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Xehetasun bibliografikoak
Egile nagusia:	Singh Khalsa, Sardar Dilbag
Formatua:	Recurso digital
Hizkuntza:
Argitaratua:	Zenodo 2025
Gaiak:	Quantum physics Adiabatic perturbation Theoretical physics
Sarrera elektronikoa:	https://doi.org/10.5281/zenodo.15687285
Etiketak:	Etiketa erantsi Etiketarik gabe, Izan zaitez lehena erregistro honi etiketa jartzen!

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\documentclass{article} \usepackage{graphicx} % Required for inserting images \usepackage{authblk} % For \affiliation command \title{Adiabatic method to compute ground state configuration: An Optimisation Problem} \author{Sardar Dilbag Singh Khalsa}   \begin{document} \maketitle \begin{document} \textbf{School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Khordha 752050 Odisha, India} \begin{center}     {\Large \textbf{Adiabatic Hamiltonian Evolution in Quantum Electrodynamics}}\\[1em]           \end{center} \vspace{1em} \section*{Problem Statement} Consider a system described by Quantum Electrodynamics (QED) on a background of a slowly varying (classical) electromagnetic field $A_{\mu}(t)$ in the Coulomb gauge (or another suitable gauge of your choice). The full Hamiltonian of the fermion-photon system can be written schematically as \[     \hat{H}(t) \;=\; \hat{H}_{0} + \hat{H}_\mathrm{int}\!\bigl(A_{\mu}(t)\bigr), \] where \begin{itemize}     \item $\hat{H}_{0}$ is the free Hamiltonian (free Dirac field + free photon field),     \item $\hat{H}_\mathrm{int}\!\bigl(A_{\mu}(t)\bigr)$ is the interaction term coupling the fermion (electron) field to the (classical) background field $A_\mu(t)$ (e.g.\ the minimal coupling $ \bar{\psi}\gamma^\mu \psi \,A_\mu(t)$ in the interaction representation, ignoring self-interactions for simplicity). \end{itemize} Suppose the background field $A_{\mu}(t)$ evolves very slowly compared to typical QED timescales (for instance, the inverse of the electron mass or the inverse of typical photon frequencies in the system). More concretely, let $\tau$ be a characteristic timescale for the external field variation, such that \[     \tau \; \gg \; \frac{1}{m_e c^2} \quad \text{and} \quad \tau \; \gg \; \frac{1}{\omega_{\gamma}}, \] where $m_e$ is the electron mass, and $\omega_\gamma$ is a characteristic frequency of the relevant photon modes.  In the interaction picture, define the time evolution operator \[     U_I(t, t_0) \;=\; \mathcal{T} \exp \!\Bigl\{         -\, i \int_{t_0}^{t} dt' \,\hat{H}_\mathrm{int}^{(I)}(t')     \Bigr\}, \] where $\hat{H}_\mathrm{int}^{(I)}(t')$ is the interaction Hamiltonian in the interaction picture.  You are to show that in the \emph{adiabatic limit} ($\tau \to \infty$, i.e.\ extremely slow variation of $A_{\mu}(t)$), the evolution of the state is governed by the instantaneous eigenstates of the full Hamiltonian (including $\hat{H}_\mathrm{int}$). In particular, if the system is initially in the ground state (or a particular eigenstate) of $\hat{H}(t_0)$, then to leading order in the slow evolution, the system remains in the instantaneous eigenstate of $\hat{H}(t)$, up to a phase factor.  Concretely, address the following tasks: \begin{enumerate}     \item {\bf Instantaneous eigenstates:} Let $|\Psi_n(t)\rangle$ be an instantaneous eigenstate of $\hat{H}(t)$ with eigenvalue $E_n(t)$:     \[         \hat{H}(t)\,|\Psi_n(t)\rangle \;=\; E_n(t)\,|\Psi_n(t)\rangle.     \]     Describe the adiabatic theorem in the context of QED and discuss why, if the evolution of $A_{\mu}(t)$ is extremely slow, the system remains approximately in $|\Psi_n(t)\rangle$ (assuming it started in $|\Psi_n(t_0)\rangle$) with only a time-dependent phase. What conditions on the energy gaps and the timescale $\tau$ are necessary for this to hold?     \item {\bf Geometrical (Berry) phase:} Identify the Berry connection associated with the slow variation of the background field $A_{\mu}(t)$ and show how one would compute the Berry phase for the path $A_{\mu}(t_0)\to A_{\mu}(t)$ in the parameter space of external electromagnetic field configurations.     \item {\bf Leading-order correction:} Estimate the magnitude of non-adiabatic transitions (i.e.\ transitions to other instantaneous eigenstates) and show that it is suppressed by powers of $1/\tau$ (the inverse of the slow timescale).     \item {\bf Physical interpretation:} Provide a physical interpretation of why the adiabatic approximation applies here (i.e.\ the field acts slowly and the quantum system has time to `adjust’ to its instantaneous configuration). In particular, comment on any subtleties that might arise in a QED context (such as unbounded Fock spaces, vacuum polarization, etc.) and how these subtleties might or might not affect the adiabatic argument in practice. \end{enumerate} \vspace{1em} \section*{Solution Sketch} \noindent \textbf{1. Instantaneous eigenstates and the adiabatic theorem in QED.} \subsection*{Basic adiabatic requirement} The adiabatic theorem states that if a Hamiltonian $\hat{H}(t)$ depends on time so slowly that level crossings are avoided and transitions between different instantaneous eigenstates are negligible, then a system initially in an eigenstate $|\Psi_n(t_0)\rangle$ of $\hat{H}(t_0)$ will remain in the corresponding instantaneous eigenstate $|\Psi_n(t)\rangle$ of $\hat{H}(t)$ for all $t$, up to a phase factor. In formulas, define the instantaneous eigenbasis: \[     \hat{H}(t)\,|\Psi_m(t)\rangle = E_m(t)\,|\Psi_m(t)\rangle. \] The adiabatic condition typically requires \[     \frac{\max_{t}\left| \langle \Psi_m(t) | \frac{d\hat{H}(t)}{dt} | \Psi_n(t) \rangle \right|}{\bigl|E_m(t)-E_n(t)\bigr|^2} \;\ll\; 1, \] for all $m\neq n$. Physically, this means the coupling between eigenstates due to the time-dependence of the Hamiltonian is small compared to the energy separation of those eigenstates, ensuring minimal non-adiabatic transitions. In a QED setting, $E_m(t)$ might label energies of the combined electron-photon system in the presence of the background field $A_\mu(t)$. The slow variation of $A_\mu(t)$ implies a slow variation of the interaction term $\hat{H}_\mathrm{int}(t)$, thus $\frac{d\hat{H}}{dt}$ remains small when $\tau$ is large. \subsection*{Conditions on energy gaps} A crucial requirement is that $E_m(t)-E_n(t)$ remains finite and non-zero for all $m\neq n$ over the entire range of $t$. Level crossings or degeneracies could invalidate the adiabatic assumption by increasing the chance of transitions between states. In practice, for QED in external fields, one needs to ensure no \emph{instanton} or \emph{pair-creation} resonance crosses the energy of the initial state. For moderate field strengths, these pathological scenarios are avoidable; for extremely strong fields (comparable to the Schwinger limit), one must check pair-creation thresholds more carefully. \subsection*{State evolution in the adiabatic regime} If $|\Psi(t)\rangle$ satisfies the time-dependent Schr\"odinger equation, in the adiabatic limit we have \[     |\Psi(t)\rangle \;\approx\; e^{-\,i\int_{t_0}^t E_n(t')\,dt'}     e^{i\,\gamma_n(t)}\,|\Psi_n(t)\rangle, \] where $\gamma_n(t)$ is the Berry phase (geometric phase) discussed below. \vspace{1em} \noindent \textbf{2. Geometrical (Berry) phase in slowly varying QED fields} The Berry phase arises because the instantaneous eigenstates $|\Psi_n(t)\rangle$ themselves depend on the parameters of the Hamiltonian, in this case the background field $A_\mu(t)$, viewed as a point in a space of classical field configurations. Define the Berry connection: \[     \bm{\mathcal{A}}_n(A) \;=\; i \,\langle \Psi_n(A) | \nabla_A \Psi_n(A)\rangle, \] where $\nabla_A$ denotes the functional derivative with respect to $A_\mu(x)$ at each point in spacetime (or, more heuristically, with respect to the small set of external parameters if $A_\mu$ is parameterized by a few coordinates in parameter space). Then the Berry phase associated with traversing a path $A_\mu(t)$ from $t_0$ to $t_1$ is \[     \gamma_n \;=\; \int_{\text{path}} \bm{\mathcal{A}}_n \cdot d\bm{A}. \] In practice, one often uses a simpler parameterization (e.g.