Nonlinear quantum effects and squeezing in cavity optomechanics Nichtlineare Quanteneffekte und gequetschte Zustände in der kavitätsbasierten Optomechanik Andreas Kronwald Nonlinear quantum effects and squeezing in cavity optomechanics Nichtlineare Quanteneffekte und gequetschte Zustände in der kavitätsbasierten Optomechanik Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Andreas Kronwald aus Dülmen Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 17.04.2015 Vorsitzende/r des Promotionsorgans: Prof. Dr. Jörn Wilms Gutachter/in: Prof. Dr. Florian Marquardt Gutachter/in: Prof. Dr. Oleg Pankratov CONTENTS 1 Introduction 1 1.1 Quantum effects......................................... 1 1.2 Squeezing............................................. 6 1.3 Cavity optomechanics ..................................... 11 1.4 Outline of this thesis ...................................... 35 2 Mechanical squeezing 39 2.1 Brief overview of mechanical squeezing........................... 39 2.2 Arbitrarily large steady-state bosonic squeezing via dissipation............. 43 2.3 Supplementary material: Generating squeezed states with a parametric amplifier . 59 3 Optical Squeezing 61 3.1 Brief overview of optical squeezing.............................. 61 3.2 Dissipative optomechanical squeezing of light....................... 67 3.3 Supplementary material: Generation of squeezed output light with nonlinear media 83 4 First signatures of the nonlinear quantum regime of optomechanics 85 4.1 Brief introduction........................................ 85 4.2 Optomechanically Induced Transparency in the Nonlinear Quantum Regime . 89 5 Photon statistics in optomechanical systems 97 5.1 Brief introduction to photon statistics............................ 97 5.2 Full photon statistics of a light beam transmitted through an optomechanical system 103 5.3 Supplementary material ....................................115 6 Measurements of x2 and their applications in optomechanics 123 6.1 Brief overview of position squared measurements.....................123 6.2 Challenges for optomechanical measurements of x2 and the phonon number . 127 7 Conclusion 153 Bibliography 157 v Abstract In this thesis we investigate nonlinear quantum effects and squeezing in cavity optomechanical systems, where light interacts with mechanical motion. In the first part of this thesis we analyze how to generate squeezed mechanical states and squeezed output light with state-of-the-art optomechanical setups via dissipation. We predict that arbitrary large steady-state bosonic squeezing can be generated. Furthermore, we show that our dissipative output light squeezing scheme can be used directly to enhance the intrinsic measure- ment sensitivity of an optomechanical cavity. In the second part, we explore the so-called “single-photon strong coupling regime” of optome- chanics. In this regime, the nonlinear quantum nature of the optomechanical interaction becomes important. We work out the first signatures of this nonlinear quantum interaction. We also propose how to observe these signatures with near-future optomechanical experiments. In the following, we analyze how an even stronger quantum interaction between photons and phonons modifies the statistics of photons which are transmitted through an optomechanical system. In the last part of this thesis, we discuss how to verify energy quantization of a mechanical de- gree of freedom. We propose to make use of an optomechanical setup where the position squared of a mechanical degree of freedom is coupled to the light field. We predict that energy quantization could be observable e.g. with nanometer-sized dielectric spheres. vii Zusammenfassung In dieser Arbeit werden nichtlineare Quanteneffekte und gequetschte Zustände in der kavitätsba- sierten Optomechanik studiert. In diesem Feld wird die Wechselwirkung zwischen Licht und me- chanischer Bewegung analysiert. Im ersten Teil der Arbeit wird untersucht, wie gequetschte mechanische Zustände und ge- quetschtes Licht mit heute verfügbaren optomechanischen Experimenten generiert werden kön- nen. Dabei nutzt die vorgeschlagene Methode Dissipation explizit aus. Es wird vorhergesagt, dass beliebig stark gequetschte, stationäre mechanische Zustände erzeugt werden können. Des Weite- ren wird gezeigt, dass die dissipative Methode zur Erzeugung von gequetschtem Licht, welches aus der Kavität austritt, direkt genutzt werden kann, um die intrinsische Messsensitivität der optome- chanischen Kavität zu erhöhen. Im zweiten Teil der Arbeit wird das sogenannte “single-photon strong coupling regime” der Op- tomechanik betreten, in dem die nichtlineare Wechselwirkung zwischen Licht und Mechanik auf