Table of contents

Controlling the dynamics or measurement of quantum systems via the manipulation of external parameters is a most important phenomenon that lies at the heart of several fields including atomic and optical physics, molecular chemistry and quantum information. As quantum technologies have matured, a host of practical applications of quantum control have been realized in quantum optics, cavity QED, atomic spin ensembles, ion trapping, and Bose--Einstein condensation. As a result, quantum control theory is a rapidly growing research field.

The aim of this special issue is to give an idea of the present status of research in quantum control, and to stimulate further activity. The papers chosen cover a great variety of ideas in this field. To aid the reader, we have divided the papers into four broad sections: quantum filtering and feedback control; open-loop control; quantum information applications; optical and related applications. Of course there are many papers that cross the boundaries between the categories we have identified, so we encourage readers to peruse the whole issue. While some may quibble with our classification scheme, we think it will be useful, especially to those who are new to the area. In each section the papers are ordered by date of submission.

The first section is on quantum filtering and feedback control. Quantum filtering means determining estimates for some (or all) observables of the system from classical measurement results obtained gradually over time from the output of the quantum system. The conditioned quantum state is one way to generate such estimates. This filtering of the measurement results is useful for feedback control (also known as closed-loop control), because those estimates can be used as the basis for varying the external control parameters. This section begins with a review article (the one exception to the ordering of papers by submission date).

The second section is on open-loop control in the broad sense. This is concerned with how to drive a quantum system from an initial given state to a pre-determined target state. This includes the question of controllability: whether a controller can drive a quantum system to a desired state, for which the main tools are group theory and graph theory. Quantum optimal control is concerned with finding the best control fields (according to some cost function) to achieve the desired target for a controllable system. Coherent control is a particularly powerful method for guiding the state of a quantum system (typically a molecule) towards the target by applying semiclassical potentials and exploiting quantum interference effects. Another technique is learning control (which can be considered a kind of closed-loop control) in which the experiment is run many times, and each time the output from the sample is used to modify the control fields for the next (independently prepared) sample.

The third and fourth sections are devoted more to applications. The third section comprises applications of ideas and techniques from quantum control (primarily coherent control) to quantum information. These include new schemes for and novel analyses of quantum logic operations, quantum error correction, quantum communication, and quantum algorithms. Finally, the fourth section contains papers on optical or atom-optical (i.e. matter wave) implementations of quantum control ideas, and related applications such as tomography.

In closing, we take the opportunity to express our gratitude to the authors of the papers and to the reviewers, for their respective efforts in preparing and ensuring the high quality of the work and its presentation.

The goal of this article is to provide a largely self-contained introduction to the modelling of controlled quantum systems under continuous observation, and to the design of feedback controls that prepare particular quantum states. We describe a bottom-up approach, where a field-theoretic model is subjected to statistical inference and is ultimately controlled. As an example, the formalism is applied to a highly idealized interaction of an atomic ensemble with an optical field. Our aim is to provide a unified outline for the modelling, from first principles, of realistic experiments in quantum control.

In this paper we formulate a risk-sensitive optimal control problem for continuously monitored open quantum systems modelled by quantum Langevin equations. The optimal controller is expressed in terms of a modified conditional state, which we call a risk-sensitive state, that represents measurement knowledge tempered by the control purpose. One of the two components of the optimal controller is dynamic, a filter that computes the risk-sensitive state. The second component is an optimal control feedback function that is found by solving the dynamic programming equation. The optimal controller can be implemented using classical electronics. The ideas are illustrated using an example of feedback control of a two-level atom.

Since quantum feedback is based on classically accessible measurement results, it can provide fundamental insights into the dynamics of quantum systems by making available classical information on the evolution of system properties and on the conditional forces acting on the system. In this paper, the feedback induced interaction dynamics between a pair of quantum systems is analysed. It is pointed out that any interaction Hamiltonian can be simulated by local feedback if the levels of decoherence are sufficiently high. The boundary between genuine entanglement generating quantum interactions and non-entangling classical interactions is identified and the nature of the information exchange between two quantum systems during an interaction is discussed.

