Quantum Plasmonics

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(1)Quantum Plasmonics.

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(3) Quantum Plasmonics. Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K.C.A.M Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op 27 november 2012 om 12:30 uur door. Reinier Willem HEERES natuurkundig ingenieur geboren te Heemskerk..

(4) Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. L. P. Kouwenhoven Copromotor: Dr. V. Zwiller. Samenstelling van de promotiecommissie: Rector Magnificus Voorzitter Prof. dr. ir. L.P. Kouwenhoven Technische Universiteit Delft, promotor Dr. V. Zwiller Technische Universiteit Delft, copromotor Prof. dr. ir. P. Kruit Technische Universiteit Delft Prof. dr. rer. nat. H. Giessen Universität Stuttgart Prof. dr. S.A. Maier Imperial College, London Prof. dr. L. Kuipers AMOLF, Amsterdam / Universiteit Twente Dr. R.H. Hadfield Heriot-Watt University, Edinburgh Prof. dr. ir. L.M.K. Vandersypen Technische Universiteit Delft, reservelid. Published by: Cover design: Printed by:. Reinier Heeres Michiel Voˆ ute Ipskamp Drukkers, Enschede. ISBN: 978-90-8593-144-7 Casimir PhD Series, Delft-Leiden 2012-32 Copyright © 2012 by Reinier Heeres Electronic version available at http://www.library.tudelft.nl/dissertations.

(5) “It is the facts that matter, not the proofs. Physics can progress without the proofs, but we can’t go on without the facts... if the facts are right, then the proofs are a matter of playing around with the algebra correctly.” Richard P. Feynman. 1. “All the fifty years of conscious brooding have brought me no closer to answer the question ‘What are light quanta?’. Nowadays every rascal thinks he knows the answer, but he is deluding himself.” Albert Einstein. 1 in. 2 In. Feynman Lectures on Gravitation a letter to Michele Besso (1951). 2.

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(7) Contents. 1 Introduction 1.1 Plasmonics . . . . . . . . . . . . . . . . . 1.2 Quantum plasmonics . . . . . . . . . . . . 1.3 Classical and quantum communication . . 1.4 Superconducting Single-Photon Detectors 1.5 Thesis outline . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 1 1 1 3 4 5 6. 2 Theory 2.1 The optical response of metals . . . . . . 2.1.1 Conductivity . . . . . . . . . . . . 2.1.2 Scattering times . . . . . . . . . . 2.2 Plasmons . . . . . . . . . . . . . . . . . . 2.2.1 Bulk and surface plasmons . . . . 2.2.2 More complex modes . . . . . . . . 2.2.3 Mode symmetries . . . . . . . . . . 2.2.4 Localized resonances . . . . . . . . 2.3 Quantum description of a beam splitter . 2.3.1 Quantum interference . . . . . . . 2.4 Spontaneous Parametric Down-Conversion 2.4.1 Classical non-linear interactions . . 2.4.2 Parametric fluorescence . . . . . . 2.4.3 Phase matching . . . . . . . . . . . 2.4.4 Absolute efficiency measurement . 2.5 Simulations . . . . . . . . . . . . . . . . . 2.5.1 Finite-Difference Time-Domain . . 2.5.2 Eigenmode solver . . . . . . . . . . 2.5.3 Mode area calculations . . . . . . . 2.5.4 Absorption calculations . . . . . . 2.6 Directional couplers . . . . . . . . . . . . 2.7 Superconducting single photon detectors . References . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. 9 9 11 12 13 13 14 14 15 15 16 18 18 18 18 19 19 19 20 20 21 21 22 24. vii.

(8) Contents 3 Methods 3.1 Fabrication recipe . . . . . . . . 3.1.1 Superconducting detectors 3.1.2 Plasmon waveguides . . . 3.2 Measurement setup . . . . . . . . 3.2.1 Electronics . . . . . . . . 3.2.2 Cryogenic setup . . . . . 3.2.3 Down-conversion source . 3.2.4 Software . . . . . . . . . . References . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 27 27 27 27 28 28 29 30 30 31. 4 On-chip Single Plasmon Detection 4.1 Introduction . . . . . . . . . . . . . . 4.2 Experiment . . . . . . . . . . . . . . 4.2.1 Electrical plasmon detection . 4.2.2 Propagation length . . . . . . 4.2.3 More complex geometries . . 4.2.4 Single plasmon detection . . 4.2.5 Additional information . . . . 4.3 Conclusions . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 33 34 34 34 36 37 38 39 42 43. 5 Quantum Interference of Surface Plasmons 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Beam splitter theory . . . . . . . . . . . . . 5.3 Device . . . . . . . . . . . . . . . . . . . . . 5.4 Beam splitter characterization . . . . . . . . 5.4.1 Excitation polarization dependence . 5.4.2 Coupling length dependence . . . . . 5.5 Interference measurements . . . . . . . . . . 5.6 Simulations . . . . . . . . . . . . . . . . . . 5.6.1 Mode area . . . . . . . . . . . . . . . 5.6.2 Coupled modes . . . . . . . . . . . . 5.6.3 SSPD absorption . . . . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 45 46 46 47 49 50 50 53 55 57 57 57 60 61. . . . . . . . . .. 6 Sub-wavelength Focusing of Light with Orbital Angular 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Angular momentum of light . . . . . . . . . . . . . 6.1.2 OAM beams . . . . . . . . . . . . . . . . . . . . . 6.2 Optical transition probabilities . . . . . . . . . . . . . . . 6.3 Plasmonic antennas . . . . . . . . . . . . . . . . . . . . . 6.4 Simulation details . . . . . . . . . . . . . . . . . . . . . . 6.4.1 OAM beam from Gaussian sources . . . . . . . . . 6.4.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . 6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii. Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63 64 64 64 65 67 68 68 69 71.

(9) Contents 6.6 Applications for optical tweezers . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 74 75. 7 Efficiency Enhancement of Superconducting Single Photon Detectors using Nano-antennas 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Optical antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77 78 78 79 80 81 81 82 85. 8 Photon-pair detection with superconducting 8.1 Introduction . . . . . . . . . . . . . . . . . . . 8.1.1 Pump-probe spectroscopy . . . . . . . 8.2 Detection probabilities . . . . . . . . . . . . . 8.3 Quantum Pump-Probe source . . . . . . . . . 8.4 Experiment . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .. 87 88 88 89 91 91 94 95. detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 9 Capacitive readout and gating of superconducting single photon detectors 97 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 9.2 Capacitor readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 9.2.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.3 Gating of SSPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 10 Measurement of g-factor ment of exciton spins 10.1 Introduction . . . . . . 10.2 Model . . . . . . . . . 10.3 Experiment . . . . . . 10.4 Charged Exciton . . . 10.5 Neutral Exciton . . . . 10.6 Discussion . . . . . . . 10.7 Conclusion . . . . . . References . . . . . . . . . .. tensor in a quantum dot and disentangle105 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 ix.

(10) Contents 11 Conclusions and outlook 11.1 Conclusions . . . . . . 11.2 Future directions . . . 11.3 Final words . . . . . . References . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 121 121 122 122 124. Summary. 127. Samenvatting. 129. Acknowledgements. 131. Curriculum Vitae. 132. List of Publications. 134. x.

