# 4.2: Decoherence

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The density operator allows us to consider the phenomenon of decoherence. Consider the pure state $$|+\rangle$$. In matrix notation with respect to the basis $$\{|+\rangle,|-\rangle\}$$, this can be written as

$\rho=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \tag{4.8}$

The trace is 1, and one of the eigenvalues is 1, as required for a pure state. We can also write the density operator in the basis $$\{|0\rangle,|1\rangle\}$$:

$\rho=|+\rangle\langle+|=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) \times \frac{1}{\sqrt{2}}(1,1)=\frac{1}{2}\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right).\tag{4.9}$

Notice how the outer product (as opposed to the inner product) of two vectors creates a matrix representation of the corresponding projection operator.

Let the time evolution of $$|+\rangle$$ be given by

$|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt{2}} \rightarrow \frac{|0\rangle+e^{i \omega t}|1\rangle}{\sqrt{2}}.\tag{4.10}$

The corresponding density operator becomes

$\rho(t)=\frac{1}{2}\left(\begin{array}{cc} 1 & e^{i \omega t} \\ e^{-i \omega t} & 1 \end{array}\right).\tag{4.11}$

The “population” in the state $$|+\rangle$$ is given by the expectation value

$\langle+|\rho(t)|+\rangle=\frac{1}{2}+\frac{1}{2} \cos (\omega t).\tag{4.12}$This oscillation is due to the off-diagonal elements of $$\rho(t)$$, and it is called the coherence of the system (see Fig. 2). Figure 2: Population in the state $$|+\rangle$$ with decoherence (solid curve) and without (dashed curve).

The state is pure at any time $$t$$. In real physical systems the coherence often decays exponentially at a rate $$\gamma$$, and the density matrix can be written as

$\rho(t)=\frac{1}{2}\left(\begin{array}{cc} 1 & e^{i \omega t-\gamma t} \\ e^{-i \omega t-\gamma t} & 1 \end{array}\right).\tag{4.13}$

The population in the state $$|+\rangle$$ decays accordingly as

$\langle+|\rho(t)|+\rangle=\frac{1}{2}+\frac{e^{-\gamma t} \cos (\omega t)}{2}.\tag{4.14}$

This is called decoherence of the system, and the value of $$\gamma$$ depends on the physical mechanism that leads to the decoherence.

The decoherence described above is just one particular type, and is called dephasing. Another important decoherence mechanism is relaxation to the ground state. If the state $$|1\rangle$$ has a larger energy than $$|0\rangle$$ there may be processes such as spontaneous emission that drive the system to the ground state. Combining these two decay processes, we can write the density operator as

$\rho(t)=\frac{1}{2}\left(\begin{array}{cc} 2-e^{-\gamma_{1} t} & e^{i \omega t-\gamma_{2} t} \\ e^{-i \omega t-\gamma_{2} t} & e^{-\gamma_{1} t} \end{array}\right).\tag{4.15}$

The study of decoherence is currently one of the most important research areas in quantum physics.

This page titled 4.2: Decoherence is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.