# Taming the Rotating Wave Approximation

Daniel Burgarth  Center for Engineered Quantum Systems, Macquarie University,
2109 NSW, Australia  Paolo Facchi  Dipartimento di Fisica, Università di Bari,
I-70126 Bari, Italy, and INFN, Sezione di Bari, I-70126 Bari, Italy  Robin
Hillier  Department of Mathematics and Statistics, Lancaster University,
Lancaster LA1 4YF, UK  Marilena Ligabò  Dipartimento di Matematica, Università
di Bari, I-70125 Bari, Italy

###### Abstract

The interaction between light and matter is one of the oldest research areas
of quantum mechanics, and a field that just keeps on delivering new insights
and applications. With the arrival of cavity and circuit quantum
electrodynamics we can now achieve strong light-matter couplings which form
the basis of most implementations of quantum technology. But quantum
information processing also has high demands requiring total error rates of
fractions of percentage in order to be scalable (fault-tolerant) to useful
applications. Since errors can also arise from modelling, this has brought
into center stage one of the key approximations of quantum theory, the
Rotating Wave Approximation (RWA) of the quantum Rabi model, leading to the
Jaynes-Cummings Hamiltonian. While the RWA is often very good and incredibly
useful to understand light-matter interactions, there is also growing
experimental evidence of regimes where it is a bad approximation. Here, we ask
and answer a harder question: for which experimental parameters is the RWA,
although perhaps qualitatively adequate, already not good enough to match the
demands of scalable quantum technology? For example, when is the error at
least, and when at most, 1%? To answer this, we develop rigorous non-
perturbative bounds taming the RWA.

We find that these bounds not only depend, as expected, on the ratio of the
coupling strength and the oscillator frequency, but also on the average number
of photons in the initial state. This confirms recent experiments on photon-
dressed Bloch-Siegert shifts. We argue that with experiments reporting
controllable cavity states with hundreds of photons and with quantum error
correcting codes exploring more and more of Fock space, this state-dependency
of the RWA is increasingly relevant for the field of quantum computation, and
our results pave the way towards a better understanding of those experiments.

The Rotating Wave Approximation (RWA) is one of the oldest and most important
approximations in Quantum Theory. The starting point is at the birthplace of
Nuclear Magnetic Resonance (NMR) in 1938, when Rabi and co-authors realized
that rather than using rotating fields, “it is more convenient experimentally
to use an oscillating field, in which case the transition probability is
approximately the same for weak oscillating fields near the resonance
frequency” [1]. This was significant: Rabi had shown earlier that the
Schrödinger equation for rotating fields is easily solved analytically [2].
This approximation was a crucial step in understanding driven quantum
dynamics, as the time-dependent Schrödinger equation is notoriously hard to
solve.

Perhaps this is the key reason for the popularity [3] of the RWA: it provides
understanding and intuition of resonant driving. In fact, the importance of
these ideas and the resulting techniques of NMR led to Rabi being awarded the
Nobel Laureate in Physics in 1944. But what justified the approximation, and
how did Rabi get to it? Primarily reporting an experimental finding, Rabi
himself does not provide justification, but over the last 80 years many
different theoretical methods were used to provide justification and deeper
understanding of the RWA (the literature is extensive, but see for instance
[4, 5, 6, 7, 8]).

Rabi described the atom as a two-level system and the field classically. In
the full quantum description of light-matter interaction the situation is much
more complicated. By the 1960s Quantum Electrodynamics was well established,
and the electromagnetic field is now itself a quantum system described by
unbounded operators. Jaynes and Cummings [9] developed the full quantum
mechanical version of the Rabi model (now called Quantum Rabi Model)

$H=\frac{\Omega}{2}\sigma_{z}+\omega
a^{\dagger}a+\lambda\sigma_{x}(a+a^{\dagger}),$ (1)

and applied the RWA to obtain the Jaynes-Cummings model

$H_{\textrm{RWA}}=\frac{\Omega}{2}\sigma_{z}+\omega
a^{\dagger}a+\lambda(\sigma_{+}a+\sigma_{-}a^{\dagger}).$ (2)

Here, $\Omega$ is the energy difference between the two states of the atom,
$\omega$ the light frequency and $\lambda$ the strength of the light-matter
coupling; we always use $\hbar=1$.

Figure 1: Light-matter interactions have been a major driver in quantum
physics for half a decade. Often, atoms are placed into cavities to amplify
their effective coupling strength with photons. Here, we show that the
rotating wave approximation is not only determined by such coupling strength
and the frequency of the driving, but also by the number of photons (naively
depicted as golden spheres) in the cavity.

Due to its simplicity and wide range of applicability, the Jaynes-Cummings
model is the main work horse of light-matter interactions and, by extension,
quantum technology. For an excellent overview of its scope see [10]. While at
the time of the original paper the RWA was rather natural, given that the bare
coupling between matter and light tends to be extremely weak, in cavity and
circuit QED nowadays it is well understood that the effective coupling can be
enhanced to a level where the RWA breaks down. This is often referred to as
the Ultrastrong Coupling regime. For examples of experiments, see [11] and
[12], for a recent review see [13].

While there is no rigorous derivation of the RWA for the Jaynes-Cummings model
till date, the common lore is that the ratio $g\equiv\lambda/\omega$ between
the light-matter coupling and the light frequency is the key parameter [10].
This is motivated by perturbative arguments and of course backed up by
extensive numerical studies and simulations. For a summary of the different
regimes see Table 1.1 in [10], where it is argued that for $g\approx 0.1$ the
RWA breaks down.

Figure 2: Bounding the error of the RWA. We consider a Fock state evolving
under the quantum Rabi and the Jaynes-Cummings model, respectively. We show
our analytical upper and lower bounds and the exact numerical norm difference
between the two models. We see that the error grows with the photon number,
and that the bounds provide a good understanding of the scaling (other
parameters here $g=\frac{\lambda}{\omega}=\frac{1}{100}$,$\Delta=0$ ,
$t\approx 0.04/\omega$).

