We investigate quantum dynamics of a generic model of light-matter interaction in the context of high impedance waveguides, focusing on the behavior of the emitted photonic states, in the framework of the spin-boson model Quantum quenches as well as scattering of an incident coherent pulse are studied using two complementary methods. First, we develop an approximate ansatz for the electromagnetic waves based on a single multimode coherent state wavefunction; formally, this approach combines ideas from adiabatic renormalization, the Born-Markov approximation, and input-output theory. Second, we present numerically exact results for scattering of a weak intensity pulse by using NRG calculations. NRG provides a benchmark for any linear response property throughout the ultra-strong coupling regime. We find that in a sudden quantum quench, the coherent state approach produces physical artifacts, such as improper relaxation to the steady state. These previously unnoticed problems are related to the simplified form of the ansatz that generates spurious correlations within the bath. In the scattering problem, NRG is used to find the transmission and reflection of a single photon, as well as the inelastic scattering of that single photon. Simple analytical formulas are established and tested against the NRG data that predict quantitatively the transport coefficients for up to moderate environmental impedance. These formulas resolve pending issues regarding the presence of inelastic losses in the spin-boson model near absorption resonances, and could be used for comparison to experiments in Josephson waveguide QED. Finally, the scattering results using the coherent state wavefunction approach are compared favorably to the NRG results for very weak incident intensity. We end our study by presenting results at higher power where the response of the system is nonlinear.
We demonstrate a new approach to dissipation engineering in microwave quantum optics. For a single mode, dissipation usually corresponds to quantum jumps, where photons are lost one by one. Here, we are able to tune the minimal number of lost photons per jump to be two (or more) with a simple dc voltage. As a consequence, different quantum states experience different dissipation. Causality implies that the states must also experience different energy shifts. Our measurements of these Lamb shifts are in good agreement with the predictions of the Kramers-Kronig relations for single quantum state…
We measure the resistance and frequency-dependent gate capacitance of carbon nanotube (CNT) thin films in ambient, vacuum, and under low-pressure (10E-6 torr) analyte environments. We model the CNT film as an RC transmission line and show that changes in the measured capacitance as a function of gate bias and analyte pressure are consistent with changes in the transmission line impedance due to changes in the CNT film resistivity alone; the electrostatic gate capacitance of the CNT film does not depend on gate voltage or chemical analyte adsorption. However, the CNT film resistance is enormous…
Elastic waves propagating in piezoelectric materials are accompanied by a time-varying electric potential, which is of critical importance for acousto-electronic applications. The spatial mapping of such a potential at microwave frequencies is challenging since the characteristic length scale is determined by the acoustic wavelength of several micrometers. In this work, we report the visualization of surface acoustic waves (SAWs) on ferroelectric samples by transmission-mode microwave impedance microscopy (T-MIM). The SAW potential launched by the interdigital transducer is detected by the tip…
The characteristic impedance of a microwave transmission line is typically constrained to a value $Z_0$ = 50 $ Ω$, in-part because of the low impedance of free space and the limited range of permittivity and permeability realizable with conventional materials. Here we suggest the possibility of constructing high-impedance transmission lines by exploiting the plasmonic response of edge states associated with the quantum Hall effect in gated devices. We analyze various implementations of quantum Hall transmission lines based on distributed networks and lumped-element circuits, including a detail…
Fast, high-fidelity single and two-qubit gates are essential to building a viable quantum information processor, but achieving both in the same system has proved challenging for spin qubits. We propose and analyze an approach to perform a long-distance two-qubit controlled phase (CPHASE) gate between two singlet-triplet qubits using an electromagnetic resonator to mediate their interaction. The qubits couple longitudinally to the resonator, and by driving the qubits near the resonator's frequency they can be made to acquire a state-dependent geometric phase that leads to a CPHASE gate independ…
