Why don't the oscillator coherent states disperse in time?

Question

A Gaussian wavepacket is made of a continuum of frequencies (or energies) and stretches in time due to the phenomenon of dispersion: the different plane wave components with different frequencies travel at different velocities. However, there is an exception!

The position space representation $\psi_\alpha(x)\equiv\langle x|\alpha\rangle$ of a Coherent state $|\alpha\rangle$ is also a Gaussian and consists of infinite many frequencies. The only difference with a generic Gaussian is that the frequencies are now countable and equally spaced. Such a wavefunction doesn't change disperse under time evolution: the wavepacket moves as a whole without changing its shape mimicking the motion of a classical oscillator.

What is so special about the Coherent states which make these Gaussians dispersion-free?

Answer 1

The coherent structure of the state is not preserved by a general quantum evolution.

It is preserved only in very special cases, i.e. when the evolution is generated by the second quantization of a one-particle operator. For quantum mechanical systems, this amounts only to the evolution generated by the harmonic oscillator.

Let us recall that coherent states are most conveniently studied in second quantization formalism. Let $\mathscr{H}$ be the one-particle Hilbert space. Then the (symmetric) Fock space over $\mathscr{H}$ is defined as $$\Gamma_\mathrm{s}(\mathscr{H})=\bigoplus_{n=0}^\infty \underbrace{\mathscr{H}\otimes_{\mathrm{s}}\mathscr{H}\dotsm \otimes_{\mathrm{s}} \mathscr{H}}_{n}=:\bigoplus_{n=0}^\infty \mathscr{H}_n\;,$$ where $\otimes_\mathrm{s}$ stands for the symmetric tensor product, and for $n=0$ one takes the complex numbers in the direct sum.

In quantum mechanics, one takes $\mathscr{H}=\mathbb{C}$, and thus $$\Gamma_\mathrm{s}(\mathbb{C})=\bigoplus_{n=0}^\infty \mathbb{C}\cong L^2(\mathbb{R})$$ (identifying each complex space in the direct sum as the span of a Hermite function).

In the second quantization formalism, the creation and annihilation operators are well-known, and the coherent state is usually written as $$\lvert\alpha\rangle = e^{a^*(\alpha)-a(\alpha)}\Omega\; ,$$ where $\alpha\in \mathscr{H}$, and $\Omega$ is the vacuum vector.

The only dynamics that preserves the coherent structure is the one generated by the following Hamiltonian. Let $\omega$ be a self-adjoint operator on $\mathscr{H}$, and $e^{-it\omega}$ the associated one-particle evolution. Then the second quantizations $\mathrm{d}\Gamma(\omega)$ and $\Gamma(e^{-it\omega})=e^{-it\mathrm{d}\Gamma(\omega)}$ are defined by the action on each $n$-particle subspace $\mathscr{H}_n$: $$\bigl(\mathrm{d}\Gamma(\omega)\psi\bigr)_n= \bigl(\sum_{j=1}^n 1\otimes \dotsm\otimes 1\otimes\omega_j\otimes 1\otimes\dotsm\otimes 1\bigr)\psi_n\;,$$ where $\omega_j$ is the operator $\omega$ acting only on the $j$-th variable; $$\bigl(\Gamma(e^{-it\omega})\psi\bigr)_n=\bigl(\prod_{j=1}^n e^{-it\omega_j}\bigr)\psi_n\; .$$

Now, it is not difficult to prove that for all $\alpha\in\mathscr{H}$, $$\Gamma(e^{-it\omega})\lvert\alpha\rangle=\lvert\alpha_t\rangle=\lvert e^{-it\omega}\alpha\rangle\; .$$ Therefore, the coherent structure is preserved by a $\Gamma$ evolution. This is, however, not true for other evolutions! A $\Gamma$ evolution is usually what models a free theory, but not interacting ones.

In quantum mechanics, since $\mathscr{H}=\mathbb{C}$, $\omega$ can only be a real number, and $\mathrm{d}\Gamma(\omega)$ is thus a (rescaled) harmonic oscillator (in fact $\mathrm{d}\Gamma(1)$ is the number operator). Therefore, in quantum mechanics only the harmonic oscillator dynamics leaves the coherent structure invariant.

Answer 2

The standard argument for coherent states in non-dimensionalized language is that $$ |\alpha\rangle \equiv e^{-|\alpha|^2/2 }~~ e^{\alpha ~ a^\dagger} | 0\rangle = e^{-|\alpha|^2/2 } \sum_{n=0}^\infty \frac {\alpha^n }{\sqrt{n!}} | n\rangle $$ when evolved by the operator $e^{-i\omega N t }$ picks up rigidly regimented phases which naturally absorb into the evident redefinition of $$ \alpha(t)= e^{-i\omega t} \alpha(0) ~. $$ This is visibly a consequence of the operators $a^\dagger$ laddering up the spectrum of $N$, and would fail, e.g. for the addition of higher powers of N in the hamiltonian. The amplitude (oscillating maximum of the Gaussian wavepacket) then remains invariant, $$ \langle \hat{x}(t) \rangle = |\alpha(0)| \sqrt{\frac{2}{\omega}} \cos (\omega t)~, $$ and its width maintains the nondispersive rigid value of its construction.

However, as in Schroedinger's original paper, E. Schrödinger, Naturwissenschaften 14, 664 (1926), they are also equivalently defined by the Minimum-uncertainty coherent states (MUCS) requirement that they minimize the position-momentum uncertainty relation, subject to the restriction that the ground state be in the set. The corresponding phases then also slosh in lockstep.

One may then canonically map other confining potentials to oscillator problems and systematically account of dispersion, as in the cornerstone paper by Nieto and Simmons, Phys Rev D20 (1979) 1321 and define suitable ladder operators, even if the spectrum is not spaced equally! The systematic group theoretical treatment then, further produces coherent states for potentials such as the Poeschl-Teller one.

This is a voluminous cottage industry, of course, Coherent States: Applications In Physics And Mathematical Physics, Klauder John R, Skagerstam Bo-sture, (1985) World Scientific, isbn 9971966522 .