The Half-Josephson Laser: Essentials and Applications

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The half-Josephson laser

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The half-Josephson laser

Essentials and applications

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 donderdag 9 januari 2014 om 12:30 uur

door

Franciscus Godschalk

Master of Science in de natuurkunde

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. Yu. V. Nazarov

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. Yu. V. Nazarov, Technische Universiteit Delft, promotor Prof. dr. ir. L. P. Kouwenhoven Technische Universiteit Delft

Prof. dr. F. W. J. Hekking Université Joseph Fourier

Prof. dr. F. Hassler RWTH Aachen Universität

Prof. dr. L. Kuipers Universiteit Twente/ AMOLF

Prof. dr. H. W. Zandbergen Technische Universiteit Delft

Dit proefschrift kwam tot stand met steun van de Stichting voor Fundamenteel On-derzoek der Materie (FOM), dat deel uit maakt van de Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO).

Printed by: Ipskamp Drukkers B.V.

Cover: Denise Godschalk What the uninitiated understands of the half-Josephson laser - in sand, latex and wool.

An electronic version of this dissertation is available at

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Preface

When I started to work on the ‘Josephson laser’ in 2010, I did not have much ex-perience in the field of mesoscopic physics, as my Master education was oriented at high energy physics. Superficially, high energy physics and condensed matter physics seemed two very different things to me. But what surprised me from the start of my research, was the resemblance between the concepts and theoretical techniques used in both fields: I found much more similarities than differences. The differences were found in an unexpected direction: condensed matter physics has a much richer phe-nomenology than high energy physics and yields endless possibilities to experiment on, and to engineer interesting states of matter. In the past four years I have learned many things about that other, fascinating world of physics. Together with my thesis, I consider this to be the most important fruit of four years of work.

For finishing this thesis and for having a good time in the Theoretical Physics group, in the department of Quantum Nanoscience, I am indebted to many people. Some of them I want to thank explicitly. Firstly, thank you, Yuli, for being my super-visor. You have taught me many things. Mainly about physics, but also about other things, like scientific writing and the virtues of bureaucracy. My way of thinking has been shaped by your example. I am always amazed how you dive into the essence of a problem, using - as it seems to me - only a small amount of relatively simple principles, and by careful and intuitive thinking. Also thank you, Fabian, for your involvement and your clear explanations in the first year of my PhD. Because of that I could start quickly on my research and we made a lot of progress, soon leading to my first publication. I enjoyed being part of the theory group. From the beginning in 2010, the atmosphere has been kind and inviting and I have appreciated that, even though I did not always join you in the multitude of informal activities. In partic-ular, thank you Alina, Ciprian, Chris and Fateme for the many, long and enriching conversations.

During the past four years I was fortunate to have my family close by, something which is indeed very different for most of my collegues. I am grateful for all your support. Most of all, thank you, Denise, for your love and patience.

Everything I have received during my PhD I believe is a blessing from my Lord and Saviour. I continuously thank him for that.

Frans Godschalk Delft, November 2013 vii

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I

Essentials

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2 Proposal for an optical laser producing light at half the Josephson frequency 29 2.1 Setup and model . . . 31

2.2 Semiclassics . . . 32

2.2.1 Scales . . . 33

2.3 Toy two-state model . . . 33

2.3.1 Two-level system . . . 33

2.3.2 Driving mechanism . . . 34

2.3.3 Stationary states of radiation . . . 35

2.4 Lasing . . . 36

2.5 Switching . . . 38

2.5.1 Decoherence . . . 38

2.6 Average power and current . . . 39 ix

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x Contents

2.7 Feasibility . . . 42

2.7.1 Many Josephson LEDS in a single cavity . . . 42

2.8 Conclusion . . . 43

References . . . 44

3 Lasing at half the Josephson frequency with exponentially long coherence times 45 3.1 Model . . . 49

3.1.1 The HJL with a single emitter . . . 49

3.1.2 The HJL with many emitters . . . 51

3.1.3 The average value of the dipole moment . . . 52

3.1.4 Phase dependence of the energy . . . 54

3.1.5 Energy . . . 55

3.1.6 Fokker-Planck equation . . . 57

3.2 Lasing . . . 58

3.3 Noise and its frequency dependence . . . 62

3.4 Large Fluctuations . . . 69

3.4.1 Trajectories and relation to Kramers’ escape problem . . . 69

3.4.2 Optimal paths and the principle of least action . . . 71

3.4.3 The action for the HJL . . . 74

3.4.4 Estimation of the action . . . 75

3.4.5 Numerical results . . . 76

3.5 Conclusions . . . 80

References . . . 82

II

Applications

85

4 Optical stabilization of voltage fluctuations in half-Josephson lasers 87 4.1 The half-Josephson laser . . . 88

4.2 Feedback . . . 90

4.3 Variance and stability . . . 92

4.4 Extended feedback scheme . . . 93

4.5 Locking two HJLs . . . 96

4.6 Conclusions . . . 98

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Contents xi

5 Light-superconducting interference devices 101

5.1 Introduction: The half-Josephson laser . . . 102

5.2 setups . . . 104

5.3 Single mode LSID . . . 106

5.4 Two-mode LSID . . . 109

5.4.1 Weak coupling limit . . . 109

5.4.2 Strong coupling limit . . . 110

5.4.3 Symmetric equations . . . 110

5.5 Periodic lasing cycles . . . 114

5.5.1 Stability . . . 114 5.5.2 Perturbative analysis . . . 116 5.5.3 Numerics . . . 117 5.6 Conclusions . . . 120 References . . . 121 Summary 123 Samenvatting 125 Curriculum Vitæ 129 List of Publications 131

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1

Introduction

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2 1. Introduction

1.1 _Coherence

Quantum physics has made the understandable inapprehensible. Until the twentieth century, physics used to be in harmony with the intuitions of everyday life. Problems like throwing a ball of a tower and calculating how it would accelerate and at what velocity it would hit the ground, were already for a long time convincingly solved. In the twentieth century, it turned out that the laws of physics change when the size of a ‘ball’ is decreased. The laws of the world of electrons and atoms are alien to the ones of everyday life. Even though the mathematical description of these microscopic laws is understandable, their intuitive meaning is inapprehensible. The language of everyday is not nearly adequate to make sense of an idea like the probability ampli-tude. The exact location of, for instance, an electron is fundamentally unknown until it is measured. The probability amplitude only gives the likelihood to find it at a par-ticular spot. Things get stranger. It is fundamentally impossible to know both where the electron is and how fast it is going. There is a trade off between these quantities. The better you know where the electron is the less you know its velocity. Things get stranger even more. The probability amplitude is like a wave. It has an amplitude and a phase and can therefore interfere with other probability amplitudes, or even with itself, and it can extend across barriers, or walls. The best way to circumscribe this weird reality is by a paradox called ‘wave-particle duality’. But this is hardly an explanation. It is only a description.

Even though the laws of quantum physics are fundamental, they are only required in the description of the microscopic world. Their effect on our everyday world is only visible indirectly, if at all. Many everyday phenomena are therefore perfectly understandable in terms of classical descriptions of physics. Sometimes however, the quantum laws become directly visible in ‘macroscopic’ objects. This happens for instance in a superconductor, which is known for its capability to carry electrical current without resistance. This is a direct result of quantum physics. The super-conductor is a macroscopic quantum object. A large number of particles occupies a single ‘macroscopic’ quantum state. These particles are coherent, meaning that their phases are fixed with respect to each other, so that also the macroscopic state carries a phase. The wave-particle duality applies to superconductors.

Another example of a macroscopic ’quantum object’ is the laser. Its operation is only understandable in terms of quantum physical concepts. It requires knowledge of the quantum mechanical structure of atoms and concepts like spontaneous and stim-ulated emission. Importantly, the light of a laser is a coherent state of light particles, photons, carrying a phase. Also here the wave-particle duality applies.

