Time-Dependent Quantum Mechanics of Two-Level Systems

The following sections are included:

• Preface
• Contents

Time dependence involves changes in the occupation of the states of a physical system. For example, the electron in a hydrogen atom may be excited from a 1s state into a 2p state due to a perturbation such as an electromagnetic field. (The states of the hydrogen atom are derived in Quantum Mechanics texts such as Konishi and Paffuti (2009, Chap. 6).) Generally, we say a transition has occurred when the physical system changes its state. It is simplest to view this as a change from the system being in state i to the system being in state j. This chapter starts with basic ideas on time dependence in Quantum Mechanics and the time evolution operator. These are applied to spin precession of a particle. Then the Einstein coefficients and rate equations are introduced. The following chapters are concerned with calculating the probability p_i(t) for the occupation of state i of a system as a function of the time t. These predictions are compared with experiments whenever possible…

Chapter 1 shows how spin precession occurs for a spin-$\frac{1}{2}$ particle in a constant magnetic field. This leads to time-dependent expectation values of the components of the spin vector. More generally, time dependence in Quantum Mechanics is concerned with transitions between the states, or energy levels, of a system such as an atom, or a molecule. The Time-Dependent Schrödinger Equation (TDSE) provides a description of these transitions. But its solution is usually numerical and this limits insight into how the system develops over time. When we are able to restrict our considerations to only two energy levels, we can reduce the TDSE to two coupled first-order differential equations in time. And these we can solve! The two states may correspond to the spin up and spin down states of a spin-$\frac{1}{2}$ particle. Or the states may correspond to the energy levels of a double-well potential. Such two-state systems are quite useful in understanding magnetic resonance phenomena and the energy levels for molecules with inversion spectra such as ammonia. In addition, numerous settings arise where two energy levels provide a good approximation to a system with many energy levels. Such systems include atoms, molecules, nanostructures and even defects in solids…

Chapter 2 treated two-level systems with a time-independent coupling potential between the two energy levels. We now add time dependence to the coupling potential. This introduces a rich variety of phenomena when the time-dependent coupling potential represents the interaction of a classical electromagnetic field with our two-level system. For example, many atomic or molecular systems may be simplified to two energy levels if the interaction of other energy levels with the time-dependent electromagnetic field may be ignored. In addition, a static magnetic field splits an energy level into its magnetic sublevels. This leads to a two-level system for a spin-$\frac{1}{2}$ electron or nucleon, and this two-level system may be probed with a time-dependent electromagnetic field in search of electron spin or nuclear magnetic resonance, respectively…

We now apply the formalism and machinery of Chap. 3 to several examples of two-level systems that evolve with time. Chapter 3 shows how Rabi oscillations (or Rabi flopping) arise and includes a physical example. Experimental demonstrations of Rabi oscillations are quite involved. The interested reader should consult the cited references to supplement the material that follows. We need to prepare a two-level system in a known state and we need a method to measure which state the system is in as a function of time. These requirements prove to be challenging and how they are satisfied varies with the two-level system. Historically, the first prediction and first observation of such oscillations were made by Rabi and involved radio frequencies (Rabi, 1937; Rabi et al., 1938a). This work led to the use of magnetic resonance to probe materials. Here, a frequency or a field is adjusted to produce data analogous to Fig. 3.5. Further work with microwaves led to the demonstration of a MASER (Microwave Amplification by Stimulated Emission of Radiation) (Gordon et al., 1954)…

The following sections are included:

• Why the Density Matrix?
• The Density Operator and the Density Matrix
• An Equation for the Time-Dependent Density Matrix
• The Density Matrix Applied to a Two-Level System
• The Introduction of Relaxation into the Density Matrix
• A Solvable Two-Level System with Relaxation Included
• The Bloch Equations
• A Solution to the Bloch Equations
• More on the Density Matrix
• Ramble 5: A Solution of the Time-Dependent Equations for the Density Matrix Elements
• References

We return to optical excitation and optical emission, and we explore experiments that focus on the photons emitted when a two-level system returns to its ground state. Such experiments are examples of resonance fluorescence and we find the density matrices of Chap. 5 useful in untangling the experiments. Our discussions involve systems such as defects in diamonds, quantum dots and molecules in condensed matter. Much of this work is aimed at developing single-photon sources, which would be useful for quantum computing and integrated circuits. For the latter, photons replace electrons for some of the “communication” levels on-chip and between chips. This requires controllable excitation of the system and the emission of a single photon upon demand. Aharonovich et al. (2011) review singlephoton emission by diamond defects, while Migdal et al. (2013) and Predojević and Mitchell (2015) provide additional information on a variety of systems. Kuhlmann et al. (2015) and Ding et al. (2016) are examples of the ongoing research into single-photon sources…

The following sections are included:

• Appendix 1. Physical Constants
• Appendix 2. The Electromagnetic Field
• Appendix 3. Diagonalization of a Matrix in Chapter 2
• Appendix 4. The Time Evolution Operator and the Sudden Approximation
• Appendix 5. Beyond the Rotating Wave Approximation
• Appendix 6. The Second-Order Correlation Function for a Two-Level System

The following sections are included:

• Index