Principles of Superconducting Quantum Computers

6 Resonators: Quantum Treatment

The basic requirement for a quantum mechanical system to be used as a qubit is for it to have two distinct states that can be unambiguously controlled and measured. In this chapter we will discuss how qubits can be realized from superconducting microwave circuits, drawing on our understanding of resonators from Chapter 5.

However, before we can jump right into understanding superconducting circuits, we need to lay a foundation with some concepts in mathematics and physics.

Several formulations of classical physics have been developed over the past several centuries. Newton’s Laws are probably the most widely known, but alternatives of particular interest in making the transition from classical to quantum formulations are Lagrange’s equations and Hamilton’s equations. More detailed discussions of these can be found in texts on classical mechanics such as [39]. We begin with Lagrangian Mechanics and Harmonic Oscillators.

6.1 Lagrangian Mechanics

6.1.1 Hamilton’s Principle

Of particular interest to us for our discussion of circuits are Lagrange’s equations, which are based on Hamilton’s principle.

According to Hamilton’s Principle, the motion of a system from time t₁ to time t₂ is the path along which the quantity

  (6.1)

is stationary, where $\mathcal{L}$ is called the Lagrangian and is given by the difference between the kinetic and potential energies of ...