be clear later, let us consider the von Neumann representa- tion [121, 122] of the operator Oˆ, defined, at time t =0, by the following relation:

Oˆ = d dµ (, µ)ˆ(, µ, t =0), (3.28) O Z where (, µ) is the kernel of the operator Oˆ and ˆ(, µ, t =0)=exp[ixˆ + iµpˆ] is O the generator of the Weyl algebra, also called characteristic or Heisenberg-Weyl operator [122]. The reduced operator Oˆ at time t is obtained by taking the unitary ˆ 1ˆ time evolution of the extended operator O B with respect to the total Hamil- ˆ 1ˆ ⌦ tonian HT of the system plus bath, where B is the bath identity operator, and by tracing over the degrees of freedom of the bath. In terms of the von Neumann representation, this reads

Oˆ = d dµ (, µ)ˆ , (3.29) t O t Z where we introduced the characteristic operator at time t:

(B) ˆ 1ˆ ˆ ˆt =Tr ⇢ˆB t†(ˆ(, µ, 0) B) t , U ⌦ U (3.30) (B) ⇣ixˆ(t)+iµpˆ(t) ⌘ =Tr ⇥⇢ˆBe , ⇤ ⇥ ⇤ and ˆ =exp( i Hˆ t), xˆ(t) and pˆ(t) are the position and momentum operators of Ut ~ T the system S evolved by the unitary evolution ˆ and ⇢ˆ is defined in Eq. (3.4). Ut B

In order to obtain the explicit expression of xˆ(t) and pˆ(t), we rewrite the bath and interaction Hamiltonians defined in Eq. (3.3) in terms of the creation and ˆ ˆ ˆ ˆ ˆ annihilation operators bk† and bk of the k-th bath oscillator: HB = k ~!kbk† bk and Hˆ = xˆBˆ(0), where Bˆ(t) is defined as I P

ˆ ~ ˆ i!kt ˆ i!kt B(t)= Ck bke + bk† e . (3.31) 2mk!k k r X ⇣ ⌘ We solve the Heisenberg equations of motions for xˆ(t) and pˆ(t) by using the Laplace transform:

pˆ 1 t xˆ(t)=G1(t)ˆx + G2(t) + ds G2(t s)Bˆ(s), (3.32a) M M 0 tZ pˆ(t)=MG˙ (t)ˆx + G˙ (t)ˆp + ds G˙ (t s)Bˆ(s), (3.32b) 1 2 2 Z0 where xˆ and pˆ denote the operators at time t =0, and the two Green functions G1(t) and G2(t) are defined as

d 1 M G1(t)= G2(t),G2(t)= 2 2 (t), (3.33) dt L M(s + !R ) [D(t)](s)/~  L Chapter 3. Quantum Brownian Motion 26 where denotes the Laplace transform, and D(t) is the dissipation kernel defined L in Eq. (3.8). Given Eqs. (3.32), since the operators of the system and of the bath commute at the initial time, it follows that:

i↵1(t)ˆx+i↵2(t)ˆp (B) ˆt = e Tr ⇢ˆBˆ(t) , (3.34) ⇥ ⇤ where ↵1(t) and ↵2(t) are defined as

↵1(t)=G1(t)+µMG˙ 1(t), (3.35a)

↵2(t)=G2(t)/M + µG˙ 2(t), (3.35b)

and the operator ˆB(t) refers only to the degrees of freedom of the environment:

t ˆ (t)=exp i ds Bˆ(s)↵ (t s) . (3.36) B 2  Z0 Under the assumption of a thermal state for the bath [cf. Eq. (3.5)], the trace over (B) (t) ˆB(t) gives a real and positive function of time Tr ⇢ˆBˆB(t) = e , where the ex- plicit form of (t) can be obtained exploiting the definition of the spectral density ⇥ ⇤ in Eq. (3.6). In Appendix A.1 we present the explicit form of (t), written as the 2 2 sum of three terms: (t)= 1(t)+µ 2(t)+µ3(t). The time derivative of ˆt gives

dˆt i~ = i↵˙ (t)ˆx + i↵˙ (t)ˆp + ↵˙ (t)↵ (t) ↵ (t)˙↵ (t) + ˙(t) ˆ ; (3.37) dt 1 2 2 1 2 1 2 t h ⇥ ⇤ i after substituting this expression in:

d dˆ Oˆ = d dµ (, µ) t , (3.38) dt t O dt Z we obtain the adjoint master equation for the operator Oˆt. We underline that ˆt depends also on the two parameter and µ.

The integral in Eq. (3.38) depends on the choice of the kernel (, µ). On the other O hand, we want an equation that can be directly applied to a generic operator Oˆ without having first to determine its kernel. This means that we want to rewrite Eq. (3.37) in the following time-dependent form

2 dˆt ˜ i ˜ˆ ˜ ˆ ˆ 1 ˆ ˆ = Lt⇤ [ˆt]= Heff(t), ˆt + Kab(t) LaˆtL† LaL†, ˆt , (3.39) dt ~ b 2 b h i a,bX=1  n o ˜ˆ ˜ where the effective Hamiltonian Heff(t), the hermitian Kossakowski matrix Kab(t) and the Lindblad operators Lˆa do not depend on the parameters and µ. Thus, the linearity of Eq. (3.38) will allow to extend Eq. (3.39) to any operator Oˆt.To reach this goal we must rewrite the the coefficients ↵i and (t), appearing in Eq. (3.37), in a way that do not depend on the parameters and µ. To achieve Chapter 3. Quantum Brownian Motion 27

this, let us consider the commutation relations among xˆ, pˆ and ˆt:

ˆt, xˆ = ~↵2(t)ˆt and ˆt, pˆ = ~↵1(t)ˆt. (3.40) ⇥ ⇤ ⇥ ⇤ Exploiting Eqs. (3.35), we can express ˆt and µˆt as a linear combination of the above commutators. By using this result, we can rewrite Eq. (3.37) in the structure given by Eq. (3.39), where

2 A ˆ pˆ (t) 1 A 2 H˜ (t)= + (ˆxpˆ +ˆpxˆ)+ M (t)ˆx , (3.41) eff 2M 2 2

A the Lindblad operators are Lˆ1 =ˆx and Lˆ2 =ˆp. The time dependent function (t), A (t), D1(t) and the elements of the Kossakowski matrix K˜ab(t) are reported in Appendix A.2. An important note: one of the elements of the Kossakowski ma- ˜ trix vanishes K22(t)=0, and the corresponding term proportional to [ˆp, [ˆp, ⇢ˆS]] is absent. In the case of Caldeira-Leggett master equation [29], this implied that the dynamics was not completely positive. In the case under study, complete posi- tivity is instead automatically satisfied, as it is explicitly shown in Sec. 3.3.3. This result is in agreement with previous results [54, 55, 116, 123].

Since Eq. (3.39) is linear in ˆt and does not depend on and µ, it holds for any ˆ d ˆ ˜ ˆ ˜ operator Ot: dt Ot = Lt⇤[Ot]. This is the adjoint master equation for QBM and Lt⇤ is the generator of the dynamics for the operators. The corresponding adjoint dynamical map is given by

t ˜ t⇤[ ]= exp ds Ls⇤ [ ]. (3.42) · T · ✓Z0 ◆ The result here obtained is very general and depends only on: the form of the total ˆ Hamiltonian HT together with the separability of the initial total state [cf. Eq. (3.4)], but does not depend on the particular initial state of the system S. We now de- rive the master equation for the density matrix, starting from the adjoint master equation, and we show that we recover known results in the literature.

3.3.2 The Master Equation for the statistical operator

Let us consider the dynamical map t for the states:

:ˆ⇢ (0) ⇢ˆ (t), (3.43) t S 7! S which is the adjoint map of t⇤ defined in Eq. (3.42), and similarly to it, the map t can be written as t t[ ]= exp ds Ls [ ]. (3.44) · T · ✓Z0 ◆ Chapter 3. Quantum Brownian Motion 28

The adjointness, denoted here by the -symbol, has to be understood in the fol- ⇤ lowing sense:

(S) (S) ˆ =Tr ⇤ [ˆ(0)]⇢ ˆ (0) =Tr ˆ(0) [ˆ⇢ (0)] . (3.45) h ti t S t S Let us consider the time derivative⇥ of ˆ ⇤and let⇥ us express it as⇤ follows: h ti

d (S) (S) ˆ =Tr ⇤⇤ [ˆ(0)]⇢ ˆ (0) =Tr ˆ(0)⇤ [ˆ⇢ (0)] . (3.46) dt h ti t S t S ⇥ ⇤ ⇥ ⇤ The above equation defines the two maps ⇤t and ⇤t⇤:

⇤t [ˆ⇢S(0)] = Lt [ˆ⇢S(t)] = Lt t [ˆ⇢S(0)] , (3.47a) ˜ ˜ ⇤t⇤ [ˆ(0)] = Lt⇤ [ˆt]=Lt⇤ t⇤ [ˆ(0)] . (3.47b) If one considers a time independent adjoint master equation, switching to the master equation for the states is straightforward: the two dynamical maps t⇤ ˜ and t, respectively defined in Eq. (3.42) and Eq. (3.44), reduce to exp(tL⇤) and ˜ exp(tL). In particular, the map t⇤ and its generator L⇤ commute, yielding to ˜ ⇤t⇤ [ˆ(0)] = t⇤ L⇤ [ˆ(0)] . (3.48) By taking the adjoint of the latter expression, and comparing it with the definition given in Eq. (3.47a), we obtain

˜ 1 ˜ L = L t t = L. (3.49) A similar procedure is applied in the time dependent case. We start by consid- ering the adjoint of the definition given in Eq. (3.47b), which takes the following form ˜ ⇤t [ˆ⇢S(0)] = t Lt [ˆ⇢S(0)] , (3.50) and we compare it the definition in Eq. (3.47a). Thus, we obtain

˜ 1 Lt = t Lt t . (3.51) In terms of this latter expression, Eq. (3.47b) becomes

⇤t⇤ [ˆ(0)] = t⇤ Lt⇤ [ˆ(0)] . (3.52) Accordingly, when the generator is time dependent, in order to construct the mas- ter equation for the states one needs to derive explicitly the form of Lt⇤. The ex- plicit calculations are reported in Appendix A.3 and the final result is:

2 i ˆ ˆ ˆ 1 ˆ ˆ Lt⇤ [ˆ(0)] = Heff(t), ˆ(0) + Kab(t) Laˆ(0)L† LaL†, ˆ(0) , (3.53) ~ b 2 b h i a,bX=1  n o Chapter 3. Quantum Brownian Motion 29 where pˆ2 A(t) 1 Hˆ (t)= (ˆxpˆ +ˆpxˆ)+ MA(t)ˆx2, (3.54) eff 2M 2 2 and the elements of Kab(t) are reported in Appendix A.3. Finally, by merging Eq. (3.46) with Eq. (3.52), we obtain

d (S) (S) ˆt =Tr Lt⇤ [ˆ(0)] t [ˆ⇢S(0)] =Tr ˆ(0)⇤t [ˆ⇢S(0)] , (3.55) dt h i

d ⇥ ⇤ ⇥ ⇤ where ⇤t[ˆ⇢S(0)] = dt ⇢ˆS(t) This can be simply done by exploiting the cyclic prop- erty of the trace Tr (S) , that applied on the expression in Eq. (3.53) leads to the · master equation for the states of the system S: ⇥ ⇤ 2 dˆ⇢S(t) i ˆ ˆ ˆ 1 ˆ ˆ = Heff(t), ⇢ˆS(t) + Kab(t) L†⇢ˆS(t)La LaL†, ⇢ˆS(t) , (3.56) dt ~ b 2 b h i a,bX=1  n o This is the desired result, which coincides with the master equation in (3.26).

3.3.3 Complete Positivity

We now discuss the complete positivity of the dynamical map t⇤ generated by ˜ Lt⇤ defined in Eq. (3.39). The action of this dynamical map on the generic operator Oˆ of the system S is

(B) ⇤[Oˆ]=Oˆ =Tr ⇢ˆ ˆ†(Oˆ 1ˆ ) ˆ , (3.57) t t B Ut ⌦ B Ut ⇥ ⇣ ⌘⇤ which is the combination of two completely positive maps: the unitary evolution provided by the total Hamiltonian of system plus environment, and the trace over the environment. Therefore, by construction the dynamical map is com- pletely positive. However, in a situation where approximations are needed in order to compute explicitly the coefficients of the (adjoint) master equation, the verification of the complete positivity of the dynamics becomes a fundamental point of interest. When the generator L of the dynamics is time independent, the sufficient and nec- essary condition for the complete positivity of the dynamical map is the positivity of the Kossakowski matrix [61, 124]. For a time dependent generator Lt, instead, a positive Kossakowski matrix is only a sufficient condition for complete positivity. An example is precisely the HPZ model under study, whose Kossakowski matrix is not positive for all times, nevertheless the dynamics is completely positive. For a time dependent generator, a necessary and sufficient condition instead is given by the following theorem [125, 126], under the assumption of a Gaussian channel. Chapter 3. Quantum Brownian Motion 30

The action of the dynamical map t on the characteristic operator ˆ is

1 ⇤ :ˆ(0) = exp i ⇠ R ˆ =exp(i ⇠ X R )exp ⇠ Y ⇠ , (3.58) t h | i 7! t h | t| i 2 h | t| i ⇣ ⌘ where X and Y are 2 2 matrices describing the evolution of the characteristic t t ⇥ operator

G1(t) G2(t)/M 21(t) 3(t) Xt = , Yt = , (3.59) MG˙ (t) G˙ (t) (t) 2 (t) ✓ 1 2 ◆ ✓ 3 2 ◆ and ⇠ =(, µ) and R =(ˆx, pˆ). In terms of Xt, Yt and of the symplectic matrix h |01 h | ⌦ = 10, we can define the following matrix t: i~ i~ T = Y + ⌦ X ⌦X . (3.60) t t 2 2 t t The theorem states that the necessary and sufficient condition for the dynamical map t to be completely positive (CP) is the positivity of t for any t>0. Since the matrix is a 2 2 matrix, the request of its positivity reduces to the request t ⇥ of positivity of its trace and determinant:

Tr = 2 (t)+ (t) , (3.61a) t 1 2 2 ⇥ ⇤ ⇣ 2 ⌘ 1 det t =41(t)2(t) 3(t) ~ F (t) . (3.61b) 4 ⇥ ⇤ ⇣ ⌘ The condition of positivity of the trace, Eq. (3.61a), is easily verified for all phys- ical spectral densities: the spectral density is positive by definition [cf. Eq. (3.6)] and this implies the negativity of 1(t) and 2(t) [cf. Eqs. (A.1)] for any tempera- ture. On the other hand, the second condition, Eq. (3.61b), cannot be easily ver- ified in general. Once a specific spectral density J(!) is chosen, one can check explicitly whether det[ ] 0. For example, the spectral density J(!) !, orig- t / inally chosen in [29] to describe the quantum brownian motion, does not satisfy the above condition also in the simple case of no external potentials. It is in fact well known that the Caldeira-Leggett master equation is not CP.

As already remarked, the QBM model automatically guarantees complete posi- tivity. On the other hand, in particular cases one is not able to compute explicitly the time dependent coefficients of the Kossakowski matrix. Approximations are needed, in which case complete positivity is not automatically guaranteed any- more. This can be checked in a relatively easy way by assessing the positivity of det( t). Chapter 3. Quantum Brownian Motion 31

3.3.4 Time evolution of relevant quantities

The original QBM master equation (3.26) is expressed in terms of functions, whose explicit expression is not easy to derive, even if one considers the solution given in [116]. They are solutions of complicated differential equations, difficult to solve except for very simple situations. More important, expectation values are not easy to compute: one has to determine the state of the system at time t, which is in general a formidable problem also in a particularly simple situation. In our derivation, instead, the adjoint master equation provides a much easier tool for the computation of expectation values. The evolution is expressed in the Heisen- berg picture, therefore it does not depend on the state of the system S but only on the properties of the adjoint evolution t. For example, by plugging the expression of xˆ2(t) (obtained from Eq. (3.32)) in Eq. (3.39) we obtain an equation for the expectation value xˆ2 : h t i

d 2 2 2 2 xˆ =2G˙ 1(t)G˙ 2(t) xˆ +2G˙ 1(t)G˙ 2(t) pˆ /M + dt h t i h i h i (3.62) +(G (t)G˙ (t)+G˙ (t)G (t)) x,ˆ pˆ /M 2˙ (t). 1 2 1 2 h{ }i 1 The time dependence in the right hand side of the latter equation is only in the functions G1(t), G2(t) and 1(t): the expectation values of the operators here ap- pearing are computed at time t =0. This implies that Eq. (3.62) can be directly solved without having to consider the full system of differential equations (con- sisting of d xˆ2 , d pˆ2 and d xˆ , pˆ ), as it is necessary when one deals with dt h t i dt h t i dt h{ t t}i the problem in the Schrödinder picture [24], as well as for the case of the Wigner function approach [116, 127, 128]. To make an explicit example, we provide the general solution of some physi- cal quantities of interest for a specific spectral density. We consider: the diffu- sion function ⇤dif(t)= xˆ2 xˆ 2 and the average energy of the system E(t)= h t ih ti pˆ2 /2M + 1 M!2 xˆ2 , whose explicit expressions are given in Appendix A.4.We h t i 2 S h t i also consider the decoherence function dec(t), which is defined as follows. We consider a particle which, at time t =0, is described by a state (t =0) = | i [ ↵ + ], where ↵ and are two gaussian wave packets with spread equal N | i | i | i | i to , centered respectively in x = ↵ xˆ ↵ and x = xˆ and is the nor- 0 ↵ h | | i h | | i N malization constant. The probability density in position at time t is [24]:

2 dec (x, t)= ⇢aa(x, t)+2 ⇢↵↵(x, t)⇢(x, t)exp[ (t)] cos '(x, t) , P N ( ) a=↵ q X ⇥ ⇤ (3.63) (B) where ⇢ (x, t)= x Tr ( ↵ ) † x . The latter expression has two contri- ↵ h | Ut | ih | Ut | i ⇢ (x, t) butions: the incoherent sum⇥ a=↵ aa ⇤ which describes the two populations of the system, and the last term which determines the interference pattern of the superposition. The latter is modulatedP by the phase '(x, t) and the reduction of the interference contrast is determined by the decoherence function dec(t) < 0, Chapter 3. Quantum Brownian Motion 32 which takes the following form:

4 2 2 2 2 40p + ~ x M 1(t) dec (3.64) (t)= 2 2 2 2 2 2 4 2 . ~ · 8M 01(t) ~ G2(t) 4M 0G1(t)

Here x and p are the distances between the two guassians in position and momentum, and the function 1(t) is defined in Eq. (A.1a). As a concrete example, we consider the case of the Drude-Lorentz spectral den- sity, which is commonly used for example to describe light-harvesting systems [129, 130]: 2 ! J(!)= M ⌦2 , (3.65) ⇡ m (!2 +⌦2) where ⌦ is the characteristic frequency of the bath. The corresponding dissipation and noise kernels, defined in Eq. (3.8), are:

2 ⌦ t D(t)=2Mm~⌦ e | |sign(t), (3.66a)

2 2⇡ t 2⇡ t 2Mm~⌦ | | ~⌦ | | ~⌦ D (t)= e ~ , 1, + e ~ , 1, + 1 ⇡ L 2⇡ L 2⇡  ✓ ◆ ✓ ◆ (3.66b) 2 ⌦ t ~⌦ +2Mm~⌦ e | | cot , 2 ✓ ◆ + n where the function L is the Hurwitz-Lerch function L(z,s,a)= n=01 z (n + s a) . The two Green functions are: P 3 (⌦+ C )eCit G (t)= i , (3.67) 2 D i=1 i X d and G1(t)= dt G2(t), where C1, C2 and C3 are the complex roots of the polynomial 2 2 2 3 y(s)=(y + !S +2⌦)(y +⌦) 2⌦ and Di = j=1,j=i(Ci Cj). In terms of these 6 functions, we can compute the functions i(t) with the help of Eqs. (A.1) as well as the three relevant quantities previously discussed,Q whose explicit expressions are displayed in Eq. (3.64), Eq. (A.17) and Eq. (A.18).

