26.5 Complex structures for d=1 and squeezed states

26.5 Complex structures for d=1 and squeezed states

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · complexification

To get a better understanding of what happens for other complex structures than $J_0$ , in this section we’ll examine the case $d=1$ . We can generalize the choice $J=J_0$ , where a basis of ${\mathcal{M}}_{J_0}^{+}$ is given by

$$z=\frac{1}{\sqrt{2}}(q-ip)$$

by replacing the i by an arbitrary complex number $\tau$. Then the condition that $q-\tau p$ be in ${\mathcal{M}}_J^{+}$ and its conjugate in ${\mathcal{M}}_J^{-}$ is

$$J(q-\tau p)=J(q)-\tau J(p)=i(q-\tau p)$$ $$J(q-\overline{\tau }p)=J(q)-\overline{\tau }J(p)=-i(q-\overline{\tau }p)$$

Subtracting and adding the two equations gives

$$J(p)=-\frac{1}{\mathrm{Im}(\tau )}q+\frac{\mathrm{Re}(\tau )}{\mathrm{Im}(\tau )}p$$

and

$$J(q)=-\frac{\mathrm{Re}(\tau )}{\mathrm{Im}(\tau )}q+\left(\mathrm{Im}(\tau )+\frac{(\mathrm{Re}(\tau ){)}^2}{\mathrm{Im}(\tau )}\right)p$$

respectively. Generalizing 26.4, the matrix for $J$ is

$$J=\left(\begin{array}{cc}-\frac{\mathrm{Re}(\tau )}{\mathrm{Im}(\tau )}& \mathrm{Im}(\tau )\frac{(\mathrm{Re}(\tau ){)}^2}{\mathrm{Im}(\tau )}\\ -\frac{1}{\mathrm{Im}(\tau )}& \frac{\mathrm{Re}(\tau )}{\mathrm{Im}(\tau )}\end{array}\right)=\frac{1}{\mathrm{Im}(\tau )}\left(\begin{array}{cc}-\mathrm{Re}(\tau )& |\tau {|}^2\\ -1& \mathrm{Re}(\tau )\end{array}\right)\text{\hspace{1em}(26.17)}$$

and it can easily be checked that det $J=1$ , so $J\in SL(2,\mathbf{R})$ and is compatible with Ω.

The positivity condition here is that $\Omega (\cdot ,J\cdot )$ is positive on $\mathcal{M}$ , which in terms of matrices (see 16.2) becomes the condition that the matrix

$$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)J^T=\frac{1}{\mathrm{Im}(\tau )}\left(\begin{array}{cc}|\tau {|}^2& \mathrm{Re}(\tau )\\ \mathrm{Re}(\tau )& 1\end{array}\right)$$

$\mathrm{gi}$ ves a positive quadratic form. This will be the case when $\mathrm{Im}(\tau )>0$ . We have thus constructed a set of J that are positive, compatible with $\Omega ,$ and parametrized by an element $\tau$ of the upper half-plane, with $J_0$ corresponding to $\tau =i$

To construct annihilation and creation operators satisfying the standard commutation relations

$$[a_{\tau },a_{\tau }]=[a_{\tau }^{†},a_{\tau }^{†}]=0,[a_{\tau },a_{\tau }^{†}]=1$$

set

$$a_{\tau }=\frac{1}{\sqrt{2\mathrm{Im}(\tau )}}(Q-\overline{\tau }P),a_{\tau }^{†}=\frac{1}{\sqrt{2\mathrm{Im}(\tau )}}(Q-\tau P)$$

The Hamiltonian

$$H_{\tau }=\frac{1}{2}(a_{\tau }{a}_{\tau }^{†}+a_{\tau }^{†}{a}_{\tau })=\frac{1}{2\mathrm{Im}(\tau )}(Q^2+|\tau {|}^2{P}^2-\mathrm{Re}(\tau )(QP+PQ))\text{\hspace{1em}(26.18)}$$

will have eigenvalues $n+\frac{1}{2}$ for $n=0,1,2,\cdots$ . Its lowest energy state will satisfy

$$a_{\tau }|0{⟩}_{\tau }=0\text{\hspace{1em}(26.19)}$$

which in the Schr¨odinger representation is the diferential equation

$$(Q-\overline{\tau }P)\psi (q)=\left(q+i\overline{\tau }\frac{d}{dq}\right)\psi (q)=0$$

which has solutions

$$\psi (q)\propto e^{i\frac{\tau }{2|\tau {|}^2}{q}^2}\text{\hspace{1em}(26.20)}$$

This will be a normalizable state for Im $.(\tau )>0$ , again showing the necessity of the positivity condition.

