We present a simple formalism describing evolution of a qubit in the process of its measurement in a circuit QED setup. When a phase-sensitive amplifier is used, the evolution depends on only one output quadrature, and the formalism is the same as for a broadband setup. When a phase-preserving amplifier is used, the qubit evolution depends on two output quadratures. In both cases a perfect monitoring of the qubit state and therefore a perfect quantum feedback is possible.
We investigate the equilibrium behaviour of a superconducting circuit QED system containing a large number of artificial atoms. It is shown that the currently accepted standard description of circuit QED via an effective model fails in an important aspect: it predicts the possibility of a superradiant quantum phase transition, even though a full microscopic treatment reveals that a no-go theorem for such phase transitions known from cavity QED applies to circuit QED systems as well. We generalize the no-go theorem to the case of (artificial) atoms with many energy levels and thus make it more …
A Reply to the Comment by C. Ciuti and P. Nataf [arXiv:1112.0986v1] on our Letter "Superradiant Phase Transitions and the Standard Description of Circuit QED" [Phys. Rev. Lett. 107, 113602 (2011)].
We theoretically study a circuit quantum electrodynamics (QED) architecture with superconducting flux qubits. The qubit is coupled to the transmission line resonator by an ac current originating from the current mode of the resonator. Ultrastrong coupling can be obtained by varying the capacitance between the qubit and the resonator. We propose a scalable design where the two-qubit coupling can be achieved.
The Jaynes-Cummings model describes the coupling between photons and a single two-level atom in a simplified representation of light-matter interactions. In circuit QED, this model is implemented by combining microwave resonators and superconducting qubits on a microchip with unprecedented experimental control. Arranging qubits and resonators in the form of a lattice realizes a new kind of Hubbard model, the Jaynes-Cummings-Hubbard model, in which the elementary excitations are polariton quasi-particles. Due to the genuine openness of photonic systems, circuit QED lattices offer the possibilit…
We investigate the behavior of a circuit QED device when the resonator is initially populated with a mesoscopic coherent field. The strong coupling between the cavity and the qubit produces an entangled state involving mesoscopic quasi-pointer states with respect to cavity dissipation. The overlap of the associated field components results in collapse and revivals for the Rabi oscillation. Although qubit relaxation and dephasing do not preserve these states, a simple analytical description of the dissipative dynamics of the circuit QED device including cavity relaxation as well as qubit dissip…
By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In Cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the $A^2$ contribution. Although in atomic Cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit QED system consisting of a single superconducting transmon qubit coupled to a multimode …
We have studied the low-lying excitations of a chain of coupled circuit-QED systems, and report several intriguing properties of its two nearly degenerate ground states. The ground states are Schrödinger cat states at a truly large scale, involving maximal entanglement between the resonator and the qubit, and are mathematically equivalent to Majorana bound states. With a suitable design of physical qubits, they are protected against local fluctuations and constitute a non-local qubit. Further, they can be probed and manipulated coherently by attaching an empty resonator to one end of the circu…
We show how a quantum Ising spin chain in a time-dependent transverse magnetic field can be simulated and experimentally probed in the framework of circuit QED with current technology. The proposed setup provides a new platform for observing the nonequilibrium dynamics of interacting many-body systems. We calculate its spectrum to offer a guideline for its initial experimental characterization. We demonstrate that quench dynamics and the propagation of localized excitations can be observed with the proposed setup and discuss further possible applications and modifications of this circuit QED q…
Superconducting qubits connected in an array can form quantum many-body systems such as the quantum Ising model. By coupling the qubits to a superconducting resonator, the combined system forms a circuit QED system. Here, we study the nonlinear behavior in the many-body state of the qubit array using a semiclassical approach. We show that sudden switchings as well as a bistable regime between the ferromagnetic phase and the paramagnetic phase can be observed in the qubit array. A superconducting circuit to implement this system is presented with realistic parameters .
We propose a one-step scheme to generate GHZ states for superconducting flux qubits or charge qubits in a circuit QED setup. The GHZ state can be produced within the coherence time of the multi-qubit system. Our scheme is independent of the initial state of the transmission line resonator and works in the presence of higher harmonic modes. Our analysis also shows that the scheme is robust to various operation errors and environmental noise.