On the other hand, this picture changes for high photon numbers. Indeed, Walls
showed [14] that the Bloch-Siegert shift (taken as a sign of the breakdown of
the RWA) scales with the number of photons. This was also observed
experimentally [15]. See also [16] for a perturbative argument that
$\lambda\sqrt{\langle a^{\dagger}a\rangle}\ll\omega$ is a more relevant
condition in that regime.

What this means is that the quality of the RWA does not only depend on the
parameters of the model, but also on the initial state of the system. See Fig.
2 for a numerical example. Indeed, we prove that there are short times [17]
$t\leq\pi/\omega$ for which

$\|e^{-itH}-e^{-itH_{\textrm{RWA}}}\|\geq\frac{1}{6}$ (3)

for $any$ parameter value. This should be considered as a big error, because
the biggest difference between two unitaries is $2$ and because modern quantum
technology demands errors well below $1\%$ (see below). Does this mean that
the RWA is wrong? No, because we also show that for any state $\varphi$ and
any time $t$,

$e^{-itH}\varphi-e^{-itH_{\textrm{RWA}}}\varphi\to 0,\quad\text{as }g\to 0.$
(4)

This is our main result, providing a rigorous justification to the RWA. It
does not contradict Eq. (3), but is a typical phenomenon of unbounded
Hamiltonians such as $H$ and $H_{\textrm{RWA}}$: there is no norm convergence,
only state-dependent convergence. This is one the key technicalities that make
it hard to apply standard perturbative arguments for the RWA.

Let us discuss the relevance of this photon-dependence in the context of
quantum technology. For fault-tolerant quantum computation, very high fidelity
with error rates $<10^{-3}$ are required [18]. Moreover, modern qubit designs
such as GKP [19] and CAT qubits use cavity states and explore high numbers of
photons. In particular, CAT states have been created with about 100 photons
[20]. It is therefore necessary to have a good handle of the error of the RWA.
Since quantum algorithms also invoke dynamics, it is not sufficient to simply
match spectral properties, as it is usually done, but we need to bound the
difference in evolution operators. The interesting evolution time regime here
are short times up to $\pi/\omega$: already there, the RWA dynamics can
deviate substantially. We show that the maximal error $\epsilon_{n}$ that the
RWA has for an $n-$photon Fock state in a short time interval up to
$\pi/\omega$ is bounded between

$5g\sqrt{n+3}\geq\epsilon_{n}\geq\frac{1}{6}-\frac{1}{216g^{2}n}-\frac{7}{12n},$
(5)

proving that the RWA becomes good for small $g$ but bad for large $n$. Tighter
and more general bounds and the full proofs of our results are provided in the
Appendix. See also Fig. 2 for numerical examples of these refined bounds.
These bounds prove that $g\sqrt{n}$ is the right parameter (as anticipated by
the perturbative argument [16]) for the validity of the RWA for Fock states.
For more general states, see the Appendix. These bounds will be useful for
experimentalists in quantum information to judge if they should apply the RWA
or not.

We would now like to explain the idea which allows us to tame the RWA.
Although there are many different conceptual ideas trying to justify the RWA,
almost all of them agree that ‘highly oscillatory terms’ in a Hamiltonian may
sometimes be discarded to a good approximation. But why? Interestingly, some
have argued that such terms are not observable, since measurements take finite
time. This is plausible; however it turns out that even if measurements are
instantaneous, the RWA can be taken. Others argue on the basis of first order
perturbation theory, when the term involves an integral over the Hamiltonian.
This gives a good qualitative picture but makes it impossible to compute a
rigorous and precise picture. In a more recent work [8] a different route was
taken: by an integration by part, the difference between two evolutions can
indeed be written in terms of an integral over the difference of their
generating Hamiltonians, where fast oscillations average out. This allows one
to prove and provide bounds for the RWA, but only in the finite dimensional
case. Here, we develop an integration by part to unbounded operators. In the
general case, this is hard, so we are employing several structures of the
specific problem of the quantum Rabi model to simplify the analysis. First,
both $H$ and $H_{\textrm{RWA}}$ are time-independent, so we can use the rich
theory of semigroups. Secondly, $H_{\textrm{RWA}}$ has many conserved
quantities and can only increase and decrease the photon number by one.
Finally, all involved quantities are well-defined on the subspace of rapidly
decreasing functions and leave it invariant, which allows us to work on that
subspace. We refer to the Appendix for the mathematical details.

To summarize, after decades of work and conjectures around the RWA for the
highly relevant quantum Rabi model, we have now got a rigorous proof and in
addition a complete quantitative measure in terms of lower and upper bounds on
the error of the approximation. In particular, this confirms the experimental
and numerical findings that the error becomes large for large ratio $g$
between light-matter coupling and light frequency or for large photon numbers
and hence the dependence on the state of the system. In practice, for given
fixed photon number and given maximally permissible error this tells us how
small $g$ has to be in order for the RWA to work. Since experiments are
working with ever growing systems, our results will be of immediate relevance
to the understanding and setup of those experiments and further developments
in quantum technology. We expect that the methods developed for our proof can
be applied to tame the RWA for other interesting models, such as systems with
multiple modes, nonlinearities and other descendants of the Jaynes-Cummings
model [10].

### Acknowledgements

DB acknowledges funding by the Australian Research Council (project numbers
FT190100106, DP210101367, CE170100009). PF and ML were partially supported by
the Italian National Group of Mathematical Physics (GNFM-INdAM), by Istituto
Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM”, and by
Regione Puglia and QuantERA ERA-NET Cofund in Quantum Technologies (GA No.
731473), project PACE-IN.

