S11 S33 S24 Anyon Colliders A time-dependent quantum Hall particle collider to reveal fractional statistics in the Laughlin sequence Master’s thesis in Erasmus Mundus Master in Nanoscience and Nanotechnology SUSHANTH VARADA DEPARTMENT OF MICROTECHNOLOGY AND NANOSCIENCE - MC2 CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2023 www.chalmers.se www.chalmers.se Master’s thesis 2023 Anyon Colliders A time-dependent quantum Hall particle collider to reveal fractional statistics in the Laughlin sequence SUSHANTH VARADA Department of Microtechnology and Nanoscience - MC2 Applied Quantum Physics Laboratory Dynamics and Thermodynamics of Nanoscale devices group Chalmers University of Technology Gothenburg, Sweden 2023 Anyon Colliders A time-dependent quantum Hall particle collider to reveal fractional statistics in the Laughlin sequence SUSHANTH VARADA © SUSHANTH VARADA, 2023. Supervisors: Dr. Christian Spånslätt1, Dr. Matteo Acciai1, Examiner: Prof. Janine Splettstößer1 Co-Supervisor: Dr. George Simion3 Co-Promotor: Prof. Kristiaan De Greve2,3 1Applied Quantum Physics Laboratory, MC2, Chalmers University, Sweden 2Department of Electrical Engineering, KU Leuven, Belgium 3IMEC, Leuven, Belgium Master’s Thesis 2023 Department of Microtechnology and Nanoscience - MC2 Applied Quantum Physics Laboratory Dynamics and Thermodynamics of Nanoscale devices group Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Schematic of a time-dependent anyon collider setup Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2023 iv Anyon Colliders A time-dependent quantum Hall particle collider to reveal fractional statistics in the Laughlin sequence SUSHANTH VARADA Department of Microtechnology and Nanoscience - MC2 Applied Quantum Physics Laboratory Chalmers University of Technology Abstract Elementary particles in nature (3+1 dimensions) are classified into bosons and fermions based on their exchange statistics. However, more general statistics, intermediate be- tween fermionic and bosonic, are possible in 2+1 dimensions. Quasiparticles obeying this intermediary statistics are called anyons. A particularly relevant phase of matter hosting anyons is the fractional quantum Hall effect, where anyonic statistics has recently been demonstrated. Generally, exchange statistics is expected to be accessible in interference experiments, such as in the Hong-Ou-Mandel effect. In this setup, fermions show van- ishing current correlations due to anti-bunching caused by the Pauli exclusion principle. Bosons, instead, bunch together due to Bose-Einstein statistics causing a surge in the current correlations. Can Hong-Ou-Mandel interferometry be extended to probe the frac- tional statistics of anyons? In this thesis, we investigate this question in a fractional quantum Hall setup in the Laughlin sequence (filling factor ν = 1/(2n + 1), n ∈ Z+), where two anyons collide at a quantum point contact with a tunable time delay. Previous studies investigating sim- ilar systems relate current correlations of quasiparticle collisions with braiding between injected anyons and quasi-particle-hole excitations at the tunneling quantum point con- tact, which emerge due to thermal or quantum fluctuations. However, it remains unclear whether the presently studied Hong-Ou-Mandel effect probes the universal exchange phase (ϑ) picked up by the quasiparticles or other parameters, such as the non-universal scaling dimension (δ). We show that ϑ accumulated by the incoming anyons due to interaction with quasi-particle-hole pairs at the quantum point contact cancel out in time-sensitive two-particle interferometry. Instead, the key quantity measured through current correla- tions is the non-universal δ of the quasi-particle-hole excitations. Keywords: Anyons, Edge states, Fractional quantum Hall effect, Topological quantum matter, Quantum interference effects. v Acknowledgements I am grateful to my supervisors, Christian Spånslätt, Matteo Acciai, and Janine Splettstößer, for their unwavering support and guidance throughout my master’s thesis project. Their expertise, encouragement, and valuable insights have shaped my academic career. I ex- press my sincere gratitude for their mentorship during my PhD applications and for being the Giants who helped me see further in my research journey. Furthermore, I thank my co-promoters, Kristiaan De Greve and George Simion, for their constructive feedback on my thesis. Finally, I extend my heartfelt appreciation to the Applied Quantum Physics (AQP) and Quantum Device Physics (QDP) lab members for productive discussions and the opportunity to learn about their research. Sushanth Varada, Gothenburg, August 2023 vii List of Acronyms Below is the list of acronyms that have been used throughout this thesis listed in alpha- betical order: 2DEG Two-Dimensional Electron Gas AC Alternating Current DC Direct Current DOS Density of States FQH Fractional Quantum Hall HBT Hanbury Brown-Twiss HOM Hong-Ou-Mandel IQH Integer Quantum Hall LL Landau Levels QPC Quantum Point Contact ix Nomenclature Below is the nomenclature of parameters and variables that have been used throughout this thesis. Parameters h Planck’s constant kB Boltzmann constant q Charge of particle ν Filling factor v Velocity of the edge mode ϑ Statistical exchange phase (or) Braiding angle δ Scaling dimension of tunneling quasiparticles excited at the QPC θ Temperature Λ Tunneling amplitude α UV or short distance cut-off ωc Energy cut-off Ω Frequency of applied voltage pulse T Time period of applied voltage pulse τd Delay between injection/arrival of two input signals or particles Variables GR,L Equilibrium bosonic Green’s function G+,− Green’s function pl,m Photoassisted coefficient xi Contents List of Acronyms ix Nomenclature xi List of Figures xv 1 Introduction 1 1.1 Partition Noise and Two-particle Interferometry . . . . . . . . . . . . . . . 2 1.2 Integer Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Experimental Realization of Anyon Colliders . . . . . . . . . . . . . . . . . 8 1.5 Goal of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory 11 2.1 Time Evolution Pictures in Quantum Mechanics . . . . . . . . . . . . . . . 11 2.2 Landau Levels and Linear Dispersion Model . . . . . . . . . . . . . . . . . 13 2.3 1D Chiral Fermions and Bosonization . . . . . . . . . . . . . . . . . . . . . 14 2.4 Bosonic Hamiltonian and Quasiparticle Operator . . . . . . . . . . . . . . 17 2.5 Temporal Voltage Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Particle collider in the FQH regime 21 3.1 Tunneling current in a weak backscattering QPC . . . . . . . . . . . . . . 21 3.2 Zero-frequency backscattered noise . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Two input sources and photoassisted coefficients . . . . . . . . . . . . . . . 27 3.4 Analysis in the DC regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 AC Analysis: Hong-Ou-Mandel Effect . . . . . . . . . . . . . . . . . . . . . 30 4 Exchange phase erasure in anyon time domain interferometry 33 4.1 Auxiliary state and Tunneling operator . . . . . . . . . . . . . . . . . . . . 33 4.2 Tunneling current in HOM configuration . . . . . . . . . . . . . . . . . . . 35 4.3 Exchange phase erasure in HOM noise ratio . . . . . . . . . . . . . . . . . 38 4.4 Interpreting the braiding phase erasure . . . . . . . . . . . . . . . . . . . . 42 5 Conclusion 47 6 Outlook 49 Bibliography 51 xiii Contents A Bosonic Green’s Function I B Fourier Transform of the Green’s Function IV B.1 Finite temperature Green’s function . . . . . . . . . . . . . . . . . . . . . . IV B.2 Temperature independent Green’s function . . . . . . . . . . . . . . . . . . VI C Photoassisted Coefficients VII D Integral of the Equilibrium Green’s Function IX xiv List of Figures 1.1 Pictorial representation of particle exchange statistics in (3+1)D and (2+1)D 1 1.2 A particle scattered by a potential barrier . . . . . . . . . . . . . . . . . . 2 1.3 The fermionic anti-bunching and bosonic bunching in the Hong-Ou-Mandel interference of indistinguishable particles . . . . . . . . . . . . . . . . . . . 3 1.4 Illustrations detailing the integer quantum Hall effect . . . . . . . . . . . . 5 1.5 Pictorial examples showcasing the formation of chiral edge modes in the integer quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Diagrams elucidating the occupation of the density of states in the integer and fractional quantum Hall effect and a plot of Hall resistance . . . . . . 7 1.7 Schematic of the three-QPC mesoscopic anyon collider setup with the mea- surement circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 A naive expectation of the Hong-Ou-Mandel noise ratio for anyons at ν = 1/3 10 2.1 Linearizing the Landau levels near the Fermi points to obtain a linear dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 A four terminal Laughlin fractional quantum Hall setup operating in the Hanbury Brown-Twiss configuration . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Plots of backscattered current and zero-frequency noise for a DC bias as a function of qνVDCω −1 c for zero and finite temperature cases . . . . . . . . . 30 3.3 A four terminal Laughlin fractional quantum Hall setup operating in the Hong-Ou-Mandel configuration . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 A plot of the standard Hong-Ou-Mandel noise ratio R as a function of the time delay τd/T between the input signals . . . . . . . . . . . . . . . . . . 32 4.1 The four terminal Laughlin FQH setup operating in the HOM configuration with ideal time-resolved anyon sources. . . . . . . . . . . . . . . . . . . . . 34 4.2 Pictorial representation of different cases arising in the calculation of the tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 A plot of the anyonic HOM noise ratio R as a function of delay τd/T between the arrival times of two injected anyons at the collider QPC . . . 41 4.4 Diagrams representing different subprocesses occurring between the in- jected anyons and the quasi-particle-hole pairs excited at the QPC . . . . . 43 4.5 Time domain braiding between the injected anyons and QPC quasi-particle- hole excitations caused due to thermal or quantum fluctuations. . . . . . . 44 xv List of Figures xvi 1 Introduction Quantum Mechanics is a very successful theoretical framework, tested by numerous ex- periments, that describes the fundamental constituents of matter and their interactions. Its conceptions were formulated by Schrödinger, Heisenberg, and Born in 1926-1927 [1] and further developed as elaborated in Ref. [2]. Quantum mechanics classifies indis- tinguishable particles in 3+1 dimensional (D) nature (3 spatial + 1 time dimension) into bosons and fermions. This limitation to only two possible types of particles in (3+1)D can be understood by examining their exchange statistics. Imagine interchang- ing two identical particles twice. This scenario is equivalent to encircling one particle around the other by forming a loop that changes the wavefunction by an arbitrary phase1 [Ψ(x1, x2) → eiΦΨ(x2, x1) → e2iΦΨ(x1, x2)]. Permitting only local deformations2, this loop can be shrunk to a point in three space dimensions, as illustrated in Fig. 1.1a. (a) Topological deformation of winding loop in (3+1)D. (b) Well-defined braiding of particles in (2+1)D. Figure 1.1: Pictorial representation of particle exchange statistics in three and two dimensions. (a) The winding loop (red trajectory) formed by moving one particle around the other is shrunk to a point through continuous deformations in the third dimension, as depicted by the orange arrows. (b) In two dimensions, the winding loop cannot be topologically deformed to a point because one particle intersects the path of the other particle. This constraint, exclusive to (2+1)D, leads to fractional exchange statistics. 