Input-output Formalism Research Articles

Fisher informations and local asymptotic normality for continuous-time quantum Markov processes

We consider the problem of estimating an arbitrary dynamical parameter of an open quantum system in the input–output formalism. For irreducible Markov processes, we show that in the limit of large times the system-output state can be approximated by a quantum Gaussian state whose mean is proportional to the unknown parameter. This approximation holds locally in a neighbourhood of size in the parameter space, and provides an explicit expression of the asymptotic quantum Fisher information in terms of the Markov generator. Furthermore we show that additive statistics of the counting and homodyne measurements also satisfy local asymptotic normality and we compute the corresponding classical Fisher informations. The general theory is illustrated with the examples of a two-level system and the atom maser. Our results contribute towards a better understanding of the statistical and probabilistic properties of the output process, with relevance for quantum control engineering, and the theory of non-equilibrium quantum open systems.

Quantum noise of a white-light cavity using a double-pumped gain medium

Laser interferometric gravitational-wave detectors implement Fabry-Perot cavities to increase their peak sensitivity. However, this is at cost of reducing their detection bandwidth, which origins from the propagation phase delay of the light. The "white-light-cavity" idea, first proposed by Wicht et al. [Optics Communications 134, 431 (1997)], is to circumvent this limitation by introducing anomalous dispersion, using double-pumped gain medium, to compensate for such phase delay. In this article, starting from the Hamiltonian of atom-light interaction, we apply the input-output formalism to evaluate the quantum noise of the system. We find that apart from the additional noise associated with the parametric amplification process noticed by others, the stability condition for the entire system poses an additional constraint. Through surveying the parameter regimes where the gain medium remains stable (not lasing) and stationary, we find that there is no net enhancement of the shot-noise limited sensitivity. Therefore, other gain mediums or different parameter regimes shall be explored for realizing the white light cavity.

Input-output formalism for few-photon transport: A systematic treatment beyond two photons

We provide a systematic treatment of $N$-photon transport in a waveguide coupled to a local system, using the input-output formalism. The main result of the paper is a general connection between the $N$-photon S matrix and the Green functions of the local system. We also show that the computation can be significantly simplified, by exploiting the connectedness structure of both the S matrix and the Green function, and by computing the Green function using an effective Hamiltonian that involves only the degrees of freedom of the local system. We illustrate our formalism by computing $N$-photon transport through a cavity containing a medium with Kerr nonlinearity, with $N$ up to 3.

Single-photon frequency conversion and multi-mode entanglement via constructive interference on Sagnac loop

Based on constructive interference in Sagnac waveguide loop, an efficient scheme is proposed for selective frequency conversion and multifrequency modes W entanglement via input–output formalism. We can adjust the probability amplitudes of output photons by choosing parameter values properly. The tunable probability amplitude will lead to the generation of output photon with a selectable frequency and W photonic entanglement of different frequencies modes in a wide range of parameter values. Our calculations show the present scheme is robust to the deviation of parameters and spontaneous decay.

Nonclassical light from few emitters in a cavity

We study the characteristics of the light generated by few emitters in a cavity at strong light-matter coupling. By means of the Glauber $g^{(2)}$-function we can identify clearly distinguished parameter regimes with super-Poissonian and sub-Poissonian photon statistics. We establish a relation between the emission characteristics for one and multiple emitters, and explain its origin in terms of the photon-dressed emitter states. Cooperative effects lead to the generation of nonclassical light already at reduced light-matter coupling if the number of emitters is increased. Our results are obtained with a full input-output formalism and master equation valid also at strong light-matter coupling. We compare the behavior obtained with and without counter-rotating light-matter interaction terms in the Hamiltonian, and find that the generation of nonclassical light is robust against such modifications. Finally, we contrast our findings with the predictions of the quantum optical master equation and find that it fails entirely at predicting regimes with different photon statistics.

Semiclassical Maxwell-Bloch equations in superradiance theory by input-output formalism of quantum stochastic differential equations

Collective spontaneous relaxation of identical atoms without any quantum correlation has been shown to give a semiclassical description of superradiance. The Maxwell- Bloch equation follows from the N atomic master equation which was derived within the framework of quantum stochastic differential equations.

Stationary discrete solitons in a driven dissipative Bose-Hubbard chain

We demonstrate that stationary localized solutions (discrete solitons) exist in a one dimensional Bose-Hubbard lattices with gain and loss in the semiclassical regime. Stationary solutions, by defi- nition, are robust and do not demand for state preparation. Losses, unavoidable in experiments, are not a drawback, but a necessary ingredient for these modes to exist. The semiclassical calculations are complemented with their classical limit and dynamics based on a Gutzwiller Ansatz. We argue that circuit QED architectures are ideal platforms for realizing the physics developed here. Finally, within the input-output formalism, we explain how to experimentally access the different phases, including the solitons, of the chain.