## Appendix

###  1 Time evolution of the Rabi and the Jaynes-Cummings models

We consider the infinite dimensional Hilbert space $L^{2}(\mathbb{R})$, the
creation operator $a^{\dagger}=\frac{1}{\sqrt{2}}\left(x-\frac{d}{dx}\right)$
and the annihilation operator
$a=\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)$ on Schwartz space
$\SS(\mathbb{R})$. A fundamental feature of these two operators is that their
commutator is the identity operator, i.e.

$[a,a^{\dagger}]=I.$ (6)

Now we consider the following two Hamiltonians

$H=\frac{\Omega}{2}\sigma_{z}\otimes I+I\otimes\omega
a^{\dagger}a+\lambda\sigma_{x}\otimes(a+a^{\dagger})$ (7)

and

$H_{\textrm{RWA}}=\frac{\Omega}{2}\sigma_{z}\otimes I+I\otimes\omega
a^{\dagger}a+\lambda(\sigma_{+}\otimes a+\sigma_{-}\otimes a^{\dagger})$ (8)

on $\mathbb{C}^{2}\otimes\SS(\mathbb{R})$, and we also denote their closures
by $H$ and $H_{\textrm{RWA}}$, with suitable dense domains. Here
$\lambda,\omega,\Omega\in\mathbb{R}$,

$\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (9)

are the Pauli matrices and

$\sigma_{+}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right),\quad\sigma_{-}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right).$ (10)

In what follows, we usually work on $\mathbb{C}^{2}\otimes\SS(\mathbb{R})$
without saying so explicitly every time, as this subspace of
$\mathbb{C}^{2}\otimes L^{2}(\mathbb{R})$ forms a common invariant core of the
operators we are studying in this section; this can be shown following the
standard methods in [22, Sec.X.6]. To simplify the notation, we usually use
the same notation for an operator on this core and its closure on the whole
domain.

####  1.1 Interaction picture: time dependent Hamiltonians $H_{1}(t)$ and
$H_{2}(t)$

We consider the following Hamiltonian

$H_{0}=\frac{\omega}{2}\sigma_{z}\otimes I+I\otimes\omega a^{\dagger}a,$ (11)

and define for all $t\in\mathbb{R}$

$U_{1}(t)=e^{itH_{0}}e^{-itH},\quad
U_{2}(t)=e^{itH_{0}}e^{-itH_{\textrm{RWA}}}.$ (12)

We have that for all $j\in\\{1,2\\}$: $U_{j}(0)=I$ and

$i\frac{dU_{j}(t)}{dt}=H_{j}(t)U_{j}(t),$ (13)

where for all $t\in\mathbb{R}$:

$H_{1}(t)=e^{itH_{0}}(H-H_{0})e^{-itH_{0}},\quad
H_{2}(t)=e^{itH_{0}}(H_{\textrm{RWA}}-H_{0})e^{-itH_{0}},$ (14)

We define the detuning between field and atom as $\Delta=\Omega-\omega$ and
compute $H_{1}(t)$:

$\displaystyle H_{1}(t)$ $\displaystyle=$ $\displaystyle
e^{itH_{0}}(H-H_{0})e^{-itH_{0}}$ (15) $\displaystyle=$ $\displaystyle
e^{itH_{0}}\left(\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda\sigma_{x}\otimes(a+a^{\dagger})\right)e^{-itH_{0}}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda(e^{it\omega\sigma_{z}/2}\sigma_{x}e^{-it\omega\sigma_{z}/2})\otimes(e^{it\omega
a^{\dagger}a}(a+a^{\dagger})e^{-it\omega a^{\dagger}a})$

We get

$\displaystyle
e^{it\omega\sigma_{z}/2}\sigma_{x}e^{-it\omega\sigma_{z}/2}=\cos\left(t\omega\right)\sigma_{x}-\sin\left(t\omega\right)\sigma_{y},$
(16)

and

$e^{it\omega a^{\dagger}a}ae^{-it\omega a^{\dagger}a}=e^{-it\omega}a,\qquad
e^{it\omega a^{\dagger}a}a^{\dagger}e^{-it\omega
a^{\dagger}a}=e^{it\omega}a^{\dagger}.$ (17)

Therefore,

$\displaystyle H_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda\left(\cos\left(t\omega\right)\sigma_{x}-\sin\left(t\omega\right)\sigma_{y}\right)\otimes\left(e^{-it\omega}a+e^{it\omega}a^{\dagger}\right)$
$\displaystyle=$ $\displaystyle\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda\left(\sigma_{+}\otimes a+\sigma_{-}\otimes
a^{\dagger}+e^{2it\omega}\sigma_{+}\otimes
a^{\dagger}+e^{-2it\omega}\sigma_{-}\otimes a\right).$

In a similar way we compute $H_{2}(t)$:

$\displaystyle H_{2}(t)$ $\displaystyle=$ $\displaystyle
e^{itH_{0}}(H_{\textrm{RWA}}-H_{0})e^{-itH_{0}}$ (19) $\displaystyle=$
$\displaystyle e^{itH_{0}}\left(\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda(\sigma_{+}\otimes a+\sigma_{-}\otimes
a^{\dagger})\right)e^{-itH_{0}}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}\sigma_{z}\otimes I+\lambda(\sigma_{+}\otimes
a+\sigma_{-}\otimes a^{\dagger}).$

We notice that $H_{2}$ is time-independent, and again we use the same symbol
for its closure.

