Chapter 2: Transformations and Pictures

2.1 Continuous Unitary Transformations

Unitary transformations are central in describing the evolution of quantum systems. They are also a key concept for describing symmetries in physical systems, an important step to understanding the emergence of various dynamical variables. These transformations can be intuitively understood as rotations within the Hilbert space. The defining characteristic of a unitary transformation, represented by $\hat{U}$ , is that it preserves inner products. This means that for any two states $|\psi\rangle$ and $|\phi\rangle$ in the Hilbert space, the inner product before and after the application of the unitary transformation remains unchanged:

\langle\psi|\hat{U}^{\dagger}\hat{U}|\phi\rangle=\langle\psi|\hat{1}|\phi\rangle

This implies that $\hat{U}^{\dagger}\hat{U}=\hat{1}$ .

Consider unitary transformations that can be parametrized continuously. The parameter may be time, but can also be something else, for example the rotation angle, a displacement, or the amount of charge present on an island of a superconductor. Taking $s$ for the parameter, such a continuously parameterized unitary transformation is given by $\hat{U}(s)$ for each $s$ , such that $\hat{U}(s)$ is unitary. Moreover, we require that the transformation is the multiplicative over the parameter,

\hat{U}(s_{1}+s_{2})=\hat{U}(s_{1})\hat{U}(s_{2}),

and that $\hat{U}(0)=\hat{1}$ . These properties make the family of transformations into a group.

Let’s consider the case of an infinitesmial transformation. For a small change in the parameter $s$ by $\delta s$ , the transformation can be expressed as:

\hat{U}(s+\delta s)=\hat{U}(s)\hat{U}(\delta s)

For very small $\delta s$ , the transformation $\hat{U}(\delta s)$ can be approximated as:

\hat{U}(\delta s)=\hat{1}+\frac{\mathrm{d}}{\mathrm{d}s}\hat{U}(0)\delta s+O(% \delta s^{2})

Applying the condition $\hat{U}^{\dagger}(\delta s)\hat{U}(\delta s)=\hat{1}$ , we find that:

\frac{\mathrm{d}}{\mathrm{d}s}\hat{U}(0)+\frac{\mathrm{d}}{\mathrm{d}s}\hat{U}% ^{\dagger}(0)=0

This leads to the conclusion that the derivative $\frac{\mathrm{d}}{\mathrm{d}s}\hat{U}(0)$ is anti-Hermitian, or in other words equivalent to $i\hat{K}$ , where $\hat{K}$ is a Hermitian operator. The operator $\hat{K}$ is called the generator of the group of transformations $\hat{U}(s)$ . Since $\hat{K}$ is Hermitian, it corresponds to a physical observable. Notice that we obtain a differential equation

\frac{\mathrm{d}}{\mathrm{d}s}\hat{U}(s)=\hat{U}(s)i\hat{K},

(2.1)

resulting in the expression

\hat{U}(s)=\exp(i\hat{K}s).

2.1.1 Momentum as the Generator of Displacement

We now apply some of the definitions above to the the single particle in one dimension, to see how momentum as a dynamical variable arises from considering transformations in position space.

Definition 2.1.1 (Translation operator).

Consider the particle in one dimension. The eigenstates of the position operator $\hat{X}$ were defined to be $|x\rangle$ for real numbers $x$ . A translation in space has the following effect:

\hat{U}(a)|x\rangle=|x+a\rangle

We denote the generator of this transformation as $-\hat{P}/\hbar$ , so

\hat{U}(a)=\exp\left(-\frac{i\hat{P}}{\hbar}a\right)

We considered above the action of the transformation $\exp\left(-\frac{i\hat{P}a}{\hbar}\right)$ on states $|x\rangle$ . But how does it act on operators? Consider the position operator $\hat{X}$ ,

\hat{X}=\int_{-\infty}^{\infty}x|x\rangle\langle x|\mathrm{d}x

\hat{U}(a)\hat{X}\hat{U}^{\dagger}(a)=\int_{-\infty}^{\infty}x\hat{U}(a)|x% \rangle\langle x|\hat{U}^{\dagger}(a)\mathrm{d}x=\int_{-\infty}^{\infty}x|x+a% \rangle\langle x+a|\mathrm{d}x=\int_{-\infty}^{\infty}(x-a)|x\rangle\langle x|% \mathrm{d}x,

from which we obtain:

\hat{U}(a)\hat{X}\hat{U}^{\dagger}(a)=\exp\left(-\frac{i\hat{P}}{\hbar}a\right% )\hat{X}\exp\left(\frac{i\hat{P}}{\hbar}a\right)=\hat{X}-a\hat{1}

Exercise 2.1.2 (Commutation relation between momentum and position).

Show that the above relation implies $[\hat{X},\hat{P}]=i\hbar$ .

2.1.2 The Hamiltonian as the Generator of Time Evolution

Here we follow a similar logic as above, and consider how the state of a system evolves in time, changing from $|\psi(t_{0})\rangle$ to $|\psi(t_{1})\rangle=|\psi(t_{0}+t)\rangle$ . We modify the notation slightly so

|\psi(t)\rangle=\hat{U}(t,t_{0})|\psi(t_{0})\rangle.

Then from equation 2.1, we find that

\frac{\mathrm{d}}{\mathrm{d}t}\hat{U}(t,t_{0})=-\frac{i\hat{H}}{\hbar}\hat{U}(% t,t_{0}),

(2.2)

where $\hat{H}$ is Hermitian. We find that equation 2.2 leads to

\hat{U}(t,t_{0})=\exp\left(-i(t-t_{0})\hat{H}/\hbar)\right).

Note that the above equations hold only for when the Hamiltonian is not changing in time. We will often be faced with system where the Hamiltonian itself is time dependent, i.e., $\hat{H}(t)$ . Then as we will study later, the differential equation for $\hat{U}$ would still hold, but the solution would need to be modified.

2.2 Time evolution and pictures

2.2.1 Equations of Motion

The quantum states of a system are transformed by time evolution – this is referred to as dynamics. In cases where this evolution is deterministic and closed, i.e., we know the dynamics completely and the components of the system are not interacting with external degrees of freedom¹¹ 1 Later we will consider 1) time evolution that is classically stochastic, for example due to fluctuations in parameters, and 2) open quantum systems where we study the time evolution of a smaller subsystem interacting with or within a larger quantum system., then this evolution is described by the unitary operator $\hat{U}(t,t_{0})$ , such that

|\psi(t)\rangle=\hat{U}(t,t_{0})|\psi(t_{0})\rangle

We can take unitary time evolution to be a fundamental postulate of quantum mechanics. We call generator of the this unitary operator the Hamiltonian. We obtain Schrödinger’s equation by taking the time derivative of the $|\psi(t)\rangle$ with respect to time. To see this, we remind ourselves that saying $\hat{U}(t,t_{0})$ is generated by a (time-dependent) Hamiltonian is simply another way of stating:

\frac{\mathrm{d}}{\mathrm{d}t}\hat{U}(t,t_{0})=-\frac{i}{\hbar}\hat{H}(t)\hat{% U}(t,t_{0}).