We propose a scheme for controlling the gradiometric flux qubit (GFQ) by applying an ac bias current in a circuit-QED architecture. The GFQ is insensitive to the magnetic flux fluctuations, which at the same time makes it challenging to manipulate the qubit states by an external magnetic field. In this study, we demonstrate that an ac bias current applied to the $α$-junction of the GFQ can control the qubit states. Further, the present scheme is robust against the charge fluctuation as well as the magnetic flux fluctuations, promising a long coherence time for quantum gate operations. We intro…
We explore the photon population dynamics in two coupled circuit QED systems. For a sufficiently weak inter-cavity photon hopping, as the photon-cavity coupling increases, the dynamics undergoes double transitions first from a delocalized to a localized phase and then from the localized to another delocalized phase. The latter delocalized phase is distinguished from the former one; instead of oscillating between the two cavities, the photons rapidly quasi-equilibrate over the two cavities. These intrigues are attributed to an interplay between two qualitatively distinctive nonlinear behaviors …
We theoretically propose a scheme for a spin quantum bit based on a double quantum dot contacted to ferromagnetic elements. Interface exchange effects enable an all electric manipulation of the spin and a switchable strong coupling to a superconducting coplanar waveguide cavity. Our setup does not rely on any specific band structure and can in principle be realized with many different types of nanoconductors. This allows to envision on-chip single spin manipulation and read-out using cavity QED techniques.
We study several dynamical properties of a recently proposed implementation of the quantum transverse-field Ising chain in the framework of circuit QED. Particular emphasis is placed on the effects of disorder on the nonequilibrium behavior of the system. We show that small amounts of fabrication-induced disorder in the system parameters do not jeopardize the observation of previously-predicted phenomena. Based on a numerical extraction of the mean free path of the system, we also provide a simple quantitative estimate for certain disorder effects on the nonequilibrium dynamics of the circuit …
We investigate the influence of temperature and dissipation on the Landau-Zener transition probability in circuit QED. Dissipation is modelled by coupling the transmission line to a bath of harmonic oscillators. The reduced description for the density operator is treated within Bloch-Redfield theory. A phase-space representation allows an efficient numerical implementation of the resulting master equation. It provides reliable results which are valid even for rather low temperatures. We find that the spin-flip probability as a function of temperature and dissipation strength exhibits a non-mon…
We propose a method to get experimental access to the physics of the ultrastrong (USC) and deep strong (DSC) coupling regimes of light-matter interaction through the quantum simulation of their dynamics in standard circuit QED. The method makes use of a two-tone driving scheme, using state-of-the-art circuit-QED technology, and can be easily extended to general cavity-QED setups. We provide examples of USC/DSC quantum effects that would be otherwise unaccessible.
We study a qubit undergoing Landau-Zener transitions enabled by the coupling to a circuit-QED mode. Summing an infinite-order perturbation series, we determine the exact nonadiabatic transition probability for the qubit, being independent of the frequency of the QED mode. Possible applications are single-photon generation and the controllable creation of qubit-oscillator entanglement.
We present an indirect two-qubit parity meter in planar circuit quantum electrodynamics, realized by discrete interaction with an ancilla and a subsequent projective ancilla measurement with a dedicated, dispersively coupled resonator. Quantum process tomography and successful entanglement by measurement demonstrate that the meter is intrinsically quantum non-demolition. Separate interaction and measurement steps allow commencing subsequent data qubit operations in parallel with ancilla measurement, offering time savings over continuous schemes.
