Small Quantum Systems Research Articles

Experimental Repetitive Quantum Error Correction

The computational potential of a quantum processor can only be unleashed if errors during a quantum computation can be controlled and corrected for. Quantum error correction works if imperfections of quantum gate operations and measurements are below a certain threshold and corrections can be applied repeatedly. We implement multiple quantum error correction cycles for phase-flip errors on qubits encoded with trapped ions. Errors are corrected by a quantum-feedback algorithm using high-fidelity gate operations and a reset technique for the auxiliary qubits. Up to three consecutive correction cycles are realized, and the behavior of the algorithm for different noise environments is analyzed.

‘Return to Equilibrium’ for Weakly Coupled Quantum Systems: A Simple Polymer Expansion

Recently, several authors studied small quantum systems weakly coupled to free boson or fermion fields at positive temperature. All the approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. We show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation \emph{aka} Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation.

Classical Boolean logic gates with quantum systems

An analytical method is proposed to implement any classical Boolean function in a small quantum system by taking the advantage of its electronic transport properties. The logical input, \U0001d6c2 = {α1, …, αN}, is used to control well-identified parameters of the Hamiltonian of the system noted . The logical output is encoded in the tunneling current intensity passing through the quantum system when connected to conducting electrodes. It is demonstrated how to implement the six symmetric two-input/one-output Boolean functions in a quantum system. This system can be switched from one logic function to another by changing its structural parameters. The stability of the logic gates is discussed, perturbing the Hamiltonian with noise sources and studying the effect of decoherence.

Linear Circuit Models for On-Chip Quantum Electrodynamics

We present equivalent circuits that model the interaction of microwave resonators and quantum systems. The circuit models are derived from a general interaction Hamiltonian. Quantitative agreement between the simulated resonator transmission frequency, qubit Lamb shift and experimental data will be shown. We demonstrate that simple circuit models, using only linear passive elements, can be very useful in understanding systems where a small quantum system is coupled to a classical microwave apparatus.

Information encoding of a qubit into a multilevel environment

I consider the interaction of a small quantum system (a qubit) with a structured environment consisting of many levels. The qubit will experience a decoherence process, which implies that part of its initial information will be encoded into correlations between system and environment. I investigate how this information is distributed on a given subset of levels as a function of its size, using the mutual information between both entities, in the spirit of the partial-information plots studied by Zurek and co-workers. In this case we can observe some differences, which arise from the fact that I am partitioning just one quantum system and not a collection of them. However, some similar features, like redundancy (in the sense that a given amount of information is shared by many subsets), which increases with the size of the environment, are also found here.

Work exchange between quantum systems: The spin-oscillator model

With the development of quantum thermodynamics it has been shown that relaxation to thermal equilibrium and with it the concept of heat flux may emerge directly from quantum mechanics. This happens for a large class of quantum systems if embedded into another quantum environment. In this paper, we discuss the complementary question of the emergence of work flux from quantum mechanics. We introduce and discuss two different methods to assess the work source quality of a system, one based on the generalized factorization approximation, the other based on generalized definitions of work and heat. By means of those methods, we show that small quantum systems can, indeed, act as work reservoirs. We illustrate this behavior for a simple system consisting of a spin coupled to an oscillator and investigate the effects of two different interactions on the work source quality. One case will be shown to allow for a work source functionality of arbitrarily high quality.

Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates

This work describes the statistics for the occupation numbers of quantum levels in a large isolated quantum system, where all possible superpositions of eigenstates are allowed provided all these superpositions have the same fixed energy. Such a condition is not equivalent to the conventional microcanonical condition because the latter limits the participating eigenstates to a very narrow energy window. The statistics is obtained analytically for both the entire system and its small subsystem. In a significant departure from the Boltzmann-Gibbs statistics, the average occupation numbers of quantum states exhibit in the present case weak algebraic dependence on energy. In the macroscopic limit, this dependence is routinely accompanied by the condensation into the lowest-energy quantum state. This work contains initial numerical tests of the above statistics for finite systems and also reports the following numerical finding: when the basis states of large but finite random matrix Hamiltonians are expanded in terms of eigenstates, the participation of eigenstates in such an expansion obeys the newly obtained statistics. The above statistics might be observable in small quantum systems, but for the macroscopic systems, it rather re-enforces doubts about self-sufficiency of nonrelativistic quantum mechanics for justifying the Boltzmann-Gibbs equilibrium.