(11) CHAPTER 1. Introduction 1.1. Plasmonics. In electronics, structures have become smaller and smaller since the first transistor (Fig. 1.1a) was developed in 1947 at Bell labs [1]. Nowadays, state-of-the-art transistors have gate sizes of about 20 nm (Fig. 1.1b), which allows to pack millions of them on a single computer chip. The ultimate limit is set by the scale of the size of a few atoms. In optics the experiments have typically been very large and realized on optical tables (Fig. 1.1c). Small on-chip waveguide structures, such as the one shown in Fig. 1.1d, have been developed in recent years, but they are fundamentally limited in size by diffraction. This implies that if structures are made smaller than the diffraction limit, given roughly by half the wavelength in a medium, light will leak out and therefore become less localized. This Thesis describes experiments with surface plasmon polaritons, optical fields bound to a metal-dielectric interface [2]. The main motivation is that plasmonic structures allow to shrink optical components to sizes much smaller than the diffraction limit [3]. However, as we shall see, there is a trade-off between confinement and losses that will prove longer and more complex circuits hard to operate. It is therefore likely that future devices will be hybrid, i.e. use plasmonics if strongly localized fields are required and dielectrics for interconnects where long propagation lengths are necessary.. 1.2. Quantum plasmonics. Although most aspects of plasmonics are purely classical and can be described by Maxwell’s equations, we are interested in the quantum regime of single plasmonic excitations. The aim is to perform quantum-optical experiments on a chip, as schematically indicated in figure Fig. 1.2a. Usually single-photon emitters and single-photon detectors are connected using free-space optics or optical fibers, resulting in bulky measurement setups and relatively poor collection and coupling efficiencies. We want to explore the possibilities of coupling these emitters and detectors through plasmonic waveguides in the near field. 1.

(12) Chapter 1. Introduction . . . . Figure 1.1: Device length scales in electronics and optics. a, The first transistor developed at Bell labs in 1947. b, Typical current transistor size. c, Free-space bulk optical setup on an optical table. d, Integrated optical circuit based on silicon waveguides (IBM).. Several experiments with respect to quantum properties of plasmons have been performed by other groups. First, it was shown that the plasmon-mediated enhanced light transmission through arrays of holes in a metal surface conserves the photon’s polarization properties, including quantum superpositions and entangled polarization states [4]. Similar work showed that energy-time entanglement is also preserved [5]. Second, it was shown that coupling single emitters to silver nanowires in the nearfield allows excitation of single plasmons [6, 7]. All these schemes relied on conversion of the plasmon to a free photon and subsequent far-field detection with traditional single photon detectors. In contrast, we will use integrated detectors to remove this conversion and expect that this can greatly enhance the detection efficiency. These quantum experiments have mostly relied on chemically synthesized silver wires. It has resulted in small ‘plasmon networks’ of up to three connected wires [8], but lacks real scalability. For more classical fields others have already taken the flexible approach of waveguides defined by lithographic techniques [9]. Our addition to the field is to realize lithographically defined circuits based on strongly confined plasmon modes and to perform experiments on the quantum level. Our key experiment is schematically shown in Fig. 1.2b, and realizes a miniaturized plasmon beam splitter to show quantum interaction between pairs of plasmons. Next to the propagating plasmons in small metallic circuits, we will also explore so-called localized surface plasmon resonances. These occur when a metallic structure is small compared to the propagation length or wavelength. In such a small structure the electromagnetic field can cause a resonant oscillation of the conduction electrons, 2.

(13) 1.3. Classical and quantum communication . . Figure 1.2: Plasmonic integration. a, An impression of plasmonic integration of single photon emitters and single photon detectors. The emitter could be an electrically contacted nanowire with a quantum dot inside. The detector consists of a superconducting meander that is sensitive to single photons, or plasmons. b, A schematic representation of the plasmon interference experiment. Two waveguides in a directional coupler geometry allow to create a compact beam splitter; detectors are integrated on-chip.. which can result in very intense localized electromagnetic fields and interactions. One example is the ‘spaser’ [10, 11], a resonant metallic particle coated with a gain medium that can function as a very small source of coherent radiation. For extremely small particles the wavefunctions of the electrons also become relevant. This results in quantum corrections to the resonances because the electron plasma can no longer be treated classically [12], a kind of quantum effect we will not explore in this Thesis. The most interesting application of localized resonances is in creating optical nanoantennas [13], structures which concentrate optical fields and allow enhancement of optical interactions with matter at sub-wavelength length-scales. Typical examples are high-resolution optical imaging [14, 15, 16] and optical lithography [17, 18]. Antennas are also particularly useful to overcome the intrinsic size difference between nanoscale systems such as quantum dots and free-space optical modes. In this work they will be used in two ways: first to increase the absorption efficiency of superconducting single-photon detectors. Second, we will describe simulations with antennas that allow to focus light with orbital angular momentum to the nanoscale. This could allow solid-state manipulation of this special quantum property of a photon and is an excellent example of the power of plasmonics to overcome a length-scale difference.. 1.3. Classical and quantum communication. The main reason why it is interesting to bring optical elements to the size of electronic components is in communication technology. A large fraction of the information that is transmitted through the internet travels through optical fibers because of the exceptionally high bandwidth and low losses. At the end of each fiber, however, the optical information is usually still converted into electrical signals for interpretation and routing; this is one of the limiting factors in communication speed and quite en3.

(14) Chapter 1. Introduction ergy intensive [19]. If, on the other hand, signals could be processed in optical form this would take out an important bottleneck. A big problem is that light usually does not interact very strongly with matter and is hard to integrate densely due to the diffraction limit; plasmonics could provide an extra tool to improve these aspects. Next to this classical motivation, the prospect of quantum communication is also an important driver. It offers intrinsically secure connections between two parties, where the actions of an eavesdropper can be detected [20]. At present, key distribution speeds are quite low, and up-scaling is a major challenge; better integration with single photon sources and detectors is certainly required. Next to key distribution, quantum computation [21] will probably also require transmission of quantum information in the form of qubits through quantum networks [22]. It seems that photons, because of their relatively weak interactions with matter and each other, are the ideal candidates to preserve coherence over large distances. The interface between solid-state qubits and these ’flying qubits’ will have to be improved and is a point in which plasmonics could play a key role.. 1.4. Superconducting Single-Photon Detectors. Superconducting Single-Photon Detectors (SSPDs) are one of the work horses in this Thesis. They were first realized by Golt’sman et al. in 2001 [23] after earlier theoretical work [24] proposed that the hotspot mechanism could be used for photodetection. The working principle of these detectors is extremely simple and elegant: a very thin (∼ 5 nm) and narrow (∼ 100 nm) superconducting wire is biased with a current close to its critical current. Upon absorption of a single photon, the wire switches to the normal, resistive state locally. This creates an easily measurable voltage pulse, after which the detector restores to the superconducting state quickly. Simply counting the voltage pulses gives the number of detected photons. There are several important advantages to these detectors over traditional silicon avalanche photodiodes (APDs): • Broad wavelength range. SSPDs are sensitive over a very broad wavelength range (ultraviolet - near infrared up to several μm). • Short dead time. The time that an SSPD requires to recover from a detection event and become ready for operation again is several nanoseconds, whereas this is typically more than 30 nanoseconds for silicon APDs. • Small timing jitter. The SSPD detection process has a very large intrinsic timing accuracy and therefore allows to determine the detection time with a very small uncertainty, the jitter. It has been measured to be at most 60 ps [25], but values as low as 20 ps have also been reported [26, 27]. • Low dark-count rate. The dark-count rate of a typical SSPD can be < 100 Hz, especially when operated at lower temperatures. • No after-pulsing. Silicon APDs sometimes show after-pulsing, which means that spurious detection events are more likely a short period after detection of a photon; this is not the case for SSPDs. 4.