Note the slight generalization over the discussion in the last section²² 2 For dynamics that have time-dependent Hamiltonians, the postulate $\hat{U}(s_{1}+s_{2})=\hat{U}(s_{1})\hat{U}(s_{2})$ is replaced by $\hat{U}(t_{1},t_{0})=\hat{U}(t_{1},t_{m})\hat{U}(t_{m},t_{0})$ . where we have taken $\hat{H}(t)$ to be itself time-dependent – so the Hamiltonian at a certain time $t$ generates the time evolution at that instance at time $t$ . This generalization means that our original $\exp(\cdots)$ solution no longer holds and we will study the more general solutions to the above differential equation in chapter LABEL:ch:perturbation. For now we see that the above equation implies Schrödinger’s equation:

i\hbar\frac{\mathrm{d}}{\mathrm{d}t}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle

Time evolution of a density matrix is given by

\hat{\rho}(t)=\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}^{\dagger}(t,t_{0}),

which leads to Schrödinger’s equation for density matrices:

\frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}(t)=-\frac{i}{\hbar}[\hat{H}(t),\hat{% \rho}(t)]

(2.3)

When connecting the density matrix to experiment, we saw in chapter LABEL:ch:quantum_states that the only connection to experiment comes from expressions of the form

O(t)=\text{Tr}[\hat{\rho}(t)\hat{O}]

where $\hat{O}$ is an observable, and $\hat{\rho}$ is the state. This means that a completely equivalent theory can be obtained by assuming that $\hat{\rho}$ is constant in time, and the operator $\hat{O}(t)$ has the time dependence needed to produce the same $O(t)$ . We can easily obtain this equivalent representation by using the cyclic property of trace:

O(t)=\text{Tr}[\rho(t)\hat{O}]=\text{Tr}[\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat% {U}^{\dagger}(t,t_{0})\hat{O}]=\text{Tr}[\hat{\rho}(t_{0})\hat{U}^{\dagger}(t,% t_{0})\hat{O}\hat{U}(t,t_{0})].

In this way, we find an equivalent representation of the theory where the time evolution is completely in the observables, so $\hat{O}_{\mathrm{H}}(t)=\hat{U}^{\dagger}(t,t_{0})\hat{O}\hat{U}(t,t_{0})$ .

Taking the time derivative of $\hat{O}_{\mathrm{H}}(t)$ , we obtain Heisenberg’s equation³³ 3 Notice that here we have assumed that the operator in the Schrödinger picture has no time-dependence. If it were to have time dependence, an additional term $\hat{U}^{\dagger}(t,t_{0})(d\hat{O}/dt)\hat{U}^{\dagger}(t,t_{0})$ which is sometimes written as $\partial\hat{O}_{\mathrm{H}}/\partial t$ would need to be added on the right side.:

\frac{\mathrm{d}}{\mathrm{d}t}\hat{O}_{\mathrm{H}}(t)=\frac{i}{\hbar}[\hat{H}(% t),\hat{O}_{\mathrm{H}}(t)]

(2.4)

This representation is called the Heisenberg picture, in contrast to the Schrödinger picture, which we considered by default. These pictures form only two of a family of many different possible pictures or frames, all of which are related to one other by unitary transformations as we will see below.

2.2.2 Generalizing pictures

We can further generalize the discussion in the previous section to go beyond the Schrödinger and Heisenberg pictures. We saw above that the expected value of a time-independent observable at time $t$ is given by

O(t)=\text{Tr}[\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}^{\dagger}(t,t_{0})\hat% {O}].

Let us now write the time evolution operator as

\hat{U}(t,t_{0})=\hat{U}_{\mathrm{T}}(t,t_{0})\hat{U}_{\mathrm{I}}(t,t_{0}).

(2.5)

This allows us to write

O(t)=\text{Tr}[\hat{U}_{\mathrm{I}}(t,t_{0})\hat{\rho}(t_{0})\hat{U}_{\mathrm{% I}}^{\dagger}(t,t_{0})\hat{U}_{\mathrm{T}}^{\dagger}(t,t_{0})\hat{O}\hat{U}_{% \mathrm{T}}(t,t_{0})].

This expression shows that we can write our equations in a picture where operators evolve according a unitary operator $\hat{U}_{\mathrm{T}}$ (generated by $\hat{H}_{\mathrm{T}}(t)$ ), and our states evolve by an operator $\hat{U}_{\mathrm{I}}$ (generated by $\hat{H}_{\mathrm{I}}(t)$ ).

In other words, we can choose a picture by choosing whatever $\hat{H}_{\mathrm{T}}(t)$ we find convenient, and then calculating $\hat{H}_{\mathrm{I}}(t)$ so that eqn. 2.5 is satisfied. The Schrödinger and Heisenberg pictures are now just special cases, obtained for $\hat{H}_{\mathrm{T}}{}=\hat{1}$ and $\hat{H}_{\mathrm{T}}{}=\hat{H}$ respectively. Other pictures can be found by choosing other $\hat{H}_{\mathrm{T}}(t)$ .

Theorem 2.2.1.

Given the decomposition of the unitary time evolution operator $\hat{U}(t,t_{0})=\hat{U}_{\mathrm{T}}(t,t_{0})\hat{U}_{\mathrm{I}}(t,t_{0})$ , and choosing the generator for the transformation of the operator to be $\hat{H}_{\mathrm{T}}(t)$ , we find that generator of the time-evolution for the state $\hat{H}_{\mathrm{I}}(t)$ is related the original and the transformation Hamiltonian as

\hat{H}(t)=\hat{H}_{\mathrm{T}}(t)+\hat{U}_{\mathrm{T}}(t,t_{0})\hat{H}_{% \mathrm{I}}(t)\hat{U}_{\mathrm{T}}^{\dagger}(t,t_{0}).

(2.6)

Exercise 2.2.2.

Prove the above expression.

Example 2.2.3 (Interaction picture).

we often split our Hamiltonian into two parts:

\hat{H}(t)=\hat{H}_{0}(t)+\hat{V}(t)

The first part may describe and unperturbed part of the system where the dynamics are well understood, or have already been solved. The second part denotes interactions that have been now imposed on the system.

The idea behind the Interaction Picture is to move the well-understood part of the dynamics onto the operators, such that anytime evolution of the state is only generated by the interaction part.

As such we choose:

\hat{H}_{\mathrm{T}}(t)=\hat{H}_{0}(t).

We find then that

\hat{H}_{\mathrm{I}}(t)=\hat{U}_{\mathrm{T}}^{\dagger}(t,t_{0})\hat{V}(t)\hat{% U}_{\mathrm{T}}(t,t_{0}).

2.2.3 Why Move to a Different Frame?

We saw in the previous section that the unitary time evolution operator may be applied to either the states or the observables to obtain the Schrödinger and Heisenberg pictures respectively. These pictures or frames are equivalent in the sense that they predict the same physics. Moreoever, these and other equivalent representations can be used for simplifying the interpretation and solutions of the equations of motion in quantum mechanics. The question arises: why choose one picture over another? A particularly beneficial feature of moving to a different frame is that it may make it significantly easier to unmask dynamics that are of primary interest.