####  1.2 Computation of $U_{2}(t)$

Notice that, since $H_{2}$ is time-independent,
$\\{U_{2}(t)\\}_{t\in\mathbb{R}}$ is a unitary group: for all
$t\in\mathbb{R}$,

$U_{2}(t)=e^{itH_{0}}e^{-itH_{\textrm{RWA}}}=e^{-it\left(\frac{\Delta}{2}\sigma_{z}\otimes
I+\lambda(\sigma_{+}\otimes a+\sigma_{-}\otimes
a^{\dagger})\right)}=e^{-itH_{2}}.$ (20)

We observe that $\mathbb{C}^{2}\otimes\SS(\mathbb{R})\subset D(H_{2})$ is a
set of analytic vectors for $K$. Let

$e_{1}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\qquad
e_{2}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$ (21)

be the canonical orthonormal basis of $\mathbb{C}^{2}$. Using the following
identities

$\sigma_{+}e_{1}=\left(\begin{array}[]{c}0\\\
0\end{array}\right),\quad\sigma_{+}e_{2}=e_{1},\quad\sigma_{-}e_{1}=e_{2},\quad\sigma_{-}e_{2}=\left(\begin{array}[]{c}0\\\
0\end{array}\right),$ (22)

we compute the even and the odd powers of $H_{2}$ on vectors
$e_{1}\otimes\psi$ and $e_{2}\otimes\psi$, with $\psi\in\SS(\mathbb{R})$. For
all $j\in\mathbb{N}$,

$H_{2}^{2j}(e_{1}\otimes\psi)=\sum_{\ell=0}^{j}\left(\begin{array}[]{c}j\\\
\ell\end{array}\right)\lambda^{2\ell}\left(\frac{\Delta}{2}\right)^{2(j-\ell)}e_{1}\otimes(aa^{\dagger})^{\ell}\psi,$
(23)
$H_{2}^{2j}(e_{2}\otimes\psi)=\sum_{\ell=0}^{j}\left(\begin{array}[]{c}j\\\
\ell\end{array}\right)\lambda^{2\ell}\left(\frac{\Delta}{2}\right)^{2(j-\ell)}e_{2}\otimes(a^{\dagger}a)^{\ell}\psi,$
(24)
$H_{2}^{2j+1}(e_{1}\otimes\psi)=\sum_{\ell=0}^{j}\left(\begin{array}[]{c}j\\\
\ell\end{array}\right)\left[\lambda^{2\ell}\left(\frac{\Delta}{2}\right)^{2(j-\ell)+1}e_{1}\otimes(aa^{\dagger})^{\ell}\psi+\lambda^{2\ell+1}\left(\frac{\Delta}{2}\right)^{2(j-\ell)}e_{2}\otimes
a^{\dagger}(aa^{\dagger})^{\ell}\psi\right],$ (25)
$H_{2}^{2j+1}(e_{2}\otimes\psi)=\sum_{\ell=0}^{j}\left(\begin{array}[]{c}j\\\
\ell\end{array}\right)\left[-\lambda^{2\ell}\left(\frac{\Delta}{2}\right)^{2(j-\ell)+1}e_{2}\otimes(a^{\dagger}a)^{\ell}\psi+\lambda^{2\ell+1}\left(\frac{\Delta}{2}\right)^{2(j-\ell)}e_{1}\otimes
a(a^{\dagger}a)^{\ell}\psi\right].$ (26)

######  Lemma 1.1.

$U_{2}(t)(\mathbb{C}^{2}\otimes\SS(\mathbb{R}))\subset\mathbb{C}^{2}\otimes\SS(\mathbb{R})$
for all $t\in\mathbb{R}$.

###### Proof.

By the $N$-representation theorem for $\SS(\mathbb{R})$ [21, Thm.V.13], we
have that $\psi\in\SS(\mathbb{R})$ if and only if for all $m\in\mathbb{N}$:

$\sup_{n\in\mathbb{N}}|\langle\varphi_{n}|\psi\rangle|n^{m}<+\infty,$ (27)

where $\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ is the orthonormal eigenbasis of
the number operator $a^{\dagger}a$, i.e.
$a^{\dagger}a\varphi_{n}=n\varphi_{n}$ for all $n\in\mathbb{N}$. We have that
for all $n\in\mathbb{N}$ and $t\in\mathbb{R}$:

$U_{2}(t)(e_{1}\otimes\varphi_{n})=e_{1}\otimes
a_{n}(t)\varphi_{n}+e_{2}\otimes b_{n}(t)\varphi_{n+1}$ (28)

and

$U_{2}(t)(e_{2}\otimes\varphi_{n})=e_{1}\otimes
c_{n}(t)\varphi_{n-1}+e_{2}\otimes d_{n}(t)\varphi_{n}$ (29)

where

$a_{n}(t)=\cos\left(t\sqrt{\lambda^{2}(n+1)+\left(\frac{\Delta}{2}\right)^{2}}\right)-\frac{i\Delta}{2}\frac{\sin\left(t\sqrt{\lambda^{2}(n+1)+\left(\frac{\Delta}{2}\right)^{2}}\right)}{\sqrt{\lambda^{2}(n+1)+\left(\frac{\Delta}{2}\right)^{2}}},$
(30)
$b_{n}(t)=-i\lambda\sqrt{n+1}\frac{\sin\left(t\sqrt{\lambda^{2}(n+2)+\left(\frac{\Delta}{2}\right)^{2}}\right)}{\sqrt{\lambda^{2}(n+2)+\left(\frac{\Delta}{2}\right)^{2}}},$
(31)
$c_{n}(t)=-i\lambda\sqrt{n}\frac{\sin\left(t\sqrt{\lambda^{2}n+\left(\frac{\Delta}{2}\right)^{2}}\right)}{\sqrt{\lambda^{2}n+\left(\frac{\Delta}{2}\right)^{2}}}$
(32)

and

$d_{n}(t)=\cos\left(t\sqrt{\lambda^{2}n+\left(\frac{\Delta}{2}\right)^{2}}\right)+\frac{i\Delta}{2}\frac{\sin\left(t\sqrt{\lambda^{2}n+\left(\frac{\Delta}{2}\right)^{2}}\right)}{\sqrt{\lambda^{2}n+\left(\frac{\Delta}{2}\right)^{2}}}.$
(33)