(15) 1.5. Thesis outline The main disadvantage is that SSPDs are made out of a very thin film, and therefore are not able to absorb all of the incoming light. This limits their overall efficiency, but can be circumvented in several ways, one of which involvis plasmonics and will be explored in this Thesis. A second practical issue is that SSPDs require low operating temperatures of <= 4 K. Advances in cryogen-free refridgeration technology have largely solved this problem by making continuous operation much easier and economical; an important factor towards their wider adoption.. 1.5. Thesis outline. Chapter 2 introduces the basic theoretical concepts relevant for this Thesis. Chapter 3 describes details of the recurring measurement setups and sample fabrication techniques. Chapters 4 and 5 are the main experimental results involving experiments performed at the single plasmon and plasmon pair level respectively. Chapter 6 introduces a new plasmonic antenna that can be used to focus light with orbital angular momentum to a sub-wavelength length-scale. Chapter 7 describes an experiment where optical antennas are fabricated directly on an SSPD to increase the detection efficiency. In Chapter 8 the temporal dynamics of SSPDs are studied using a photon pair source and in Chapter 9 we describe how such a detector can be timegated. Chapter 10 is a somewhat unrelated experiment, in which the g-factor tensor is determined of excitons in a quantum dot embedded in an InP nanowire. Finally, Chapter 11 briefly discusses the main conclusions and outlook for the future.. 5.

(16) References. References [1] J. Bardeen and W. H. Brattain, The transistor, a semi-conductor triode, Phys. Rev. 74(2), 230 (1948). [2] W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optics, Nature 424(6950), 824–830 (2003). [3] J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Plasmonics for extreme light concentration and manipulation, Nature Mater. 9(3), 193–204 (2010). [4] E. Altewischer, M. P. Van Exter, and J. P. Woerdman, Plasmon-assisted transmission of entangled photons, Nature 418(6895), 304–306 (2002). [5] S. Fasel, F. Robin, E. Moreno, D. Erni, N. Gisin, and H. Zbinden, Energy-time entanglement preservation in plasmon-assisted light transmission, Phys. Rev. Lett. 94(11), 110501 (2005). [6] A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, Generation of single optical plasmons in metallic nanowires coupled to quantum dots, Nature 450(7168), 402–406 (2007). [7] R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stöhr, A. A. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Wave–particle duality of single surface plasmon polaritons, Nature Phys. 5, 470–474 (2009). [8] H. Wei, Z. Wang, X. Tian, M. Käll, and H. Xu, Cascaded logic gates in nanophotonic plasmon networks, Nature Commun. 2, 387 (2011). [9] R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons, Opt. Express 13(3), 977–984 (2005). [10] D. J. Bergman and M. I. Stockman, Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems, Phys. Rev. Lett. 90(2), 27402 (2003). [11] M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, Demonstration of a spaser-based nanolaser, Nature 460(7259), 1110–1112 (2009). [12] J. A. Scholl, A. L. Koh, and J. A. Dionne, Quantum plasmon resonances of individual metallic nanoparticles, Nature 483(7390), 421–427 (2012). [13] L. Novotny and N. van Hulst, Antennas for light, Nature Photon. 5(2), 83–90 (2011). [14] J. Wessel, Surface-enhanced optical microscopy, JOSA B 2(9), 1538–1541 (1985). [15] L. Novotny and S. Stranick, Near-field optical microscopy and spectroscopy with pointed probes, Annu. Rev. Phys. Chem. 57, 303–331 (2006). 6.

(17) References [16] C. H¨ oppener and L. Novotny, Antenna-based optical imaging of single Ca2+ transmembrane proteins in liquids, Nano letters 8(2), 642–646 (2008). [17] W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, Flying plasmonic lens in the near field for high-speed nanolithography, Nature Nano. 3(12), 733–737 (2008). [18] L. Pan, Y. Park, Y. Xiong, E. Ulin-Avila, Y. Wang, L. Zeng, S. Xiong, J. Rho, C. Sun, and D. B. Bogy, Maskless plasmonic lithography at 22 nm resolution, Sci. Rep. 1 (2011). [19] R. S. Tucker, Green optical communications—Part II: Energy limitations in networks, IEEE J. Sel. Topics in Quantum Electron. 17(2), 261–274 (2011). [20] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74(1), 145–195 (2002). [21] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature 464(7285), 45–53 (2010). [22] H. J. Kimble, The quantum internet, Nature 453(7198), 1023–1030 (2008). [23] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Picosecond superconducting single-photon optical detector, Appl. Phys. Lett. 79, 705 (2001). [24] A. M. Kadin and M. W. Johnson, Nonequilibrium photon-induced hotspot: A new mechanism for photodetection in ultrathin metallic films, Appl. Phys. Lett. 69, 3938 (1996). [25] H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors, Nature Photon. 1(6), 343–348 (2007). [26] A. Verevkin, A. Pearlman, W. Slysz, J. Zhang, M. Currier, A. Korneev, G. Chulkova, O. Okunev, P. Kouminov, K. Smirnov, et al., Ultrafast superconducting single-photon detectors for near-infrared-wavelength quantum communications, J. Mod. Opt. 51(9-10), 1447–1458 (2004). [27] W. Pernice, C. Schuck, O. Minaeva, M. Li, G. N. Goltsman, A. V. Sergienko, and H. X. Tang, High speed travelling wave single-photon detectors with near-unity quantum efficiency, Arxiv preprint arXiv:1108.5299 (2011).. 7.

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(19) CHAPTER 2. Theory This chapter introduces the theoretical concepts that are recurring throughout this Thesis. It starts with treating surface plasmons, a completely classical result based on Maxwell’s equations. Then we go on to the quantum mechanical aspects of this work, with a quantum description of a beam splitter and a treatment of spontaneous parametric down-conversion. Finally the operating principle of a key component, the superconducting single photon detector, is introduced.. 2.1. The optical response of metals. The most important factor determining the response of metals to electromagnetic radiation are the free conduction electrons. These can be described well using the Drude free-electron model, in which the n valence electrons per atom can move freely through the material, except for a characteristic relaxation time τ (or rate γ = τ −1 ) after which the momentum is lost. For gold and silver this time is approximately τ ≈ 9 × 10−15 sec. The individual electrons are free to move and an incident harmonic electric field E(t) = E0 e−iωt will cause each of the electrons to be displaced by a distance: x(t) =. eE(t) m0 (ω 2 + iγω). (2.1). With N the free electron density. This results in a collective polarization P = −N ex of the plasma. The dielectric function that characterizes the optical response is found from the definition of the electric displacement: D = 0 E + P = r 0 E. (2.2). ωp2 ω 2 + iγω. (2.3). and gives r (ω) = 1 −. 9.

(20) Chapter 2. Theory Here ωp is the plasma frequency, given by N e2 ωp = 0 m0. (2.4). with N the electron density, 0 the dielectric permeability of vacuum and m0 the electron mass. The plasma frequency is the natural resonant frequency of the electron plasma, driven by the restoring force of the collective polarization. Below the plasma frequency the electrons can respond fast enough to an electric field and will therefore create a polarization that is π out of phase, which will result in reflection of the incoming wave. Metals, with a plasma frequency in the ultra-violet due to the large electron density, are therefore highly reflective in the range of visible wavelengths as shown in Fig. 2.3.. . . Figure 2.1: Metal bandstructures. a, Calculations for gold, from [1]. The bold arrow indicates a direct interband transition. b, Calculation for NbN [2]. The dotted arrow indicates an indirect inter- or intra-band transition.. The Drude model is a relatively good approximation at low energies (for gold <∼ 2 eV, for silver <∼ 4 eV), but at higher energies it is required to take into account the fact that metals have a complex band-structure as well. For example, the bandstructure for gold [3, 1] is shown in Fig. 2.1a. Interband transitions, indicated in the figure by an arrow, modify the response by introducing extra loss channels already far below the plasma frequency. Photons predominantly drive vertical transition in this band diagram as they have negligible momentum compared to the electrons. Interband transitions are clearly seen in the experimentally determined dielectric functions of gold and silver, as shown in Fig. 2.2a and b. For gold they are strongest above 2 eV, but a significant broad peak contributes around 1.5 eV [4, 5] as well. These transitions can be modeled as damped, driven Harmonic oscillators, adding a resonant response of r (ω) = 10. n ˜ e2 1 m0 ω02 − ω 2 − iγω. (2.5).