We investigate photon-mediated transport processes in a hybrid circuit-QED structure consisting of two double quantum dots coupled to a common microwave cavity. Under suitable resonance conditions, electron transport in one double quantum dot is facilitated by the transport in the other dot via photon-mediated processes through the cavity. We calculate the average current in the quantum dots, the mean cavity photon occupation, and the current cross-correlations with both a full numerical simulation and a recursive perturbation scheme that allows us to include the influence of the cavity order-…
We propose a scheme for the quantum non-demolition (QND) measurement of Fock states of a nanomechanical resonator via feedback control of a coupled circuit QED system. A Cooper pair box (CPB) is coupled to both the nanoresonator and microwave cavity. The CPB is read-out via homodyne detection on the cavity and feedback control is used to effect a non-dissipative measurement of the CPB. This realizes an indirect QND measurement of the nanoresonator via a second-order coupling of the CPB to the nanoresonator number operator. The phonon number of the Fock state may be determined by integrating th…
We theoretically propose an efficient way to generate and detect squeezed light by a single qubit in circuit QED. By tuning the qubit energy splitting close to the fundamental frequency of the first harmonic mode (FHM) in a transmission line resonator and placing the qubit at the nodal point of the third harmonic mode, one can generate the resonantly enhanced squeezing of the FHM upon pumping with the second harmonic mode. In order to investigate the photon number splitting for the squeezed FHM, we have numerically calculated the qubit absorption spectrum, which exhibits regularly spaced peaks…
The driven-damped Jaynes-Cummings model in the regime of strong coupling is found to exhibit a coexistence between the quantum photon blockaded state and a quasi-coherent bright state. We characterize the slow time scales and the basin of attraction of these metastable states using full quantum simulations. This form of bistability can be useful for implementing a qubit readout scheme that does not require additional circuit elements. We propose a coherent control sequence that makes use of a simple linear chirp of drive amplitude and frequency as well as qubit frequency. By optimizing the par…
Discrete time-translation symmetry breaking can be observed in periodically-driven systems oscillating at a fraction of the frequency of the driving force. However, with the exception of the parametric instability in period-doubling, multi-periodic driving does not lead to an instability threshold. In this paper, we point out that quantum vacuum fluctuations can be generically employed to induce period multiplication. In particular, we discuss the period-tripled states in circuit QED and propose a microwave setup. We show that for weak dissipation or strong driving, the system exhibits a non-e…
We present a feasible protocol to mimic topological Weyl semimetal phase in a small one-dimensional circuit-QED lattice. By modulating the photon hopping rates and on-site photon frequencies in parametric spaces, we demonstrate that the momentum space of this one-dimensional lattice model can be artificially mapped to three dimensions accompanied by the emergence of topological Weyl semimetal phase. Furthermore, via a lattice-based cavity input-output process, we show that all the essential topological features of Weyl semimetal phase, including the topological charge associated with each Weyl…
Quantum optical photodetection has occupied a central role in understanding radiation-matter interactions. It has also contributed to the development of atomic physics and quantum optics, including applications to metrology, spectroscopy, and quantum information processing. The quantum microwave regime, originally explored using cavities and atoms, is seeing a novel boost with the generation of nonclassical propagating fields in circuit quantum electrodynamics (QED). This promising field, involving potential developments in quantum information with microwave photons, suffers from the absence o…
In this work we propose a microwave photon detector which successfully reaches 100% efficiency with only one absorber. Our design consists of a metastable quantum circuit coupled to a semi-infinite transmission line which performs highly efficient photodetection in a simplified manner as compared to previous proposals. We extensively study the scattering properties of realistic wavepackets against this device, thereby computing the efficiency of the detector. We find that the detector has many operating modes, can detect detuned photons, is robust against design imperfections and can be made b…
We investigate theoretically how the spectroscopy of an ancillary qubit can probe cavity (circuit) QED ground states containing photons. We consider three classes of systems (Dicke, Tavis-Cummings and Hopfield-like models), where non-trivial vacua are the result of ultrastrong coupling between N two-level systems and a single-mode bosonic field. An ancillary qubit detuned with respect to the boson frequency is shown to reveal distinct spectral signatures depending on the type of vacua. In particular, the Lamb shift of the ancilla is sensitive to both ground state photon population and correlat…
