Photon statistics of a double quantum dot micromaser: Quantum treatment

A semiconductor single-atom micromaser consists of a microwave cavity coupled to a gain medium, a double quantum dot driven out of equilibrium by a bias voltage. The masing threshold of this system was recently probed by measuring photon statistics in the cavity [Y-Y. Liu \textit{et al},  (2017)]. In this paper, we develop an in-depth, rigorous understanding of this experiment and related works. First, we use a semiclassical theory and study transmission spectroscopy. This approach allows us to derive the masing threshold condition for arbitrary temperature and voltage bias, and expose microscopic principles required for realizing photon gain and thereby a photon amplifier. Next, by employing the quantum master equation approach we extend the Scully-Lamb quantum theory of a laser to the present setup, and investigate the statistics of emitted photons below and above the masing threshold as a function of experimentally tunable parameters. Although our focus is primarily on hybrid quantum dot circuit - quantum electrodynamics systems, our approach is adaptable to other light-matter systems where the gain medium consists of a mesoscopic structure.

Related papers:

title: Threshold Dynamics of a Semiconductor Single Atom Maser.
abstract: We demonstrate a single atom maser consisting of a semiconductor double quantum dot (DQD) that is embedded in a high-quality-factor microwave cavity. A finite bias drives the DQD out of equilibrium, resulting in sequential single electron tunneling and masing. We develop a dynamic tuning protocol that allows us to controllably increase the time-averaged repumping rate of the DQD at a fixed level detuning, and quantitatively study the transition through the masing threshold. We further examine the crossover from incoherent to coherent emission by measuring the photon statistics across the masing transition. The observed threshold behavior is in agreement with an existing single atom maser theory when small corrections from lead emission are taken into account.

title: Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems
abstract: Hybrid quantum circuits combine two or more physical systems, with the goal of harnessing the advantages and strengths of the different systems in order to better explore new phenomena and potentially bring about novel quantum technologies. This article presents a brief overview of the progress achieved so far in the field of hybrid circuits involving atoms, spins and solid-state devices (including superconducting and nanomechanical systems). How these circuits combine elements from atomic physics, quantum optics, condensed matter physics, and nanoscience is discussed, and different possible approaches for integrating various systems into a single circuit are presented. In particular, hybrid quantum circuits can be fabricated on a chip, facilitating their future scalability, which is crucial for building future quantum technologies, including quantum detectors, simulators, and computers.

title: Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip.
abstract: We demonstrate a hybrid architecture consisting of a quantum dot circuit coupled to a single mode of the electromagnetic field. We use single wall carbon nanotube based circuits inserted in superconducting microwave cavities. By probing the nanotube dot using a dispersive readout in the Coulomb blockade and the Kondo regime, we determine an electron-photon coupling strength which should enable circuit QED experiments with more complex quantum dot circuits.

title: Photon-mediated interaction between distant quantum dot circuits.
abstract: Engineering the interaction between light and matter is an important goal in the emerging field of quantum opto-electronics. Thanks to the use of cavity quantum electrodynamics architectures, one can envision a fully hybrid multiplexing of quantum conductors. Here we use such an architecture to couple two quantum dot circuits. Our quantum dots are separated by 200 times their own size, with no direct tunnel and electrostatic couplings between them. We demonstrate their interaction, mediated by the cavity photons. This could be used to scale up quantum bit architectures based on quantum dot circuits or simulate on-chip phonon-mediated interactions between strongly correlated electrons.

title: Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua
abstract: Microwave cavities have been widely used to investigate the behavior of closed few-level systems. Here, we show that they also represent a powerful probe for the dynamics of charge transfer between a discrete electronic level and fermionic continua. We have combined experiment and theory for a carbon nanotube quantum dot coupled to normal metal and superconducting contacts. In equilibrium conditions, where our device behaves as an effective quantum dot-normal metal junction, we approach a universal photon dissipation regime governed by a quantum charge relaxation effect. We observe how photon dissipation is modified when the dot admittance turns from capacitive to inductive. When the fermionic reservoirs are voltage biased, the dot can even cause photon emission due to inelastic tunneling to/from a Bardeen-Cooper-Schrieffer peak in the density of states of the superconducting contact. We can model these numerous effects quantitatively in terms of the charge susceptibility of the quantum dot circuit. This validates an approach that could be used to study a wide class of mesoscopic QED devices.

