Unitary transformation (quantum mechanics)

In quantum mechanics, the Schrödinger equation describes how a system changes with time.

All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.

[1][2] 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

Solutions to the untransformed and transformed equations are also related by

Beginning with the Schrödinger equation, we can therefore insert the identity

and rearranging, we get Finally, combining (1) and (2) above results in the desired transformation: If we adopt the notation

to describe the transformed wave function, the equations can be written in a clearer form.

Unitary transformations can be seen as a generalization of the interaction (Dirac) picture.

In the latter approach, a Hamiltonian is broken into a time-independent part and a time-dependent part, In this case, the Schrödinger equation becomes The correspondence to a unitary transformation can be shown by choosing

which couples the two states, and that the time-dependent driven Hamiltonian is for some complex drive strength

), it is difficult to anticipate the effect of the drive (see driven harmonic motion).

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

, resonance will occur and then the equation above reduces to From this it is apparent, even without getting into details, that the dynamics will involve an oscillation between the ground and excited states at frequency

We can figure out the dynamics in that case without solving the Schrödinger equation directly.

Suppose the system starts in the ground state

This can also be expressed by saying that an off-resonant drive is rapidly rotating in the frame of the atom.

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

In addition to the microwave cavities, the experiment also involved a transmon qubit,

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 (H.c. is shorthand for the Hermitian conjugate.)

For carefully chosen amplitudes, this transformation will cancel

This leaves us with Expanding this expression and dropping the rapidly rotating terms, we are left with the desired Hamiltonian, It is common for the operators involved in unitary transformations to be written as exponentials of operators,

Further, the operators in the exponentials commonly obey the relation

By now introducing the iterator commutator, we can use a special result of the Baker-Campbell-Hausdorff formula to write this transformation compactly as, or, in long form for completeness,