(21) 2.1. The optical response of metals a. b. 0. 30. Ag Au Ag / Au (D) NbN (D). 20. { }. { }. 20. 10. Ag Au Ag / Au (D) NbN (D). 40. 60 0.0. 2.5. 5.0 Energy (eV). 7.5. 10.0. 0 0.0. 2.5. 5.0 Energy (eV). 7.5. 10.0. Figure 2.2: Dielectric functions. Real (a) and imaginary (b) part of the dielectric functions of gold and silver (from Palik [6]) and Drude approximations for gold/silver and NbN. 1.0. Reflectivity. 0.8 0.6 0.4. Ag Au Ag / Au (D) NbN (D). 0.2 0.0. 0. 2. 4 6 Energy (eV). 8. 10. Figure 2.3: Reflectivity. The reflectivity of several materials as a function of energy. The √ point where the reflectivity drops to zero sharply corresponds to ωp/ 2. This is only clear for the Drude models; for real metals the interband transitions lower the reflectivity earlier.. with n ˜ the bound electron density.. 2.1.1. Conductivity. It is sometimes easier to work with the conductivity than the permittivity, for example when calculating absorption. The current is given by the time derivative of the plasma polarization P = −N ex(t). Again, in response to a harmonic driving field E(t) and using Eq. (2.1) we find: j(t). = =. N 2 e2 1 dE(t) dP =− = −iωP dt m0 ω 2 + iγω dt e2 τ 1 E(t) = σE(t) m0 1 − iωτ. (2.6) (2.7) 11.

(22) Chapter 2. Theory The relation between the conductivity and the dielectric function can be found from the electric displacement in Eq. (2.2):. 2.1.2. σ(ω). =. r (ω). =. iω0 (1 − r (ω)) iσ(ω) 1+ 0 ω. (2.8) (2.9). Scattering times. A crucial parameter in the Drude model is the momentum relaxation time τ . Experiments on gold films show that this parameter is usually not independent of frequency [7]. However, after annealing the frequency dependence can be drastically lowered and τ improved. This experimental fact can be explained by employing a two-carrier model [8], where it is assumed that one part of the electrons is located in the highly disordered areas between crystals (region a) and the rest in a perfect lattice (region b). The relaxation times for the electrons in both areas are not equal: in (a) it will be dominated by scattering due to disorder (mean-free path ∼ 0.5 nm), whereas electron-phonon scattering is the most important factor in (b) (mean-free path ∼ 15 nm). It should be noted that in this model scattering of electrons in volume (b) from a grain-boundary can largely be treated as specular, i.e. momentum conserving. Annealing reduces the amount of disordered volume at the grain boundaries and therefore increases the effective scattering time τef f . Lowering the temperature decreases the electron-phonon interaction time by reducing the number of available phonons, and can therefore increase τb for the electrons in (b). It will, however, not be possible to make τef f arbitrarily large, because the electrons in region (a) will contribute an almost temperature independent τa . Experimentally determined temperature dependences [5, 9] match this model well, although the temperature where it becomes relevant depends strongly on the fraction of disorder and therefore annealing. One can separate the defect and surface scattering rate (γd ) from the electronphonon scattering rate (γp ) because the latter has a known dependence on scaled temperature T/θ, where θ is the Debye temperature [4]. In a thick (1 μm) film that was annealed at 500 K for several hours, at room temperature the values γP = 4.3 × 1013 sec−1 and γd = 1.2×1013 sec−1 were found from near IR reflection measurements. This means that the scattering rate can be reduced by about a factor 5 by going from room temperature to absolute zero. This paper also shows data for less well annealed films, suggesting that they can have larger scattering rates by more than a factor 2. This also means that going to lower temperatures will have a much smaller effect. It is often noted that quickly evaporated thin films (100 ˚ Asec−1 ) have the same quality as bulk material [10]. However, in our experiments the equipment usually limits the rate to 1 − 2 ˚ Asec−1 , where grain structure or a rough surfaces could influence the film quality. Our e-beam fabricated structures are not annealed because the mobility of the gold atoms could deform our structure significantly. We therefore expect the dielectric constant as well as the optically induced losses not to change drastically when cooled to 4 K, the operating temperature of our experiments. 12.

(23) 2.2. Plasmons. 2.2 2.2.1. Plasmons Bulk and surface plasmons. The Drude response in Eq. (2.3) gives a natural response frequency ωp for collective density oscillations of the electron plasma in the bulk [11]. These oscillations are quantized with an energy ωp ≈ 10 eV for typical metals, which can be observed in Electron Energy Loss Spectroscopy [12]. At a metal-dielectric interface, the surface charge associated with the plasma os√ cillations modifies the natural resonance frequency to be ωsp = ωp/ 1 + d . Bound electromagnetic surface waves are possible up to this frequency and follow the dispersion relation: ω d m (2.10) ksp = c d + m With d and m the complex dielectric constants of the dielectric and metal respectively. We will denote the real part of by a single prime and the imaginary part by a double prime. If d is real, usually the case for a dielectric, and m < |m | the wavevector is complex, and describes a bound mode with: ω d m ksp ≈ (2.11) c d + m 3/2 ω m d m ≈ (2.12) ksp c d + m 22 m Fig. 2.4 displays the dispersion curve for an interface between gold or silver and air and between air and a Drude free-electron model of gold, neglecting the interband contribution to . It is clear that a real metal has a much more complicated response than an idealized Drude model. When the real part of the dielectric constant is negative, the wavevector lies to the right of the light-line that describes the dispersion of free-space light. This means that a plasmon has more momentum than a free photon and the two can therefore not couple efficiently. Both the real and the imaginary part of ksp diverge when m = d , the surface plasmon resonance. For a Drude free-electron √ response this occurs when ω = ωp/ 1 + d . When approaching this frequency the group dω velocity vg = /dk goes to zero and the wave slows down. The complex part of the wavevector described a characteristic loss-distance, or the propagation length: LSP =. 1 c = 2kSP ω. . m + d m d. 3/2. 2m m. (2.13). The imaginary part of the dielectric function is the origin of the losses, and causes Ohmic heating by scattering of electrons. The spatial extent into the dielectric and the metal can be found from the z-component of the wavevector: ω 2 − kx2 (2.14) kz = c 13.

(24) Chapter 2. Theory a 10.0. 5.0. Propagation length (m). 7.5 E (eV). b 10. Ag Au Ag / Au (D) NbN (D) light line. 2.5 0.0. 0. 10. 20 (. 30 ). 40. 3. Ag Au Ag / Au (D) NbN (D). 10 4 10 5 10 6 10 7 10 8 10. 9 0.0. 2.5. 5.0. 7.5. Energy (eV). Figure 2.4: Surface modes. Dispersion relation (a) and propagation length (b) of surface plasmons on a metal-air interface for several commonly used materials.. Since kx > (ω/c)2 this is an imaginary number and the wave decays exponentially from the surface, i.e. is evanescent. Typically the decay is several tens of nanometers in the metal and several hundreds in the dielectric.. 2.2.2. More complex modes. Before the plasmon resonances described above were studied, it was actually very well known that metallic structures could exhibit bound modes in the microwave range, known as Sommerfeld waves [13, 14]. For a metallic cylinder an analytic solution was found that was more strongly confined for smaller wires, at the cost of increased loss. The extrapolation to optical frequencies, however, had not been performed. The surface plasmon described by dispersion relation Eq. (2.10) should be seen as a special case of a bound mode with a straight-forward analytic solution. The boundary conditions set by the geometry play a very important role in determining the fielddistribution of such a mode, and in fact there is a whole zoo. In general such modes are described by their complex effective index, where the real part determines the phase velocity and the complex part the attenuation or loss-rate. There typically is a trade-off between confinement and losses: increasing the confinement of a mode causes larger electric fields in the metallic structure, and correspondingly higher Ohmic losses.. 2.2.3. Mode symmetries. In an asymmetric environment (a metal slab on a dielectric substrate), there are 2 sets of modes: one on the air/metal interface and the other at the dielectric/metal interface. The former are normally referred to as the ’leaky plasmon mode’, because the dispersion relation of this mode crosses the light line of the dielectric at some point, and therefore has radiative losses into the dielectric. This also causes the mode to be excitable in the Kretschmann geometry, i.e. by illuminating from the substrate side. The mode on the metal/dielectric interface does not intersect the light line anywhere, and is therefore referred to as the ’bound plasmon mode’. The only losses that are present for this mode are the Ohmic losses caused by the field in the metal. 14. 10.0.