Scattering processes in quantum optics

The derivation of a quantum Markovian model for an optomechanical system consisting of a quantum mechanical mirror interacting with quantum optical input fields via radiation pressure is a difficult problem which ultimately involves the scattering process of quantum stochastic calculus. We show that, while the scattering process may be approximated in a singular limit by regular processes using different schemes, the limit model is highly sensitive to the mathematical interpretation of the approximation scheme. We find two main types of stochastic limits of regular models, and illustrate the origin of this difference at the level of one-particle scattering. As an alternative modeling scheme, we consider models of mirrors as nontrivial dielectric media with boundaries that are themselves quantized. Rather than treating the plane waves for the electromagnetic field, we take the actual physical modes and quantize these. The input-output formalism is then obtained in the far zone where the plane-wave approximation is valid. Several examples are considered, and the quantum stochastic model is derived. We also consider the quantum trajectories problem for continual measurement of the reflected output fields, and derive the stochastic master equations for homodyning and photon-counting detection to estimate the mirror observables.

Quantum field tomography

We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of unknown quantum fields based on data of correlation functions. At the basis of the analysis is the concept of continuous matrix product states (cMPS), a complete set of variational states grasping states in one-dimensional quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used in the context of compressed sensing such as Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation functions. In the absence of a phase reference, we highlight how specific higher order correlation functions can still be predicted. We exemplify the functioning of the approach by reconstructing randomized cMPS from their correlation data and study the robustness of the reconstruction for different noise models. Furthermore, we apply the method to data generated by simulations based on cMPS and using the time-dependent variational principle. The presented approach is expected to open up a new window into experimentally studying continuous quantum systems, such as those encountered in experiments with ultra-cold atoms on top of atom chips. By virtue of the analogy with the input–output formalism in quantum optics, it also allows for studying open quantum systems.

Ultra-wide single-photon frequency conversion and entanglement via constructive interference

We investigate theoretically a single-photon frequency conversion with a Sagnac waveguide loop by input-output formalism. The results show that an ultra-wide tunable frequency conversion can be achieved via constructive interference and a flexible adjustment of parameters. For instance, we discuss the generation of ultraviolet photons in detail, which corresponds to the up-conversion process of frequency. We also can obtain entangled states of photons and atoms under suitable parameter conditions. Our calculations prove that a single-photon frequency conversion with high efficiency can be realized, even if effected by spontaneous decay and deviation of parameters.

Dephasing in single-electron generation due to environmental noise probed by Hong-Ou-Mandel interferometry

We consider the effect of dephasing on a quantum dot which injects single electrons on a chiral edge channel of the quantum Hall effect. Dephasing is described by the coupling of the dot to a bosonic bath which represents the electromagnetic environment. Using the input-output formalism of quantum optics, we derive the density matrix of the edge degrees of freedom. Results are illustrated by computing the zero frequency current-current correlations when two such single electron emitters achieve a collision at the location of a quantum point contact, in the same spirit as the Hong Ou Mandel experiment of quantum optics. Such correlations are directly linked to the quantum mechanical purity. We show that as observed in a recent experiment, the effect of dephasing leads to a non-vanishing of the Hong Ou Mandel dip when the time delay between the two electron wave packets is zero. Generalizations to time filtered wave packets as well as to asymmetric, detuned injection between opposite edges are obtained.

Single-photon frequency conversion for generation of entanglement via constructive interference in Sagnac interferometers

A single-photon frequency conversion process with a Sagnac interferometer (Bradford et al 2012 Phys. Rev. Lett. 108 103902 and Bradford and Shen 2012 Phys. Rev. A 85 043814) is described by the input–output formalism. The result shows that a perfect frequency conversion can be obtained in an appropriate parameter regime via constructive interference. Based on the frequency-conversion process, an alternative scheme is provided for the generation of entanglement between distant particles including two-body entanglement and multi-body Greenberger–Horne–Zeilinger entanglement. We study the effect of the parameters’ deviation and spontaneous decay of the excited state and waveguide mode on the transmission amplitudes, and also on the frequency-conversion efficiency. In addition, the influence of the experimental parameters on the fidelity of the entangled state is discussed.