In this thesis the combination of these two coherent objects, superconductors and lasers, is studied. With the coupling between these two, their phases become locked to each other. This simple feature provides a rich playground for new physics and

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1.2. Superconductivity 3

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new devices.

This chapter provides a background for some important concepts in this thesis. The two kinds of coherence relevant in this work, are treated. First, a short intro-duction is given on superconductivity, including an overview of BCS theory and the physics of Josephson junctions. This is based, among others, on the standard book on superconductivity by Michael Tinkham [1]. After that, the results of a recent study are summarized in Sec. 1.3. This concerns a device proposal which is of particular relevance. In Sec. 1.4, an introduction is given to laser physics. The semiclassical approach is explained and exemplified, using a simple model. Furthermore, noise and coherence are discussed. We conclude with an overview of the thesis.

1.2 _{Superconductivity}

During several decades after the discovery of superconductivity, in 1911 [2, 3], a satisfactory theory has remained elusive and progress towards that theory was slow. Only in 1957 John Bardeen, Leon Cooper and John Schrieffer (BCS) presented their ground-braking microscopic theory and revolutionized the understanding of super-conductivity [4]. In 1911, however, the language needed for this explanation, quan-tum mechanics, was only in its infancy. The basic concept of a single particle wave function had yet to be invented, let alone the idea of many particle wave functions.

Phenomenologically, superconductivity is known for two hallmarks. The first one has delivered it its name: upon cooling below a certain critical temperature some ma-terials become a perfect conductor: they conduct without dissipation. This is what Heike Kamerlingh Onnes found in his original series of experiments. The second hallmark is an effect discovered some 22 years after the discovery of superconductiv-ity, by Walther Meissner and Robert Ochsenfeld [5]. They found superconductors to be perfectly diamagnetic. Magnetic fields are expelled from the superconductor, even when the sample is originally normal and then cooled through the critical tempera-ture. This latter property can not be explained by perfect conductivity only.

A first explanation to the phenomenology of the Meissner effect was given shortly after by the London brothers [6]. They developed an electrodynamic theory for super-conductors, in which magnetic field can only penetrate the superconductor from the outside for some characteristic length λL. An important implication of the Meissner

effect is that a critical magnetic field exists above which the superconductor becomes a normal conductor. Later Alexei Abrikosov showed [7] that another type of super-conductor can exist with two critical magnetic fields. For the magnetic field between these critical values, magnetic flux penetrates the superconductor in flux tubes. At the location of a flux tube the superconductivity vanishes, while it remains elsewhere in the material. The magnetic field is forced into flux tubes by vortices of supercurrent. An important step in the understanding of superconductors was made by the

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4 1. Introduction

modynamic description of Vitaly Ginzburg and Lev Landau [8] in 1950, seven years before BCS. They introduced a complex order parameter, ψ(x), that can be thought of as the wavefunction of the superconducting electrons, where the local density of electrons is |ψ(x)|2_{and the phase of ψ(x) is the macroscopic superconducting phase}

φ. Later, it was shown by Lev Gor’kov [9] that this theory is in fact a limiting form of the BCS theory, being valid near the transition temperature of superconductivity.

Bardeen, Cooper and Schrieffer were awarded a Nobel price in 1972 for their microscopic theory of superconductivity. A year later, Brian Josephson was awarded one for his ideas on coupled superconductors in what is now known as the Josephson effect. In the remainder of this section, an overview is given of the BCS theory and the physics of Josephson junctions.

1.2.1 The BCS theory

The BCS theory of superconductivity is a microscopic, quantum mechanical descrip-tion of a large number of electrons in the superconducting state. It builds on an im-portant idea of Leon Cooper [10], that the Fermi sea is unstable against the formation of bound pairs (Cooper pairs) in the presence of an attractive interaction, no matter how weak. The simplest case of s-wave pairing was considered, where the lowest energy configuration is a spin singlet with zero total momentum. The origin of the attractive interaction in BCS superconductors is the electron-phonon interaction. It is in competition with the repulsive Coulomb interaction between electrons. Below (above) the critical temperature the net interaction between the electrons is attractive (repulsive). Because of the attractive interaction, the superconductor minimizes en-ergy by forming Cooper pairs, up to the point that the binding enen-ergy for an extra pair is zero. We note that also other kinds of superconductors were found, exhibiting other kinds of mechanisms to generate the attractive interaction [11] and with other kinds of pairing symmetries [12].

The groundstate of the superconductor is a superfluid, or condensate, of Cooper pairs. The Cooper pairs occupy a single quantum state. In the BCS theory, a mean field approach is used to find this state. This approach entails that the occupancy of a Cooper pair state, represented by momentum k, is only determined by the average occupancy of all other states. This results in the groundstate [1]

|ψGi= Y k uk+ vkˆc † k,↑ˆc † −k,↓ |Φ₀i, (1.1)

with |Φ0i the vacuum state and ˆc†_k,σ, the creation operator of an electron with spin

σ =↑, ↓ and momentum k. The coherence factors ukand vkdiffer by a phase φ, which

is the phase of the macroscopic condensate wavefunction and they satisfy |uk|2 +

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with (k ↑, −k ↓) is (un)occupied with a probability (|uk|2) |vk|2. Using a variational,

self-consistent approach, BCS showed that the coherence factors are given by |uk|2= 1 2 " 1+Ek ξk # = 1 − |vk|2, ξk= q E2 k+ |∆k| 2. _(1.2)

The norm of∆k can be understood as an energy gap, while its phase is the earlier

introduced superconducting phase φ.∆krepresents the “mean field” influence of the

superconducting condensate on the Cooper pair state k. The energy Ekis the single

particle energy, as counted from the chemical potential, µ. As might be expected from the expression of |ψGi, the number of Cooper pairs in the condensate, N, is

not fixed. In fact, it is conjugate to the phase of the condensate, φ. They satisfy an uncertainty relation∆N∆φ & 1. Hence, a well defined number of particles, means a completely uncertain phase, and vice versa. Fluctuations about the mean-field so-lution are expected to be very small considering the macroscopic number of Cooper pairs involved.

The BSC superconductor can be accurately described using a mean field Hamil-tonian

HSC=

X

k,σ

Ekˆc†_k,σˆck,σ+ ∆kˆc†_k,↑ˆc†_−k,↓+ ∆∗kˆc−k,↓ˆck,↑, (1.3)

where the “mean field”∆k= PlVklhψG|ˆc−k,↓ˆck,↑|ψGi is nonzero because of the

coher-ence in the superconductor. Here, Vklis an attractive potential in the superconducting

state, which is taken constant in the BCS theory,∆k= ∆. The definition for ∆ is in fact

a self-consistency condition, since |ψGi depends on∆. The phase of the condensate

is assumed to be well defined in this approach, thus leaving the number indefinite.

Quasiparticle excitations

The mean field Hamiltonian is particularly useful when considering excitations in the superconductor. Using the canonical Bogoliubov transformation [13] it can be diagonalized HSC= X k Ek−ξk+ ξk h ˆγ†_k0ˆγk0+ ˆγ † k1ˆγk1i , (1.4)

with Bogoliubov operators ˆγ†_k0= u∗_kˆc†_k↑−v∗_kˆc−k↓and ˆγ†_k1= u∗_kˆc†_−k↓+v∗_kˆck↑. The operators

ˆγk0 and ˆγk1 both have |ψGi as the vacuum state, while their conjugates create an

excitation. These excitations are a superposition of an electron and a hole. The weight of the electron part and the hole part in the quasiparticle excitation depends on its energy. If this energy is much larger than |∆|, an excitation at positive (negative)

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6 1. Introduction

energy is predominantly of an electron (a hole) character, because then |uk| ≈ 1 (|vk| ≈

1). As a result of this mixing of electrons and holes, the charge of the quasiparticle is not a well defined quantity and depends on the energy of the excitation. To ensure overall charge conservation Cooper pairs have to be added to or removed from the superconductor additionally.

