The natural choice of physical qubit for quantum communication is the photon: a photon can be transmitted quickly from the sender to a distant receiver and the technology for creating, manipulating, distributing, and measuring light pulses is well established. Many of these classical techniques can also be employed for quantum communication protocols. We will therefore focus our attention on optical setups for quantum communication.

We will study a number of optical setups in the next few sections and use the conventions introduced in Chapter 4 to analyze them. In particular, we will assume that the qubits are encoded in the degrees of freedom of a single photon. Some of the main technical challenges in quantum communication arise from this need to work with single photon pulses. For instance, photons do not interact with each other in vacuum or linear media, and interactions between two photons in non-linear optical media are relatively small, so that realizing entangling two photon gates via coherent interactions is a challenging task.

The quantum communication schemes discussed here circumvent this technical problem to a large extent. They only use non-linear optical materials for parametric down-conversion to create Bell-pairs of photons and then exploit standard linear optical devices and photo-detectors to manipulate them. No further non-linear entangling gates are required.

Parametric down-conversion

In non-linear optical media a single photon can be down-converted into a pair of photons. In this coherent process the incoming photon is destroyed and two photons of lower energy are created.

Trapped ions, trapped atoms, and NMR spin systems are all fine ways of building small “toy” quantum computers, each with its own advantages and disadvantages. There are also many other techniques which have been suggested, although these three so far remain in the lead for general-purpose quantum computing. However, the most powerful general-purpose quantum computers constructed to date have only about a dozen qubits, and this is not nearly large enough to make quantum computers useful rather than merely interesting. (Larger systems have been used to demonstrate particular quantum information processing techniques, but these cannot as yet be used to implement arbitrary quantum algorithms.)

Although it is not completely clear how complex a general-purpose quantum computer needs to be, it is clear that such a device will involve thousands or even millions of qubits, rather than the dozens involved today. It is, therefore, important to consider whether there is any hope of scaling up these technologies to useful sizes, and we will consider each of the three approaches in turn, before turning briefly to alternative technologies which have not been discussed so far.

Trapped ions

Trapped ions initially look very promising as a candidate for scaling up, as it is possible to trap thousands of ions while keeping a reasonable distance between them. Early experiments relied on particular tricks which only work with systems of two ions, but this is not true of more recent work, and there is no reason in principle why these large strings of ions could not be controlled.

Deutsch's algorithm and Grover's quantum search are theoretically interesting, but it is not obvious that these two algorithms are actually useful for anything important. We next consider a selection of more advanced algorithms, several of which may have real-life applications. Some of these will be too complicated to explain fully, and their properties will only be sketched briefly.

The Deutsch–Jozsa algorithm

Deutsch's algorithm is simple, but important, as it shows that a quantum algorithm can find a property of an unknown function (its parity) with a smaller number of queries than any possible classical algorithm (one rather than two). For this reason we can say that quantum computing is more efficient than classical computing within the oracle model of function evaluation. (It is widely believed that quantum computing is more efficient than classical computing in general, but this is a surprisingly hard thing to prove.) The simplicity of the algorithm is also an advantage, as it can be implemented on very primitive quantum computers. Beyond this, however, Deutsch's algorithm is also important as the simplest member of a large family of quantum algorithms, including most notably Shor's quantum factoring algorithm.

The second simplest algorithm in the family is the Deutsch–Jozsa algorithm, which solves a very closely related problem. Consider an unknown binary function with n input bits, giving N = 2n possible inputs, and a single output bit.

In recent years, intense laser fields have become available, over a wide frequency range, in the form of short pulses. Such laser fields are strong enough to compete with the Coulomb forces in controlling the dynamics of atomic systems. As a result, atoms in intense laser fields exhibit new properties that have been discovered via the study of multiphoton processes. After some introductory remarks in Section 1.1, we discuss in Section 1.2 how intense laser fields can be obtained by using the “chirped pulse amplification” method. In the remaining sections of this chapter, we give a survey of the new phenomena discovered by studying three important multiphoton processes in atoms: multiphoton ionization, harmonic generation and laser-assisted electron–atom collisions.