Recent realizations of single-atom trapping and tracking in cavity QED open the door for feedback schemes which actively stabilize the motion of a single atom in real time. We present feedback algorithms for cooling the radial component of motion for a single atom trapped by strong coupling to single-photon fields in an optical cavity. Performance of various algorithms is studied through simulations of single-atom trajectories, with full dynamical and measurement noise included. Closed loop feedback algorithms compare favourably to open loop 'switching' analogues, demonstrating the importance of applying actual position information in real time. The high optical information rate in current experiments enables real-time tracking that approaches the standard quantum limit for broadband position measurements, suggesting that realistic active feedback schemes may reach a regime where measurement backaction appreciably alters the motional dynamics.

Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case). One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a strongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.

We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton–Jacobi–Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton–Pontryagin set-up. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback. A controlled qubit with a feedback is considered as example.

Photon counting induces an effective non-linear optical phase shift in certain states derived by linear optics from single photons. Although this non-linearity is non-deterministic, it is sufficient in principle to allow scalable linear optics quantum computation (LOQC). The most obvious way to encode a qubit optically is as a superposition of the vacuum and a single photon in one mode—so-called 'single-rail' logic. Until now this approach was thought to be prohibitively expensive (in resources) compared to 'dual-rail' logic where a qubit is stored by a photon across two modes. Here we attack this problem with real-time feedback control, which can realize a quantum-limited phase measurement on a single mode, as has been recently demonstrated experimentally. We show that with this added measurement resource, the resource requirements for single-rail LOQC are not substantially different from those of dual-rail LOQC. In particular, with adaptive phase measurements an arbitrary qubit state can be prepared deterministically.

In a partially observed quantum or classical system the information that we cannot access results in our description of the system becoming mixed, even if we have perfect initial knowledge. That is, if the system is quantum the conditional state will be given by a state matrix ρ_r(t), and if classical, the conditional state will be given by a probability distribution P_r(x,t), where r is the result of the measurement. Thus to determine the evolution of this conditional state, under continuous-in-time monitoring, requires a numerically expensive calculation. In this paper we demonstrate a numerical technique based on linear measurement theory that allows us to determine the conditional state using only pure states. That is, our technique reduces the problem size by a factor of N, the number of basis states for the system. Furthermore we show that our method can be applied to joint classical and quantum systems such as arise in modelling realistic (finite bandwidth, noisy) measurement.

Control of multiphoton transitions is demonstrated for a multilevel system by generalizing the instantaneous phase of any chirped pulse as individual terms of a Taylor series expansion. In the case of a simple two-level system, all odd terms in the series lead to population inversion while the even terms lead to self-induced transparency. The results hold for multiphoton transitions that do not have any lower-order photon resonance or any intermediate virtual state dynamics within the laser pulse width.

We study the dynamics of potassium atoms in intense laser fields using femtosecond phase-locked pulse pairs in order to extract physical mechanisms of strong field quantum control. The structure of the Autler–Townes (AT) doublet in the photoelectron spectra is measured to analyse transient processes. The analysis shows that the physical mechanism is based on the selective population of dressed states (SPODS). Experimental results of closed loop optimization of SPODS are presented in addition. Applications to decoherence measurements with implications for quantum information are also proposed.

As is well known, when an SU(2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including (with ) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quantum interfaces. By employing nonunitary control operations, we extend the idea of controllability to a set of points which are not necessarily connected by unitary transformations. The second application shows how the orbit of the coherence vector can be used to assess the entangling power of Heisenberg exchange interaction.

A unified theory is presented of dynamically modified decay and decoherence in driven multilevel quantum systems that are weakly coupled to arbitrary zero-temperature reservoirs. Examples of different phase and amplitude modulations are given for two-level systems (qubits). Analysis of modulations on multilevel systems is detailed with a numerical example using quasiperiodic impulsive phase jumps. The merits and disadvantages of the different modulation types are discussed.

We consider the degrees of controllability of multi-partite quantum systems, as well as necessary and sufficient criteria for each case. The results are applied to the problem of simultaneous control of an ensemble of quantum dots with a single laser pulse. Finally, we apply optimal control techniques to demonstrate selective excitation of individual dots for a simultaneously controllable ensemble of quantum dots.