A quintessential example of this advantage emerges when we try to solve equations for a driven atom or a two-level system. Take, for instance, a system with a ground state $|g\rangle$ and an excited state $|e\rangle$ . These states might have energy levels with a separation of $2$ eV or roughly 500 THz ( $\omega_{eg}\sim 10^{15}\leavevmode\nobreak\ 1/\text{s}$ ). By applying an electromagnetic field (i.e.light) at close to this transition frequency, we induce terms such as $\Omega|g\rangle\langle e|+\mathrm{h.c.}$ in the Hamiltonian. The strength of this interaction, called the Rabi frequency $\Omega$ might be significantly smaller than the transition frequency. For example, a typical value may be on the order of $1$ MHz ( $\Omega\sim 10^{7}\leavevmode\nobreak\ 1/\text{s}$ ) – more than 8 orders of magnitude smaller.

This disparity causes complications when we attempt to solve the equations numerically in a naive way by just plugging the equations into numerical integrator:

1.

The necessary time-step, $\delta t$ , required to solve the equation would need to be considerably smaller than $1/\omega_{o}$ , for example around $\delta t\sim 10^{-17}$ seconds. Then, roughly $10$ billion steps would be needed to accurately capture the dynamics induced by the term proportional to $\Omega$ .
2.

In the resulting equations of motion, we would be combining terms proportional to the various terms in the Hamiltonian, i.e.those proportional to $\omega_{o}$ and $\Omega$ . On a digital computer, adding and subtracting small and large numbers often leads to difficulties. If the computer’s precision in representing these values is inadequate, the cumulative effects of this imprecision over multiple steps will lead to unreliable results.

Moving into a rotating frame on the other hand, would allow us to obtain accurate results with only $10-100$ steps of numerical integration – an 8 to 9 order of magnitude improvement over the naive approach. It is clear that even if we are only interested in solving quantum mechanical equations numerically, finding the correct picture is an essential step that must be taken with care.

In the following sections, we will study techniques centered around moving between different frames and representations and obtain some practice. We will see that moving to a new frame often offers an elegant solution to the challenges outlined above. In addition to helping us find reliable computational solutions to the equations in question, moving to a new frame can also improve our understanding of the dynamics, remove time-dependencies that complicate analytic analysis, and allow us to make approximations in a controlled fashion.

2.2.4 A Derivation of the Interaction Picture for Time-independent $\hat{H}_{0}$ in terms of States and Coefficients

In the section 2.2.2, a very general description of pictures beyond Schrödinger and Heisenberg was provided. For clarity, we consider below the derivation of the Interaction Picture (given as example 2.2.3) that is somewhat more explicitly performed at the level of states and coefficients.

Let us begin by considering a system governed by a time-independent Hamiltonian, $\hat{H}_{0}$ . This Hamiltonian, responsible for the system’s dynamics, can be diagonalized to yield its eigenstates and eigenenergies $\{|\phi_{k}\rangle,\hbar\omega_{k}\}$ . As these eigenstates form a complete basis, any quantum state at a given time $t$ can be expressed as:

|\psi(t)\rangle_{\textrm{S}}=\sum_{k}d_{k}(t)|\phi_{k}\rangle,

where $d_{k}(t)$ are time-dependent coefficients and the subscript S indicates that this state is described in the Schrödinger picture.

We can derive the time evolution of $d_{k}(t)$ by utilizing the Schrödinger equation:

\frac{d}{dt}|\psi(t)\rangle_{\text{S}}=-\frac{i}{\hbar}\hat{H}_{0}|\psi(t)% \rangle_{\text{S}}.

Taking the inner product of both sides with $\langle\phi_{k}|$ , we find the set of differential equations:

\frac{d}{dt}d_{k}(t)=-i\omega_{k}d_{k}(t)\Rightarrow d_{k}(t)=c_{k}e^{-i\omega% _{k}t}.

This time evolution of the coefficients is captured by an operator $\hat{U}_{0}(t)$ such that $|\psi(t)\rangle_{\text{S}}=\hat{U}_{0}(t)|\psi(0)\rangle_{\text{S}}$ , where

\hat{U}_{0}(t)\equiv\sum_{k}|\phi_{k}\rangle\langle\phi_{k}|e^{-i\omega_{k}t}=% \exp\left(-i\hat{H}_{0}t/\hbar\right).

However, in many cases, our Hamiltonian $\hat{H}$ is a combination of a primary part, $\hat{H}_{0}$ , and an additional, possibly time-dependent, interaction part $\hat{V}(t)$ . Here, $\hat{H}_{0}$ is diagonal in its own eigenbasis, making it easier to work with, while $\hat{V}(t)$ may not be.

One common situation is when the dynamics of our quantum system are influenced by some external interaction, represented by $\hat{V}(t)$ . In such cases, we’re interested in understanding the additional dynamics induced by this interaction term.

To handle this, it’s convenient to describe our system in the eigenbasis of $\hat{H}_{0}$ , transforming the Schrödinger picture to what is known as the Dirac or interaction picture. We relate the state vectors in the interaction and Schrödinger pictures, $|\psi(t)\rangle_{\text{I}}$ and $|\psi(t)\rangle_{\text{S}}$ , to each other by the rotation $\hat{U}_{0}(t)$ in Hilbert space:

|\psi(t)\rangle_{\text{S}}=\sum_{k}c_{k}(t)e^{-i\omega_{k}t}|\phi_{k}\rangle=% \hat{U}_{0}(t)|\psi(t)\rangle_{\text{I}}\quad\Longleftrightarrow\quad|\psi(t)% \rangle_{\text{I}}=\sum_{k}c_{k}(t)|\phi_{k}\rangle.

When our Hamiltonian is only $\hat{H}=\hat{H}_{0}$ , we saw above that the amplitude for $|\phi_{k}\rangle$ in the Schrödinger picture evolve as $d_{k}(t)=c_{k}e^{-i\omega_{k}t}$ . The $c_{k}(t)$ in the equations above are constants and all of the time evolution of $|\psi(t)\rangle_{\text{I}}$ has been absorbed into its definition, so that it too remains constant in time.

For a Hamiltonian that includes an interaction term $\hat{H}=\hat{H}_{0}+\hat{V}(t)$ , $c_{k}(t)$ are time-dependent coefficients and their time-dependence is due solely to the interaction term $\hat{V}(t)$ .

Theorem 2.2.4 (Relation between Schrödinger Picture and Interaction Picture).

The transformation between the Schrödinger picture and the interaction picture is mediated by the operator $\hat{U}_{0}(t)=\exp\left(-iH_{0}t/\hbar\right)$ . This relationship can be summarized as follows:

	$\displaystyle\hat{H}=\hat{H}_{0}+\hat{V}$	$\displaystyle\quad\Longleftrightarrow\quad\hat{H}_{\text{I}}(t)=\hat{U}_{0}^{% \dagger}(t)\hat{V}\hat{U}_{0}(t),$
	$\displaystyle\|\psi(t)\rangle_{\text{S}}=\sum_{k}d_{k}(t)\|\phi_{k}\rangle$	$\displaystyle\quad\Longleftrightarrow\quad\|\psi(t)\rangle_{\text{I}}=\sum_{k}c% _{k}(t)\|\phi_{k}\rangle,$
	$\displaystyle\hat{O}_{\text{S}}$	$\displaystyle\quad\Longleftrightarrow\quad\hat{O}_{\text{I}}=\hat{U}_{0}^{% \dagger}(t)\hat{O}_{\text{S}}\hat{U}_{0}(t),$

where $\hat{O}_{\text{S}}$ and $\hat{O}_{\text{I}}$ denote any quantum operator in the Schrödinger and interaction pictures, respectively, and we have defined

|\psi(t)\rangle_{\text{S}}=\hat{U}_{0}(t)|\psi(t)\rangle_{\text{I}},% \leavevmode\nobreak\ \leavevmode\nobreak\ d_{k}(t)=c_{k}(t)e^{-i\omega_{k}t}.