Introduction

If radiation fields of sufficient intensity interact with atoms, processes of higher order than the single-photon absorption or emission play a significant role. These higher-order processes, called multiphoton processes, correspond to the net absorption or emission of more than one photon in an atomic transition. It is interesting to note that, in the first paper he published in Annalen der Physik in the year 1905, his “Annus mirabilis,” Einstein [1] not only introduced the concept of “energy quantum of light” – named “photon” by Lewis [2] in 1926 – but also mentioned the possibility of multiphoton processes occurring when the intensity of the radiation is high enough, namely “if the number of energy quanta per unit volume simultaneously being transformed is so large that an energy quantum of emitted light can obtain its energy from several incident energy quanta.”

In this chapter we turn to the formulation of the theory of the interaction of intense laser fields with atoms in the important case where the laser photon energy is much smaller than the ionization potential of the initial atomic state. When the intensity is sufficiently high and the frequency sufficiently low, ionization proceeds as if the laser electric field were quasi-static. In this regime, it is appropriate to make the “strong-field approximation,” or SFA, in which one assumes that an active electron, after having been ionized, interacts only with the laser field and not with its parent core. Using this approximation, Keldysh [1] showed that analytical expressions for the rate of ionization can be obtained when the electric-field amplitude, the laser frequency and the binding energy of the initial state are such that the Keldysh parameter γK defined by Equation (1.8) is much less than unity and the photoelectron does not escape by over-the-barrier ionization (OBI). However, the applicability of the SFA extends beyond this regime and, more importantly, it can be used to investigate high-order ATI and high-order harmonic generation. The SFA also provides a framework in which the physical origin of these processes, embodied in the semi-classical three-step recollision model introduced in Section 1.3, can be understood.

We begin in Section 6.1 by examining the low-frequency limit of the Floquet theory and showing how the total ionization rate of the atom can be obtained using the adiabatic approximation.

In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively.

We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies, which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.

The availability of intense laser fields over a wide frequency range, in the form of short pulses of coherent radiation, has opened a new domain in the study of light–matter interactions. The peak intensities of these laser pulses are so high that the corresponding laser fields can compete with the Coulomb forces in controlling the dynamics of atomic systems. Atoms interacting with such intense laser fields are therefore exposed to extreme conditions, and new phenomena occur which are known as multiphoton processes. These phenomena generate in turn new behaviors of bulk matter in strong laser fields, with wide-ranging applications.

The purpose of this book is to give a self-contained and unified presentation of high- intensity laser–atom physics. It is primarily aimed at physicists studying the interaction of laser light with matter at the microscopic level, although it is hoped that any scientist interested in laser–matter interactions will find it useful.

The book is divided into three parts. The first one contains two chapters, in which the basic concepts are presented. In Chapter 1, we give a general overview of the new phenomena discovered by studying atomic multiphoton processes in intense laser fields. In Chapter 2, the theory of laser–atom interactions is expounded, using a semi-classical approach in which the laser field is treated classically, while the atom is described quantum mechanically.

In this chapter, we shall discuss the theory of laser–atom interactions, using a semi-classical method in which the laser field is treated classically, while the atom is studied by using quantum mechanics. This semi-classical approach constitutes an excellent approximation for intense laser fields, since in that case the number of photons per laser mode is very large [1, 2]. In addition, spontaneous emission can be neglected. We begin therefore by giving in Section 2.1 a classical description of the laser field in terms of electric- and magnetic-field vectors satisfying Maxwell's equations. We start by considering plane wave solutions of these equations. Then general solutions describing laser pulses are introduced. The dynamics of a classical electron in the laser field, and in particular the ponderomotive energy and force, are discussed in Section 2.2. Neglecting first relativistic effects, we write down in Section 2.3 the time-dependent Schrödinger equation (TDSE), which is the starting point of the theoretical study of atoms in intense laser fields, and introduce the dipole approximation. In the subsequent two sections, we study the behavior of the TDSE under gauge transformations and the Kramers frame transformation. In view of the central role that the time evolution operator plays in the development of the theory of laser–atom interactions, some general properties of this operator are reviewed in Section 2.6.