1This representation is Abelian, meaning that the wavefunction is a scalar quantity that only acquires a phase after a braiding operation (exchange of two particles). There are more complicated types of anyons, where a non-abelian representation applies: in that case, the wavefunction is a multi-component object that transforms according to a unitary matrix [3–5]. In this case, the order in which consecutive braiding operations are performed is important [6]. 2Continuous deformations of a geometry or structure that only include stretching, bending, and shrink- ing. These transformations do not involve operations like tearing and cutting. 1 1. Introduction Therefore, winding one particle around the other in (3+1)D is topologically equivalent to not moving the particles. Thus, the wavefunction of the particles remains unaltered under two such exchanges. Consequently, the particle wavefunction can only change by a phase factor eiΦ of either +1 or −1 under a single exchange [Ψ(x1, x2) → eiΦΨ(x2, x1)]. Bosons, defined by a many-body wavefunction that is symmetric when exchanging two particles, can be described by an accumulated exchange phase of Φ = 0. In contrast, fermions with an anti-symmetric wave function pick up an exchange phase of Φ = π, such that eiπ = −1 and e2iπ = 1. Two-dimensional systems lack the extra dimension to topologically deform this winding loop to a point, as depicted in Fig. 1.1b. Therefore, encircling one particle around the other is non-trivial in (2+1)D systems and the wavefunction can acquire any phase Φ = ϑ. Particles described by these wavefunctions are termed any-ons and were predicted in the late 1970s [7–9]. Anyons are neither bosons nor fermions and are governed by fractional exchange statistics that is possible for point particles only in 2 spatial + 1 time dimensions [10, 11]. The renewed interest in exploring the properties of anyons stems from the experimental observation of their fractional statistics with potential applications in topological quantum computing [6, 12]. Generally, the exchange statistics of indistinguishable particles is expected to be accessible through interference experiments. The fractional statistics of Abelian anyons were detected in two seminal experiments [13, 14], each utilizing a different type of interferometer. These experiments attracted much attention and instigated several theoretical studies. Moreover, the anyon collider (anyon interferometry) experiment was reproduced independently by three different experimental groups [15–17]. In this chapter, we begin by describing two-particle interferometry for particles in (3+1)D and discuss the prospects of extending the idea to anyons in (2+1)D. The remainder of the chapter introduces key concepts required throughout the thesis. 1.1 Partition Noise and Two-particle Interferometry Consider a source emitting a stream of N particles impinging onto a potential barrier. The particles are either transmitted or reflected independently, with a probability of T and 1 − T , respectively, as depicted in Fig. 1.2. The detector on the right-hand side clicks only when a particle successfully tunnels through the barrier. The average number of particles the detector measures is then represented by ⟨N1⟩ = NT . The deviations from this average manifest as fluctuations, described by ∆N1 = N1 − ⟨N1⟩. The variance is obtained by auto-correlating ∆N1 as ⟨∆N1∆N1⟩ = ⟨∆N2 1 ⟩ = ⟨(N1 − ⟨N1⟩)2⟩ = ⟨N1⟩(1 − T ). (1.1) Equation (1.1) represents the partition noise that occurs due to the random partitioning Barrier Detector Particle Figure 1.2: Particle scattered by a potential barrier into a transmitted or reflected signal with probabilities T and 1 − T , respectively. 2 1. Introduction Fermions Bosons Beam Splitter S1 S2 D2 D1 Beam Splitter S1 S2 D2 D1 Figure 1.3: Illustration of the fermionic anti-bunching and bosonic bunching in the Hong-Ou-Mandel interference. The beam splitter is symmetric with a transmission (and reflection) probability of 1/2. A different behavior in fluctuations emerges when two bosons or fermions arrive simultaneously (τd = 0) at the beam splitter. Fluctuations are suppressed for fermions because they avoid each other due to the Pauli exclusion principle. Conversely, fluctuations surge for bosons as they tend to occupy the same state because of their underlying Bose-Einstein statistics. of the stream of particles into transmitted and reflected signals. Likewise, the fluctuations in the occupation number of the reflected beam are given by ∆N2 = N2 −⟨N2⟩. The cross- correlation ⟨N1N2⟩ between the number of particles tunneling through the barrier (T ) and those being deflected (1−T ) is always zero because each particle can be either transmitted (N1 = 1, N2 = 0) or reflected (N1 = 0, N2 = 1). Using this, we can extrapolate the following relation: ⟨∆N1∆N2⟩ = −⟨∆N2 1 ⟩ = −⟨∆N2 2 ⟩. (1.2) Thus, both auto- and cross-correlations contain identical information about the partition noise that describes the statistical fluctuations in the detected number of particles. This relation remains valid regardless of whether the particle (as shown in Fig. 1.2) is a fermion or boson. However, to reveal the quantum statistics of indistinguishable particles, we need to consider two-particle interferometry [18, 19]. Figure 1.3 represents a Hong-Ou-Mandel (HOM) interferometer [20] with two sources and detectors separated by a beam splitter. A non-zero time delay (τd ̸= 0) between particles arriving at the beam splitter from Source 1 (S1) and Source 2 (S2) results in a 50% cross-correlation of fluctuations at the two detectors (averaged over a long time). ⟨∆N1∆N2⟩ = 0.5, holds true if the transmission of the beam splitter is 1/2, irrespec- 3 1. Introduction tive of the particles being either fermions or bosons. However, a very different behavior arises for a vanishing time delay τd = 0. If two identical fermions arrive at the beam splitter simultaneously, they will avoid each other because they cannot occupy the same quantum state due to the Pauli exclusion principle [21–24]. Therefore, each detector will always measure a particle definitively without fluctuations: ∆N1 = ∆N2 = 0. The cross-correlation ⟨∆N1∆N2⟩ = 0 indicates uncorrelated fluctuations that do not vary in time. This phenomenon observed in HOM interferometry is called fermion antibunching. It is characterized by the Pauli dip, which is the suppression of partition noise3 as a function of τd, shown in Fig. 1.3. Bose-Einstein statistics show that multiple bosons can occupy the same quantum state. Consequently, two indistinguishable bosons at τd = 0 always leave the beam splitter together towards either of the detectors. Hence, the prob- ability of finding two bosons at Detector 1 (D1) and zero bosons at Detector 2 (D2) or vice versa is equal. This phenomenon of boson bunching surges the fluctuations in one detector ∆N1/2 = 1 − 0 = 1, while concurrently reducing the fluctuations in the other ∆N2/1 = 1 − 2 = −1. Therefore, the fluctuations at the two detectors are perfectly anti- correlated with ⟨∆N1∆N2⟩ = −1. As illustrated in Fig. 1.3, this correlation |⟨∆N1∆N2⟩| is characterized by a peak in the partition noise as a function of τd. Hence, the underlying exchange statistics of particles is made manifest in noise measurements. Can this concept be extended to probe the fractional statistics of anyons? To answer this, we require (i) sources that emit anyons, (ii) channels to guide them towards a (iii) beam splitter for quasiparticles. Fractional Quantum Hall systems are a promising testbed that hosts anyons [25]. Furthermore, the components required to assemble a HOM setup can be implemented in (2+1)D quantum Hall systems [23, 24, 26–28]. Hence, in the following sections, we briefly introduce the integer quantum Hall effect and extend the concept to the fractional Hall regime before discussing anyon interferometry. 1.2 Integer Quantum Hall Effect When a current-carrying conductor is placed in a perpendicular magnetic field (B⊥), the electrons deflect from their trajectory due to the Lorentz force. This mechanism gives rise to the classical Hall effect in which the transverse resistivity ρxy is directly proportional to the strength of the applied magnetic field B⃗. In contrast, the longitudinal resistivity ρxx is independent of the B⃗-field and assumes a constant value depending on the scattering time τ (as τ → ∞ ρxx → 0) [29]. Increasing the strength of the B⃗-field at low temperature causes a phase transition in the 2D classical Hall system. It results in the quantization of the relationship between the magnetic field (B⃗) and the Hall resistivity ρxy as plotted in Fig. 1.4a. It is the integer quantum Hall (IQH) effect discovered by von Klitzing in 1980, using samples prepared by Dorda and Pepper [30] for which he was awarded the 1985 Nobel Prize. The experimental data shows that the longitudinal resistivity ρxx = 0, as long as ρxy lies on a plateau with the value ρxy = h e2 1 ν , ν ∈ Z+, (1.3) 3For electrons, ⟨N1⟩ is given by ⟨I1⟩(t/e) where ⟨I1⟩ is the average current measured over the time interval t, and so ⟨∆N1∆N1⟩ is related to the current fluctuations, or shot noise ⟨S11⟩. 4 1. Introduction (a) Image adapted from Ref. [31]. (b) 2DEG with Lx, Ly sample dimensions in the IQH. Figure 1.4: (a) A plot of the longitudinal resistivity ρxx and transverse resistivity ρxy as a function of the magnetic field (B⃗) in the IQH effect. ρxx is independent of the B⃗-field and spikes to a finite value only when ρxy transits onto the next plateau as a function of the B⃗-field. (b) Vxx and Vy are the measured voltages in the longitudinal and transverse directions of the sample in the IQH regime. The sample’s bulk exhibits insulating behavior because the electrons are trapped in closed circular orbits. Concurrently, electricity is conducted at the edges without any dissipation due to the formation of skipping orbits. where h is Planck’s constant and e is the charge of an electron, ν is measured to be an integer with remarkable accuracy. Moreover, ρxx spikes to a finite value only during the transition of ρxy to the next plateau as a function of the B⃗-field. The vanishing ρxx for ρxy ̸= 0 indicates the presence of a perfect dissipationless conductor that does not oppose the flow of electrons. However, examining the conductivity tensor σ reveals the existence of an insulator with a vanishing longitudinal conductivity σxx = ρxx ρ2 xx + ρ2 xy = 0, for ρxx = 0, ρxy ̸= 0. (1.4) Therefore, in the IQH regime, the system is an insulator and conductor concurrently. Classically, the strong perpendicular magnetic field causes the electrons in the sample bulk to move in circular orbits with an angular frequency ωB (cyclotron frequency4), as depicted in Fig. 1.4b. As the electrons are trapped in these closed orbits, they do not conduct electricity transforming the bulk into an insulator. However, at the edge of the sample, the closed orbits form connected skipping orbitals that facilitate electron transport in only one direction. Therefore, the 1D boundary of the sample acts as a chiral conductor. Quantum mechanics shows that the strong magnetic field discretizes the energy spectrum into energy levels equally spaced by the cyclotron energy5 ℏωB. These levels that encompass a macroscopic number of degenerate energy states are called Landau levels (LL). A confining potential Vconfine(y) arises due to the finite boundary of the sample. This potential is zero within the bulk and increases towards the edge of 4The cyclotron frequency is given by ωB = eB/m, where m is the mass of an electron. 5Energy associated with the cyclotron motion of charged particles in a magnetic field. 