Input–output description of microwave radiation in the dynamical Coulomb blockade

We study microwave radiation emitted by a small voltage-biased Josephson junction connected to a superconducting transmission line. An input–output formalism for the radiation field is established, using a perturbation expansion in the junction's critical current. Using output field operators solved up to the second order, we estimate the spectral density and the second-order coherence of the emitted field. For typical transmission line impedances and at frequencies below the main emission peak at the Josephson frequency, radiation occurs predominantly due to two-photon emission. This emission is characterized by a high degree of photon bunching if detected symmetrically around half of the Josephson frequency. Strong phase fluctuations in the transmission line make related nonclassical phase-dependent amplitude correlations short lived, and there is no steady-state two-mode squeezing. However, the radiation is shown to violate the classical Cauchy–Schwarz inequality of intensity cross-correlations, demonstrating the nonclassicality of the photon pair production in this region.

Single-photon time-dependent spectra in coupled cavity arrays

We show how the input-output formalism for cascaded quantum systems combined with the quantum trajectory approach yields a compact and physically intuitive description of single photons propagating through a coupled cavity array. As a new application we obtain the time-dependent spectrum of such a single photon, which directly reflects the fact that only certain frequency components of single-photon wavepackets are trapped inside the cavities and hence are delayed in time. We include in our description the actual generation of the single photon, by assuming we have a single emitter in one of the resonators.

Outgoing Cuntz scattering system for a coisometric lifting and transfer function

We study a coisometry that intertwines Popescu’s presentations of minimal isometric dilations of a given operator tuple and of a coisometric lifting of the tuple. Using this we develop an outgoing Cuntz scattering system which gives rise to an input–output formalism. A transfer function is introduced for the system. We also compare the transfer function and the characteristic function for the associated lifting.

Nonclassical Photon Pair Production in a Voltage-Biased Josephson Junction

We investigate electromagnetic radiation emitted by a small voltage-biased Josephson junction connected to a superconducting transmission line. At frequencies below the well-known emission peak at the Josephson frequency (2eV/h), extra radiation is triggered by quantum fluctuations in the transmission line. For weak tunneling couplings and typical Ohmic transmission lines, the corresponding photon-flux spectrum is symmetric around half the Josephson frequency, indicating that the photons are predominately created in pairs. By establishing an input-output formalism for the microwave field in the transmission line, we give further evidence for this nonclassical photon pair production, demonstrating that it violates the classical Cauchy-Schwarz inequality for two-mode flux cross correlations. In connection to recent experiments, we also consider a stepped transmission line, where resonances increase the signal-to-noise ratio.

Non-Markovian quantum input-output networks

Quantum input-output response analysis is a useful method for modeling the dynamics of complex quantum networks, such as those for communication or quantum control via cascade connections. Non-Markovian effects have not yet been studied in such networks. Here we extend the Markovian input-output network formalism developed in optical systems to non-Markovian cascaded networks which can be used, e.g., to analyze the input-output response of mesoscopic quantum networks. We use this formalism to explore the behavior of superconducting qubit networks, where we examine the effect of finite cavity bandwidths. We also discuss its application to open- and closed-loop control networks, and show how these networks create effective Hamiltonians for the controlled system.

Dissipation and detection of polaritons in the ultrastrong-coupling regime

We have investigated theoretically a dissipative polariton system in the ultrastrong light-matter coupling regime without using the rotating-wave approximation on system-reservoir coupling. Photons in a cavity and excitations in matter respectively couple two large ensembles of harmonic oscillators (photonic and excitonic reservoirs). Inheriting the quantum statistics of polaritons in the ultrastrong coupling regime, in the ground state of the whole system, the two reservoirs are not in the vacuum states but they are squeezed and correlated. We presume the reservoirs to be in this nonvacuum state in the master equation and in the input-output formalism with Langevin equations. Both approaches consistently guarantee the decay of the polariton system to its ground state; no photon detection is also obtained when the polariton system is in the ground state.

Few-Photon Single-Atom Cavity QED With Input-Output Formalism in Fock Space

The Jaynes-Cummings (JC) system, which describes the interaction between a cavity and a two-level atom, is one of the most important systems in quantum optics. We obtain analytic solutions for the one- and two-photon transport in a waveguide side-coupled to the JC system using input-output formalism in Fock space. With these results, we discuss the conditions under which the JC system functions as a photon switch for waveguide photons in both the strong and weak coupling regimes

Input–output Gaussian channels: theory and application

Setting off from the classic input–output formalism, we develop a theoretical framework to characterize the Gaussian quantum channels relating the initial correlations of an open bosonic system to those of properly identified output modes. We then proceed to apply our formalism to the case of quantum harmonic oscillators, such as the motional degrees of freedom of trapped ions or nanomechanical oscillators, interacting with travelling electromagnetic modes through cavity fields and subject to external white noise. We thus determine the degree of squeezing that can be transferred from an intra-cavity oscillator to light and show that the intra-cavity squeezing can be transformed into distributed optical entanglement if one can access both output fields of a two-sided cavity.