Within the framework of optimal control theory we develop a simple iterative scheme to determine optimal laser pulses with spectral and fluence constraints. The algorithm is applied to a one-dimensional asymmetric double well where the control target is to transfer a particle from the ground state, located in the left well, to the first excited state, located in the right well. Extremely high occupations of the first excited state are obtained for a variety of spectral and/or energetic constraints. Even for the extreme case where no resonance frequency is allowed in the pulse the algorithm achieves an occupation of almost 100%.

A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllable nor completely controllable. Moreover, a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.

We present a scheme of control for the arbitrary interplay between a stationary qubit and a flying qubit (carried by a single-photon wavepacket) at a quantum interface composed of a three-level system coupled to a continuum through a cavity. It can be used for generation or reception of an arbitrarily shaped single-photon wavepacket. The generation process can also be controlled to create entanglement between the stationary qubit and flying qubit. The generation and reception operation can be combined to perform quantum network operations such as transfer, swap and entanglement creation for qubits at distant nodes.

We explore the connections between the constraints on the precision of quantum logical operations that arise from a conservation law, and those arising from quantum field fluctuations. We show that the conservation-law-based constraints apply in a number of situations of experimental interest, such as Raman excitations, and atoms in free space interacting with the multimode vacuum. We also show that, for these systems, and for states with a sufficiently large photon number, the conservation-law-based constraint represents an ultimate limit closely related to the fluctuations in the quantum field phase.

We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, unlike in the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows one to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.

We introduce a general scheme to realize massive quantum memories in simple systems of interacting qubits. Such systems are described by spin rings with XXZ intersite couplings of suitably time-periodically controlled amplitudes. We show that initially localized excitations undergo perfect periodic revivals, allowing for the simultaneous storage of arbitrary sets of different local states. This novel approach to the problem of storing quantum information hints at a new way to control and suppress the effect of decoherence on a quantum computer realized in a system with nonvanishing interactions between the constituent qubits.

We propose a scheme for quantum logic with neutral atoms stored in an array of holographic dipole traps where the positions of the atoms can be rearranged by using holographic optical tweezers. In particular, this allows for the transport of two atoms to the same well where an external control field is used to perform gate operations via the molecular interaction between the atoms. We show that optimal control techniques allow for the fast implementation of the gates with high fidelity.

We describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation. The method is based on the motion reversal of random unitary operators. In the simplest case our method enables direct estimation of the average gate fidelity. The more general fidelities are characterized by a universal exponential rate of fidelity loss. In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation, quantified by the trace of the superoperator describing the non-unitary dynamics. While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible), our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device. By varying over different experimental arrangements and error-correction strategies, additional information about the noise can be determined.

We show that the multidimensional Zeno effect combined with non-holonomic control allows one to efficiently protect quantum systems from decoherence by a method similar to classical random coding. The method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems. The quantum encoding approaches the Hamming upper bound for large dimension increases. Applicability of the method is demonstrated with a seven-qubit toy computer.

We shall consider a quantum communication channel constituted by two open spin chains with nearest neighbour interactions and fixed coupling strengths, connected to several users. Assuming that the users of this quantum data bus can apply simple gates on sites of the chain that they are connected to, we will see that it is possible to gain information on the position of the qubit during the transmission, and to use this information to accelerate the communication process. We will also see that performing frequent measurements on the system affects the dynamics of propagation of quantum information in an interesting way.

Many-particle confinement (localization) is studied for a 1D system of spinless fermions with nearest-neighbour hopping and interaction, or equivalently, for an anisotropic Heisenberg spin-1/2 chain. This system is frequently used to model quantum computers with perpetually coupled qubits. We construct a bounded sequence of site energies that leads to strong single-particle confinement of all states on individual sites. We show that this sequence also leads to a confinement of all many-particle states in an infinite system for a time that scales as a high power of the reciprocal hopping integral. The confinement is achieved for strong interaction between the particles while keeping the overall bandwidth of site energies comparatively small. The results show the viability of quantum computing with time-independent qubit coupling.

There are several known schemes for entangling trapped ion quantum bits for large-scale quantum computation. Most are based on an interaction between the ions and external optical fields, coupling internal qubit states of trapped ions to their Coulomb-coupled motion. In this paper, we examine the sensitivity of these motional gate schemes to phase fluctuations introduced through noisy external control fields, and suggest techniques for suppressing the resulting phase decoherence.