5 1. Introduction Figure 1.5: Bending of LLs at the edges (0, Ly) of the 2DEG sample due to the confining potential Vconfine(y). n edge modes emerge when the Fermi level EF intersects the LLs at 2n points that result in n filled LLs corresponding to the filling factor ν = n in the IQH regime. The cases for ν = 1, 2 are depicted to the left and right, respectively. the sample to restrict the electrons. Consequently, the energy states close to the edge are raised by Vconfine(y) that bend the associated LL at the sample boundary. The band situated in the bulk of the sample has filled states, making it an insulator. Whereas the bent LL having empty states available above the Fermi energy give rise to 1D conductors at the edge of the sample. The two Fermi points (intersection of EF with the bent energy spectrum) shown in Fig. 1.5 lead to two chiral edge states with opposite group velocities (v): right moving state at y = Ly, with positive v, and left moving state at y = 0 with negative v. The spatial separation of the chiral modes suppresses the backscattering of electrons from right-moving to left-moving or vice versa, forming perfect chiral conductors. Therefore, the edge states are immune to scattering caused by impurities that “smoothly deform” LL [31]. Furthermore, EF lying between the gap of nth and n+ 1th LL intersects them at 2n points, spawning ν = n [where n ∈ Z+] chiral modes at each edge. Notably, the filling factor ν corresponds to the number of filled LL and IQH effect manifests at integer-number values of ν. Hence, the chiral edge modes in the quantum Hall systems facilitate robust one-way channels to guide particles for interferometry experiments. 1.3 Fractional Quantum Hall Effect In a realistic sample with disorder due to underlying lattice impurities, the density of states (DOS) n(E) and the corresponding LL do not assume a perfect train of Dirac δ functions. They instead have a Gaussian or Lorentzian profile, as illustrated in Fig. 1.6a. If the strength of the random potential introduced by the disorder is smaller than the splitting of LL: ℏωB > Vdisorder, it helps stabilize the edge modes and makes the plateaus better discernable in the IQH effect [31]. If the B⃗-field is further increased (ℏωB increases) in very pristine samples, other plateaus emerge at fractional values ν ∈ Q, as plotted in Fig. 1.6b. The most prominent and simple states lie on plateaus with odd denominators ν = 1/(2n+1), n ∈ Z+. These states are explained by the emergence of a single-edge mode and are termed the Laughlin sequence [32]. The edge states for other filling factors are more complicated [33, 34] and fall outside the scope of this study. The fractional quantum Hall effect [35] can be understood by considering the Coulomb interaction between electrons. These interactions lift the degeneracy of the macroscopic states embedded in a LL, leading to a spectrum of states of width proportional to Ecoulomb. The DOS corresponding to different energy scales do not overlap as long as ℏωB > Ecoulomb > Vdisorder. This spectrum consists of partially filled LL that exhibit small excitation gaps at fractional filling factors 6 1. Introduction (a) DOS and occupation of LL in a B⃗-field ignoring the particle spin. (b) Image taken from Ref. [36]. Figure 1.6: (a) Transition of DOS from IQH to FQH regime. ℏωB is the cyclotron energy, and Vdisorder represents the strength of the random potential introduced by the disorder. The width of the integer filled LLs (ν = n, where n ∈ Z+) are proportional to Vdisorder. These levels spread out as the magnetic field increases, making the plateaus more discernable in the limit ℏωB > Vdisorder in the IQH regime. The Coulomb interaction Ecoulomb between the electrons becomes prominent as we transition to the FQH regime. These interactions lift the degeneracy of the macroscopic states resulting in a spectrum of states whose width is proportional to Ecoulomb. This spectrum has gaps at fractional values, and the filled states are discernable when ℏωB > Ecoulomb > Vdisorder. The case for ν = 1/3 is shown here. (b) A plot depicting the Hall resistance RH as a function of the magnetic field B⃗ in the context of the FQH effect. where the Hall states are observed. Figure 1.6a depicts a Laughlin state with a single gap at ν = 1 3 . Fractionally charged quasiparticle excitations with q∗ = qν above such collective ground state of correlated electrons have been predicted [25, 32, 37] and recently confirmed [13, 14] to be emergent Abelian anyons. These fractional charges can be detected by implementing a Quantum Point Contact (QPC) [38, 39] in the FQH system, as initially suggested by Kane and Fisher in Ref. [40]. The QPC is a narrow constriction that partially distorts the trajectory of chiral edge modes by imposing a negative voltage 7 1. Introduction polarization that depletes the underlying 2DEG. Tuning the voltage polarization varies the transmission of the QPC by closing or opening the edge modes. Therefore, a QPC is analogous to a beam splitter that transmits or reflects impinging particles with specific probabilities. Tunneling of quasiparticles through the quantum Hall liquid between the edge modes corresponds to a backscattering event. The transmission probability through the QPC in the weak-backscattering limit is T ≈ 1. Therefore, the tunneling events corresponding to the reflection probability 1 − T ≪ 1 are so rare that the quasiparticles backscatter without any correlation between them. This stochastic tunneling generates zero-temperature shot noise S = 2q∗⟨IT ⟩, in the limit eV ≫ kBθ, (1.5) where IT is the tunneling current through the QPC, q∗ is the effective charge, kB is the Boltzmann constant, and θ is the temperature. The shot noise measurements proved the existence of fractionally charged quasiparticles in FQH systems at filling factors ν = 1/3 and ν = 2/3 [41, 42]. The FQH system with a QPC in the weak backscattering limit can generate a dilute beam of quasiparticles. Therefore, it acts as a stochastic anyon source [40] to implement the HOM interferometer to collide anyons. 1.4 Experimental Realization of Anyon Colliders The direct observation of fractional statistics is much more subtle than detecting the fractional charge of anyons. Combining the elements discussed in Sections 1.2 and 1.3, a two-particle anyon collider at filling factor ν = 1/3 was proposed in Ref. [43] and realized in Ref. [14]. QPC1 and QPC2 of the setup shown in Fig. 1.7 are DC biased into a weak backscattering regime with transmission probabilities Ts = T1 = T2 by VDC1 and VDC2, respectively. Due to the tunneling of q∗ = q/3 quasiparticles at random, these QPCs serve as sources that emit anyons following a Poisson distribution. This random emission implies that the source QPCs are not time-resolved and do not permit control over the emission times of the quasiparticles. The tunneling currents I1 and I2 carrying the quasiparticles interfere at the collider cQPC with a transmission probability T . The shot noise accompanied by I3 and I4 embeds the information about the exchange statistics of anyons. The cross-correlation between the output currents describes the shot noise as SI3I4(classical) = −2qν(1 − p)TsT (1 − T )(I1 + I2) , (1.6) where p is the exclusion (quasi)probability. Equation 1.6 originates from an intuitive classical model, which suggests an interpretation of the exchange statistics in terms of the exclusion probability. A fermionic behavior with p = 1 results in vanishing shot noise, as discussed in Sec. 1.1. At ν = 1/3, the fractional exchange phase Φ = π/3 is closer to bosons (Φ = 0) than fermions (Φ = π). Therefore, p is predicted to be negative (p < 0), ensuing a negative cross-correlation SI3I4 < 0. Thus, the classical model generalizes the Pauli dip and associates the negative value of the exclusion probability p with the fractional statistics of anyons in a non-rigorous manner. A quantum mechanical treatment of the current cross-correlations for abelian anyons with an exchange phase Φ = π/m (for m ≥ 3) leads to the following result for the shot noise: SI3I4(quantum) = 2qν −2 m− 2T (I1 + I2) =⇒ P = SI3I4 2qνT (I1 + I2) = −2 m− 2 . (1.7) 8 1. Introduction Figure 1.7: False color scanning micrograph of the mesoscopic anyon collider setup with the measurement circuitry (image taken from Ref. [14]). The quasiparticle tunneling currents I1 and I2 originating from the DC biased QPC 1 and 2 interfere at the collider cQPC with a transmission probability T . Contacts 3 and 4 collect the output signals I3 and I4 to compute the current cross-correlations. Equation (1.7) describes a generalized Fano factor P that is only dependent on the ex- change phase of anyons. The experimental demonstration measured P = −2 ± 0.1, which agrees with the prediction of the theoretical model that P = −2 at ν = 1/m = 1/3 [43]. Later theoretical investigations [44, 45] have re-interpreted the Fano factor P to be directly associated with braiding between anyons interfering at the cQPC of the setup depicted in Fig. 1.7. This implies that the experiment in Ref. [14] is the first evidence of Abelian anyon fractional statistics through shot noise measurements at ν = 1/3. 1.5 Goal of the thesis Probing the fractional statistics of anyons through Hong-Ou-Mandel interferometry is the primary focus of this thesis. We expect that the HOM interference will reveal the underlying quantum statistics of anyons through current correlations that depend on the tunable time delay (τd) between particles arriving at the beam splitter. This expectation aligns with the HOM effect obtained for fermions and bosons, as explained in Sec. 1.1. Earlier descriptions of anyon colliders [43] employed DC biased Poissonian sources of anyons that generate a random stream of quasiparticles without any control over their emission times. However, we require a time-resolved emission of anyons that enables us to modulate the time delay (τd) between the arrival of quasiparticles at the collider 9 1. Introduction Figure 1.8: Visual representation of a naive expectation regarding the fluctuations in HOM interferometry of anyons in the Laughlin sequence at ν = 1/3. This depiction showcases an accumulated exchange phase of ϑ = π/3 as described by Eq. (1.8). QPC to perform the HOM interferometry. The correlations describing the HOM curves of fermions and bosons represented in Fig. 1.3 are defined by [18] |⟨∆N1∆N2⟩| = 1 2 [ 1 + |J |2 cos (ϑ) ] , (1.8) where |J | represents the spatial overlap between two incident wavefunctions at the beam splitter, and ϑ denotes the acquired exchange phase. Taking |J | = 1 at null delay τd = 0, the fluctuations described in Eq. (1.8) entirely depend on the accumulated statistical exchange phase ϑ. Building upon the success of the HOM interference in uncovering the exchange statistics of fermions and bosons, we investigate: • How would the fluctuations in the HOM interferometry manifest due to the frac- tional statistics of anyons? • How would the HOM noise curves presented in Fig. 1.3 be influenced by the acquired exchange phase of the colliding anyons? • Would anyons generate intermediate noise fluctuations, corresponding to ϑ = π/(2n+ 1), where n ∈ Z+ in the Laughlin sequence? (as portrayed in Fig. 1.8 for n = 1). • What distinguishes the interference between colliding anyons in the HOM interfer- ometry from those in fermions and bosons? Providing insights into these questions is the main motivation for this work. 10 2 Theory This chapter briefly overviews the theoretical tools necessary to describe edge transport of anyons in the Laughlin fractional quantum Hall states. Initially, we digress to discuss the concept of quantum time evolution pictures, which is a foundational framework utilized throughout this thesis. We then discuss the formulation of Landau levels and chiral edge modes in the IQH regime before generalizing it to Laughlin FQH edges. Subsequently, we describe the technique of bosonization of 1D systems and demonstrate the utility of the introduced theory toolbox in describing an out-of-equilibrium Laughlin FQH edges. 