title: Tunable hybrid quantum electrodynamics from nonlinear electron transport
abstract: Recent advances in quantum electronics have allowed to engineer hybrid nano-devices comprising on chip a microwave electromagnetic resonator coupled to an artificial atom, a quantum dot. These systems realize novel platforms to explore non-equilibrium quantum impurity physics with light and matter. Coupling the quantum dot system to reservoir leads (source and drain) produces an electronic current as well as dissipation when applying a bias voltage across the system. Focusing on a standard model of biased quantum dot coupled to a photon mode we elucidate the signatures of the electronic correlations in the phase of the transmitted microwave signal. In addition, we illustrate the effect of the electronic degrees of freedom on the photon field, giving rise to non-linearities, damping and dissipation, and discuss how to control these effects by means of gate and bias voltages.

title: Quantum-dot circuit-QED thermoelectric diodes and transistors
abstract: Recent breakthroughs in quantum-dot circuit-quantum-electrodynamics (circuit-QED) systems are important both from a fundamental perspective and from the point of view of quantum photonic devices. However, understanding the applications of such setups as potential thermoelectric diodes and transistors has been missing. In this paper, via the Keldysh nonequilibrium Green's function approach, we show that cavity-coupled double quantum-dots can serve as excellent quantum thermoelectric diodes and transistors. Using an enhanced perturbation approach based on polaron-transformations, we find non-monotonic dependences of thermoelectric transport properties on the electron-photon interaction. Strong light-matter interaction leads to pronounced rectification effects for both charge and heat, as well as thermal transistor effects in the linear transport regime, which opens up a cutting-edge frontier for quantum thermoelectric devices.

title: Many-body quantum electrodynamics networks: Non-equilibrium condensed matter physics with light
abstract: We review recent developments concerning non-equilibrium quantum dynamics and many-body physics with light, in superconducting circuits and Josephson analogues. We start with quantum impurity models summarizing the effect of dissipation and of driving the system. We mention theoretical and experimental efforts to characterize these non-equilibrium quantum systems. We show how Josephson junction systems can implement the equivalent of the Kondo effect with microwave photons. The Kondo effect is characterized by a renormalized light-frequency and a peak in the Rayleigh elastic transmission of a photon. We also address the physics of hybrid systems comprising mesoscopic quantum dot devices coupled to an electromagnetic resonator. Then, we discuss extensions to Quantum Electrodynamics (QED) Networks allowing to engineer the Jaynes-Cummings lattice and Rabi lattice models. This opens the door to novel many-body physics with light out of equilibrium, in relation with the Mott-superfluid transition observed with ultra-cold atoms in optical lattices. Then, we summarize recent theoretical predictions for realizing topological phases with light. Synthetic gauge fields and spin-orbit couplings have been successfully implemented with ultra-cold atoms in optical lattices --- using time-dependent Floquet perturbations periodic in time, for example --- as well as in photonic lattice systems. Finally, we discuss the Josephson effect related to Bose-Hubbard models in ladder and two-dimensional geometries. The Bose-Hubbard model is related to the Jaynes-Cummings lattice model in the large detuning limit between light and matter. In the presence of synthetic gauge fields, we show that Meissner currents subsist in an insulating Mott phase.

title: Semiconductor double quantum dot micromaser
abstract: Tunnel through and emit coherently The generation of coherent light (lasers and masers) forms the basis of a large optics industry. Liu et al. demonstrate a type of laser that is driven by the tunneling of single electrons in semiconductor double-quantum dots. Distinct from other existing semiconductor lasers, the emission mechanism is driven by tunneling of single charges between discrete energy levels that are electrically tunable. The ability to tune the levels by single-electron charging would allow their laser (or maser) to be turned on and off rapidly. Science, this issue p. 285 A coherent microwave source that is driven by the tunneling of single electrons is demonstrated. The coherent generation of light, from masers to lasers, relies upon the specific structure of the individual emitters that lead to gain. Devices operating as lasers in the few-emitter limit provide opportunities for understanding quantum coherent phenomena, from terahertz sources to quantum communication. Here we demonstrate a maser that is driven by single-electron tunneling events. Semiconductor double quantum dots (DQDs) serve as a gain medium and are placed inside a high-quality factor microwave cavity. We verify maser action by comparing the statistics of the emitted microwave field above and below the maser threshold.

title: Phonon-assisted gain in a semiconductor double quantum dot maser.
abstract: We develop a microscopic model for the recently demonstrated double-quantum-dot maser. In characterizing the gain of this device we find that, in addition to the direct stimulated emission of photons, there is a large contribution from the simultaneous emission of a photon and a phonon, i.e., the phonon sideband. We show that this phonon-assisted gain typically dominates the overall gain, which leads to masing. Recent experimental data are well fit with our model.

