Coherent States

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

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Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

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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 are the eigenstates of the (non-Hermitian) lowering operator ${\hat {a}},\!$

$a|\alpha \rangle =\alpha |\alpha \rangle ,\!$

where $\alpha \!$ 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 ${\hat {a}}^{\dagger }$ because $a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle ;$ this fact means that application of ${\hat {a}}^{\dagger }$ 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 $\alpha \!$ is given by

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

We may see that this is a coherent state with the given eigenvalue as follows:

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

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'.