2.1 Time Evolution Pictures in Quantum Mechanics There are three different equivalent approaches to treating time dependence in quantum mechanics [46, 47], and they find utility in distinct contexts. It is also possible to transform between these time evolution pictures. Schrödinger picture: The quantum states |φ(t)⟩S evolve with time, while the operators are fixed at an initial time t0, OS(t) = O(t0). One can introduce a time evolution operator acting on the quantum states |φ(t)⟩S = U(t, t0) |φ(t0)⟩ and evolve them according to the Schrödinger equation, iℏ∂t |φ(t)⟩S = H |φ(t)⟩S, from which we obtain iℏ∂tU(t, t0) |φ(t0)⟩ = HU(t, t0) |φ(t0)⟩ . (2.1) For a time-dependent Hamiltonian H, Eq. (2.1) gives us U(t, t0) = T [ e − i ℏ ∫ t t0 dt′H(t′) ] , where T is the time ordering operator that orders a product of time-dependent operators in descending sequence of time. When the Hamiltonian is independent of time, Eq. (2.1) simplifies to U(t, t0) = e− i ℏ (t−t0)H. Heisenberg picture: The quantum states are stationary in time |φ(t)⟩H = |φ(t0)⟩, while the operators procure a time dependence through the time evolution operator OH(t) = U †(t, t0)O(t0)U(t, t0). The equation of motion for an observable in the Heisenberg picture is given by iℏ∂tO(t) = [O(t), H(t)] + (∂tO)(t). (2.2) Suppose the operator does not explicitly depend on time; its time evolution boils down to iℏ∂tO(t) = [O(t), H(t)], where the expression [O(t), H(t)] represents the commutator between the operators O(t) and H(t) defined as [O(t), H(t)] = O(t)H(t) −H(t)O(t). 11 2. Theory Interaction picture: It is also termed1 the mixed picture because both the quantum states and operators carry part of the time dependence. This picture is beneficial for constructing perturbative expansions and dealing with the time evolution of observables caused by interactions. Consider the Hamiltonian of the form H = H + V(t), where H is a known time-independent operator and V(t) is a non-diagonalizable complex operator carrying the time dependence. In the interaction picture, the operator O(t0) evolves with the trivial H like in the Heisenberg picture: OI(t) = e i ℏ (t−t0)HO(t0)e− i ℏ (t−t0)H = O (0) H (t)2. Whereas UI(t, t0) = e i ℏ (t−t0)HU(t, t0), acting on the quantum states evolve according to the Schrodinger equation only with respect to the non-trivial interaction term VI(t) = e i ℏ (t−t0)HV(t)e− i ℏ (t−t0)H as follows: ∂t |φ(t)⟩I = ∂tUI(t, t0) |φ(t0)⟩ = − i h VI(t)UI(t, t0) |φ(t0)⟩ . (2.3) Equation (2.3) is analogous to Eq. (2.1) with which we can express UI(t, t0) as a function of the interaction part VI(t) as: UI(t, t0) = T [ e − i ℏ ∫ t t0 dt′ VI(t′) ] . This expression is useful to establish a bridge between operators in the Heisenberg and interaction pictures to construct perturbative expansions OI(t) = e i ℏ (t−t0)HO(t0)e− i ℏ (t−t0)H = UI(t, t0)OH(t)U † I (t, t0). (2.4) Equation (2.4) spawns a perturbative expansion for OH(t): OH(t) = T̃ [ e i ℏ ∫ t t0 dt′ VI(t′) ] OI(t)T [ e − i ℏ ∫ t t0 dt′ VI(t′) ] , OH(t) = T̃ [ 1 + i ℏ ∫ t t0 dt′ VI(t′) + 1 2 ( i ℏ )2 ∫ t t0 dt′ VI(t′) ∫ t t0 dt′′ VI(t′′) + . . . ] ×O (0) H (t), × T [ 1 − i ℏ ∫ t t0 dt′ VI(t′) + 1 2 ( − i ℏ )2 ∫ t t0 dt′ VI(t′) ∫ t t0 dt′′ VI(t′′) + . . . ] , OH(t) = O (0) H (t) − i ℏ ∫ t t0 dt′ [ O (0) H (t),VI(t′) ] + ( − i ℏ )2 ∫ t t0 dt′ ∫ t t0 dt′′ [[ O (0) H (t),VI(t′) ] ,VI(t′′) ] + . . . . (2.5) Taking the expectation value of Eq. (2.5) over an equilibrium state characterized by the equilibrium density matrix ρ0(t) retrieves the Kubo formula as the first-order perturbation term in the following series [48]. ⟨OH(t)⟩0 = 〈 O (0) H (t) 〉 0 − i ℏ ∫ t t0 dt′ 〈[ O (0) H (t),VI(t′) ]〉 0 − 1 ℏ2 ∫ t t0 dt′ ∫ t t0 dt′′ 〈[[ O (0) H (t),VI(t′) ] ,VI(t′′) ]〉 0 + . . . . (2.6) 1The interaction picture is alternatively referred to as the Dirac picture. 2O (0) H (t) is the time evolution of O(t0) with a time-independent Hamiltonian in the Heisenberg picture. 12 2. Theory 2.2 Landau Levels and Linear Dispersion Model We mathematically model the LL restricted by a confining potential Vconfine(y) [31, 49], as discussed in Sec. 1.2. We fix a Landau gauge with vector potential A = yBx⃗, such that ∇ × A = −Bz⃗, describes a magnetic field acting perpendicular to the x-y plane. With momentum pµ = −iℏ∂µ, the Hamiltonian is given by H = 1 2m ( px + e c By )2 + 1 2mp2 y + Vconfine(y). (2.7) Because the system exhibits translational invariance in the y-direction, the energy eigen- states of py can be expressed as plane waves, and the composite eigenstate of the system can be written using separation of variables as φ(x, y) = eikxΦ(y). Considering the slow spatial variation of the confining potential ∂yVconfine(y) ≪ ℏωc/lB, where lB is the mag- netic length3 defined as √ ℏ/eB, we can approximate Vconfine(y) by a constant potential. It transforms the Hamiltonian into − ℏ2 2m∂2 y + 1 2mω 2 c (y − kl2B)2 + Vconfine(y0). (2.8) Equation (2.8) resembles the Hamiltonian for a harmonic oscillator in the y-direction with a center displaced from the origin by y0. The wavefunctions satisfying the time-dependent Schrödinger equation are given by φ(x, y) = eikx−ωktΦ(y − kl2B), (2.9) with corresponding eigenenergies ℏωk = ( n+ 1 2 ) ℏωc + Vconfine(kl2B). (2.10) The above equations imply that the wavefunctions are localized at y0 = kl2B, and the spatial localization of LLs depend on k (y ∝ k). As described in Sec. 1.2, the low energy excitations of the LL lie on the chiral edges with ℏωk ≈ EF . We only consider these low energy excitations close to the two Fermi points at k = ±kF , where EF intersects the LLs, and ignore the rest of the energy spectrum to model the chiral edge states. Hence, it is acceptable to linearize the spectrum around the Fermi points, as shown in Fig. 2.1. This process is analogous to the standard linearization procedure of non-interacting 1D electron systems with two Fermi points, as detailed in Ref. [50]. We then establish the linear dispersion relations with the right/left moving branches as ϵR/L(k) = ±ℏv(k ∓ kF ), with v = ∂ωk/∂k ∣∣∣ k=kF . To formulate the Hamiltonian in the framework of second quantization, consider a right-moving chiral edge mode corresponding to a LL with zero momentum at the Fermi point kF = 0. By setting ℏ = 1, we obtain ϵR(k) = vk, such that Hedge,R = v ∑ k k c† k,R ck,R , (2.11) where ck,R annihilates an electron with momentum k on the right-moving edge mode. However, an infinite number of fermions in the Dirac sea occupy the states k ∈ (−∞, 0], 3lB the is characteristic length scale that determines the spatial extent of the Landau levels in B⊥ 13 2. Theory Figure 2.1: The Hamiltonian in Eq. (2.8) implies that the spatial localization of the LLs depend on k, as y ∝ k. To model the physics of the edge modes, we only consider the low energy states close to the Fermi points where the Fermi level EF intersects the LLs. Following the standard linearization procedure of a non-interacting 1D electron gas model [50], we linearize the low energy spectrum of the LLs into the linear dispersion model near the Fermi points at k = ±kF . making the expectation value of the number operator ∑k c † k,Rck,R divergent. Since this is an artifact of the linearization procedure, the physically relevant quantity is the excess number operator with respect to the ground state |0⟩0. It is obtained by introducing the normal ordering as : c† k,Rck,R :≡ c† k,Rck,R − ⟨c† k,Rck,R⟩0. Therefore, we only examine the finite excess number of fermions with respect to the ground state |0⟩0. Now, we introduce the fermionic field operators in real space that satisfy the usual anti-commutation relation:{ ψR(x), ψ† R(y) } = δ(x− y), and are defined as ψR(x) = 1√ L +∞∑ k=−∞ eikxck,R ψ† R(x) = 1√ L +∞∑ k=−∞ e−ikxc† k,R , (2.12) where L is the size of the 1D system. Applying the above identities in Eq. (2.11) leads to the following Hamiltonian in the thermodynamic limit of L → ∞: Hedge,R = ∫ ∞ −∞ dx : ψ† R(x)(−iv∂x)ψR(x) : . (2.13) Repeating the same contruction for the left-moving edge, one obtains Hedge,L = ∫ ∞ −∞ dx : ψ† L(x)(iv∂x)ψL(x) : (2.14) and the total edge hamiltonian is therefore Hedge = Hedge,R + Hedge,L 2.3 1D Chiral Fermions and Bosonization The chirality of the 1D edge modes can be demonstrated by evolving the fermionic field operators in time with Hedge,R in the Heisenberg picture to obtain their equation of motion. Considering the fermionic creation operator ψ† R on the right-moving edge in equilibrium ∂tψ † R(x, t) = i [ Hedge,R(y), ψ† R(x, t) ] = v ∫ ∞ −∞ dy [ ψ† R(y)∂yψR(y), ψ† R(x, t) ] . (2.15) 14 2. Theory Using the identity [A,BC] = {A,B}C−B {A,C} [51], withA = ψ† R(x, t), B = ψ† R(y), C = ∂yψR(y), and applying the anti-commutation relation { ψ† R(x), ψ† R(y) } = 0, we obtain ∂tψ † R(x, t) = v ∫ ∞ −∞ dy ψ† R(y) { ψ† R(x, t), ∂yψR(y) } . (2.16) Changing the order of operations { ψ†(x, t), ∂yψ(y) } = ∂y { ψ†(x, t), ψ(y) } , and using the commutation relation { ψ†(x), ψ(y) } = δ(x− y) ∂tψ † R(x, t) = v ∫ ∞ −∞ dy ψ† R(y) ∂yδ(x− y) = −v∂xψ † R(x). (2.17) To obtain the above expression we used the identity: ∫ dz f(z) ∂zδ(z) = −∂zf(z) ∣∣∣ z=0 . Performing similar calculations for ψR, the equation of motions are: ∂tψ † R(x, t) + v∂xψ † R(x) = 0 ∂tψR(x, t) + v∂xψR(x) = 0. (2.18) The above expressions represent right-moving chiral waves propagating at speed v. In the absence of boundary conditions, their solutions take the form ψ†(x − vt) and ψ(x − vt), respectively. Performing the above calculations for the left-moving edge will produce the following equations of motion: ∂tψ † L(x, t) − v∂xψ † L(x) = 0 ∂tψL(x, t) − v∂xψL(x) = 0 , (2.19) with solutions of the form ψ† L(x+vt) and ψL(x+vt), propagating in the opposite direction. The linear 1D fermionic system allows an exact description in terms of bosonic density fluctuations that are unique to the one-dimensional setting. To develop this description, we introduce the density fluctuation operator [50] that creates particle-hole excitations in the infinite Dirac sea. We concentrate on the right-moving particles and drop the subscript R in the subsequent calculations for brevity. A similar approach can be followed to develop an equivalent description for left-movers. ρ(p)(l) = ∑ k : c† k+lck : (for l ̸= 0 only). (2.20) Note that ρ(p) does not alter the particle count of the system (i.e., |N⟩0 to |N + n⟩0, where n ∈ Z)4 and ∂tρR/L have corresponding right and left chiral evolutions in time. Using the identities in Eq. (2.12), ρ(p)(x) =: ψ†(x)ψ(x) : can be re-written as ρ(p)(x) = 1 L ∑ k : c† kck : + 1 L ∑ l ̸=0 e−ilx ∑ k : c† k+lck : , = N L + 1 L ∑ l>0 ( e−ilxρ(p)(l) + eilxρ(p)(−l) ) , (2.21) where N is the number operator defined as N = ∑ k : c† kck : = ∑ k ( c† kck − ⟨c† kck⟩0 ) . Using the identities from the derivation of Eq. (2.16) and by omitting the superscript (p) for brevity, the commutation relations of the particle density operators are expressed as [ρ(m), ρ(l)] = ∑ kk′ [ c† k+mck, c † k′+lck′ ] = ∑ k ( c† k+m+lck − c† k+lck−m ) . (2.22) 4Connecting Hilbert spaces with different particle numbers is taken care by the Klein factors that alter |N⟩0 by one as F † |N⟩0 = |N + 1⟩0 and F |N⟩0 = |N − 1⟩0. 