(25) 2.3. Quantum description of a beam splitter In a symmetric environment (e.g. a metal slab in a uniform dielectric) the modes on opposite interfaces can couple, giving rise to new modes that are symmetric and anti-symmetric combinations. This nomenclature can be somewhat confusing, since one should always specify the field component one is referring to. Here we will follow the suggestion by Berini [15], who considers the major electric field component. For waveguides which have a width w in the x direction and a thickness t in y (w/t 1), this will Ey . This field component can be anti-symmetric or symmetric with respect to the y and the x axes, which will be denoted by an ‘a’ or an ‘s’ respectively. For example, the mode ‘as’ is anti-symmetric with respect to the y axis and symmetric with respect to the x axis. In general the symmetric modes are not so well confined whereas the anti-symmetric ones are.. 2.2.4. Localized resonances. The previous sections described propagating plasmon modes. Reducing the dimensionality further can lead to localized plasmon resonances. In the special case of a particle much smaller than the wavelength, the response can easily be calculated √ exactly: the resulting surface charges gives a particle resonance ωpp = ωp/ 2 + d [16]. If the particle size becomes appreciable compared to the plasmon wavelength, the problem can not be treated as stationary anymore and the resonances become somewhat harder calculate [17]. At such a resonance the field intensity near the particle is strongly enhanced and the scattering cross-section increased. Such structures can effectively be used as antennas for light. By coupling a string of particles with localized resonances together, one can construct a propagating mode again [18]. It is also possible to create a resonance that strongly focuses the electromagnetic energy by combining particles of different sizes [19].. 2.3. Quantum description of a beam splitter. A beam splitter transforms input modes {a1 , a2 } into output modes {b1 , b2 } in the following way: a ˆ†1 |vac → tˆb†1 + rˆb†2 |vac (2.15) (2.16) a ˆ†2 |vac → rˆb†1 + tˆb†2 |vac Alternatively this transformation can be written in matrix notation: t r b= a r t Where the components of a and b correspond to (1 0) = ˆ |1, 0 and (0 1) = ˆ |0, 1. This matrix has to be unitary to conserve probabilities, which directly leads to several restrictions: |r| = |r |, |t| = |t | |r|2 + |t|2 = 1, |r |2 + |t |2 = 1 r∗ t + t∗ r = 0. (2.17) (2.18) (2.19) 15.

(26) Chapter 2. Theory Combining Eq. (2.17) with Eq. (2.19) leads to: |r||t|e−φr +φt + |r||t|e−φt +φr = 0, which corresponds to φt + φt − φr − φr = ±π. A lossless beam splitter has to obey this relation of phase shifts. Exactly where the phase-shift takes place depends on the implementation of the beam splitter, e.g. a simple dielectric plate beam splitter introduces a π phase-shift upon reflection on the air-side of the air-dielectric interface and a directional coupler is more symmetric with φr = φr = π/2. . . . . . . . . . . . . . . . . . . . Figure 2.5: Beam splitter interference. a - d, Four possible outcomes when a particle is created in mode a1 and a2 simultaneously. The physical picture of Hong-Ou-Mandel interference is that two of these possibilities carry an opposite phase (a and c) and therefore cancel out.. 2.3.1. Quantum interference. The output of a symmetric beam splitter when creating a photon in both inputs a1 and a2 is given by: √ √ √ √ a ˆ†1 a ˆ†2 |vac = T ˆb†1 + i Rˆb†2 i Rˆb†1 + T ˆb†2 |vac (2.20) √ (2.21) = i RT ˆb†1ˆb†1 + ˆb†2ˆb†2 + (T − R) ˆb†1ˆb†2 |vac √ = 2RT (|2, 0 − |0, 2) + (T − R) |1, 1 (2.22) In the case of a 50/50 beamsplitter, with R = T = 1/2, the output state |b1 , b2 reduces to: 1 (2.23) Ψout = √ (|2, 0 − |0, 2) 2 Where the overall phase-factor i has been dropped. The lack of the |1, 1 term is called Hong-Ou-Mandel interference. Fig. 2.5 schematically shows why this occurs: there are 4 possible options for the photons to exit the beam splitter. Quantum mechanically these possibilities have to be added coherently. Since two of them (a and c) carry an opposite phase, they cancel out. Without interference one observes a click at both outputs when the two incoming photons are both reflected or both transmitted. The probability P (|1, 1) is therefore R2 + T 2 . When HOM interference is occurring, the output state is given by Eq. (2.22) and the probability P (|1, 1) is reduced to R2 + T 2 − 2RT . This results in an interference visibility of: VHOM = 16. 2RT P (|1, 1) − P (|1, 1)) = 2 P (|1, 1) R + T2. (2.24).

(27) 2.3. Quantum description of a beam splitter 1.0. 1.0 N = 1 N = 2. 0.6. 0.4. 0.2. 0.6. 0.4. 0.2. 0.0 0.0. 0.8. N = 3. Peak ratio. Contrast. 0.8. 0.0 0.2. 0.4. 0.6. 0.8. 1.0. 0.0. 0.2. 0.4. 0.6. 0.8. 1.0. Figure 2.6: Beam splitter properties. a, HOM-interference visibility versus beam splitter ratio for several different number of modes per input and output. b, Time-resolved correlation measurements show two distinct peaks when sending one photon into one input arm and a delayed photon in the other. The first peak comes from both photons being reflected (R2 ), the second from both being transmitted (T 2 ). The result is that time-resolved correlation data shows a strongly imbalanced peak ratio of T 2/(1 − T )2 upon deviation from a 50/50 splitter.. which is plotted in Fig. 2.6a. From this figure we see that the visibility does not depend so strongly on the beam splitter ratio. In a time-resolved correlation measurement the cases ‘2 photons reflected’ and ‘2 photons transmitted’ can be distinguished for large time delays between the photons. The relative occurrence of these two events is strongly dependent on the beam splitting ratio (Fig. 2.6b), and therefore an excellent method to characterize a beam splitter.. Classical version of HOM interference It is possible to construct an experiment that will give a HOM-like interference dip with classical light [20]. When sending a classical light-beam into a Mach-Zehnder interferometer, the two output ports show interference as a function of the time delay between the two paths. If the time delay is smaller than the first order coherence time τc , this will result in intensities I1 ∼ sin(Δφ) and I2 ∼ cos(Δφ) in arms 1 and 2 respectively. If the phase in one of the arms is now rapidly randomized, the output intensities will have to be averaged over Δφ and are seen to be constant. The intensity correlation function I1 (t)I2 (t + τ ), however, will still show an effect of the first order coherence with an interference visibility of half the second order interference visibility, giving a maximum contrast of one half. However, this does not mean that every HOM-interference dip with a visibility smaller than 1/2 is not of quantum mechanical origin [21]. In the experiments considered here, for example, we will see interference of photons from a parametric down-conversion source which already represent a highly non-classical state in themselves [22]: each of the two beams individually obeys Poissonian photon statistics, but since the photons are created in pairs they are very strongly correlated. This feature is clearly visible if one of the inputs to the beam splitter is blocked: the correlations disappear completely and Poissonian statistics are found again; no classical field can mimic this property [22]. 17.

