Coherent States

A coherent state $\left\vert \alpha \right>$ is an eigenstate of the annihilation operator ${\hat{a}}$ with respect to the eigenvalue $\alpha\in\mathbb{C}$:

$${\hat{a}}\left\vert \alpha \right> \; = \; \alpha\left\vert \alpha \right>.$$ (A.24)

These states $\left\vert \alpha \right>$ are especially useful when dealing with operators that are expanded in terms of the ladder operators ${\hat{a}}^\dagger$, ${\hat{a}}$: an operator in this form can easily be applied to a coherent state by treating ${\hat{a}}$ as the c-number $\alpha$, i.e. by evaluating expressions of the form $f({\hat{a}})\left\vert \alpha \right>=f(\alpha)\left\vert \alpha \right>$ and $\left< \alpha \right\vert g({\hat{a}}^\dagger)=\left< \alpha \right\vert g(\alpha^*)$. Situations like this are frequently encountered in quantum optics. An example for the application of coherent states in this field is the modelling of stationary vibrational states of a laser-driven ion in an ion trap [DMFV96a,DMFV96b].

An explicit formula for coherent states can be found by expanding $\left\vert \alpha \right>$ in terms of the eigenstates of the harmonic oscillator, or number states $\left\vert n \right>$, $n\in\mathbb{N}_0$, substituting into the definition (A.34) and equating the coefficients of the $\left\vert n \right>$. The result is

$$\left\vert \alpha \right> \; = \; e^{-\frac{1}{2}\vert\alpha... ...n=0}^\infty \frac{1}{\sqrt{n!}} \alpha^n \left\vert n \right>$$ (A.25)

for all $\alpha\in\mathbb{C}$. Here, the factor $e^{-\frac{1}{2}\vert\alpha\vert^2}$ is chosen in such a way that $\left\vert \alpha \right>$ is normalized, since for all $\alpha,\beta\in\mathbb{C}$ one has:

$$\left< \alpha \left\vert \beta \right> \right. \; = \; e^{\... ...\alpha\vert^2+\vert\beta\vert^2)}, % \quad \alpha,\beta\in\CC$$ (A.26)

and therefore $\left< \alpha \left\vert \alpha \right> \right.=1$. Equations (A.34) and (A.35) show that every $\alpha\in\mathbb{C}$ is an eigenvalue of ${\hat{a}}$ and thus defines a coherent state $\left\vert \alpha \right>$. Note that the $\left\vert \alpha \right>$ are not pairwise orthogonal, as the scalar product $\left< \alpha \left\vert \beta \right> \right.$ is not zero for $\alpha\neq\beta$.

In order to come to a more intuitive understanding of coherent states it is useful to consider WEYL's unitary displacement operator [MM71]

$$\hat{D}(\alpha) \; := \; e^{\alpha{\hat{a}}^\dagger-\alpha^*{\hat{a}}} \; = \; e^{\frac{i}{\hbar}(p'{\hat{x}}-x'{\hat{p}})}$$ (A.27)

with the real parameters $x'$ and $p'$, which are essentially the real and imaginary parts of $\alpha$:

$$\alpha \; = \; \alpha(x',p') \; =: \; \frac{1}{\sqrt{2\hbar m_0\omega_0}} \left( m_0\omega_0 x'+ip' \right).$$ (A.28)

Formally, $\alpha$ is obtained from the definition of the annihilation operator (A.24) by exchanging the operators ${\hat{x}}$, ${\hat{p}}$ for the scalars $x'$, $p'$.