The quantity∆ acts like a gap in the excitation spectrum of the superconduc-tor. In an isolated superconductor, electron conservation demands that excitations are created or destroyed in pairs, of which the simplest example is the equivalent of an electron hole pair. The corresponding excitation energy is ξk+ ξk0 ≥ 2|∆|. For that

reason |∆| is interpreted as an energy gap in the superconductor.

Real superconductors always contain a finite density of quasiparticles. One could expect that these consist purely of thermally excited quasiparticles. For temperature going to zero, however, the quasiparticle density saturates, in stead of going to zero [14]. The mechanism for this is still the topic of debate, but it is thought that non-equilibrium processes may cause this saturation [15].

Proximity effect

Superconducting order is not purely confined to the superconductor. The wave func-tion of the superconducting groundstate extends some range outside the supercon-ducting material, in the form of an evanescent wave. If a normal metal is within this range, its density of states is affected by the superconductor. Close to the interface an energy gap exists in the metal, falling of with the distance from the interface [16]. The gap carries the phase of the superconductor and is of the order of the Thouless energy ETh ' ~/τ, with τ the typical time an electron dwells in the normal metal, before it

reaches the superconductor [17]. This extension of superconducting correlations into non-superconducting materials is called the proximity effect [18].

1.2.2 Josephson effect

We have considered an isolated superconductor and described it using a single com-plex parameter,∆. What happens if we let two or more superconductors interact with each other, by bringing them into close proximity?

From a fundamental point of view, superconductivity can be understood as a phe-nomenon where the global U(1) symmetry is spontaneously broken in the region of space occupied by the superconductor. With an unbroken U(1) symmetry, in the nor-mal state, the phase of a wavefunction is arbitrary. It can be chosen at will without changing the Hamiltonian of the theory. This is in contrast to the ‘broken case’, where the choice of phase determines the form and parameters of the Hamiltonian. When considering a single superconductor, this is not very relevant, because the physics of the superconductor must still be independent of the choice of phase. In the case of multiple coupled superconductors, however, the relative phases become important.

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One of the phases can be chosen at will by using the local U(1) gauge symmetry,

associated to the electromagnetic field1_{. The other phases have their value with}

re-spect to this single phase. Generally, the Hamiltonian is thus phase dependent. Only because of the interaction terms, which contain phases of multiple superconductors, this results in a measurable dependency on phase differences.

Brian Josephson was ‘fascinated by the idea of broken symmetry, and wondered whether there could be any way of observing it experimentally.’ Inspired by experi-ments on tunnel barriers he ‘realized that (. . . ) a supercurrent should be a function of ∆φ.’ [20] He was the first to predict the phenomenology of two superconductors that are coupled through a tunnel barrier [21]. He reasoned that depending on the phase difference, Cooper pairs tunnel through the barrier and argued that, as a result of that, a dc supercurrent flows without the application of a bias voltage. Additionally, he argued that with the application of a bias voltage, an ac supercurrent flows. The fre-quency of the current being 2eV/~. He showed that a dc supercurrent can also be obtained when an ac voltage is applied with frequency ν, in addition to a dc voltage, V, such that they satisfy ~ν = 2eV. As mentioned before, the impact of his work has delivered Josephson the Nobel prize for Physics in 1973.

Josephson relations

Richard Feynman provided an intuitive, phenomenological derivation of the Joseph-son relations [22], which is largely followed here. Microscopically, the Josephson junction can be understood in terms of Andreev bound states, [18,23]. Here, we con-sider two superconducting leads described by an order parameter, ψi, with i = 1, 2.

They are coupled by a tunnel barrier. This setup is now known as a Josephson junc-tion. Furthermore, we assume a bias, V, across this juncjunc-tion. Choosing the reference point of the potential conveniently at the center of the tunnel barrier, the Schrödinger equation of this junction reads,

i~dψ1

dt = eVψ1+ kψ2, i~ dψ2

dt = −eVψ2+ kψ1, (1.5)

with the coupling between the superconductors given by k, which’ value is char-acteristic for the tunnel junction. The prefactors eV correspond to the electrostatic potential of the Cooper pair with respect to the reference point. These equations can be rewritten in terms of the Cooper pair densities ni= |ψi|2and the phase difference,

χ = φ1−φ2. The supercurrent is given by the charge passing through the barrier per

unit time. It is proportional to the densities, I = ˙Q = D˙n1 = −D˙n2, with some

con-stant D that depends on the properties of the tunnel barrier used. Defining the critical

1_{The spontaneous breaking of the global U(1) symmetry implies that the excitations of the electromagnetic}

field become massive. This is known as the Higgs mechanism. Photons only penetrate the superconductor on a finite length scale, which explains the Meissner effect [19].

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8 1. Introduction current IC = (2kD/~) √ n1n2, we find I= ICsin χ, χ = eV/~,˙ (1.6)

which are respectively the first and second Josephson relation. As predicted by Josephson, at zero voltage bias, a constant supercurrent flows through the junction, depending on the sine of the phase difference across the junction. This is also called the dc Josephson effect. The critical current is the maximum current bias that can be applied to the junction without dissipation. If a larger current is applied, a dc voltage can be measured over the junction.

Josephson oscillations

The Josephson junction also supports ac supercurrents. With a dc voltage bias ap-plied, the phase grows linearly in time, resulting in an oscillating supercurrent at a rather high frequency, ' eV/~ [Eq. (1.6)]. These Josephson oscillations are also re-ferred to as the ac Josephson effect. There is also an inverse ac Josephson effect, where a dc current is generated using an ac voltage on top of a dc voltage. One way to apply this ac voltage, is by irradiating the Josephson junction with microwave ra-diation, with frequency, ν. With the voltage being a simple sinusoid, the supercurrent can be rewritten in terms of Bessel functions and sinusoids. The result is proportional to a sum over n of the n-th order Bessel function times sin[χ+ 2eVt/~ − nνt]. This current has a dc component if ν= 2eV/n~. The supercurrent is expected to synchro-nize to the irradiation, leading to steps in the I − V characteristic of the Josephson junction. This was first shown by Sidney Shapiro, who studied experimentally the ac Josephson effect [24].

In the inverse ac Josephson effect, the Josephson junction absorbs radiation co-herently. Then, microscopic reversibility requires it to also emit radiation [25]. In a coherent transfer from the superconductor at the high potential to the one at the low potential, a Cooper pair has an excess in energy of 2eV which is emitted as a single photon [1]. This is called Josephson radiation. Naturally, the energy of Josephson radiation is limited by the superconducting gap |∆|, since above this gap, quasiparti-cles can excited incoherently. For a gap on the order of 1 meV, photons range up to Thz frequencies. Interestingly, Shapiro steps are not only visible in the absorption of radiation, but also in its emission [26].

The electromagnetic environment of a Josephson junction causes decoherence of the superconducting phase difference. Generally, the potential difference, V, between two weakly coupled superconductors depends on both the electrostatic potential and the vector potential [1], which are subject to noise. Because of the second Josephson relation, the noise transfers to the superconducting phase difference, which drifts as a result.

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Hysteresis

The Josephson relations describe the essential behavior of the Josephson junction. This is, however, not enough to explain the varying phenomenology of the different types of Josephson junctions that are experimentally realized. A more realistic de-scription also accounts for (intrinsic) capacitance and resistance in the junction. This is done in the ‘Resistively and Capacitively Shunted Junction model’ (RCSJ model) [27], consisting of a parallel circuit of a Josephson junction, a resistor, R, and a ca-pacitor, C. Using the law of current conservation and the second Josephson relation, to relate voltage to phase, a differential equation for the phase is derived. The model is equivalent to that of a driven and damped swinging pendulum. In this model, the junction capacitance is responsible for hysteretic behavior. With a linear sweep of bias current, starting above the critical current and decreasing, the voltage does not drop to zero at IC, but at a significantly smaller value of the current. However, upon

increasing the bias current again, a dc voltage will only appear at IC. A hysteretic

junction is called underdamped and occurs when ICR2C > ~/2e. There is also

an-other mechanism for hysteretic behavior: it was demonstrated that it can also be of thermal origin [28].