The interaction picture is particularly helpful when the interaction part of the Hamiltonian $\hat{V}(t)$ is not diagonal in the eigenbasis of $\hat{H}_{0}$ . In such situations, this picture simplifies the dynamics by isolating the effect of the interaction term.

Example 2.2.5 (Two-level System in the Interaction Picture (coefficients)).

Consider a two-level quantum system with energies $\hbar\omega_{0}$ and $0$ respectively. The system’s time-independent Hamiltonian, $\hat{H}_{0}$ , is given by:

\hat{H}_{0}=\begin{pmatrix}\hbar\omega_{0}&0\\ 0&0\end{pmatrix}.

Let’s suppose that this system is driven by a time-dependent field with strength $\Omega(t)$ . The interaction Hamiltonian, $\hat{V}(t)$ , can then be expressed as:

\hat{V}(t)=\frac{\hbar}{2}\begin{pmatrix}0&\Omega(t)\\ \Omega(t)&0\end{pmatrix}.

The state of the system in the Schrödinger picture, $|\psi(t)\rangle_{\text{S}}$ , can be written as:

|\psi(t)\rangle_{\text{S}}=d_{0}(t)|\phi_{0}\rangle+d_{1}(t)|\phi_{1}\rangle,

where $d_{0}(t)$ and $d_{1}(t)$ are time-dependent coefficients.

The time evolution of these coefficients in the Schrödinger picture is governed by the Schrödinger equation:

	$\displaystyle i\hbar\frac{d}{dt}d_{0}(t)$	$\displaystyle=\hbar\omega_{0}d_{0}(t)+\frac{\hbar\Omega(t)}{2}d_{1}(t),$
	$\displaystyle i\hbar\frac{d}{dt}d_{1}(t)$	$\displaystyle=\frac{\hbar\Omega(t)}{2}d_{0}(t).$

In the absence of the interaction term $\hat{V}(t)$ , the differential equations governing the evolution of $d_{0}(t)$ and $d_{1}(t)$ simplify to:

	$\displaystyle i\hbar\frac{d}{dt}d_{0}(t)$	$\displaystyle=\hbar\omega_{0}d_{0}(t),$
	$\displaystyle i\hbar\frac{d}{dt}d_{1}(t)$	$\displaystyle=0.$

These equations can be solved easily:

	$\displaystyle d_{0}(t)$	$\displaystyle=d_{0}(0)e^{-i\omega_{0}t}$
	$\displaystyle d_{1}(t)$	$\displaystyle=d_{1}(0).$

Now, let’s transform this system into the interaction picture. The state in this picture, $|\psi(t)\rangle_{\text{I}}$ , can be written as:

|\psi(t)\rangle_{\text{I}}=c_{0}(t)|\phi_{0}\rangle+c_{1}(t)|\phi_{1}\rangle,

where $c_{0}(t)$ and $c_{1}(t)$ are time-dependent coefficients that represent the probability amplitudes of the system being in the lower and upper energy level respectively. They are related to the Schrödinger picture coefficients by

	$\displaystyle d_{0}(t)$	$\displaystyle=c_{0}(0)e^{-i\omega_{0}t}$
	$\displaystyle d_{1}(t)$	$\displaystyle=c_{1}(0).$

The time evolution of these coefficients in the interaction picture is governed by the Schrödinger equation:

	$\displaystyle i\hbar\frac{dc_{0}(t)}{dt}$	$\displaystyle=\frac{\hbar\Omega(t)}{2}e^{i\omega_{0}t}c_{1}(t),$
	$\displaystyle i\hbar\frac{dc_{1}(t)}{dt}$	$\displaystyle=\frac{\hbar\Omega(t)}{2}e^{-i\omega_{0}t}c_{0}(t).$

Example 2.2.6 (Two-level System in the Interaction Picture (vectors and operators)).

Consider a two-level system governed by a time-independent Hamiltonian, $\hat{H}_{0}=\frac{\hbar\omega_{0}}{2}(\hat{\sigma}_{z}+\hat{1})$ , and an interaction Hamiltonian $\hat{V}(t)=\frac{\hbar\Omega(t)}{2}\hat{\sigma}_{x}$ , where $\Omega(t)$ is a time-dependent driving field.

The Schrödinger picture wavefunction is $|\psi(t)\rangle_{S}=d_{0}(t)|\phi_{0}\rangle+d_{1}(t)|\phi_{1}\rangle=\hat{U}_% {0}(t)|\psi(0)\rangle_{S}$ , with $\hat{U}_{0}(t)=\exp(-i\hat{H}_{0}t/\hbar)$ . In the absence of interaction, its coefficients evolve as $d_{0}(t)=d_{0}(0)e^{-i\omega_{0}t}$ and $d_{1}(t)=d_{1}(0)$ .

We then transfer to the interaction picture, where the wavefunction is $|\psi(t)\rangle_{S}=\hat{U}_{0}(t)|\psi(0)\rangle_{I}$ . The interaction Hamiltonian in this picture becomes $\hat{V}_{I}(t)=\hat{U}_{0}^{\dagger}(t)\hat{V}(t)\hat{U}_{0}(t)$ .

In the Schrödinger picture, the additional Hamiltonian representing the drive is written in terms of vectors as $\hat{V}(t)=\frac{\hbar\Omega(t)}{2}(|\phi_{0}\rangle\langle\phi_{1}|+|\phi_{1}% \rangle\langle\phi_{0}|)$ . Now, transitioning into the interaction picture, we have to compute the interaction Hamiltonian as $\hat{V}_{I}(t)=\hat{U}_{0}^{\dagger}(t)\hat{V}(t)\hat{U}_{0}(t)$ . This gives us

\hat{V}_{I}(t)=\frac{\hbar\Omega(t)}{2}(e^{i\omega_{0}t}|\phi_{0}\rangle% \langle\phi_{1}|+e^{-i\omega_{0}t}|\phi_{1}\rangle\langle\phi_{0}|).

This is the interaction picture Hamiltonian which gives us the same differential equations derived in example 2.2.5.

2.2.5 Pictures related by unitary transformations

The Schrödinger and Interaction pictures as defined above are related to one another by the unitary transformation $\hat{U}_{0}(t)$ . Other unitary transformation can also be used and often valuable for the computational and conceptual simplifications that they provide. It is therefore helpful to to derive the equations of motion and definition of states and operators for general unitary transformations.

Theorem 2.2.7 (Relation between Schrödinger Picture and a Frame Defined by a General Unitary Transformation).