Fig. 3.1 and Fig. 3.2 show the evolution of the diffusion function ⇤dif(t), of the energy E(t) and of the decoherence function dec(t), and we compare their time evolution according to the quantum Brownian motion model as described here above (QBM), with that of the Caldeira-Leggett (CL) master equation (3.13). We also consider the evolution given by the Modification of the Caldeira-Leggett (MCL) master equation, which is obtained from Eq. (3.13) by adding the term m [ˆp, [ˆp, ⇢ˆ (t)]] to guarantee the complete positivity of the dynamics [24, 112, 8M S 113]. As for the initial state, in Fig. 3.1 we considered the first excited state of the harmonic oscillator with frequency ! initially centred in the origin: xˆ =0= pˆ . S h i h i Chapter 3. Quantum Brownian Motion 33

Figure 3.1: Figure taken from [120]. Time evolution of the energy E(t) (top panel) and diffusion in space ⇤dif(t) (bottom panel) for the first excited state of the har-

monic oscillator with frequency !S = 100 centred in the origin ( xˆ =0= pˆ ) with 1 h i h i parameters M =1, m =0.3, ⌦= 2000, = 10 and ~ =1. The plot shows the behavior of E(t) and ⇤dif(t) for the QBM model with the Drude-Lorentz spectral

density, for the Caldeira-Leggett model (CL) and for its modification (MCL). Eth,Q, dif dif Eth,C, ⇤th,Q and ⇤th,C are also plotted.

The asymptotic value of E(t) is given by the equilibrium energy of the thermal state ⇢ˆ exp( H ): th / S ~!S ~!S Eth,Q = + , (3.68) 2 e~!S 1 which the high temperature limit coincides with the classical value Eth,C =1/.

For high temperatures the difference between the two thermal energies, Eth,Q and

Eth,C, is negligible; in this case the three dynamics lead to the same asymptotic value. This is expected since both CL and MCL are derived in the high tempera- ture limit, and our result is exact. However at low temperatures, as Fig. 3.1 shows, the difference between the quantum and classical cases becomes important and shows the quantum properties of the system S: the zero-point energy ~!S/2 is the minimal allowed energy. The CL dynamics, at low temperatures, fails to cap- ture this feature since its asymptotic value is lower. The MCL dynamics leads to an asymptotic energy which is different from both the classical and the quantum value. This is due to the correction to the C-L master equation. As mentioned before, the latter is needed to satisfy complete positivity, however it leads to un- physical effects, e.g. the system is overheated. Only the QBM model displays the correct quantum behavior. A similar situation is found for the diffusion in position ⇤dif(t). According to the well-known result of equilibrium quantum statistical physics, its asymptotic value is given by [29, 109]:

dif ~ ~!S ⇤ = coth , (3.69) th,Q 2M! 2 S ✓ ◆ which is the diffusion for an harmonic oscillator in the thermal state ⇢ˆth. In the dif 2 high temperature limit Eq. (3.69) gives ⇤th,C =1/(M!S ), which is the classical Chapter 3. Quantum Brownian Motion 34

Figure 3.2: Figure taken from [120]. Decoherence function exp(dec(t)) with param-

eters M =1, m =0.3, ⌦= 200, !S = 100, ~ =1, 0 = ~/(2M!S), x =20 and =0. The plot shows the behavior of the decoherence function exp((t)) for the p p Drude-Lorentz spectral density (QBM), for the Caldeira-Leggett model (CL) and 1 4 for its modification (MCL) at two different temperatures: = 10 and = 10 . 4 For = 10 the differences between the three models are minimal and the three curves coincide with the dotted line. asymptotic value. Again, for high temperatures the difference between the clas- sical and quantum thermal diffusion can be neglected, and the three dynamics give the same result. For low temperatures the difference becomes important. The MCL asymptotic value differs both from the classical and quantum equilib- rium values. Fig. 3.2 shows how dec(t) decays in time. For high temperatures, exp(dec(t)) reaches rapidly its asymptotic value, i.e. the decoherence time ⌧D is very short. In the low temperature case instead ⌧D is higher. Notice that the asymptotic value in both cases is not zero but, in agreement with the literature [24], it saturates at a finite value: dec( )= 1 2/2. 1 8 x 0

3.3.5 Non-Gaussian initial state

As already stated at the beginning of Sec. 3.3, the Wigner function approach pro- vides a particular simple way to compute expectation values when Gaussian states are considered [cf. Eq. (3.27)]. However, when non-Gaussian states are studied, the Wigner function approach is not adeguate, thus one needs to seek alternative solutions. The following example will make clear the advantage of the present approach in such cases. Consider a system initially (t =0) confined by the square potential V (x)=0for x [0,a] and V (x)=+ otherwise, at rest 2 1 Chapter 3. Quantum Brownian Motion 35 in the ground state. The initial state then is:

2/a sin(⇡x/a) x [0,a], (x)= 2 (3.70) (p0 otherwise. The corresponding initial expectation values for the quadratic operators are:

a xˆ = 2 , pˆ =0, h i h 2i 2 (3.71) xˆ2 = 1 (2 3 )a2, pˆ2 = ⇡ ~ , x,ˆ pˆ =0. h i 6 ⇡2 h i a2 h{ }i The system is then subject to an harmonic potential, that drives its evolution. The time evolution of the diffusion function ⇤dif(t) and energy E(t) is easy to obtain, as one can see from Eq. (A.17) and Eq. (A.18). In fact in our approach the only quantities that might change, when changing the state of the system, are the ini- tial expectation values. The functional dependence of the physical quantities on the initial values instead does not change. Then, by plugging in Eq. (A.17) and Eq. (A.18) the initial expectation values for the non-gaussian state [cf. Eq. (3.71)], one directly obtains the time evolution of ⇤dif(t) and E(t), which are plotted2 in Fig. 3.3. While the time evolution of E(t) is qualitatively the same as in the ex-

180

160

140 first excited state h.o. ground state box 120 Eth,Q

E(t) 100

80

60

40 0 1 2 3 4 5 6 7 8 9 10 Time t

Figure 3.3: Figure taken from [120]. Comparison of the solutions of the QBM model for two different initial states: the first excited state of the harmonic oscillator at

frequency !S (dashed line) and the ground state of the square potential displayed in

Eq. (3.70) (continuous line). The chosen parameters are: M =1, m =0.3, ⌦= 2000, 1 !S = 100, = 10 , a =0.23 and ~ =1. Top panel: Evolution of the energy E(t) for the two systems, compared with the equilibrium energy Eth,Q for the quantum thermal state. Bottom panel: Evolution of the diffusion function ⇤diff(t) for the two diff systems, compared with the equilibrium diffusion ⇤th,Q for the quantum thermal state. ample previously considered, the diffusion function ⇤dif(t) shows high frequency oscillations when the initial state is taken equal to Eq. (3.70). These oscillations

2The ground state of the square potential was preferred to its first excited since the initial values of energy and diffusion function are more compatible with those of the first excited state of the harmonic oscillator. However, a similar time dependence is shown when the initial state is taken equal to the first excited state of the square potential. Chapter 3. Quantum Brownian Motion 36 arise from the choice of the initial state and are present also when the system is isolated. With no bath, the diffusion function is equal to

sin2(! t) ⇤dif(t)=cos2(! t) xˆ2 xˆ 2 + S pˆ2 pˆ 2 + S h ih i M 2!2 h ih i S (3.72) 2cos(! t)sin( ! t) x,ˆ pˆ + S S h{ }i xˆ pˆ . M! 2 h ih i S ✓ ◆ By plugging into this expression the expectation values for the ground state of the square potential [cf. Eq. (3.71)] we obtain an oscillatory behaviour, while for the eigenstates of the harmonic oscillator Eq. (3.72) the diffusion is constant. When the bath is switched on, ⇤dif(t) simply decays exponentially as plotted in Fig. 3.3). As we have shown, the evolution of the expectation values is easy to obtain by using our approach. Once the functional dependence of the physical quantities on the initial values is computed, we obtain their time dependence for different initial states simply by inserting the initial expectation values. On the other hand, when working in the Schrödinger picture, as typically done in the literature [24, 55], or with the Wigner formalism [116, 127, 128], one has to find the explicit time evolution of the initial state, which changes depending on the initial state. 37

Chapter 4 Gravitational time dilation

Recently, the existence of a new source of decoherence was proposed by Pikovski et al. [56]. This effect is due to Gravitational Time Dilation (GTD) of a system in a gravitational potential. The authors consider a system consisting of a large num- ber N of harmonic oscillators in thermal equilibrium, in a gravitational poten- tial. They show that the superposition in space of two center-of-mass wave pack- ets decoheres when the two wave packets are centered in two positions, which have different gravitational potential, for example at two different heights on the Earth’s surface [cf. Fig. 4.1]. As we will describe in Sec. 4.1, this is an example of

Figure 4.1: Figure taken from [131]. Graphical representation of the gravitational decoherence effect. The center of mass state of a system for simplicity, we consider a sphere of radius r is initially prepared in a vertical spatial superposition with separation x. The coupling of the internal and center-of-mass degrees of freedom, mediated by the gravitational potential, decoheres the system. ⌧G is the decoherence time. entanglement between relative and center-of-mass degrees of freedom, mediated by the gravitational potential, and has already given rise to an intense debate [132–144]. The aim of this chapter is to compare this effect with other sources of decoherence, and to understand under which conditions it can be detected, at least in principle. The analysis done in [56], which relies on the comparison between gravitational decoherence against decoherence due to thermal emission, shows that, for low Chapter 4. Gravitational time dilation 38 temperatures and small superposition distances, the former is dominant. How- ever, as pointed-out in [141], in typical interferometric experiments also colli- sional decoherence with the residual gas should be taken into account, although the gas can be very diluted. We do this in Sec. 4.3.

4.1 Model for universal decoherence

Consider a system of mass M in a gravitational field. The motion of its center of ˆ mass is described by the Hamiltonian HCM, while the internal d.o.f. are governed by Hˆ0. Because of the mass-energy equivalence [145], gravity couples to the total ˆ 2 rest mass MTOT = M + H0/c and not only to the bare mass M of the system. Thus, gravity also couples to internal energy, and the Hamiltonian governing the ˆ ˆ ˆ ˆ system becomes HTOT = HCM + H0 + Hint, where 1 Hˆ = Hˆ (xˆ), (4.1) int c2 0 where (xˆ) is the gravitational field and xˆ the center of mass position. Let us con- sider the internal d.o.f. as a set of Nm molecules, described by three-dimensional harmonic oscillators. We can set

N ˆ H0 = ~!inˆi, (4.2) i=1 X where N =3Nm is the total number of harmonic oscillators, nˆi is the number operator for the i-th internal mode of the system, oscillating at the frequency !i.

The time evolution for the density matrix ⇢ˆT for the total system is determined by

dˆ⇢T(t) i ˆ = HTOT, ⇢ˆT(t) . (4.3) dt ~ ⇥ ⇤ By tracing over the internal d.o.f., under the Born-Markov approximations, we find out the master equation for the center of mass

dˆ⇢ (t) i E 2 CM = Hˆ + (xˆ)E¯ /c2, ⇢ˆ (t) 0 t (xˆ), (xˆ), ⇢ˆ (t) , (4.4) dt CM 0 CM c2 CM ~ ✓ ~ ◆ ⇥ ⇤ ⇥ ⇥ ⇤⇤ 2 ˆ 2 ˆ 2 where E0 = H0 H0 describes the amplitude of the energy fluctuations, h ih i 2 which can be characterized by the heat capacity CV =E0 /kBT for a system in thermal equilibrium. Near to the Earth’s surface, the gravitational field in the vertical x-direction can be 2 approximated by (ˆx) gxˆ, where g 9.81 ms is the gravitational acceleration ' ' on Earth, and the above master equation takes the same form of the Joos and Zeh Chapter 4. Gravitational time dilation 39 master equation (2.19), with

E g2 2 ⇤=2 0 t. (4.5) c2 ✓ ~ ◆ The decoherence time due to GTD is then quantified by

p2~c2 ⌧G = , (4.6) pkBCVgTx where x is the vertical distance of a superposition of center-of-mass states. One crucial issue is how to express CV appearing in ⌧G in terms of the parameters of the system. This depends on the model one uses to describe the structure of the system. Next we consider two of such a models, which become of interest depending on the temperature regime one considers. In particular, following the analysis performed in [56], we focus on the low temperature regime being the most favourable in terms of GTD effects, since in general this is the regime where standard decoherence effects are minimized.

4.2 Heat capacity for gravitational decoherence: com- parison of Einstein and Debye model

A first simple model for the heat capacity CV is provided by Einstein, who de- scribed the structure of a crystal as made of independent harmonic oscillators, all having the same frequency [146]. The associated heat capacity gives an accurate description for high temperatures. In this limit (T>TD, where TD is the Debye CL temperature of the crystal) it reduces to the well known classical value CV = NkB [146]. This is the expression considered in [56], which leads to the following for- mula for ⌧G: p 2 E 2~c ⌧G = . (4.7) pNgkBT x For high temperatures, Einstein’s model works in good agreement with the ex- perimental data, however for low temperatures the predictions deviate for the observations [147]. A better model at low temperatures is provided by Debye. It assumes the crystal as made of independent harmonic oscillators distributed according to the Bose-

Einstein statistics [148]. The Debye heat capacity, in the limit T TD of low 4 3 ⌧ temperatures, is [146] CV =4⇡ /5 NkB(T/TD) , which yields to the following ex- pression for ⌧G: 2 3/2 D 1 5 ~c TD ⌧G = 2 5/2 . (4.8) ⇡ r2N gkBT x Chapter 4. Gravitational time dilation 40

104 100 E G

τ 1000 / D G τ 100 10 E G 0.000 0.002 0.004 0.006 0.008 τ / T/TD D G τ 1

0.1

0.0 0.1 0.2 0.3 0.4

T/TD

Figure 4.2: Figure taken from [131]. Gravitational decoherence times as given by the E D D E Einstein (⌧G) and Debye (⌧G ) models. The plot shows the ratio ⌧G /⌧G as a function of T/TD. This ratio is independent from the superposition distance x and from D E the specifications of the system. The vertical purple line shows when ⌧G and ⌧G are equal.

This is the gravitational decoherence time we will use for our (low temperature) analysis1. Figure 4.2 compares the two expressions for the gravitational decoherence time. D E As ⌧G /⌧G depends only on the ratio T/TD, it is independent from the specifications of the system, at least within the limits of validity of the models here considered. For temperatures smaller than T /T 0.2, Einstein’s model overestimates the eq D ⇠ gravitational decoherence effect, while it underestimates it for T>Teq. Therefore, conceiving an experiment at very low temperatures to detect this gravitational effect is harder than originally estimated. As an explicit example, a sphere of radius r =1µm made of sapphire (T =1047K) containing N 1011 molecules, D m ⇠ at an equilibrium temperature T =1.0 K and delocalized over a distance x = 3 10 m (as considered in [56], see their Fig. 3), has a gravitational decoherence time ⌧ D 6.9 105 s according to Eq. (4.8), which is three orders of magnitude G ⇠ ⇥ longer than ⌧ E 1.8 102 s, as given by Eq. (4.7). G ⇠ ⇥

4.3 Competing effects

In order to be detectable, gravitational decoherence must be stronger than the other competing decoherence sources. We consider the two most common de- coherence sources in experiments involving quantum superpositions of material systems: thermal and collisional decoherence. In both cases, the decay of the

1There could be further contributions to the decoherence time, which are not captured by the Debye model, e.g. due to the size and shape of the system, or to processes, which do not contribute to CV. Nevertheless the Debye model is expected to give a reliable estimate of the effect [149]. Chapter 4. Gravitational time dilation 41 off-diagonal elements of the density matrix is well described by the following expression [68]: t/⌧TC ⇢CM(x, y, t)=⇢CM(x, y, 0)e , (4.9) where the decoherence time ⌧TC takes into account both thermal and collisional decoherence. This expression works well for recoil-free collisions (infinite mass limit) and low pressures; both conditions are satisfied in typical experiments as those here considered. We consider a spherical crystal of radius r made of Nm = 4⇡r3n/3 molecules, where n is the number density of molecules. Two different limits for ⌧ are relevant [24, 25, 30, 150]. In the long wavelength limit (2⇡ TC x ⌧ dB), where x = x y and dB is the de Broglie wavelength of the system, 2 | | ⌧TC =1/⇤x where ⇤ is the localization parameter characterizing the decoherence 1 mechanism. In the short wavelength limit (2⇡ ) instead, ⌧ = , where x dB TC is rate of events. A reasonable ansatz for ⌧TC, connecting the two limiting cases, is [131]: 1 2 ⌧TC = i tanh(x⇤i/i) , (4.10) i ! X where the sum runs over all decoherence mechanisms (thermal and collisional, in our case). Thermal decoherence includes three processes; scattering with the thermal envi- ronmental photons [30, 68, 150]

8!8⇠(9)cr6 k T 9 ✏ 1 2 ⇤ = B , (4.11) scatt 9⇡ c < ✏ +2 ✓ ~ ◆ ✓  ◆ where ⇠ is the Riemann zeta function and ✏ is the complex dielectric constant of the crystal; absorption of the same environmental photons ⇤abs and spontaneous emission of photons ⇤em, where ⇤abs =⇤em [150]. For the last two processes, we found two different models in the literature. The first, most widely used model (which we will refer to as Model 1), describes the system as a homogeneous par- ticle of radius r, without taking its internal structure into account, in which case we have [150, 151]: 16⇡5cr3 k T 6 ✏ 1 ⇤(1) = B . (4.12) em 189 c = ✏ +2 ✓ ~ ◆  The second model (Model 2) also takes the internal structure of the system into account, and for this reason shows an explicit dependence on the heat capacity. In this case:

4cr3 k T 6 ✏ 1 ⇤(2) = B 3 em ⇡ c = ✏ +2 · ✓ ~ ◆  2( +1)( +8)+1/2(2 +10 +15)e/2 erfc( /2) (4.13) · h p i z t2 where = C /k , and erfc(z)=1 2/p⇡ dt e is the complementary error V B 0 R Chapter 4. Gravitational time dilation 42

function. In the following analysis we use both expressions for ⇤em and conse- quently for ⇤abs. The rate of events, for all thermal processes, is the same and is given by [25]: 2 k T 3 = ⇠(3)cr2 B . (4.14) th ⇡ c ✓ ~ ◆ Beside decoherence due to thermal photons, one typically has to consider also decoherence due to collisions with the residual gas particles, whose localization parameter is given by:

2 8p2⇡⇠(3) r 3/2 ⇤coll = pmgasngas (kBT ) , (4.15) 3⇠(3/2) ~2 where ngas is the number density of the gas, which can be related to the pressure P and temperature T , under the assumption of a dilute gas, using the ideal gas law: ngas = P/(kBT ). The rate instead is given by:

Pr2 coll =16p3⇠(3/2) . (4.16) mgaskBT

These are the four main effects one typicallyp takes into account when devising an interferometric experiment aimed at detecting quantum interference, or the lack of. Appendix B contains further details about the decoherence formulas, in particular the explicit derivation of the expressions in Eq. (4.13), Eq. (4.15) and Eq. (4.16).

4.3.1 Comparison of the effects

We compare the strength of the different decoherence sources. In Table 4.1 we consider some of the best known interferometric experiments, either already per- formed, or at the stage of proposal. The first column shows the duration of the ex- periment, the second one the gravitational decoherence time ⌧G, computed using the Debye model; the third column displays the combined thermal+collisional decoherence time ⌧TC, where we chose Model 1 for the computation of the ther- mal decoherence. For simplicity, since we are interested in an order-of-magnitude analysis, we model all material systems as a homogeneous spheres of suitable ra- dius. The numbers show that in all experiments so far performed [cf. Fig. 4.1], decoherence times are much longer that the experimental time, which is the rea- son why interference was detected (i.e. no decoherence). However, in all cases the gravitational decoherence time is several orders of magnitude longer than the thermal+collisional decoherence time. This means that the current setups have to be significantly improved, to be used as detectors of gravitational decoherence.

In Fig. 4.3 we explore the parameter space temperature T vs. x, to see for which values gravitational decoherence dominates the other effects. Following [56], Chapter 4. Gravitational time dilation 43 we choose sapphire, because of its low microwave emission at low tempera- tures [56, 156, 157] and therefore for its suppressed thermal emission. The rel- evant parameters for sapphire are: Debye temperature TD =1047K, density 3 3 9 µ =4.0 10 kg m and constant dielectric constant✏ =10+10 i. Again, 0 ⇥ we model the system as a homogeneous sphere of radius r. For the residual gas, we consider air, i.e. a mixture of nitrogen N2 at 78% and oxygen O2 at 22% at the 17 very low pressure of P =10 mbar, which is more o less the lowest pressure, which can be reached with existing technology [158].

