Simulating a quantum system exposed to some explicitly time-dependent influence differs from that of quantum systems without time dependence in the Hamiltonian. In the latter case, one can, as in Chapter 2, study the full time evolution by means of a relatively simple time-evolution operator, whereas small time steps must be imposed to study the more dynamic case in which also the Hamiltonian changes in time. The first examples of such address the comparatively simple cases of one and two spin-½ particles exposed to magnetic fields. In this context, the rotating wave approximation is introduced. Later, the spatial wave function of a one-dimensional model of an atom exposed to a laser pulse is simulated. To this end, so-called Magnus propagators are used. It is also outlined how the same problem may be recast as an ordinary differential equation by expanding the wave function in the so-called spectral basis consisting of the eigenstates of the time-independent part of the Hamiltonian. The time evolution in this context may be found by more standard methods for ordinary differential equations. Also, the two-particle case if briefly addressed before what is called the adiabatic theorem is introduced. Its validity is checked by implementing a specific, dynamical system.

This chapter aims to illustrate how quantum theory provides useful technological solutions – applications that may be more integrated in our everyday lives than we tend to think. Some applications lend themselves to a particularly straightforward outline through examples already seen in the preceding chapters. These include scanning tunnelling microscopy and emission spectroscopy, which utilize tunnelling and energy quantization, respectively. Prior knowledge and readymade implementations allow these applications to be studied in a quantitative manner. Also, nuclear magnetic resonance is, albeit in a somewhat simplified model, studied quantitatively – within the framework of an oscillating spin-½ particle developed in Chapter 5. The remainder of the chapter is dedicated to quantum information technology. Also in this context, the notion of one or two spin-½ particles is applied frequently. A spin-½ particle is one possible realization of a quantum bit, and it serves well as a model even in cases when quantum bits are implemented differently. After having introduced some basic notions, two specific protocols for quantum communication are studied in some detail. The last part of the chapter addresses adiabatic quantum computing. This technology is studied in a manner that lies close to the last example of Chapter 5.

In Chapter 2, we will review basic facts of quantum measurement that are usually discussed in basic texts on quantum mechanics. These include a motivating experiment – the Stern–Gerlach effect – and discussions of measurement results, statistics, the Born rule, and wavefunction collapse.

In Chapter 10 we discuss feedback and control as an advanced topic. We introduce how to use the measurement results to control the quantum system, via applying conditional unitary operator. A number of experimental systems are discussed, including active qubit phase stabilization, adaptive phase measurements, and continuous quantum error correction.

Chapter 3 takes a step beyond textbook measurements and introduces generalized measurements, beginning with the motivating experiment of an optical polarization measurement with a calcite crystal. In this case, wavefunction collapse is imperfect, and we will discuss how to describe and predict the statistics of outcomes and how to assign postmeasurement states. This topic is closely related to Bayesian probability theory, and we discussed a “Quantum Bayes Rule.”

The book concludes in Chapter 11, where we give in our epilogue a more philosophical reflection on the state of the field. We discuss what it all means, where the field is going, how quantum computers are the ultimate test of quantum mechanics, and speculate on a future post-quantum science.

In Chapter 4, we take a limit where the coupling of the measurement apparatus to the quantum system is very small, and in this limit, discuss weak measurements and weak values. The later involved a sequence of a weak and a strong measurement. Generalizations of these effects are discussed using the concepts in Chapter 3, and we discuss generalized eigenvalues of quantum observables that can exceed the eigenvalue range and reincorporate the concept of observables in generalized measurement theory.

In Chapter 7, we discuss the fundamental limits of quantum amplification. When quantum effects are amplified to classical signals, noise is added to the signal (there are exceptions, but other trade-offs come into play). A detailed discussion of linear response theory is given, which is applicable to many kinds of quantum-limited measurements. This theory is applied to mesoscopic charge detectors and resonant optical cavities. While fundamental bounds quantum mechanics gives to amplification are important, they do not tell you how to invent a quantum-limited amplifier.

Chapter 5 considers the case of diffusive continuous measurements, where the measurement outcomes and quantum state dynamics are analogous to a Brownian noise process. We motivate this type of measurement by considering the example of a double quantum dot system being measured by a quantum point contact. The intrinsic shot noise of the measurement naturally brings about an effective time-continuous measurement. A second example of a superconducting circuit made from Josephson junctions readout with a microwave frequency electromagnetic wave is also discussed in detail. The mathematics of quantum trajectory theory is then pedagogically built up, resulting in the stochastic Schrodinger equation, the stochastic master equation, and the stochastic path integral. We also discuss experimental data and its comparison with this theoretical formalism. These experiments allow us to peer into the inner workings of wavefunction collapse, giving an empirical handle on the many philosophical issues that arise in quantum measurement.