The transformation between the Schrödinger picture and a frame defined by a general unitary transformation is provided by the operator $\hat{U}_{\text{T}}(t)$ . This connection is captured in the following equations:

	$\displaystyle\hat{H}(t)$	$\displaystyle\quad\Longleftrightarrow\quad\hat{H}_{\text{I}}(t)=\hat{U}_{\text% {T}}^{\dagger}(t)\hat{H}(t)\hat{U}_{\text{T}}(t)-i\hbar\hat{U}_{\text{T}}^{% \dagger}(t)\partial_{t}\hat{U}_{\text{T}}(t),$
	$\displaystyle\|\psi(t)\rangle_{\text{S}}$	$\displaystyle\quad\Longleftrightarrow\quad\|\psi(t)\rangle_{\text{T}}=\hat{U}_{% \text{T}}^{\dagger}(t)\|\psi(t)\rangle_{\text{S}},$
	$\displaystyle\hat{O}_{\text{S}}$	$\displaystyle\quad\Longleftrightarrow\quad\hat{O}_{\text{T}}=\hat{U}_{\text{T}% }^{\dagger}(t)\hat{O}_{\text{S}}\hat{U}_{\text{T}}(t),$

where $\hat{O}_{\text{S}}$ and $\hat{O}_{\text{T}}$ represent any quantum operator in the Schrödinger and transformed pictures, respectively.

Proof 2.2.8.

The proof follows directly from discussion in 2.2.2, but we will provide an alternate proof here that starts with the Schrödinger equation.

We start with the Schrödinger equation for state evolving under the influence of a possibly time-dependent Hamiltonian:

\displaystyle i\hbar\partial_{t}|\psi(t)\rangle_{\text{S}}

\displaystyle=

\displaystyle\hat{H}(t)|\psi(t)\rangle_{\text{S}}.

We denote our general unitary transformation, $\hat{U}_{\text{T}}(t)$ (T stands for transformation). This transformation defines a new frame of reference where hopefully the physics, which is equivalent to that described in any other frame, is a bit easier to analyze. Acting on a state in the new basis, the transformation maps every vector in the new frame to its corresponding Schrödinger picture state:

\displaystyle|\psi(t)\rangle_{\text{S}}

\displaystyle\equiv

\displaystyle\hat{U}_{\text{T}}(t)|\psi(t)\rangle_{\text{T}}.

Using this definition, we can see what the original equation implies:

$\displaystyle i\hbar\partial_{t}\|\psi(t)\rangle_{\text{S}}$	$\displaystyle=$	$\displaystyle\hat{H}(t)\|\psi(t)\rangle_{\text{S}}$
$\displaystyle\Rightarrow i\hbar\partial_{t}\left(\hat{U}_{\text{T}}(t)\|\psi(t)% \rangle_{\text{T}}\right)$	$\displaystyle=$	$\displaystyle\hat{H}(t)\left(\hat{U}_{\text{T}}(t)\|\psi(t)\rangle_{\text{T}}\right)$
$\displaystyle\Rightarrow i\hbar\left(\left[\partial_{t}\hat{U}_{\text{T}}(t)% \right]\|\psi(t)\rangle_{\text{T}}+\hat{U}_{\text{T}}(t)\partial_{t}\|\psi(t)% \rangle_{\text{T}}\right)$	$\displaystyle=$	$\displaystyle\hat{H}(t)\left(\hat{U}_{\text{T}}(t)\|\psi(t)\rangle_{\text{T}}\right)$

leading to

\displaystyle i\hbar\partial_{t}|\psi(t)\rangle_{\text{T}}=\underbrace{\left(% \hat{U}_{\text{T}}^{\dagger}(t)\hat{H}(t)\hat{U}_{\text{T}}(t)-i\hbar\hat{U}_{% \text{T}}^{\dagger}(t)\left[\partial_{t}\hat{U}_{\text{T}}(t)\right]\right)}_{% \hat{H}_{\text{I}}(t)}|\psi(t)\rangle_{\text{T}}.

Note that by setting $\hat{H}(t)=\hat{H}_{0}+\hat{V}$ and choosing the transformation $\hat{U}_{\text{T}}(t)=\hat{U}_{0}(t)\equiv\exp\left(-\frac{i\hat{H}_{0}t}{% \hbar}\right)$ , we arrive at the interaction picture as discussed in section 2.2.4. This is due to the fact that $-i\hbar\hat{U}_{\text{T}}^{\dagger}(t)\partial_{t}\hat{U}_{\text{T}}(t)=-\hat{% H}_{0}$ , while $\hat{U}_{\text{T}}^{\dagger}(t)\hat{H}(t)\hat{U}_{\text{T}}(t)=\hat{H}_{0}+% \hat{U}_{\text{T}}^{\dagger}(t)\hat{V}\hat{U}_{\text{T}}(t)$ .

We arrive at the Heisenberg picture by setting $\hat{U}_{\text{T}}(t)$ to be the unitary evolution operator from the Schödinger picture, generated by the full Hamlitonian $\hat{H}$ .

We summarize the resulting relationship between the three pictures in table 2.1.

Schrödinger

Picture

Interaction

Picture

Heisenberg

Picture

Hamiltonian

\hat{H}=\hat{H}_{0}+\hat{V}(t)

\hat{H}_{I}(t)=\hat{V}_{I}(t)

\hat{V}_{I}(t)=\hat{U}_{0}^{\dagger}(t)\hat{V}(t)\hat{U}_{0}(t)

\hat{H}_{H}=0

States

\hat{\rho}(t)=\hat{U}(t)\hat{\rho}(0)\hat{U}^{\dagger}(t)

\hat{\rho}_{I}(t)=\hat{U}_{I}(t)\hat{\rho}_{I}(0)\hat{U}_{I}^{\dagger}(t)

\hat{\rho}(t)=\hat{\rho}(0)

Observables

\hat{O}(t)

\hat{O}_{I}(t)=\hat{U}_{0}^{\dagger}(t)\hat{O}(t)\hat{U}_{0}(t)

\hat{O}_{H}(t)=\hat{U}^{\dagger}(t)\hat{O}(t)\hat{U}(t)

\hat{O}_{H}(t)=\hat{U}_{I}^{\dagger}(t)\hat{O}_{I}(t)\hat{U}_{I}(t)

Propagators

i\hbar\partial_{t}\hat{U}(t)=\hat{H}(t)\hat{U}(t)

i\hbar\partial_{t}\hat{U}_{I}(t)=\hat{H}_{I}(t)\hat{U}_{I}(t)

\hat{U}(t)=\hat{U}_{0}(t)\hat{U}_{I}(t)

\hat{U}_{H}(t)=\hat{1}

\hat{U}_{\text{T}}(t)

\hat{1}

\hat{U}_{0}(t)\equiv\exp\left(-\frac{i\hat{H}_{0}t}{\hbar}\right)

\hat{U}(t)

Table 2.1: Relations between Schödinger, Interaction, and Heisenberg pictures.

2.2.6 Examples

In the following example, we examine the system that we considered in the introductory paragraph of this chapter. A two level system is driven by an external field oscillating at the frequency close to its transition.

Example 2.2.9 (Periodic Driving of a Two-Level System).

Consider a two-level system with states $|0\rangle$ and $|1\rangle$ . For this system, the Hamiltonian consists of two parts: a time-independent part, $\hat{H}_{0}$ , and a time-dependent driving term, $\hat{V}(t)$ .