title: Circuit quantum electrodynamics with a spin qubit
abstract: Electron spins trapped in quantum dots have been proposed as basic building blocks of a future quantum processor. Although fast, 180-picosecond, two-quantum-bit (two-qubit) operations can be realized using nearest-neighbour exchange coupling, a scalable, spin-based quantum computing architecture will almost certainly require long-range qubit interactions. Circuit quantum electrodynamics (cQED) allows spatially separated superconducting qubits to interact via a superconducting microwave cavity that acts as a ‘quantum bus’, making possible two-qubit entanglement and the implementation of simple quantum algorithms. Here we combine the cQED architecture with spin qubits by coupling an indium arsenide nanowire double quantum dot to a superconducting cavity. The architecture allows us to achieve a charge–cavity coupling rate of about 30 megahertz, consistent with coupling rates obtained in gallium arsenide quantum dots. Furthermore, the strong spin–orbit interaction of indium arsenide allows us to drive spin rotations electrically with a local gate electrode, and the charge–cavity interaction provides a measurement of the resulting spin dynamics. Our results demonstrate how the cQED architecture can be used as a sensitive probe of single-spin physics and that a spin–cavity coupling rate of about one megahertz is feasible, presenting the possibility of long-range spin coupling via superconducting microwave cavities.

title: Dipole coupling of a double quantum dot to a microwave resonator.
abstract: We demonstrate the realization of a hybrid solid-state quantum device, in which a semiconductor double quantum dot is dipole coupled to the microwave field of a superconducting coplanar waveguide resonator. The double dot charge stability diagram extracted from measurements of the amplitude and phase of a microwave tone transmitted through the resonator is in good agreement with that obtained from transport measurements. Both the observed frequency shift and linewidth broadening of the resonator are explained considering the double dot as a charge qubit coupled with a strength of several tens of MHz to the resonator.

title: Microwave Emission from Hybridized States in a Semiconductor Charge Qubit.
abstract: We explore the microwave radiation emitted from a biased double quantum dot due to the inelastic tunneling of single charges. Radiation is detected over a broad range of detuning configurations between the dot energy levels, with pronounced maxima occurring in resonance with a capacitively coupled transmission line resonator. The power emitted for forward and reverse resonant detuning is found to be in good agreement with a rate equation model, which considers the hybridization of the individual dot charge states.

title: Vacuum Rabi splitting in a semiconductor circuit QED system.
abstract: Vacuum Rabi splitting is demonstrated in a GaAs double quantum dot system coupled with a coplanar waveguide resonator. The coupling strength g, the decoherence rate of the quantum dot γ, and the decay rate of the resonator κ are derived, assuring distinct vacuum Rabi oscillation in a strong coupling regime [(g,γ,κ)≈(30,25,8.0) MHz]. The magnitude of decoherence is consistently interpreted in terms of the coupling of electrons to piezoelectric acoustic phonons in GaAs.

title: A Quantum Electrodynamics Kondo Circuit with Orbital and Spin Entanglement
abstract: Recent progress in nanotechnology allows to engineer hybrid mesoscopic devices comprising on chip an artificial atom or quantum dot, capacitively coupled to a microwave (superconducting) resonator. These systems can then contribute to explore non-equilibrium quantum impurity physics with light and matter, by increasing the input power on the cavity and the bias voltage across the mesoscopic system. Here, we build such a prototype system where the artificial atom is a graphene double quantum dot (DQD). Controlling the coupling of the photon field and the charge states of the DQD, we measure the microwave reflection spectrum of the resonator. When the DQD is at the charge degeneracy points, experimental results are consistent with a Kondo impurity model entangling charge, spin and orbital degrees of freedom. The light reveals the formation of the Kondo or Abrikosov-Suhl resonance at low temperatures. We then study the complete phase diagram as a function of gate voltages, bias voltage and microwave input power.