We suggest a scheme to probe critical phenomena at a quantum phase transition (QPT) using the quantum correlation of two photonic modes simultaneously coupled to a critical system. As an experimentally accessible physical implementation, a circuit QED system is formed by a capacitively coupled Josephson junction qubit array interacting with one superconducting transmission line resonator (TLR). It realizes an Ising chain in the transverse field (ICTF) which interacts with the two magnetic modes propagating in the TLR. We demonstrate that in the vicinity of criticality the originally independen…
An implementation of a small quantum absorption refrigerator in a circuit QED architecture is proposed. The setup consists of three harmonic oscillators coupled to a Josephson unction. The refrigerator is autonomous in the sense that it does not require any external control for cooling, but only thermal contact between the oscillators and heat baths at different temperatures. In addition, the setup features a built-in switch, which allows the cooling to be turned on and off. If timing control is available, this enables the possibility for coherence-enhanced cooling. Finally, we show that signi…
Quantum simulations is a promising field where a controllable system is used to mimic another system of interest, whose properties one wants to investigate. One of the key issues for such simulations is the ability to control the environment the system couples to, be it to isolate the system or to engineer a tailored environment of interest. One strategy recently put forward for environment engineering is the use of metamaterials with negative index of refraction. Here we build on this concept and propose a circuit-QED simulation of many-body Hamiltonians using superlattice metamaterials. We g…
We propose a scheme for quantum estimation by means of parametric amplification in circuit Quantum Electrodynamics. The modulation of a SQUID interrupting a superconducting waveguide transforms an initial thermal two-mode squeezed state in such a way that the new state is sensitive to the features of the parametric amplifier. We find the optimal initial parameters which maximize the Quantum Fisher Information. In order to achieve a large number of independent measurements we propose to use an array of non-interacting resonators. We show that the combination of both large QFI and large number o…
Open classical and quantum systems with effective parity-time ($\mathcal{PT}$) symmetry, over the past five years, have shown tremendous promise for advances in lasers, sensing, and non-reciprocal devices. And yet, how such effective $\mathcal{PT}$-symmetric non-Hermitian models emerge out of Hermitian quantum mechanics is not well understood. Here, starting from a fully Hermitian microscopic Hamiltonian description, we show that a non-Hermitian Hamiltonian emerges naturally in a double-quantum-dot-circuit-QED (DQD-circuit QED) set-up, which can be controllably tuned to the $\mathcal{PT}$-symm…
Quantum transmission lines are a central to superconducting and hybrid quantum computing. Parallel to these developments are those of left-handed meta-materials. They have a wide variety of applications in photonics from the microwave to the visible range such as invisibility cloaks and perfect flat lenses. For classical guided microwaves, left-handed transmission lines have been proposed and studied on the macroscopic scale. We combine these ideas in presenting a left-handed/right-handed hybrid transmission line for applications in quantum optics on a chip. The resulting system allows circuit…
We study the cavity field's and atomic asymptotic mean excitation numbers due to anti-rotating term (AMENDART) in the circuit Quantum Electrodynamics (circuit QED) system, composed of a two-level atom and a single cavity field mode, subject to Markovian damping and dephasing mechanisms. We show that the AMENDART are above the thermal values, and their behavior is analyzed analytically and numerically for typical parameters in circuit QED implementations described by the Rabi Hamiltonian. We point out that "parasitic elements", such as other cavity modes or eventual off-resonant atoms, also con…
We propose a method for fast, deterministic resonator reset based on tunable dissipative modes. The dissipator is based on a Josephson junction with relatively low quality factor. When the dissipator is tuned into resonance with a high quality microwave resonator, resonator photons are absorbed by the dissipator at a rate orders of magnitude faster than the resonator relaxation rate. We determine the optimal parameters for realization of the tunable dissipator, and examine application of the dissipator to removing spurious photon population in the qubit readout resonator in circuit quantum ele…
Using a circuit QED device, we present a theoretical study of real-time quantum state estimation via quantum Bayesian approach. Suitable conditions under which the Bayesian approach can accurately update the density matrix of the qubit are analyzed. We also consider the correlation between some basic and physically meaningful parameters of the circuit QED and the performance of the Bayesian approach. Our results advance the understanding of quantum Bayesian approach and pave the way to study quantum feedback control and adaptive control.