As indicated by its name, the displacement operator acts on a state by shifting it in phase space. Using the BAKER-CAMPBELL-HAUSDORFF formula (A.4) one can derive the product representation

$$\hat{D}(\alpha) \; = \; e^{-\frac{i}{2\hbar} x' p'} e^{ \frac{i}{ \hbar} p'{\hat{x}}} e^{-\frac{i}{ \hbar} x'{\hat{p}}}$$ (A.29)

of $\hat{D}(\alpha)$, which turns out to be the composition of the two translation operators $e^{-\frac{i}{\hbar}x'{\hat{p}}}$ and $e^{\frac{i}{\hbar}p'{\hat{x}}}$ -- in addition to these there is also the phase factor $e^{-i\frac{x'p'}{2\hbar}}$ which is irrelevant for the interpretation of $\hat{D}(\alpha)$. The translation operators each move a wave packet in position and momentum space by $x'$ and $p'$, respectively,

such that in phase space

$$\hat{D}(x',p') \; := \; \hat{D}\big(\alpha(x',p')\big)$$ (A.29)

moves the wave packet by the vector $(x',p')^t$. The translation operator $\hat{T}(\cdot)$ of equation (A.11) is a special case of the more general $\hat{D}(\cdot,\cdot)$: $\hat{T}(x')=\hat{D}(x',0)$.

Using the displacement operator one can now write a coherent state as

$$\left\vert \alpha \right> \; = \; e^{-\frac{1}{2}\vert\alpha... ...t\vert 0 \right> \; = \; \hat{D}(\alpha)\left\vert 0 \right>.$$ (A.30)

This implies that all coherent states can be generated by $\hat{D}(\alpha)$ acting on the coherent state defined by $\alpha=0$. This particular coherent state $\left\vert \alpha=0 \right>$ is identical with the ground state $\left\vert n=0 \right>$ of the harmonic oscillator, as can be concluded from equation (A.35).

With this result it is now clear how the set of all coherent states $\left\{ \left\vert \alpha \right>,\alpha\in\mathbb{C}\right\}$ is to be interpreted: it is obtained by moving the ground state of the harmonic oscillator to all points of the phase plane. Therefore the well-known properties of $\left\vert n=0 \right>$,

$$\left< x \left\vert 0 \right> \right. \; = \; \sqrt[4]{\frac... ...2} \quad \mbox{with} \quad \sigma^2=\frac{\hbar}{m_0\omega_0}$$ (A.31)

(see equation (2.39)), carry over to each of the coherent states. With the parameters $x'$, $p'$ of equation (A.38) one then has for all states $\left\vert \alpha \right>$:

In particular, the coherent states are minimum uncertainty states, i.e. they are characterized by the smallest possible product of standard deviations of the position and momentum operators as given by the HEISENBERG uncertainty relation:

$$(\Delta{\hat{x}})(\Delta{\hat{p}}) \; = \; \frac{\hbar}{2}.$$ (A.31)

In addition, setting $m_0\omega_0=1$ (for example by a suitable scaling as in section 2.1) one obtains coherent states for which the standard deviations of position and momentum are the same:

$$\Delta{\hat{x}}\; = \; \Delta{\hat{p}}\; = \; \sqrt{\frac{\hbar}{2}}.$$ (A.32)

In the form (A.35) coherent states were first constructed by SCHRÖDINGER, who intended to use them to demonstrate the ``continuous transition from micro- to macromechanics'' [Sch26]. He showed that a wave packet (A.35), when specified as the initial state submitted to the potential of the harmonic oscillator, does not broaden during the dynamics.^A.8With the according Hamiltonian

$$\H_{\rm ho} \; = \; \frac{1}{2m_0}{\hat{p}}^2+\frac{1}{2}m_... ...r\omega_0\left( {\hat{a}}^\dagger{\hat{a}}+\frac{1}{2} \right)$$ (A.33)

($\H_{\rm ho}$ is identical with $\H_{\mbox{\scriptsize free}}$ of equations (2.29a) and (2.31) before scaling) one obtains for the time evolution of $\left\vert \alpha \right>$ after time $t$:

$$e^{-\frac{i}{\hbar}\H_{\rm ho}t}\left\vert \alpha \right> \;... ...rac{i}{2}\omega_0 t}\left\vert e^{-i\omega_0 t}\alpha \right>,$$ (A.34)

that is, again a (different) coherent state, except for a phase factor. Therefore the uncertainties $\Delta {\hat{x}}$, $\Delta {\hat{p}}$ are conserved, which is just the expected behaviour for a classical particle.^A.9Further information about the time evolution of coherent states may be found in [Ger92].

