In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.
Often, however, the Schrödinger equation is difficult to solve (even with a computer). Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original.
A unitary transformation (or frame change) can be expressed in terms of a time-dependent Hamiltonian and unitary operator . Under this change, the Hamiltonian transforms as:
The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by . Specifically, if the wave function satisfies the original equation, then will satisfy the new equation.
for some complex drive strength . Because of the competing frequency scales (, , and ), it is difficult to anticipate the effect of the drive (see driven harmonic motion).
Without a drive, the phase of would oscillate relative to . In the Bloch sphere representation of a two-state system, this corresponds to rotation around the z-axis. Conceptually, we can remove this component of the dynamics by entering a rotating frame of reference defined by the unitary transformation . Under this transformation, the Hamiltonian becomes
If the driving frequency is equal to the g-e transition's frequency, , resonance will occur and then the equation above reduces to
Without getting into details[why?], we can already predict that the dynamics will involve an oscillation between the ground and excited states at frequency .
As another limiting case, suppose the drive is far off-resonant, . We can figure out the dynamics in that case without solving the Schrödinger equation directly. Suppose the system starts in the ground state . Initially, the Hamiltonian will populate some component of . A small time later, however, it will populate roughly the same amount of but with completely different phase. Thus the effect of an off-resonant drive will tend to cancel itself out. This can also be expressed by saying that an off-resonant drive is rapidly rotating in the frame of the atom.
These concepts are illustrated in the table below, where the sphere represents the Bloch sphere, the arrow represents the state of the atom, and the hand represents the drive. (Note that a real drive is continuous. The "poking" motion in the animations is meant to make the effects clearer.)
Resonant drive in the lab frame
Resonant drive in a frame rotating with the atom
Off-resonant drive in the lab frame
Off-resonant drive in a frame rotating with the atom
The example above could also have been analyzed in the interaction picture. The following example, however, is more difficult to analyze without the general formulation of unitary transformations. Consider two harmonic oscillators, between which we would like to engineer a beam splitter interaction,
This was achieved experimentally with two microwave cavity resonators serving as and . Below, we sketch the analysis of a simplified version of this experiment.
In addition to the microwave cavities, the experiment also involved a transmonqubit, , coupled to both modes. The qubit is driven simultaneously at two frequencies, and , for which .
In addition, there are many fourth-order terms coupling the modes, but most of them can be neglected. In this experiment, two such terms which will become important are