1.
The Two-Level System:

$\displaystyle\hat{H}_{0}$ $\displaystyle=\hbar\omega_{1}|1\rangle\langle 1|$

Where:
- •
  
  $\omega_{1}$ is the energy difference between the two states.
2.
The Driving Term: The system is subjected to an external drive given by:

$\displaystyle\hat{V}(t)$ $\displaystyle=\hbar\Omega[|1\rangle\langle 0|+|0\rangle\langle 1|]\cos(\omega_% {d}t)$

Where:
- •
  
  $\Omega$ is the Rabi frequency which determines the strength of the drive.
- •
  
  $\omega_{d}$ is the driving frequency.
3.

The Total Time-Dependent Hamiltonian:

$\displaystyle\hat{H}(t)$ $\displaystyle=\hat{H}_{0}+\hat{V}(t)$
4.

Interaction Picture Hamiltonian: By moving to the interaction picture, the Hamiltonian becomes:

$\displaystyle\hat{H}_{I}(t)$ $\displaystyle=\hat{U}_{0}^{\dagger}(t)\hat{V}(t)\hat{U}_{0}(t)$

With:

$\displaystyle\hat{U}_{0}(t)$ $\displaystyle=\exp[-i\hat{H}_{0}t/\hbar]=|0\rangle\langle 0|+e^{-i\omega_{1}t}% |1\rangle\langle 1|$

Though this is a perfectly valid frame to work in, we will find it even more convenient to move into a slightly different frame, one that oscillates at the drive frequency.
5.

Moving to the Frame of the Drive: We define a transformation

$\displaystyle\hat{U}_{T}(t)$ $\displaystyle=\exp[-i\omega_{d}t|1\rangle\langle 1|]=|0\rangle\langle 0|+e^{-i% \omega_{d}t}|1\rangle\langle 1|$

that rotates with the drive. The new Hamiltonian $\hat{H}_{\text{I}}=\hat{U}_{\mathrm{T}}^{\dagger}(t)\hat{H}\hat{U}_{\mathrm{T}% }(t)-\hbar\omega_{d}|1\rangle\langle 1|$ results in:

$\displaystyle\hat{H}_{\text{I}}$ $\displaystyle=-\hbar\delta|1\rangle\langle 1|+\hbar\Omega[e^{i\omega_{d}t}|1% \rangle\langle 0|+|0\rangle\langle 1|e^{-i\omega_{d}t}]\cos(\omega_{d}t)$

Where $\delta=\omega_{d}-\omega_{1}$ is the detuning between the drive and the system.
6.

Making the Rotating Wave Approximation (RWA): In the RWA, rapidly oscillating terms are ignored. Observe the terms in $\hat{H}_{\text{T}}$ that oscillate rapidly.

$\displaystyle\hat{H}_{I}$ $\displaystyle=-\hbar\delta|1\rangle\langle 1|+\frac{\hbar\Omega}{2}\left[|1% \rangle\langle 0|+|0\rangle\langle 1|\right]+\frac{\hbar\Omega}{2}\left[e^{2i% \omega_{d}t}|1\rangle\langle 0|+e^{-2i\omega_{d}t}|0\rangle\langle 1|\right]$

$\displaystyle=\hat{H}_{TI}+\hat{H}_{TD}$

where the time-independent ( $\hat{H}_{TI}$ ) and time-dependent ( $\hat{H}_{TD}$ ) parts of the transformed Hamiltonian are defined as:

$\displaystyle\hat{H}_{TI}$ $\displaystyle\equiv-\hbar\delta|1\rangle\langle 1|+\frac{\hbar\Omega}{2}\left[% |1\rangle\langle 0|+|0\rangle\langle 1|\right]$

$\displaystyle\hat{H}_{TD}$ $\displaystyle\equiv\frac{\hbar\Omega}{2}\left[e^{2i\omega_{d}t}|1\rangle% \langle 0|+e^{-2i\omega_{d}t}|0\rangle\langle 1|\right]$

We make the rotating wave approximation by noting that when $\Omega\ll\omega_{d}$ , $\hat{H}_{TD}$ only minutely affects the system’s evolution since the rapidly oscillating part averages out. This gives us

$\displaystyle\hat{H}_{I}$ $\displaystyle\approx\hat{H}_{TI}\equiv-\hbar\delta|1\rangle\langle 1|+\frac{% \hbar\Omega}{2}\left[|1\rangle\langle 0|+|0\rangle\langle 1|\right]$

We see that by moving into the frame of the drive, we can eliminate rapidly oscillating terms and obtain a time-independent Hamiltonian – a significant simplification of our equations of motion.

The two following examples require some familiarity with the Harmonic Oscillator. You may wish to come back to these two after Chapter LABEL:chap:Qumodes.

Example 2.2.10 (Linear Driving of a Harmonic Oscillator).

Consider a spring-block system that is subject to an external driving force $F(t)$ . In quantum mechanics, our harmonic oscillator will have a Hamiltonian, $\hat{H}_{0}$ , and an external driving term, $\hat{V}$ .

1.
The Free Harmonic Oscillator:

$\displaystyle\hat{H}_{0}$ $\displaystyle=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega_{o}^{2}\hat{x}^{2}$

Where:
- •
  
  $\hat{p}$ is the momentum operator.
- •
  
  $m$ is the mass of the particle.
- •
  
  $\omega_{o}$ is the natural frequency of the oscillator.
- •
  
  $\hat{x}$ is the position operator.
2.
The Driving Term: The system is driven by an external force which is periodic in time:

$\displaystyle\hat{V}(t)$ $\displaystyle=-F(t)\cdot\hat{x}=2F_{\omega}\cos(\omega_{d}t)\hat{x}$

Where we chose $F(t)=-2F_{\omega}\cos(\omega_{d}t)$ with
- •
  
  $F_{\omega}$ is the amplitude of the external force.
- •
  
  $\omega_{d}$ is the driving frequency.
- •
  
  The factor of 2 is for normalization convenience.
3.

The Total Time-Dependent Hamiltonian:

$\displaystyle\hat{H}(t)$ $\displaystyle=\hat{H}_{0}+\hat{V}(t)$
4.

Expression in terms of Creation and Annihilation Operators:
As shown in Chapter LABEL:chap:Qumodes, it is often convenient to express the position and momentum operators in terms of creation and annihilation operators $\hat{a}^{\dagger}$ and $\hat{a}$ defined as:

$\displaystyle\hat{x}$ $\displaystyle=x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})$

$\displaystyle\hat{p}$ $\displaystyle=ip_{\text{zpf}}(\hat{a}^{\dagger}-\hat{a})$

Where:
- •
  
  $x_{\text{zpf}}$ is the standard deviation of the oscillator’s position when in its ground state; $x_{\text{zpf}}=\sqrt{\hbar/2m\omega_{o}}$ .
- •
  
  $p_{\text{zpf}}$ is the standard deviation of the oscillator’s momentum when in its ground state; $p_{\text{zpf}}=\sqrt{\hbar m\omega_{o}/2}$ .
Then for the quantum harmonic oscillator, we find⁴⁴ 4 Actually $\hat{H}_{0}=\hbar\omega_{o}(\hat{a}^{\dagger}\hat{a}+1/2)$ , but we drop the constant offset of $\hbar\omega_{o}/2$ in the energy for simplicity.

