23.1 Coherent states and the Heisenberg group action

23.1 Coherent states and the Heisenberg group action

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · eigenvalue · Lie algebra · Lie algebra representation · unitary representation · irreducible representation · Lie group · group action · Lie bracket · complexification

Since the Hamiltonian for the $d=1$ harmonic oscillator does not commute with the operators a or $a^{†}$ which give the representation of the Lie algebra ${\mathfrak{h}}_3$ on the state space ${\mathcal{F}}_{:}$ , the Heisenberg Lie group and its Lie algebra are not symmetries of the system. Energy eigenstates do not break up into irreducible representations of the group but rather the entire state space makes up such an irreducible representation. The state space for the harmonic oscillator does however have a distinguished state, the lowest energy state |0⟩, and one can ask what happens to this state under the Heisenberg group action.

Elements of the complexified Heisenberg Lie algebra ${\mathfrak{h}}_3\otimes \mathbf{C}$ can be written as

$$i\alpha z+\beta \overline{z}+\gamma$$

for $\alpha ,\beta ,\gamma$ in $\mathbf{C}$ (this choice of $\alpha$ simplifies later formulas). The Lie algebra ${\mathfrak{h}}_3$ is the subspace of real functions, which will be those of the form

$$i\alpha z-i\overline{\alpha }z+\gamma$$

for $\alpha \in \mathbf{C}$ and $\gamma \in \mathbf{R}$ . The Lie algebra structure is given by the Poisson bracket

$$\{i{\alpha }_{1}z-i{\overline{\alpha }}_1\overline{z}+{\gamma }_1,i{\alpha }_{2}z-i{\overline{\alpha }}_2\overline{z}+{\gamma }_2\}=2\mathrm{Im}({\overline{\alpha }}_1{\alpha }_2)$$

Here ${\mathfrak{h}}_3$ is identified with $\mathbf{C}\oplus \mathbf{R}$ , and elements can be written as pairs $(\alpha ,\gamma )$ with the Lie bracket

$$[({\alpha }_1,{\gamma }_1),({\alpha }_2,{\gamma }_2)]=(0,2\mathrm{Im}({\overline{\alpha }}_1{\alpha }_2))$$

This is just a variation on the labeling of ${\mathfrak{h}}_3$ elements discussed in chapter 13, and one can again use exponential coordinates and write elements of the Heisenberg group $H_3$ also as such pairs, with group law

$$({\alpha }_1,{\gamma }_1)({\alpha }_2,{\gamma }_2)=({\alpha }_1+{\alpha }_2,{\gamma }_1+{\gamma }_2+\mathrm{Im}({\overline{\alpha }}_1{\alpha }_2))$$

Quantizing using equation 22.5, one has a Lie algebra representation ${\Gamma }^{\prime }$ , with operators for elements of ${\mathfrak{h}}_3$

$${\Gamma }^{\prime }(\alpha ,\gamma )={\Gamma }^{\prime }(i\alpha z-i\overline{\alpha }z+\gamma )=\alpha a^{†}-\overline{\alpha }a-i\gamma 1\text{\hspace{1em}(23.1)}$$

and exponentiating these will give the unitary representation

$$\Gamma (\alpha ,\gamma )=e^{\alpha a^{†}-\overline{\alpha }a-i\gamma }$$

We define operators

$$D(\alpha )=e^{\alpha a^{†}-\overline{\alpha }a}$$

which satisfy (using Baker-Campbell-Hausdorf)

$$D({\alpha }_1)D({\alpha }_2)=D({\alpha }_1+{\alpha }_2)e^{-i\mathrm{Im}({\overline{\alpha }}_1{\alpha }_2)}$$

Then

$$\Gamma (\alpha ,\gamma )=D(\alpha )e^{-i\gamma }$$

and the operators $\Gamma (\alpha ,\gamma )$ give a representation, since they satisfy

$$\Gamma ({\alpha }_1,{\gamma }_1)\Gamma ({\alpha }_2,{\gamma }_2)=D({\alpha }_1+{\alpha }_2)e^{-i({\gamma }_1+{\gamma }_2+\mathrm{Im}({\overline{\alpha }}_1{\alpha }_2))}=\Gamma (({\alpha }_1,{\gamma }_1)({\alpha }_2,{\gamma }_2))$$