SQUID

The Josephson junction is the basis for various important devices. For instance, the nonlinearity in this junction makes it very suitable for several kinds of superconduct-ing qubits [18]. One particular device, which is relevant for this thesis, is the ‘Su-perconducting Quantum Interference Device’ (SQUID), in which interference of two supercurrents occurs. The simplest realization of this device is called a dc SQUID. Essentially this is a parallel superconducting circuit of two Josephson junctions. Su-percurrent that enters the SQUID splits up and divides over both arms. After passing the Josephson junctions, the currents combine again. In a classical device, the total current that flows through the parallel setup depends only on ‘local’ properties of the elements in the circuit. In the SQUID, also quantum effects play an important role, because of the coherence of the superconductors, yielding a ‘global’ contribution to the current. A magnetic flux in the superconducting loop gives rise to a geometrical phase. A Cooper pair going through one arm picks up a different phase value than one going through the other arm. Upon combining the supercurrents from the two arms in the SQUID, the total geometrical phase is simply given by the total magnetic flux,Φ, through the superconducting loop, 2πΦ/Φ0. Here,Φ0= π~/e is the quantum

of flux. The geometrical phase comes in addition to the phase drops associated to the Josephson junctions,∆φ. The total phase acquired by a supercurrent going once around the loop must be zero, so that∆φ = φ1−φ2 = −2πΦ/Φ0.

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10 1. Introduction

is I= IC[sin φ1+ sin φ2], which becomes

I= 2ICcos[πΦ/Φ0] sin[φ2+ πΦ/Φ0]. (1.7)

The form of this relation is equivalent to the first Josephson relation, with the critical current given by 2ICcos[πΦ/Φ0]. Hence, this device can be seen as a flux tunable

Josephson junction.

The SQUID is extremely sensitive to small changes in magnetic fields. Assuming a uniform magnetic field, B, penetrating the superconducting loop, the flux is given by BA, with A the area of the loop. The peak to peak distance of the flux sensitive critical current corresponds to a change in magnetic field of δB= 2Φ0/A. Hence, a

larger loop area yields a higher sensitivity. For an area of ∼ 0.1 cm2, δB ∼ 10−10 Tesla.

1.3 _{Josephson Light Emitting Diode}

The work of this thesis builds on a device proposed in 2009, named the Josephson light emitting diode [29]. Because of the importance of this device for the work of this thesis, we summarize its essentials in this section.

It is instructive to briefly provide a context in which this device proposal has arisen. Since the proposal of the Josephson junction and its first realization, the research on the physics and applications of this device has expanded enormously. Overviews of the advances in Josephson physics are found in Refs [30,31]. The combination of superconductors and semiconductors is very desirable. The advan-tage of such combination lies in particular in the ability of the modern day technology to engineer all kinds of semiconductor devices and nanostructures. Incorporated in a Josephson junction, these structures determine the transport properties of the junc-tion, allowing for instance to design new kinds of qubits [33] or exotic states of matter [34]. It has not been possible for quite some time to combine super- and semicon-ductors. The challenges lie in creating superconductor-semiconductor interfaces with a high transparency, which is crucial to access properties related to superconductiv-ity [32]. Solutions to these issues have been found first in the context of quantum wells [35] and nanowires [36]. In these first proposals, n-type semiconductors were coupled to superconducting leads, with low Schottky barriers. Recently, a d-wave superconductor - p-type semiconductor tunnel diode was demonstrated [37], paving the way for proximity induced superconducting gaps in p-type semiconductors [38]. The paper covered in this section is about the combination of a Josephson junc-tion and a p-n semiconductor light emitting diode (LED). The ability to engineer semiconductor devices that emit light by electron-hole recombination [39], is crucial in modern day semiconductor technology. Current research efforts have a focus on

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1.3. Josephson Light Emitting Diode 11

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increasing the control on the quantum properties of the emitted light. For the

pur-poses of quantum information, semiconductor quantum dots are used to generate on demand single photons or entangled photon pairs [40]. Also in these light emitting devices, the coupling to superconductors proves beneficial. Recent proposals [41] and experiments [42] have shown that the superconducting coherence can facilitate superradiance effects in LEDs. In these cases, only the n-type semiconductor of the LED was coupled to a superconductor. In the proposal discussed in this section, both the n-type and p-type parts of the LED are coupled to superconductors.

In short, the Josephson light emitting diode (JoLED) is envisioned to be a super-conductor - LED - supersuper-conductor heterostructure. The LED is a p-n semisuper-conductor nanowire containing two single level quantum dots (QDs) in the depletion layer [43]. The n-type QD accommodates up to two electrons, with opposite spins, while the p-type QD accommodates up to two holes with opposite spins. Owing to the proxim-ity effect, superconducting correlations are induced in the QDs. The device is biased with a voltage V close to the band gap of the semiconductor, which is in the optical frequency range. In a regular Josephson junction, such a bias would undo the ef-fects of the superconducting correlations on the device, as the voltage is much larger than the superconducting gap in the leads, |∆SC|. In such case, Cooper pairs from

the superconductor at the high chemical potential, would go to the other supercon-ductor incoherently as quasiparticle excitations. This is not the case in the JoLED. Because of the potential barriers induced by the LED, charge transfer is only possi-ble by electron-hole recombination, resulting in the emission of a photon with energy ' eV. Since, the electrons and holes are correlated with the superconductors the emit-ted light can also carry such correlation. We will refer to the emission at energy ' eV as being “red”. In this setup it is also possible to obtain “blue” photons at an energy ' 2eV. In a particular process the “blue” photons are coherently generated Josephson radiation with energy 2eV.

1.3.1 Hamiltonian

The JoLED is modeled using an effective Hamiltonian consisting of three parts. The first two parts describe respectively the electron and hole QDs, while the third part represents photon emission and absorption by electron hole annihilation and creation. The QD Hamiltonians have the same form. For the electron QD we have

HQDe = Ee X σ ˆc†_σˆcσ+ ∆eˆc†_↑ˆc†_↓+ ∆∗eˆc↑ˆc↓+ Ueˆne↑ˆn e ↓, (1.8)

with cσ the annihilation operator and ˆne

σ the number operator of an electron with

spin σ = ±1/2, Eethe energy required to put an electron on the QD counted from

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12 1. Introduction

interaction between two electrons, and∆ethe proximity induced pair potential at the

QD, which has the same phase, φe, as the corresponding superconducting condensate,

but a much smaller amplitude, |∆e/∆SC 1. We note that each QD only couples to

one superconducting lead. The energies Ee, Ue, |∆e|, are assumed to be of the same

order of magnitude. The Hamiltonian He

QD can be diagonalized by a Bogoliubov transformation, as

described in previous section. The expression for the energy in the coherence factors, in Eq. (1.2), is different though, Ek→ ˜Ee≡ Ee+Ue/2. Diagonalization results in two

degenerate single particle states, |σie= ˆc†σ|0ie, with energy Ee, and two superposition

states of the unoccupied and doubly occupied states. The ground state singlet reads |gie= −e−iφe|ue||0ie+ |ve||2ie, (1.9)

with ueand vethe superconducting coherence factors as in Eq. (1.2). The eigenenergy

is εg= ˜Ee− ˜E2e+ |∆e|21/2. For the excited state the eigenenergy is εex= ˜Ee+ ˜E2e+

|∆_e|21/2_{, while the corresponding singlet state reads}

|eie= e−iφe|ve||0ie+ |ue||2ie. (1.10)

The effective Hamiltonian for the holes is similar to that of the electrons. It is obtained by replacing the indices e → h and operators ˆcσ→ ˆh†−σ. The unoccupied state of the

single level hole QD is therefore defined as being occupied by two electrons. An attractive interaction between electrons and holes is neglected in this scheme.

