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Adiabatic theorem
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== Mathematical statement == Under a slowly changing Hamiltonian <math>H(t)</math> with instantaneous eigenstates <math>| n(t) \rangle</math> and corresponding energies <math>E_n(t)</math>, a quantum system evolves from the initial state <math display="block">| \psi(0) \rangle = \sum_n c_n(0) | n(0) \rangle</math> to the final state <math display="block">| \psi(t) \rangle = \sum_n c_n(t) | n(t) \rangle ,</math> where the coefficients undergo the change of phase <math display="block">c_n(t) = c_n(0) e^{i \theta_n(t)} e^{i \gamma_n(t)}</math> with the '''dynamical phase''' <math display="block">\theta_m(t) = -\frac{1}{\hbar} \int_0^t E_m(t') dt'</math> and '''[[geometric phase]]''' <math display="block">\gamma_m(t) = i \int_0^t \langle m(t') | \dot{m}(t') \rangle dt' .</math> In particular, <math>|c_n(t)|^2 = |c_n(0)|^2</math>, so if the system begins in an eigenstate of <math>H(0)</math>, it remains in an eigenstate of <math>H(t)</math> during the evolution with a change of phase only. === Proofs === :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Sakurai in ''Modern Quantum Mechanics''<ref name="Modern Quantum Mechanics">{{Cite book|last1=Sakurai|first1=J. J.| url=https://www.cambridge.org/highereducation/books/modern-quantum-mechanics/DF43277E8AEDF83CC12EA62887C277DC#contents |title=Modern Quantum Mechanics |last2=Napolitano|first2=Jim |date=2020-09-17 |publisher=Cambridge University Press| isbn=978-1-108-58728-0| edition=3 |doi=10.1017/9781108587280|bibcode=2020mqm..book.....S }}</ref> |- | This proof is partly inspired by one given by Sakurai in ''Modern Quantum Mechanics''.<ref name="Modern Quantum Mechanics"/> The instantaneous eigenstates <math>| n(t) \rangle</math> and energies <math>E_n(t)</math>, by assumption, satisfy the time-independent Schrödinger equation <math display="block">H(t) | n(t) \rangle = E_n(t) | n(t) \rangle</math> at all times <math>t</math>. Thus, they constitute a basis that can be used to expand the state <math display="block">| \psi(t) \rangle = \sum_n c_n(t) | n(t) \rangle</math> at any time <math>t</math>. The evolution of the system is governed by the time-dependent Schrödinger equation <math display="block">i \hbar |\dot{\psi}(t) \rangle = H(t) | \psi(t) \rangle,</math> where <math>\dot{} = d / dt</math> (see {{slink|Notation for differentiation|Newton's notation}}). Insert the expansion of <math>| \psi(t) \rangle</math>, use <math>H(t) | n(t) \rangle = E_n(t) | n(t) \rangle</math>, differentiate with the product rule, take the inner product with <math>| m(t) \rangle</math> and use orthonormality of the eigenstates to obtain <math display="block">i \hbar \dot{c}_m(t) + i \hbar \sum_n c_n(t) \langle m(t) | \dot{n}(t) \rangle = c_m(t) E_m(t) .</math> This coupled first-order differential equation is exact and expresses the time-evolution of the coefficients in terms of inner products <math>\langle m(t) | \dot{n} (t) \rangle</math> between the eigenstates and the time-differentiated eigenstates. But it is possible to re-express the inner products for <math>m \neq n</math> in terms of matrix elements of the time-differentiated Hamiltonian <math>\dot{H}(t)</math>. To do so, differentiate both sides of the time-independent Schrödinger equation with respect to time using the product rule to get <math display="block">\dot{H}(t)|n(t)\rangle + H(t)|\dot{n}(t)\rangle = \dot{E}_n(t) |n(t)\rangle + E_n(t) |\dot{n}(t)\rangle .</math> Again take the inner product with <math>| m(t) \rangle</math> and use <math>\langle m(t) | H(t) = E_m(t) \langle m(t) |</math> and orthonormality to find <math display="block">\langle m(t) | \dot{n}(t) \rangle = - \frac{\langle m(t) | \dot{H}(t) | n(t) \rangle}{E_m(t) - E_n(t)} \qquad (m \neq n).</math> Insert this into the differential equation for the coefficients to obtain <math display="block">\dot{c}_m(t) + \left(\frac{i}{\hbar} E_m(t) + \langle m(t) | \dot{m}(t) \rangle \right) c_m(t) = \sum_{n \neq m} \frac{\langle m(t) | \dot{H} | n(t) \rangle}{E_m(t) - E_n(t)} c_n(t).