The demand for a fast high-frequency read-out of high impedance devices, such as quantum dots, necessitates impedance matching. Here we use a resonant impedance matching circuit (a stub tuner) realized by on-chip superconducting transmission lines to measure the electronic shot noise of a carbon nanotube quantum dot at a frequency close to 3 GHz in an efficient way. As compared to wide-band detection without impedance matching, the signal to noise ratio can be enhanced by as much as a factor of 800 for a device with an impedance of 100 k$Ω$. The advantage of the stub resonator concept is the e…
We demonstrate how the dissipative interaction between a superconducting qubit and a microwave photonic crystal can be used for quantum bath engineering. The photonic crystal is created with a step-impedance transmission line which suppresses and enhances the quantum spectral density of states, influencing decay transitions of a transmon circuit. The qubit interacts with the transmission line indirectly via dispersive coupling to a cavity. We characterize the photonic crystal density of states from both the unitary and dissipative dynamics of the qubit. When the qubit is driven, it dissipates …
The statistical model proposed in an accompanying paper is generalized to treat multiport scattering problems. Attention is first focused on two-port lossless systems and the model is shown to be consistent with Random Matrix Theory. The predictions are then tested by direct numerical simulation for a specific two-port cavity. Formula are derived for the average transmission and reflection coefficients in terms of the port radiation impedance. The cases of cavity with multiple ports, and with a single port and distributed losses are compared.
We study tunneling through a resonant level connected to two dissipative bosonic baths: one is the resistive environment of the source and drain leads, while the second comes from coupling to potential fluctuations on a resistive gate. We show that several quantum phase transitions (QPT) occur in such a model, transitions which emulate those found in interacting systems such as Luttinger liquids or Kondo systems. We first use bosonization to map this dissipative resonant level model to a resonant level in a Luttinger liquid, one with, curiously, two interaction parameters. Drawing on methods f…
Quantum Hall edge states have some characteristic features that can prove useful to measure and control solid state qubits. For example, their high voltage to current ratio and their dissipationless nature can be exploited to manufacture low-loss microwave transmission lines and resonators with a characteristic impedance of the order of the quantum of resistance $h/e^2\sim 25\mathrm{kΩ}$. The high value of the impedance guarantees that the voltage per photon is high and for this reason high impedance resonators can be exploited to obtain larger values of coupling to systems with a small charge…
Coulomb interaction has important consequences on the physics of quantum spin Hall edge states, weakening the topological protection via two-particle scattering and renormalizing both the velocity and charge of collective plasmon modes compared to that of free electrons. Despite these effects, interactions remain difficult to quantify. We propose here simple and robust edge resonator geometries to characterize Coulomb interaction by means of high-frequency measurements. They rely on a transmission line approach, and take advantage of the impedance mismatch between the edge states and their mic…
We report the realization of a bonded-bridge on-chip superconducting coil and its use in impedance-matching a highly ohmic quantum dot (QD) to a $\rm{3~GHz}$ measurement setup. The coil, modeled as a lumped-element $LC$ resonator, is more compact and has a wider bandwidth than resonators based on coplanar transmission lines (e.g. $λ/4$ impedance transformers and stub tuners) at potentially better signal-to-noise ratios. In particular for measurements of radiation emitted by the device, such as shot noise, the 50$\times$ larger bandwidth reduces the time to acquire the spectral density. The res…
We study the microwave impedance of extremely high aspect ratio (length/width ~ 5,000) superconducting niobium nitride nanowires. The nanowires are fabricated in a compact meander geometry that is in series with the center conductor of a 50 ohm coplanar waveguide transmission line. The transmission coefficient of the sample is measured up to 20 GHz. At high frequency, a peak in the transmission coefficient is seen. Numerical simulations show that this is a half-wave resonance along the length of the nanowire, where the nanowire acts as a high impedance, slow wave transmission line. This resona…
