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I am struggling with the concept of deriving an effective Hamiltonian using perturbation theory. Say we have

$$ \hat{H} = \hat{H}_0 + \hat{V} $$

and suppose we know the energies $E_n^{(0)}$ and eigenvectors $|n^{(0)}\rangle$. We also know the matrices $H$ and $V$

\begin{align} (H)_{ij} &= \langle i^{(0)}|\hat{H}|j^{(0)}\rangle\\ (V)_{ij} &= \langle i^{(0)}|\hat{V}|j^{(0)}\rangle\\ \end{align}

This means that we can find a perturbation series for the true eigenvectors, $|n\rangle$ of the full Hamiltonian $\hat{H}$. For example, Wikipedia gives a formula for $|n\rangle$ out to second order and higher.

In general we have

$$ |n\rangle = \sum_{m=0}^\infty |n^{(m)}\rangle $$

Define

$$ |n^{[k]}\rangle = \sum_{m=0}^k |n^{(m)}\rangle $$

Excuse the confusing notation, I'm trying not to conflict with Wikipedia's notation.

The Question

My question concerns using perturbation theory to come up with an effective Hamiltonian (maybe in a lower energy subspace, maybe not...)

We have (generally) that

$$ \hat{H} = \sum_{ij} |i\rangle\langle i|\hat{H}|j\rangle \langle j| = \sum_{ij} |i\rangle(H)_{ij} \langle j| $$

What I would like to be the case is that I can come up with an effective Hamiltonian (to order $k$) for the system by simply replacing $|i\rangle \rightarrow |i^{[k]}\rangle$

$$ \hat{H}_{eff}^{[k]} = \sum_{ij} |i^{[k]}\rangle\langle i^{[k]}|\hat{H}|j^{[k]}\rangle \langle j^{[k]}| $$

In general if we replace $\hat{V} \rightarrow \lambda \hat{V}$ We will find that the expression for $\hat{H}_{eff}^{[k]}$ (or the matrix representing it) will have terms up to order $O(\lambda^{4k})$. Terms up to some order need to be eliminated but I'm not exactly sure what is the correct order. One guess is that you should truncate terms up to order $O(\lambda^{2k})$...

Is this procedure correct? It seems simple and straightforward but I haven't been able to find any references spelling out the approach this clearly.. I would appreciate any insight or references on the topic of effective Hamiltonians and how/if they can be directly derived from perturbation theory.

Example System

My main question is the general question given above. What follows below is a related but possibly different question about a specific system. I add this example to help clarify what I am thinking about but I'd also like to hear if there is something specific in the way I am thinking about this example that is causing my confusion or if I'm doing something wrong.. If this counts as too much of a separate question I can delete it here and ask it as a new question.

The three level system I am looking at is governed by

$$ H = \begin{bmatrix} 0 & 0 & 0\\ 0 & \delta & 0\\ 0 & 0 & \Delta \end{bmatrix} $$ $$ V = \begin{bmatrix} 0 & 0 & \Omega_1\\ 0 & 0 & \Omega_2\\ \Omega_1 & \Omega_2 & 0 \end{bmatrix} $$

I can calculate $\hat{H}_{eff}^{[1]}$ and keep terms up to $O(\lambda^2)$. If I do this then for $\delta, \Omega \ll \Delta$ I see that the exact dynamics (found by diagonalizing the full Hamiltonian) are closely approximated by the dynamics of the effective Hamiltonian in the low-energy subspace. However, the dynamics of the excited state are not well approximated by the effective Hamiltonian. Should this be expected?

I have tried going to higher order in perturbation theory by calculating $\hat{H}_{eff}^{[2]}$ and eliminating terms to different orders in $\lambda$ but there I see results which seem to be even more incorrect..