The region where gravitational decoherence is stronger, i.e. ⌧G <⌧TC, is high- lighted in grey. The are two types of regions. The one filled in grey corresponds to choosing Model 1 for thermal decoherence [cf. Eq. (4.12)], while the region marked with diagonal grey lines to Model 2 [cf. Eq. (4.13)]. As we can see, there are two disjoint grey regions. This is due to the different dependence of the decoherence rate with respect to the delocalization distance x. For gravitational decoherence, the dependence is linear, while for thermal and collisional decoherence it is quadratic at short distances (long wavelength limit) and constant at large distances (short wavelength limit); see Eq. (4.7) and Eq. (4.10). More specifically, the lower grey region, present in all four panels, correspond to the regime where the long wavelength limit applies: ⌧TC scales with 2 x, while ⌧G scales with x. The upper grey region, which appears only in the two bottom panels, correspond to the short wavelength limit: in this case ⌧TC is independent of x, while ⌧G still scales with x. For this reason, there is a gap between the two grey regions in the bottom panels. As Figure 4.3 shows, there are regions where gravity is the dominant decoherence

D Interferometric experiment (parameters) texp (s) ⌧G (s) ⌧TC(s) 5 29 3 Atoms (r 100 pm, x 54 cm) 10 10 10 ⇠ ⇠ 2 6 1 Fullerenes (r 500 pm, x 100 nm) 10 10 10 ⇠ ⇠ 12 Micro-particles (r =1µm, x 500 nm) 10 1 ⇠ 13 8 2 Diamonds (r 500 nm, x 10 pm) 10 10 10 ⇠ ⇠ 3 19 Macro-particles (r =2cm, 1 nm) 10 10 x ⇠ D Table 4.1: Experiments’ time texp, gravitational decoherence time ⌧G as given by Eq. (4.8) and thermal+collisional decoherence time ⌧TC for some interferometric ex- periments with: atoms [152], fullerenes [153], micro-particles [154], diamonds [45], macro-pacrticles [155]. In all cases, we grossly simplified the system by shaping it as a homogeneous sphere of radius r. For thermal decoherence, we considered Model 1 of Section 4.3. The radius r and delocalization distance x are shown in the table. The other parameters of the experiments are as follows. For 87Rb atoms: 17 9 3 ✏ 0.3+0.1i, P = 10 mbar, T = 10 K. For C60 fullerenes: ✏ 4.4 + 10 i, ⇠ 17 ⇠ P =5 10 mbar, environmental T = 300 K, fullerenes’ T = 900 K. For micro- ⇥ 4 17 particles of Nb: ✏ 41 + 10 i, P = 10 mbar and T = 30 mK. For diamonds: 4 ⇠ 17 ✏ 5.7 + 10 i, P = 10 mbar and T = 300 K. For macro-particles of SiO2: ⇠ 3 7 ✏ 3.9 + 10 i, P =5 10 mbar and T =4K. In the case of micro and macro- ⇠ ⇥ particles, the experimental time texp is not reported since these are theoretical pro- posals. In the case of fullerenes, the temperature of the experiment is well above

the Debye temperature of the system (TD = 185 K). In this case one has to use the Einstein’s model, which gives ⌧ E 108 s. G ⇠ Chapter 4. Gravitational time dilation 44 Gravitational decoherence vs Competing effects Gravitational decoherence vs Competing effects

Largest delocalization distance 0 Largest delocalization distance 0 Largest delocalization distance 0 Largest delocalization distance 0 (Atomic interferometry) (Atomic interferometry) (Atomic interferometry) (Atomic interferometry)

6 6 τG=10 s τG=10 s -5 -5 -5 -5 10 6 τG=10 s τG=10 s 10 τG=10 s

Smallest resolution distance Smallest resolution distance Smallest resolution distance Smallest resolution distance -10 -10 -10 -10

(Trapped nanoparticles) 6 )] (Trapped nanoparticles) )] (Trapped nanoparticles) )] (Trapped nanoparticles) )] τ =10 s 14 G m m m 14 m ( 10 ( τG=10 s ( ( x τG=10 s x x τG=10 s x Δ Δ Δ Δ [ [ [ [

10 10 10 18 10 14 τG=10 s 10 τG=10 s τG=10 s -15 Log -15 Log 18 -15 Log -15 Log τG=10 s

18 14 τG=10 s τG=10 s -20 -20 -20 -20

18 τG=10 s (r=10-3m, N≅1020) (r=10-6m, N≅1011) (r=10-8m, N≅105) (r=10-9m, N≅102) -25 -25 -25 -25 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 T(K) T(K) T(K) T(K)

Figure 4.3: Figure taken from [131]. Gravitational decoherence time ⌧G vs. ther-

mal+collisional decoherence ⌧TC as a function of the temperature T and delocaliza- tion distance x. We took a spherical crystal of sapphire (as considered in [56]) 3 6 8 of radius r = 10 m (first panel), r = 10 m (second panel), r = 10 m (third 9 panel) and r = 10 m (fourth panel). The regions where gravitational decoherence

is stronger, i.e. ⌧G <⌧TC, are colored in grey. We considered two models for ther- mal emission used in the literature, as discussed in Sec. 4.2: the region filled with grey refers to Model 1, that marked with diagonal grey lines to Model 2. We show also the maximum delocalization distance x currently achievable (54 cm for atom interferometry [152], purple dot-dashed line) and the minimum resolvable distance 10 ( 10 m[159] in purple dashed line). The colored lines (orange, red, blue and ⇠ green) show some numerical value of the gravitational decoherence times, as given by Eq. (4.8). mechanisms. However, these regions are not easy to access experimentally with the present-day technology, due to the quite extreme conditions required. The first difficulty is related to the distances one has to resolve. For example, if we 6 consider an optomechanical setup with a microparticle of radius r =10 m or larger, gravitational decoherence dominates over the other effects if one is able to 16 resolve distances 10 m, which is smaller than the radius of the proton’s x ⇠ charge. 8 9 For smaller radii, as shown in the two right panels (r =10 m and r =10 m), the numbers seem more favourable. But then one has to take another problem into account: the gravitational decoherence time (106 1010 s) is much longer ⇠ than the typical time-scales of interferometric experiments ( 1 ms for matter- ⇠ wave interferometry [40] and 100 ms for optomechanics [160]). ⇠ 45

Chapter 5 Collapse Models

Collapse models (CM) are phenomenological models aiming at describing the transition from the micro-world, well described by quantum mechanics, and the macro-world, where systems are never observed in superpositions. In order to do so, CM add non-linear terms to the Schrödinger equation in such a way that the collapse of the quantum superposition state is embedded in the dynamics. This solves the quantum measurements problem without the necessity of intro- ducing, beside the usual Schrödinger dynamics, a second evolution, given by the wavepacket reduction postulate. The terms modifying the standard quantum dynamics must be stochastic [161, 162] and they must act at the wavefunction level. The latter requirement together with non-linearity, are required to prevent an apparent collapse of the wavefunction: the coherences are suppressed, but the process leaves the individual wavefunctions spatially extended. Thus, in this case the system does not collapse in a localized state. On the other hand, when non-linearity is included in the model, the stochasticity is required in order to avoid superluminal signalling [161]. The modifications made by CM must be small for microscopic systems in order to agree with known and experimentally verified results about quantum mechanics; at the same time they must act strongly on macroscopic systems, where super- positions are suppressed and systems behave classicaly [163]. For an extensive discussion on the properties of the dynamical reduction models we refer to [37].

In this Chapter we introduce the Continuous Spontaneous Localization (CSL) model, which is the most known and studied among the CMs. Moreover, we show how optomechanical experiments can provide strong bounds on the values of the parameters of the model.

5.1 Continuous Spontaneous Localization Model

The Continuous Spontaneous Localization (CSL) model is constructed in a way that the localization happens in position and that the following requirements are satisfied [37]: 1. Macroscopic objects are always localized in space. 2. Microscopic objects evolve without appreciable difference from the stan- dard quantum mechanical dynamics. Chapter 5. Collapse Models 46

3. The energy increase due localization is negligible. 4. The model is able to describe systems of identical particles. The CSL collapse equation reads [36]

i pCM d = Hˆ dt + dx Mˆ (x) Mˆ (x) dW (x)+ | ti m h it t  ~ 0 Z ⇣ ⌘ 2 CM ˆ ˆ 2 M(x) M(x) t dt t , (5.1) 2m0 h i | i ⇣ ⌘ ˆ where H is the Hamiltonian of the system, CM is the constant determining the strength of the collapse, m0 is a reference mass, chosen equal to the nucleon mass, and Wt is a standard Wiener process satisfying:

dW (x) =0, and dW (x)dW (y) = (3)(x y)dt. (5.2) h t i h t t i Here Mˆ (x) is the locally averaged density operator of the form

Mˆ (x)= m dy g(x y)ˆa†(y,s)ˆa (y,s), (5.3) j j j j s X X Z with mj being the mass of the particle of type j and spin s, which is created and annihilated by the operators aˆj†(y,s) and aˆj(y,s), and

(x y)2 1 2 2rC g(x y)= 2 3/2 e (5.4) (2⇡rC ) is a smearing function imposing a spatial correlation of the noise Wt characterized by the length rC. Equation (5.1) describes a noise field inducing the collapse that acts in the same way on all particles of the same type. There are strong motivations to believe that the collapse should be mass proportional. In fact, the original CSL equa- tion, which was not mass proportional and which can be simply obtained from Eq. (5.1) by setting all mj = m0, has been falsified by the experiments on the spontaneous X-ray emission from a Germanium sample [164]. In the mass pro- portional CSL model, the contributions due to electrons are negligible and we can consider only the nucleons as relevant for the localization process, consequently the sums j s in Eq. (5.3) can dropped. The masterP equationP for the statistical operator can be easily obtained by consid- ering that ⇢ˆS(t) is given by the stochastic average E[ ] over the Wiener processes · in Eq. (5.1) of . Then, it follows that | tih t|

dˆ⇢S(t)=E d t t + t d t + d t d t . (5.5) | ih | | ih | | ih | h i Chapter 5. Collapse Models 47

By merging this equation with the stochastic averages in Eq. (5.2) we obtain the mass proportional CSL master equation [37]

d i ⇢ˆ (t)= H,ˆ ⇢ˆ (t) CSL dz Mˆ (z), Mˆ (z), ⇢ˆ (t) , (5.6) dt S S 2r3⇡3/2m2 S ~ C 0 Z ⇥ ⇤ ⇥ ⇥ ⇤⇤ where we introduced CM CSL = 3/2 3 , (5.7) 8⇡ rC and 2 (z qˆn) 2r2 Mˆ (z)=m0 e C , (5.8) n X has been rewritten in the first quantization formalism, with the sum n running on all the nucleons of the system and qˆn being the position operator of the n-th nucleon. Equation (5.6) not only is accountable for the collapse-inducedP suppres- sion of coherences, but it also describes a diffusion mechanism.

The CSL master equation1 in (5.6) exactly describes the process of localization in position, but, for all practical purposes, is difficult to handle. However, when the spread of the center-of-mass wave-function is much smaller than rC, which is the case for all the experiments considered in this thesis, one can approxi- mate Eq. (5.6) with the Quantum Mechanics with Universal Position Localization (QMUPL) master equation [32, 165]. This can be done by considering the position operator of the n-th nucleon written as follows [45, 47]:

(0) qˆn = qn +qˆn + qˆ, (5.9)

(0) where qn is the classical equilibrium position of the n-th particle, qˆn measures the quantum displacement of the n-th particle with respect to its classical equilib- rium position and qˆ measures the fluctuations of the center of mass of the body where the n-th nucleon belongs. Then, under the assumption of rigid body, the latter fluctuations are the same for all the particles and qˆn can be neglected.

When the spread of the center of mass wavefunction is much smaller than rC, Eq. (5.8) can be Taylor expanded up to the first order in qˆ:

(z x)2 dx 2r2 Mˆ (z) M (z)+ µ(x)e C (z x) qˆ, (5.10) ⇡ 0 r2 · Z C (3) (0) where M (z) is a c-function, and µ(x)=m (x qn ) is the system mass 0 0 n distribution. Introducing Eq. (5.10) in Eq. (5.6) we obtain the QMUPL master P 1For the sale of simplicity, from now we will always refer to the mass proportional CSL model as the CSL model. Chapter 5. Collapse Models 48 equation [166, 167]

d i 1 ⇢ˆ (t)= H,ˆ ⇢ˆ (t) ⌘ qˆ , qˆ , ⇢ˆ (t) , (5.11) dt S S 2 ij i j S ~ i,j=x,y,z ⇥ ⇤ X ⇥ ⇥ ⇤⇤ where qˆi is the i-th component of qˆ, and the CSL diffusion rate is given by

3 CSLrC 2 k2r2 ⌘ = dk µ˜(k) e C k k , (5.12) ij ⇡3/2m2 | | i j 0 Z where µ˜(k) is the Fourier transform of the mass density µ(x).

Despite the simplicity of the QMUPL master equation (5.11), whose structure is identical to that of the Joos-Zeh dynamics [cf. Eq. (2.19)], its predictions are practically equivalent to those of the CSL master equation (5.6). Because of this relation, Eq. (5.11) is used to set bounds to the CSL collapse parameters CSL and rC, on whose possible values there is no consensus so far. Ghirardi, Rimini and 16 1 7 Weber (GRW) originally set [32] CSL =10 s and rC =10 m. Later, Adler 7 8 2 1 suggested different values [38, 39] namely rC =10 m with CSL =10 ± s and 6 6 2 1 rC =10 m with CSL =10 ± s .

So far we focused on linear diffusion [cf. Eq. (5.11)]. Very recent technological developments allow to achieve better and better control of torsional motion of non-spherical objects [92], thus paving the way to testing torsional CSL-induced diffusion [58, 168–170]. Consider a system whose shape is not spherical and thus one can identify with ⌦ its angular configuration, see also Sec. 2.2. Consequently, Eq. (5.9) is substituted by [170] ˆ (0) qˆn = R(⌦)[qn ]+qˆn + qˆ, (5.13) where Rˆ(⌦) takes into account the angular configuration of the system. By ap- plying a Taylor expansion on Eq. (5.8) with Eq. (5.13), one finds an equation de- scribing a CSL-induced linear and angular diffusion. If one monitors the motion along and around only one direction (say the x axis, cf. Fig. 5.1), the master equa- tion reads

d i ˆ ⌘xx ⌘T ˆ ˆ ⇢ˆS(t)= H,⇢ˆS(t) x,ˆ x,ˆ ⇢ˆS(t) , , ⇢ˆS(t) , (5.14) dt ~ 2 2 ⇥ ⇤ ⇥ ⇥ ⇤⇤ ⇥ ⇥ ⇤⇤ where ˆ is the angular operator describing rotations along the x axis, such that ˆ ˆ ˆ , Lx = i~, with Lx the angular momentum operator along the same direction, and ⇥ ⇤ 3 rC r2 k2 2 ⌘ = dk e C k @ µ˜(k) k @ µ˜(k) , (5.15) T ⇡3/2m2 | y kz z ky | 0 Z is the CSL angular momentum diffusion coefficient. Chapter 5. Collapse Models 49

Figure 5.1: Figure taken from [58]. Graphical representation of the cylinder with respect to the chosen Cartesian axes. The monitored motions are the vibration along the x axis and the torsion around it (represented in blue and red respectively).

5.1.1 Imaginary noise trick

A very useful mathematical trick can be applied for computing the action of the CSL noise. This is the imaginary noise trick: the dynamics in Eq. (5.6) is mimicked by a standard Schrödinger equation with an additional stochastic potential of the form [52] p ˆ ~ ˆ VCSL(t)= dz M(z)w(z,t), (5.16) ⇡3/4r3/2m C 0 Z where w(z,t) is a white noise with w(z,t) =0and w(z,t)w(y,s) = (t h i h i s)(3)(z y). Such a stochastic potential acts on the system as a stochastic force and a stochastic torque, which in the same limit of validity of the expansion in Eq. (5.14), can be recast in the following simplified form

Fx(t)= ~p⌘xxw(t), (5.17) ⌧x(t)= ~p⌘Tw(t), where the x axis is taken as reference and the noise w(t) is fully described by: w(t) =0and w(t)w(s) = (t s). h i h i

5.2 Optomechanical probing collapse models

The CSL equation (5.1) predicts the collapse of the wavefunction, however it also describes a diffusion process, which becomes evident when one considers the corresponding master equation (5.6). Thus, two different experimental tests can be considered in order to falsify the model or infer bounds on its parameters. The first class regards interferometric experiments, where a superposition of a sys- tem is created and the collapse of the wavefunction is probed. Examples of this type are molecular interferometry [40–43] and entanglement experiment with di- amonds [44, 45]. The second class of experiments includes non-interferometric tests, where the above described diffusion process is investigated. Chapter 5. Collapse Models 50

Figure 5.2: Figure taken from Editors’ Suggestion of [51]. Cartoon of the cantilever experiment [cf. Sec. 5.4]. The motion of the cantilever is determined by the ther- mal and the non-thermal forces acting on it. Its characterization is used to infer constrains on CSL.

Recent works [46–53, 171, 172] show that non-interferometric experiments can place strong bounds on collapse models. In particular, optomechanical experi- ments can reach very low noise levels by cooling the mechanical oscillator close to its lowest energy state [173], allowing for the enhancement of any quantum feature emerging from the system dynamics. Thus optomechanics is the most promising experimental setup to look at for the quantum-to-classical debate and infer bounds on the CSL parameters.

Optomechanical systems are composed by two constituents and can profit of the advantageous properties of both the components: a mechanical resonator which is strictly correlated to guided photons. Since the mechanical part can be fabri- cated with high accuracy and with a high flexibility in design, there exists a vast zoo of optomechanical systems. For an extensive discussion on the properties and features of optomechanical system we refer to [174–176].

Here we focus on an optomechanical setup consisting of a system whose vibra- tional and torsional degrees of freedom are monitored by a laser. For the sake of simplicity, we consider the system of cylindrical shape [cf. Fig. 5.1], which is har- monically trapped, both in position and in angle. The monitored motions are the center of mass vibrations along the x axis and the torsions around it [cf. Fig. 5.1]. The Hamiltonian, describing the vibrational motion of the system harmonically trapped at frequency !0 by interacting with a cavity field, is given by [177, 178]

2 ˆ pˆ 1 2 2 HV = ~!Caˆ†aˆ + + M!0xˆ ~aˆ†aˆx.ˆ (5.18) 2M 2 In Eq. (5.18) the first term describes the free evolution of the cavity mode at fre- quency !C, with aˆ† and aˆ denoting photons creation and annihilation operators; the next two terms describe the oscillatory motion of the mass M in the cavity, where xˆ and pˆ are, respectively, its position and momentum operators. The last Chapter 5. Collapse Models 51 term describes the interaction between the cavity field and the vibrational motion of the system, with coupling constant . If we include also torsions of the cylin- der along the direction of propagation of the radiation field (x axis) we need to consider the additional term [75, 76]

ˆ2 ˆ Lx 1 2 ˆ2 ˆ HT = + I! ~gaˆ†aˆ, (5.19) 2I 2 which is the torsional Hamiltonian, characterized by a moment of inertia I and torsional frequency !; the third term, proportional to the coupling constant g, accounts for the laser interaction with the torsional degrees of freedom.

