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I have often heard the statement that non-Hermitian Hamiltonians can be used to describe open systems, since the dynamics are non-unitary. However, I have not been able to find any examples of a non-Hermitian Hamiltonian actually arising in a systematic way starting from a system coupled to an environment. The closest analogy I have seen is the Lindblad equation, which allows one to describe the continuous time evolution of the density matrix with Lindblad operators, whose form can be derived from the full Hamiltonian of the system. This is non-Hermitian in a sense, since it describes non-unitary time evolution, but the picture I would have expected was something like

$$ \frac{d\rho}{dt} = - i[H, \rho]$$

where $H$ is a non-Hermitian operator. Perhaps I'm misunderstanding the utility of non-Hermitian Hamiltonians, but is there a way to start with a full system where dynamics are controlled by a Hermitian Hamiltonian, and then obtain a non-Hermitian Hamiltonian that describes the dynamics of the subsystem?

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A decaying particle has this. We are only looking at part of the total hamiltonian (we drop the terms that describe the decay products). Our particle has sates $i$ that have their own energy and decay widths $E_i$ and $\Gamma_i$ respectively.

The probability is a decaying exponential, $$ P(t) = e^{-\Gamma t} $$ This admits wavefunctions, $$\psi_i(t) = Ae^{-iEt -\frac{\Gamma t}{2}}$$

This time evolution is given by a diagonal matrix with entries, $$H_{ii} = E_i + i\frac{\Gamma_i}{2}$$ which on complex conjugation is not hermitian.

Though this can be divided into two matrices, a mass matrix, and a decay matrix both of which are hermitian, $$H = M + i\Gamma$$

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    $\begingroup$ Can you explain exactly how you go from the total Hamiltonian to this? This makes sense if you give me a particle and tell me its states and decay widths, but I'd like to understand it more systematically. Specifically, I'm imagining the full Hamiltonian as being something like $H = H_A + H_B + H_{decay}$, where $H_A$ and $H_B$ are the Hamiltonians describing particles A and B, and $H_{decay}$ has transition elements from particle A to B. I can toss out $H_B$ easily, but I don't know a general way of restricting $H_{decay}$ to subsystem A. $\endgroup$
    – Henry Shackleton
    Jun 21, 2019 at 16:42
  • $\begingroup$ Actually, I think that makes sense because there are standard ways of calculating decay widths. I guess I was still thinking about it from a subsystem point of view. So does this hold more generally when we're talking about a system and an environment? $\endgroup$
    – Henry Shackleton
    Jun 21, 2019 at 16:50
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    $\begingroup$ That is exactly correct. All the terms $H_A$, $H_B$, and $H_{decay}$ are hermitian themselves. I would prefer to call $H_{decay}$ as $H_{interaction}$. The key is that when studying the dynamics of system $A$ we are interested in the process $A\rightarrow{}B$ but $H_{interaction}$ includes also $B\rightarrow{}A$. In ignoring that we are only taking a part of the that term and we end up with non-unitary time evolution (or non-hermitian effective Hamiltonian) $\endgroup$
    – TEH
    Jun 21, 2019 at 16:53
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    $\begingroup$ As a final note the probabilities $P(A\rightarrow{}B) = P(B\rightarrow{}A)$, however that does not mean that $\Gamma_{A\rightarrow{}B}$ is the same as $\Gamma_{B\rightarrow{}A}$ because the decay widths take into account the phase space and multiplicities of the states. $\endgroup$
    – TEH
    Jun 21, 2019 at 16:56
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Once you un-physically drop the jump term in your open quantum system master equation or Lindblad equation, you can obtain a non-Hermian type of equation.

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