\ $A(t) = \lambda(t)\,A^{(\mathrm{shape})}$) and integrates in $\lambda$ space.  Hence, the total phase of the state is the sum of the dynamical phase $-\int E_n(t')\,dt'$ and the geometric (Berry) phase $\gamma_n$. \vspace{1em} \noindent \textbf{3. Leading-order correction and non-adiabatic transitions} To estimate corrections to the adiabatic approximation, one can treat the time-derivative of the Hamiltonian in perturbation theory. The typical scale of non-adiabatic corrections is set by \[     \frac{1}{\tau}\,\frac{1}{E_m - E_n} \quad\text{and higher powers of}\quad \frac{1}{\tau}. \] If $\tau \to \infty$, these corrections vanish at leading order. For finite but large $\tau$, the amplitude for transition from the $n$-th instantaneous eigenstate to a different state $m \neq n$ is roughly \[     c_{m \leftarrow n}      \;\sim\; \frac{\langle \Psi_m(t) | \dot{\hat{H}}(t) | \Psi_n(t) \rangle}{\bigl(E_m(t)-E_n(t)\bigr)^2}. \] Since $\dot{\hat{H}}(t)$ is of order $1/\tau$, this factor is suppressed by $1/\tau$.  Hence, the slower the variation (larger $\tau$), the smaller the probability of non-adiabatic transitions. The leading correction to the wavefunction is of order $1/\tau$. \vspace{1em} \noindent \textbf{4. Physical interpretation and QED subtleties} In QED, the electromagnetic field is dynamical. However, in this problem, $A_\mu(t)$ is treated as a classical external field (i.e.\ not quantized). The approximation that the quantum state of the system follows the instantaneous eigenstate is valid provided: \begin{itemize}     \item The external field changes slowly (large $\tau$).     \item Energy level crossings or resonances (e.g.\ pair creation) are absent in the range of $A_\mu(t)$.     \item The system starts in a well-defined eigenstate of the initial Hamiltonian. \end{itemize} In full QED (with quantized fields), additional subtleties can arise: \begin{itemize}     \item \textit{Vacuum polarization:} The vacuum itself is modified by external fields, leading to renormalized effective couplings and energies. If these changes are also slow, the adiabatic argument can still hold, with suitably redefined instantaneous eigenstates.     \item \textit{Unbounded photon Fock space:} In principle, there are infinitely many photon excitations. Yet for an adiabatic background field, the relevant excitations can still be well-approximated by an instantaneous vacuum or low-lying excitations if the field strength is not excessive. \end{itemize} These do not usually invalidate the adiabatic theorem but can complicate its rigorous proof. The key remains that the timescale of change in $A_\mu$ is slow compared to all intrinsic timescales for excitations, ensuring that transitions between eigenstates remain exponentially or power-law suppressed in $\tau$. \vspace{2em} \hrule \vspace{0.8em} \noindent \textbf{Conclusion:} Under the stated conditions (large $\tau$, no level crossings, and moderate background field strengths), the system will stay in the corresponding instantaneous eigenstate of the time-dependent QED Hamiltonian, accumulating only a dynamical phase and the Berry (geometric) phase associated with the slow evolution of $A_\mu(t)$. Non-adiabatic corrections scale down with $1/\tau$, justifying the adiabatic approximation for slowly varying electromagnetic fields. \vspace{1em} \hrule \vspace{1em} \begin{center}      \end{center} {\it Acknowledgement:~} We acknowledges CSIR (Grant No. 09/1059(11052)/2021-EMR-I ) for funding to support this research.I am thankful to IIT Bhubaneswar for providing an advanced lab with a high-end computational research facility.\\ {\it E: mail :  s21ph09010@iitbbs.ac.in}\\ {\it Declaration: I Declare no competing interest between authors and all the guidelines of publishing ethics are followed} \end{document} \end{document}

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