The dynamics can be obtained by merging the Hamiltonians in Eq. (5.18) and in Eq. (5.19). To the corresponding equations of motion, we add the CSL and en- vironmental influences. The CSL-induced diffusions are described by Eq. (5.17). The dampings and thermal noises due to environment [179] reads aˆ + p2aˆ , in pˆ + ⇠ˆV and D Lˆ /I + ⇠ˆT, respectively, for the cavity modes, the vibrational m x and the torsional motion of the system. Explicitly, we get the equations [75]:

dˆx pˆ dˆ Lˆ = , = x , dt M dt I dˆa ˆ = i(0 g i)ˆa + iaˆxˆ + p2aˆin, dt (5.20) dˆp 2 ˆV = M!0xˆ + ~aˆ†aˆ mpˆ + ⇠ ~p⌘xxwV, dt dLˆx D 2 ˆ ˆ ˆT = I! + ~gaˆ†aˆ Lx + ⇠ ~p⌘TwT, dt I where = ! ! is the detuning of the laser frequency ! from the cavity 0 C 0 0 resonance; , m and D are the damping rates for the cavity, for the vibrations and torsions of the system respectively; aˆin is a noise operator describing the in- 2 cident laser field, defined by the input power Pin = ~!C ↵ , with ↵ = aˆin , and | | h i delta-correlated fluctuations aˆ (t)aˆ† (s) = (t s), where aˆ = ↵ + aˆ . The h in in i in in noise operators ⇠ˆV and ⇠ˆT describe the thermal action of the surrounding environ- ment (supposed to be in equilibrium at temperature T ), which is assumed to act independently on vibrations and torsions. They are assumed to be Gaussian with zero mean and correlation function [75]

ˆj ˆj ⇠t ⇠s d! i!(t s) h i = e ![1 + coth(!˜ )] (j = T,V), (5.21) ✏ 2⇡ ~ j Z ˜ with = ~/2kBT , ✏T = D, and ✏V = mm. As already discussed, the CSL noise acts as a source of stochastic noise, whose influence on the dynamics of the system is encompassed by the addition, in Eqs. (5.20), of the force terms ~p⌘jwj (j = T,V) with w =0and w (t)w (s) = (t s) [46, 47]. h ji h i j i ij From Eq. (5.20) and Eq. (5.21) we can derive the density noise spectrum (DNS) Chapter 5. Collapse Models 52

DNS Parameter ! Gj j,eff j,eff j Vibration !0,eff m,eff m Torsion g !,eff D,eff/I I

Table 5.1: Explicit form of the parameters entering the DNS of the fluctuations of the torsional and vibrational degrees of freedom of the system [cf. Eq. (5.22)]. associated to x˜(!) and ˜(!), which are the fluctuations of the position and angle operators in Fourier space respectively, (!)= 1 d⌦ O˜ (!),O˜ (⌦) (j = Sj 4⇡ h j j i T,V). R Through a lengthy but straightforward calculation, the explicit form of both (!) SV and (!) can be calculated and recast under the Lorentzian form ST 2 2 2 2 2 ˜ 2 2~ ↵  j +[ +( !) ] ~!✏j coth(!)+~ ⌘j (!)= | | G . (5.22) Sj 2 [2 +( !)2][(!2 h !2)2 +2 !2] i j j,eff j,eff The parameters that appear in Eq. (5.22)( ,! , and ) are given in Ta- Gj j,eff j,eff j ble 5.1. We have introduced the effective frequencies !0,eff and !,eff and damp- ing constants m,eff and D,eff [75, 180], whose explicit expressions are presented in Appendix C.2, and = g ˆ xˆ . The CSL contributions are encom- 0 h i h i passed by the diffusion constant ⌘j, which enters j(!) as an additional heating S ˜ term akin to the environment-induced one ~!✏j coth(!). In the high temperature limit (˜ 0), which is in general valid for typical low-frequency optomechanical ! ˜ experiments, the latter takes the form ~✏j/. Therefore, in such a limit, we have

˜ 2 1 ~⌘j ~✏j ~!✏j coth(!)+~ ⌘j ~✏j + , (5.23) ! ˜ ✏ ⌘ ˜ ✓ j ◆ j,eff ˜ where we have defined the j-dependent effective inverse temperature j,eff, thus showing that the different degrees of freedom of the system thermalise to differ- ent, in principle distinguishable, CSL-determined temperatures. This means that CSL gives an enhancement of the vibrational and rotational temperatures, which read 2 2 V ~ ⌘xx R ~ ⌘T TCSL = and TCSL = . (5.24) 2kBmm 2kBD Upon subtracting the optomechanical contribution to the temperature embodied by the first term2 in Eq. (5.22), the experimental measurement of the temperature of the system is given by T T, where T is the experimental measurement m ± accuracy. Unless one observes a temperature excess of unknown origin (as it is

2The contribution arising from the driving field can be accurately calibrated experimentally due to of the relatively large intensity of the field (which also makes any uncertainty negligible with respect to the nominal signal). Such well characterized contribution can then be subtracted from the density noise spectrum to let the features linked to the mechanical motion emerge. We also note that the optomechanical contribution can be strongly suppressed by implementing a stroboscopic measurement strategy. Chapter 5. Collapse Models 53 reported [51] and discussed here in Sec. 5.4), the outcome of the experiment will be T T, thus setting a bound on the collapse parameters. CSL 

5.3 Gravitational wave detectors bound collapse pa- rameters space

AURIGA, Advanced LIGO and LISA Pathfinder represent the state of the art in their class, respectively ground-based interferometric detectors, precursors of space-born detectors and resonant mass GW detectors. A GW detector monitors the deformation of space-time produced by gravitational waves. The strain noise spectrum Shh(!) quantifies the strength of such a deformation. Advanced LIGO and LISA Pathfinder monitor the optical distance between pairs of nominally free masses, while AURIGA is based on a single cylindrical bar me- chanical oscillator [cf. Fig. 5.3]. In the first case the CSL noise acts on the relative distance between the two masses; in the second case it causes a driving force on the bar oscillator. We will consider both types of experiments.

z x x a a

R x R L

a L L

y

R x

L

Figure 5.3: Figure taken from [52]. Graphical representation of the three exper- iments here considered; the images are not in scale. LIGO on the top, LISA Pathfinder on the middle and AURIGA is on the bottom. In LIGO, four identical cylindrical masses (radius R, length L) are arranged as in Figure; a is the distance between the center-of-mass of two masses on each arm of the interferometer. The arms are oriented along the x and y directions. LISA Pathfinder features two cubic (length L) masses, displaced along the x direction with relative distance between their center-of-mass equal to a. AURIGA features a cylindrical single mass (radius R, length L), aligned with respect to the direction x of measurement.

We consider the x direction of the motion of each mass of the system, modelled by an harmonic oscillator of mass M↵ and resonant frequency !↵. The corresponding Chapter 5. Collapse Models 54 quantum Langevin equations read:

d pˆ↵(t) d 2 xˆ↵(t)= , and pˆ↵(t)= M↵!↵xˆ↵(t) ↵pˆ↵(t)+F↵(t), (5.25) dt M↵ dt where pˆ↵(t) is the momentum of the ↵-th mass distribution and F↵(t) is the stochas- tic force acting on it, both along the x direction. We have added as usual a dis- sipative term pˆ (t), which can be expressed in terms of the quality factor of ↵ ↵ the system Q↵ = !↵/↵. A more general treatment [cf. Eq. (5.20)] should include additional noise terms to take into account the action of the environment and the measurement apparatus. However, since we are primarily interested in estimat- ing the effect of the CSL noise, we neglect all other noise sources. Furthermore, the actual noise of the systems here considered is the sum of several noise sources (thermal, quantum, seismic, gravity gradient, etc.) and it is typically difficult to accurately distinguish and characterize each of them. This is typically the case for interferometric detectors. In order to set an upper limit on the CSL parame- ters we will take a conservative approach by assuming that all the experimentally measured noise is attributed to CSL. The physical quantity we are interested in + is the force noise spectral density S (!)= 1 1 F˜(!), F˜(⌦) , expressed in FF 4⇡ h i 2 1 ˜ 1 N Hz , where F (!) is the Fourier transformR of the x component of stochastic force. Since the single arm of LIGO and LISA Pathfinder are composed by two masses, the expression for the stochastic force can be computed by starting from the CSL stochastic potential in Eq. (5.16), and it reads

(z x)2 ~pCSL dz dx 2 2rC F↵(t)= 3/4 7/2 µ↵(x)e (z x) w(z,t). (5.26) ⇡ m0 r Z C Notice that here the noise w(z,t) is spatially uncorrelated: it acts randomly and (z x)2 independently on every nucleon of the system. The smearing function exp( 2r2 ) C will introduce a spatial correlation. In the case of LISA Pathfinder and one arm of LIGO, there are two equal masses at an average distance a and the monitored motion is the relative one, which is described by the following Langevin equations:

d 2ˆp (t) d M xˆ (t)= rel , and pˆ (t)= !2xˆ (t) pˆ (t)+F (t), (5.27) dt rel M dt rel 2 0 rel m rel rel where F (t)= 1 (F (t) F (t)). The corresponding force noise spectral density is rel 2 1 2 given by 2 3 2 2 L ~ CSLrC 2 iakx 2 r k S (!)= dk µ˜(k) 1 e k e C , (5.28) FF 2⇡3/2m2 | | x 0 Z where µ˜(k) is the Fourier transform of µ(x), and the correlation for the Fourier transformed white noise is w˜(z,!)˜w(y, ⌦) =2⇡(! +⌦)(3)(z y). Here, there h i are two CSL contributions to the motion: the incoherent action on the single mass Chapter 5. Collapse Models 55

(first term in parenthesis) and the correlation between the two masses (second term), the latter being relevant when a

2 2 2 L2 R2 2 2 LIGO 8~ CSLM rC 4r2 2r2 R R S (!)= 1 e C + f 1 e C I +I , FF L2m2R2 corr 0 2r2 1 2r2 0 !" ✓ ✓ C ◆ ✓ C ◆◆# (5.29a) 2 2 2 4 L2 L2 LISA 16~ CSLM rC 4r2 4r2 L L S (!)= 1 e C + f 1 e C p⇡ erf , FF m2L6 corr 2r 2r 0 ! C ✓ C ◆! (5.29b) where I0 and I1 denote the first two modified Bessel functions of the first kind, and fcorr describes the correlations of the forces. Because of the particular geometry of LIGO and LISA Pathfinder, fcorr have the same form for both the experiments:

(a+L)2 aL L(2a+L) 1 2 2 2 4r r 4r fcorr = e C 1+e C 2e C . (5.30) 2 !

The effect of the correlations is to suppress the CSL effect in the relative motion of two equal masses when rC >a. In the case of LIGO, an extra factor 2 appears in Eq. (5.29a) to take into account the two arms of the interferometer. In the case of AURIGA, we have a single mass, and the monitored motion is the deformation of the resonant bar. The system can be modeled as two half- cylinders of mass M/2 and length L/2, connected by a spring and oscillating in counterphase with the same elongation of the bar extrema. The disposition of the two cylinders is the same as that of the single arm of the LIGO experiment, with a = L/2 so that the two cylinders touch each other. The Langevin equations de- scribing the relative motion of the two masses are described by Eq. (5.27), where M/2 replaces M in the first of the two equations. Since the single arm of LIGO and our modeling of AURIGA have the same disposition, Eq. (5.29a) describes also the force noise spectral density for AURIGA, after replacing both a and L with L/2 in fcorr [cf. Eq. (5.30)]. The expression must also be divided by a factor 2 Chapter 5. Collapse Models 56 since there is only one arm. Eq. (5.29a) becomes:

2 2 2 L2 L2 AURIGA 4~ CSLM rC 3 1 2 2 4rC 16rC SFF (!)= 2 2 2 2 2 e e L m0R ! R2 2 R2 R2 2rC 1 e I0 2r2 +I1 2r2 , (5.31) · " C C # ⇣ ⇣ ⌘ ⇣ ⌘⌘ where M and L are the mass and the length of the AURIGA cylinder. A final note: since the experimentally measured spectral densities refer only to positive frequencies, one has to multiply the expressions in Eq. (5.29) and Eq. (5.31) by a factor 2 to take into account the conversion from the two-side to one-side spectra.

5.3.1 Interferometric GW detectors: LIGO

Interferometric GW detectors, such as LIGO [181] (as well as Virgo [182]), are es- sentially Michelson interferometers in which the two arms are configured as a Fabry-Perot cavity. A passing gravitational wave induces a differential change of the arm lengths, resulting in a phase change of the output light. Each arm in- cludes two suspended test masses acting as end mirrors, placed at several km to each other (4 km for LIGO, 3 km for Virgo) to maximize the response to the grav- itational wave strain h. The suspensions are made of actively controlled multi- stage pendulum systems, with resonant frequency !0/2⇡ below 1 Hz, designed to heavily filter seismic noise. The last stage is designed for ultrahigh quality factor 8 (Q = !0/> 10 ) in order to suppress as much as possible the thermal noise. The actual frequency band sensitive to gravitational waves is roughly above 10 Hz, ⇠ implying that the test masses can be considered to a good approximation in the free-mass limit ! ! . 0 Given the arm length a [cf. Fig. 5.3], the differential change a = a a | x y| of the two arm lengths induced by an optimally oriented strain h is predicted by General Relativity to be a = ha. It follows immediately that any displacement LIGO noise spectral density Sxx (!) of one of the two arms will cause an equivalent LIGO LIGO 2 strain noise Shh (!)=Sxx (!)/a . The former can be derived with the usual approach from Eqs. (5.27) by solving them in the frequency domain [183]:

4 SLIGO(!) SLIGO(!)= FF , (5.32) xx m2 (!2 !2)2 +(!!0 )2 0 Q where SLIGO(!) is defined in Eq. (5.29). In this way, in the free-mass limit ! ! , FF 0 we can derive the expression for the equivalent strain induced by the CSL noise.

4 From Eq. (5.32), it follows that the CSL contribution to Shh(!) features a 1/! de- 2 pendence, or a 1/! dependence when the square root spectrum Sh(!) is consid- ered. The minimum force noise and therefore the strongest upper bound on the Chapter 5. Collapse Models 57

CSL parameters will be achieved at a well-defined frequency !¯/2⇡. As the typi- cally measured Sh(!) is convex [181], !¯/2⇡ can be graphically inferred from the spectrum as the frequency at which Sh(!), displayed in log-log scale, is tangent to a straight line with slope equal to 2.

For Advanced LIGO at the time of the first detection [184, 185], Sh(¯!) is in the 23 1 range of 10 Hz 2 . From the published spectrum we infer that the effective 1 force noise reaches a minimum S (¯!) 95 fN Hz 2 at !¯/2⇡ 30 35 Hz. We F ⇡ ⇠ have used the numerical values m =40kg for the test mass and a =4km for the arm length. For the design sensitivity of Advanced LIGO, not yet reached, 1 one can estimate from the design curves a minimum force S (¯!) 25 fN Hz 2 at F ⇡ !¯/2⇡ 15 20 Hz [181, 184, 185]. Each test mass is a cylinder of fused silica ⇠ 3 (density µ0 =2200kg/m ) with radius R =17cm and length L =20cm. By plugging the test mass parameters and the measured force noise in Eq. (5.29), we obtain the exclusion region for the CSL parameters shown in blue in Fig. 5.4. The achievable upper bound from the foreseen design sensitivity is shown with dashed blue line.

5.3.2 Space-based experiments: LISA Pathfinder

The second system we consider is LISA Pathfinder. This space mission has been recently launched as a technology demonstrator of the proposed space-based gravitational wave detector LISA. LISA concept is similar to terrestrial interfero- metric detectors, but will exploit a much longer baseline 106 km and the more ⇠ favourable conditions of operation in space. The detector will be sensitive to gravitational waves in the mHz range, thus providing different and complemen- tary informations compared to ground-based detectors. LISA test masses will be in nearly ideal free-fall and essentially free from the vibrational, seismic and grav- ity gradient disturbances which unavoidably affect any terrestrial low-frequency experiment. The main goal of LISA Pathfinder is to demonstrate the technology required by LISA, in particular to assess the accuracy of the achievable free-fall condition. The core of LISA Pathfinder consists in a pair of test masses [cf. Fig. 5.3] in free- fall, protected by a satellite which follows the mass trying to minimize the stray disturbance. The overall objective is to demonstrate the performance required for the test masses of LISA Pathfinder in terms of acceleration noise. Thus, the output of the experiment is directly expressed as a relative acceleration noise spectrum

Sgg(!), which is related to the relative force noise spectral density by the relation: 4 SLISA(!)= SLISA(!). (5.33) gg M 2 FF The geometry of each test mass is straightforward: a cube of side L =4.6 cm, made of an alloy of AuPt, with a mass M =1.928 kg, and the distance between the two masses a =37.6 cm. Thanks to the space operation, it is possible to Chapter 5. Collapse Models 58 achieve a force sensitivity better than ground-based experiments. The current best experimental figure reaches a minimum acceleration noise of Sgg(!)=2.7 29 2 4 ⇥ 10 m s Hz [186]. Using Eq. (5.29) and Eq. (5.33) and plugging in the numerical values of the pa- rameters and the best force noise, as given above, we obtain the green exclusion area in Fig. 5.4. The force noise in LISA Pathfinder is steadily improving with time likely because of progressive outgassing of the spacecraft, and is already significantly better than the published data [187]. Assuming a reasonable im- provement by factor of 2 we get the dashed green line in Fig. 5.4. Notice that this result would overcome the bound set by the ultracold cantilever experiment [50] 7 for the standard value taken for rC =10 m.

5.3.3 Resonant GW detectors: AURIGA

The principle of resonant-mass GW detectors is to monitor the deformation of an elastic body, typically a massive ton-scale resonant bar or sphere, induced by a gravitational wave. The main drawback compared to interferometers, see above, is the smaller bandwidth and the shorter characteristic length 1 m. However, as ⇠ these detectors have been operated at cryogenic temperature and have achieved 20 1 impressive displacement noise pS 10 m Hz 2 , it is worth considering their xx ⇠ sensitivity to CSL effects. As best case we consider AURIGA [188, 189], which is 3 based on a aluminum (density µ0 =2700kg/m ) cylinder with length L =3m, radius R =0.3 m and mass M =2300kg cooled to T =4.2 K, schematically rep- resented in Fig. 5.3. Other detectors of the same class feature similar parameters. The fundamental longitudinal mode of deformation at ! /2⇡ 900 Hz is moni- 0 ⇠ tored by a sensitive SQUID-based readout [189]. The system, as described above, is model as two masses M/2 connected by a spring and oscillating in counter- phase. We expect this procedure to yield a crude but reasonable estimate of the

CSL effect, within a factor of 2. The equivalent force noise spectrum SFF(!) of the reduced system is related to the strain noise spectrum Shh(!) by the relation [188, 190]: M!2L 2 SAURIGA(!)= 0 SAURIGA(!). (5.34) FF ⇡2 hh ✓ ◆ For the AURIGA detector in the current scientific run, the minimum strain noise 21 1 at resonance is S (¯!)= S (¯!)=1.6 10 Hz 2 at !¯/2⇡ =931Hz (in the h hh ⇥ following we will use single index to represent square rooted spectral densities). p An independent absolute calibration was performed, based on the fluctuation- dissipation theorem, demonstrating that the noise at resonance is dominated by thermal noise [188]. The calibration accuracy was of the order of 10% in en- ⇠ ergy. Taking this into account, we estimate the minimum unknown force noise, 1 2 which could be attributed to CSL, as SF = 12 pN Hz . Note that AURIGA (as well as NAUTILUS [191]) has been also operated in previous runs at lower tem- peratures 100 mK [192]. The minimum strain noise at resonance was actually ⇠ Chapter 5. Collapse Models 59

Figure 5.4: Figure taken from [52]. Upper and lower bounds on the CSL collapse

parameters CSL and rC . Blue, green and red lines (and respective shaded regions): upper bounds (and exclusion regions) from LIGO, LISA Pathfinder and AURIGA. Blue and green dashed lines: upper bounds from foreseen improved sensitivity respectively of LIGO and LISA Pathfinder. Purple line: upper bound from ultracold cantilever experiments [50]. Light blue line: upper bound from X-ray experiments [49]. Other weaker bounds [42, 43, 45, 193, 194] are not reported. Gray line: lower bound based on theoretical arguments [42]. The GRW [32, 36] and Adler [57] values and ranges are indicated in black. lower, but an accurate thermal noise calibration in that case was not performed. This amounts to a value for the minimum unknown force noise comparable to that given above. The comparison of the CSL prediction with the experimental data leads to the red line and exclusion area in Fig. 5.4. The three exclusion regions computed here have a very similar shape, achieving a minimum for rC of the order of the test mass relevant length. For rC > 1 m the bounds are roughly comparable, with the one set by LIGO slightly better. Theo- retically, such values of rC are not much interesting, since already excluded by as- suming the effective collapse of macroscopic objects [42] (gray region in Fig. 5.4).

For smaller values of rC, the best bound is set by LISA Pathfinder. Chapter 5. Collapse Models 60

We observe that these bounds are the best so far, for rC ranging from roughly 1 µm up to the macroscopic scale, thereby excluding a substantial region of the parameter space. While this rC interval is not the one usually considered relevant or theoretically favoured, we point out that, as long as rC and CSL are free param- eters, unambiguous exclusion of a given region of parameter space can be done only through experiments.