title: Coherent coupling of a single spin to microwave cavity photons
abstract: Making hybrid quantum information systems Different physical implementations of qubits—quantum bits—each have their pros and cons. An appealing idea is to combine them into hybrid architectures, taking advantage of their respective strengths. Tabuchi et al. placed a ferromagnetic sphere and a superconducting qubit in a cavity and used an electromagnetic mode of the cavity as the mediator between the two. They achieved strong coupling between a collective magnetic mode of the sphere and the qubit. Viennot et al. coupled a single spin in a double quantum dot to photons in a cavity. Both approaches hold promise for future applications. Science, this issue pp. 405 and 408 Noncollinear magnets hybridize the spin and charge states of a double quantum dot, enabling spin-photon coupling in a cavity. Electron spins and photons are complementary quantum-mechanical objects that can be used to carry, manipulate, and transform quantum information. To combine these resources, it is desirable to achieve the coherent coupling of a single spin to photons stored in a superconducting resonator. Using a circuit design based on a nanoscale spin valve, we coherently hybridize the individual spin and charge states of a double quantum dot while preserving spin coherence. This scheme allows us to achieve spin-photon coupling up to the megahertz range at the single-spin level. The cooperativity is found to reach 2.3, and the spin coherence time is about 60 nanoseconds. We thereby demonstrate a mesoscopic device suitable for nondestructive spin readout and distant spin coupling.

title: Mesoscopic admittance of a double quantum dot
abstract: We calculate the mesoscopic admittance G({omega}) of a double quantum dot (DQD), which can be measured directly using microwave techniques. This quantity reveals spectroscopic information on the DQD and is also directly sensitive to a Pauli spin blockade effect. We then discuss the problem of a DQD coupled to a high quality photonic resonator. When the photon correlation functions can be developed along a random-phase-approximation-like scheme, the response of the resonator gives an access to G({omega}).

title: Photon emission from a cavity-coupled double quantum dot.
abstract: We study a voltage biased InAs double quantum dot (DQD) that is coupled to a superconducting transmission line resonator. Inelastic tunneling in the DQD is mediated by electron phonon coupling and coupling to the cavity mode. We show that electronic transport through the DQD leads to photon emission from the cavity at a rate of 10 MHz. With a small cavity drive field, we observe a gain of up to 15 in the cavity transmission. Our results are analyzed in the context of existing theoretical models and suggest that it may be necessary to account for inelastic tunneling processes that proceed via simultaneous emission of a phonon and a photon.

title: Cavity-coupled double-quantum dot at finite bias: Analogy with lasers and beyond
abstract: We present a theoretical and experimental study of photonic and electronic transport properties of a voltage biased InAs semiconductor double quantum dot (DQD) that is dipole coupled to a superconducting transmission line resonator. We obtain the master equation for the reduced density matrix of the coupled system of cavity photons and DQD electrons accounting systematically for both the presence of phonons and the effect of leads at finite voltage bias. We subsequently derive analytical expressions for transmission, phase response, photon number, and the nonequilibrium steady-state electron current. We show that the coupled system under finite bias realizes an unconventional version of a single-atom laser and analyze the spectrum and the statistics of the photon flux leaving the cavity. In the transmission mode, the system behaves as a saturable single-atom amplifier for the incoming photon flux. Finally, we show that the back action of the photon emission on the steady-state current can be substantial. Our analytical results are compared to exact master equation results establishing regimes of validity of various analytical models. We compare our findings to available experimental measurements.

title: Electron-photon interaction in a quantum point contact coupled to a microwave resonator
abstract: We study a single-mode cavity weakly coupled to a voltage-biased quantum point contact. In a perturbative analysis, the lowest order predicts a thermal state for the cavity photons, driven by the emission noise of the conductor. The cavity is thus emptied as all transmission probabilities of the quantum point contact approach one or zero. Two-photon processes are identified at higher coupling, and pair absorption dominates over pair emission for all bias voltages. As a result, the number of cavity photons, the cavity damping rate and the second order coherence $g^{(2)}$ are all reduced and exhibit less bunching than the thermal state. These results are obtained with a Keldysh path integral formulation and reproduced with rate equations. They can be seen as a backaction of the cavity measuring the electronic noise. Extending the standard $P(E)$ theory to a steady-state situation, we compute the modified noise properties of the conductor and find quantitative agreement with the perturbative calculation.

title: Shuttling a single charge across a one-dimensional array of silicon quantum dots
abstract: Significant advances have been made towards fault-tolerant operation of silicon spin qubits, with single qubit fidelities exceeding 99.9%, several demonstrations of two-qubit gates based on exchange coupling, and the achievement of coherent single spin-photon coupling. Coupling arbitrary pairs of spatially separated qubits in a quantum register poses a significant challenge as most qubit systems are constrained to two dimensions with nearest neighbor connectivity. For spins in silicon, new methods for quantum state transfer should be developed to achieve connectivity beyond nearest-neighbor exchange. Here we demonstrate shuttling of a single electron across a linear array of nine series-coupled silicon quantum dots in ~50 ns via a series of pairwise interdot charge transfers. By constructing more complex pulse sequences we perform parallel shuttling of two and three electrons at a time through the array. These experiments demonstrate a scalable approach to physically transporting single electrons across large silicon quantum dot arrays.As quantum computers grow in complexity the challenge of moving information between physically separated qubits becomes more pressing. Mills et al. demonstrate transfer of single electrons across an array of nine silicon quantum dots three orders of magnitude faster than spin qubit decoherence times.