Non-Markovian quantum repeated interactions and measurements

A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.

Dynamics of mesoscopic systems: Non-equilibrium Green's functions approach

Recent developments in technologies and experiments enable us to observe dynamical behavior of very small quantum systems under conditions when it is sensitive to system parameters, internal interactions, the environment and the external time-dependent fields. This contribution deals with the status and perspectives of description of such truly non-equilibrium quantum many body systems using the non-equilibrium Green's functions (NGF). The basic aim of this approach is to describe time development of the many-body system out of equilibrium from its initial state over its transient dynamics to its long time asymptotic evolution. The early stages of the transient evolution will be characterized for a broad class of the initial conditions generated in the course of Keldysh switch-on states. Gradual loss of initial correlations due to interactions and related renormalization processes in the system may, under favorable conditions, give rise to the evolution stage expressible in terms of non-equilibrium quasi-particles. This permits to reduce the description of the system behavior to a quantum transport equation. The consistency of such approximations for NGF may be judged by checking the non-equilibrium version of transport Ward identities. These identities follow from the gauge invariance of the 1st kind for NGF. To cover the whole time domain of the transient, we obtain the non-equilibrium Ward identities for NGF with initial conditions included.

Real-time renormalization group in frequency space: A two-loop analysis of the nonequilibrium anisotropic Kondo model at finite magnetic field

We apply a recently developed nonequilibrium real-time renormalization group (RG) method in frequency space to describe nonlinear quantum transport through a small fermionic quantum system coupled weakly to several reservoirs via spin and/or orbital fluctuations. Within a weak-coupling two-loop analysis, we derive analytic formulas for the nonlinear conductance and the kernel determining the time evolution of the reduced density matrix. A consistent formalism is presented how the RG flow is cut off by relaxation and dephasing rates. We apply the general formalism to the nonequilibrium anisotropic Kondo model at finite magnetic field. We consider the weak-coupling regime, where the maximum of voltage and bare magnetic field is larger than the Kondo temperature. In this regime, we calculate the nonlinear conductance, the magnetic susceptibility, the renormalized spin relaxation and dephasing rates, and the renormalized $g$ factor. All quantities are considered up to the first logarithmic correction beyond leading order at resonance. Up to a redefinition of the Kondo temperature, we confirm previous results for the conductance and the magnetic susceptibility in the isotropic case. In addition, we present a consistent calculation of the resonant line shapes, including the determination whether the spin relaxation or dephasing rate cuts off the logarithmic divergence. Furthermore, we calculate quantities characterizing the exponential decay of the time evolution of the magnetization. In contrast to the conductance, we find that the derivative of the spin relaxation (dephasing) rate with respect to the magnetic field is logarithmically enhanced (suppressed) for voltages smaller (larger) than the renormalized magnetic field, and that the logarithmic divergence is cut off by the opposite rate. The renormalized $g$ factor is predicted to show a symmetric logarithmic suppression at resonance, which is cut off by the spin relaxation rate. We propose a three-terminal setup to measure the suppression at resonance. For all quantities, we analyze also the anisotropic case and find additional nonequilibrium effects at resonance.

Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom

In this review we summarize and amplify recent investigations of coupled quantum dynamical systems with few degrees of freedom in the short-wavelength, semiclassical limit. Focusing on the correspondence between quantum and classical physics, we mathematically formulate and attempt to answer three fundamental questions. (i) How can one drive a small dynamical quantum system to behave classically? (ii) What determines the rate at which two single-particle quantum-mechanical subsystems become entangled when they interact? (iii) How does irreversibility occur in quantum systems with few degrees of freedom? These three questions are posed in the context of the quantum-classical correspondence for dynamical systems with few degrees of freedom, and we accordingly rely on two short-wavelength approximations to quantum mechanics to answer them: the trajectory-based semiclassical approach on the one hand, and random matrix theory on the other hand. We construct novel investigative procedures towards decoherence an...

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.

