Coherent States
The general states of a harmonic oscillator can be expressed as a superpostion of the energy eigenstates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n\rangle.\!} A class of states that is of particular importance are the eigenstates of the (non-Hermitian) lowering operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a},\!}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|\alpha\rangle=\alpha|\alpha\rangle,\!}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\!} can be any complex number.
These states are known as coherent states. The term, "coherent", reflects their important role in optics and quantum electronics.
Note that it is not possible to construct an eigenstate of the raising operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}^{\dagger}} because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle;} this fact means that application of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}^{\dagger}} to any superposition of harmonic oscillator eigenstates eliminates the lowest-energy state that was present in the superposition.
The following are some properties of coherent states.
Construction of Coherent States
The coherent state with eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\!} is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle=e^{\alpha\hat{a}^\dagger}|0\rangle.}
We may see that this is a coherent state with the given eigenvalue as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}|\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\hat{a}|n\rangle=\sum_{n=1}^{\infty} \frac{\alpha^n}{\sqrt{n!}}\sqrt{n}|n-1\rangle=\sum_{n=1}^{\infty}\frac{\alpha^n}{\sqrt{(n-1)!}}|n-1\rangle= \alpha\left(\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle\right)=\alpha|\alpha\rangle}
This state, however, is not normalized, so we will now normalize it. Let us introduce a normalization constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N,\!} into the coherent state:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=Ne^{\alpha a^{\dagger}} |0\rangle }
We now determine what value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\!} yields a normalized state:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=\langle\alpha|\alpha\rangle=\langle 0|Ne^{\alpha^*a} Ne^{\alpha a^{\dagger}} |0\rangle = N^2\langle 0|e^{\alpha^*a} e^{\alpha a^{\dagger}} |0\rangle }
We now use the fact that, for any two operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B}} that both commute with their commutator, the following formula, known as the Campbell-Baker-Hausdorff formula, holds:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{A} e^\hat{B} = e^{\hat{A}+\hat{B}} e^{[\hat{A},\hat{B}]/2}\!}
Similarly,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{B} e^\hat{A} = e^{\hat{B}+\hat{A}} e^{[\hat{B},\hat{A}]/2} = e^{\hat{A}+\hat{B}} e^{-[\hat{A},\hat{B}]/2}.}
Combining the above two formulas, we obtain
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{A} e^\hat{B} = e^\hat{B} e^\hat{A} e^{[\hat{A},\hat{B}]}.\!}
This result applies for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}=\alpha^\ast \hat{a}\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B}=\alpha \hat{a}^{\dagger}\!} because the commutator for these two operators is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{A},\hat{B}]=|\alpha|^2,\!} , which is a constant. We thus obtain
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle\alpha|\alpha\rangle &= N^2\langle 0|e^{\alpha^\ast\hat{a}}e^{\alpha\hat{a}^{\dagger}}|0\rangle=N^2\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}e^{[\alpha^\ast\hat{a},\alpha\hat{a}^{\dagger}]} |0\rangle \\ &=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}|0\rangle=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}} |0\rangle \\ &=N^2e^{|\alpha|^2}\langle 0|0\rangle=N^2e^{|\alpha|^2} \end{align} }
We have thus determined the normalization constant,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rightarrow N=e^{-\frac{1}{2}|\alpha|^2}.}
The normalized coherent state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle} is therefore
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=e^{-|\alpha |^2/2}e^{\alpha\hat{a}^{\dagger}}|0\rangle.}
Inner Product of Two Coherent States
We have shown that, for any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,\!} there is an eigenstate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle\!} of the lowering operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}.} Therefore, we have a complete set of coherent states. However, this is not an orthogonal set. Indeed, the inner product of two coherent states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\beta\rangle\!} can be calculated as follows:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle \beta|\alpha \rangle &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}\langle 0|e^{\beta^\ast\hat{a}} e^{\alpha\hat{a}^\dagger} |0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}} e^{[\beta^\ast\hat{a},\alpha\hat{a}^\dagger]}|0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}}|0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*} \end{align} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\langle\beta|\alpha\rangle |^2 = e^{-|\alpha-\beta|^2}.}
Hence, the set of coherent states is not orthogonal and the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha-\beta|\!} in the complex plane measures the degree to which the two eigenstates are "approximately orthogonal".