We study experimentally the transmission of an electro-magnetic waveguide in the frequency range from 160 to 300 GHz. Photo-mixing is used to excite and detect the fundamental TE10 mode in a rectangular waveguide with two orders-of-magnitude lower impedance. The large impedance mismatch leads to a strong frequency dependence of the transmission, which we measure with a high-dynamic range of up to 80 dB, and with high frequency-resolution. The modified transmission function is directly related to the information rate of the waveguide, which we estimate to be about 1 bit per photon. We suggest t…
Frequency-resolved decoherence spectroscopy of a semiconductor charge qubit coupled to a high-impedance resonator
ETH Zurich（苏黎世联邦理工学院）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Microsoft Quantum Lab Delft（微软量子实验室代尔夫特）；Lund University（隆德大学）；Delft University of Technology（代尔夫特理工大学）；Leiden University（莱顿大学）；Norwegian University of Science and Technology（挪威科技大学）
We discuss a model for studying the statistical properties of the impedance ($Z$) and scattering ($S$) matrices of open electromagnetic cavities with several transmission lines or waveguides connected to the cavity. In this paper, we mainly discuss the single port case. The generalization to multiple ports is treated in a companion paper. The model is based on assumed properties of chaotic eigenfunctions for the closed system. Analysis of the model successfully reproduces features of the random matrix model believed to be universal, while at the same time incorporating features which are speci…
Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.
We systematically investigate how to quantize a transmission line resonator (TLR) in a mesoscopic electrical circuits in the presence of the resistance and the conductance of the dielectric media. Developed from the quantum bath based effective Hamiltonian method for single mode harmonic oscillator, the approach we presented in this article is a microscopic theory integrating quantum fluctuation-dissipation relation. To qualitatively check the condition under which the TLR can behave as a quantum object we study the classical-quantum boundary characterized by the standard quantum limit.
We demonstrate impedance-matched low-loss transmission lines with a signal wavelength more than 150 times smaller than the free space wavelength using superconducting nanowires on high permittivity substrates. A niobium nitride thin film is patterned in a coplanar waveguide (CPW) transmission line geometry on a bilayer substrate consisting of 100 nm of epitaxial strontium titanate on high-resitivity silicon. The use of strontium titanate on silicon enables wafer-scale fabrication and maximizes process compatibility. It also makes it possible to realize a $50$ $Ω$ characteristic impedance acros…
We consider the influence of nonradiative damping of qubits on the microwave transport of photons, propagating in an open one-dimensional microstrip line. Within the framework of the formalism of a non-Hermitian Hamiltonian we obtained the expressions for the transmission and reflection coefficients for two qubits which explicitly account for the indirect interaction between qubits due to nonradiative decay into common bath. It is shown that this interaction leads to the results that are significantly different from those already known
We report on radio-frequency measurements of the charge-phase qubit being under continuous microwave irradiation in the state of weak coupling to a radio-frequency tank circuit. We studied the rf impedance dependence on the two important parameters such as power of microwave irradiation whose frequency is close to the gap between the two lowest qubit energy levels, and temperature of the internal heat bath. We have found that backaction effects of the qubit on the rf tank, and vice versa, tank on the qubit, lead to a negative as well as a positive real part of the qubit impedance Re$Z(ω)$ seen…
We analyze the current-voltage characteristic of a quantum conduction channel coupled to an electromagnetic environment of arbitrary frequency-dependent impedance. In the weak blockade regime the correction to the ohmic behavior is directly related to the channel current fluctuations vanishing at perfect transmission in the same way as shot noise. This relation can be generalized to describe the environmental Coulomb blockade in a generic mesoscopic conductor coupled to an external impedance, as the response of the latter to the current fluctuations in the former.
The RF-response of a three-terminal carbon nanotube resonator coupled to RF-transmission lines is studied by means of perturbation theory and direct numerical integration. We find three distinct oscillatory regimes, including one regime capable of exhibiting very large hysteresis loops in the frequency response. Considering a purely capacitive transduction, we derive a set of algebraic equations which can be used to find the output power (S-parameters) for a device connected to transmission lines with characteristic impedance $Z_0$.