Electron-hole pairs, or excitons, can recombine radiatively. This happens most efficiently in direct band gap materials. The holes are usually considered to be of the “heavy” type [44], for which the spin quantum number is ±3/2. Radiative recombina-tion requires the total exciton spin to be ±1. Excitons with spin ±2 are called dark, as they do not couple to the electromagnetic field. The polarization of the emitted light depends on the symmetry properties of the material used. For instance [45], a cylin-drical QD has a D2d point symmetry group which results in a degenerate excitonic

doublet, resulting in the emission of circularly polarized light, | ± 1i, upon exciton recombination. For a lower symmetry, the doublet splits, resulting in linearly polar-ized emission, with the states (anti)symmetric combinations of | ± 1i. This happens, for instance, owing to the exchange interaction related to QD asymmetry. Ref. [29] assumes emission of circularly polarized light, described by Hamiltonian

Hint= G X q ˆb† q,−ˆh↓ˆc↑+ ˆb†q,+ˆh↑ˆc↓ e−ieVt+ h.c. (1.11)

Here ˆb†q,± creates a photon in mode q with polarization ±1, and G is the coupling

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1.3. Josephson Light Emitting Diode 13

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potential of the two leads. In the models introduced in chapters 2 and 3 the photon

emission is assumed to be linearly polarized in a single direction. The corresponding Hamiltonian reads Hlin= G Pqˆb

†

q[ˆh↓ˆc↑+ ˆh↑ˆc↓] exp(−ieVt)+ hermitian conjugate.

The effective Hamiltonian conserves the parity of the number of excitations in the QDs. If the total number of electrons and holes is even (odd), it will remain so. In reality, however, the parity is not conserved. Rare processes occur, where photon emission is accompanied by the excitation of a quasiparticle in one of the superconducting leads. Necessarily, the parity of the QD state changes in the course of such a process. Compared to ‘normal’ parity conserving photon emission, the rate of this process is suppressed by a factor |∆e,h/∆SC| 1.

1.3.2 Emission cycles

The mixing of the unoccupied and doubly occupied QD states, by the superconduc-tors, enables emission cycles. Since parity is conserved in the Hamiltonian, we refer to these cycles as ‘even’ or ‘odd’ parity emission cycle, depending on the parity con-sidered. We consider the “red” even parity emission cycle, of which an illustration is shown at the left hand side of Fig. 1.1. Here, the state on the QDs is one of the four singlet states |αie|βih, with α, β = g, e, or one of the degenerate doublet states,

indicated by |1ie|1ih, which are assumed to be radiative, and not dark.

In a single emission cycle, two “red” photons are emitted, with opposite polar-ization. Assuming it to start in one of the four singlet states, the QD state changes respectively to one of the doublet states and again to one of the four singlet states. This cycle is reminiscent of a biexciton-exciton cascade in a normal QD [46]. Nor-mally, the QD is in the ground state after such a cascade. A new emission cascade first requires the excitation of two excitons. Continuous emission therefore requires pumping. In the case of the JoLED no such excitation is required, as all singlet states have a transition dipole moment to and from the doublet states. This results from the mixing of states in the QDs by the superconductors. The emission cycle can therefore continue without pumping.

Generally, all optical transitions have a different frequency, leading to many dis-tinct sharp peaks in the emission spectrum of the device. The above mentioned parity changing processes lead to a low intensity continuous spectrum at frequencies below ' eV − |∆SC|.

1.3.3 Josephson radiation

The JoLED is capable of emitting light at exactly the Josephson frequency, ωJ =

2eV/~. In the case of the “red” emission, each Cooper pair transfer through the JoLED is accompanied by the emission of two photons, each associated to the annihilation of one electron hole pair. To obtain Josephson radiation, only a single photon has to be emitted. This requires nonradiative annihilation of one exciton. In a sense,

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14 1. Introduction

ω, G

ω

', H

ω

J

2eV

eV

0 E

Figure 1.1: Even parity emission cycle. Horizontal lines represent energy levels of the eigenstates of the QDs. Four singlet states and two doublet states are involved in this cycle, as described in the main text. The vertical dashed lines represent emission processes. On the left hand side, two red photons are emitted in a single emission cycle, with opposite polarization, ±, ∓, and frequencies ω, ω0' eV/~. The right hand side shows the emission cycle for coherent Josephson radiation. The initial and final states are the same, while the intermediate state, |1ie|1ih, is only occupied virtually. The solid (dashed) arrows going to and

coming from the intermediate state represent the emission of a real (zero-frequency) photon.

one of the electron hole pairs emits a photon at ' ωJ, while the other emits a

zero-frequency photon. This is possible in the presence of a static in-plane electric field. If the initial and final states of this emission process are different, the photon frequency is not exactly ωJ, and the photons are incoherent. Emission processes with the same

initial and final states are always coherent, with the photon frequency exactly ωJ[see

Fig. 1.1]. This process produces Josephson radiation at optical frequencies.

The probability amplitude for Josephson radiation depends on the phase di ffer-ence between the superconducting leads, φe−φh. In a setup with two JoLEDs in the

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1.4. Laser physics 15

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depending on the magnetic flux through the superconducting loop.

1.4 _{Laser physics}

In this thesis, a new kind of laser is investigated. It is therefore sensible to first provide an introduction to the ‘common’ laser and explain its essentials. The founda-tional principle of the laser was first devised by Albert Einstein, when he considered the possibility of spontaneous and stimulated emission [47]. The first device that uti-lized stimulated emission, the ammonia maser, was reauti-lized by Charles Townes in 1953. It was a microwave amplifier, using ammonia molecules as a gain medium, and it yielded a pulsed output. Here, the word ‘maser’ is an acronym for ‘Microwave Amplification by Stimulated Emission of Radiation’. In the same time, Nikolaj Basov en Aleksandr Prokhorov worked on a continuous wave maser, which was realized in 1954. In 1964, these three gentleman received the Nobel price in physics for their achievements.

The ‘optical maser’ (or laser) was first considered in 1958 by Arthur Schawlow and Charles Townes [48]. The first laser was realized by Theodore Maiman who then worked in the Hughes Research Laboratories. His device used Ruby as a gain medium and delivered light pulses at a wavelength of 694 nm. Since that first realization, many kinds of lasers have been realized, based on many kinds of gain media and for a wide range of radiation frequencies and power densities. Because of its versatility, the laser has become an indispensable tool, not only for the practices of science, but also in industry and in common electrical appliances.

Despite the versatility in its appearance, the fundamental operation principles of the laser are always the same [49]. Generally, there are two kinds of lasers. The ones with a continuous wave output and the ones with pulsed output. For now we consider the operation of a single mode, continuous wave laser. Here, an ensemble of atoms populates a single electromagnetic mode with photons, by stimulated emission. To obtain steady state lasing, there always needs to be a majority of atoms which is in the excited state. This is known as ‘population inversion’. It is achieved by a process called pumping, where the atoms are forced to undergo transitions to states with a higher energy. The optical field in the mode is represented by a standing wave electric field. From the point of view of quantum mechanics, the field is a coherent state, of which the coherence time is proportional to the average number of photons in the mode [50].

To isolate a single electromagnetic mode for lasing, a resonator, or cavity, is used, in which the gain medium is placed. The resonator alters the mode density of the electromagnetic field in a certain volume L3. In the ideal case of a bare cavity, only photons are allowed of which a half integer multiple of the wavelength fits the dimen-sion of the cavity, L. This results in a cavity spectrum of delta function peaks at half

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16 1. Introduction

integer multiples of the minimum frequency νm= (m/2)ν0= mc/4L, with c the speed

of light and m an integer. In the non-ideal case, these peaks are broadened, because photons escape from the cavity. The peaks are Lorentzians with full width at half maximumΓm. This is the decay rate at which photons escape from the resonator. It is

related to the quality factor, which is defined as Q= νm/Γm. An alternative quantity,

the finesse of the cavity, is related to Q. It is a measure for the distinguishability of the peaks in the spectrum of the resonator and is given by∆ν/Γm, with∆ν the free

spectral range. A high value of the finesse means clearly distinguishable peaks. The mode density in the resonator is also determined by the volume, V. This is relevant when considering the coupling strength of a dipole, d, to the resonant mode. The coupling strength is given by the energy of the dipole with respect to the vacuum expectation value of the electric field in the resonator, G ' d√~ν/2εV, with ε the permittivity of the gain medium. Therefore, a smaller mode volume yields a larger coupling strength. This makes microcavities, which have a small volume, important tools in the study of strong coupling phenomena in resonators [51]. One particularly relevant effect, related to the enhancement of the mode density, is the Purcell effect [52]. It describes the enhancement of spontaneous emission by a factor P = (3/4π2_)(λ/n

r)3(Q/V), with λ the wavelength, and nr the refractive index of the

gain medium. A decrease in volume corresponds to an increase in the spontaneous emission rate.