</math> This differential equation describes the time-evolution of the coefficients, but now in terms of matrix elements of <math>\dot{H}(t)</math>. To arrive at the adiabatic theorem, neglect the right hand side. This is valid if the rate of change of the Hamiltonian <math>\dot{H}(t)</math> is small '''and''' there is a finite gap <math>E_m(t) - E_n(t) \neq 0</math> between the energies. This is known as the '''adiabatic approximation'''. Under the adiabatic approximation, <math display="block">\dot{c}_m(t) = i \left(-\frac{E_m(t)}{\hbar} + i \langle m(t) | \dot{m}(t) \rangle \right) c_m(t)</math> which integrates precisely to the adiabatic theorem <math display="block">c_m(t) = c_m(0) e^{i \theta_m(t)} e^{i \gamma_m(t)}</math> with the phases defined in the statement of the theorem. The dynamical phase <math>\theta_m(t)</math> is real because it involves an integral over a real energy. To see that the geometric phase <math>\gamma_m(t)</math> is purely real, differentiate the normalization <math>\langle m(t) | m(t) \rangle = 1</math> of the eigenstates and use the product rule to find that <math display="block">0 = \frac{d}{dt} \Bigl ( \langle m(t) | m(t) \rangle \Bigr ) = \langle \dot{m}(t) | m(t) \rangle + \langle m(t)) | \dot{m}(t) \rangle = \langle m(t)) | \dot{m}(t) \rangle^* + \langle m(t)) | \dot{m}(t) \rangle = 2 \, \operatorname{Re} \Bigl ( \langle m(t)) | \dot{m}(t) \rangle \Bigr ) . </math> Thus, <math>\langle m(t)) | \dot{m}(t) \rangle </math> is purely imaginary, so the geometric phase <math>\gamma_m(t) </math> is purely real. |} :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Adiabatic approximation<ref name="Zwiebach">{{Cite web |last=Zwiebach |first=Barton |url=https://www.youtube.com/watch?v=pgEFvhkEp-c |archive-url=https://ghostarchive.org/varchive/youtube/20211221/pgEFvhkEp-c |archive-date=2021-12-21 |url-status=live| title=L16.1 Quantum adiabatic theorem stated| date=Spring 2018| publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref><ref name="MIT 8.06 Quantum Physics III">{{Cite web|title=MIT 8.06 Quantum Physics III| url=https://ocw.mit.edu/8-06S18}}</ref> |- | Proof with the details of the adiabatic approximation<ref name="Zwiebach"/><ref name="MIT 8.06 Quantum Physics III"/> We are going to formulate the statement of the theorem as follows: : For a slowly varying Hamiltonian <math>\hat{H}</math> in the time range T the solution of the Schrödinger equation <math>\Psi(t)</math> with initial conditions <math>\Psi(0) = \psi_{n}(0)</math> : where <math>\psi_{n}(t)</math> is the eigenvector of the instantaneous Schrödinger equation <math>\hat{H}(t)\psi_{n}(t)=E_{n}(t)\psi_{n}(t)</math> can be approximated as: <math display="block">\left\| {\Psi(t)-\psi_\text{adiabatic}(t)} \right\| \approx O(\frac{1}{T})</math> where the adiabatic approximation is: <math display="block"> |\psi_\text{adiabatic}(t)\rangle = e^{i\theta_{n}(t)}e^{i\gamma_{n}(t)}|\psi_n(t)\rangle</math> and <math display="block">\theta_{n}(t) = - \frac{1}{\hbar} \int_{0}^{t}E_{n}(t') dt'</math> <math display="block">\gamma_{n}(t) = \int_{0}^{t}\nu_{n}(t')dt'</math> also called [[Berry phase]] <math display="block">\nu_{n}(t) = i \langle\psi_{n}(t) | \dot{\psi}_{n}(t)\rangle</math> And now we are going to prove the theorem. Consider the ''time-dependent'' [[Schrödinger equation]] <math display="block">i \hbar{\partial \over \partial t} |\psi(t)\rangle = \hat{H}(\tfrac{t}{T}) |\psi(t)\rangle</math> with [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>\hat{H}(t).</math> We would like to know the relation between an initial state <math>|\psi(0)\rangle</math> and its final state <math>|\psi(T)\rangle</math> at <math>t = T</math> in the adiabatic limit <math>T \to \infty.</math> First redefine time as <math>\lambda = \tfrac{t}{T} \in [0,1]</math>: <math display="block">i \hbar{\partial \over \partial \lambda} |\psi(\lambda)\rangle = T \hat{H}(\lambda) |\psi(\lambda)\rangle.