title: Nonlocal transport properties of nanoscale conductor-microwave cavity systems
abstract: Recent experimental progress in coupling nanoscale conductors to superconducting microwave cavities has opened up for transport investigations of the deep quantum limit of light-matter interactions, with tunneling electrons strongly coupled to individual cavity photons. We have investigated theoretically the most basic cavity-conductor system with strong, single photon induced nonlocal transport effects: two spatially separated double quantum dots (DQDs) resonantly coupled to the fundamental cavity mode. The system, described by a generalized Tavis-Cummings model, is investigated within a quantum master equation formalism, allowing us to account for both the electronic transport properties through the DQDs as well as the coherent, nonequilibrium cavity photon state. We find sizable nonlocally induced current and current cross-correlations mediated by individual photons. From a full statistical description of the electron transport we further reveal a dynamical channel blockade in one DQD lifted by photon emission due to tunneling through the other DQD. Moreover, large entanglement between the orbital states of electrons in the two DQDs is found for small DQD-lead temperatures.

title: Electroluminescence Caused by the Transport of Interacting Electrons through Parallel Quantum Dots in a Photon Cavity
abstract: This work was financially supported by the Research 
Fund of the University of Iceland, the Icelandic Research 
Fund, grant no. 163082-051, and the Icelandic Instruments 
Fund. HSG acknowledges support from Ministry 
of Science and Technology of Taiwan, under grant no. 
103-2112-M-002-003-MY3.

title: Current correlations for the transport of interacting electrons through parallel quantum dots in a photon cavity
abstract: We calculate the current correlations for the steady-state electron transport through multi-level parallel quantum dots embedded in a short quantum wire, that is placed in a non-perfect photon cavity. We account for the electron–electron Coulomb interaction, and the para- and diamagnetic electron–photon interactions with a stepwise scheme of configuration interactions and truncation of the many-body Fock spaces. In the spectral density of the temporal current–current correlations we identify all the transitions, radiative and non-radiative, active in the system in order to maintain the steady state. We observe strong signs of two types of Rabi oscillations.

title: Phase locking of a semiconductor double quantum dot single atom maser
abstract: We experimentally study the phase stabilization of a semiconductor double quantum dot (DQD) single atom maser by injection locking. A voltage-biased DQD serves as an electrically tunable microwave frequency gain medium. The statistics of the maser output field demonstrate that the maser can be phase locked to an external cavity drive, with a resulting phase noise of -99 dBc/Hz at a frequency offset of 1.3 MHz. The injection locking range, and the phase of the maser output relative to the injection locking input tone are in good agreement with Adler's theory. Furthermore, the electrically tunable DQD energy level structure allows us to rapidly switch the gain medium on and off, resulting in an emission spectrum that resembles a frequency comb. The free running frequency comb linewidth is ~8 kHz and can be improved to less than 1 Hz by operating the comb in the injection locked regime.

title: One-atom maser.
abstract: The exchange of photons between single Rydberg atoms and a single mode of a superconducting cavity with a quality factor $Q=8\ifmmode\times\else\texttimes\fi{}{10}^{8}$ at 2 K was observed. Signals could still be detected with an average number of only 0.06 atom simultaneously in the cavity. With one Rydberg atom the linewidth of the maser transition at about 21 GHz was power broadened and at higher densities asymmetry of the transition was observed, which is ascribed to an ac Stark effect.

title: Experimental realization of a one-atom laser in the regime of strong coupling
abstract: Conventional lasers (from table-top systems to microscopic devices) typically operate in the so-called weak-coupling regime, involving large numbers of atoms and photons; individual quanta have a negligible impact on the system dynamics. However, this is no longer the case when the system approaches the regime of strong coupling for which the number of atoms and photons can become quite small. Indeed, the lasing properties of a single atom in a resonant cavity have been extensively investigated theoretically. Here we report the experimental realization of a one-atom laser operated in the regime of strong coupling. We exploit recent advances in cavity quantum electrodynamics that allow one atom to be isolated in an optical cavity in a regime for which one photon is sufficient to saturate the atomic transition. The observed characteristics of the atom–cavity system are qualitatively different from those of the familiar many-atom case. Specifically, our measurements of the intracavity photon number versus pump intensity indicate that there is no threshold for lasing, and we infer that the output flux from the cavity mode exceeds that from atomic fluorescence by more than tenfold. Observations of the second-order intensity correlation function demonstrate that our one-atom laser generates manifestly quantum (nonclassical) light, typified by photon anti-bunching and sub-poissonian photon statistics.

