18. Decoherence#
18.1. Example: two-dimensional system interacting with a large-dimensional environment#
18.1.1. State of the system#
A 2-dimensional system is interacting with a large-dimensional environment. The state of the 2-dimensional system is identified by the wave function
\[| \Psi_S \rangle = c_0 | 0 \rangle + c_1 | 1 \rangle \ ,\]
the state of the environment by the wave function \(| \Psi_E \rangle\), and the state of the whole system (S+E) by the tensor product of the two,
\[| \Psi_{S+E} \rangle = | \Psi_S \rangle \otimes | \Psi_E \rangle \ .\]
18.1.2. Evolution of the system#
The evolution of the system is governed by the Schrodinger equation, with the Hamiltonian of the whole system
\[i \hbar \dfrac{d}{dt} | \Psi_{S+E} \rangle = \hat{H}_{S+E} | \Psi_{S+E} \rangle \ ,\]
whose evolution can be represented by an unitary operator \(\hat{U}_{S+E; t,0}\),
\[| \Psi_{S+E} \rangle_t = U_{S+E; t,0} | \Psi_{S+E} \rangle_0 = \dots = \alpha | 0 \rangle \otimes | \Psi_{E;0} \rangle + \beta | 1 \rangle \otimes | \Psi_{E;1} \rangle\]
Density operator reads
\[\hat{\rho} := | \Psi_{S+E} \rangle \langle \Psi_{S+E} | \ .\]
Tracing out the environment states,
\[\begin{split}\begin{aligned}
\hat{\rho}_E
& = \text{Tr}_E \left( \hat{\rho} \right) = \\
& = \sum_{k} \left( \alpha | 0 \rangle \langle \phi_k | \Psi_{E;0} \rangle + \beta | 1 \rangle \langle \phi_k | \Psi_{E;1} \rangle \right) \left( \alpha^* \langle 0 | \langle \Psi_{E;0} | \phi_k \rangle + \beta^* \langle 1 | \langle \phi_k | \Psi_{E;1} \rangle \right) = \\
& = |\alpha|^2 | 0 \rangle \langle 0 | \langle \Psi_{E;0} | \underbrace{\sum_k | \phi_k \rangle \langle \phi_k}_{= \hat{\mathbf{1}}} | \Psi_{E;0} \rangle +
\alpha \beta^* | 0 \rangle \langle 1 | \langle \Psi_{E;0} | \underbrace{\sum_k | \phi_k \rangle \langle \phi_k |}_{= \hat{\mathbf{1}} } \Psi_{E;1} \rangle + \\
& \quad +
\alpha^* \beta | 1 \rangle \langle 0 | \langle \Psi_{E;1} | \underbrace{\sum_k | \phi_k \rangle \langle \phi_k}_{= \hat{\mathbf{1}}} | \Psi_{E;0} \rangle +
|\beta|^2 | 1 \rangle \langle 1 | \langle \Psi_{E;1} | \underbrace{\sum_k | \phi_k \rangle \langle \phi_k |}_{= \hat{\mathbf{1}} } \Psi_{E;1} \rangle = \\
& = |\alpha|^2 | 0 \rangle \langle 0 | + \alpha \beta^* \langle \Psi_{E;0} | \Psi_{E;1} \rangle | 0 \rangle \langle 1 | +
+ \alpha^* \beta \langle \Psi_{E;1} | \Psi_{E;0} \rangle | 1 \rangle \langle 0 | + |\beta|^2 | 1 \rangle \langle 1 | \ .
\end{aligned}\end{split}\]
The components of the reduced density operator in the system basis are
\[\begin{split}\begin{bmatrix} |\alpha|^2 & \alpha \beta^* \langle \Psi_{E;0} | \Psi_{E;1} \rangle \\ \alpha^* \beta \langle \Psi_{E;1} | \Psi_{E;0} \rangle & |\beta|^2 \end{bmatrix} \ .\end{split}\]
18.1.3. Decoherence time#
todo check and uncomment