Compact-time validity of the rotating wave approximation
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Problem
Does the Jaynes–Cummings evolution approximate the Rabi evolution strongly and uniformly on each compact time interval in the high-frequency limit? Work on \(\mathbb C^2\otimes L^2(\mathbb R)\). Let \(a,a^\dagger\) be oscillator operators with \([a,a^\dagger]=1\) and \(N=a^\dagger a\). Write \(\sigma_x,\sigma_y,\sigma_z\) for the Pauli matrices and \(\sigma_\pm=(\sigma_x\pm i\sigma_y)/2\). Fix \(\lambda>0\) and \(\Delta\in\mathbb R\). For \(\omega>\max\{0,-\Delta\}\), define
Tensor factors of the identity are implicit. Use the self-adjoint closures of Eq. (1) from \(\mathbb C^2\otimes\mathcal S(\mathbb R)\). Here \(\mathcal S(\mathbb R)\) is the Schwartz space. Is it true that every normalized \(\psi\in\mathbb C^2\otimes L^2(\mathbb R)\) and every finite \(T>0\) satisfy
The vector, time horizon, coupling, and detuning in Eq. (2) are fixed as \(\omega\) grows.
Source
This compact-time formulation follows from the rotating-wave question resolved by Burgarth, Facchi, Hillier, and Ligabò, Theorem 2.1 [BFHL24]. It makes explicit the uniformity that follows from their time-dependent estimate.
Progress
Theorem 2.1, Eq. (45), gives for every \(\psi\in\mathbb C^2\otimes\mathcal S(\mathbb R)\)
\begin{equation} \begin{aligned} \|(e^{-itH_\omega}-e^{-itJ_\omega})\psi\| \leq\frac{\lambda}{\omega}\bigl[ &(1+|t||\Delta|)\|(N+2)^{1/2}\psi\|\\ &+3|t|\lambda\|((N+2)(N+3))^{1/2}\psi\|\bigr]. \end{aligned} \tag{3} \end{equation}Equation (3) tends to zero uniformly for \(|t|\leq T\). The interaction-picture change used in the paper leaves the norm difference unchanged [BFHL24].
For an arbitrary normalized \(\psi\), choose a Schwartz vector \(\phi\) close to it. Unitarity bounds the contribution of \(\psi-\phi\) by \(2\|\psi-\phi\|\), independently of \(\omega\) and \(t\). Apply Eq. (3) to \(\phi\) and then let \(\|\psi-\phi\|\) tend to zero. This proves Eq. (2) without an energy assumption on \(\psi\).
Comment
The resolving theorem is peer-reviewed. The compact time interval is essential to the stated consequence of Eq. (3). It does not assert an error tending to zero uniformly over all times or all normalized input vectors.