(28) Chapter 2. Theory. 2.4. Spontaneous Parametric Down-Conversion. 2.4.1. Classical non-linear interactions. The polarizability of a material is given by [23]: P (t). = =. χ(1) E(t) + χ(2) E 2 (t) + . . . P1 (t) + P2 (t). (2.25) (2.26). The first term is the well-known linear susceptibility. The second term is the secondorder non-linear susceptibility. Materials where χ(2) is non-zero provide possibilities for second-harmonic generation or parametric down-conversion; this is only possible if the crystal structure does not posses inversion symmetry. Consider a field described by: E(ω, t) = E0 e−iωt + c.c.. (2.27). The second order polarization P2 (t) is then given by: P2 (t) = 2χ(2) E0 E0∗ + (χ(2) E02 e−2iωt + c.c.). (2.28). and contains a DC component as well as at 2ω0 , the second harmonic. When driving with two different frequencies ω1 and ω2 : P 2 (t) = χ(2) (E12 e−2iω1 t + E22 e−2iω2 t + 2E1 E2 e−i(ω1 +ω2 )t + 2E1 E2∗ e−i(ω1 −ω2 )t + c.c.) + 2χ(2) (E1 E1∗ + E2 E2∗ ). (2.29). Next to the DC component and the second harmonics of the fundamental frequencies, this contains the sum and differences ω1 + ω2 and ω1 − ω2 as well. Difference frequency generation can be used to amplify a weak input field in a process called ‘parametric amplification’ by mixing it with a strong, higher frequency pump in a non-linear crystal.. 2.4.2. Parametric fluorescence. When treating the interaction of optical fields in non-linear crystals quantum mechanically, it turns out that a difference frequency generation process can also occur without the input of a field at a second frequency [24]. This process is called ‘parametric fluorescence’. In difference frequency generation a high energy photon effectively splits in two lower energy photons (the ‘signal’ and ‘idler’ photons), catalyzed by the non-linear medium. Parametric fluorescence in the quantum treatment can basically be viewed as an interaction with the vacuum fluctuations resulting in parametric amplification of a field with an effective input intensity of one photon per mode.. 2.4.3. Phase matching. To efficiently perform parametric down-conversion, i.e. use parametric fluorescence to create photon pairs, it is important that the fields at the down-converted frequencies 18.

(29) 2.5. Simulations in the non-linear crystal build up coherently. For this to happen the crystal needs to be phase-matched. Because energy and momentum should be maintained in the downconversion process, the following requirements hold: ωp kp. =. ω1 + ω2. (2.30). =. k1 + k2. (2.31). We perform degenerate type-II collinear phase-matching, which means that we will optimize the process for creation of two photons of half the pump energy, one H and one V polarized, all propagating in the same direction. For these waves to satisfy the criteria Eq. (2.4.3), we now require ne (ωp ) = 12 ne (ω) + 12 no (ω), where ne and no are the extra-ordinary and ordinary index of refraction of the birefringent crystal. In our case we KTP, which allows phase-matching the desired process by cutting it at the right angle and then rotating the crystal to fine-tune. In a collinear process the intensity at the signal and idler frequency will only build up throughout the crystal when phase-matching condition is satisfied. As a function of Δk is the intensity is given by: L sin2 (Δk L/2) eiΔkz dz = L2 = sinc(Δk L/2) (2.32) I(Δk) = (Δk L/2)2 0 where L is the crystal length. Converting Δk into a wavelength range gives: 2π 2 λ (2.33) L We therefore find that the bandwidth for a 20 mm long crystal with central wavelength 1064 nm will be approximately 0.36 nm. Δλ ≈. 2.4.4. Absolute efficiency measurement. When detecting photons produced by a photon-pair source, the count rates N1 and N2 on detector 1 and 2, and the correlated count rate Nc are given by: N1 = η1 N, N2 = η2 N, Nc = η1 η2 N. (2.34). Where η1 and η2 correspond to the overall efficiency, i.e. both collection and detection efficiency. These relations can be used in a straight-forward way to perform an absolute efficiency measurement and to determine the number of produced photon pairs: η1 =. Nc N1 N2 Nc , η2 = , N= N2 N1 Nc. (2.35). However, this does not allow to determine the collection and detection efficiencies separately.. 2.5 2.5.1. Simulations Finite-Difference Time-Domain. Finite-difference time-domain simulations solve for Maxwell’s equations in time by discretizing space into a grid and time into small steps [25]. An optical excitation 19.

(30) Chapter 2. Theory pulse is inserted at the start of the simulation and the evolution is calculated using the time-dependent Maxwell’s equations in partial differential form. Because the excitation is usually a short pulse, it necessarily has a broad spectral content. By placing Fourier-transform monitors in the simulation domain the response of a system can be determined over a broad range of frequencies. The technique is, however, limited in application to small structures due to it’s computational requirements. Especially when a system involves metallic structures, the spatial mesh needs to be very fine and quickly results in exhaustion of the available computer memory. In our work we make use of Lumerical FDTD [26], a commercial software package which allows easy visual construction of a simulation model and basic analysis.. 2.5.2. Eigenmode solver. Waveguides typically support eigenmodes, a particular configuration of the optical field that does not change along the waveguide except for an overall amplitude and phase factor. The optical field is therefore an eigen-solution of the propagation operator given by Maxwell’s equations in case of an invariant structure in the propagation direction. Big advantages are that this technique only requires 2-dimensional meshing and that it operates in the frequency domain. Discretizing space results in a set of linear equations that can be solved numerically, as described in [27]. We use a solver based on this method and implemented in Matlab to find the dispersion relation of the modes for our waveguides and their magnetic field components Hx and Hy . From these solutions the remaining field components can be calculated. The Matlab code was provided by Rashid Zia.. 2.5.3. Mode area calculations. The mode area is a measure of the spatial extend of a waveguide mode, and can be defined in several different ways. We use a well-known formula used in fiber optics [28], which corresponds to a weighted average:. 2 |F (x, y)|dA Aef f =. |F (x, y)|2 dA. (2.36). Here F (x, y) is the quantity of interest, in our case the magnetic field intensity |H(x, y)|2 . We use this quantity because our mode solver seemed to produce slightly exagerated E-fields from the determined H-fields, and therefore signifcantly overestimates the energy in the electric field. We also think that the very strong E-field enhancements at the sharp edges of our waveguides are overestimated because the edges will be less sharp in real structures. The H-field effective mode area should be considered a conservative estimate, as the electric energy is usually more confined to the metallic structure. One could also use the total field energy [29, 30, 31], which for an absorbing material with dispersion is given by: 2ωnκ 0 n2 + |E|2 (2.37) W = 2 γe where γe again is the electron scattering rate. 20.

(31) 2.6. Directional couplers An alternative measure for the mode area is. (x, y)|E(x, y)|2 dA Aef f,alt = max((x, y)|E(x, y)|2 ). (2.38). This, however, does not always give a good estimate of the extent of the field profile in plasmonics, because the maximum field intensity can occur in a very small area [32] for structures with sharp features.. 2.5.4. Absorption calculations. The work by a field on a charge distribution ρ(r) is given by: · v dtdV F · dldV = ρ(r)E (2.39) W = δW · v dV = E · ρ(r)v dV = E · JdV · σ EdV = ρ(r)E = E (2.40) δt and J will have a net contribution to the work, Only the in-phase components of E so averaged over time:

(32) δW · Re(σ E)dV = Re(E) (2.41) δt (2.42) = E0 cos(−ωt) · ω0 r E0 cos(−ωt)dV 1 2 |E0 | ω0 r dV = (2.43) 2. 2.6. Directional couplers. In a directional coupler, the eigenmodes of two individual waveguides interact. This results in two new eigenmodes, or supermodes, which are the symmetric and antisymmetric combinations of the individual waveguide modes. Since these supermodes have different field distributions, their effective index will in general be different. At the input of a directional coupler, one starts with all power in one of the arms. This state can be written as a superposition of the symmetric and the anti-symmetric modes: |L =. |S + |A √ 2. (2.44). The transformation from the {|L, |R} basis to the {|S, |A} basis can also be described by the matrix: 1 1 1 vSA = √ vLR 2 1 −1 Which is its own inverse. 21.