15 2. Theory It generates the following two cases: [ρ(m), ρ(l)] = 0 for m ̸= −l, = ∑ k ( ⟨c† kck⟩0 − ⟨c† k−mck−m⟩0 ) = −Lm 2π for m = −l, that can be formulated in a compact way as [ρ(m), ρ(l)] = −Lm 2π δ(m+ l). (2.23) Equation (2.23) is analogous to a bosonic commutation relation that allows us to define bona fide bosonic creation and annihilation operators for l > 0 bl = √ 2π Ll ρ(−l) = √ 2π Ll ∑ k : c† k−lck : b† l = √ 2π Ll ρ(l) = √ 2π Ll ∑ k : c† k+lck : . (2.24) Akin to fermions, we define bosonic field operators in real space φ(x) = i√ L ∑ l>0 eilx √ l bl e −αl/2 φ†(x) = − i√ L ∑ l>0 e−ilx √ l b† l e −αl/2, (2.25) where the factor e−αl/2, in which α is a UV or short distance cut-off, is introduced by hand to regulate divergences. It is convenient to define a new field ϕ, also called a compact boson. This bosonic field is constrained within a spatial dimension, with its values wrapping around periodically as they traverse through this dimension ϕ(x) = φ(x) + φ†(x) = i√ L ∑ l>0 1√ l e−αl/2 ( ble ilx − b† l e −ilx ) . (2.26) Combining the results from Eqs. (2.21), (2.24), and (2.26), we can express the density fluctuation operator (particle density) as ρ(p)(x) =: ψ†(x)ψ(x) := N L − 1√ 2π ∂xϕ(x). (2.27) In the thermodynamic limit L → ∞, we omit the first term in the above equation. We can further define the total charge density by introducing the charge q in Eq. (2.27) as ρ(x) = −q 1√ 2π ∂xϕ(x). (2.28) From the commutation of ψ(x) with bl, it can be shown that ψ(x) |N⟩0 is an eigenstate of the bona fide bosonic annihilation operator with an eigenvalue αl(x), l > 0. We can describe ψ(x) ∝ e ∑ l>0 αl(x)b† l |N − 1⟩0, because coherent states are known to be the eigenstates of the bosonic annihilation operator [52]. Generalizing it to any state |N⟩, we obtain the Mattis-Mandelstam formula [53, 54] ψ(x) |N⟩ = F√ L ei 2πNx L e ∑ l>0 αl(x)b† l e− ∑ l>0 α∗l(x)bl |N⟩ . (2.29) 16 2. Theory Finally, using the identity eAeB = eA+BeC/2, if C = [A,B] and [A,C] = [B,C] = 0 [51], and Eqs. (2.26), (2.29) we can write ψ(x) = F√ L ei 2πNx L e−i √ 2πφ†(x)e−i √ 2πφ(x), ψ†(x) = F † √ L e−i 2πNx L ei √ 2πφ†(x)ei √ 2πφ(x), (2.30) ψ(x) = F√ 2πα ei 2πNx L e−i √ 2πϕ(x), ψ†(x) = F † √ 2πα e−i 2πNx L ei √ 2πϕ(x), (2.31) where the operators in Eq. (2.30) are normal ordered (cf. Sec. 2.2), while the operators in Eq. (2.31) are not. 2.4 Bosonic Hamiltonian and Quasiparticle Operator The operators presented in Eq. (2.31), which are exponentiated bosons, are known as vertex operators [55]. They describe electrons propagating in a chiral edge mode and are instrumental in rewriting the Hamiltonian defined in Eq. (2.13) in terms of bosonic fields. Hedge,R = vπ L NR(NR + 1) + v 2 ∫ L 2 − L 2 dx : [∂xϕR(x)]2 : . (2.32) The zero-mode contribution (first term in the above Hamiltonian) will be dropped in the thermodynamic limit of L → ∞. This bosonic Hamiltonian is complemented by the Kac-Moody commutation relation [50] that governs the bosonic fields as [ϕR(x, t), ϕR(y, t)] = i 2sgn(x− y), (2.33) where sgn(x−y) is the signum function. Equations (2.31), (2.32), and (2.33) are useful to develop a description for a wide range of interacting 1D systems, including the FQH edge modes [56]. The pre-factor operators (Klein factors and exponentiated number operators) in Eq. (2.31) can be ignored because they are not important for the calculations in this thesis. To determine the charge associated with the fermionic operators, we compute its commutation with ρ(x) using the identity [eA, B] = [A,B]eA and Eq. (2.33) [57, 58].[ ρR(x), ψ† R(y) ] = q 2π √ α ∂x [ ei √ 2πϕR(y), ϕR(x) ] = iq∂x [ϕR(y), ϕR(x)]ψ† R(y), = q 2∂xsgn(x− y)ψ† R(y) = qδ(x− y)ψ† R(y). (2.34) Equation (2.34) implies that ψ† R(x) and ψR(x) describe the creation and annihilation of fermions with a charge q. The statistical exchange phase accumulated by these fermionic excitations is determined by exchanging the vertex operators defined in Eq. (2.31) at dif- ferent spatial coordinates x, y. To proceed, we use the Baker-Campbell-Hausdorff identity eAeB = eBeAe[A,B] and Eq. (2.33). ψR(x)ψR(y) = ψR(y)ψR(x)e−2π[ϕR(x),ϕR(y)] =⇒ ψR(x)ψR(y) = ψR(y)ψR(x)e−iπsgn(x−y). (2.35) 17 2. Theory Here, the acquired phase is given by Φ = −π for x > y and Φ = π for x < y. Hence, the fermionic excitations described by the vertex operators accumulate a statistical exchange phase of |Φ| = π, effectively retrieving the fermionic anticommutation relation. We now introduce the filling factor ν into the charge density operator defined in Eq. (2.28) and the vertex operators from Eq. (2.31), in the following manner ρR(x) = −q 1√ 2π √ ν∂xϕR(x), (2.36) ψqp(R)(x) = 1√ 2πα e−i √ 2π √ νϕR(x) ψ† qp(R)(x) = 1√ 2πα ei √ 2π √ νϕR(x). (2.37) By following a similar approach as we used to examine the charge and statistical phase associated with fermionic vertex operators, we deduce[ ρR(x), ψ† qp(R)(y) ] = q √ ν 2π √ α ∂x [ ei √ 2π √ νϕR(y), ϕR(x) ] = qνδ(x− y)ψ† qp(R)(y), (2.38) ψqp(R)(x)ψqp(R)(y) = ψqp(R)(y)ψqp(R)(x)e−2π[ϕR(x),ϕR(y)] = ψqp(R)(y)ψqp(R)(x)e−iπνsgn(x−y). (2.39) Hence, the operators presented in Eq. (2.37) create and annihilate a fractional charge of qν and acquire a statistical exchange phase of |Φ| = πν. Therefore, these quasiparticle creation and annihilation operators describe the anyonic excitations in the FQH edge modes. To simplify the derived results, we rescale the bosonic field √ 2πνϕR(x) → ϕR(x) to obtain (presented equations are extended to encompass both left and right edges) Hedge(L/R) = v 4πν ∫ ∞ −∞ dx [ (∂xϕL/R)2 ] , (2.40) ψqp(L/R)(x) = 1√ 2πα e−iϕL/R(x), (2.41) [ ϕL/R(x, t), ϕL/R(y, t) ] = ∓ iπνsgn(x− y), (2.42) ρL/R(x) = ± q ∂xϕL/R(x) 2π . (2.43) Until now, we utilized a single boson field description of an FQH edge state hosting one kind of quasiparticle. However, a broader description of the edge states exists, employing multiple boson fields and offering a precise definition of the statistical exchange phase or braiding angle ϑ. Following the Haldane-Halperin hierarchal description of quantum Hall states [34], Wen derived an effective theory for generic Abelian FQH edge modes hosting distinct kinds of quasiparticles [59]. The theory describes the edge states with n-boson fields ϕ(x, t) = (ϕ1, ϕ2, . . . , ϕn)T and a charge vector q = (q1, q2, . . . , qn)T that determines the units of charge carried by quasiparticles of each kind l = (l1, l2, . . . , ln)T . The K-matrix consisting of integer elements is given by K =  p1 0 0 . . . 0 p2 0 . . . 0 0 . . . . . . ... ... ... pn  , (2.44) 18 2. Theory where pj, j = 1, . . . , n are odd integers. It defines each edge mode’s filling factor νj = 1/pj and governs the commutation relation between the bosonic fields. ν = qTK−1q , (2.45) e∗ l = qTK−1l , (2.46)[ ϕi(L/R)(x, t), ∂yϕj(L/R)(y, t) ] = ∓ 2iπ(K−1)ij δ(x− y). (2.47) This is the so-called symmetric basis in which all charge vector elements qj = 1. The commutation relation in Eq. (2.47) together with the Hamiltonian Hedge(L/R) = 1 4π n∑ i,j=1 ∫ ∞ −∞ dx ∂xϕi(L/R)vij∂xϕj(L/R) , (2.48) where vij are elements of a positive definite matrix V , generalizes the single bosonic edge mode Hamiltonian in Eq. (2.40) to multiple edge modes. The diagonal elements vij|i=j of the matrix V assign a velocity to each edge mode, whereas the off-diagonal elements vij|i ̸=j describe the short-range interactions between these edge modes. The statistical exchange phase accumulated by the interaction of such quasiparticles is given by ϑ = πlTK−1l . (2.49) We adopt the above conventions for all the subsequent calculations in the thesis. 2.5 Temporal Voltage Drive We now analyze the influence of a generic time-dependent voltage pulse V (t) on the time evolution of the edge modes. Our methodology closely follows the approach presented in Ref. [60]. Here, we consider that a voltage pulse is applied to the right-moving chiral edge mode at x = xR. The voltage source is described by the function UR(x, t) = Θ(−x+ xR)V (t). The Heaviside function ensures that the voltage source is only defined within the contact, i.e., the region x ∈ (−∞, xR). The voltage pulse couples to the edge mode via the Hamiltonian Hg = ∫ dx UR(x, t)ρR(x). (2.50) At t = −∞, the system is in equilibrium, and the edge modes’ time evolution is only attributed to the Hamiltonian Hedge,R. When the applied voltage drive couples to the system at t = −∞ + ϵ, the system is driven out of equilibrium, and the time evolution is also carried by Hg. We use the Heisenberg picture introduced in Sec. 2.1 to derive the equation of motion of the bosonic field operator with respect to the non-equilibrium Hamiltonian H = Hedge,R + Hg. ∂tϕR(x, t) = i [Hedge,R(y), ϕR(x, t)] + i [Hg(y), ϕR(x, t)] , = i v 4πν ∫ ∞ −∞ dy [ (∂yϕR(y))2, ϕR(x, t) ] − i q 2π ∫ ∞ −∞ dy UR(y, t) [∂yϕR(y), ϕR(x, t)] . (2.51) 19 2. Theory By using the identity [AB,C] = A [B,C] + [A,C]B [51], Kac Moody commutation rela- tions, and following a similar procedure as outlined in Sec. 2.3, we simplify the integrals in Eq. (2.51) to obtain ∂tϕR(x, t) = −v ∫ ∞ −∞ dy ∂yϕR(y)δ(x− y) + qν ∫ ∞ −∞ dy UR(y, t)δ(x− y) , ∂tϕR(x, t) = −v∂xϕR(x) + qνUR(x, t). (2.52) The above equation can be solved with Green’s function approach. To this end, we define a differential operator L = (∂t + v∂x), and define a general solution of the form ϕ(x, t) = ϕ0(x − vt, 0) + A(x, t), where ϕ0(x − vt, 0) is the solution of Eq. (2.52) when UR(x, t) = 0. This substitution gives us an equation relating the ansatz function A(x, t) with the voltage drive as (∂t + v∂x)A(x, t) = qνUR(x, t) =⇒ LA(x, t) = qνUR(x, t). (2.53) We now introduce Green’s function, which is the impulse response of the differential operator L LG(x, x′; t, t′) = δ(x− x′)δ(t− t′). (2.54) By its definition [61], G(x, x′; t, t′) encodes all the output responses of a linear time- invariant system for all input frequencies. Therefore, convoluting our generic input UR(x, t) with G(x, x′; t, t′) will fetch us the output response A(x, t) A(x, t) = qν ∫ ∫ dx′dt′ G(x, x′; t, t′)UR(x′, t′). (2.55) Equation (2.53) can be recovered by applying the differential operator L on Eq. (2.55), and using the property (2.54). We now substitute the Green’s function of the operator L in Eq. (2.55) to obtain G(x, x′; t, t′) = Θ(t− t′)δ (v(t− t′) − (x− x′)) , (2.56) A(x, t) = qν ∫ t −∞ dt′ UR(x− v(t− t′), t′) , (2.57) ϕR(x, t) = ϕR0(x− vt, 0) + qν ∫ t −∞ dt′ UR(x− v(t− t′), t′) . (2.58) Using the bosonization identity from Eq. (2.41), the time evolution of the quasiparticle field operator can be derived by a simple substitution ψqp(R)(x) = 1√ 2πα e−iϕR0(x−vt,0)e −iqν ∫ t −∞ dt′ UR(x−v(t−t′),t′) . (2.59) Notably, the field operators evolve chirally despite an arbitrary voltage pulse driving the system out of equilibrium. The chiral evolution of the applied time-dependent voltage drive is a direct consequence of the chirality of the edge modes. It is an intrinsic property of quantum Hall edge states and can be validated in both the fermionic and bosonic descriptions. 20 3 Particle collider in the FQH regime In the previous chapters, we introduced the essential ingredients required to describe time- resolved two-particle interferometry for anyons. To reiterate, the Hamiltonian Hedge(L/R) described in Eq. (2.40) models the quantum Hall edge modes that act as transmission lines for the left or right-moving quasiparticles. Equation (2.50) describes the Hamilto- nian that couples time-dependent input voltage sources to these edge modes propagating in either direction. In this chapter, we model the effects of a QPC (taking the role of a beam splitter) introduced in an FQH setup in the Laughlin sequence, i.e., states with filling factors ν = 1/(2n + 1), where n ∈ Z+. In the weak backscattering regime1, the QPC topologically deforms the edge modes without entirely depleting the quantum Hall fluid and forms a narrow constriction, as shown in Fig. 3.1. The QPC allows tunnel- ing between the counter-propagating edge modes at the position x = xQP C . Here, the quasiparticles stochastically tunnel between the opposite edge modes with a small, and for simplicity assumed energy independent, amplitude |Λ| ≪ 1. A tunneling Hamiltonian that effectively describes the most relevant tunneling process in this configuration reads HΛ = Λψ† qp(L)(xQP C , t)ψqp(R)(xQP C , t) + h.c., (3.1) where h.c. is the hermitian conjugate [Λ∗ψ† qp(R)(xQP C)ψqp(L)(xQP C)] and ψqp(x, t), ψ† qp(x, t) are the quasiparticle creation and annihilation operators introduced in Sec. 2.4. As fol- lows, we will drop the subscript notation of qp from the operators for all the subsequent calculations for brevity. The space-time coordinate (x, t) denotes arbitrary points in the setup at the space coordinate x and time coordinate t. Note that the chiral evolution of the edge modes induces the following relation between these coordinates: ψR(x, t) → ψR(x− vt, 0) ψL(x, t) → ψL(x+ vt, 0). (3.2) We now analyze the collider setup in the FQH regime operated in different configurations by computing the tunneling current and backscattered noise. A similar analysis was performed in the Refs. [62, 63] applying Schwinger-Keldysh contour formalism. 