7 At the standard characteristic length rC =10 m, the bound from LISA Pathfinder 8 1 < 3 10 s is still interesting, falling about a factor of 2 from the best limit CSL ⇥ obtained so far with mechanical techniques at ultralow temperature [50]. The 7 strongest bound so far for rC =10 m is still provided by X-ray experiments [49], although the latter bound is less stable to possible modification of the CSL model. We also note that the inferred bounds are conservative, at least for the LIGO and LISA Pathfinder cases, as we have assumed that all measured noise is attributed to CSL. Actually, the interferometer noise can be to a good extent characterized and attributed to well-defined sources. Subtraction of well-characterized noise may enable in principle a slight improvement of the bounds. For the AURIGA case, the noise at resonance is almost entirely due to thermal noise, and an abso- lute calibration based the fluctuation-dissipation theorem was performed. In this case a noise subtraction within the calibration uncertainty is entirely legitimate. For interferometers such as LIGO this task might be more difficult, as several noise sources combine together to yield the measured spectrum, depending on frequency. Some of them, such thermal noise, can be in principle fully charac- terized, but for others, like newtonian or seismic noise, the task is much more complicated. For the LISA Pathfinder case, there is evidence that thermal noise from the residual gas is dominating the residual force noise. Unfortunately, un- certainty in the pressure and composition of the gas make it hard to perform an independent calibration based on the fluctuation-dissipation theorem [186]. We also mention that there is another class of macroscopic mechanical resonators, namely torsion balances, which have been the most sensitive force sensors since the time of Cavendish. We have not considered explicitly this class of experiments 2 because typical sensor size (10 1 m) and frequency band (mHz) are very simi- lar to the those of LISA Pathfinder. In fact, ground testing of LISA technology has been primarily done by means of torsion pendulum experiments. However, the actual performances achieved by LISA Pathfinder have arguably improved over ground-based tests by at least 2 orders of magnitude [186]. As an immediate consequence of our analysis, we can also exclude a quantum gravity induced decoherence model proposed long time ago by Ellis et al. [195, 196]. Briefly, the model estimates the decoherence induced by the interaction with a background of wormholes. As long as the wavelength of the wormholes is much longer than the characteristic magnitude of the motion of the system, deco- herence can be effectively described by the one-dimensional version of Eq. (5.11), with diffusion coefficient 4 2 (cm0) M ⌘Ellis = 3 , (5.35) (~mPl) Chapter 5. Collapse Models 61

where mPl is the Planck mass and c the speed of light. A recent analysis [197] shows that this model is incompatible with the latest atom interferometry ex- periment of Kasevich’s group, performing a spatial separation of 0.5 m[152]. ⇠ However in the latter case the negative result is not very strong, as the experi- mentally measured rate ⌘ is just 25 times smaller than ⌘ . In our case, data exp ⇠ Ellis from LISA Pathfinder show that ⌘ is 1012 times smaller than ⌘ , thus setting exp ⇠ Ellis a significantly stronger bound.

5.4 Ultra-cold cantilever detection of non-thermal ex- cess noise

The cantilever experiment measures the possible temperature enhancements dic- tated by CSL [cf. Eq. (5.24)]. The experiment we here report is an updated and improved version of a previous cantilever experiment [198], which was already used to infer upper bounds on the CSL parameters [50]. The scheme of the ex- periment is similar to the previous one and is shown in Fig. 5.5. A cantilever with a spherical ferromagnetic load is continuously monitored by a SQUID. The displacement x is converted into magnetic flux by a linear coupling x =d/ dx which depends on the magnet position and orientation. A novel feature is that the mechanical quality factor Q = !0/m is much higher than in previous exper- iments, and heavily temperature dependent. Moreover, we observe a dynamical SQUID-induced magnetic spring effect, analog to optical spring effects in op- tomechanics, which modifies the quality factor from its intrinsic value Q to an apparent value Qa [199]. To account for this new feature, we seek a strategy to directly measure the effective force noise acting on the resonator, rather than the mean energy which may be affected by the dynamically-modified quality factor Qa. To this end, let us consider the Lorentzian spectral density associated with the cantilever displacement fluctuations:

4 SFF,0 4kBT f0 Sxx = 2 + 2 . (5.36) k k! Q 2 ff ✓ 0 ◆ (f 2 f 2) + 0 0 Qa ⇣ ⌘ Here, f0 = !0/2⇡ is the resonant frequency, k is the spring constant, kB the Boltz- mann constant. T the temperature and SFF,0 the spectral density of any nonther- mal force noise. Eq. (5.36) says that magnetic spring effects, similarly to optome- chanical ones, only affect the dynamics and thus the denominator of the resonant term, characterized by an apparent quality Qa [176, 200, 201]. Instead, the ampli- tude of the Lorentzian curve is proportional to the total force noise. The thermal contribution scales with T/Q, where Q is the intrinsic quality factor. Our strat- egy is to characterize the thermal noise by accurate measurements as function of T/Q. Any excess nonthermal noise, included the CSL-induced one, will cause a Chapter 5. Collapse Models 62

Figure 5.5: Figure taken from [51]. Simplified measurement scheme. The funda- mental bending mode of a cantilever loaded with a ferromagnetic microsphere with magnetic moment µ is continuously monitored by a SQUID susceptometer. The SQUID measures the magnetic flux =xx coupled by the displacement x of the magnetic particle and is operated in flux locked loop (feedback electronics is not shown for simplicity). The flux-dependent circulating current J, combined with fi- nite feedback gain, causes a dynamical magnetic spring effect which modifies the apparent quality factor of the cantilever.

constant force spectral density SFF,0, independent of T/Q. The maximum nonther- mal force noise compatible with the experiment can be used to test CSL predic- tions. This requires modeling the CSL force acting on the continuous mechanical resonator, cf. Eq. (5.17). The mechanical resonator in our setup is a commercial tipless AFM silicon can- tilever, with size 450 57 2.5 µm3 and stiffness k =0.40 0.02 N/m. A hard ferro- ⇥ ⇥ ± 3 magnetic microsphere (radius R =15.5 µm, density µ0 =7.43 kg/m ) is glued to the cantilever free end and magnetized. The microsphere has a twofold function. 2 It increases the cross-section to the CSL field, which scales as µ0, while at the same time enabling a straightforward detection by means of a nearby SQUID suscep- tometer [50]. The SQUID is gradiometric and comprises two distant loops with radius RS =10µm[202]. The particle is aligned above the first loop at a height h 40 µm, with the motion of the first flexural mode orthogonal to the SQUID ' plane. The SQUID is operated in two-stage flux-locked-loop configuration with the feedback applied to the second loop. This geometry strongly suppresses di- rect coupling between the feedback signal and the cantilever. The cantilever-SQUID system is enclosed in a copper box, suspended above the mixing chamber plate of a pulse-tube dilution refrigerator (Janis Jdry-100-Astra) by means of a two-stage suspension system. The measured mechanical attenu- ation is higher than 80 dB at the cantilever frequency. The temperature of the mixing chamber is measured by a RuO2 thermometer, while the temperature of the SQUID box is measured by a SQUID noise thermometer. Both devices have Chapter 5. Collapse Models 63 been calibrated against a superconducting reference point device with accuracy better than 0.5%. The resonant frequency of the fundamental mode of the cantilever is at the fre- quency f0 =8174.01 Hz. The actual mechanical quality factor Qa is determined by ringdown measurements. The dynamical magnetic spring effect depends on the SQUID working point and scales as 1/ G where G ( G 1) is the open loop | | | | gain of the feedback electronics. Specifically, we expect a linear dependence of the actual quality factor Q as 1/Q =1/Q + c/ G , where Q is the intrinsic quality a a | | factor and c/ G is the SQUID dynamical effect. This behaviour is experimen- | | tally observed by varying the gain G , and allows to infer the intrinsic quality | | factor Q. The whole measurement procedure is repeated at each temperature T [cf. App. C.3]. The measured intrinsic Q is of the order of 6 105 at temperature of the order of ⇥ 1 K, and is observed to increase roughly as 1/T upon reducing temperature be- low T =500mK, approaching Q 107 at T 20 mK. This behaviour is consistent ' ' with measurements performed with the SQUID weakly coupled, and is reminis- cent of two-level systems dissipation [203]. Standard glassy two-level systems in a 2 nm amorphous oxide layer on the cantilever surface are able to explain the observed effect quantitatively. The noise is measured by acquiring and averaging spectra of the SQUID signal, calibrated as magnetic flux, with typical integration time of 800 - 1200 seconds. Before measuring noise and quality factor, the system is let thermalize for at least 3 hours. During the noise measurement the pulse tube is switched off, and the mixing chamber temperature actively stabilized by a PID controller. Examples of averaged flux noise spectra at three representative temperatures are shown in

Figure 5.6: Figure taken from [51]. Examples of averaged spectra with the respec- tive best fit with Eq. (5.37). Red, green and blue lines refer to the temperatures of 38 mK, 170 mK and 350 mK respectevely. Chapter 5. Collapse Models 64

Fig. 5.6. The spectra are fitted with the curve:

4 2 2 2 Bf0 + C (f f1 ) S = A + 2 . (5.37) (f 2 f 2)2 + ff0 0 Qa ⇣ ⌘ The term proportional to B is the relevant one, as it corresponds to the fluctua- tions of the cantilever induced by thermal or extra force noise, given by Eq. (5.36), converted into magnetic flux. The term proportional to A is the purely additive wideband noise of the SQUID. The last term, proportional to C arises because of the flux noise applied to the SQUID by the feedback electronics in order to compensate for the SQUID additive wideband noise. This flux noise induces a current J circulating in the SQUID through the finite responsivity J =dJ/d [199], which eventually leads to an effective back-action on the cantilever. The transfer function of this mechanism has been directly measured by injecting a calibration signal and features an antiresonance at f = f , with f f =1.1 Hz. 1 1 0 The overall effect is a small asymmetric distortion of the Lorentzian peak. All spectra have been checked by 2 tests to be consistent with Eq. (5.37), see Appendix C.3. All estimations of the SQUID parameters A and C are consistent 13 2 with each other, with mean values A =(1.23 0.05) 10 0/Hz and C = 13 2 ± ⇥ (3.78 0.05) 10 /Hz. In particular, A and C do not depend significantly ± ⇥ 0 on temperature. This is expected, as for this type of SQUID [201] the noise is saturated by hot electron effect [204] for T<400 mK, and our measurements satisfy this condition. Fig. 5.7 shows the measured symmetric amplitude B of the Lorentzian noise as function of T/Q, varied by changing the bath temperature. The uncertainty on the estimation of B is remarkably low, of the order of 1%. The x-error bar, dominated by the uncertainty on Q, is thus significant [cf. App. C.3]. The data agree with a linear behaviour over the whole T/Q range. A weighted 19 orthogonal linear fit B0 + B1T/Q yields the intercept B0 =(1.33 0.11) 10 2 19 2 ± ⇥ /Hz and the slope B =(0.292 0.002) 10 / (nK Hz). We exploit the 0 1 ± ⇥ 0 · linear dependence on T/Q to infer the actual coupling between cantilever and SQUID. Given Eq. (5.36) and Eq. (5.37) we can express the thermal slope B1 as:

2 4kB x B1 = , (5.38) !0 k

2 which allows the coupling factor x/k to be evaluated from the measured B1. The finite intercept, clearly visible in the inset of Fig. 5.7 implies that the data are not compatible with a pure thermal noise behavior, and a nonthermal excess noise is present. According to Eq. (5.36) we can convert B0 into a residual force noise:

4kBk B0 SFF,0 = . (5.39) !0 B1 The measured coupling factor and residual force noise are reported in Table 5.2, Chapter 5. Collapse Models 65

Figure 5.7: Figure taken from [51]. Symmetric amplitude of the Lorentzian noise B, as measured by the SQUID, as function of the ratio T/Q, together with the best linear fit. In the inset, the data at the lowest T/Q are zoomed in order to highlight the nonzero intercept of the fit.

2 2 Pulse tube x/k (fH) SFF,0 (aN /Hz) Off 116 1 1.87 0.16 0.1(sys) ± ± ± Off 347 3 2.12 0.20 0.1(sys) ± ± ± On 114 2 2.58 0.20 0.1(sys) ± ± ± Table 5.2: Operating conditions, measured coupling and residual force noise for the different measurement datasets. together with the same quantities inferred from the additional measurements dis- cussed in the following. The systematic error on SFF,0 arises from the uncertainty on k. We checked for possible physical sources of the excess force noise. First of all, 2 we expect a back-action force spectral density SFF, BA = SJFJ from the noise in the current circulating in the SQUID loop. Here, SJ is the current spectral density and FJ =dF/dJ is the backward current-to-force factor. Because of reciprocity, FJ must be equal to the forward displacement-to-flux factor x [198] so that SFF, BA = 2 SJx. In other words, the back-action noise leads to a finite intercept, and the corresponding force noise scales with the coupling factor. We took advantage of this property and performed additional measurements in a subsequent cooldown at a different cantilever position, with effective coupling increased by a factor 3 [cf. Tab. 5.2]. We observe again a linear behavior in very ⇠ good agreement with the experimental data, with a finite intercept, see Appendix C.3. The corresponding residual force noise is reported in the second row of Table 5.2 and is consistent within the error bar with the one at low coupling. This clearly indicates that most of the observed excess noise cannot be attributed to SQUID back-action. Chapter 5. Collapse Models 66

We can compare this result with the prediction of the Clarke-Tesche model [205] for the current noise, S = k T /R with 11 for an optimized SQUID. J B SQ SQ ' Here R =8⌦and T 400 mK are the measured shunt resistance and the SQ SQ ' typical SQUID electron temperature. From this expression we estimate a small increase of the back-action force noise S 0.6 aN2/Hz between the two FF, BA ' measurements, which is compatible within 2 with the experimental increase S =0.24 0.26 aN2/Hz. FF,0 ± In order to investigate the role of vibrational noise from the refrigerator or from the outside world, we repeated the measurements at low coupling by keeping the pulse tube on [cf. App. C.3]. The input mechanical noise provided by the pulse tube in our cryostat is known to be 2-3 orders of magnitude larger than the back- ground noise when the pulse tube is off. However, while the measured spectra with pulse tube on are significantly dirtier, we can still perform a Lorentzian fit and the residual force noise, reported in Table 5.2, is only slightly increased with respect to the measurements with pulse tube off. This confirms that the mechani- cal suspensions are working well within design specifications, and suggests that vibrational noise is not the source of the observed excess noise with pulse tube off. We have also ruled out vibrational noise from the 3He flow, by switching on and off the circulation pump without noticeable effects [cf. App. C.3]. Magnetic effects, such as fluctuations of the environmental magnetic field or fluc- tuations of the microsphere magnetization, can be also considered as possible excess noise sources. We can substantially rule out these mechanisms, based on theoretical order of magnitude estimations, and a further test which has shown the quality factor to be independent of the external static field [cf. App. C.3]. Another option is that we are actually observing thermomechanical noise, but the effective temperature of the noise source (or part of it) is higher than the one of the thermal bath because of thermal gradients along the cantilever. In this case one would expect to observe saturation effects, as observed in [50] rather than a linear behaviour with a fixed intercept. Furthermore, we have performed simple thermal modeling of the cantilever. The power dissipated in the magnet by eddy currents induced by SQUID Josephson radiation is estimated of the order of 1 fW, and would cause a temperature gradient between the magnet and the cantilever base smaller than 1 mK in the temperature range explored by this experiment. The observed finite intercept could also be a subtle artifact due to an unknown systematic error in the determination of 1/Q. We find that this is in principle possible, but the systematic error on 1/Q would have to be 10 times larger than the statistical error bar to be consistent with zero excess noise. Moreover, the data would not follow a linear behaviour anymore [cf. App. C.3]. Finally, let us compare our results with the predictions of the CSL model. By using the same method discussed in [50], we can convert the observed excess noise into the red curve in the parameter space CSL - rC of the CSL model, shown in Fig. 5.8. This curve can be considered as a conservative improved upper bound 7 6 from mechanical experiments for 10 m

Figure 5.8: Figure taken from [51]. Exclusion plot in the CSL - rC plane based on our experimental data, compared with the best experimental upper bounds reported so far and with theoretical predictions. Continuous thick (red) curve: CSL collapse rate

CSL, as function of the characteristic length rC, assuming that the observed noise is entirely due to CSL. The shaded region would be excluded by our experiment if the physical origin of the excess noise were identified. The other thin lines represent upper limits from labeled experiments: previous cantilever experiment (orange) [50], LISA Pathfinder (black) [52], cold atoms (green) [193] and X-ray spontaneous emission (dashed blue) [49]. The (dark green) bars represent the CSL collapse rate suggested by Adler [38].

7 7.7 on the curve. For the standard choice rC =10 m this would imply CSL =10 1 s , in agreement with Adler’s predictions [38]. Alternatively, if the observed noise can be eventually reduced to standard physical effects, its identification and elimination will lead to an improved upper bound on CSL, determined by the experimental error bar. The parameter region which can be potentially excluded is shaded in Fig. 5.8. A full exclusion would almost completely rule out Adler’s predictions [38].

5.5 Hypothetical bounds from torsional motion

In [58] we proposed an unattempted non-interferometric test aimed to cover the unexplored region of the CSL parameter space. We show that the angular mo- mentum diffusion predicted by CSL [cf. Eq. (5.14)] heavily constrains the para- metric values of the model. In particular, we consider the roto-vibrational motion of a cylinder coupled to the field of an optical cavity. By addressing the ensuing dynamics of the cylinder, we show that the torsional degree of freedom offers en- hanced possibilities for exploring a wide-spread region of the parameters space of the CSL model, thus contributing significantly to the ongoing quest for the va- lidity of collapse theories. We provide a thorough assessment of the experimental Chapter 5. Collapse Models 68 requirements for the envisaged test to be realized and highlight the closeness of our proposal to state-of-the-art experiments. V T We first compare the magnitude of the two temperatures TCSL and TCSL defined

10-8 Adler

10-10

10-12 ) 1 - s ( 10-14
GRW

10-16

10-18

10-20 10-8 10-7 10-6 10-5 10-4 10-3 10-2

rC (m)

Figure 5.9: Figure taken from [58]. Panel (a): In grey we show the strongest bounds presently reported in literature [42, 43, 45, 49, 50, 52, 193, 194]. The values sug- gested by GRW [32, 36] and Adler [57], and the associated ranges, are indicated in

black. The cyan lines show the values of rC we consider in our analysis, namely 7 4 rC = 10 m (dotted line) and 10 m (continuous line). Panels (b) and (c): CSL 1 7 temperature contribution TCSL against R/L, for =1s , rC = 10 m [panel 4 (b)] and rC = 10 m [panel (c)]. The blue and green lines (either dotted or solid) V denote the behavior of TCSL along the x axis and the symmetry axis, respectively. T The red lines (dotted and solid) show TCSL. The dip in the red curve occurs when the dimensions of the cylinder are similar, which makes it less sensitive to torsions. in Eq. (5.24) for different ratios of the radius R and the length L of the cylinder. T,V 1 Without loss of generality (as T ⌘ ) we set =1s . For defi- CSL / j / CSL CSL niteness we take a silica cylinder with m =10µg and vary the ratio between the radius R and the length L. For the residual gas, we consider He-4, at the temper- 13 ature of T =1K and pressure P =5 10 mbar, which can be reached with ⇥ V,R 3 7 existing technology [158]. Fig. 5.9 shows the behaviour of TCSL for rC =10 m 4 (dotted lines) and rC =10 m (continuous lines). In the latter case the strongest contribution comes from vibrations along the x axis (blue lines) at 2R L, while ⇠ in the former case it comes from torsions (red lines) for R L, thus showing that in specific circumstances torsions can substantially contribute the CSL testing. As a comparison, we also report TCSL given by vibrations along the symmetry axis (green lines). In Fig. 5.10a we compare the hypothetical upper bounds obtained from the vi- brational and torsional motion, taken individually. This is done by setting the accuracy in temperature to T =0.1 K and varying the dimensions of the cylin- der. As a case-study, we consider a thought experiment aimed at testing CSL in 7 the region rC 10 m, and exploit torsions of a coin shaped system to maximize ⇠ the CSL effect [cf. Fig. 5.9b]. As shown, the hypothetical upper bound given by the torsional motion is stronger than the vibrational one. This analysis assumes that the temperature accuracy T is the same for the detection of both the vibra- tional and torsional motion. Actually, the experimental accuracy on temperature

3 These values of rC are chosen due to their closeness to the boundaries of the unexplored CSL parameter region. Chapter 5. Collapse Models 69

10-8 Adler

10-10

10-12 ) 1 - s ( 10-14
GRW

10-16

10-18

10-20 10-8 10-7 10-6 10-5 10-4 10-3 10-2

rC (m)

Figure 5.10: Figure taken from [58]. Panels (a) and (b): Hypothetical upper bounds T V T obtained assuming TCSL T. We choose T = T in (a) and T =(3+ L 2 V V ( R ) )/48 T in (b) with T =0.1 K in both panels. The chosen system is a silica cylinder of radius R and length L, cooled at the temperature of T =1K and at 13 a pressure of P =5 10 mbar. Blue, green and red lines: Upper bounds for ⇥ the vibrational along the x axis, the symmetry axis and torsional motion around it respectively. The dotted (dotted-dashed) lines correspond to R =0.1mm and L =0.1µm(R =1cm and L = 10µm). is such that T R I!22 = , with I = m (3R2 + L2), (5.40) V 2 2 12 T m!0x and x and denoting the accuracy in positions and angles respectively, where = x/2R for 2R>Land = x/L for 2R

5.5.1 Experimental feasibility

Having assessed formally how the torsional motion of a levitated cylinder of (fairly) macroscopic dimensions can set very strong bounds on the CSL model (almost testing the GRW hypothesis), we now address the experimental feasibil- ity of our proposal, showing that the proposed experiment is entirely within the grasp of current technology. First, the cylinder has to be trapped magnetically or electrically to allow for its torsional motion around the x-axis, as in Fig. 5.1. Needless to say, we must avoid competing heating effects such those due to gas collisions and exchange of thermal photons between the environment and the trapped cylinder. More- over, one must ensure the ability to control and detect precisely enough the tor- sional motion of the trapped cylinder. The first condition can be granted by performing the experiments at low temperatures and pressures. Standard di- lution cryostats reach temperatures < 10 mK [50, 51] and the reachable pressure 17 is as low as 10 mbar (as done in the cryogenic Penning-trap experiment re- 13 ported in Ref. [158]), which is much lower than the 5 10 mbar considered ⇥ Chapter 5. Collapse Models 70 in Fig. 5.10a,b). The trapping of the cylinder can be done magnetically or elec- trically [186, 206], while the control and readout of the torsional motion can be achieved by an optical scattering technique [207]. Furthermore, a stroboscopic detection mode can be chosen to avoid any heating by the detection light of the torsional motion of the cylinder [172]. The torsional state can be prepared very reliably by feedback control [208]. The feedback is then turned off to allow for heating. Alternatively, if a magnetic cylinder is utilized, SQUID sensors could be used to read the torsional state. Although the torsional motion equivalent to a temperature of 1 mK needs to be resolved, this is possible with state of the art experiments [50, 209]. One has to take into account that mea- suring T with a high quality factor is proportionally more difficult. Therefore all requirements for this proposal can be reached in a dedicated experiment based on existing technology. Notice also that this non-interferometric test does not require the preparation of any non-classical state, which would need much advanced technology, yet to be demonstrated for such a macroscopic object. While the experimental scenario and the shape of the cylinder is the same as discussed already in [168, 169], here we find that macroscopic dimensions for the cylinder are useful for testing collapse models. This should make our proposed test far less demanding than experi- menting with a nano- or micro-scale cylinder. Clearly a big experimental chal- lenge will be the control of mechanical vibrations [210]. In this respect, torsional degrees of freedom can be decoupled from vibrational noise much more effec- tively than translational ones. A well known and paradigmatic example is given by the torsion pendulum. 71

Chapter 6 Conclusions

In this thesis, both lines of research developed during this Ph.D. were consid- ered: decoherence and collapse models. Albeit conceptually they are different, they have much in common under technical aspects: the mathematical descrip- tion of the statistical operator evolution is described by very same equations1, and thus the experimental tests one can perform to confirm of falsify one of these models are the same.