Acting on |0⟩ with $D(\alpha )$ gives:

Definition (Coherent states)

The coherent states in $H$ are the states

$$|\alpha ⟩=D(\alpha )|0⟩=e^{\alpha a^{†}-\overline{\alpha }a}|0⟩$$

where $\alpha \in \mathbf{C}$

Using the Baker-Campbell-Hausdorf formula

$$e^{\alpha a^{†}-\overline{\alpha }a}=e^{\alpha a^{†}}{e}^{-\overline{\alpha }a}{e}^{-\frac{|\alpha {|}^2}{2}}=e^{-\overline{\alpha }a}{e}^{\alpha a^{†}}{e}^{\frac{|\alpha {|}^2}{2}}$$

so

$$|\alpha ⟩=e^{\alpha a^{†}}{e}^{-\overline{\alpha }a}{e}^{-\frac{|\alpha {|}^2}{2}}|0⟩$$

and since $a|0⟩=0$ this becomes

$$|\alpha ⟩=e^{-\frac{|\alpha {|}^2}{2}}{e}^{\alpha a^{†}}|0⟩=e^{-\frac{|\alpha {|}^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}|n⟩\text{\hspace{1em}(23.2)}$$

Since $a|n⟩=\sqrt{n}|n-1⟩$

$$a|\alpha ⟩=e^{-\frac{|\alpha {|}^2}{2}}\sum _{n=1}^{\infty }\frac{{\alpha }^n}{\sqrt{(n-1)!}}|n-1⟩=\alpha |\alpha ⟩$$

and this property could be used as an equivalent definition of coherent states. In a coherent state the expectation value of a is

$$⟨\alpha |a|\alpha ⟩=⟨\alpha |\frac{1}{\sqrt{2}}(Q+iP)|\alpha ⟩=\alpha$$

so

$$⟨\alpha |Q|\alpha ⟩=\sqrt{2}\mathrm{Re}(\alpha ),⟨\alpha |P|\alpha ⟩=\sqrt{2}\mathrm{Im}(\alpha )$$

Note that coherent states are superpositions of diferent states $|n⟩$ , so are not eigenvectors of the number operator $N.$ , and do not describe states with a fixed (or even finite) number of quanta. They are eigenvectors of

$$a=\frac{1}{\sqrt{2}}(Q+iP)$$

with eigenvalue $\alpha$ so one can try and think of $\sqrt{2}\alpha$ as a complex number whose real part gives the position and imaginary part the momentum. This does not lead to a violation of the Heisenberg uncertainty principle since this is not a self-adjoint operator, and thus not an observable. Such states are however very useful for describing certain sorts of physical phenomena, for instance the state of a laser beam, where (for each momentum component of the electromagnetic field) one does not have a definite number of photons, but does have a definite amplitude and phase.

Digression (Spin coherent states). One can perform a similar construction replacing the group $H_3$ by the group $SU(2)$ , and the state |0⟩ by a highest weight vector of an irreducible representation $({\pi }_n,V^n={\mathbf{C}}^{n+1})$ of spin $\frac{n}{2}$ . Writing $|\frac{n}{2}$ for such a highest weight vector, we have

$${\pi }_n^{\prime }(S_3)|\frac{n}{2}⟩=\frac{n}{2}|\frac{n}{2}⟩,{\pi }_n^{\prime }(S_{+})|\frac{n}{2}⟩=0$$

and we can create a family of spin coherent states by acting on $|\frac{n}{2}$ by elements of $SU(2)$ . If we identify states in this family that difer only by a phase, the states are parametrized by a sphere.

For the case $n=1$ , this is precisely the Bloch sphere construction of section ${}^{7.5,}$ where we took as highest weight vector $|\frac{1}{2}⟩=(\genfrac{}{}{0ex}{}{1}{0})$ . In that case, all states in the representation space ${\mathbf{C}}^2$ were spin coherent states (identifying states that difer only by scalar multiplication). For larger values of $n,$ only a subset of the states in ${\mathbf{C}}^{n+1}$ will be spin coherent states.

Chapter contents · Next section: 23.2 Coherent states and the Bargmann-Fock state space · 中文版本

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