The remainder of this section is as follows. A semiclassical description of a con-tinuous wave laser is derived, after which an example is given of a three level laser. Then, the pulsed laser operation is shortly discussed, followed by a section on noise and coherence in the laser.

1.4.1 Semiclassical description

Let us consider a laser consisting of a resonator, with a single resonance frequency ν and a cavity decay rateΓ, that is filled with a polarizable medium. When polarized, the medium provides a dipole moment to drive the resonant mode. Typically, the number of photons in the resonator mode will be much larger than one, so that a semiclassical description is appropriate.

The semiclassical equations of motion can be derived from a quantum descrip-tion, involving the density matrix of the system, ρ. To model the laser, we consider a system consisting of a single quantum harmonic oscillator at frequency ν (the res-onator), coupled to an ensemble of N dipoles (the gain medium). The system is, through the resonator, weakly coupled to a thermal bath of quantum harmonic oscil-lators, representing the coupling to the quantized electromagnetic field in free space. The density matrix for the system-bath evolves coherently according to the equation of motion i~ ˙ρfull = [H, ρfull], with H the full Hamiltonian of the model. The dynamics

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a partial trace of the equation of motion over the degrees of freedom of the bath. The

resulting equation of motion is further simplified by applying two approximations. The Markov approximation assumes that on relevant timescales, where significant changes of the single mode occur, no correlations exist between the bath and the sys-tem. The rotating wave approximation assumes fastly rotating terms in the equation of motion to be zero. Defining ρ to be the partial trace of ρfull, we find the master

equation [53] dρ dt = −i ~ Hsyst, ρ + Γ− ˆb †_{ρˆb −}1 2 _ˆbˆb†_{, ρ} ! + Γ+ ˆbρˆb†−1₂ˆb†ˆb, ρ ! , (1.12)

with ˆb†the photon creation operator for the resonator mode and with the anticommu-tator brackets {A, B}= AB + BA. The rate Γ₊(Γ−) is the transition rate to a state with

one photon less (more) in the mode. The ratio of these is given by Boltzmann’s law, Γ+/Γ−= exp(~ν/kBT). The difference relates to the cavity decay rate

Γ = Γ+−Γ−=

2π ~

|g(ν)|2D(ν). (1.13)

Here, g(ν) is the coupling strength of the resonator to the bath and D(ν) the density of states of the bath, both taken at the resonance frequency.

The Hamiltonian Hsyst consists of three parts. The free evolution of the mode

and of the gain medium, which consists of N quantum emitters, and the interaction between these. In the most general form, it reads

Hsyst= ~νˆb

†_ˆb₊X

jk

Ejkdˆ†_jkdˆjk+ Gjk(ˆb†dˆjk+ ˆd†_jkˆb). (1.14)

Here, ˆd†_jkraises the state k of the quantum emitter j to k+1. Generally, the interaction depends on the strength of the electric field at the position of the quantum emitter, and at the state of the emitter. Hence the indices in Gjk.

The quantity b = hˆbi is used to represent the classical electromagnetic field in the resonator mode. The magnitude of the electric field is E ' √~ν/εV Im b, while that of the magnetic field is cB ' √~ν/εV Re b, with c the speed of light. The total energy in the mode corresponding to these fields is ' ~ν|b|2= ~νn, with n the number of photons. This definition n= |b|2is valid for coherent states. Indeed classical laser fields are coherent states [54]. Often in this thesis, we refer to b as the ‘electric field in the resonator mode’ or ‘the optical field’.

Let us derive the semiclassical equation of motion for b. To this end, we apply a time derivative to b and use Eq. (1.12), with Eq. (1.14), and the canonical

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commuta-{ {

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18 1. Introduction

tion relations for ˆb, to derive ˙b= − iν +Γ 2 ! b − iX j Gj ~ hdji, Gjhdji= X k Gjkh ˆdjki. (1.15)

Here, hdji is the average dipole moment of emitter j and Gj = PkGjk. A second

equation exists which is complex conjugate to this one. In the context of lasers, the dipole moment is proportional to the electric polarization of the gain medium, which is proportional to the electric field in the resonator, b. On the other hand, the very same dipole moment supports the electric field b where it depends on. Therefore, these equations are usually referred to as self-consistency equations.

A steady lasing state is found when ˙b = 0 while b , 0. This is only possible when the cavity losses, Γ, are replenished by the dipole moment. In particular, it requires ~Γ = 2P

jGjIm[hdji/b] ≡ ~gd(n): the losses being equal to the gain. Let

us now suppose that the resonator starts being empty, but is filled with photons by the gain medium. If hdji is linear in b the photon number grows exponentially with

a constant rate gd(0) −Γ. Any kind of polarizable medium is, however, expected

to saturate when the electric field becomes strong enough. With increasing photon number, when nonlinearities become important, gd(n) decreases. Steady state is then

achieved for the number of photons nssuch that gd(ns)= Γ. This value of the gain is

also called the saturated gain. The gain at n= 0 is called linear gain [55]. Generally, lasing occurs when the linear gain is larger thanΓ. The lasing threshold is therefore defined by gd(0)= Γ.

The real part of the dipole moment affects the frequency of the mode. In a steady state ~ν = −P

jGjRe[hdji/b]. Typically, if the frequency of the photon emitted by

the quantum emitter differs slightly from the mode frequency, the latter frequency shifts towards the former. This effect is called mode pulling.

Three-level laser

It is instructive to demonstrate the principles discussed above using an example of a gain medium. Additionally, this allows us to demonstrate a feature occurring often in lasers, named power broadening.

We consider a simple example with N identical quantum emitters, each with three energy levels. This is the minimum number of levels needed. When using only two levels, an electric field only causes coherent oscillations between them. For example, in case of a single dipole moment, an oscillation occurs between the excited state with no photons in the mode, and the groundstate with a single photon in the mode. With three levels a different situation can be achieved. Two of the states are related to the optical transition. Let us call these |hi and |li, for ‘high’ and ‘low’ energy. The energy difference between these states is ' ~ν. The third state is auxiliary and should

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not be involved in the optical transitions. Its use resides in the following. Suppose an

emitter is in the state |hi and because of coherent oscillations it oscillates to |li, while also an extra photon is created in the mode. If nothing else happens, the oscillation proceeds and the original state is recovered again. This cannot happen if, on the other hand, a transition occurs from |li to the third state. With another transition, from the third state to |hi, the emitter is in the original state again (neglecting saturation effects for the moment), while a photon is added to the mode. This to works if the transition time from |li to the third state is faster than the coherent oscillation. Furthermore, one of the transitions needs to be the result of pumping, because energy is added the resonator mode.

The three-level emitter can be modeled as an effective two level emitter. For this, we take two phenomenological rates,Γ↑ andΓ↓, for transitions |hi → |li and

|li → |hi respectively. The energy difference between these states, ~νE, is assumed

to be close to but not necessarily equal to the photon energy, ~ν. This difference results in a change of photon frequency or mode pulling. Assuming equal coupling constants, the interaction between dipole moment and resonator mode reads Hint =

GP

j(b∗dˆje−iνlht+ b ˆd†eiνlht. This Hamiltonian is taken in a rotating frame of reference,

such that the free Hamiltonian of the two level system is zero while the frequency of the mode has reduced to ν −Ω. Here, Ω is the oscillation frequency of the electric field in the mode. In this frame νlh= Ω − νE.