3.1 Tunneling current in a weak backscattering QPC A time-dependent voltage source VR(t) is coupled to the right-moving edge mode via Source 2 terminal at the spatial coordinate x = xR, in the setup shown in Fig. 3.1. The voltage source VL(t) connected to the left-moving edge mode at x = xL is switched off, and its terminal Source 1 is grounded. The considered setup is geometrically symmetric 1In the strong backscattering regime, the QPC fully depletes the underlying quantum Hall fluid, permitting only stochastic electron tunneling through the depleted region. 21 3. Particle collider in the FQH regime I2 S11 xQPC xQPC xR xL Figure 3.1: The four terminal setup driven by a single input source is dubbed to be oper- ating in the Hanbury Brown-Twiss (HBT) configuration [64, 65]. The voltage source VR(t) is defined in the region x < xR, coupled to the right-moving edge via the Source 2 termi- nal. The QPC is at the position xQP C , and the Source 1 terminal at xL is grounded. The Drain 2 terminal collects the unperturbed current IR (depicted by I2 in the schematic), and the Drain 1 terminal collects the tunneling current. All the terminals are equidistant from the QPC by a distance d. such that the Source and Drain terminals are equidistant from the position of QPC. We assign the parameter d to measure the distance between the components of the setup as follows: xL − xR = 2d; xL − xQP C = d; xQP C − xR = d. The terminal Drain 2 is used to collect the transmitted (or unperturbed) current I(0) driven by VR(t) in the right-moving lower edge mode of the setup. The tunneling current at the QPC, which is also utilized to compute the backscattered noise (refer to Sec. 3.2 for more details), is measured by the Drain 1 terminal coupled to the left-moving upper edge mode. As our first step, we compute the current operator I from the continuity equation relating the charge density ρ to the current density J . Due to the inherent one-dimensionality of the current carrying edge modes, we have ∂xJ(x, t) + ∂tρ(x, t) = 0, and I(x, t) = J(x, t). (3.3) For a right-moving chiral edge mode ∂tρR(x, t) = −v∂xρR(x, t), cf. Sec. 2.3. Therefore, ∂xIR(x, t) = v∂xρR(x, t) → IR(x, t) = vρR(x, t) = −qv∂xϕR(x, t) 2π . (3.4) The QPC is tuned to operate in the weak backscattering regime with a weak tunneling amplitude (|Λ| ≪ 1). This diminutive amplitude allows us to treat the tunneling at x = xQP C as a perturbation to the system being driven out of equilibrium by VR(t). Therefore, we can utilize the perturbative expansion introduced in Sec. 2.1 [cf. Eq. (2.5)] to compute the tunneling current IT (t). While the time-dependent current operator is constructed as a power series in perturbation theory [48], components of the expansion consisting of powers of Λ greater than three can be neglected because of the small tunneling 22 3. Particle collider in the FQH regime amplitude |Λ| ≪ 1. As mentioned earlier, the Hamiltonian Hedge(L/R) defined in Eq. (2.40) (cf. Sec. 2.5) models the FQH edge modes propagating in either direction. Therefore, we use two copies of Eq. (2.40) defined in both the left-moving (L) and right-moving (R) subspaces (cf. Sec. 2.2) to model the transport phenomenon in the Laughlin FQH setup depicted in Fig. 3.1. We now consider the Hamiltonian of the form H = H + V(t), where H is the initial unperturbed non-equilibrium Hamiltonian described as H = Hedge + Hg = v 4πν ∫ ∞ −∞ dx [ (∂xϕL)2 + (∂xϕR)2 ] + ∫ ∞ −∞ dx UR(x, t)ρR(x) . (3.5) The time dependence is now carried by the tunneling Hamiltonian introduced in Eq. (3.1), which gives us V(t) = HΛ(t). Using Eq. (2.5), the time evolution of the current operator in a single QPC setup for a second-order perturbation at the space-time coordinates (y, t), given that y > xQP C can be written as IR(y, t) = I(0)(y, t) + i ∫ t −∞ dt′ [ H (0) Λ (xQP C , t ′), I(0) R (y, t) ] + i2 ∫ t −∞ dt′ ∫ t′ −∞ dt′′ [ H (0) Λ (xQP C , t ′′), [ H (0) Λ (xQP C , t ′), I(0) R (y, t) ]] + O(Λ3) . (3.6) We proceed with a piece-wise calculation of IR(y, t) = I(0)(y, t) + I(1)(y, t) + I(2)(y, t). Zeroth order : It is the time evolution of the right-moving current operator IR(y, t) with the initial unperturbed non-equilibrium Hamiltonian (H) which is already given by Eq. (3.4). We use the general formula of the right-moving chiral boson from Eq. (2.58) ϕR(y, t) = ϕR0(y − vt, 0) + qν ∫ t −∞ dt′ UR(y − v(t− t′), t′) . The voltage source is coupled to the system at xR, such that UR(y, t) = Θ(−y + xR)VR(t). I(0)(y, t) = −qv∂yϕR0(y, t) 2π − q2ν ∫ t −∞ dt′ ∂yΘ(−y + xR + (t− t′)v)VR(t′) , = −qv∂yϕR0(y, t) 2π + q2ν ∫ t −∞ dt′ δ ( t− t′ + −y + xR v ) VR(t′) , = −qv∂yϕR0(y, t) 2π + q2νVR ( t+ −y + xR v ) . (3.7) First order : As HΛ(xQP C , t ′) = H† Λ(xQP C , t ′), we apply the identity [A + A†, B] = [A,B] − ([A,B])†, if B = B† [51] and bosonize the tunneling Hamiltonian using the identities in Eq. (2.41) to simplify the calculations. We then have I(1)(y, t) = i ∫ t −∞ dt′ [ Λψ† L(xQP C , t ′)ψR(xQP C , t ′) + h.c., − qv ∂yϕR0(y, t) 2π + q2νVR ( t+ −y + xR v ) ] = i ∫ t −∞ dt′ [ Λψ† L(xQP C , t ′)ψR(xQP C , t ′),−qv∂yϕR0(y, t) 2π ] − h.c., = − iqvΛ (2π)2α ∫ t −∞ dt′ e −iqν ∫ t′+(xR−xQP C )/v −∞ dτ VR(τ) [ eiϕL0(xQP C ,t′)e−iϕR0(xQP C ,t′), ∂yϕR0(y, t) ] − h.c. . 23 3. Particle collider in the FQH regime Taking into account that the operators in the left-moving and right-moving edge modes commute, we apply the identities [eA, B] = [A,B]eA and [AB,C] = A[B,C] + [A,C]B and the Kac-Moody commutation relation [cf. Eq. (2.50)] to obtain =⇒ [ eiϕL0(xQP C ,t′)e−iϕR0(xQP C ,t′), ∂yϕR0(y, t) ] = eiϕL0(xQP C ,t′) [ e−iϕR0(xQP C ,t′), ∂yϕR0(y, t) ] , = eiϕL0(xQP C ,t′) [−iϕR0(xQP C , t ′), ∂yϕR0(y, t)] e−iϕR0(xQP C ,t′), = −ieiϕL0(xQP C ,t′) [−2iπνδ (xQP C − y + v(t− t′))] e−iϕR0(xQP C ,t′), = −2πνδ (xQP C − y + v(t− t′)) eiϕL0(xQP C ,t′)e−iϕR0(xQP C ,t′). Using the above result and by reabsorbing the pre-factors, I(1)(y, t) can be rewritten in terms of quasiparticle annihilation and creation operators as I(1)(y, t) = iqν ∫ t −∞ dt′ δ ( t− t′ + xQP C − y v ) ( Λψ† L(xQP C , t ′)ψR(xQP C , t ′) − h.c. ) . (3.8) The region of interest is after the pulse injection point, i.e., at y > xQP C > xR. It implies xQP C − y < 0, ensuring t′ = t + (xQP C − y)/v falls within the time interval (−∞, t). We enforce this condition using a Heaviside function and apply the identity∫ dt f(t)δ(t− t0) = f(t0) to obtain I(1)(y, t) = iqνΘ(y − xQP C) ( Λψ† L(2xQP C − y + vt, 0)ψR(y − vt, 0) − h.c. ) . (3.9) Second order : To compute the contribution of the second-order perturbation term, we utilize the previously derived expression of the commutator [ H (0) Λ (xQP C , t ′), I(0) R (y, t) ] in the calculation of I(1)(y, t) as follows I(2)(y, t) = i2 ∫ t −∞ dt′ ∫ t′ −∞ dt′′ [ H (0) Λ (xQP C , t ′′), qνδ ( t− t′ + xQP C − y v ) ( Λψ† L(xQP C , t ′)ψR(xQP C , t ′) − h.c. ) ] . (3.10) To reduce the number of operators appearing during the expansion of Eq. (3.10), we ne- glect the terms that zero out when we evaluate the expectation value of I(2)(y, t). Upon careful analysis, it is clear that components containing an equal number of quasiparticle creation and annihilation operators in the left-moving and right-moving subspaces will contribute to ⟨I(2)(y, t)⟩. We only retain non-vanishing contributions and drop the nota- tion of denoting xQP C from the space-time coordinate (xQP C , t) in subsequent calculations. I(2)(y, t) = i2qν ∫ t −∞ dt′ ∫ t′ −∞ dt′′ δ ( t− t′ + xQP C − y v ) [ H (0) Λ (t′′),Λψ† L(t′)ψR(t′) − h.c. ] , = qν|Λ|2Θ(y − xQP C) ∫ t̃ −∞ dt′′ [ ψ† L(t′′)ψR(t′′), ψ† R(t̃)ψL(t̃) ] − h.c., (3.11) where t̃ = t + (xQP C − y)/v and ψL/R(t′′) = ψL/R(xQP C ± vt′′, 0). We next compute the expectation value of the time-evolved current operator with respect to the density matrix ρ0(t0) over the ground state of the system (by omitting the subscript notation ⟨.⟩0 24 3. Particle collider in the FQH regime for brevity). Consider the ground state in equilibrium at t0 = −∞, and that the input voltage source, tunneling at the QPC (treated as perturbation), transpired at later times. ⟨IR(y, t)⟩ = ⟨I(0)(y, t)⟩ + ⟨I(1)(y, t)⟩ + ⟨I(2)(y, t)⟩ [cf. Eq. (2.6)] , (3.12) ⟨I(0)(y, t)⟩ = −qv 〈 ∂yϕR0(y, t) 2π 〉 + q2ν 〈 VR ( t+ −y + xR v )〉 , ⟨I(0)(y, t)⟩ = q2νVR ( t+ xR − y v ) . (3.13) By definition, the compact boson field ϕ ∝ bq − b† q [cf. Eq. (2.26)] creates and destroys particle-hole pairs. Therefore, the expectation value of the first term in the above equa- tion reduces to zero without any counter-balancing operators acting on the equilibrium state. Furthermore, the vanishing expectation value of the quasiparticle creation and annihilation operators ⟨ψqp⟩ = 0, result in a zero contribution from I(1)(y, t) ⟨I(1)(y, t)⟩ = iqνΛ 〈 ψ† L(2xQP C − y + vt, 0)ψR(y − vt, 0) − h.c. 〉 = 0. (3.14) To calculate the expectation value of the second order contribution, we expand the com- mutator in Eq. (3.11) and split the current as I(2)(y, t) = ∫ dt′′ (a − b) − h.c., where a = ψ† L(t′′)ψR(t′′)ψ† R(t̃)ψL(t̃) and b = ψ† R(t̃)ψL(t̃)ψ† L(t′′)ψR(t′′). We compute each term separately by bosonizing the quasiparticle operators. The evaluation of ⟨a⟩ is detailed in the subsequent calculations ⟨I(2)(y, t)⟩ = qν|Λ|2Θ(y − x) ∫ t̃ −∞ dt′′ 〈 ψ† L(t′′)ψR(t′′)ψ† R(t̃)ψL(t̃) 〉 − ⟨b⟩ − h.c., ⟨a⟩ = 〈 eiϕL0(t′′)e−iϕL0(t̃) 〉 〈 e−iϕR0(t′′)eiϕR0(t̃) 〉 e iqν ∫ ť −∞ dτ VR(τ)−iqν ∫ t̄ −∞ dτ VR(τ) , where ť = t + (xR − y)/v and t̄ = t′′ + (xR − xQP C)/v. Applying the property of time translational invariance and the exponentiated bosonic correlation formula [55]〈 n∏ k=1 eOk 〉 = e ∑n j kBθ. S(11) = (qνΘ(y − x)|Λ|)2 lim T →∞ 2 T ∫ T/2 −T/2 dt ∫ ∞ −∞ ds 〈 ψ† L(t+ s)ψR(t)ψ† R(t+ s)ψL(t) 〉 + h.c. . The backscattered noise has a similar mathematical form as that of ⟨I(2)(y, t)⟩, entailing a similar calculation method (cf. Sec. 3.1). Therefore, we state the final generalized noise expressions ignoring the intermediate computation steps: S(11) = (qν|Λ|)2 lim T →∞ 2 T ∫ T/2 −T/2 dt ∫ ∞ −∞ ds ⟨IT (t+ s)IT (t)⟩ , (3.24) S(11) = (qν|Λ|)2 lim T →∞ 4 T ∫ T/2 −T/2 dt ∫ ∞ −∞ ds G2 +(0, 0, s, 0) cos ( qν ∫ t+s t dτ VR(τ) ) . (3.25) 3.3 Two input sources and photoassisted coefficients So far, we have derived results for a single source setup. However, we must now extend our analysis to a two-source case because we aim to model the HOM interferometry. In addition to the previously defined input source UR(y, t) in the region x < xR, we introduce another voltage source UL(y, t) = Θ(y−xL)VL(t) in the region x > xL that couples to the left-moving chiral edge mode modifying the quasiparticle and bosonic field operators as ϕL(y, t) = ϕL0 (y + vt, 0) + qν ∫ t −∞ dt′ UL(y + v(t− t′), t′) , (3.26) ψL(t) = 1√ 2πα e−iϕL0(y+vt,0)e −iqν ∫ t+y/v−xL/v −∞ dt′ VL(t′) . (3.27) It is apparent that the additional source term augments a phase factor corresponding to the applied voltage VL(t) altering the previously computed observables as (results stated after