Let $\psi\in\SS(\mathbb{R})$ and $t\in\mathbb{R}$, then

$U_{2}(t)(e_{1}\otimes\psi)=e_{1}\otimes\psi_{1}+e_{2}\otimes\psi_{2},\quad
U_{2}(t)(e_{2}\otimes\psi)=e_{1}\otimes\psi_{3}+e_{2}\otimes\psi_{4},$ (34)

where

$\psi_{1}=\sum_{n=0}^{+\infty}\langle\varphi_{n}|\psi\rangle
a_{n}(t)\varphi_{n},\quad\psi_{2}=\sum_{n=0}^{+\infty}\langle\varphi_{n}|\psi\rangle
b_{n}(t)\varphi_{n+1},$ (35)

and

$\psi_{3}=\sum_{n=1}^{+\infty}\langle\varphi_{n}|\psi\rangle
c_{n}(t)\varphi_{n-1},\quad\psi_{4}=\sum_{n=0}^{+\infty}\langle\varphi_{n}|\psi\rangle
d_{n}(t)\varphi_{n}.$ (36)

Notice that for all $m\in\mathbb{N}$ and for all $j\in\\{1,2,3,4\\}$:

$\sup_{n\in\mathbb{N}}|\langle\varphi_{n}|\psi_{j}\rangle|n^{m}<+\infty,$ (37)

hence $\psi_{j}\in\SS(\mathbb{R})$ and therefore
$U_{2}(t)(\mathbb{C}\otimes\SS(\mathbb{R}))\subset\mathbb{C}\otimes\SS(\mathbb{R})$.
∎

######  Lemma 1.2.

For all $t\in\mathbb{R}$ and $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$, we
have

$\int_{0}^{t}(H_{2}-H_{1}(s))\Psi\,ds=-\frac{\lambda\sin\left(t\omega\right)}{\omega}\left(e^{it\omega}\sigma_{+}\otimes
a^{\dagger}+e^{-it\omega}\sigma_{-}\otimes a\right)\Psi,$ (38)

and we denote the closure of this operator by $S_{21}(t)$. Moreover:

  * •

$S_{21}(t)(\mathbb{C}^{2}\otimes\SS(\mathbb{R}))\subset\mathbb{C}^{2}\otimes\SS(\mathbb{R})$
for all $t\in\mathbb{R}$;

  * •

for all $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$ and for all
$t\in\mathbb{R}$: $\frac{d}{dt}S_{21}(t)\Psi=(H_{2}(t)-H_{1}(t))\Psi$.

###### Proof.

On $\mathbb{C}^{2}\otimes\SS(\mathbb{R})$, we have

$\displaystyle S_{21}(t)$ $\displaystyle:=$
$\displaystyle\int_{0}^{t}(H_{2}-H_{1}(s))\,ds$ (39) $\displaystyle=$
$\displaystyle-\lambda\int_{0}^{t}\left(e^{2is\omega}\sigma_{+}\otimes
a^{\dagger}+e^{-2is\omega}\sigma_{-}\otimes a\right)\,ds$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{2i\omega}\left[e^{2is\omega}\right]_{s=0}^{s=t}\sigma_{+}\otimes
a^{\dagger}+\frac{\lambda}{2i\omega}\left[e^{-2is\omega}\right]_{s=0}^{s=t}\sigma_{-}\otimes
a$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{2i\omega}\left(e^{2it\omega}-1\right)\sigma_{+}\otimes
a^{\dagger}+\frac{\lambda}{2i\omega}\left(e^{-2it\omega}-1\right)\sigma_{-}\otimes
a$ $\displaystyle=$
$\displaystyle-\frac{\lambda\sin\left(t\omega\right)}{\omega}\left(e^{it\omega}\sigma_{+}\otimes
a^{\dagger}+e^{-it\omega}\sigma_{-}\otimes a\right).$

∎

######  Lemma 1.3.

For all $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$:

$\displaystyle i(U_{2}(t)-U_{1}(t))\Psi$ $\displaystyle=$ $\displaystyle
S_{21}(t)U_{2}(t)\Psi+$
$\displaystyle+i\int_{0}^{t}U_{1}(t)U_{1}(s)^{\dagger}(S_{21}(s)H_{2}-H_{1}(s)S_{21}(s))U_{2}(s)\Psi\,ds,$

###### Proof.

Let $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$, by Lemmas 1.1 and 1.2 we
have that for all $s\in\mathbb{R}$:

$U_{2}(s)\Psi,S_{21}(s)H_{2}U_{2}(s)\Psi,H_{1}(s)S_{21}(s)U_{2}(s)\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R}).$
(41)

By equation (13) we have that

$\displaystyle i(U_{2}(t)-U_{1}(t))\Psi$ $\displaystyle=$ $\displaystyle
iU_{1}(t)(U_{1}(t)^{\dagger}U_{2}(t)-I)\Psi$ (42) $\displaystyle=$
$\displaystyle iU_{1}(t)\left[U_{1}(s)^{\dagger}U_{2}(s)\right]_{0}^{t}\Psi$
$\displaystyle=$ $\displaystyle
iU_{1}(t)\int_{0}^{t}\frac{d}{ds}U_{1}(s)^{\dagger}U_{2}(s)\Psi\,ds$
$\displaystyle=$ $\displaystyle
U_{1}(t)\int_{0}^{t}U_{1}(s)^{\dagger}(H_{2}-H_{1}(s))U_{2}(s)\Psi\,ds.$

We observe that for all $s\in\mathbb{R}$:

$\displaystyle\frac{d}{ds}\left(U_{1}(s)^{\dagger}S_{21}(s)U_{2}(s)\Psi\right)$
$\displaystyle=$ $\displaystyle
iU_{1}(s)^{\dagger}(H_{1}(s)S_{21}(s)-S_{21}(s)H_{2})U_{2}(s)\Psi+$ (43)
$\displaystyle+U_{1}(s)^{\dagger}(H_{2}-H_{1}(s))U_{2}(s)\Psi,$

therefore

$\displaystyle i(U_{2}(t)-U_{1}(t))\Psi$ $\displaystyle=$ $\displaystyle
U_{1}(t)\int_{0}^{t}U_{1}(s)^{\dagger}(H_{2}-H_{1}(s))U_{2}(s)\Psi\,ds$ (44)
$\displaystyle=$ $\displaystyle S_{21}(t)U_{2}(t)\Psi$
$\displaystyle+i\int_{0}^{t}U_{1}(t)U_{1}(s)^{\dagger}(S_{21}(s)H_{2}-H_{1}(s)S_{21}(s))U_{2}(s)\Psi\,ds.$

∎

Notice that a similar Lemma might hold with $U_{1}(t)$ and $U_{2}(t)$
interchanged. This would however be much harder to prove, as our current prove
relies on the simple structure of the Jaynes-Cummings interaction through
Lemma 1.1.

###  2 Computation of bounds and the rotating wave approximation

Without loss of generality, we can assume $\lambda,\Omega,\omega>0$.

####  2.1 Upper bound for generic vectors

######  Theorem 2.1.

For all $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$ and $t\in\mathbb{R}$:

$\left\|(U_{2}(t)-U_{1}(t))\Psi\right\|\leq\frac{\lambda}{\omega}\biggl{[}\|(N+2)^{1/2}\Psi\\\
|+|t|\Bigl{(}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|\bigl{(}(N+2)(N+3)\bigr{)}^{1/2}\Psi\|\Bigr{)}\biggr{]},$
(45)

where $N=I\otimes a^{\dagger}a$. Moreover for all
$\Psi\in\mathbb{C}^{2}\otimes L^{2}(\mathbb{R})$:

$\lim_{\omega\to+\infty}\left\|(e^{-itH_{\textrm{RWA}}}-e^{-itH})\Psi\right\|=0,$
(46)

uniformly for $t$ in compact sets.

This theorem proves (4). In particular, (46) shows that mathematically the
rotating wave approximation is correct in the limit $\omega\to\infty$. How
appropriate the approximation is in practice with finite parameters can be
computed in (45), which provides a concrete upper bound on the norm difference
of the time evolution of an initial state under the actual time evolution and
under the rotating wave approximation.

###### Proof.

First we prove (45). Let $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$ and
$t\in\mathbb{R}$. First of all we observe that

$\|(I\otimes a)\Psi\|^{2}=\langle(I\otimes a)\Psi|(I\otimes
a)\Psi\rangle=\langle\Psi|(I\otimes
a^{\dagger}a)\Psi\rangle=\langle\Psi|N\Psi\rangle=\|N^{1/2}\Psi\|^{2},$ (47)
$\|(I\otimes
a^{\dagger})\Psi\|^{2}=\langle\Psi|(N+1)\Psi\rangle=\|(N+1)^{1/2}\Psi\|^{2}.$
(48)

Moreover, the conservation law

$[H_{2},\mathcal{N}]=0,\qquad\mathcal{N}=P_{+}+N,\qquad
P_{+}=\sigma_{+}\sigma_{-}\otimes I,\qquad$ (49)

implies that

$\displaystyle U_{2}(t)^{\dagger}NU_{2}(t)$ $\displaystyle=$ $\displaystyle
U_{2}(t)^{\dagger}\mathcal{N}U_{2}(t)-U_{2}(t)^{\dagger}P_{+}U_{2}(t)=\mathcal{N}-U_{2}(t)^{\dagger}P_{+}U_{2}(t)$
$\displaystyle=$ $\displaystyle
N+\bigl{(}P_{+}-U_{2}(t)^{\dagger}P_{+}U_{2}(t)\bigr{)}$ $\displaystyle\leq$
$\displaystyle N+1$

on $\mathbb{C}^{2}\otimes\SS(\mathbb{R})$.

Start from the equality

$\bigl{(}U_{2}(t)-U_{1}(t)\bigr{)}\Psi=-iS_{21}(t)U_{2}(t)\Psi+\int_{0}^{t}U_{1}(t)U_{1}(s)^{\dagger}\bigl{(}S_{21}(s)H_{2}-H_{1}(s)S_{21}(s)\bigr{)}U_{2}(s)\Psi\,ds\,.$
(51)

Let $u(t)=U_{2}(t)\Psi$. One gets

$\displaystyle\|S_{21}(t)U_{2}(t)\Psi\|^{2}$ $\displaystyle=$
$\displaystyle\langle S_{21}(t)u(t)|S_{21}(t)u(t)\rangle$ (52)
$\displaystyle=$
$\displaystyle\left(\frac{\lambda\sin\left(t\omega\right)}{\omega}\right)^{2}\langle
u(t)|\bigl{(}\sigma_{+}\sigma_{-}\otimes
a^{{\dagger}}a+\sigma_{-}\sigma_{+}\otimes aa^{\dagger}\bigr{)}u(t)\rangle$
$\displaystyle=$
$\displaystyle\left(\frac{\lambda\sin\left(t\omega\right)}{\omega}\right)^{2}\|\bigl{(}\sigma_{+}\sigma_{-}\otimes
n^{1/2}+\sigma_{-}\sigma_{+}\otimes(n+1)^{1/2}\bigr{)}u(t)\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{\lambda^{2}}{\omega^{2}}\|(N+1)^{1/2}u(t)\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{\lambda^{2}}{\omega^{2}}\|(N+2)^{1/2}\Psi\|^{2}.$

Moreover, let

$V(t)=H_{2}-H_{1}(t)=-\lambda\left(e^{2it\omega}\sigma_{+}\otimes
a^{{\dagger}}+e^{-2it\omega}\sigma_{-}\otimes a\right).$ (53)