The density matrix of a single emitter evolves according to the usual nonhermitian evolution equation of an open system. The Hamiltonian and the density matrix of the single emitter read

H= G 0 be iνlht b∗_e−iνlht ₀ ! , ρ = ph phleiνlht plhe−iνlht pl ! ,

with ph(pl) the probability to be in state |hi (|li). The transition probabilities plhand

phl are related by complex conjugation. DefiningΓ± ≡ (Γ↑+ Γ↓)/2, the evolution

equations are d ph dt = − d pl dt = −iG(bplh− b ∗_p hl)+ Γ↑pl−Γ↓ph, d dt h

phleiνlhti = i(G/~)b(ph− pl)+ iνlhphl−Γ+phl eiνlht.

(1.16)

Assuming stationary operation, expressions for the population difference and the average dipole moment are readily found. The population difference is

ph− pl=Γ − Γ+· 1 1+ I · L(νlh) , I ≡ 2πG 2_n ~2Γ+ . (1.17)

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20 1. Introduction

Here, L(νlh) = π−1Γ+/(Γ2₊+ ν2lh) is a Lorentzian. The intensity, I, is proportional to

the number of photons in the mode, n. The dependence of the population difference on I is strongest when |νlh| Γ+. The average dipole moment of the single emitter

reads hdi= Tr(ρ ˆd) = phl= iGb(ph− pl) ~(Γ+− iνlh) =iG ~ Γ− Γ+ Γ++ iνlh ν2 lh+ Γ 2 +[1+ (I/πΓ+)]b. This expression demonstrates the power broadening of the width of the dipole mo-ment, from the valueΓ₊at small I to the valueΓ₊[1+ (I/πΓ₊)]1/2at large I.

The condition for the saturated gain, which depends on the imaginary part of the dipole moment, leads to the steady state intensity. For N identical quantum emitters it reads I= 2π NΓ− Γ+ G2 ~2Γ − [L(νlh)]−1 (1.18)

Taking I = 0 yields the lasing threshold, N(Γ−/Γ+) = ~2Γ/2πG2L(νlh). Since the

right hand side is positive, it follows that lasing requiresΓ−> 0, and thus ph− pl> 0.

This demonstrates the necessity to have an inverted population to obtain lasing. We note that the lasing threshold is most efficiently reached when |νlh| Γ+, where

L(νlh) is largest.

The above analysis was based solely on the gain equations and is about the inten-sity of the laser. The frequency dependent equation leads to mode pulling. First, we note that ~(ν − Ω)/NG = −Re[hdi/b] = (νlh/Γ+)Im[hdi/b]= (νlh/Γ+)~Γ/2NG. The

latter follows from the condition for the saturated gain. Using the expression for νlh,

we derive

Ω = νE(Γ/2) + νΓ+

Γ/2 + Γ+ (1.19)

Hence, the frequency of the electric field in the mode,Ω, is pulled from the resonator mode frequency, ν, towards the frequency difference between the emitter states, νE,

in something like a center-of-mass formula. The refractive index of the gain medium is changed by the electric field, resulting in a shift of the resonator frequency [55].

We make a remark about the case b= 0. Since the dipole moment is proportional to b, a steady state solution is found at b= 0, so that one might think that lasing never occurs spontaneously, even if the laser is in the right parameter regime, where the linear gain is larger thanΓ. In practice, however, noise is always present. Quantum noise of the electric field, causes spontaneous emission of photons, triggering the exponential grow of the lasing field. Hence, in the lasing regime, the solution at b= 0 is a steady state one, but is unstable against small perturbations.

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In some lasers, many steady state lasing solutions are possible. Using linear

sta-bility analysis, the stasta-bility of these solutions can be studied. For this, the equations of motion, (1.15), are expanded about the steady state in consideration. This expan-sion represents the behavior of small perturbations to the steady state optical field. Only if the solutions to this linearized equation are all exponentially decaying, the steady state is stable against small perturbations.

Pulsed lasers

The example of previous section concerns a continuous wave laser. Often, however, lasers have a pulsed emission. Essentially, pulsed lasers emit radiation at various frequencies, resulting in a beat in the intensity. Usually, the emission at multiple frequencies results in mode competition, because the modes are all driven by the same set of quantum emitters. In some cases the modes synchronize. In some other cases, competition results in a frequency comb, where many modes are regularly spaced in frequency space.

Frequency comb lasers are of particular importance as they yield a regular, pulsed output. Let us consider such a laser spectrum, where many narrow peaks, regularly separated with mode spacing δ, are multiplied with a Gaussian envelope, with width σ. In the time domain, this corresponds to Gaussian shaped pulses, of width ' σ−1_,

separated in time by ' δ−1[55]. Frequency comb lasers are also used as a ‘frequency ruler’, allowing accurate measurements of the frequency of another optical source by comparing it to the frequency comb.

1.4.2 Noise en Coherence

The analysis Sec. 1.4.1 is based on averages of the electric field and the dipole mo-ment. Any realistic laser is subjected to various noise sources, for instance due to the pumping, because of thermal fluctuations and because of quantum fluctuations. In this section we briefly discuss noise in the common classical laser. Unfortunately, there are no short ways to derive the coherence time of a laser that are both clear and simple and that contain all relevant contributions. Therefore, this section is limited to heuristic and qualitative arguments.

Noise causes phase diffusion in the laser, resulting in a finite coherence time. The linewidth of the laser directly relates to this. It is proportional to the inverse of the coherence time. In a common laser, all phases are equally likely to occur. In the example of previous section, the lasing state was found to be independent of the optical phase. Similar to the superconductor, the laser can be understood in terms of a spontaneously broken global U(1) symmetry. In the unbroken case, the U(1) symmetry implies that there is no preference for any value of the optical phase. Absorption or spontaneous emission of a photon is phase independent. When the symmetry is spontaneously broken in the resonator, the electric field obtains a finite

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22 1. Introduction

expectation value, with a particular value of the phase. This phase value now has a preference over the other phase values. Because of stimulated emission, photons are very likely to have the same phase value. The value of the phase that is actually realized, upon the symmetry breaking, is arbitrary, since each is equally likely to be realized. This equivalence of phases is a remnant of the former symmetry.

Because of the equivalence of all phase values, the common laser undergoes phase diffusion. Linear stability analysis shows that fluctuations in the amplitude of the electric field decay to zero. In contrast to this, phase fluctuations are neither growing nor decaying. The laser is critically stable against phase fluctuations. Therefore, a series of fluctuations might drive the phase away from its initial value. After a typical time it is no longer clear whether the phase is close to or far from its initial value. This is the coherence time, the time in which phase information is lost because of fluctuations.

Many sources exist that produce phase fluctuations. They result from any kind of decay process by which energy is lost, either in the gain medium or in the optical field. These processes cause homogeneous broadening of the quantum emitter eigenstates and, hence, in the emission spectrum of the gain medium. The importance and effect of particular noise sources depend on the type of laser and its design [56]. Usually, phase fluctuations are induced both directly and indirectly, via intensity fluctuations. The contribution of the indirect kind can be much larger than that of the direct kind. Intensity fluctuations change the saturation of the gain medium and thereby the index of refraction. Typically, the phase then evolves slightly, according to the equations of motion. When the intensity fluctuation has faded out, the phase is shifted.

The laser noise and coherence times are fundamentally limited by quantum ef-fects. The lower bound is set by spontaneous emission of photons. Even though the laser field is coherently created by stimulated emission, some photons are emitted spontaneously, at a rateΓ, as a result of quantum noise in the mode. Since these pho-tons are not coherent with the other ones, they randomly change the optical phase. To fully randomize the phase, the number of spontaneously emitted photons has to be of the same order of magnitude as the photons in the mode, being n. This results in a coherence time ' n/Γ. The corresponding linewidth, Γ/n, is found in literature as the fundamental linewidth of the laser [50].