enforcing the symmetric conditions of the setup in Fig. 3.1 and by writing G±(xQP C , y, t, t ′′) as G± and G+(0, 0, s, 0) as G+(0, s) for brevity): ⟨IT (t)⟩ = 2iqν|Λ|2 ∫ t −∞ dt′′ ( G2 − −G2 + ) sin ( qν ∫ t−d/v t′′ dτ [VR(τ) − VL(τ)] ) , (3.28) S(11) = (qν|Λ|)2 lim T →∞ 4 T ∫ T/2 −T/2 dt ∫ ∞ −∞ ds G2 +(0, s) cos ( qν ∫ t+s t dτ [VR(τ) − VL(τ)] ) (3.29) By coupling two voltage sources to both the chiral edge modes, we can now operate the Laughlin FQH setup in both the HOM (collider regime) [20] and HBT configurations (by setting either of the inputs to 0) [64, 65]. To extend our examination with relevance to experiments, we assume that the voltage source drives the system periodically with a time-period T (angular frequency Ω). This approach stems from the practical challenges associated with achieving single-pulse detection, leading to the frequent use of periodic sources in experimental setups. Consequently, the tunneling current satisfies: ⟨IT (t)⟩ = ⟨IT (t + T )⟩, because the sinusoidal function in Eq. (3.28) is periodic in time. Therefore, 27 3. Particle collider in the FQH regime averaging the tunneling current over one cycle of T would suffice to further our analysis. We split the applied effective voltage bias into AC [VAC(τ) = V AC R (τ) − V AC L (τ)] and DC [VDC(τ) = V DC R (τ) − V DC L (τ)] components [68] to rearrange the sine term as follows: ⟨IT (t)⟩ = 2iqν|Λ|2 1 T ∫ T 0 dt ∫ t −∞ dt′′ ( G2 − −G2 + ) sin ( qν ∫ t−d/v t′′ dτ [VAC(τ) + VDC ] ) , = qν|Λ|2 1 T ∫ T 0 dt ∫ t −∞ dt′′ ( G2 − −G2 + ) eiqν ∫ t−d/v t′′ dτVAC(τ)eiqνVDC(t−t′′−d/v) − h.c. . Due to the periodicity of VAC(τ), the exponential can be expressed as a Fourier series by introducing the photoassisted coefficients [69, 70] elaborated in Appendix C. ⟨IT (t)⟩ = qν|Λ|2 T ∫ T 0 dt ∫ t −∞ dt′ ( G2 − −G2 + ) ∞∑ l=−∞ ∞∑ m=−∞ p∗ l pme ilΩte−imΩt′′ eiQΩ(t−t′′) − h.c., where QΩ = qνVDC , G2 ± = G2 ±(xQP C , y, t, t ′′), and pl/m is the photoassisted coefficient. Performing a change of variables t− t′′ = −t′ by subtracting t′′ from the time arguments transforms the observable into2 ⟨IT (t)⟩ = qν|Λ|2 T ∫ T 0 dt ∫ 0 −∞ dt′ ( G2 − −G2 + ) ∞∑ l=−∞ ∞∑ m=−∞ p∗ l pme iΩt(l−m)e−iΩt′(l+Q) − h.c., where G2 ± = G2 ±(xQP C , y,−t′, 0). We drop the space coordinates from Green’s function and change the order of operations to obtain (1/T ) ∫ T 0 dte−iΩt(l−m) = 1, which yields l = m within a single period T of the applied periodic voltage pulses. ⟨IT (t)⟩ = qν|Λ|2 ∫ 0 −∞ dt′ ( G2 +(−t′, 0) −G2 −(−t′, 0) ) ∞∑ l=−∞ |pl|2 [ eiΩ(t′)(l+Q) − e−iΩ(t′)(l+Q) ] , ⟨IT (t)⟩ = 2iqν|Λ|2 ∞∑ l=−∞ |pl|2 ∫ 0 −∞ dt′ ( G2 +(−t′, 0) −G2 −(−t′, 0) ) sin (Ωt′ (l +Q)). (3.30) As G2 −(−t′, 0) = G2 +(t′, 0), we can simplify Eq. (3.30) by treating the integral as follows: ⟨IT (t)⟩ = 2iqν|Λ|2 ∞∑ l=−∞ |pl|2 [ ∫ 0 −∞ dt′ G2 +(−t′, 0) sin (Ωt′ (l +Q)) − ∫ 0 −∞ dt′ G2 +(t′, 0) sin (Ωt′ (l +Q)) ] . Applying the substitution t′ = −t′, to the first integral in the above equation, we obtain ⟨IT (t)⟩ = −2iqν|Λ|2 ∞∑ l=−∞ |pl|2 ∫ ∞ −∞ dt′ G2 +(t′, 0) sin (Ωt′ (l +Q)). (3.31) Expressing the sinusoidal in exponents leads to the Fourier transform of Green’s function calculated in Appendix B. It reformulates ⟨IT (t)⟩ into summation of pl ⟨IT (t)⟩ = qν|Λ|2 ∞∑ l=−∞ |pl|2 [∫ 0 −∞ dt′ G2 +(t′)e−iΩt′(l+Q) − ∫ 0 ∞ dt′ G2 +(t′)e−iΩt′(l+Q) ] , = qν|Λ|2 ∞∑ l=−∞ |pl|2 [P2ν (Ω (l +Q)) − P2ν (−Ω (l +Q))] . (3.32) 2The constant global phase factor e−iΩ(l+Q)d/v appearing in ⟨IT (t)⟩ can be ignored in our calculation. 28 3. Particle collider in the FQH regime Following a similar procedure, we connect with the photoassisted coefficients to compute the photoassisted shot noise [71, 72]. We expand the cosine term in Eq. (3.29) to obtain S(11) = lim T →∞ 2 (qν|Λ|)2 T ∫ T/2 −T/2 dt ∫ ∞ −∞ ds G2 +(s)e−iQΩs ∞∑ l=−∞ ∞∑ m=−∞ p∗ l pme iΩt(l−m)e−i(l+m)Ωs/2 + h.c. . Considering the periodic nature of the voltage pulses, the above equation boils down to the Fourier transform of G2 +(s) (cf. Appendix B). S(11) = 2 (qν|Λ|)2 ∞∑ l=−∞ |pl|2 ∫ ∞ −∞ ds G2 +(s) [ e−i(l+Q)Ωs + ei(l+Q)Ωs ] , = 2 (qν|Λ|)2 ∞∑ l=−∞ |pl|2 [P2ν (Ω (l +Q)) + P2ν (−Ω (l +Q))] . (3.33) 3.4 Analysis in the DC regime Driving the Laughlin FQH setup with a pure DC bias Q ̸= 0, where Q = qνVDC/Ω by ze- roing out the AC component W = 0, where W = qνVAC/Ω, transforms the photoassisted coefficient to pl(W = 0) = −Jl(0) = δl,0 = δ(l) (cf. Appendix C). The noise and current expressions in the DC regime read as ⟨IT (t)⟩ = ⟨IT (t)⟩ = qν|Λ|2 ∞∑ l=−∞ |δ(l)|2 [P2ν (Ω (l +Q)) − P2ν (−Ω (l +Q))] , = 2qν|Λ|2 Γ(2ν)ωc ( 2πkBθ ωc )2ν−1 ∣∣∣∣Γ(ν + iqνVDC 2πkBθ )∣∣∣∣2 sinh ( qνVDC 2kBθ ) . (3.34) S(11) = 2 (qν|Λ|)2 ∞∑ l=−∞ |δ(l)|2 [P2ν (Ω (l +Q)) + P2ν (−Ω (l +Q))] , = (2qν|Λ|)2 Γ(2ν)ωc ( 2πkBθ ωc )2ν−1 ∣∣∣∣Γ(ν + iqνVDC 2πkBθ )∣∣∣∣2 cosh ( qνVDC 2kBθ ) . (3.35) The temperature-dependent behavior of the system at ν = 1/3 for a sweep of the dimen- sionless DC parameter (qνVDCω −1 c , where ωc is the energy cut-off, cf. Appendix A) from -0.5 to 0.5 are plotted in Fig. 3.2. As discussed in Sec. 3.2, our computations pertain to low temperatures bounded by the limit Vapplied > kBθ. The perfectly overlapping curves Fig. 3.2 indicate the negligible impact of finite temperature effects within this regime. However, this limit is no longer valid for a diminishing applied voltage VDC → 0 where the relatively significant temperature diverges the current and noise responses of the system. Notably, this divergent behavior is persistent in a multiple QPC setup as both Vapplied and θ tend towards zero [73]. The DC analysis reveals another aspect of the nature of the tunneling particles at the QPC. In the temperature independent regime (cf. Appendix B.2), we can derive ⟨IT (t)⟩ ∝ V 2ν−1 applied from Eq. (B.16) which suggests the power law gov- erning the relation between tunneling current and the applied input voltage. ν = 1 refers to the tunneling current caused by electrons that dwindle to zero as Vapplied → 0. In con- trast, the Laughlin sequence with ν = 1/(2n+ 1), where n ∈ Z+, exhibits an asymptotic current due to the tunneling of quasiparticles at the QPC. It illustrates the dominance of 29 3. Particle collider in the FQH regime (a) ⟨IT (t)⟩ at ν = 1/3 in the DC regime (b) S(11) at ν = 1/3 in the DC regime Figure 3.2: Plots of backscattered current and zero-frequency noise as a function of the dimensionless parameter qνVDCω −1 c at zero (kBθ = 0, in red) and finite temperature (kBθ = 0.01ωc, in blue). The curves overlap perfectly within the limit VDC > kBθ. This limit is no longer valid in the region where the temperature-independent curves diverge. quasiparticle tunneling over electron tunneling as a low-energy perturbation to the FQH system [74]. Furthermore, we can establish a relationship between the photoassisted shot noise and tunneling current as the following: S(11) = 2qν⟨IT (t)⟩ coth ( qνVDC 2kBθ ) . (3.36) Sending θ to the infinitesimal limit (θ → 0) at VDC > 0 retrieves S = 2qν⟨IT (t)⟩. As discussed in Sec. 1.3, this theoretical argument was instrumental in discovering fractional charges through current-noise measurements. 3.5 AC Analysis: Hong-Ou-Mandel Effect We aim to realize the HOM effect by colliding two identical excitations at the QPC in the Laughlin FQH setup. The Source 1 and Source 2 terminals in Fig. 3.3 are driven by two periodic voltage pulses identical in amplitude but separated in time by a tunable delay τd. When we drive the system with two voltage pulses of equal amplitudes, it zeroes out the contribution from the DC components due to the effective voltage seen by the system ∆V = VR(t) − VL(t + τd) = VAC(t) + ���VDC − VAC(t + τd) − ���VDC , leading to Q = qνVDC/Ω = 0. Furthermore, the tunable time delay alters the general form of the photoassisted coefficient presented in Eq. (C.4) in Appendix C as follows: pl(HOM) = ∫ T/2 −T/2 dt 1 T eilΩte−iqν ∫ t 0 dτVAC(τ)eiqν ∫ t 0 dτVAC(τ+τd). (3.37) 30 3. Particle collider in the FQH regime I2 S11 xQPC xQPC xR xL Figure 3.3: The four terminal Laughlin FQH setup in the HOM configuration. The voltage sources VR(t) and VL(t) are defined in the regions x < xR and x > xL at the Source terminals 2 and 1, respectively. Identical periodic voltage pulses are applied at these Source terminals in the HOM configuration that drive the FQH setup out-of-equilibrium. The Drain terminals measure the backscattered current and zero-frequency noise. By changing the order of operations and considering the periodic nature of input voltage, pl(HOM) = e−iqν ∫ td 0 dτVAC(τ) ∫ T/2 −T/2 dt 1 T eilΩt ∞∑ m=−∞ ∞∑ n=−∞ pme −imΩtp∗ ne inΩ(t+τd), = e−iqν ∫ td 0 dτVAC(τ) ∞∑ n=−∞ pn+l p ∗ ne inΩτd = ∞∑ n=−∞ pn+l p ∗ ne inΩτd . (3.38) Hence, the photoassisted coefficient for HOM pl(HOM) 3 is a function of the generic pho- toassisted coefficient pn and time delay τd. Although a broader class of HOM collisions exist that involve input voltages with distinct amplitudes and temporal shapes [62, 66], we only consider sinusoidal input for our analysis. Detailed calculations of pn for a sinusoidal input are presented in Appendix C. A substantial time delay between the input signals in the HOM experiment is equivalent to driving the system independently through Sources 1 and 2 in the HBT configuration. Therefore, it is a standard practice to normalize the HOM noise with the HBT noise by defining a ratio R that is only a function of the time delay τd. Typically, the noise caused by particles tunneling at the QPC is overshadowed by the equilibrium fluctuation S(0) that is independent of Vapplied. For this reason, we focus on the excess noise ∆S = S(11) − S(0) by subtracting the background fluctuations to define the standard HOM noise ratio as R(τd) = S (11) HOM − S(0) S (11) HBT/R + S (11) HBT/L − 2S(0) . (3.39) 3As Eqs. (3.31) and (3.33) involve the square modulus of pn, we can safely ignore the constant global phase factor e −iqν ∫ td 0 dτVAC (τ) appearing in pl(HOM) 31 3. Particle collider in the FQH regime in Figure 3.4: A plot of the standard HOM noise ratio R as a function of the dimensionless delay parameter τd/T . Identical sinusoidal input pulses with frequency Ω (in GHz range) are considered at temperature kBθ = 0.01Ω for filling factors ν = 1 and ν = 1/3. Note that we observe a vanishing HOM noise ratio even at a fractional filling factor of ν = 1/3 in Fig. 3.4. This behavior can be justified by examining the nature of excitations induced by the voltage sources in the FQH regime. As discussed in Sec. 1.1, a control- lable source of quasiparticles is a primary ingredient required to perform anyonic HOM. However, the conventional voltage sources (marked by orange boxes in Fig. 3.3) cannot excite a single fractionally charged quasiparticle using any of the Lorentzian, sinusoidal, or square voltage drives [66]. Moreover, even the minimal excitation4 in the FQH regime corresponds to an integer number of electrons instead of a fractional charge [62]. Thus, the HOM dip observed at fractional filling factors in Fig. 3.4 should not be interpreted as stemming from the fractional statistics of quasiparticle excitations. Therefore, the feasibility of employing controllable sources emitting single quasiparticles restricts the possibility of performing HOM for anyons to probe their fractional statistics. In the next Chapter, we explore this possibility by formally considering a prepared auxiliary state descibing the time-controlled injection of a single quasiparticle excitation. 4Minimal excitations in a non-interacting system are typically characterized by a single particle exci- tation above the Fermi level, free from any additional particle-hole pair excitations that generate noise [75]. This notion can be extended to the context of strongly correlated FQH states by imposing that these minimal excitations do not generate any excess noise apart from thermal noise. Interestingly, the required voltage drive remains a Lorentzian pulse carrying an integer charge [62]. 