$\displaystyle\hat{H}_{0}$ $\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}$

$\displaystyle\hat{V}(t)$ $\displaystyle=2F_{\omega}\cos(\omega_{d}t)x_{\text{zpf}}(\hat{a}^{\dagger}+% \hat{a})$

Using the identity $\cos(\omega_{d}t)=\frac{1}{2}\left(e^{i\omega_{d}t}+e^{-i\omega_{d}t}\right)$ , we express the driving term as:

$\displaystyle\hat{V}(t)$ $\displaystyle=F_{\omega}x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})(e^{i\omega_{% d}t}+e^{-i\omega_{d}t})$
5.

Going into the frame of the drive field: Our goal is to attempt to remove the time-dependence in the equations above – this would greatly simplify their analysis. It is generally not possible to find a time-independent description for a system that contains a time-dependent force. But in the case of the harmonically driven oscillator, we can succeed to within an approximation described below. First we move into a rotating frame by choosing the correct transformation:

$\displaystyle\hat{U}_{T}(t)$ $\displaystyle=\exp(-i\omega_{d}\hat{a}^{\dagger}\hat{a}t).$

Notice, that this is slightly different than $\hat{U}_{0}$ which we use to go the interaction picture. Applying this transformation to our Hamiltonian, we find⁵⁵ 5 We use the operator identity $\exp(-i\phi\hat{a}^{\dagger}\hat{a})\hat{a}\exp(i\phi\hat{a}^{\dagger}\hat{a})% =e^{i\phi}\hat{a}$ which is discussed in Chapter LABEL:chap:Qumodes.:

$\displaystyle\hat{H}_{\text{I}}(t)=\hat{U}_{\text{T}}^{\dagger}(t)\hat{H}(t)% \hat{U}_{\text{T}}(t)-i\hbar\hat{U}_{\text{T}}^{\dagger}(t)\partial_{t}\hat{U}% _{\text{T}}(t)$

which gives us

$\displaystyle\hat{H}_{I}(t)$ $\displaystyle=\hat{H}_{0}-\hbar\omega_{d}\hat{a}^{\dagger}\hat{a}+F_{\omega}x_% {\text{zpf}}(\hat{a}^{\dagger}e^{i\omega_{d}t}+\hat{a}e^{-i\omega_{d}t})(e^{i% \omega_{d}t}+e^{-i\omega_{d}t})$

.

Expanding the driving term in $\hat{H}_{T}(t)$ , we have:

$\displaystyle F_{\omega}x_{\text{zpf}}$ $\displaystyle(\hat{a}^{\dagger}e^{i\omega_{d}t}+\hat{a}e^{-i\omega_{d}t})(e^{i% \omega_{d}t}+e^{-i\omega_{d}t})$

$\displaystyle=F_{\omega}x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})+F_{\omega}x_% {\text{zpf}}(\hat{a}^{\dagger}e^{2i\omega_{d}t}+\hat{a}e^{-2i\omega_{d}t})$

Now, splitting $\hat{H}_{I}(t)$ into time-independent and time-dependent parts, we find:

$\displaystyle\hat{H}_{TI}$ $\displaystyle=\hat{H}_{0}-\hbar\omega_{d}\hat{a}^{\dagger}\hat{a}+F_{\omega}x_% {\text{zpf}}(\hat{a}^{\dagger}+\hat{a})$

$\displaystyle\hat{H}_{TD}(t)$ $\displaystyle=F_{\omega}x_{\text{zpf}}(\hat{a}^{\dagger}e^{2i\omega_{d}t}+\hat% {a}e^{2i\omega_{d}t})$
6.

Making the rotating wave approximation (RWA): The crux of the RWA lies in observing that certain terms in the Hamiltonian will oscillate rapidly, and under some conditions will effectively average to zero over time and hence can be neglected.

Looking at $\hat{H}_{TD}(t)$ , we notice the terms are oscillating at twice the driving frequency. On the other hand, the rest of the Hamiltonian in this picture is generating time evolution at frequencies on the order of $|\omega_{d}-\omega_{0}|$ and $|F_{\omega}x_{\text{zpf}}|$ . If $2\omega_{d}$ is much faster than $|\omega_{d}-\omega_{0}|$ and $|F_{\omega}x_{\text{zpf}}|$ , then its effect will be averaged out and we can ignore it. This is called the RWA.

Within the RWA, the Hamiltonian is approximated as:

$\displaystyle\hat{H}_{I}$ $\displaystyle\approx\hbar(\omega_{o}-\omega_{d})\hat{a}^{\dagger}\hat{a}+F_{% \omega}x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})$

Example 2.2.11 (Parametric Driving of a Harmonic Oscillator).

Consider a parametric oscillator where the oscillation frequency is modulated, for example, by changing the spring constant periodically in time.

1.
The Parametrically Driven Hamiltonian:

$\displaystyle\hat{H}{}(t)$ $\displaystyle=\frac{\hat{p}{}^{2}}{2m}+\frac{1}{2}m\omega_{o}^{2}(1+\epsilon(t% ))\hat{x}{}^{2}$

$\displaystyle\epsilon(t)$ $\displaystyle=\epsilon_{0}\cos(2\omega_{d}t)$

Here:
- •
  
  $\epsilon(t)$ represents the time-dependent modulation of the spring constant.
- •
  
  $\epsilon_{0}$ is the amplitude of the modulation.
- •
  
  $2\omega_{d}$ is the frequency of the parametric drive.

Hamiltonian in terms of Creation and Annihilation Operators: Using the definitions above for the Hamiltonian and the drive term, and the relation between the position and momentum operators, and the creation and annihilation operators, we find:

	$\displaystyle\hat{H}(t)$	$\displaystyle=\frac{p_{\text{zp}}^{2}}{2m}(\hat{a}^{\dagger}-\hat{a})^{2}+% \frac{1}{2}m\omega_{o}^{2}(1+\epsilon(t))x_{\text{zp}}^{2}(\hat{a}+\hat{a}^{% \dagger})^{2}$
		$\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}+\frac{1}{4}\hbar\omega_{% o}\epsilon(t)(\hat{a}+\hat{a}^{\dagger})^{2}$
		$\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}+\frac{1}{4}\hbar\omega_{% o}\epsilon_{0}\cos(2\omega_{d}t)(\hat{a}+\hat{a}^{\dagger})^{2}$
		$\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}+\frac{1}{8}\hbar\omega_{% o}\epsilon_{0}(e^{2i\omega_{d}t}+e^{-2i\omega_{d}t})(\hat{a}+\hat{a}^{\dagger}% )^{2}$

Now we define $\beta\equiv\frac{1}{8}\omega_{o}\epsilon_{0}$ , and find:

\displaystyle\hat{H}{}(t)

\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}{}+\hbar\beta\left(e^{-2i% \omega_{d}t}\hat{a}^{\dagger 2}+e^{2i\omega_{d}t}\hat{a}^{2}\right)+\text{(% other terms)}

Here:

•

$\beta\equiv\frac{1}{8}\omega_{o}\epsilon_{0}$ is a parameter related to the amplitude of the parametric drive.