A simple, approximate method for analysis of Kerr-Newman black hole dynamics and thermodynamics

In this work we present a simple approximate method for analysis of the basic dynamical and thermodynamical characteristics of Kerr-Newman black hole. Instead of the complete dynamics of the black hole self-interaction, we consider only the stable (stationary) dynamical situations determined by condition that the black hole (outer) horizon 'circumference' holds the integer number of the reduced Compton wave lengths corresponding to mass spectrum of a small quantum system (representing the quantum of the black hole self-interaction). Then, we show that Kerr-Newman black hole entropy represents simply the ratio of the sum of static part and rotation part of the mass of black hole on one hand, and the ground mass of small quantum system on the other hand. Also we show that Kerr-Newman black hole temperature represents the negative value of the classical potential energy of gravitational interaction between a part of black hole with reduced mass and a small quantum system in the ground mass quantum state. Finally, we suggest a bosonic great canonical distribution of the statistical ensemble of given small quantum systems in the thermodynamical equilibrium with (macroscopic) black hole as thermal reservoir. We suggest that, practically, only the ground mass quantum state is significantly degenerate while all the other, excited mass quantum states, are non-degenerate. Kerr-Newman black hole entropy is practically equivalent to the ground mass quantum state degeneration. Given statistical distribution admits a rough (qualitative) but simple modeling of Hawking radiation of the black hole too.

Nonlocal effects in semiconductor nanostructure transport

A small quantum system, such as a quantum dot, can be consideredto be mainly closed, in which case current flow follows by tunneling,or can be mainly open, in which case the conductance is several times2e2/h, the quantum unit of conductance. Even in this case, however, there are trapped stateswithin the quantum system which impact transport via phase space tunneling. When wehave multiple systems, either an array of dots or of quantum point contacts, or of acombination of these, then there is the chance for a more nonlocal interactionto occur, and to affect the transport. Some of these cases are discussed in thispaper.

Magnetic resonance in iron oxide nanoparticles: Quantum features and effect of size

In order to better understand the transition from quantum to classical behavior in spin system, electron magnetic resonance (EMR) is studied in suspensions of superparamagnetic magnetite nanoparticles with an average diameter of ∼9 nm and analyzed in comparison with the results obtained in the maghemite particles of smaller size (∼5 nm). It is shown that both types of particles demonstrate common EMR behavior, including special features such as the temperature-dependent narrow spectral component and multiple-quantum transitions. These features are common for small quantum systems and not expected in classical case. The relative intensity of these signals rapidly decreases with cooling or increase of particle size, marking gradual transition to the classical ferromagnetic resonance (FMR) behavior.

Work and work fluctuations in quantum systems

With the development of quantum thermodynamics [1], it turned out that the existence of a thermal equilibrium can be derived directly from quantum mechanics. This finding has raised the question, what other thermodynamic concepts could be applied to quantum systems and how they might emerge from quantum mechanics. Here, we discuss how the concept of work translates to quantum systems and how its emergence can be understood. Moreover, we show that even for small and simple quantum systems, work may be a meaningful concept. We then address the question of work fluctuations in quantum systems. We discuss the Jarzynski relation and its quantum counterparts and we show that corresponding relations hold even for open quantum systems.

Theory of Non-Equilibrium Stationary States as a Theory of Resonances

We study a small quantum system (e.g., a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states parametrized by two numbers, T 1 and T 2 (‘reservoir temperatures’). If T 1 ≠ T 2, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production and are dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T 1 and T 2, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T 1,T 2) > g 2 + α, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions.

Information Erasure and the Generalized Second Law of Black Hole Thermodynamics

We consider the generalized second law of black hole thermodynamics in the light of quantum information theory, in particular information erasure and Landauer’s principle (namely, that erasure of information produces at least the equivalent amount of entropy). A small quantum system outside a black hole in the Hartle-Hawking state is studied, and the quantum system comes into thermal equilibrium with the radiation surrounding the black hole. For this scenario, we present a simple proof of the generalized second law based on quantum relative entropy. We then analyze the corresponding information erasure process, and confirm our proof of the generalized second law by applying Landauer’s principle.

Qubits in a random environment

Decoherence phenomena in a small quantum system coupled to a complex environment can be modelled with random matrices. We propose a simple deterministic model in the limit of a high dimensional environment. The model is investigated numerically and some analytically addressable questions are singled out.

Return to equilibrium for small quantum systems interacting with environment

Rigorous results about the return to equilibrium for small quantum systems interacting with environment are described.