As a first step, we provided the basic instruments to describe the evolution of a system undergoing the dynamics of the above mentioned classes of models. We started by considering the framework of decoherence models, where the system interacts with its environment. If one focuses only on the degrees of freedom of the system, its evolution can be depicted in terms of a master equation, which is the main tool of the theory of open quantum systems. The same can be done for collapse models. In fact, for all practical purposes, the non-linear evolution, de- scribing the collapse at the level of the wave-function, is substituted with a linear one governing the statistical operator.

Afterwards, we illustrated how the scattering of a gas particle on the system pro- duces the suppression of the interference terms in a superposition of a system in two different positions. This effect is named decoherence, and a first model de- scribing it, is the well known one by Joos and Zeh [30]. Furthermore, we applied the model to a system prepared in a superposition of different angular configu- rations rather than positions.

A more sophisticated model describing decoherence due to the interaction with the surrounding environment is the quantum Brownian motion model, that al- lows to take into account also dissipative effects of the dynamics. We provided an exact and analytic equation for the time evolution of the operators and we showed that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is obtained regardless of the strength of the coupling between the system and the environment. This derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations do. Moreover, we are not bound to compute the

1This is particularly true when it comes to the comparison of the non-dissipative decoherence model by Joos and Zeh [cf. Sec. 2.1] with the short length limit of the Continuous Spontaneous Localization (CSL) model [cf. Sec. 5.1]. Chapter 6. Conclusions 72 time evolution of the state of the system, which in general is a complicated task. The explicit dependence on the initial state appears only in the initial expectation values and not in the dynamics. This simplifies the derivation of the evolution of expectation values and makes this possible also for nontrivial states, e.g. non- Gaussian state.

The last decoherence model we considered in this thesis is a recent model [56]. Based on the mass-energy equivalence, it acts universally on every system whose superposition is extended on positions experiencing different gravitational po- tentials. We studied the conditions under which this mechanism becomes the dominant decoherence effect in typical interferometric experiments. The follow- ing competing sources were considered: spontaneous emission of light, absorp- tion, scattering with the thermal photons and collisions with the residual gas. We quantified all these effects and showed that current experiments are off by sev- eral orders of magnitude. New ideas are needed in order to achieve the necessary requirements.

Finally, we come to collapse models. Since a broad literature already exists, we only gave a basic introduction of the most known of these models, the Contin- uous Spontaneous Localization (CSL) model. In particular we focused on how we can benefit from optomechanical setups, which are the best experimental tool that current technology can provide us, to probe such a model. Three experi- ments were considered.

In the first example, we computed the upper bounds on the CSL parameters, which can be inferred by the gravitational wave detectors LIGO, LISA Pathfinder, and AURIGA. We showed that these experiments exclude a huge portion of the CSL parameter space, the strongest bound being set by the recently launched space mission LISA Pathfinder. We also ruled out a proposal for quantum-gravity induced decoherence.

In the second example, we reported new results from an experiment based on a high quality cantilever cooled to millikelvin temperature, potentially able to im- prove by one order of magnitude the current bounds on the CSL model. High accuracy measurements of the cantilever thermal fluctuations reveal a nonther- mal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the ob- served noise have been ruled out. Further investigations are needed in order to probe other possible explanations. Above all, this experiment neatly illustrates the fundamental challenge of collapse models testing. Negative results are ro- bust, but positive claims require extremely careful and systematic work in order to exclude any conceivable alternative physical explanation. Chapter 6. Conclusions 73

In the last example we studied and proposed an unattempted non-interferometric test aimed to investigate the still unexplored region of the CSL parameter space. The novelty with respect to the tests that have already been suggested in liter- ature and performed is twofold. First, our proposal exploits torsional degrees of freedom rather than the usual vibrational ones. Second, the scheme focuses on objects of macroscopic dimensions instead of micro-scale ones. Both aspects offer considerable advantages that were at the basis of the reduction of the pa- rameter space mentioned above. Although the features above have already been discussed and studied individually, an investigation combining such advantages together is unique of our proposal. We believe that this test, which has been shown to adhere well to the current experimental state of the art, provides a new avenue of great potential for testing the CSL model. 74

Appendix A Quantum Brownian Motion master equation

A.1 Explicit form of (t)

With reference to Eq. (3.36), it is easy to see that for ⇢ˆB as in Eq. (3.5) the trace over

ˆB(t) gives a real and positive function of time. Using the definition of the spectral 2 2 density in Eq. (3.6) one immediately derives: (t)= 1(t)+µ 2(t)+µ3(t), where the explicit form of i(t) is:

t t D1(t0 s) (t)= dt0 ds G (t t0)G (t s), (A.1a) 1 4M 2 2 2 Z0 Z0 1 t t (t)= dt0 dsD (t0 s)G (t t0)G (t s), (A.1b) 2 4 1 1 1 Z0 Z0 G (t) t (t)= 2 dsD (s)G (t s), (A.1c) 3 2M 1 2 Z0 with D1(t) denoting the noise kernel defined in Eq. (3.8).

A.2 Explicit form of the adjoint master equation

Starting from Eqs. (3.35) for ↵1(t) and ↵2(t), linear combinations of these relations give the following relations:

MG˙ (t) G˙ (t) ˆ = 1 [ˆ , xˆ] 2 [ˆ , pˆ] , (A.2a) t F (t) t F (t) t G (t) G (t) µˆ = 1 [ˆ , xˆ]+ 2 [ˆ , pˆ] , (A.2b) t F (t) t MF(t) t where we defined ˙ ˙ F (t)=~ G1(t)G2(t) G1(t)G2(t) . (A.3) ⇣ ⌘ By combining the results in Eq. (3.37) and Eqs. (A.2), one immediately can check that Eq. (3.37) takes the Lindblad time-dependent form described in Eq. (3.39). In Appendix A. Quantum Brownian Motion master equation 75

ˆ particular, the effective Hamiltonian Heff is given by Eq. (3.41), where

G1(t)G¨2(t) G¨1(t)G2(t) A ~ (t)= , (A.4a) 2 ⇣ F (t) ⌘

G˙ 1(t)G¨2(t) G¨1(t)G˙ 2(t) A(t)=~ , (A.4b) ⇣ F (t) ⌘ and the elements of the Kossakowski matrix Kab(t) are:

1 t K11(t)= dsD1(s) G1(t)G˙ 2(t s) G˙ 1(t)G2(t s) , ~F (t) 0 Z t ⇣ ⌘ 1 A(t) K12(t)= dsD1(s) G1(t s)G2(t) G1(t)G2(t s) i , 2M~F (t) 0 ~ Z ⇣ ⌘ (A.5) and K22(t)=0.

A.3 Derivation of the master equation for the states

˜ To construct Lt, we start from the derivative with respect to the parameters and µ of the characteristic operator ˆt, see Eq. (3.34) of the main text:

@ G (t) ˆ = iG (t)ˆxˆ + i 2 pˆˆ + A(t)ˆ , (A.6) @ t 1 t M t t and @ ˆ = iMG˙ (t)ˆxˆ + iG˙ pˆˆ + B(t)ˆ , (A.7) @µ t 1 t 2 t t where: i A(t)= F (t)+ (t) µ +2 (t), 2 3 1 ✓ ◆ (A.8) i B(t)= F (t)+ (t) +2 (t)µ, 2 3 2 ✓ ◆ where F (t) is defined in Eq. (A.3). By linearly combining Eq. (A.6) and Eq. (A.7) we arrive at the following expressions:

i~ @ i~ @ i~ G2(t) xˆˆ = G (t) ˆ G˙ (t) ˆ + G˙ (t)A(t) B(t) ˆ , t MF(t) 2 @µ t F (t) 2 @ t F (t) 2 M t  i~ @ i~ @ i~ pˆˆ = G (t) ˆ + MG˙ (t) ˆ MG˙ (t)A(t) G (t)B(t) ˆ , t F (t) 1 @µ t F (t) 1 @ t F (t) 1 1 t h (A.9)i which we use to replace the terms proportional to xˆ and pˆ in Eq. (3.37) of the main ˜ ˜ text; the right-hand side gives Lt⇤[ˆt]=Lt⇤ t⇤[ˆ(0)]. By multiplying it from the Appendix A. Quantum Brownian Motion master equation 76

1 left with (t⇤) we obtain

1 ˜ (t⇤) Lt⇤ t⇤ [ˆ(0)] = Lt⇤ [ˆ(0)] = 1 A @ A @ =(⇤) M (t)µ A(t) + +2µ (t) B(t) + t @ M @µ (A.10)  ✓ ◆ ✓ ◆✓ ◆ i~ + ↵˙ (t)↵ (t) ↵ (t)˙↵ (t) + ˙(t) ⇤ [ˆ(0)] , 2 1 2 1 2 t ⇥ ⇤ where A(t) and A(t) are defined in Eqs. (A.4) of the main text. Now, the ex- pression within the square brackets contains no operator, therefore the action of 1 the inverse map (t⇤) and of the direct map t⇤ cancel each other. Moreover, because of Eqs. (A.6) and (A.7) we have

@ ~ @ ~ ˆ(0) = i xˆ + µ ˆ(0), and ˆ(0) = i pˆ ˆ(0), (A.11) @ 2 @µ 2 ✓ ◆ ✓ ◆ and therefore Eq. (A.10) becomes

A ~ Lt⇤ [ˆ(0)] = M (t)µ A(t) i xˆ + µ + 2  ✓ ✓ ◆◆ A ~ + +2µ (t) i pˆ B(t) + M 2 ✓ ◆✓ ✓ ◆ ◆ i~ + ↵˙ (t)↵ (t) ↵ (t)˙↵ (t) + ˙(t) ˆ(0). (A.12) 2 1 2 1 2 ⇥ ⇤ Now we want to rewrite the above relation without any explicit dependence on and µ. In order to do so, we use the same procedure used in passing from Eq. (3.37) to Eq. (3.39) of the main text. According to Eq. (3.40):

ˆ(0), xˆ = ~µˆ(0) and ˆ(0), pˆ = ~ˆ(0), (A.13) ⇥ ⇤ ⇥ ⇤ which, together with the Eq. (A.12) and Eq. (A.11), gives the explicit form of Lt⇤ reported in Eq. (3.53) of the main text, where

2 A A K11(t)= ˙2(t) 42(t) (t)+M (t)3(t) , (A.14a) ~2 K22(t)=0, h i (A.14b) 1 2 i K (t)= ˙ (t) (t)+2MA(t) (t) 2 (t)A(t) A(t). (A.14c) 12 2 3 M 2 1 3 ~  ~

A.4 Explicit expression for ⇤dif(t) and E(t)

Following the procedure described in the main text, we can derive the solutions for the quadratic combinations of the position and momentum operators. Start- ˜ ˆ ing from Eq. (3.39), one applies Lt⇤ to the unitary evolved operator O(t) written Appendix A. Quantum Brownian Motion master equation 77 in terms of xˆ and pˆ. Then, one applies in Eq. (3.39) the commutation relations be- ˜ ˆ tween the operators at time t =0and finds Lt⇤[Ot] depending only from operators at time t =0. For example, in the case of xˆ2 this reads:

˜ 2 ˙ ˙ 2 ˙ ˙ 2 2 Lt⇤[ˆxt ]=2G1(t)G2(t)ˆx +2G1(t)G2(t)ˆp /M + (A.15) +(G (t)G˙ (t)+G˙ (t)G (t)) x,ˆ pˆ /M 2˙ (t). 1 2 1 2 { } 1

One integrates the obtained expression and finds the evolution of Oˆt under the reduced dynamics. In the case of the quadratic combinations of the position and momentum operators the solutions are:

G (t)G (t) 1 xˆ2 = G2(t)ˆx2 + 1 2 x,ˆ pˆ + G2(t)ˆp2 2 (t), t 1 M { } M 2 2 1 2 x,ˆ pˆ =2MG˙ (t)G˙ (t) xˆ2 + G (t)G (t) pˆ2 + h{ }ti 1 2 h i M 1 2 h i (A.16) + G (t)G˙ (t)+G˙ (t)G (t) x,ˆ pˆ 2 (t), 1 2 1 2 h{ }i 3 pˆ2 = ⇣M 2G˙ 2(t) xˆ2 + MG˙ (t)G˙⌘(t) x,ˆ pˆ + G˙ 2(t) pˆ2 2 (t). h t i 1 h i 1 2 h{ }i 2 h i 2 Then we can compute how the system diffuses in space ⇤dif(t)= xˆ2 xˆ 2: h t ih ti G2(t) ⇤dif(t)=G2(t) xˆ2 xˆ 2 + 2 pˆ2 pˆ 2 + 1 h ih i M 2 h ih i (A.17) 2G1(t)G2(t) p,ˆxˆ + h{ }i pˆ xˆ 2 (t). M 2 h ih i 1 ✓ ◆ The energy E(t)= pˆ2 /2M + 1 M!2 xˆ2 of the system S is: h t i 2 S h t i 2 2 ˙ 2 !S G2(t)+G2(t) M 2 2 ˙ 2 2 2 E(t)= !S G1(t)+G1(t) xˆ + pˆ + 2 h i ⇣ 2M ⌘ h i (A.18) 1 ⇣ ⌘ (t) + !2G (t)G (t)+G˙ (t)G˙ (t) x,ˆ pˆ 2 + M!2 (t) . 2 S 1 2 1 2 h{ }i M S 1 ⇣ ⌘ ✓ ◆ 78

Appendix B Gravitational time dilation

(2) With reference to Sec. 4.3, here we derive ⇤em in Eq. (4.13), ⇤coll in Eq. (4.15) and

coll in Eq. (4.16) in the low temperature limit, where the momentum distribution cannot be oversimplified by the Maxwell-Boltzmann law, as usually done in the literature. (All other quantities used in the main text take expressions, which are standard.)

(2) The localization rate ⇤em for thermal emission is defined as follows [24, 30, 56]:

+ 1 2 ⇤em = c dkk N(k)g(k)eff(k), (B.1) Z0 where N(k) is the number of photons with wave vector k, g(k) is the density of modes and eff(k) is the effective scattering cross section of the process. For black- 2 2 body radiation [211] the mode density is g(k)=⇡ k (see [151] for a further discussion). The number of photons N(k) is given by [41, 212, 213]:

2 ~ck kB ~ck N(k)=2exp , (B.2) k T 2C k T " B V ✓ B ◆ # where the heat capacity CV conveys the information about the internal structure. The cross section for spontaneous emission from a sphere of radius r, in the limit kr 1, is given by [211, 212]: ⌧ (k)=4⇡ [(✏(k) 1)(✏(k)+2)]kr3, (B.3) eff = where ✏(k) is the complex dielectric constant of the crystal, which can be assumed not to change with k,(✏(k) ✏). The two assumption (kr 1 and ✏(k) ✏) are ' ⌧ ' well justified; in fact the dominant contribution to the integral in Eq. (B.1) is given by small values of k. Eq. (B.1) then reduces to Eq. (4.13).

The general expression for localization rate ⇤coll for collisional decoherence by a residual gas particles is [24, 25, 150]

+ 2 ngas 1 3 dˆn dˆn0 2 2 ⇤ = dp⌫(p)p sin (✓/2) F (pn,ˆ pnˆ0) (B.4) coll 3 m 2 4⇡ | | gas~ Z0 Z where ngas is the gas density, ✓ is the angle between the unitary vectors nˆ and nˆ0, which define the directions of motion of the gas molecule before and after the Appendix B. Gravitational time dilation 79

scattering (with incoming momentum p), F (pn,ˆ pnˆ0) is the scattering amplitude of the process and ⌫(p) describes the momentum distribution of the particles. For our low temperature analysis, we have to consider the Bose-Einstein distribu- tion instead of the Maxwell-Boltzmann distribution, which is usually used in the literature [24, 25, 150]. We have:

2 1 p2 ⌫(p)= 2 , (B.5) 3/2 p /(2mgaskBT ) ⇡ ⇠(3/2)(mgaskBT ) e 1 r + gas with 1 dp⌫(p)=1. In the limit we can use the geometric cross 0 x ⌧ dB section and Eq. (B.4) reduces to: R 2 ⇡r ngas 3 ⇤coll = 2 p ⌫ , (B.6) 3~ mgas h i where p3 is computed with respect to the distribution ⌫(p): h i⌫ + 1 2 ⇠(3) p3 = dp⌫(p)p3 =8 (m k T )3/2. (B.7) h i⌫ ⇡ ⇠(3/2) gas B Z0 r We then obtain the expression in Eq. (4.15) which, expressed in terms of the pres- sure P , becomes: 8p2⇡⇠(3) r2 ⇤coll = P mgaskBT. (B.8) 3⇠(3/2) ~2 p Using the same distribution ⌫(p), we derive the rate coll from [150]:

16⇡p2⇡ Pr2 coll = , (B.9) p3 p h i⌫ where ⇡p2⇡ p = m k T. (B.10) h i⌫ 3⇠(3/2) gas B Combining these two expressions, we obtainp (4.16). 80

Appendix C Collapse Models

C.1 CSL Diffusion coefficients

The CSL diffusion coefficients have been computed in [46, 47, 170]. Given the mass density µ(r), their expression is given by Eq. (5.12) and Eq. (5.15) and here are reported

3 CSLrC r2 k2 2 C ⌘ij = 3/2 2 dk e kikj µ˜(k) , ⇡ m0 | | 3 Z (C.1) CSLrC r2 k2 2 ⌘ = dk e C k @ µ˜(k) k @ µ˜(k) , T ⇡3/2m2 | y kz z ky | 0 Z where µ˜(k) is the Fourier transform of µ(r). Here ⌘T refers to rotations around the x axis. For a cylinder of length L and radius R we have [47, 170]

2 2 R2 L2 8M r 2 2 2 (cyl) C CSL R 2r Lp⇡ L 4r ⌘ = I1 2 e C erf( ) 1+e C , xx 2 2 2 2r 2rC 2rC L m0R C ! ⇣ ⌘ 2 2 L2 R2 (cyl, sym) 8M rC CSL 2 2 R2 R2 4rC 2rC ⌘xx = 2 2 2 1 e 1 e I0 2r2 +I1 2r2 , L m0R ! C C ! ⇣ ⇣ ⌘ ⇣ ⌘⌘ 4 2 L2 2 (cyl) 2CSLrC M 4r2 R 2Lp⇡ L ⌘ = 1 e C 8+ erf T 2 2 2 2 2rC L R m0 (" ! rC rC # ✓ ◆ ⇣ ⌘ (C.2) R2 L2 2 2 2 2 R Lp⇡ R 2r 4r L 2I0 2 e C 1 e C 4+ erf 2rC 2 2rC " ! rC rC # ⇣ ⌘ ✓ ◆ ⇣ ⌘ R2 L2 2 L2 2 2 2 2 L 2 3R 1 R 2r 4r 4r I1 2 e C 3 e C +2 1 e C 14 + 3 2rC 2 2 " ! rC ! rC ⇣ ⌘ ✓ ◆ Lp⇡ L2 L 24 + 2 erf , r 2rC 2rC C ⇣ ⌘ ⇣ ⌘ which are the diffusion coefficients for vibrations with respect to the x, symmetry axis and rotations around the x axis respectively [cf. Fig. 5.1]. Here In denotes the Appendix C. Collapse Models 81 n-th modified Bessel function. We also need the following coefficient

2 2 6 L (cube) 8CSL M rC 4r2 p⇡L L ⌘ = 1 e C erf T 2rC 3 m0 L 2rC ! ✓ ◆ ⇣ ⌘ ⇣ ⌘ L2 L2 2 L2 4r2 4r2 L 4r2 1 e C 2 3 e C +32 1 e C (C.3) ⇥ r ( !" ! ✓ C ◆ ! p⇡L L 2 L 2 24 + erf( L ) +3⇡ erf2( L ) , r r 2rC r 2rC C " ✓ C ◆ # # ✓ C ◆ ) which refers to rotations of a cube of side L.