title: Low threshold electrically pumped quantum dot-micropillar lasers
abstract: We report on low threshold lasing in electrically pumped quantum dot-micropillar cavities. Lasing action associated with threshold currents as low as 8μA at 10K is observed for micropillar cavities with quality factors exceeding 10.000. Due to an optimized contact scheme lasing is achieved for pillar structures with diameters as small as 1.5μm, containing on average less than 100 quantum dots in the active layer. The transition from spontaneous to stimulated emission is confirmed by autocorrelation measurements which reveal pronounced photon bunching near threshold.

title: Quantum to classical transition in a single-ion laser
abstract: ‘Quantum’ lasers — with one atom interacting with a single optical mode — do not exhibit a conventional-laser-like threshold in the regime of strong light–matter coupling. A study now shows that a threshold is seen when the coupling strength is reduced, which represents the onset of classical lasing.

title: Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary
abstract: Microwave photons trapped in a superconducting cavity constitute an ideal system to realize some of the thought experiments imagined by the founding fathers of quantum physics. The interaction of these trapped photons with Rydberg atoms crossing the cavity illustrates fundamental aspects of measurement theory. The experiments performed with this ``photon box'' at Ecole Normale Sup\'erieure (ENS) belong to the domain of quantum optics called ``cavity quantum electrodynamics.'' We have realized the nondestructive counting of photons, the recording of field quantum jumps, the preparation and reconstruction of ``Schr\"odinger cat'' states of radiation and the study of their decoherence, which provides a striking illustration of the transition from the quantum to the classical world. These experiments have also led to the demonstration of basic steps in quantum information processing, including the deterministic entanglement of atoms and the realization of quantum gates using atoms and photons as quantum bits. This lecture starts by an introduction stressing the connection between the ENS photon box and the ion-trap experiments of David Wineland, whose accompanying lecture recalls his own contribution to the field of single particle control. I give then a personal account of the early days of cavity quantum electrodynamics before describing the main experiments performed at ENS during the last 20 years and concluding by a discussion comparing our work to other researches dealing with the control of single quantum particles.

title: Single artificial-atom lasing
abstract: Solid-state superconducting circuits are versatile systems in which quantum states can be engineered and controlled. Recent progress in this area has opened up exciting possibilities for exploring fundamental physics as well as applications in quantum information technology; in a series of experiments it was shown that such circuits can be exploited to generate quantum optical phenomena, by designing superconducting elements as artificial atoms that are coupled coherently to the photon field of a resonator. Here we demonstrate a lasing effect with a single artificial atom—a Josephson-junction charge qubit—embedded in a superconducting resonator. We make use of one of the properties of solid-state artificial atoms, namely that they are strongly and controllably coupled to the resonator modes. The device is essentially different from existing lasers and masers; one and the same artificial atom excited by current injection produces many photons.

title: Tunable photonic cavity coupled to a voltage-biased double quantum dot system: Diagrammatic nonequilibrium Green's function approach
abstract: We investigate gain in microwave photonic cavities coupled to voltage-biased double quantum dot systems with an arbitrarily strong dot-lead coupling and with a Holstein-like light-matter interaction, by employing the diagrammatic Keldysh nonequilibrium Green's function approach. We compute out-of-equilibrium properties of the cavity: its transmission, phase response, mean photon number, power spectrum, and spectral function. We show that by the careful engineering of these hybrid light-matter systems, one can achieve a significant amplification of the optical signal with the voltage-biased electronic system serving as a gain medium. We also study the steady-state current across the device, identifying elastic and inelastic tunneling processes which involve the cavity mode. Our results show how recent advances in quantum electronics can be exploited to build hybrid light-matter systems that behave as microwave amplifiers and photon source devices. The diagrammatic Keldysh approach is primarily discussed for a cavity-coupled double quantum dot architecture, but it is generalizable to other hybrid light-matter systems.