32 4 Exchange phase erasure in anyon time domain interferometry As outlined in previous chapters, we require time-resolved sources capable of injecting fractional excitations into the Laughlin FQH setup to explore anyon correlations in HOM interferometry. In the following, we theoretically model such ideal quasiparticle sources to describe the interference between anyon collisions at the QPC to probe their fractional statistics. Our theoretical description relies on an auxiliary state, as detailed in Sec. 4.1. Despite the experimental impracticability of such ideal quasiparticle sources, a mapping has been derived in Ref. [63], showing that an infinitely narrow, δ-like voltage pulse V (t) = 2πδ(t)/e (where e is the charge of an electron) is formally equivalent to the description of an ideal quasiparticle source, as described in Sec. 4.1. As a result, from an experimental point of view, the features we describe in this chapter can be approximately mimicked by driving the FQH collider with extremely narrow voltage pulses carrying an average fractional charge. 4.1 Auxiliary state and Tunneling operator In the absence of excitations, we denote the ground state of the Laughlin FQH system in Fig. 4.1 with |0⟩. To model an ideal time-resolved generic source of anyons capable of exciting any kind l = (l1, l2, . . . , ln)T of quasiparticle, we adopt the K-matrix formalism introduced in Sec. 2.4. A single quasiparticle injection is then denoted by an auxiliary state |φ⟩, which is just the system’s ground state dressed by a single quasiparticle exci- tation [43, 76]. It is defined as |φ⟩ = ψ† l1 (x, t) |0⟩, where ψ† l1 is the quasiparticle creation operator (cf. Sec. 2.4) that adds a single quasiparticle of kind l1 1 to the system at position x and time t. As our system hosts a single type of quasiparticles, we omit the subscript l1 from the operators and switch to the following edge modes description: left-moving edge (L) → upper edge (u) right-moving edge (R) → lower edge (d). This notation facilitates a succinct description of various subprocesses arising at the QPC, which will be elucidated later in the chapter. We next introduce an operator A(t) describ- ing tunneling quasiparticles at the QPC to distinguish various quantities in the theory clearly. Considering the weak tunneling amplitude to be real, i.e., Λ ∈ R, we compactly express the tunneling Hamiltonian as HΛ = Λ [ A(t) + A†(t) ] , (4.1) 1A non-interacting FQH system with a single edge mode hosts identical quasiparticles of a single kind. 33 4. Exchange phase erasure in anyon time domain interferometry SHOM xQPC xQPC xu xd Figure 4.1: The four terminal Laughlin FQH setup in HOM configuration with ideal time-resolved anyon sources. Temporal Anyon Source is modeled by an auxiliary state |φ⟩ = ψ† u(xu, tu)ψ† d(xd, td) |0⟩ that injects anyons in the upper (u) and lower (d) edges at positions xu, xd and times tu, td, respectively. The resulting HOM noise due to anyon collisions at the QPC (x = xQP C) is measured by the Drain terminals. where A(t) = ψ† u(xQP C , t)ψd(xQP C , t). From the bosonization formalism introduced in Sec. 2.3, it is apparent that the chiral edge modes host quasiparticles of the form ψ ∝ e−ilϕ. The tunneling operator can thus be written as A(t) = eil1ϕu(xQP C ,t)e−il1ϕd(xQP C ,t), which describes the creation of a quasi-particle-hole pair at the QPC. These tunneling quasi- particle-hole pairs can be attributed to either thermal excitations or quantum fluctuations occurring at different times, and their correlations are determined by the scaling dimension δ. The scaling dimension appears as a power-law exponent that governs the decay of the equilibrium Green’s function of the tunneling quasiparticles (quasiholes) taken as ⟨A(t)A†(t′)⟩0 = ⟨A†(t)A(t′)⟩0 = [ πkBθα sinh(πkBθ|t− t′|) ]4δ e−i2πδsgn(t−t′). (4.2) Note that Eq. (4.2) is analogous to G2 ± quantity from Eq. (3.16) derived (cf. Appendix A) in Sec. 3.1. The term consisting of the UV cut-off parameter can be conformally mapped to the exponential with sgn function by selecting a suitable contour in the complex plane [77]. By comparing Eqs. (3.16) and Eq. (4.2) we arrive at the relation ν = 2δ which fur- ther implies that ϑ = 2πδ for the Laughlin edge states. Here, ϑ is the braiding phase given by πν. This equivalence holds for non-interacting edges hosting Abelian anyons. However, in general, the scaling dimension δ is a non-universal parameter affected by neutral modes or 1/f noise [74, 78–82]. In contrast, the filling factor ν and the statistical exchange phase ϑ are universal parameters that are intrinsic characteristics of the FQH system insensitive to such external influences. Additionally, distinguishing between ϑ from the non-universal effects of 2πδ is of high experimental relevance. If left unaccounted for, this coincidence 34 4. Exchange phase erasure in anyon time domain interferometry between δ and ν, dictated by the theory of non-interacting Abelian edges, could lead to misinterpretations about detecting fractional statistics through standard HOM noise ratio measurements. To compute observable quantities such as average backscattered current and noise, we replace the system’s ground state |0⟩ with the prepared auxiliary state |φ⟩. As shown in Fig. 4.1, the Temporal Anyon Sources operating in HOM configuration drive the Laughlin FQH setup out-of-equilibrium by injecting two time-resolved quasiparticles into the system. It is modeled by dressing the ground state of the system with a quasi- particle excitation on both the upper and lower edges as |φ⟩ = ψ† u(xu, tu)ψ† d(xd, td) |0⟩. 4.2 Tunneling current in HOM configuration The tunneling current operator from Eq. (3.19) derived in Sec. 3.1 using Heisenberg’s equation of motion can be written in the framework of tunneling operators as IT (t) = iqνΛ [ A(t) − A†(t) ] . Using Eq. (3.20) expressed in terms of the tunneling operators, we calculate the expectation value of the tunneling current with respect to the auxiliary state ⟨φ| . |φ⟩ = ⟨.⟩qp as follows: ⟨IT (t)⟩ = qνΛ2 ∫ t −∞ dt′ 〈[ ψ† u(t)ψd(t), ψ† d(t′)ψu(t′) ] + [ ψ† u(t′)ψd(t′), ψ† d(t)ψu(t) ]〉 qp , = qνΛ2 ∫ t −∞ dt′ [ ⟨A(t)A†(t′)⟩qp + ⟨A(t′)A†(t)⟩qp − ⟨A†(t)A(t′)⟩qp − ⟨A†(t′)A(t)⟩qp ] . (4.3) We are interested in injecting a quasiparticle into the upper and lower edges at the lo- cations xu > xQP C and xd < xQP C at times tu and td in the non-equilibrium driving of the Laughlin FQH setup in HOM configuration. The quasiparticle creation opera- tor eil1ϕu/d(xu/d,tu/d) acting on the edges creates a stable localized disturbance (soliton) in each bosonic field. Consequently, the bosonic fields evolving chirally with a velocity v accumulate a phase due to the Kac Moody commutation relations (cf. Sec. 2.4)[83] ϕu(x, tu) → ϕu(x, tu) + 2πK−1l1Θ(−(x+ v(t− tu)) + xu) , (4.4) ϕd(x, td) → ϕd(x, td) + 2πK−1l1Θ(x− v(t− td) − xd) . (4.5) The injected quasiparticles interfere at the position x = xQP C when there is zero time delay τd = 0. Enforcing the symmetric conditions of the setup in Fig. 4.1 generates constant offset components (d/v) that correspond to the relative position of the QPC with respect to the anyon sources in the setup. By absorbing the offset terms into the injection times, the accumulated phase can be simplified to a function of time arguments. However, tu and td will now represent the arrival times of the injected anyons at the QPC that is inherently controlled by the injection times (tu, td). Hence, we continue with the same notation without loss of generality ϕu/d(tu/d) → ϕu/d(tu/d) + 2πK−1l1Θ(tu/d − t) . (4.6) Therefore, from the bosonic description of the tunneling operator, it is clear that the exponentiated phase factors out of the non-equilibrium correlation function leading to the relation [holds for all combinations of observables appearing in Eq. (4.3)] ⟨A†(t)A(t′)⟩qp = ⟨A†(t)A(t′)⟩0e −i2πl1K−1l1[−Θ(td−t)+Θ(tu−t)+Θ(td−t′)−Θ(tu−t′)]. (4.7) 35 4. Exchange phase erasure in anyon time domain interferometry The obtained phase component ϑ = πl1K −1l1, is the standard definition of the statistical braiding angle between two quasiparticles of the same kind introduced in Sec. 2.4. Equa- tion (4.7) demonstrates that the injected quasiparticles acquire a non-trivial exchange phase by interacting (braiding) with the quasi-particle-hole pairs created at the QPC due to thermal or quantum fluctuation. In the HOM configuration, the product accumulates a braiding phase ϑ only when both the arrival times of the quasiparticles td and tu fall within the range of the QPC quasi-particle-hole pair creation times t and t′. Introducing Φ = 2ϑ [−Θ(td − t) + Θ(tu − t) + Θ(td − t′) − Θ(tu − t′)] for brevity. ⟨IT (t)⟩ = qνΛ2 ∫ t −∞ dt′ ( ⟨A(t)A†(t′)⟩0 − ⟨A†(t′)A(t)⟩0 ) eiΦ + ( ⟨A(t′)A†(t)⟩0 − ⟨A†(t)A(t′)⟩0 ) e−iΦ, = qνΛ2 ∫ t −∞ dt′ [ πkBθα sinh(πkBθ|t− t′|) ]4δ ( e−i2πδsgn(t−t′) − e−i2πδsgn(t′−t) ) ( eiΦ − e−iΦ ) , = −4qνΛ2 ∫ t −∞ dt′ [ πkBθα sinh(πkBθ|t− t′|) ]4δ sin (Φ) sin (2πδ)sgn(t− t′) . (4.8) The integral over dt′ in Eq. (4.8) ranges from −∞ to t, implying that t′ is limited to values less than t (t′ < t). This condition simplifies the integral to ⟨IT (t)⟩ = −4qνΛ2 sin (2πδ) ∫ t −∞ dt′ [ πkBθα sinh(πkBθ(t− t′)) ]4δ sin (Φ) . (4.9) We attempt to simplify the integral in Eq. (4.9), focusing on the sinusoidal function and by assuming td > tu. However, we will accommodate the converse case in subsequent cal- culations. The controllable injection (arrival) times tu and td impose different conditions on the temporal parameter t, modifying the integration bounds where the sinusoidal func- tion is nonvanishing. We proceed with a piece-wise calculation of the conditions depicted pictorially in Fig. 4.2. and denote the equilibrium Green’s function with J (t− t′). t > td > tu: By definition, the Heaviside functions Θ(td − t) and Θ(tu − t) in Φ vanish due to the imposed condition. The leftover terms generate a finite tunneling current only when the argument t′ is between the arrival times such that td > t′ > tu.∫ t −∞ dt′ sin (2ϑ [−Θ(td − t) + Θ(tu − t) + Θ(td − t′) − Θ(tu − t′)]) = ∫ td tu dt′ sin (2ϑ) . td > t > tu: The function Θ(tu − t) zeroes out while −Θ(td − t) + Θ(td − t′) cancel out each other. The residual argument ensures a non-trivial tunneling current for t′ < tu.∫ t −∞ dt′ sin (2ϑ [−Θ(td − t) + Θ(tu − t) + Θ(td − t′) − Θ(tu − t′)]) = − ∫ tu −∞ dt′ sin (2ϑ) . td > tu > t: It results in a null tunneling current because the quasiparticles never arrive at the QPC. All the Heaviside functions in Φ cancel out each other, giving ⟨IT (t)⟩ = 0. Considering the three discussed scenarios, it is clear that the tunneling current is finite only when t > tu. Modeling alternative cases based on conditions imposed by td. ⟨IT (t)⟩ = sin (2ϑ)Θ(t− tu) [ Θ(t− td) ∫ td tu dt′J (t− t′) − Θ(td − t) ∫ tu −∞ dt′J (t− t′) ] , = sin (2ϑ)Θ(t− tu) [ Θ(t− td) ∫ td −∞ dt′J (t− t′) − ∫ tu −∞ dt′J (t− t′) ] . (4.10) 36 4. Exchange phase erasure in anyon time domain interferometry Figure 4.2: Pictorial representation of different cases arising in the calculation of ⟨IT (t)⟩. The figure portrays a sliding window of controllable quasiparticle injection times that vary in the integration region over dt′ ranging from −∞ to t. The shaded area represents the region outside the integration limits. Entailing similar computations, the arrival times in Eq. (4.10) would swap for tu > td ⟨IT (t)⟩ = sin (2ϑ)Θ(t− td) [ Θ(t− tu) ∫ tu −∞ dt′J (t− t′) − ∫ td −∞ dt′J (t− t′) ] . (4.11) We introduce a tunable time delay τd = td − tu corresponding to the difference between injection (arrival) times of quasiparticles into the lower and upper edges. We focus on the integrals and perform a change of varia