Coherent States

The general states of a harmonic oscillator can be expressed as a superpostion of the energy eigenstates $|n\rangle \!$. A class of states that is of particular importance consists of the eigenstates of non-Hermitian lowering operator $a\!$, with eigenvalue $\alpha \!$:

$a|\alpha \rangle =\alpha |\alpha \rangle \!$ where $\alpha \!$ can be any complex number.

Such states are called coherent states. The term coherent reflects their important role in optics and quantum electronics. The following are some properties of coherent states.

Note that it is not possible to construct an eigenstate of $a^{\dagger }$ because $a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle$.

I. Coherent states construction.

$|\alpha \rangle =\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle =e^{\alpha a^{+}}|0\rangle$

$a|\alpha \rangle =\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}a|n\rangle =\sum _{n=1}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}{\sqrt {n}}|n-1\rangle =\sum _{n=1}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {(n-1)!}}}|n-1\rangle =\alpha \left(\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle \right)=\alpha |\alpha \rangle$

II. Coherent states normalization.

$|\alpha \rangle =Ne^{\alpha a^{\dagger }}|0\rangle$ where $N$ is normalization constant.

$1=\langle \alpha |\alpha \rangle =\langle 0|Ne^{\alpha ^{*}a}Ne^{\alpha a^{\dagger }}|0\rangle =N^{2}\langle 0|e^{\alpha ^{*}a}e^{\alpha a^{\dagger }}|0\rangle$

For any operators A and B which both commute with their commutator, we have:

$e^{A}e^{B}=e^{A+B}e^{{\frac {1}{2}}[A,B]}$

and similarly, $e^{B}e^{A}=e^{B+A}e^{{\frac {1}{2}}[B,A]}=e^{A+B}e^{-{\frac {1}{2}}[A,B]}$

therefore: $e^{A}e^{B}=e^{B}e^{A}e^{[A,B]}\!$

Apply this result for $A=\alpha ^{*}a\!$ and $B=\alpha a^{\dagger }\!$ ( A and B both commute with their commutator because $[A,B]=|\alpha |^{2})\!$, we have:

$1=\langle \alpha |\alpha \rangle =N^{2}\langle 0|e^{\alpha ^{*}a}e^{\alpha a^{\dagger }}|0\rangle$

${\begin{aligned}N^{2}\langle 0|e^{\alpha a^{\dagger }}e^{\alpha ^{*}a}e^{[\alpha ^{*}a,\alpha a^{\dagger }]}|0\rangle &=N^{2}e^{|\alpha |^{2}}\langle 0|e^{\alpha a^{\dagger }}e^{\alpha ^{*}a}|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\langle 0|e^{\alpha a^{\dagger }}|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\langle 0|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\end{aligned}}$

$\rightarrow N=e^{-{\frac {1}{2}}|\alpha |^{2}}$

$\rightarrow {\mbox{Normalized coherent states:}}|\alpha \rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}e^{\alpha a^{\dagger }}|0\rangle$

III. Inner product of two coherent states

There is an eigenstate $|\alpha \rangle \!$ of lowering operator $a\!$ for any complex number $\alpha \!$. Therefore, we have a set of coherent states. This is NOT an orthogonal set. Indeed, the inner product of two coherent states $|\alpha \rangle \!$ and $|\beta \rangle \!$ can be calculated as follows:

${\begin{aligned}\langle \beta |\alpha \rangle &=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}\langle 0|e^{\beta ^{*}a}e^{\alpha a^{+}}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}\langle 0|e^{\alpha a^{+}}e^{\beta ^{*}a}e^{[\beta ^{*}a,\alpha a^{+}]}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}e^{\alpha \beta ^{*}}\langle 0|e^{\alpha a^{+}}e^{\beta ^{*}a}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}e^{\alpha \beta ^{*}}\end{aligned}}$

$\rightarrow |\langle \beta |\alpha \rangle |^{2}=e^{-|\alpha -\beta |^{2}}$

Hence, the set of coherent states is not orthogonal and the distance $|\alpha -\beta |\!$ in a complex plane measures the degree to which the two eigenstates are 'approximately orthogonal'.