(33) Chapter 2. Theory After propagating a distance l the state has evolved to: |T =. |S + ei √. lΔn/λ. |A. (2.45). 2. When transforming this back into the |L and |R basis one obtains: |T =. (1 + ei. lΔn/λ. )|L + (1 − ei √ 2. lΔn/λ. )|R. (2.46). The amplitude in the left and right output arms are now given by 2πlΔn/λ 1 + e2πilΔn/λ 2 = 1 + cos √ I(L) = 2 2 2πlΔn/λ 1 − e2πilΔn/λ 2 = 1 − cos √ I(R) = 2 2. (2.47) (2.48). In case of a beam splitter based on a plasmonic structure, one should also take into − 2π Im(n)l ˜ /λ account the losses. This modifies the equations by adding a term e for both the symmetric and anti-symmetric mode.. 2.7. Superconducting single photon detectors. SSPDs are essentially local energy or heating detectors. In that sense they are similar to bolometric detectors. Unlike bolometers that produce an analog output, however, the SSPD discretizes the disturbance into a binary signal which results in a click if enough local energy (compared to the thermal energy) is detected. The clicks from the detector are in the form of voltage pulses which can simply be counted to give the number of detected photons. It is quite an exceptional feature that the balance of heating and cooling in these detectors is such that the local, resistive region can be quickly restored, as has been modeled extensively [33, 34]. These detectors are becoming more and more common in applications involving single-photon detection outside of the silicon detector range, where no good alternative is available [35]. Of primary importance is the absorption process in the superconducting material. In our experiments mostly niobium nitride is used, a very lossy material at optical frequencies. Absorption is governed both by the Drude free-electron model through scattering and by interband transitions. However, in this case the typical scattering time has been measured to be τN bN = 8.3 × 10−15 sec, about an order of magnitude smaller than in gold. The result is that scattering processes of the free electrons involving phonons are relevant already at lower energies and transitions in the band diagram (Fig. 2.1b) are less restricted to be momentum conserving, i.e. vertical. The description of the SSPD detection mechanism usually follows the so-called hotspot model [36, 37], and is schematically shown in Fig. 2.7. It starts with a narrow nanowire carrying a supercurrent close to the critical current. Photon absorption creates of a highly excited quasi-particle (electron and/or hole) (a). Electron-electron and electron-phonon interactions cause a growing hot-spot and expel the supercurrent outwards. The carriers reach thermal equilibrium again on a characteristic relaxation 22.

(34) 2.7. Superconducting single photon detectors .

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(36). .

(37). .

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(39). Figure 2.7: The hot-spot model. a, Absorption of a photon results in a hot quasiparticle. b, Electron-electron and electron-phonon interactions cause a growing hot-spot by which the supercurrent is expelled outwards. The quasi-particles relax through electronphonon interactions and thermalize with the substrate. c, The supercurrent exceeds the critical current density in at the sides of the nanowire. d, The nanowire segment is in the resistive state over the whole width.. time τr (b). If the disturbance and the biasing current were large enough, the current density outside the hot spot can exceed the critical current density and cause a normal, resistive region over the whole width of the nanowire (c). The current running through the nanowire is dissipated in the resistive region on a time τd = L/Rhotspot and gives rise to a measurable voltage pulse. The inductance L is dominated by the kinetic inductance of the superconducting wire, i.e. the inertia of the Cooper pairs, and is typically of the order of 100 nH for the short meanders in our experiments, or ∼ 400 nH for typical fiber-coupled devices [38]. The hotspot resistance Rhotspot is not exactly known, but estimated to be of order ∼ 6 kΩ. The rise time of the voltage pulse is therefore very short, ∼ 17 ps. After reaching the superconducting state again, the supercurrent is restored. Now, however, the impedance that matters is the one of the load, which in our case this corresponds to Z = 50 Ω. The recovery time scale L/50 Ω is therefore much longer than the rise-time, typically ∼ 2 ns for our devices (∼ 8 ns for long meanders).. 23.

(40) References. References [1] P. Romaniello and P. L. de Boeij, The role of relativity in the optical response of gold within the time-dependent current-density-functional theory., J. Chem. Phys. 122(16), 164303 (2005). [2] L. F. Mattheiss, Electronic band structure of niobium nitride, Phys. Rev. B 5(2), 315 (1972). [3] N. E. Christensen and B. O. Seraphin, Relativistic band calculation and the optical properties of gold, Phys. Rev. B 4(10), 3321 (1971). [4] J. N. Hodgson, The optical properties of gold, J. Phys. Chem. Solids 29(12), 2175–2181 (1968). [5] G. P. Pells and M. Shiga, The optical properties of copper and gold as a function of temperature, J. Phys. C: Solid State 2, 1835 (1969). [6] E. D. Palik, Handbook of optical constants of solids, Volume 1, Academic Press, 1985. [7] M. L. Thèye, Investigation of the optical properties of Au by means of thin semitransparent films, Phys. Rev. B 2(8), 3060 (1970). [8] S. R. Nagel and S. E. Schnatterly, Frequency dependence of the Drude relaxation time in metal films, Phys. Rev. B 9(4), 1299 (1974). [9] P. Winsemius, M. Guerrisi, and R. Rosei, Splitting of the interband absorption edge in Au: Temperature dependence, Phys. Rev. B 12(10), 4570 (1975). [10] P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B 6(12), 4370 (1972). [11] P. Nozières and D. Pines, Correlation energy of a free electron gas, Phys. Rev. 111(2), 442 (1958). [12] C. J. Powell, The origin of the characteristic electron energy losses in ten elements, P. Phys. Soc. 76, 593 (1960). [13] A. Sommerfeld, Ueber die fortpflanzung elektrodynamischer wellen l¨ angs eines drahtes, Annalen der Physik 303(2), 233–290 (1899). [14] G. Goubau, Surface waves and their application to transmission lines, J. Appl. Phys. 21(11), 1119–1128 (1950). [15] P. Berini, Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures, Phys. Rev. B 61(15), 10484 (2000). [16] R. H. Ritchie, Plasma losses by fast electrons in thin films, Phys. Rev. 106(5), 874 (1957). [17] L. Novotny, Effective wavelength scaling for optical antennas, Phys. Rev. Lett. 98(26), 266802 (2007). 24.

(41) References [18] S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, A. A. Requicha, et al., Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides, Nature Mater. 2(4), 229–232 (2003). [19] K. Li, M. I. Stockman, and D. J. Bergman, Self-similar chain of metal nanospheres as an efficient nanolens, Phys. Rev. Lett. 91(22), 227402 (2003). [20] Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, Fourth-order interference technique for determining the coherence time of a light beam, JOSA B 6(1), 100–103 (1989). [21] J. H. Shapiro and K. X. Sun, Semiclassical versus quantum behavior in fourthorder interference, JOSA B 11(6), 1130–1141 (1994). [22] L. Mandel, Non-Classical States of the Electromagnetic Field, Phys. Scr. T 12, 34–42 (1986). [23] R. W. Boyd, Nonlinear optics, Academic Press, 1992. [24] A. Yariv, Quantum Electronics, Wiley, third edition, 1989. [25] K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE T. Antenn. Propag. 14(3), 302–307 (1966). [26] Lumerical Solutions Inc., Vancouver, BC, Canada, FDTD Solutions. [27] P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, Analysis of vectorial mode fields in optical waveguides by a new finite difference method, J. Lightwave Technol. 12(3), 487–494 (1994). [28] G. P. Agrawal, Nonlinear fiber optics, Academic Press, 2001. [29] R. Loudon, The propagation of electromagnetic energy through an absorbing dielectric, J. Phys. A: Gen. Phys. 3(3), 233 (1970). [30] R. Ruppin, Electromagnetic energy density in a dispersive and absorptive material, Phys. Lett. A 299(2), 309–312 (2002). [31] S. A. Maier, Plasmonic field enhancement and SERS in the effective mode volume picture, Opt. Express 14(5), 1957–1964 (2006). [32] R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, Confinement and propagation characteristics of subwavelength plasmonic modes, New J. Phys. 10(10), 105018 (2008). [33] J. K. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M. Rosfjord, and K. K. Berggren, Modeling the electrical and thermal response of superconducting nanowire single-photon detectors, IEEE T. Appl. Supercon. 17(2), 581–585 (2007). 25.