der Quanten-Ebene wichtig wird. Zunächst werden erste Anzeichen dieser nichtlinearen Quan- ten-Wechselwirkung herausgearbeitet und es wird vorgeschlagen, wie diese Anzeichen mit Experi- menten beobachtet werden könnten, die in naher Zukunft entwickelt werden. Anschließend wird analysiert, welchen Einfluss eine sehr starke Wechselwirkung zwischen einzelnen Photonen und Phononen auf die Statistik von Photonen hat, die durch das optomechanische System transmit- tiert wurden. Im letzten Teil der Arbeit wird diskutiert, wie man die Quantisierung der Energie eines me- chanischen Freiheitsgrades messen kann. Dabei wird die quadratische Position des mechanischen Freiheitsgrades an das Lichtfeld gekoppelt. Es wird vorhergesagt, dass Energiequantisierung mit dielektrischen Kügelchen beobachtet werden könnte, die einen Durchmesser in der Größenord- nung einiger Nanometer haben. viii HAPTER C 1 INTRODUCTION 1.1 Quantum effects Have you ever seen a plane being at two places at the same time, i.e., being in a spatial quantum superposition? Interestingly, such a quantum superposition state could in principle exist accord- ing to the laws of quantum mechanics, cf. Fig. 1.1(a). Initially a plane is localized at airport A. As quantum dynamics evolves, a quantum superposition state could be generated. If the plane was able to take off from airport A, cruising for some time T and landing safely at airport B, the plane would be localized at airport B after time T . Alternatively, it could have happened that the plane did not take off from airport A due to some technical failure. In that case, the plane would still be located at airport A after time T . In the quantum world, without further measurements, this sce- nario would correspond to a superposition of the plane being at airport A and B simultaneously. Only after a measurement of the plane’s position (or the plane’s technical status) the plane’s wave- function collapses and the plane’s position would be revealed. Then, the plane is indeed localized at a single airport only. Of course, no one has ever observed a plane being in such a “strange” quantum superposition state, even though from a quantum-mechanical perspective such a superposition state could exist. The mechanism preventing the observation of such effects is called decoherence (see e.g. [1,2]). In our plane-example mentioned above, the mechanism which destroys quantum coherence can be understood as follows: Let us assume a human being observes the plane at airport B after landing. This observation rules out the possibility that the plane is still located at airport A. This in turn “destroys” the quantum superposition state. Note that even if no human being observes the plane, one would always find that the environment (e.g. air molecules at airport B etc.) will observe the plane’s landing (or it’s absence), and, hence, localizes the plane’s wavefunction. Thus, in physical terms, any spatial quantum superposition state of a plane being at airport A and B simultaneously will decohere due to a (possibly unwanted) continuous measurement of the plane’s position. This decoherence in turn transfers the wavefunction describing the spatial quantum superposition state to a wavefunction which describes the plane being localized at airport B (or A, if it did not take off) only. Thus, the only way to generate such a quantum superposition state of a plane would be to make the impact of decoherence smaller. However, as we all know, this cannot be done as otherwise planes would not behave as classical as they do. 1 1. Introduction (a) airport A airport B (b) Figure 1.1: (a) Initially, a plane is localized at airport A. In the next step, the plane is prepared in a spatial quantum superposition of being at airport A and airport B. Measuring the position of the plane with outcome “Airport B” (or, equivalently, not Airport A) will localize the plane at airport B. (b) Interference experiment. Sending an object through a double slit reveals its wave- like (quantum) nature if interference patterns (blue line) are observed. This discussion raises the following general question: do experiments exist in which such quan- tum phenomena can be observed with macroscopic objects? For instance, has someone ever pre- pared an object into a spatial quantum superposition state and verified this state before decoher- ence has destroyed the quantum superposition state?1 Indeed, such spatial quantum superposition states have been verified experimentally in so- called double-slit experiments, cf. Fig. 1.1(b). In essence, the double slit can be thought of as local- izing the particle at the two slits simultaneously. The double slit, hence, prepares the object into