Then

$\displaystyle X(t)$ $\displaystyle=$ $\displaystyle
S_{21}(t)H_{2}-H_{1}(t)S_{21}(t)$ (54) $\displaystyle=$
$\displaystyle[S_{21}(t),H_{2}]+V(t)S_{21}(t)$ $\displaystyle=$
$\displaystyle-\frac{\lambda\sin\left(t\omega\right)}{\omega}\biggl{[}\Delta\left(-e^{it\omega}\sigma_{+}\otimes
a^{{\dagger}}+e^{-it\omega}\sigma_{-}\otimes a\right)$
$\displaystyle\qquad\qquad\quad\;\;+\lambda\sigma_{+}\sigma_{-}\otimes\left(e^{it\omega}a^{{\dagger}2}-e^{-it\omega}a^{2}-e^{it\omega}a^{{\dagger}}a\right)$
$\displaystyle\qquad\qquad\quad\;\;+\lambda\sigma_{-}\sigma_{+}\otimes\left(-e^{it\omega}a^{{\dagger}2}+e^{-it\omega}a^{2}-e^{-it\omega}aa^{{\dagger}}\right)\biggr{]}.$

We want to estimate $\|X(t)u(t)\|$. First we observe that

$\|\left(-e^{it\omega}\sigma_{+}\otimes
a^{{\dagger}}+e^{-it\omega}\sigma_{-}\otimes
a\right)u(t)\|\leq\|(N+1)^{1/2}u(t)\|.$ (55)

Moreover for all $\psi\in\SS(\mathbb{R})$:

$\displaystyle\|\left(e^{it\omega}a^{{\dagger}2}-e^{-it\omega}a^{2}-e^{it\omega}a^{{\dagger}}a\right)\psi\|$
$\displaystyle\leq$
$\displaystyle\|a^{{\dagger}2}\psi\|+\|a^{2}\psi\|+\|a^{{\dagger}}a\psi\|$
(56) $\displaystyle\leq$ $\displaystyle
3\|((a^{\dagger}a+1)(a^{\dagger}a+2))^{1/2}\psi\|$

and

$\displaystyle\|\left(-e^{it\omega}a^{{\dagger}2}+e^{-it\omega}a^{2}-e^{-it\omega}aa^{{\dagger}}\right)\psi\|$
$\displaystyle\leq$
$\displaystyle\|a^{{\dagger}2}\psi\|+\|a^{2}\psi\|+\|aa^{{\dagger}}a\psi\|$
(57) $\displaystyle\leq$ $\displaystyle
3\|((a^{\dagger}a+1)(a^{\dagger}a+2))^{1/2}\psi\|.$

Therefore,

$\displaystyle\|X(t)u(t)\|$ $\displaystyle\leq$
$\displaystyle\frac{\lambda}{\omega}\Bigl{[}|\Delta|\|(N+1)^{1/2}u(t)\|+3\lambda\|((N+1)(N+2))^{1/2}u(t)\|\Bigr{]}$
(58) $\displaystyle\leq$
$\displaystyle\frac{\lambda}{\omega}\Bigl{[}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|((N+2)(N+3))^{1/2}\Psi\|\Bigr{]}.$

Taking things together we get

$\left\|(U_{2}(t)-U_{1}(t))\Psi\right\|\leq\frac{\lambda}{\omega}\biggl{[}\|(N+2)^{1/2}\Psi\|+|t|\Bigl{(}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|\bigl{(}(N+2)(N+3)\bigr{)}^{1/2}\Psi\|\Bigr{)}\biggr{]}.$
(59)

Therefore

$\lim_{\omega\to+\infty}\left\|(e^{-itH_{\textrm{RWA}}}-e^{-itH})\Psi\right\|=\lim_{\omega\to+\infty}\left\|(U_{2}(t)-U_{1}(t))\Psi\right\|=0,$
(60)

for all $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$, and since the latter is
dense in $\mathbb{C}^{2}\otimes L^{2}(\mathbb{R})$, (46) follows.

∎

####  2.2 Lower bound

######  Theorem 2.2.

For all $\Psi\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$ and for all $0\leq
t\leq\pi/\omega$:

$\displaystyle\|(U_{2}(t)-U_{1}(t))\Psi\|$ $\displaystyle\geq$
$\displaystyle\frac{\lambda}{\omega}\sin(t\omega)\|(N-1)_{+}^{1/2}\Psi\|$
$\displaystyle-\frac{\lambda}{\omega^{2}}(1-\cos(t\omega))\Bigl{(}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|\bigl{(}(N+2)(N+3)\bigr{)}^{1/2}\Psi\|\Bigr{)},$

where $(N-1)_{+}$ denotes the positive part of the operator $N-1$.

This theorem reveals the limitation of the rotating wave approximation in
practice: the error grows with the photon number, so for larger systems the
rotating wave approximation may no longer be justified.

###### Proof.

Start from the equality

$\bigl{(}U_{2}(t)-U_{1}(t)\bigr{)}\Psi=-iS_{21}(t)U_{2}(t)\Psi+\int_{0}^{t}U_{1}(t)U_{1}(s)^{\dagger}\bigl{(}S_{21}(s)H_{2}-H_{1}(s)S_{21}(s)\bigr{)}U_{2}(s)\Psi\,ds\,.$
(61)

We have that

$\displaystyle\|(U_{2}(t)-U_{1}(t))\Psi\|$ $\displaystyle\geq$
$\displaystyle\|S_{21}(t)\Psi\|-\int_{0}^{t}\|(S_{21}(s)H_{2}-H_{1}(s)S_{21}(s))U_{2}(s)\Psi\|\,ds$