C.2 Effective frequencies and damping constants

The effective frequencies !0,eff and !,eff and damping constants m,eff and D,eff in- troduced in Table 5.1 take the following form

2~2 ↵ 2(2 +2 !2) !2 = !2 | | , 0,eff 0 m (2 +( !)2)(2 +(+!)2) (C.4) 2~g2 ↵ 2(2 +2 !2) !2 = !2 | | , ,eff I (2 +( !)2)(2 +(+!)2) 4~2 ↵ 2 = + | | , m,eff m m (2 +( !)2)(2 +(+!)2) (C.5) 4~g2 ↵ 2 D = D + | | . ,eff (2 +( !)2)(2 +(+!)2)

The damping constants m and D can be expressed in terms of the parameters of the system [214]

P 2⇡m 3L = gas R2 1+ 1+ ⇡ , m m k T 2R 6 r B  2 3 (C.6) ⇡mgas 4 ⇡ L 1 L 1 L ⇡ D = P R 1+ + + + 1+ , 2kBT " 4 R 2 R 4 R 6 # r ✓ ◆ ✓ ◆ ⇣ ⌘ where P is the pressure of the surrounding gas of particles of mass mgas. For the vibrational motion along the symmetry axis the damping rate m must be substituted by the following expression [214]

P 8⇡m ⇡ L sym = gas R2 1+ + . (C.7) m m k T 4 2R r B ✓ ◆ This gives the green lines in Fig. 5.9 and Fig. 5.10. Appendix C. Collapse Models 82

C.3 Cantilever

The expression for the CSL-induced force noise spectral density can be obtained directly from the correlations of the CSL force on the system [50, 186]. In terms of the CSL parameters and the mass density distribution µ(r) of the system, the one-sided spectral density SFF reads:

2 SFF =2~ ⌘xx, (C.8) where ⌘xx is given in Eq. (5.12). Here, the factor 2 originates from the fact that in the experiment we use the one-sided definition of spectral density. For the cantilever microsphere system the integration can be carried out exactly as done in Ref. [50], as the geometry is the same. The contributions to ⌘xx are three: ⌘xx =

⌘xx,sphere + ⌘xx,cantilever + ⌘xx,mix, which are the one given by the attached sphere, the cantilever and the term coming from the interference between the two. Their explicit form is given by [50]

2 2 2 R2 2 3CSLrC M 2rC r2 2rC ⌘ = 1 + e C 1+ , xx,sphere R4m2 R2 R2 0 ✓ ◆! 2 2 4 2 R1 R2 32CSLrC M 2 2 p⇡R2 R 4rC 4rC 2 ⌘xx,cantilever = 1 e 1 e erf R2R2R2m2 2r 2rC 1 2 3 0 ! C ! (C.9) 2 ⇣ ⌘ R3 4r2 p⇡R3 R 1 e C erf 3 , 2rC ⇥ 2rC ! 3 ⇣ ⌘ 2CSLrC k2r2 2 ⌘ = dk [˜µ (k)˜µ⇤ (k)] e C k , xx,mix ⇡3/2m2 < sphere cantilever x 0 Z where R, R1, R2 and R3 are the radius of the sphere, the x, y and z length of the cantilever respectively, µ˜sphere(k) and µ˜cantilever(k) are the Fourier transform of the sphere and cantilever mass densities. Since ⌘xx,mix is small compared to the other contributions, it can be safely neglected; we refer to [50] for its explicit form. For a given measured residual force noise SFF,0 the exclusion plot in the CSL parameter space is obtained by comparing SFF,0 with Eq. (C.8).

Additional information on fitting the noise spectra

The power spectra of the SQUID output signal are experimentally obtained by av- eraging a number, typically nav =120, of power FFT periodograms of the SQUID signal. The sampling frequency was set to fs =100kHz and the length of each 20 dataset was 2 samples, corresponding to a frame period of tf =10.49 s and a frequency resolution of 95.36 mHz. Each averaged spectrum is fitted with a weighted nonlinear Levenberg - Mar- quardt procedure on the fixed interval 8100 - 8240 Hz, using Eq. (5.37) as template and setting the relative error bar of each bin as 1/pnav. As it is usually difficult to Appendix C. Collapse Models 83

Figure C.1: Figure taken from [51]. Normalized fit residuals for the three repre- sentative spectra shown in Fig. 5.6. From bottom to top, T = 43, 171 and 351 mK respectively. The residuals at 171 and 351 mK are shifted from 0 for a better visual- ization.

fit very narrow resonance peaks, we fix the parameters f0,f1 and Q to the values independently determined by ringdown and calibration measurements, leaving only A, B and C as free parameters. A and C are mainly determined by the SQUID noise, while B is mainly determined by the tails of the resonant peak. The fits are typically good, according to a standard 2 test. In Fig. C.1 we plot the residuals of the fits, normalized by the error bar, for the three representative spectra shown in Fig. 5.6. In general, no systematic discrepancy between the fitted function and the data is observed, except for a small reproducible imperfect fitting exactly around the resonance frequency of 8174.0 Hz. This feature can be explained as a data processing artifact due to spectral leakage. In fact, the width of the resonance peak f = f /Q 50 mHz is comparable to the spectrum 0 a ' resolution, so that there is a slight broadening, typically non Lorentzian, of the 3 - 5 points at the very top of the peak. This localized imperfection leads to a slight increase of the value of 2, however it does not significantly affect the fit results. In fact, with the parametrization of Eq. (5.37), the B parameter is determined by the whole tails of the resonant peak, while the top part is determined by the fixed Qa factor. In other words, by fitting with Eq. (5.37) we use the information on the cantilever noise available over the whole noise bandwidth (roughly 20 Hz). This would not be the case for a measurement strategy aiming at measuring the total integrated cantilever noise. In Fig. C.2 we plot the reduced 2 (with 1460 d.o.f) of all fits, obtained after exclu- sion of the 5 points around the resonance. The gray region represents the theo- retical two-sided 2 interval of the reduced 2 distribution. The experimental 2 is essentially in agreement with the theoretical distribution, with only one point Appendix C. Collapse Models 84

Figure C.2: Figure taken from [51]. Reduced 2 of the fits, compared with the theo- retical two-sided 2 interval (light gray region) of the reduced 2 distribution with the same number of degrees of freedom. slightly beyond the 2 limit.

Estimation of the intrinsic quality factor

The knowledge of the intrinsic quality factor Q is essential in order to evaluate the factor T/Qand therefore to predict the thermal force noise at a given temperature. Unfortunately, we have direct experimental access only to the apparent quality factor Qa, which is affected by the SQUID-induced magnetic spring. In general, we have 1/Qa =1/Q +1/QSQ, where 1/QSQ represents the SQUID effect. It is theoretically expected that, in the limit of large feedback gain: 1 c = , (C.10) Q G SQ | | where G is the open loop gain of the SQUID feedback electronics, and c is a cou- | | pling constant depending on the SQUID working point. This relation is easily derived by noting that the magnetic spring kSQ induced by the SQUID arises from the effective flux =xx applied to the SQUID by the cantilever motion, which in turns generates a current J circulating around the SQUID loop through the re- sponsivity J =dJ/d and a back-action force through the coupling FJ =dF/dJ.

In absence of flux feedback, kSQ can be written as : dF k = = F J = J 2, (C.11) SQ dx J x x where FJ =x because of reciprocity, and J is the only quantity depending on the SQUID working point. When feedback is applied, the effective flux applied to Appendix C. Collapse Models 85 the SQUID, and therefore the magnetic spring, is reduced by a factor 1/ [1 + G (!)] where G (!) is the open loop gain. In general G (!) is complex, so that the spring features both a real and an imaginary part, leading respectively to a frequency shift f and a dissipation 1/Q . Both components are proportional to 1/ G for SQ SQ | | large G and fixed argument. Thus, by varying the magnitude G it is possible | | | | to distinguish 1/Q , according to Eq. (C.10). In particular, for 1/ G 0 the SQ | |! magnetic spring vanishes, allowing the intrinsic quality factor Q to be estimated.

The apparent quality factor Qa is measured by using a standard ringdown method. Fig. C.3 shows the measurements of Q as function of 1/ G at different temper- a | | atures, corresponding to the data points of Fig. 5.7. We could not measure Qa at larger gain (lower 1/ G ) because of the onset of the principal feedback instabil- | | ity. All datasets are in very good agreement with the expected linear behaviour. Moreover, all slopes obtained from a linear fit are consistent within the error bar, as expected given that the SQUID working point was the same across all mea- surements.

Figure C.3: Figure taken from [51]. Inverse of the apparent quality factor Qa, es- timated by ringdown measurements, as function of the inverse of the open loop gain G of the SQUID feedback electronics. Each dataset refers to a different tem- | | perature: from top to bottom 43, 54, 68, 84, 102, 132, 171, 221, 281 and 351 mK. These temperatures correspond to the noise measurements shown in Fig. 5.7. The intercept of each linear fit provides an estimate of the intrinsic quality factor at the corresponding temperature.

Uncertainty budget on T and Q and possible systematic errors

The quality factor 1/Q at a given temperature is estimated as the intercept of a linear fit to the datasets in Fig. C.3. The error bar is obtained from the standard error on the fitting parameter. Because of the low number of points, the error bar is enlarged by a factor 1.32, corresponding to the 1 (i.e. 68% probability) confidence interval of a Student’s t-distribution with 2 degrees of freedom. Appendix C. Collapse Models 86

The measurement of T is based on a SQUID-based noise thermometer, which has been further calibrated against a superconducting reference point thermometer with accuracy better than 0.5% [215]. The noise thermometer is semiprimary (as it needs only one calibration point) and is simultaneously consistent with all ref- erence points in the range 21 mK - 1.1 K, so its effective accuracy could be even better, but we take as conservative accuracy the value 0.5% set by the reference point device. The x-error bars on T/Q in Fig. 5.7 (and in Fig. C.5 and C.7) are obtained by combining in quadrature the relative error on 1/Q with the estimated accuracy T/T =0.005 on the measurement of T . The uncertainty on T is practically neg- ligible with respect to the uncertainty on Q. As the uncertainties on the x axis and the y axis in Fig. 5.7 are both significant, the linear fit of the noise data is performed as a weighted orthogonal fit which takes simultaneously into account both uncertainties. The goodness of the fit is checked by means of a standard 2- test, which gives a regular value 2 =9.27 with 8 d.o.f. This is a good indication that the error bars are correctly estimated. Concerning the estimation of 1/Q described above, any possible systematic error on the measurement procedure based on varying the open loop gain would be the same for any dataset, as all ringdown measurements of Qa are performed in the same way at the same settings of the SQUID electronics. One may ask whether a common unknown constant bias on 1/Q would be potentially able to explain the nonzero intercept in the noise data of Fig. 5.7. To check this possibility, we have manually added a constant additional offset 1/Q0 to the data, and repeated the whole data analysis. It turns out that it is indeed possible to reduce the intercept 7 to 0 for 1/Q 1.1 10 , which is roughly 10 times larger than the average 0 ' ⇥ error bar on 1/Q. However, for such a choice, the data deviate significantly from the linear behaviour, as we obtain 2 =26with a relative probability of 0.001. 2 Furthermore, we find that by varying 1/Q0 the is actually minimized for a 9 much lower offset 1/Q = 7 10 , which is consistent with the error bar. In 0 ⇥ other words, under the assumptions that the data follow a linear relation, the likelihood of the observed data given an arbitrary 1/Q0 is essentially maximized by the choice 1/Q 0 (no systematic error). This is a further indication that the 0 ' accuracy of the estimation of 1/Q is well within the measurement error bar.

Measurements with higher coupling

High coupling measurements were performed in a separate cooldown. The rel- ative position of the cantilever with respect to the SQUID was carefully changed under the microscope and the system reassembled without other modifications. The measurements were performed in similar way to the main run, but with a lower number of temperature points. Because of higher coupling the effective bandwidth of the cantilever noise was larger, leading in turn to a smaller error bar on the fitting parameters. Fig. C.4 shows three representative spectra. The data are again fitted by Eq. (5.37). The 2 is well within the 2 interval of the Appendix C. Collapse Models 87 theoretical distribution for all spectra. Fig. C.5 shows the B parameter extracted

Figure C.4: Figure taken from [51]. Three representative spectra of the noise ac- quired in the high coupling run. The best fits with Eq. (5.37) are also shown. from the fits as a function of T/Q. An orthogonal linear fit of the data leads to 19 2 19 2 B =(4.3 0.4) 10 /Hz and B =(0.872 0.007) 10 / (nK Hz). The 0 ± ⇥ 0 1 ± ⇥ 0 · reduced 2 with 4 degrees of freedom is 0.21, which falls within the 2 interval of the theoretical distribution. The coupling factor and the residual force noise inferred from B0 and B1 are reported in the second row of Table 5.2.

Measurements with pulse tube on

During the same cooldown of the main measurements, we have performed ad- ditional measurements without switching off the pulse tube cryocooler. Under normal operation, the high pressure pulses at frequency 1.5 Hz generated by ⇠ the pulse tube compressor are by far the strongest source of vibrational noise in our cryostat. Typically, we observe two different effects. On the one hand there is a direct generation of vibrational noise at the mixing chamber plate level, extending up to the 10 kHz region, which can be directly measured by standard accelerometers. 5 At the cantilever frequency the acceleration noise is less than 10 g/pHz and our suspension system provides a factor 104 attenuation. As the effective mass of 10 ⇠ our cantilever is 10 Kg, this translates into a force noise 1 aN/pHz, thus ⇠ ⇠ comparable or lower than our residual measured force noise. The noise with pulse tube off is at least a factor of 10 better (a factor 100 in power). On the other hand, very high vibrational noise levels are sometimes observed at the cantilever frequency due to nonlinear upconversion of low frequency noise. Upconversion is a poorly understood and rather unpredictable effect. It is highly Appendix C. Collapse Models 88

Figure C.5: Figure taken from [51]. B parameter as a function of T/Q for the high coupling dataset. The lowest points are zoomed in the inset. A linear fit is fully consistent with the data, yielding a finite intercept. nonstationary and threshold-like, with the noise at the resonator frequency which can vary by orders of magnitude, depending on the magnitude of the low fre- quency motion. We have evidence that upconversion is related either with soft thermal links or with the SQUID braided cable. We have been able to strongly re- duce upconversion noise by implementing a passive magnetic damper in the sus- pension system, to reduce the low frequency motion, and by a proper clamping of the SQUID wiring. In particular, in the cooldown here considered, nonlinear upconversion was essentially absent even with the pulse tube on. Fig. C.6 shows three representative spectra acquired with the pulse tube on, with the same acquisition settings of the main measurements run. Several broad bumps are apparent on the right tail of the resonator peak, while the left tail is rather clean. A global fit yields a 2 unacceptably high, which confirms the presence of coloured vibrational noise. However, we obtain acceptable 2 by excluding a wide portion of the right tail from the fit, as shown in Fig. C.6. By applying the standard data analysis we obtain again a good linear behaviour of B as a function 19 of T/Q, as shown in Fig. C.7. The slope of the linear fit B1 =(0.286 0.003) 10 2 ± ⇥ 0/Hz nK is consistent with the one at pulse tube off, while the intercept B0 = · 19 2 (1.71 0.13) 10 /Hz leads to a larger residual force noise [cf. Table 5.2]. ± ⇥ 0

Measurements with pump off

Under pulse tube off operation, the stronger source of vibrational and acoustic noise is the roots mechanical pump which is employed to circulate the 3He-4He mixture in the dilution refrigerator. We have tried to investigate whether the Appendix C. Collapse Models 89

Figure C.6: Figure taken from [51]. Three representative spectra of the noise ac- quired with pulse tube on. Several bumps are apparent on the right tail. The best fit with Eq. (5.37) are also shown. The fit is restricted to f<8178 Hz and the parame- ters A and C are fixed to the values obtained with pulse tube off.

Figure C.7: Figure taken from [51]. B parameter as a function of T/Q for the mea- surements with pulse tube on. The lowest points are zoomed in the inset. A linear fit is again consistent with the data, yielding a finite intercept. pump noise can be related to the observed cantilever excess noise. The measure- ment was performed during a separate cooldown with the high coupling setting at the lowest temperature T =43mK. Unfortunately, it is not possible to maintain a stable temperature for long time after switching off the circulation pump. The cooling power drops to zero very quickly and the temperature starts drifting after a time of the order of 1 minute. Appendix C. Collapse Models 90

In contrast, we can easily operate the dilution refrigerator with pulse tube off up to half an hour while keeping the temperature of the mixing chamber actively stabilized. In order to collect significant statistics while keeping a stable temperature, we switch off the circulation pump for short periods (about 40 seconds), barely suf- ficient to wait for the low frequency suspension modes to relax and acquire one single data frame. Subsequently we switch the pump on, wait several minutes for the circulation to stabilize and then repeat the procedure. We collected a to- tal of 12 acquisitions. The averaged spectrum is then compared with a spectrum with circulation pump on. For a fair comparison, the spectrum with pump on is acquired with the same setting and the same number of averages.

Figure C.8: Figure taken from [51]. Spectra acquired with circulation pump on (red line) and off (dark green line). The pulse tube is off. Both measurements are performed under the same conditions, with the same number of averages. The best fit with Eq. (5.37) is also shown. No significant difference in the fitting parameters is observed.

The two spectra are shown in Fig. C.8, and can be hardly distinguished. The best fitting curves are also shown and are essentially coincident. The B parameters 18 2 resulting from the fits are B =(1.26 0.04) 10 0/Hz and B =(1.29 0.04) 18 2 ± ⇥ ± ⇥ 10 0/Hz for the pump on and pump off case respectively. Therefore, there is apparently no significant effect of the circulation pump on the excess noise, which at this temperature contributes to about 30% of B.