</math> At every point in time <math>\hat{H}(\lambda)</math> can be diagonalized <math>\hat H(\lambda)|\psi_n(\lambda)\rangle = E_n(\lambda)|\psi_n(\lambda)\rangle</math> with eigenvalues <math>E_n</math> and eigenvectors <math>|\psi_n(\lambda)\rangle</math>. Since the eigenvectors form a complete basis at any time we can expand <math>|\psi(\lambda)\rangle</math> as: <math display="block"> |\psi(\lambda)\rangle = \sum_n c_n(\lambda)|\psi_n(\lambda)\rangle e^{iT\theta_n(\lambda)},</math> where <math display="block">\theta_n(\lambda) = -\frac{1}{\hbar}\int_0^\lambda E_n(\lambda')d\lambda'.</math> The phase <math>\theta_n(t)</math> is called the ''dynamic phase factor''. By substitution into the Schrödinger equation, another equation for the variation of the coefficients can be obtained: <math display="block">i \hbar \sum_n (\dot{c}_n|\psi_n\rangle + c_n|\dot{\psi}_n\rangle + i c_n|\psi_n\rangle T\dot{\theta}_n)e^{iT\theta_n} = \sum_n c_n T E_n|\psi_n\rangle e^{iT\theta_n}.</math> The term <math>\dot{\theta}_n</math> gives <math>-E_n/\hbar</math>, and so the third term of left side cancels out with the right side, leaving <math display="block">\sum_n \dot{c}_n|\psi_n\rangle e^{iT\theta_n} = -\sum_n c_n|\dot{\psi}_n\rangle e^{iT\theta_n}.</math> Now taking the inner product with an arbitrary eigenfunction <math>\langle\psi_m|</math>, the <math>\langle\psi_m|\psi_n\rangle</math> on the left gives <math>\delta_{nm}</math>, which is 1 only for ''m'' = ''n'' and otherwise vanishes. The remaining part gives <math display="block">\dot{c}_m = -\sum_n c_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)}.</math> For <math>T \to \infty</math> the <math>e^{iT(\theta_n-\theta_m)}</math> will oscillate faster and faster and intuitively will eventually suppress nearly all terms on the right side. The only exceptions are when <math>\theta_n-\theta_m</math> has a critical point, i.e. <math>E_n(\lambda) = E_m(\lambda)</math>. This is trivially true for <math>m = n</math>. Since the adiabatic theorem assumes a gap between the eigenenergies at any time this cannot hold for <math>m \neq n</math>. Therefore, only the <math>m = n</math> term will remain in the limit <math>T \to \infty</math>. In order to show this more rigorously we first need to remove the <math>m = n</math> term. This can be done by defining <math display="block">d_m(\lambda) = c_m(\lambda) e^{\int_0^\lambda\langle\psi_m|\dot{\psi}_m\rangle d\lambda} = c_m(\lambda) e^{-i\gamma_m(\lambda)}.</math> We obtain: <math display="block">\dot{d}_m = -\sum_{n\neq m} d_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)}.</math> This equation can be integrated: <math display="block">\begin{align} d_m(1)-d_m(0) &= -\int_0^1 \sum_{n\neq m} d_n\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)} d\lambda\\ &= -\int_0^1 \sum_{n\neq m} (d_n-d_n(0))\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)} d\lambda - \int_0^1 \sum_{n\neq m} d_n(0)\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)-i(\gamma_m-\gamma_n)}d\lambda \end{align}</math> or written in vector notation <math display="block">\vec{d}(1)-\vec{d}(0) = -\int_0^1 \hat{A}(T, \lambda) (\vec{d}(\lambda)-\vec{d}(0)) d\lambda - \vec{\alpha}(T).</math> Here <math>\hat{A}(T, \lambda)</math> is a matrix and <math display="block">\alpha_m(T) = \int_0^1 \sum_{n\neq m} d_n(0)\langle\psi_m|\dot{\psi}_n\rangle e^{iT(\theta_n-\theta_m)- i(\gamma_m-\gamma_n)}d\lambda</math> is basically a Fourier transform. It follows from the [[Riemann–Lebesgue lemma|Riemann-Lebesgue lemma]] that <math>\vec{\alpha}(T) \to 0 </math> as <math>T \to \infty</math>. As last step take the norm on both sides of the above equation: <math display="block">\Vert\vec{d}(1)- \vec{d}(0)\Vert \leq \Vert\vec{\alpha}(T)\Vert + \int_0^1 \Vert\hat{A}(T, \lambda)\Vert \Vert\vec{d}(\lambda)-\vec{d}(0)\Vert d\lambda</math> and apply [[Grönwall's inequality]] to obtain <math display="block">\Vert\vec{d}(1)-\vec{d}(0)\Vert \leq \Vert\vec{\alpha}(T)\Vert e^{\int_0^1 \Vert\hat{A}(T, \lambda)\Vert d\lambda}.