Shared entanglement allows, under certain conditions, the remote implementation of quantum operations. We revise and extend recent theoretical results on the remote control of quantum systems as well as experimental results on the remote manipulation of photonic qubits via linear optical elements.

We address nonlocality of bipartite continuous variable systems in the presence of dissipation and noise. Three nonlocality tests have been considered, based on the measurement of displaced parity, field quadrature and pseudospin operator, respectively. Nonlocality of twin-beams has been investigated, as well as that of their non-Gaussian counterparts obtained by inconclusive subtraction of photons. Our results indicate that (i) nonlocality of twin-beams is degraded but not destroyed by noise; (ii) photon subtraction enhances nonlocality in the presence of noise, especially in the low-energy regime.

The short-time dynamics of a quantum Brownian particle in a harmonic potential is studied in phase space. An exact non-Markovian analytic approach to calculate the time evolution of the Wigner function is presented. The dynamics of the Wigner function of an initially squeezed state is analysed. It is shown that virtual exchanges of energy between the particle and the reservoir, characterizing the non-Lindblad short-time dynamics where system–reservoir correlations are not negligible, show up in phase space.

Multiphoton detachment rates for the H^− 1S ground state irradiated by a weak trichromatic ac field consisting of the fundamental frequency ω = 0.272 eV and its second, third or fourth higher harmonics were computed from first principles. The weak intensities are in the range of 10⁷–10⁸ W cm⁻². The calculations incorporated systematically electronic structure and electron correlation effects. They were done by implementing a time-independent, nonperturbative many-electron, many-photon theory (MEMPT) which obtains cycle-averaged complex eigenvalues, whose real part gives the field-induced energy shift, Δ, and the imaginary part is the multiphoton ionization rate, Γ. Through analysis, plausible arguments and computation, we show that when the intensities are weak the dependence of Γ on phase differences is simple. Specifically, Γs are depicted in the form of plane surfaces, with minor ripples due to higher order ionization paths, in terms of trigonometric functions of the phase differences. This dependence is likely to be applicable to other atomic systems as well, and to provide a definition of the weak field regime in the trichromatic case. When the field intensities are such that higher order ionization paths become important, these dependences break down and we reach the strong field regime.

A bipartite quantum state is tomographically faithful when it can be used as an input of a quantum operation acting on one of the two quantum systems, such that the joint output state carries complete information about the operation itself. Tomographically faithful states are a necessary ingredient for the tomography of quantum operations and for complete quantum calibration of measuring apparatuses. In this paper we provide a complete classification of such states for continuous variables in terms of the Wigner function of the state. For two-mode Gaussian states faithfulness simply resorts to correlation between the modes.

There exists a large body of experimental and theoretical work carried out on Bose–Einstein condensates (BECs) of dilute alkali atoms since their first creation some ten years ago. Much of this work, although not stated explicitly, involves exciting initial steps for the quantum control of coherent matter waves. In this paper, we shall discuss some examples of the experimental and theoretical work on BECs from the perspective of quantum control.

Quantum optical states which have no coherent amplitude, such as squeezed vacuum states, cannot rely on standard readout techniques to generate error signals for control of the quadrature phase. Here we investigate the use of asymmetry in the quadrature variances to obtain a phase-sensitive readout and to lock the phase of a squeezed vacuum state, a technique which we call noise locking (NL). We carry out a theoretical derivation of the NL error signal and the associated stability of the squeezed and anti-squeezed lock points. Experimental data for the NL technique both in the presence and absence of coherent fields are shown, including a comparison with coherent locking techniques. Finally, we use NL to enable a stable readout of the squeezed vacuum state on a homodyne detector.

Two non-interacting two-level atoms immersed in a common bath can become mutually entangled when evolving with a Markovian, completely positive dynamics. For an environment made of external quantum fields, this phenomenon can be studied in detail: one finds that entanglement production can be controlled by varying the bath temperature and the distance between the atoms. Remarkably, in certain circumstances, the quantum correlations can persist in the asymptotic long-time regime.