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  • $\begingroup$ My understanding of effective Hamiltonian is that you project your system to low energy subspace of interest, so it is not surprising that the excited state dynamics is not well-approximated, although effects of excited states are written into the perturbed low energy states. I see you study atomic physics, so have you consulted Atom-Photon Interaction by Cohen-Tannoudji where he shows how to project onto chosen subspace? $\endgroup$ – IamAStudent Dec 18 '18 at 20:06
  • $\begingroup$ Yes I do know that often you are concerned with dynamics in a low energy sub space and project into it. However, I don’t see where my approach would go wrong when considering the high energy state as well. Especially at high enough order in the perturbation theory. I will check the Cohen-Tannoudji book. $\endgroup$ – jgerber Dec 18 '18 at 20:15
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You make several mistakes in your derivation: (1) You assume that the spectrum of the (unperturbed and perturbed) Hamiltonian consists of eigenvalues. (2) By merely scaling the potential, you forget about what is equally important: the decay of $V_{ij}$ if $|i - j|$ becomes large. (3) You forget there are other basis, that do not diagonalize $\hat{H}_0$ that are often used (e. g. a basis of Wannier functions to derive a tight-binding model from a continuum model).

What you want to do is the following:

(1) Identify relevant states; these make up a subspace $\mathcal{H}_{\mathrm{rel}} \subset \mathcal{H}$ of your Hilbert space. These could be states from an energy window, for example.

Note that the relevant subspace must be (approximately) be left invariant by the dynamics, meaning the dynamics map relevant states onto relevant states up to a small error. This translates to the requirement that $\hat{H}$ and the orthogonal projection $\hat{P}_{\mathrm{rel}}$ onto $\mathcal{H}_{\mathrm{rel}}$ approxmiately commute, \begin{align*} \bigl [ \hat{H} , \hat{P}_{\mathrm{rel}} \bigr ] \approx 0 . \end{align*} (2) Find a suitable orthonormal basis $\{ \psi_j \}$ for your relevant subspace $\mathcal{H}_{\mathrm{rel}}$. These could be exponentially localized Wannier functions.

(3) Verify that $H_{ij} = \langle \psi_i | \hat{H} | \psi_j \rangle$ decays rapidly (e. g. exponentially) as $|i - j|$ becomes large. Then you can replace $\hat{H}$ with \begin{align*} \hat{H}_{\mathrm{eff}} = \sum_{|i - j| \leq N} H_{ij} \, |\psi_i \rangle \langle \psi_j | , \end{align*} and can now approximate the full hamiltonian by the effective hamiltonian for the relevant states, \begin{align*} \hat{H} \, \hat{P}_{\mathrm{rel}} \approx \hat{H}_{\mathrm{eff}} \, \hat{P}_{\mathrm{rel}} , \end{align*} where $P_{\mathrm{rel}} = \sum_j | \psi_j \rangle \langle \psi_j |$ is the orthogonal projection onto the relevant subspace $\mathcal{H}_{\mathrm{rel}}$.

This is because the above equality implies that also the dynamics remain close in time: using the Fundamental Theorem of Calculus, we can write the difference of the evolutions as an integral and then estimate, \begin{align*} \Bigl ( \mathrm{e}^{- \mathrm{i} t \hat{H}_{\mathrm{eff}}} - \mathrm{e}^{- \mathrm{i} t \hat{H}} \Bigr ) \, \hat{P}_{\mathrm{rel}} &= \int_0^t \mathrm{d} s \; \frac{\mathrm{d}}{\mathrm{d} s} \Bigl ( \mathrm{e}^{- \mathrm{i} s \hat{H}_{\mathrm{eff}}} \; \mathrm{e}^{- \mathrm{i} (t - s) \hat{H}} \Bigr ) \, \hat{P}_{\mathrm{rel}} \\ &\approx \int_0^t \mathrm{d} s \; \mathrm{e}^{- \mathrm{i} s \hat{H}_{\mathrm{eff}}} \, \underbrace{\bigl ( \hat{H}_{\mathrm{eff}} - \hat{H} \bigr ) \, \hat{P}_{\mathrm{rel}}}_{\approx 0} \, \mathrm{e}^{- \mathrm{i} (t - s) \hat{H}} \, \hat{P}_{\mathrm{rel}} \\ &\approx 0 , \end{align*} where we used in the second step that $\hat{H}$ and $\hat{P}_{\mathrm{rel}}$ approximately commute (i. e. that $\mathcal{H}_{\mathrm{rel}}$ is an invariant subspace).