title: Quantum statistics of a single-atom Scovil-Schulz-DuBois heat engine
abstract: We study the statistics of the lasing output from a single-atom quantum heat engine, which was originally proposed by Scovil and Schulz-DuBois [H. E. D. Scovil and E. O. Schulz-DuBois,  (1959)]. In this heat engine model, a single three-level atom is coupled with an optical cavity and is in contact with a hot and a cold heat bath together. We derive a fully quantum laser equation for this heat engine model and obtain the photon number distribution both below and above the lasing threshold. With the increase of the hot bath temperature, the population is inverted and lasing light comes out. However, we notice that if the hot bath temperature keeps increasing, the atomic decay rate is also enhanced, which weakens the lasing gain. As a result, another critical point appears at a very high temperature of the hot bath, after which the output light become thermal radiation again. To avoid this double-threshold behavior, we introduce a four-level heat engine model, where the atomic decay rate does not depend on the hot bath temperature. In this case, the lasing threshold is much easier to achieve and the double-threshold behavior disappears.

title: Full counting statistics of photons emitted by a double quantum dot
abstract: We analyze the full counting statistics of photons emitted by a double quantum dot (DQD) coupled to a high-quality microwave resonator by electric dipole interaction. We show that at the resonant condition between the energy splitting of the DQD and the photon energy in the resonator, photon statistics exhibits both a sub-Poissonian distribution and antibunching. In the ideal case, when the system decoherence stems only from photodetection, the photon noise is reduced below one-half of the noise for the Poisson distribution and is consistent with current noise. The photon distribution remains sub-Poissonian even at moderate decoherence in the DQD. We demonstrate that Josephson junction based photomultipliers can be used to experimentally assess statistics of emitted photons.

title: Bistable Photon Emission from a Solid-State Single-Atom Laser.
abstract: We predict a bistability in the photon emission from a solid-state single-atom laser comprising a microwave cavity coupled to a voltage-biased double quantum dot. To demonstrate that the single-atom laser is bistable, we evaluate the photon emission statistics and show that the distribution takes the shape of a tilted ellipse. The switching rates of the bistability can be extracted from the electrical current and the shot noise in the quantum dots. This provides a means to control the photon emission statistics by modulating the electronic transport in the quantum dots. Our prediction is robust against moderate electronic decoherence and dephasing and is important for current efforts to realize single-atom lasers with gate-defined quantum dots as the gain medium.

title: Microwave quantum optics and electron transport through a metallic dot strongly coupled to a transmission line cavity
abstract: We investigate theoretically the properties of the photon state and the electronic transport in a system consisting of a metallic quantum dot strongly coupled to a superconducting microwave transmission line cavity. Within the framework of circuit quantum electrodynamics, we derive a Hamiltonian for arbitrary strong capacitive coupling between the dot and the cavity. The dynamics of the system is described by a quantum master equation, accounting for the electronic transport as well as the coherent, nonequilibrium properties of the photon state. The photon state is investigated, focusing on, for a single active mode, signatures of microwave polaron formation and the effects of a nonequilibrium photon distribution. For two active photon modes, the intermode conversion and polaron coherences are investigated. For the electronic transport, electrical current and noise through the dot and the influence of the photon state on the transport properties are at the focus. We identify clear transport signatures due to the nonequilibrium photon population, in particular the emergence of superpoissonian shot noise at ultrastrong dot-cavity couplings.

title: Nonlocal transport properties of nanoscale conductor-microwave cavity systems
abstract: Recent experimental progress in coupling nanoscale conductors to superconducting microwave cavities has opened up for transport investigations of the deep quantum limit of light-matter interactions, with tunneling electrons strongly coupled to individual cavity photons. We have investigated theoretically the most basic cavity-conductor system with strong, single photon induced nonlocal transport effects: two spatially separated double quantum dots (DQDs) resonantly coupled to the fundamental cavity mode. The system, described by a generalized Tavis-Cummings model, is investigated within a quantum master equation formalism, allowing us to account for both the electronic transport properties through the DQDs as well as the coherent, nonequilibrium cavity photon state. We find sizable nonlocally induced current and current cross-correlations mediated by individual photons. From a full statistical description of the electron transport we further reveal a dynamical channel blockade in one DQD lifted by photon emission due to tunneling through the other DQD. Moreover, large entanglement between the orbital states of electrons in the two DQDs is found for small DQD-lead temperatures.