Magnetic effects

As the ferromagnetic microsphere is magnetized, an external magnetic field noise could be also held responsible for anomalous force driving the cantilever. Let us assume an environmental magnetic noise Bn with direction along the cantilever length and negligible spatial dependence over the magnetic sphere volume. Bn Appendix C. Collapse Models 91

9 would generate a torque µB , where µ 5 10 J/T is the microsphere mag- n ' ⇥ netic moment, which translates into an effective force noise µBn/l, where l is the effective length of the cantilever. Under these assumptions, the observed excess force noise would result from a magnetic field noise Bn with spectral density 13 1 10 T/pHz. Such noise is typical of an unshielded environment at kHz fre- ⇥ quency, but is unrealistically large for a shielded environment. The walls of the copper box hosting the cantilevers are about 20 times thicker than the penetration depth at the cantilever frequency, thus providing an attenuation of external mag- netic fields by many orders of magnitude. Thermal magnetic noise from eddy currents in the walls or other elements inside the box is estimated to be largely negligible. A related but distinct mechanism is given by fluctuations of the microsphere mag- netization. The magnetic microsphere is at finite temperature, so there will be magnetization fluctuations, which will couple to the static magnetic field yielding a finite torque and force noise. Magnetization fluctuations for a fully magnetized hard ferromagnet are expected to be very small, due to the very high anisotropy field. Experiments with rare-earth micromagnets have actually shown that at kHz frequency a larger effect is due to conductive eddy currents [216]. Along with the approach of [216] we estimate both effects to be many orders smaller than what is needed to explain the observed force noise. For instance, the eddy current dissipation in the microsphere can be calculated analytically and is 6 or- ders of magnitude smaller than the cantilever mechanical dissipation. However, we can also provide an experimental argument to rule out this mech- anism. Magnetization fluctuations would behave as thermal force noise and the same mechanism would also appear as mechanical dissipation. In particular, both noise and dissipation would scale with the square of the external magnetic field. In a separate test we have have increased the external magnetic field by a factor 4 with respect to the earth field. We did not see any significant change of the quality factor, confirming that thermal magnetization fluctuations are likely not significant in our experiment. 92

Bibliography

[1] M. Planck, The Genesis and Present State of Development of the Quantum Theory, Nobel Lectures, Physics 1901-1921 (Elsevier Publishing Company, 1967). [2] G. Kirchhoff, Annal. der Phys. 185, 275 (1860). [3] B. Stewart, Transactions of the Royal Society of Edinburgh 22, 1 (1861). [4] M. Planck, Annal. der Phys. 309, 553 (1901). [5] L. De Broglie, Recherches sur la théorie des Quanta, Ph.D. thesis, Migration - université en cours d’affectation (1924). [6] C. Jönsson, Z. Phys. 161, 454 (1961). [7] H. Hertz, Annal. der Phys. 267, 983 (1887). [8] A. Einstein, Annal. der Phys. 322, 132 (1905). [9] R. A. Millikan, Phys. Rev. 7, 355 (1916). [10] I. Estermann and O. Stern, Z. Phys. 61, 95 (1930). [11] O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991). [12] S. N. Bose, Z. Phys. 26, 178 (1924). [13] M. H. Anderson et al., Science 269, 198 (1995). [14] A. Pomyalov and D. J. Tannor, J. Chem. Phys. 123, 204111 (2005). [15] T. Guérin, O. Bénichou, and R. Voituriez, Nat. Chem. 4, 568 (2012). [16] X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, Phys. Rev. A 83, 053823 (2011). [17] B. D. Fainberg, M. Sukharev, T.-H. Park, and M. Galperin, Phys. Rev. B 83, 205425 (2011). [18] M. Thorwart et al., Chem. Phys. Lett 478, 234 (2009). [19] X.-T. Liang, Phys. Rev. E 82, 051918 (2010). [20] P. Nalbach, J. Eckel, and M. Thorwart, New J. Phys. 12, 065043 (2010). [21] G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, Nat. Chem. 3, 763 (2011). [22] M. Cho et al., J. Phys. Chem. B 109, 10542 (2005). [23] J. Piilo, K. Härkönen, S. Maniscalco, and K.-A. Suominen, Phys. Rev. A 79, 062112 (2009). BIBLIOGRAPHY 93

[24] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Ox- ford University Press, Oxford, 2002). [25] E. Joos et al., Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. (Springer-Verlag Berlin Heidelberg, 2003). [26] A. Rivas, S. Huelga, and M. Plenio, Rep. Prog. Phys. 77, 094001 (2014). [27] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016). [28] I. de Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017). [29] A. O. Caldeira and A. J. Leggett, Phys. A 121, 587 (1983). [30] E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985). [31] G. S. Engel et al., Nature 446, 782 (2007). [32] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986). [33] L. Diósi, J. Phys. A 21, 2885 (1988). [34] G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78 (1990). [35] G. C. Ghirardi, R. Grassi, and P. Pearle, Found. Phys. 20, 1271 (1990). [36] G. C. Ghirardi, R. Grassi, and F. Benatti, Found. Phys. 25, 5 (1995). [37] A. Bassi and G. C. Ghirardi, Phys. Rep. 379, 257 (2003). [38] S. L. Adler, J. Phys. A 40, 2935 (2007). [39] S. L. Adler, J. Phys. A 40, 13501 (2007). [40] S. Eibenberger et al., Phys. Chem. Chem. Phys. 15, 14696 (2013). [41] K. Hornberger, J. E. Sipe, and M. Arndt, Phys. Rev. A 70, 053608 (2004). [42] M. Torošand A. Bassi, arXiv (2016), 1601.03672 . [43] M. Torošand A. Bassi, arXiv (2016), 1601.02931 . [44] K. C. Lee et al., Science 334, 1253 (2011). [45] S. Belli et al., Phys. Rev. A 94, 012108 (2016). [46] M. Bahrami et al., Phys. Rev. Lett. 112, 210404 (2014). [47] S. Nimmrichter, K. Hornberger, and K. Hammerer, Phys. Rev. Lett. 113, 020405 (2014). [48] L. Diósi, Phys. Rev. Lett. 114, 050403 (2015). [49] C. Curceanu, B. C. Hiesmayr, and K. Piscicchia, J. Adv. Phys. 4, 263 (2015). [50] A. Vinante et al., Phys. Rev. Lett. 116, 090402 (2016). [51] A. Vinante et al., Phys. Rev. Lett. 119, 110401 (2017). BIBLIOGRAPHY 94

[52] M. Carlesso, A. Bassi, P. Falferi, and A. Vinante, Phys. Rev. D 94, 124036 (2016). [53] B. Helou, B. J. J. Slagmolen, D. E. McClelland, and Y. Chen, Phys. Rev. D 95, 084054 (2017). [54] F. Haake and R. Reibold, Phys. Rev. A 32, 2462 (1985). [55] B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992). [56] I. Pikovski, M. Zych, F. Costa, and C. Brukner, Nat. Phys. 11, 668 (2015). [57] S. L. Adler and A. Bassi, J. Phys. A 40, 15083 (2007). [58] M. Carlesso et al., ArXiv (2017), 1708.04812 . [59] B. Vacchini and G. Amato, Sci. Rep. 6, 37328 (2016). [60] G. Lindblad, Commun. Math. Phys. 48, 119 (1976). [61] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976). [62] E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963). [63] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38 (1990). [64] L. Diósi, EPL 30, 63 (1995). [65] B. Vacchini, Phys. Rev. Lett. 84, 1374 (2000). [66] K. Hornberger and B. Vacchini, Phys. Rev. A 77, 022112 (2008). [67] G. Gasbarri, Collisional Models for a Quantum Particle in a Gas, Ph.D. thesis, University of Trieste (2017). [68] M. A. Schlosshauer, Decoherence and the Quantum-To-Classical Transition, 1st ed. (Springer-Verlag Berlin Heidelberg, 2007). [69] H. Naeij, M. Carlesso, and A. Bassi, In preparation . [70] Wolfram Alpha, http://reference.wolfram.com/language/ref/EulerAngles.html. [71] T. Fischer, C. Gneiting, and K. Hornberger, New J. Phys. 15, 063004 (2013). [72] C. Zhong and F. Robicheaux, Phys. Rev. A 94, 052109 (2016). [73] J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Addison-Wesley, 2011). [74] N. Zettili, Quantum Mechanics: Concepts and Applications (Wiley, 2009). [75] M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 153603 (2007). [76] M. Bhattacharya, P.-L. Giscard, and P. Meystre, Phys. Rev. A 77, 030303 (2008). [77] O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, New J. Phys. 12, 033015 (2010). BIBLIOGRAPHY 95

[78] J. Trojek, L. Chvátal, and P. Zemánek, J. Opt. Soc. Am. A 29, 1224 (2012). [79] M. Bhattacharya, J. Opt. Soc. Am. B 32, B55 (2015). [80] B. A. Stickler et al., Phys. Rev. A 94, 033818 (2016). [81] B. A. Stickler, B. Papendell, and K. Hornberger, Phys.Rev. A 94, 033828 (2016). [82] H. Shi and M. Bhattacharya, J. Phys. B 49, 153001 (2016). [83] M. Carlesso, M. Paternostro, H. Ulbricht, and A. Bassi, ArXiv (2017), 1710.08695 . [84] L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001), http://science.sciencemag.org/content/292/5518/912.full.pdf . [85] K. D. Bonin, B. Kourmanov, and T. G. Walker, Opt. Express 10, 984 (2002). [86] W. A. Shelton, K. D. Bonin, and T. G. Walker, Phys. Rev. E 71, 036204 (2005). [87] P. H. Jones et al., ACS Nano 3, 3077 (2009). [88] L. Tong, V. D. Miljkovi´c, and M. Käll, Nano Lett. 10, 268 (2010). [89] Y. Arita, M. Mazilu, and K. Dholakia, 4, 2374 (2013). [90] S. Kuhn et al., Nano Lett. 15, 5604 (2015). [91] T. M. Hoang et al., Phys. Rev. Lett. 117, 123604 (2016). [92] S. Kuhn et al., Optica 4, 356 (2017). [93] R. Brown, Philosophical Magazine Series 2 4, 161 (1828). [94] A. Einstein, Investigations on the Theory of the Brownian Movement (Courier Corporation, 1956). [95] P. Langevin, C. R. Acad. Sci. 146, 530 (1908). [96] G. W. Ford, M. Kac, and P. Mazur, Journal of Mathematical Physics 6, 504 (1965). [97] P. Ullersma, Physica (Utrecht) 1, 27 (1966). [98] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [99] H. B. Callen and R. F. Greene, Phys. Rev. 86, 702 (1952). [100] R. Kubo, Rep. Prog. Phys. 29, 255 (1966). [101] N. Pottier and A. Mauger, Physica A 291 (2001). [102] Y. Yan and R. Xu, Ann. Rev. Phys. Chem. 56, 187 (2005). [103] W. Thirring, Quantum Mathematical Physics (Springer-Verlag Berlin Heidel- berg, 2002). BIBLIOGRAPHY 96

[104] F. Strocchi, Symmetry Breaking, 2nd ed. (Springer, Berlin, 2007). [105] W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071 (1989). [106] F. Petruccione and B. Vacchini, Phys. Rev. E 71, 046134 (2005). [107] C. H. Fleming, A. Roura, and B. L. Hu, Annal. Phys. 326, 1207 (2011). [108] R. P. Feynman and F. L. J. Vernon, Ann. Phys. (NY) 24, 118 (1963). [109] R. P. Feynman and A. R. Hibbs, Quantum Physics and Path Integrals (McGraw-Hill, 1965). [110] S. Groblacher et al., Nat. Commun. 6 (2015). [111] M. Frimmer, J. Gieseler, and L. Novotny, Phys. Rev. Lett. 117, 163601 (2016). [112] V. Ambegaokar, Ber. Bunsenges. Phys. Chem. 95, 400 (1991). [113] L. Diósi, EPL 22, 1 (1993). [114] F. Haake and M. Lewenstein, Phys. Rev. A 28, 3606 (1983). [115] U. Geigenmüller, U. Titulaer, and B. Felderhof, Phys. A 119, 41 (1983). [116] G. W. Ford and R. F. O’Connell, Phys. Rev. D 64, 105020 (2001). [117] S. Ghose and B. C. Sanders, J. Mod. Opt. 54, 855 (2007). [118] E. S. Gómez et al., Phys. Rev. A 91, 013801 (2015). [119] K. j. Huang et al., Phys. Rev. A 93, 033832 (2016). [120] M. Carlesso and A. Bassi, Phys. Rev. A 95, 052119 (2017). [121] J. von Neumann, Math. Ann. 104, 570 (1931). [122] B. J. Hiley, J. Comput. Electron. 14, 869 (2015). [123] L. Ferialdi, Phys. Rev. Lett. 116, 120402 (2016). [124] F. Benatti, Dynamics, Information and Complexity in Quantum Systems (Springer Netherlands, 2009). [125] B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys. 2, 161 (1977). [126] T. Heinosaari, A. S. Holevo, and M. M. Wolf, Quantum Inf. Comp. 10, 0619 (2010). [127] J. Halliwell and T.Yu, Phys.Rev. D D53, 2012 (1996), quant-ph/9508004 . [128] J. J. Halliwell, J. Phys. A 40, 3067 (2007). [129] M. Wendling et al., J. Phys. Chem. B 104, 5825 (2000). [130] F. Fassioli, A. Olaya-Castro, and G. D. Scholes, J. Phys. Chem. Lett. 3, 3136 (2012). [131] M. Carlesso and A. Bassi, Phys. Lett. A 380, 2354 (2016). BIBLIOGRAPHY 97

[132] M. Zych, F. Costa, I. Pikovski, and C. Brukner, Nat. Commun. 2, 505 (2011). [133] A. Bassi, Nat. Phys. 11, 626 (2015). [134] C. Gooding and W. G. Unruh, Found. Phys. 45, 1166 (2015). [135] H. D. Zeh, arXiv (2015), 1510.02239 . [136] I. Pikovski, M. Zych, F. Costa, and C.ˇ Brukner, arXiv (2015), 1509.07767 . [137] I. Pikovski, M. Zych, F. Costa, and C. Brukner, arXiv (2015), 1508.03296 . [138] L. Diósi, arXiv (2015), 1507.05828 . [139] Y. Margalit et al., Science 349, 1205 (2015). [140] Y. Bonder, E. Okon, and D. Sudarsky, Phys. Rev. D 92, 124050 (2015). [141] S. L. Adler and A. Bassi, Phys. Lett. A 380, 390 (2016). [142] B. H. Pang, Y. Chen, and F. Y. Khalili, Phys. Rev. Lett. 117, 090401 (2016). [143] I. Pikovski, M. Zych, F. Costa, and C. Brukner, Nat. Phys. 12, 2 (2016). [144] Y. Bonder, E. Okon, and D. Sudarsky, Nat. Phys. 12, 2 (2016). [145] A. Einstein, Annal. der Phys. 340, 898 (1911). [146] H. J. W. Müller-Kirsten, Basics of Statistical Physics (World Scientific Publish- ing Company, 2013). [147] L. K. Nash, Elements of Statistical Thermodynamics, 2nd ed. (Addison-Wesley, 1972). [148] F. Mandl, Statistical Physics (Wiley Library, 1988). [149] I. Pikovski, Private communications (2016). [150] O. Romero-Isart, Phys. Rev. A 84, 052121 (2011). [151] I. Pikovski, Macroscopic quantum systems and gravitational phenomena, Ph.D. thesis, University of Vienna (2014). [152] T. Kovachy et al., Nature 528, 530 (2015). [153] M. Arndt et al., Nature 401, 680 (1999). [154] H. Pino, J. Prat-Camps, K. Sinha, B. P. Venkatesh, and O. Romero-Isart, arXiv (2016), 1603.01553 . [155] R. Schnabel, Phys. Rev. A 92, 012126 (2015). [156] V. B. Braginsky, V. S. Ilchenko, and K. S. Bagdassarov, Phys. Lett. A 120, 300 (1987). [157] J. W. Lamb, Int. J. Infr. Mill. W. 17, 1997 (1996). [158] G. Gabrielse et al., Phys. Rev. Lett. 65, 1317 (1990). BIBLIOGRAPHY 98

[159] J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, Phys. Rev. Lett. 109, 103603 (2012). [160] J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, Nat. Com- mun. 5 (2014). [161] N. Gisin, Hel. Phys. Acta 62, 363 (1989). [162] A. Bassi, D. Dürr, and G. Hinrichs, Phys. Rev. Lett. 111, 210401 (2013). [163] A. Bassi et al., Rev. Mod. Phys. 85, 471 (2013). [164] Q. Fu, Phys. Rev. A 56, 1806 (1997). [165] G. C. Ghirardi, A. Rimini, and T. Weber, Quantum Probability and Appli- cations II, edited by L. Accardi and W. von Wandelfels, Lecture Notes in Mathematics, Vol. 1136 (Springer, 1985). [166] L. Diósi, Phys. Rev. A 40, 1165 (1989). [167] L. Diósi, Phys. Rev. A 42, 5086 (1990). [168] B. Collett and P. Pearle, Found. Phys. 33, 1495 (2003). [169] S. Bera, B. Motwani, T. P. Singh, and H. Ulbricht, Sci. Rep. 5, 7664 EP (2015). [170] B. Schrinski, B. A. Stickler, and K. Hornberger, J. Opt. Soc. Am. B 34, C1 (2017). [171] S. L. Adler, A. Bassi, and E. Ippoliti, J. Phys. A 38, 2715 (2005). [172] D. Goldwater, M. Paternostro, and P. F. Barker, Phys. Rev. A 94, 010104 (2016). [173] J. B. Clark et al., Nature 541, 191 (2017). [174] T. J. Kippenberg and K. J. Vahala, Opt. Express 15, 17172 (2007). [175] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). [176] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014). [177] S. Mancini and P. Tombesi, Phys. Rev. A 49, 4055 (1994). [178] D. Walls and G. Milburn, Quantum optics (Springer Science & Business Me- dia, 2007). [179] C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag Berlin Heidel- berg, 2004). [180] C. Genes et al., Phys. Rev. A 77, 033804 (2008). [181] J. Aasi et al. (LIGO Scientific Collaboration), Class. Quantum Grav. 32, 074001 (2015). [182] F. Acernese et al., Class. Quantum Grav. 32, 024001 (2015). BIBLIOGRAPHY 99

[183] M. Paternostro et al., New J. Phys. 8, 107 (2006). [184] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 131103 (2016). [185] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016). [186] M. Armano et al., Phys. Rev. Lett. 116, 231101 (2016). [187] W. J. Weber, Private communications (2016). [188] A. Vinante et al. (AURIGA Collaboration), Class. Quantum Grav. 23, S103 (2006). [189] L. Baggio et al., Phys. Rev. Lett. 94, 241101 (2005). [190] M. P. McHugh et al., Class. Quantum Grav. 22, S965 (2005). [191] P. Astone et al., Astrop. Phys 7, 231 (1997). [192] J. P. Zendri et al., in AIP conference proceedings (IOP, 2000) pp. 421–422. [193] M. Bilardello, S. Donadi, A. Vinante, and A. Bassi, Physica A 462, 764 (2016). [194] F. Laloë, W. J. Mullin, and P. Pearle, Phys. Rev. A 90, 052119 (2014). [195] J. Ellis et al., Nuc. Phys. B 241, 381 (1984). [196] J. Ellis, S. Mohanty, and D. V. Nanopoulos, Phys. Lett. B 221, 113 (1989). [197] J. Mináˇr, P. Sekatski, and N. Sangouard, Phys. Rev. A 94, 062111 (2016). [198] O. Usenko, A. Vinante, G. Wijts, and T. H. Oosterkamp, App. Phys. Lett. 98, 133105 (2011). [199] C. Hilbert and J. Clarke, J. Low Temp. Phys. 61, 263 (1985). [200] M. Poot and H. S. J. van der Zant, Phys. Rep. 511, 273 (2012). [201] P. Falferi et al., App. Phys. Lett. 93, 172506 (2008). [202] A. Vinante and P. Falferi, Phys. Rev. Lett. 111, 207203 (2013). [203] A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008). [204] F. C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B 49, 5942 (1994). [205] C. D. Tesche and J. Clarket, J. Low Temp. Phys. 37, 397 (1979). [206] A. Großardt, J. Bateman, H. Ulbricht, and A. Bassi, Phys. Rev. D 93, 096003 (2016). [207] J. Vovrosh et al., J. Opt. Soc. Am. B 34, 1421 (2017). [208] S. Kuhn et al., arXiv (2017), 1702.07565 . BIBLIOGRAPHY 100

[209] A. Vinante et al., Nat. Commun. 2, 572 EP (2011). [210] J. Prat-Camps et al., arXiv (2017), arXiv:1703.00221 . [211] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Library, 2008). [212] S. Frauendorf, Z. Phys. D 35, 191 (1995). [213] K. Hansen and E. E. B. Campbell, Phys. Rev. E 58, 5477 (1998). [214] A. Cavalleri et al., Phys. Lett. A 374, 3365 (2010). [215] “Hdl1000 measurement system,” http://hdleiden.home.xs4all.nl/srd1000. [216] B. C. Stipe et al., Phys. Rev. Lett. 87, 277602 (2001).