Eigenstates of $H_{\tau }$ for $\tau =ic,c>0$ real, are known as “squeezed states” in physics. By equation 26.20 the lowest energy state |0⟩ will have spatial dependence proportional to

$$e^{-\frac{1}{2c}{q}^2}$$

and higher energy eigenstates |n⟩ will also have such a Gaussian factor in their position dependence. For $c<1$ such states will have narrower spatial width than conventional quanta (thus the name “squeezed”), but wider width in momentum space. For $c>1$ the opposite will be true. In some sense that we won’t try to make precise, the limits as $c\to \infty$ and $c\to 0$ correspond to the Schr¨odinger representations in position and momentum space respectively (with the distinguished Bargmann-Fock state |0⟩ approaching the constant function in position or momentum space).

The subgroup $\mathbf{R}\subset SL(2,\mathbf{R})$ of equation 24.7 acts non-trivially on $J_0.$ , by

$$J_0=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\to \left(\begin{array}{cc}{e}^r& 0\\ 0& e^{-r}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{cc}{e}^{-r}& 0\\ 0& e^r\end{array}\right)=\left(\begin{array}{cc}0& e^{2r}\\ -e^{-2r}& 0\end{array}\right)$$

taking $J_0$ to the complex structure with parameter $\tau =i{e}^{2r}$

Recall from section 24.4 that changing from $q,p$ coordinates to $z,\overline{z}$ coordinates on the complexified phase space, the group $SL(2,\mathbf{R})$ becomes the isomorphic group $SU(1,1)$ , the group of matrices

$$\left(\begin{array}{cc}\alpha & \beta \\ \overline{\beta }& \overline{\alpha }\end{array}\right)$$

satisfying

$$|\alpha {|}^2-|\beta {|}^2=1$$

Looking at equation 24.11 that gives the conjugation relating the two groups, we see that

$$\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)\in SU(1,1)\leftrightarrow \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\in SL(2,\mathbf{R})$$

and as expected, in these coordinates $J_0$ acts on $z$ by multiplication by $i,$ on $\overline{z}$ by multiplication $\mathrm{by}-i$

$SU(1,1)$ matrices can be parametrized in terms of $t,\theta ,{\theta }^{\prime }$ by taking

$$\alpha =e^{i{\theta }^{\prime }}\cosh t,\beta =e^{i\theta }\sinh t$$

Such matrices will have square −1 and give a positive complex structure when ${\theta }^{\prime }=\frac{\pi }{2}$ , so of the form

$$\left(\begin{array}{cc}i\cosh t& -e^{i\theta }\sinh t\\ -e^{-i\theta }\sinh t& -i\cosh t\end{array}\right)$$

The $U(1)$ subgroup of $SU(1,1)$ preserving $J_0$ will be matrices of the form

$$\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)$$

Using the matrix from equation 24.9, the subgroup $\mathbf{R}\subset SL(2,\mathbf{R})$ of equation 24.7 takes $J_0$ to

$$\left(\begin{array}{cc}\cosh r& \sinh r\\ \sinh r& \cosh r\end{array}\right)\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)\left(\begin{array}{cc}\cosh r& -\sinh r\\ -\sinh r& \cosh r\end{array}\right)=i\left(\begin{array}{cc}-\cosh 2r& \sinh 2r\\ -\sinh 2r& \cosh 2r\end{array}\right)$$

Previous section: 26.4 Complex vector spaces with Hermitian inner product as phase spaces · Next section: 26.6 Complex structures and Bargmann-Fock quantization for arbitrary d · 中文版本

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