Even for a ‘classical’ laser, a thorough study of noise and coherence, requires a quantum mechanical analysis [57]. However, classical studies are also possible. In particular, a Fokker-Planck equation that accounts for quantum effects can be derived from a quantum master equation [58]. A Fokker-Planck equation describes the evolu-tion of the probability distribuevolu-tion of stochastic variables, ~x. It contains information about the evolution of the averages, h~xi, called ‘drift’, and about the effect of noise on correlations between the variables, called ‘diffusion’. Classical analysis is also possible using Langevin equations. A description with a quasilinear Fokker-Planck

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1.5. Outline of the thesis 23

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equation is fully equivalent to one using a Langevin equation [59]. To change the

equation of motion of Eq. (1.15) into a Langevin equation, one or more stochastic forces Lj(t) are to be added, satisfying hLji= 0 and hLj(t0)Lk(t)i= Γjδjkδ(t0− t). The

Γjrepresent the magnitude of the noise. For the quantum noise in the modeΓb = Γ

[58].

1.5 _{Outline of the thesis}

In this thesis, a ‘superconducting opto-electronic’ device called ‘half-Josephson laser’ (HJL) is investigated, in which the coherence of the superconductor and the laser are combined. The structure that facilitates such a combination is the Josephson light emitting diode.

The remainder of this thesis is divided in two parts. In the first part, the essentials of the HJL are being studied. In chapter 2, a HJL model is used, based on the JoLED of section 1.3. This model is examined for lasing and coherence. The model of chap-ter 2 is extended to that of a general gain medium, in chapchap-ter 3. Here, a minimal set of conditions is derived which any HJL needs to satisfy. The HJL model is studied in the limit of weak coupling to both the superconductors and the optical field. Particu-lar attention is paid to small and Particu-large fluctuations. The coherence time of the HJL is shown to have an exponential dependence in the photon number.

In the second part of the thesis, two possible applications of the HJL are being studied. In chapter 4 a setup is investigated, where an optical stabilization technique is used to stabilize voltage fluctuations in superconductors. Finally, in chapter 5, a device is investigated where two HJLs are combined in a SQUID-like setup. This device can operate in a single mode, where it is a flux tunable continuous wave HJL, and it can operate as a dual mode laser, with time dependent output.

References

[1] M. Tinkham, Introduction to Superconductivity, 2nd edition (McGraw-Hill, New York, 1996). [2] H. Kamerlingh Onnes,Further experiments with liquid helium. D. On the change of electric

re-sistance of pure metals at very low temperatures, etc. V. The disappearance of the rere-sistance of mercury, Comm. Phys. Lab. Univ. Leiden, 122b (1911).

[3] D. van Delft and P. Kes,The discovery of superconductivity, Phys. Today 63, 38 (2010).

[4] J. Bardeen, L.N. Cooper and J.R. Schrieffer,Microscopic Theory of Superconductivity, Phys. Rev. 106, 162 - 164 (1957).

[5] W. Meissner and R. Ochsenfeld,Ein neuer Effekt bei Eintritt der Supraleitfähigkeit, Naturwis-senschaften 21, 787 - 788 (1933).

[6] F. London and H. London,The Electromagnetic Equations of the Supraconductor, Proc. R. Soc. Lond. A 149, 71 - 88 (1935).

[7] A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) [Sov. Phys. JETP 5 1174 (1957)]. [8] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).

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24 References

[10] L.N. Cooper,Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev. 104, 1189 - 1190 (1956). [11] J. Hu and H. Ding,Local antiferromagnetic exchange and collaborative Fermi surface as key

ingre-dients of high temperature superconductors, Scientific Reports 2, 381 (2012).

[12] E. Demler and S.-C. Zhang,Quantitative test of a microscopic mechanism of high-temperature superconductivity, Nature 396, 733 - 735 (1998).

[13] N.N. Bogoliubov, On a new method in the theory of superconductivity, Nuovo Cimento 7, 794 - 805 (1958).

[14] P.J. de Visser, J.J.A. Baselmans, P. Diener, S.J.C. Yates, A. Endo and T.M. Klapwijk,Number Fluctuations of Sparse Quasiparticles in a Superconductor, Phys. Rev. Lett. 106, 167004 (2011); P.J. de Visser, J.J.A. Baselmans, S.J.C. Yates, P. Diener, A. Endo and T.M. Klapwijk, Microwave-induced excess quasiparticles in superconducting resonators measured through correlated conduc-tivity fluctuations, Appl. Phys. Lett. 100, 162601 (2012).

[15] M. Lenander, H. Wang, R.C. Bialczak, E. Lucero, M. Mariantoni1, M. Neeley, A.D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, A.N. Cleland and J.M. Martinis, Measurement of energy decay in superconducting qubits from nonequilibrium quasiparticles, Phys. Rev. B 84, 024501 (2011);

G. Catelani, J. Koch, L. Frunzio, R.J. Schoelkopf, M.H. Devoret and L.I. Glazman,Quasiparticle Relaxation of Superconducting Qubits in the Presence of Flux, Phys. Rev. Lett. 106, 077002 (2011). [16] S. Guéron, H. Pothier, Norman O. Birge, D. Esteve and M. H. Devoret,Superconducting Proximity

Effect Probed on a Mesoscopic Length Scale, Phys. Rev. Lett. 77, 3025 - 3028 (1996).

[17] W.L. McMillan,Tunneling Model of the Superconducting Proximity Effect, Phys. Rev. 175, 537 -542 (1968).

[18] Yu.V. Nazarov and Y.M. Blanter, Quantum Transport: introduction to nanoscience (Cambridge University Press, Cambridge, 2009).

[19] M.E. Peskin and D.V. Schroeder, An introduction to Quantum Field Theory (Westview Press, Perseus Books Group, 1995).

[20] B.D. Josephson,The discovery of tunnelling supercurrents, Rev. Mod. Phys. 46, 251 - 254 (1974). [21] B.D. Josephson,Possible new effects in superconductive tunnelling, Phys. Lett. 1, 251 - 253 (1962). [22] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics , Vol 3

(Addison-Wesley, 1963 - 1965).

[23] G.E. Blonder, M. Tinkham and T.M. Klapwijk,Transition from metallic to tunneling regimes in su-perconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion, Phys. Rev. B 25, 4515 - 4532 (1982).

[24] S. Shapiro,Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations, Phys. Rev. Lett. 11, 80 - 82 (1963).

[25] I.K. Yanson, V.M. Svistunov and I.M. Dmitrenko, Zh. Eksp. Teor. Fiz. 48, 976 (1965);

A.H. Dayem and C.C. Grimes,Microwave emission from superconducting point-contacts, Appl. Phys. Lett. 9, 47 (1966).

[26] D.N. Langenberg, D.J. Scalapino, B.N. Taylor and R.E. Eck,Investigation of Microwave Radiation Emitted by Josephson Junctions, Phys. Rev. Lett. 15, 294 - 297 (1965).

[27] K.K. Likharev, Dynamics of Josephson Junctions and Circuits, (Gordon and Breach, New York, 1991).

[28] H. Courtois, M. Meschke, J.T. Peltonen and J.P. Pekola,Origin of Hysteresis in a Proximity Joseph-son Junction, Phys. Rev. Lett. 101, 067002 (2008).

[29] P. Recher, Yu.V. Nazarov and L.P. Kouwenhoven,Josephson Light-Emitting Diode, Phys. Rev. Lett. 104, 156802 (2010).

[30] K. K. Likharev,Superconducting weak links, Rev. Mod. Phys. 51, 101 - 159 (1979).

[31] A.A. Golubov, M.Yu. Kupriyanov and E. Il’ichev,The current-phase relation in Josephson junctions, Rev. Mod. Phys. 76, 411 - 469 (2004).