We define $u(t)=U_{2}(t)\Psi$ and we have

$\displaystyle\left\|S_{21}(t)U_{2}(t)\Psi\right\|^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda\sin\left(t\omega\right)}{\omega}\right)^{2}\langle
u(t)|\bigl{(}\sigma_{+}\sigma_{-}\otimes
a^{{\dagger}}a+\sigma_{-}\sigma_{+}\otimes aa^{{\dagger}}\bigr{)}u(t)\rangle$
(63) $\displaystyle\geq$
$\displaystyle\left(\frac{\lambda\sin\left(t\omega\right)}{\omega}\right)^{2}\|N^{1/2}u(t)\|^{2},$

moreover, by (2.1), we have

$U_{2}(t)^{\dagger}NU_{2}(t)\geq N-1.$ (64)

Hence, one can show that

$\left\|S_{21}(t)U_{2}(t)\Psi\right\|^{2}\geq\left(\frac{\lambda\sin\left(t\omega\right)}{\omega}\right)^{2}\|(N-1)_{+}^{1/2}\Psi\|^{2}.$
(65)

Moreover, for $0\leq t\leq\pi/\omega$ we have

$\displaystyle\int_{0}^{t}\|(S_{21}(s)H_{2}-H_{1}(s)S_{21}(s))U_{2}(s)\Psi\|\,ds$
$\displaystyle\leq\frac{\lambda}{\omega}\Bigl{[}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|((N+2)(N+3))^{1/2}\Psi\|\Bigr{]}\int_{0}^{t}\sin(\omega
s)\,ds$ $\displaystyle=\frac{\lambda(1-\cos(\omega
t))}{\omega^{2}}\Bigl{[}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|((N+2)(N+3))^{1/2}\Psi\|\Bigr{]}$
(66)

and hence, for $0\leq t\leq\pi/\omega$, we get

$\displaystyle\|(U_{2}(t)-U_{1}(t))\Psi\|$ $\displaystyle\geq$
$\displaystyle\|S_{21}(t)\Psi\|-\int_{0}^{t}\|(S_{21}(s)H_{2}-H_{1}(s)S_{21}(s))U_{2}(s)\Psi\|\,ds$
$\displaystyle\geq$
$\displaystyle\frac{\lambda}{\omega}\sin(t\omega)\|(N-1)_{+}^{1/2}\Psi\|$
$\displaystyle-\frac{\lambda}{\omega^{2}}(1-\cos(t\omega))\Bigl{(}|\Delta|\|(N+2)^{1/2}\Psi\|+3\lambda\|\bigl{(}(N+2)(N+3)\bigr{)}^{1/2}\Psi\|\Bigr{)}.$

∎

####  2.3 Applying the bounds to Fock states

To understand the scaling better, let us apply the above bounds to
(normalised) Fock states
$\Phi_{j,n}=e_{j}\otimes\varphi_{n}\in\mathbb{C}^{2}\otimes\SS(\mathbb{R})$,
with $j\in\\{1,2\\}$ and $n\in\mathbb{N}$, $n>0$. For simplicity, we consider
the case $\Delta=0$, but the argument is easily generalised. From the upper
bound (45) we obtain

$\left\|(U_{2}(t)-U_{1}(t))\Phi_{j,n}\right\|\leq\frac{\lambda(n+2)^{1/2}}{\omega}\biggl{[}1+3|t|\lambda(n+3)^{1/2}\biggr{]}.$
(68)

For the lower bound we have, for $0\leq t\leq\pi/\omega$,

$\displaystyle\|(U_{2}(t)-$ $\displaystyle
U_{1}(t))\Phi_{j,n}\|\geq\frac{\lambda}{\omega}\sin(t\omega)(n-1)^{1/2}-\frac{3\lambda^{2}}{\omega^{2}}(1-\cos(t\omega))\bigl{(}(n+2)(n+3)\bigr{)}^{1/2}.$
(69)

We compute that this as a function of $t$ has a maximum at

$t_{*}=\frac{\cos^{-1}\left(\frac{3\lambda}{\sqrt{9\lambda^{2}+\frac{(n-1)\omega^{2}}{(n+2)(n+3)}}}\right)}{\omega}$
(70)

At this time, the right hand side of Eq. (69) evaluates as

$\frac{g\left(9g^{2}(n+2)(n+3)-3g\sqrt{(n+2)(n+3)\left(9g^{2}(n+2)(n+3)+n-1\right)}+n-1\right)}{\sqrt{9g^{2}(n+2)(n+3)+n-1}}.$
(71)

Here, we have set $g\equiv\frac{\lambda}{\omega}.$

We can expand this in order of $n^{-1}$ to obtain a slightly simpler exact
lower bound (valid for $n>0$)

$\displaystyle\sup_{t\in[0,\frac{\pi}{\omega}]}\|(U_{2}(t)-$ $\displaystyle
U_{1}(t))\Phi_{j,n}\|\geq\frac{1}{6}-\frac{1}{216g^{2}n}-\frac{7}{12n}.$ (72)

Focussing on the same time interval also for the upper bound and linearising
it, we can conclude that

$\displaystyle 5g\sqrt{n+3}\geq\sup_{t\in[0,\frac{\pi}{\omega}]}\|(U_{2}(t)-$
$\displaystyle
U_{1}(t))\Phi_{j,n}\|\geq\frac{1}{6}-\frac{1}{216g^{2}n}-\frac{7}{12n},$ (73)

proving (5). This bound is not necessarily a sharp bound but it shows nicely
that the short-time error becomes small for small $g$ and large for high
photon number $n$, hence it provides us with a quantitative condition on $g$
in order to reduce the error below a certain bound for given photon number.
For high photon numbers $n\to\infty$, there is a time such that the difference
becomes greater than $\frac{1}{6}$, i.e.,

$\|e^{-itH}-e^{-itH_{\textrm{RWA}}}\|\geq\frac{1}{6}$ (74)

by taking the supremum over all states in (72), which means that the rotating
wave approximation does not work for arbitrarily high photon numbers; this
proves (3).

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