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I think I have found the flaw in my reasoning and it seems pretty obvious in hindsight but it was helpful for me to go through.

My reasoning was motivated by looking at the example system and wondering: How do I get an effective Raman coupling of strength $\frac{\Omega_1 \Omega_2}{\Delta}$? I knew the answer was "something something perturbation theory."

Consider $$ |e^{[1]}\rangle = |e^{(0)}\rangle + \frac{\Omega_1}{\Delta}|1^{(0)}\rangle + \frac{\Omega_2}{\Delta -\delta}|2^{(0)}\rangle $$ If you start off towards calculating $$ |e^{(1)}\rangle\langle e^{(1)}|\hat{H}_0 + \hat{V}|e^{(1)}\rangle\langle e^{(1)}|$$ You will notice that terms like $$ \frac{\Omega_1 \Omega_2}{\Delta}|1^{(0)}\rangle \langle 2^{(0)}| $$

will appear. This is what made me think the approach I laid out above was on the right track.

However, I've straightened out my simulations a bit more and understand things better and I see what is happening.

First regarding the appropriate orders of approximation: The statement I will make here is that if we calculate $$ \hat{H}_{eff}^{[k]} = \sum_{ij} |i^{[k]}\rangle \langle i^{[k]}|\hat{H}|j^{[k]}\rangle \langle j^{[k]}| $$

Then since the kets are approximated to $O(\lambda^k)$ we can only keep terms up to this same order $O(\lambda^k)$ in $H_{eff}$. This should have been a warning sign to me because in what I described at the top of this answer I was expanding the kets to $O(\lambda)$ but the term I am looking for, $\frac{\Omega_1 \Omega_2}{\Delta}$ would come from terms which are $O(\lambda^2)$.

When I do this what I neglect the appropriate(at least for the simple 3-level system) is that the effective Hamiltonian $\hat{H}_{eff}$comes out diagonal. This is in fact not surprising since the pertubation kets should be approximating the actual eigenvectors which diagonalize the Hamiltonian!

In fact, I would expect that at high enough level in perturbation theory my $\hat{H}_{eff}$ should exactly produce $\hat{H}$. and the perturbation theory will just approximate the eigenvalues and eigenvectors of the full Hamiltonian. That is, my procedure doesn't find an effective Hamiltonian, it simply solves the existing Hamiltonian.

I had been seeing things in my simulation that looked like what I wanted when I expanded the kets to $O(\lambda)$ but kept terms in $H_{eff}$ out to $O(\lambda^2)$ but as I mentioned there was some strange behavior and I think this is to be expected because I kept some $O(\lambda^2)$ terms when I shouldn't have been, that is I may have been dropping important $O(\lambda^2)$ terms but keeping some of them which should produce weird results.

Things were also even more strange if I expanded the kets to $O(\lambda^2)$ and then kept terms of $O(\lambda^3)$ or $O(\lambda^4)$

In any case as has been pointed out in the comments and answer, to come up with an effective Hamiltonian it is necessary to do some sort of projection or something so that the Hamiltonian is actually "effective" rather than just the same Hamiltonian you started with.

As pointed out by @IAmAStudent in the comments, Atom-Photon Interaction by Cohen-Tannoudji has a good section on "Description of the Effect of a Perturbation by an Effective Hamiltonian" in which the Schrieffer-Wollf Transformation is used to find an effective Hamiltonian. The idea is to perform a unitary transformation (change of basis) which block diagonalizes the full Hamiltonian, i.e. decouples the relevant subspaces. The goal is to find the operator which performs this unitary transformation. The advantage of this over fully diagonalizing the Hamiltonian is that the operator which performs the transformation can be found in a perturbation expansion. See the reference for more details.

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