</math> Since <math>\vec{\alpha}(T) \to 0</math> it follows <math>\Vert\vec{d}(1)-\vec{d}(0)\Vert \to 0</math> for <math>T \to \infty</math>. This concludes the proof of the adiabatic theorem. In the adiabatic limit the eigenstates of the Hamiltonian evolve independently of each other. If the system is prepared in an eigenstate <math>|\psi(0)\rangle = |\psi_n(0)\rangle</math> its time evolution is given by: <math display="block">|\psi(\lambda)\rangle = |\psi_n(\lambda)\rangle e^{iT\theta_n(\lambda)}e^{i \gamma_n(\lambda)}.</math> So, for an adiabatic process, a system starting from ''n''th eigenstate also remains in that ''n''th eigenstate like it does for the time-independent processes, only picking up a couple of phase factors. The new phase factor <math>\gamma_n(t)</math> can be canceled out by an appropriate choice of gauge for the eigenfunctions. However, if the adiabatic evolution is [[Berry connection and curvature#Berry phase and cyclic adiabatic evolution|cyclic]], then <math>\gamma_n(t)</math> becomes a gauge-invariant physical quantity, known as the [[Berry phase]]. |} :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Generic proof in parameter space |- | Let's start from a parametric Hamiltonian <math>H(\vec{R}(t))</math>, where the parameters are slowly varying in time, the definition of slow here is defined essentially by the distance in energy by the eigenstates (through the uncertainty principle, we can define a timescale that shall be always much lower than the time scale considered). This way we clearly also identify that while slowly varying the eigenstates remains clearly separated in energy (e.g. also when we generalize this to the case of bands as in the [[TKNN formula]] the bands shall remain clearly separated). Given they do not intersect the states are ordered and in this sense this is also one of the meanings of the name [[topological order]]. We do have the instantaneous Schrödinger equation: <math display="block">H(\vec{R}(t))| \psi_m(t)\rangle = E_m(t)| \psi_m(t)\rangle </math> And instantaneous eigenstates: <math display="block">\langle\psi_m(t)|\psi_n(t)\rangle = \delta_{mn}</math> The generic solution: <math display="block">|\Psi(t)\rangle = \sum a_n(t)|\psi_n(t)\rangle</math> plugging in the full Schrödinger equation and multiplying by a generic eigenvector: <math display="block">\langle \psi_m(t)|i\hbar\partial_t|\Psi(t)\rangle = \langle \psi_m(t)|H(\vec{R}(t))|\Psi(t)\rangle </math> <math display="block">\dot{a}_m + \sum_n\langle \psi_m(t)|\partial_{\vec{R}} |\psi_n(t)\rangle\dot{\vec{R}}a_n = -\frac{i}{\hbar}E_m(t)a_m </math> And if we introduce the adiabatic approximation: <math display="block"> | \langle \psi_m(t)|\partial_{\vec{R}} |\psi_n(t)\rangle\dot{\vec{R}}a_n | \ll |a_m|</math> for each <math>m\ne n</math> We have <math display="block">\dot{a}_m = - \langle \psi_m(t)|\partial_{\vec{R}} |\psi_m(t)\rangle\dot{\vec{R}}a_m -\frac{i}{\hbar}E_m(t)a_m</math> and <math display="block">a_m(t) = e^{-\frac{i}{\hbar} \int_{t_0}^t E_m(t')dt'} e^{i\gamma_m(t)}a_m(t_0)</math> where <math display="block">\gamma_m(t) = i \int_{t_0}^t \langle \psi_m(t)|\partial_{\vec{R}} |\psi_m(t)\rangle\dot{\vec{R}}dt' = i \int_C \langle \psi_m(\vec{R})|\partial_{\vec{R}} |\psi_m(\vec{R})\rangle d\vec{R} </math> And C is the path in the parameter space, This is the same as the statement of the theorem but in terms of the coefficients of the total wave function and its initial state.<ref>{{Cite book| title=Topological insulators and Topological superconductors|last1=Bernevig| first1=B. Andrei|last2=Hughes|first2=Taylor L.| year=2013| pages=Ch. 1|publisher=Princeton university press}}</ref> Now this is slightly more general than the other proofs given we consider a generic set of parameters, and we see that the Berry phase acts as a local geometric quantity in the parameter space. Finally integrals of local geometric quantities can give topological invariants as in the case of the [[Gauss-Bonnet theorem]].<ref>{{Cite web | last=Haldane | title=Nobel Lecture | url=https://www.nobelprize.org/uploads/2018/06/haldane-lecture-slides.pdf}}</ref> In fact if the path C is closed then the Berry phase persists to gauge transformation and becomes a physical quantity. |}
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