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Many papers that I have read which concern the dynamics of open systems and even otherwise typically start with a Hamiltonian of the system and then apply a unitary transformation depending upon the form of their Hamiltonian and the effect that they want to bring out. The effect that these unitary transformations explicitly bring out interesting effects which, of course, are already present (or can be guessed to be present) in the system Hamiltonian.

Consider the following Hamiltonian,

$$H=\Delta a^{\dagger}a + \omega b^{\dagger}b+g_o\bigg(b^{\dagger}b+\frac{1}{2}\bigg)(a^{\dagger}+a)$$ where, $a$ and $b$ are usual photon and phonon bosonic operators respectively.

It is known that this kind of $H$ has a non-linearity in $b^{\dagger}b$ and to show this explicitly the author in this paper uses what is known as a polaron transformation: $$H'=UHU^{\dagger}$$ with $U=\exp[\big(b^{\dagger}b+\frac{1}{2}\big)(a^{\dagger}a)]$

and gets the Hamiltonian $H \propto$ $(b^{\dagger}b)^2$

Similarly, other papers have different such unitary transformations (even non-linear ones, check this paper if interested) which are used to write the hamiltonian in an effective way that clearly can indicate an interesting phenomenon (like the above non-linearity is an indicator of phonon blockade in the system ).

Finding out the 'right' unitary transformation becomes crucial for many studies and in cases where one has a system Hamiltonian which one suspects should show a certain phenomenon under optimal parameters. Say one suspects his system to show phonon blockade. He can try to coming up or building a unitary transformation that transforms the $H$ to become proportional to $(b^{\dagger}b)^2$. The question is how shall he find such a transformation assuming that such a transformation exists looking at the form of the Hamiltonian and the form he wants to bring it to?

All the papers that I have read do not show how they come up with the unitary transformation for their hamiltonian.

Are there any general approaches or studies for constructing unitary transformations for a given hamiltonian or is it just guesswork (which I doubt it cannot be)?

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  • $\begingroup$ I am not sure, but I doubt there exists general approaches as it very much depends on what you want the unitary map to do. Often you can do many of these things easier using a Coherent state path integral formalism. For example, if you want to know how the $b$ degree of freedom effectively self-interact due to the $a$-$b$ interactions, you can integrate out $a$ (which is gaussian) and get the effective theory for $b$. Which would contain $(b^\dagger b)^2$ terms. $\endgroup$
    – Heidar
    Aug 29, 2021 at 18:44
  • $\begingroup$ A very powerful method is Hubbard-Stratonovich transformation, which can be used to rewriting the theory in terms of other (many-body) degrees of freedom that might be more physically important for the problem at hand. For example, when studying superconductivity (BCS Hamiltonian) one can (using HS-transformation) work with the cooper pair degrees of freedoms rather than the electrons themselves. $\endgroup$
    – Heidar
    Aug 29, 2021 at 18:45

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Quadratic Hamiltonians
For quadratic Hamiltonians canonical transformation corresponds to the rotation of the basis in non-second quantized representation. Thus, the new operators can be represented as a linear transformation of the current operators, and the canonical transformation is easily derive dusing matrix algebra.

Bogoliubov transformation
The non-interacting procedure works for some interacting Hamiltonians as well. A notable case is Bogoliubov transformation applied to BCS Hamiltonian - in this specific case one considers linear combination of creationa nd annihilation operators, which, strictly speaking, is not simple basis rotation.

Perturbative procedures
One can obtain approximate transformation using perturbative methods, e.g., with the help of the Baker-Campbell_Houssdorff formula. The classical example here is the derivation of the Schrieffer-Wolff transformation, transforming Anderson model to Kondo model. (See the Wikipedia article or the original paper by Schriffer and Wolff for more details.)

Known transformations
There are many known transformations, obtained through educated guesses, experience, and trial and error. Such are: Holstein-Primakoff transformation, Jordan-WIgner transformation, Polaron transformation, bosonization transformation (as described by Tomonaga, Schotte&Schotte and Haldane), and others.

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  • $\begingroup$ Yourcandwer helps a lot.Thanks. Is there a systematic or some standard a way to eliminate the above list depending on the form wanted? (I mean there obviously is as that's what people do) What I am trying to ask is (since I am new to do this)...what does a person look for generally in these hamiltonians to guess the appropriate form of the transformation? Is there a reduced form of the $H$ that gives an idea of what can be used or something of that sort? $\endgroup$
    – Lost
    Aug 30, 2021 at 13:00
  • $\begingroup$ It depends on the problem: if the Hamiltonian is quadratic, one is trying to directly diagonalize it. Schrieffer-Wolff transformation is aiming to remove a small perturbation. Then in many cases one knwos what kind of excitations/behavior are in the problem, and tries to reduce it to desirable form - e.g., polaron, spin waves, etc. Symmetry arguments may work as well. But one rarely needs to actually design a transformation - there is usually a transformation that everybody uses in the field (unless the field is very new and Hamiltonian is very unusual). $\endgroup$
    – Roger V.
    Aug 30, 2021 at 13:18

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