In quantum optics coherent states introduced by Glauber have a localized probability distribution in classical phase-space with maximum following classical equations of motions. This is not a coincidence once we take a look at Hamiltonian in single-mode approximation

$$\hat{\mathcal{H}} = \hbar \omega\hat{a}^{\dagger}\hat{a} + \gamma \hat{a} + \gamma^{*}\hat{a}^{\dagger}\ \text{.}$$ Operators $\left\{\hat{a}^{\dagger}\hat{a}, \hat{a}, \hat{a}^{\dagger}, 1 \right\}$ span $h_{4}$ Lie algebra with associated Heisenberg-Weyl group $H_{4}$. From the group-theoretical construction of coherent states

$$\left| \alpha\right> = e^{\alpha \hat{a}^{\dagger} - \alpha^{*}\hat{a}}\left| 0\right>\ \text{,}$$ initial coherent state will remain coherent during quantum evolution as long as Hamiltonian is linear in generators of the Lie algebra. They proved to be useful in the study of quantum-classical correspondence.

The concept of coherent states can be generalized to any quantum system governed by some dynamical Lie group. Once again, initial coherent state will remain coherent during quantum evolution as long as Hamiltonian is linear in generators of the Lie algebra. Classical limit in this case is straightforward. But, how about nonlinear hamiltonians? What is the general construction of the classical hamiltonian?

For example, lets consider $SU(2)$ group with hamiltonian

$$\hat{\mathcal{H}} = \hat{J}_{z}^{2} + \hat{J}_{x}\text{.}$$ Husimi maps of the generators are the following

$$J_{x}(\theta, \phi) = J \sin\theta\cos\phi \\ J_{y}(\theta, \phi) = J \sin\theta\sin\phi \\ J_{z}(\theta,\phi) = J\cos\theta$$ What would be the classical hamiltonian $J \rightarrow \infty$ ?


1 Answer 1


The construction of coherent states does not depend on the Hamiltonian but rather on its dynamical algebra. In the example you give, the Hamiltonian is a polynomial function of elements of $su(2)$, so the dynamical algebra is $su(2)$ and one would construct $su(2)$ coherent states as a set of (overcomplete) basis states for your problem.

Moreover, the Husimi map (or $Q$-function) is a function on the classical phase space for your problem. This phase space is connected to coherent states as follows. The $SU(2)$ coherent states are states over $SU(2)/U(1)$ since the highest weight state is invariant under $U(1)$. You can see this because the $SU(2)$ coherent states depend on only two angles rather than three. The "missing" angle is related to the $U(1)$ transformation $e^{i\psi L_z}$ that leaves the highest weight state $\vert j,j\rangle$ invariant.

It turns out that $SU(2)/U(1)$ is nothing but the 2-dimensional sphere, explaining why your $Q-$symbols depend on two angles parametrizing a pont on the sphere.

One has to be careful in this formulation because the symbol for $\hat A^2$ is NOT in general the square of the symbol for $\hat A$. Symbols are "ordinary" commuting functions but operators do not commute so to account for the differences one must introduce a special type of composition of symbols called the $\star$-product. For spin systems this $\star$-product is quite involved and it is easier to obtain the symbol for $\hat J_z^2$ from first principles. For the Husimi function the symbol is quite generally $$ A(\theta,\phi)= \langle \theta,\phi\vert \hat A\vert \theta,\phi\rangle $$ where $\vert\theta,\phi\rangle$ is the $SU(2)$ coherent state.

Finally, the classical limit is not related directly to the symbols but to how symbols behave under the so-called Moyal bracket. This is a variation on the commutator which uses the $\star$-product rather than the usual product: $$ \{A,B\}_M:= A\star B - B\star A=\frac{1}{\sqrt{j(j+1)}}\{A,B\}_P+\hbox{corrections} $$ where $A$ and $B$ are symbols of the operators $\hat A$ and $\hat B$, and $\{A,B\}_P$ is the Poisson bracket on the sphere. In the classical limit, where $j\to\infty$, the correction terms disappear and the Moyal bracket goes to the usual Poisson bracket. The symbols have a $j$ dependence that "cancels" the prefactor in the Poisson bracket.

Nota: this is quite a long answer. It is generally correct but there could be some typos here and there so check with sources. For $SU(2)$ stuff there is a detailed exposition

  • in the book by Klimov and Chumakov,
  • in the paper of R. Gilmore, "Q and P representatives of spherical tensors", J.Phys.A vol 9 (1976) L65 (this paper is behind a paywall),
  • a closely related problem (using the Husimi function) is discussed at length in E. Romera, M. Calixto and O. Castanos, Phys.Scr. vol.89 (2014) 095103 (also behind a paywall). In particular these authors give $$ \hat J_z^2\mapsto j\,\frac{(2j+1)+(2j-1)\cos(2\theta)}{4}\, . $$

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