title: Noise spectrum of a quantum dot?resonator lasing circuit
abstract: Single-electron tunneling processes through a double quantum dot can induce a lasing state in an electromagnetic resonator which is coupled coherently to the dot system. Here we study the noise properties of the transport current in the lasing regime, i.e. both the zero-frequency shot noise and the noise spectrum. The former shows a remarkable super-Poissonian behavior when the system approaches the lasing transition, but a sub-Poissonian behavior deep in the lasing state. The noise spectrum contains information about the coherent dynamics of the coupled dot–resonator system. It shows pronounced structures at frequencies matching that of the resonator due to the excitation of photons. For strong interdot Coulomb interaction, we observe asymmetries in the auto-correlation noise spectra of the left and right junctions, which we trace back to asymmetries in the incoherent tunneling channels.

title: High-Resolution Valley Spectroscopy of Si Quantum Dots.
abstract: We study an accumulation mode Si/SiGe double quantum dot (DQD) containing a single electron that is dipole coupled to microwave photons in a superconducting cavity. Measurements of the cavity transmission reveal dispersive features due to the DQD valley states in Si. The occupation of the valley states can be increased by raising the temperature or applying a finite source-drain bias across the DQD, resulting in an increased signal. Using the cavity input-output theory and a four-level model of the DQD, it is possible to efficiently extract valley splittings and the inter- and intravalley tunnel couplings.

title: Lasing and transport in a multi-level double quantum dot system coupled to a microwave oscillator
abstract: We study a system of two quantum dots, each with several discrete levels, which are coherently coupled to a microwave oscillator. They are attached to electronic leads and coupled to a phonon bath, both leading to inelastic processes. For a simpler system with a single level in each dot it has been shown that a population inversion can be created by electron tunneling, which in a resonance situation leads to lasing-type properties of the oscillator. In the multi-level system several resonance situations may arise, some of them relying on a sequence of tunneling processes which also involve non-resonant, inelastic transitions. The resulting photon number in the oscillator and the current-voltage characteristic are highly sensitive to these properties and accordingly can serve as a probe for microscopic details.

title: Mesoscopic cavity quantum electrodynamics with quantum dots
abstract: We describe an electrodynamic mechanism for coherent, quantum-mechanical coupling between spatially separated quantum dots on a microchip. The technique is based on capacitive interactions between the electron charge and a superconducting transmission line resonator, and is closely related to atomic cavity quantum electrodynamics. We investigate several potential applications of this technique which have varying degrees of complexity. In particular, we demonstrate that this mechanism allows design and investigation of an on-chip double-dot microscopic maser. Moreover, the interaction may be extended to couple spatially separated electron-spin states while only virtually populating fast-decaying superpositions of charge states. This represents an effective, controllable long-range interaction, which may facilitate implementation of quantum information processing with electron-spin qubits and potentially allow coupling to other quantum systems such as atomic or superconducting qubits.

title: Lasing and transport in a quantum-dot resonator circuit
abstract: We study a double quantum-dot system coherently coupled to an electromagnetic resonator. A current through the dot system can create a population inversion in the dot levels and, within a narrow resonance window, a lasing state in the resonator. The lasing state correlates with the transport properties. On one hand, this allows probing the lasing state via a current measurement. On the other hand, the resulting narrow current peak allows resolving small differences in the dot properties (e.g., a small difference in the Zeeman splittings of the two dots). For realistic situations relaxation processes have pronounced consequences. Remarkably, they may even enhance the resolution between different spin states by releasing a trapped population in the off-resonant spin channel.

title: Microwave Detection of Electron-Phonon Interactions in a Cavity-Coupled Double Quantum Dot.
abstract: Quantum confinement leads to the formation of discrete electronic states in quantum dots. Here we probe electron-phonon interactions in a suspended InAs nanowire double quantum dot (DQD) that is electric-dipole coupled to a microwave cavity. We apply a finite bias across the wire to drive a steady state population in the DQD excited state, enabling a direct measurement of the electron-phonon coupling strength at the DQD transition energy. The amplitude and phase response of the cavity field exhibit oscillations that are periodic in the DQD energy level detuning due to the phonon modes of the nanowire. The observed cavity phase shift is consistent with theory that predicts a renormalization of the cavity center frequency by coupling to phonons.

title: Probing confined phonon modes by transport through a nanowire double quantum dot.
abstract: Strong radial confinement in semiconductor nanowires leads to modified electronic and phononic energy spectra. We analyze the current response to the interplay between quantum confinement effects of the electron and phonon systems in a gate-defined double quantum dot in a semiconductor nanowire. We show that current spectroscopy of inelastic transitions between the two quantum dots can be used as an experimental probe of the confined phonon environment. The resulting discrete peak structure in the measurements is explained by theoretical modeling of the confined phonon mode spectrum, where the piezoelectric coupling is of crucial importance.