We present two pulse schemes for actively depleting measurement photons from a readout resonator in the nonlinear dispersive regime of circuit QED. One method uses digital feedback conditioned on the measurement outcome while the other is unconditional. In the absence of analytic forms and symmetries to exploit in this nonlinear regime, the depletion pulses are numerically optimized using the Powell method. We shorten the photon depletion time by more than six inverse resonator linewidths compared to passive depletion by waiting. We quantify the benefit by emulating an ancilla qubit performing…
Motivated by recent experimental progress towards the measurement and manipulation of Majorana fermions with superconducting circuits, we propose a device interfacing Majorana fermions with circuit quantum electrodynamics. The proposed circuit acts as a charge parity detector changing the resonance frequency of a superconducting $λ/4$ - resonator conditioned on the parity of charges on nearby gates. Operating at both charge and flux sweet spots, this device is highly insensitive to environmental noise. It enables high-fidelity single-shot quantum non demolition readout of the state of a pair o…
We analyze the backaction of homodyne detection and photodetection on superconducting qubits in circuit quantum electrodynamics. Although both measurement schemes give rise to backaction in the form of stochastic phase rotations, which leads to dephasing, we show that this can be perfectly undone provided that the measurement signal is fully accounted for. This result improves upon that of Phys. Rev. A, 82, 012329 (2010), showing that the method suggested can be made to realize a perfect two-qubit parity measurement. We propose a benchmarking experiment on a single qubit to demonstrate the met…
We study decoherence in a circuit QED system consisting of a charge qubit and two superconducting transmission line resonators (TLRs). We show that in the dispersive regime of the circuit QED system one TLR can be used as an auxiliary subsystem to realize decoherence-free quantum dynamics of the bipartite target system consisting of the charge qubit and the other TLR conditioned on the auxiliary TLR initially being a proper number state. Our study gives new insight into control and manipulation of decoherence in quantum systems.
We use a quasi Hamiltonian formalism to describe the dissipative dynamics of a circuit QED qubit that is affected by several fluctuating two level systems with a 1/f noise power spectrum. The qubit-resonator interactions are described by the Jaynes Cummings model. We argue that the presence of pure dephasing noise in such a qubit-resonator system will also induce an energy relaxation mechanism via a fluctuating dipole coupling term. This random modulation of the coupling is seen to lead to rich physical behavior. For non-Markovian noise, the coupling can either worsen or alleviate decoherence …
We describe a method to tune, in-situ, between transverse and longitudinal light-matter coupling in a hybrid circuit-QED device composed of an electron spin degree of freedom coupled to a microwave transmission line cavity. Our approach relies on periodic modulation of the coupling itself, such that in a certain frame the interaction is both amplified and either transverse, or, by modulating at two frequencies, longitudinal. The former realizes an effective simulation of certain aspects of the ultra-strong coupling regime, while the latter allows one to implement a longitudinal readout scheme …
We report the experimental observation, and a theoretical explanation, of collective suppression of linewidths for multiple superconducting qubits coupled to a good cavity. This demonstrates how strong qubit-cavity coupling can significantly modify the dephasing and dissipation processes that might be expected for individual qubits, and can potentially improve coherence times in many-body circuit QED.