Transforming to a Rotating Frame of Reference:

We use a transformation $\hat{U}_{T}(t)=\exp\left(-i\omega_{d}\hat{a}^{\dagger}\hat{a}t\right)$ , which puts us in the frame of reference of the drive:

	$\displaystyle\hat{H}_{I}(t)$	$\displaystyle=\hat{U}_{T}^{\dagger}(t)\hat{H}(t)\hat{U}_{T}(t)-i\hbar\hat{U}_{% T}^{\dagger}(t)\frac{d\hat{U}_{T}(t)}{dt}$
		$\displaystyle=\hbar(\omega_{o}-\omega_{d})\hat{a}^{\dagger}\hat{a}{}+\hbar% \beta\left(\hat{a}^{\dagger 2}+\hat{a}^{2}\right)+\hat{U}_{T}^{\dagger}(t)% \text{(other terms)}\hat{U}_{T}(t)$

4.

Simplification to a Time-Independent Hamiltonian within RWA:

Note that in the interaction picture, if we drop the “other terms”, the Hamiltonian becomes time-dependent. Dropping these other terms may be justified within the rotating wave approximation resulting in:

$\displaystyle\hat{H}_{I}{}$ $\displaystyle\approx\hbar(\omega_{o}-\omega_{d})\hat{a}^{\dagger}\hat{a}{}+% \hbar\beta(\hat{a}^{\dagger 2}+\hat{a}^{2})$

$\displaystyle i\hbar\partial_{t}\|\psi(t)\rangle_{\text{S}}$	$\displaystyle=$	$\displaystyle\hat{H}(t)\|\psi(t)\rangle_{\text{S}}$
$\displaystyle\Rightarrow i\hbar\partial_{t}\left(\hat{U}_{\text{T}}(t)\|\psi(t)% \rangle_{\text{T}}\right)$	$\displaystyle=$	$\displaystyle\hat{H}(t)\left(\hat{U}_{\text{T}}(t)\|\psi(t)\rangle_{\text{T}}\right)$
$\displaystyle\Rightarrow i\hbar\left(\left[\partial_{t}\hat{U}_{\text{T}}(t)% \right]\|\psi(t)\rangle_{\text{T}}+\hat{U}_{\text{T}}(t)\partial_{t}\|\psi(t)% \rangle_{\text{T}}\right)$	$\displaystyle=$	$\displaystyle\hat{H}(t)\left(\hat{U}_{\text{T}}(t)\|\psi(t)\rangle_{\text{T}}\right)$

	$\displaystyle\hat{H}_{I}$	$\displaystyle=-\hbar\delta\|1\rangle\langle 1\|+\frac{\hbar\Omega}{2}\left[\|1% \rangle\langle 0\|+\|0\rangle\langle 1\|\right]+\frac{\hbar\Omega}{2}\left[e^{2i% \omega_{d}t}\|1\rangle\langle 0\|+e^{-2i\omega_{d}t}\|0\rangle\langle 1\|\right]$
		$\displaystyle=\hat{H}_{TI}+\hat{H}_{TD}$

	$\displaystyle\hat{H}_{TI}$	$\displaystyle\equiv-\hbar\delta\|1\rangle\langle 1\|+\frac{\hbar\Omega}{2}\left[% \|1\rangle\langle 0\|+\|0\rangle\langle 1\|\right]$
	$\displaystyle\hat{H}_{TD}$	$\displaystyle\equiv\frac{\hbar\Omega}{2}\left[e^{2i\omega_{d}t}\|1\rangle% \langle 0\|+e^{-2i\omega_{d}t}\|0\rangle\langle 1\|\right]$

	$\displaystyle\hat{x}$	$\displaystyle=x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})$
	$\displaystyle\hat{p}$	$\displaystyle=ip_{\text{zpf}}(\hat{a}^{\dagger}-\hat{a})$

	$\displaystyle\hat{H}_{0}$	$\displaystyle=\hbar\omega_{o}\hat{a}^{\dagger}\hat{a}$
	$\displaystyle\hat{V}(t)$	$\displaystyle=2F_{\omega}\cos(\omega_{d}t)x_{\text{zpf}}(\hat{a}^{\dagger}+% \hat{a})$

	$\displaystyle F_{\omega}x_{\text{zpf}}$	$\displaystyle(\hat{a}^{\dagger}e^{i\omega_{d}t}+\hat{a}e^{-i\omega_{d}t})(e^{i% \omega_{d}t}+e^{-i\omega_{d}t})$
		$\displaystyle=F_{\omega}x_{\text{zpf}}(\hat{a}^{\dagger}+\hat{a})+F_{\omega}x_% {\text{zpf}}(\hat{a}^{\dagger}e^{2i\omega_{d}t}+\hat{a}e^{-2i\omega_{d}t})$

	$\displaystyle\hat{H}_{TI}$	$\displaystyle=\hat{H}_{0}-\hbar\omega_{d}\hat{a}^{\dagger}\hat{a}+F_{\omega}x_% {\text{zpf}}(\hat{a}^{\dagger}+\hat{a})$
	$\displaystyle\hat{H}_{TD}(t)$	$\displaystyle=F_{\omega}x_{\text{zpf}}(\hat{a}^{\dagger}e^{2i\omega_{d}t}+\hat% {a}e^{2i\omega_{d}t})$

	$\displaystyle\hat{H}{}(t)$	$\displaystyle=\frac{\hat{p}{}^{2}}{2m}+\frac{1}{2}m\omega_{o}^{2}(1+\epsilon(t% ))\hat{x}{}^{2}$
	$\displaystyle\epsilon(t)$	$\displaystyle=\epsilon_{0}\cos(2\omega_{d}t)$

2.1 Continuous Unitary Transformations

2.1.1 Momentum as the Generator of Displacement

Definition 2.1.1 (Translation operator).

Exercise 2.1.2 (Commutation relation between momentum and position).

2.1.2 The Hamiltonian as the Generator of Time Evolution

2.2 Time evolution and pictures

2.2.1 Equations of Motion

2.2.2 Generalizing pictures

Theorem 2.2.1.

Exercise 2.2.2.

Example 2.2.3 (Interaction picture).

2.2.3 Why Move to a Different Frame?

2.2.4 A Derivation of the Interaction Picture for Time-independent H^0 in terms of States and Coefficients

Theorem 2.2.4 (Relation between Schrödinger Picture and Interaction Picture).

Example 2.2.5 (Two-level System in the Interaction Picture (coefficients)).

Example 2.2.6 (Two-level System in the Interaction Picture (vectors and operators)).

2.2.5 Pictures related by unitary transformations

Theorem 2.2.7 (Relation between Schrödinger Picture and a Frame Defined by a General Unitary Transformation).

Proof 2.2.8.

2.2.6 Examples

Example 2.2.9 (Periodic Driving of a Two-Level System).

Example 2.2.10 (Linear Driving of a Harmonic Oscillator).

Example 2.2.11 (Parametric Driving of a Harmonic Oscillator).

2.2.4 A Derivation of the Interaction Picture for Time-independent $\hat{H}_{0}$ in terms of States and Coefficients