The equations (A.39-A.43) also show that the state $\left\vert x',p' \right>$ of equation (A.29) in fact is a coherent state up to a phase factor,

$$\left< x \left\vert \alpha \right> \right. \; = \; \sqrt[4]... ...rt x',p' \right> \right. e^{-i\textstyle \frac{x'p'}{2\hbar}},$$ (A.34)

and with equation (A.30) one has for the antinormal-ordered distribution function:

$$F^{\rm AN}(x,p,t) \; = \; \frac{1}{2\pi\hbar} \; \big\vert \left< \alpha(x,p) \left\vert \psi(t) \right> \right. \big\vert^2.$$ (A.35)

For this reason the antinormal-ordered distribution function is often called the coherent state representation of the state $\left\vert \psi(t) \right>$ (see e.g. [ABB96]).

It is interesting to note that the set of coherent states $\left\{ \left\vert \alpha \right>,\alpha\in\mathbb{C}\right\}$ is overcomplete:

$$% \frac{1}{\pi} \int \Ket{\alpha}\Bra{\alpha} \dop^2\alpha ... ... := {\mbox{d}}(\mbox{Re}\alpha) \, {\mbox{d}}(\mbox{Im}\alpha)$$ (A.36)

-- in contrast, for instance, to the set $\left\{ \left\vert n \right>,n\in\mathbb{N}_0 \right\}$ of eigenstates of the harmonic oscillator, which is ``only'' complete. Overcompleteness of $\left\{ \left\vert \alpha \right> \right\}$ is indicated by the factor $1/\pi$ in equation (A.52). Overcompleteness of $\left\{ \left\vert \alpha \right> \right\}$ guarantees that every state $\left\vert \psi \right>$ can be expanded into a superposition of coherent states; but due to the nonorthogonality of these states as expressed by equation (A.36) this expansion is not unique, in general.^A.10

Using the overcompleteness property (A.52) of the coherent states and the expression (A.51) for $F^{\rm AN}$ it is easy to show that the antinormal-ordered distribution is normalized in the sense of

$$\int\limits _{-\infty}^\infty\!\! {\mbox{d}}x\!\! \int\lim... ...nfty}^\infty\!\! {\mbox{d}}p\, F^{\rm AN}(x,p,t) \; = \; 1 .$$ (A.37)

For more information on coherent states I refer the reader to the specialist literature: the monograph [KS85] remains the standard work in this field; some more recent studies are for example [WK93,KWZ94,ZK94], where among other questions the problem of coherent states in finite-dimensional HILBERT spaces is addressed. Finally, in [Nie97a] NIETO presents an interesting overview of the historic development of the theory of coherent states as well as of the squeezed states which I discuss in the following subsection.

Footnotes

... dynamics.^A.8

But note that SCHRÖDINGER's generalizing interpretation in [Sch26] of this feature of the harmonic oscillator is erroneous. See [Str01b] for some background material on this issue.

... particle.^A.9

Moreover, for the present case of the harmonic oscillator EHRENFEST's theorem allows to conclude that the dynamics of the mean values of the position and momentum operators coincide with the classical dynamics of the observables $x$ and $p$, namely a harmonic oscillation bounded by turning points which are determined by the energy $E=\hbar\omega_0\vert\alpha\vert^2$:

with the phase shift $\varphi_0=\arg(\alpha)$.

... general.^A.10

By imposing certain additional conditions on the expansion coefficients it is still possible to achieve a unique expansion in terms of coherent states. More on this GLAUBER expansion can be found in [Per93].