We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as 17% of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings model. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional i…
In this experiment, we couple a superconducting Transmon qubit to a high-impedance $645\ Ω$ microwave resonator. Doing so leads to a large qubit-resonator coupling rate $g$, measured through a large vacuum Rabi splitting of $2g\simeq 910$ MHz. The coupling is a significant fraction of the qubit and resonator oscillation frequencies $ω$, placing our system close to the ultra-strong coupling regime ($\bar{g}=g/ω=0.071$ on resonance). Combining this setup with a vacuum-gap Transmon architecture shows the potential of reaching deep into the ultra-strong coupling $\bar{g} \sim 0.45$ with Transmon q…
In superconducting circuit architectures for quantum computing, microwave resonators are often used both to isolate qubits from the electromagnetic environment and to facilitate qubit state readout. We analyze the full counting statistics of photons emitted from such driven readout resonators both in and beyond the dispersive approximation. We calculate the overlap between emitted-photon distributions for the two qubit states and explore strategies for its minimization with the purpose of increasing fidelity of intensity-sensitive readout techniques. In the dispersive approximation and at negl…
Circuit cavity quantum electrodynamics (QED) is proving to be a powerful platform to implement quantum feedback control schemes due to the ability to control superconducting qubits and microwaves in a circuit. Here, we present a simple and promising quantum feedback control scheme for deterministic generation and stabilization of a three-qubit $W$ state in the superconducting circuit QED system. The control scheme is based on continuous joint Zeno measurements of multiple qubits in a dispersive regime, which enables us not only to infer the state of the qubits for further information processin…
Superconducting electrical circuits can be used to study the physics of cavity quantum electrodynamics (QED) in new regimes, therefore realizing circuit QED. For quantum information processing and quantum optics, an interesting regime of circuit QED is the dispersive regime, where the detuning between the qubit transition frequency and the resonator frequency is much larger than the interaction strength. In this paper, we investigate how non-linear corrections to the dispersive regime affect the measurement process. We find that in the presence of pure qubit dephasing, photon population of the…
We propose a scheme for detecting noncommutative feature of the non-Abelian geometric phase in circuit QED, which involves three transmon qubits capacitively coupled to an one-dimensional transmission line resonator. By controlling the external magnetic flux of the transmon qubits, we can obtain an effective tripod interaction of our circuit QED setup. The noncommutative feature of the non-Abelian geometric phase is manifested that for an initial state undergo two specific loops in different order will result in different final states. Our numerical calculations show that this difference can b…
Transitions between quantum states by photon absorption or emission are intimately related to symmetries of the system which lead to selection rules and the formation of dark states. In a circuit quantum electrodynamics setup, in which two resonant superconducting qubits are coupled through an on-chip cavity and driven via the common cavity field, one single-excitation state remains dark. Here, we demonstrate that this dark state can be excited using local phase control of individual qubit drives to change the symmetry of the driving field. We observe that the dark state decay via spontaneous …
Implementation of discrete-time quantum walk (DTQW) with superconducting qubits is difficult since on-chip superconducting qubits cannot hop between lattice sites. We propose an efficient protocol for the implementation of DTQW in circuit quantum electrodynamics (QED), in which only $N+1$ qutrits and $N$ assistant cavities are needed for an $N$-step DTQW. The operation of each DTQW step is very quick because only resonant processes are adopted. The numerical simulations show that high-similarity DTQW with the number of step up to $20$ is feasible with present-day circuit QED technique. This pr…
We study photon condensation phenomena in a driven and dissipative array of superconducting microwave resonators. Specifically, we show that by using an appropriately designed coupling of microwave photons to superconducting qubits, an effective dissipative mechanism can be engineered, which scatters photons towards low-momentum states while conserving their number. This mimics a tunable coupling of bosons to a low temperature bath, and leads to the formation of a stationary photon condensate in the presence of losses and under continuous-driving conditions. Here we propose a realistic experim…
We present a theoretical proposal for a microwave driven circuit composed of superconducting resonators and qubits which shows a bistable behaviour, and we present a simple mechanism that allows single- or few-photon microwave pulses to work as Set- and Reset-signals that switch the circuit between its stable modes. The resulting system constitutes an ultra-low-energy Set-Reset flip-flop, and we show that its memory lifetime far exceeds the lifetime of states stored in any of its separate components.
In cavity quantum electrodynamics (QED), light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED, where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both i…
We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of three-level superconducting qubits inside a resonator constitutes a key tool to perform diverse two-qubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant two-qubit gates in circuit QED, we consider the implementation of a two-dimensional cluster state in an array of N x N superconducting qubits by using resonant controlled-phase (CPHASE) and one-qubit g…
Transmission-line waveguides can mediate long-range fluctuation-induced forces between neutral objects. We present two approaches for the description of these forces between electric components embedded in transmission-line circuits. The first, following ordinary quantum electrodynamics (QED), consists of the quantization and scattering theory of voltage and current waves inside transmission lines. The second approach relies on a simple circuit analysis with additional noisy current sources due to resistors in the circuit, as per the fluctuation-dissipation theorem (FDT). We apply the latter a…