(42) References [34] A. J. Kerman, J. K. Yang, R. J. Molnar, E. A. Dauler, and K. K. Berggren, Electrothermal feedback in superconducting nanowire single-photon detectors, Phys. Rev. B 79(10), 100509 (2009). [35] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, Superconducting nanowire single-photon detectors: physics and applications, Supercond. Sci. Technol. 25(6), 3001 (2012). [36] A. M. Kadin and M. W. Johnson, Nonequilibrium photon-induced hotspot: A new mechanism for photodetection in ultrathin metallic films, Appl. Phys. Lett. 69, 3938 (1996). [37] A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, Quantum detection by current carrying superconducting film, Physica C 351(4), 349–356 (2001). [38] A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. N. Gol’tsman, and B. Voronov, Kinetic-inductance-limited reset time of superconducting nanowire photon counters, Appl. Phys. Lett. 88(11), 111116 (2006).. 26.

(43) CHAPTER 3. Methods This chapter describes the fabrication recipe for the samples described in this Thesis, as well as details about the measurement setup.. 3.1. Fabrication recipe. All fabrication was performed by the author of this Thesis. The SSPD recipe was a modified version of the one used by Sander Dorenbos [1].. 3.1.1. Superconducting detectors. The samples are fabricated on a sapphire substrate purchased from Scontel with a ∼ 5 nm thick layer of NbN sputtered with the substrate heated to ∼ 800 ◦ C. Contacts are defined by e-beam writing in a ∼ 250 nm thick PMMA 495k layer and developed for 90 seconds in MIBK:IPA (1:3). This is followed by evaporation and lift-off of chrome (15 nm) / gold (50 nm). The SSPDs are patterned by e-beam in a 70 nm thick hydrogen silsesquioxane (HSQ) spin-on-glass layer. This negative tone e-beam resist is very well suited for high-resolution lithography [2, 3]. However, it is not always so convenient to work with because sticking to the substrate can be problematic and it has a limited shelf life, which changes the required e-beam dose over time. They are developed in TMAH (5 sec) and MF322 : H2 O 1:9 (15 sec) for the highest possible resolution and then rinsed in H2 O. The HSQ is used as an etching mask for an SF6 / He reactive ion etch (RIE) at a pressure of 10 μbar for 60 seconds. Usually the HSQ is removed by a 2 sec dip in BHF, but for the quantum interference sample the remaining layer (∼ 40 nm after etching) was left on as it seemed to damage some devices, possibly due to dirt under the NbN film.. 3.1.2. Plasmon waveguides. A thin insulating layer of ∼ 10 nm Al2 O3 is deposited by Atomic Layer Deposition (ALD) to protect the SSPDs from the subsequent steps. The waveguides require a three layer mask because the substrate is not conductive at this point, as shown in 27.

(44) Chapter 3. Methods .

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(49) . . . .

(50).

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(53). Figure 3.1: Three layer fabrication procedure. a, Resist layers, top PMMA layer patterned by e-beam lithography. b, An SF6 RIE etch transfers the pattern to the tungsten layer, which acts as a stop-mask for the subsequent O2 plasma that defines the pattern in the S1805 layer. c, A gold is deposited using e-beam evaporation. d, Lift-off is performed in acetone and a 5 sec HNO3 dip.. Fig. 3.1. We spin a ∼ 450 nm layer of Shipley S1805 photo-resist, sputter a ∼ 10 nm layer of tungsten and spin a layer of ∼ 90 nm PMMA 950k. The waveguide pattern is e-beam written in the PMMA. The fact that the layer is thin allows a high pattern resolution. After developing in MIBK:IPA 1:3, the tungsten is removed with an SF6 RIE and the S1805 using an O2 plasma. This ensures that the substrate is very clean, so a 150 nm gold layer can be evaporated without using a sticking layer, which would make the plasmonic modes much more lossy. Lift-off is performed in warm acetone and followed by an HNO3 dip (5 sec) to remove photo-resist residues. For the quantum interference measurements, the last step is to add a thick Al2 O3 layer to make the environment more symmetric. The waveguides are first covered with a thin layer of ∼ 10 nm Al2 O3 deposited by ALD. Again the sample is not conducting, so a ∼ 7 nm layer of chrome is sputtered before spinning a ∼ 900 nm thick layer of PMMA. After e-beam writing and developing, the chromium layer is removed by wet etching and a ∼ 550 nm layer of Al2 O3 is sputtered. Finally, lift-off is performed in acetone and the remaining chromium is removed by wet etching. An extremely valuable resource for developing this fabrication recipe is was reference [4].. 3.2. Measurement setup. All measurement setups were designed and built by the author of this Thesis. The electronics was mainly designed by Raymond Schouten.. 3.2.1. Electronics. The SSPDs are biased using a home-made bias tee with built-in RC filters. The high frequency output is amplified by a 1 GHz bandwidth minicircuits ZFL-1000LN+ followed by a 1.45 GHz RF-Bay LNA-1450. For pulse counting measurements the signal is converted to TTL pulses using a comparator circuit and sent to a frequency divider. The divide-by-2 output is connected to a National Instruments USB-6216 card for counting. This card is also used to provide the bias current for the SSPDs and to set the discriminator level of the comparator circuit. All the electronics are housed a rackmountable chassis, as shown in Fig. 3.2a. Time-resolved correlation measurements 28.

(54) 3.2. Measurement setup. Figure 3.2: Measurement setup. a, Two-channel SSPD driver electronics including biastee, amplifiers and counting electronics. b, Sample mounted on attocube XYZ positioners with a microscope objective above it. The PCB has 2 SMC connectors for high-frequency coax lines. c, Inside of the dipstick, (b) is a zoom-in of the bottom. d, Optical table that is placed on top of the dipstick when immersed in liquid helium. e, Spontaneous parametric down-conversion setup.. are performed by directly sending the generated TTL pulses to a Picoharp 300 time correlated single photon counter.. 3.2.2. Cryogenic setup. Measurements are performed at 4 K with a dipstick in a helium transport dewar. The inside of the ∼ 1 m long dipstick is shown in Fig. 3.2c. It can be enclosed in a steel tube and is filled with helium exchange gas. A window is present at the top to allow free-space access. The sample is mounted on a printed circuit board with SMC connectors and placed at the bottom of the dipstick on Attocube XYZ slip-stick piezo positioner stages. A high NA microscope objective (Leitz Weizlar, NA 0.95) is placed right above the sample (Fig. 3.2b) and seems to perform well even at low temperature. It has been cooled down over a hundred times without being damaged, although properties like achromaticity deteriorate. Once cooled to 4 K, an optical table (Fig. 3.2d) is positioned on top of the helium dewar and the optics aligned. The optical table contains a white light source and sensitive camera (Watec 120N+) with a 30 cm lens that allows to image the sample and find the device of interest. Two fiber-coupled input paths are present to send beams to individually controllable 29.