Coherent States and their existence

Question

In my Quantum Mechanic class, I have learned that to solve for any quantum system, we solve the time independent Schrodinger equation(for time independent Hamiltonian) and then apply the time evolution over it using any of the Schrodinger or Heisenberg picture. When we apply the time independent SE, we get the eigenstates of the Hamiltonian operator and our state ket $|\psi>$ is represented using the eigenkets of $H$ in x-basis. Now, for a harmonic oscillator, do the coherent states define some new kind of state $\psi_{1} $ or is it the same state $\psi$? If it is the same, why do we give so much importance to coherent states?

Answer 1

Preamble:

(This part taken from these lecture notes.) The idea of the coherent state originated with the study by Glauber of the coherence properties of quantized light. As the electromagnetic field quantizes as a harmonic oscillator, a single mode electric field can be written as \begin{align} E(r,t)= E_0 \boldsymbol{\epsilon} e^{-i(\vec k\cdot \vec r-\omega t)}\hat a+ E_0 \boldsymbol{\epsilon} e^{i(\vec k\cdot \vec r-\omega t)}\hat a^\dagger =\hat E^+(r,t)+\hat E^-(r,t) \end{align} with $\hat E^+(r,t)\sim \hat a e^{+i\omega t}$ and $\hat E^-(r,t)\sim \hat a^\dagger e^{-i\omega t}$ where $\boldsymbol{\epsilon}$ accounts for the polarization of the field.

For final and initial states $\vert f\rangle$ and $\vert i\rangle$ respectively, the probability of detecting a photon is proportional to $$ P_{fi}=\vert\langle f\vert \hat E^+(r, t)\vert i\rangle\vert^2 $$ (since the photon must be annihilated from the initial state) so the light intensity at a point $r$ is obtained by summing over all final states $$ I(r,t)=\sum_{f} P_{fi}=\langle i\vert \hat E^{-} (r, t)\hat E^{+}(r, t)\vert i\rangle =\hbox{Tr}\left(\rho \hat E^{-}(r,t) \hat E^{+}(r,t)\right) $$ for $\rho=\sum_{i,j} p_{ij}\vert i\rangle \langle j\vert $. The quantity $$ G^{(1)}(r,r')= \hbox{Tr}\left(\rho \hat E^{-}(r,t) \hat E^{+}(r',t)\right) $$ and more generally $$ G^{(n)}(r_1,\dots,r_n, r'_1,\ldots r'_n): = \hbox{Tr}\left(\rho \hat E^{-}(r_1,t) \ldots \hat E^{-}(r_n,t) \hat E^{+}(r_1',t)\ldots \hat E^{+}(r_n',t)\right) $$ is the $n$'th order coherence function.

For two sources located at $r_1$, $r_2$, the intensity of light at a point $r$ is thus obtained from \begin{align} I(r, t)\sim G^{(1)}(r_1,r_1)+G^{(1)}(r_1,r_1) + 2 \vert G^{(1)}(r_1,r_2)\vert \cos(\varphi(x_1,x_2)) \end{align} where the correlation functions are computed using \begin{align} \hat E^+(r, t)&= E_1^+(r, t)+ E_2^+ (r, t)\, ,\\ \hat E_i^{+}(r, t)&=E_i^+\left(r_i, t-\frac{s_i}{c}\right)\frac{e^{i(k-\omega/c)}}{s_i} \end{align} where $s_i$ is the distance between source $i$ and the point $r$ where the intensity is to be computed. Thus, if $G^{(1)}(r_1,r_2)\ne 0$, the intensity will display interference fringes. The maximum is reached when $G^{(1)}(r_1,r_2)$ factorizes into the product of two functions \begin{align} G^{(1)}(r_1,r_2)= f_1(r_1)f_2(r_2)\, . \end{align} The field is said to be coherent to $n$-th order if $G^{(n)}$ factorizes.

Glauber coherent states:

This happens when the state is an eigenstate of the harmonic oscillator. Glauber showed that the harmonic oscillator coherent states (also called Glauber coherent states) \begin{align} \vert \alpha\rangle=\sum_{n}\frac{\alpha^n}{\sqrt{n!}}\vert n\rangle \qquad \alpha\in \mathbb{C} \tag{1} \end{align} are coherent to all orders and indeed satisfy $\hat a\vert\alpha\rangle=\alpha\vert\alpha\rangle$. Note that, since $\hat a$ is not hermitian, its eigenvalues are not necessarily real.

It turns out that one can also write \begin{align} \vert \alpha \rangle &= D(\alpha)\vert 0\rangle \, , \qquad D(\alpha)=e^{\alpha \hat a^\dagger-\alpha^*\hat a} \tag{2} \end{align} where $D(\alpha)$ is a translation operator in $(x,p)$ space that will translate the ground state to a point $(x_0,p_0)$ given by the real and imaginary parts of $\alpha$. One can easily show that \begin{align} \hat a \vert\alpha\rangle = \hat a D(\alpha)\vert 0\rangle = D(\alpha)(\hat a+\alpha)\vert 0\rangle = \alpha D(\alpha)\vert 0\rangle =\alpha \vert\alpha\rangle\, . \end{align} These states have a number of interesting properties. For instance, they saturate the uncertainty relation in the sense that \begin{align} \Delta x\Delta p=\frac{\hbar}{2} \end{align} for all times. The individual average values \begin{align} \langle x(t)\rangle \sim \cos(\omega t+ \phi_0)\, ,\qquad \langle p(t)\rangle \sim \sin(\omega t+ \phi_0) \tag{3} \end{align} as for a classical harmonic oscillator. (See this post and this post for details.)

The coherent states of Eq.(1) are NOT eigenstate of the harmonic oscillator Hamiltonian, so their time evolution is non-trivial. However, it turns out that $\vert\alpha(t)\rangle =\vert e^{i\omega t}\alpha(0)\rangle$, i.e. the evolution works out to be simple as the complex parameter $\alpha$ just picks up a time-dependent phase. This is the root of the behaviour of $\langle x(t)\rangle $ and $\langle p(t)\rangle $ in Eq.(3). The coherent states are temporally stable, in the sense that the shape of the probability distribution simply sloshes back and forth in time without distorting.

Perelmov coherent states:

Perelomov capitalized on the displacement property of Eq.(2) to define the generalized coherent state as a translate of a specified (or fiducial) state. For angular momentum, the (Perelomov) coherent state is then \begin{align} \vert\theta,\varphi\rangle = R(\theta,\varphi)\vert J,J\rangle := R_z(\varphi)R_y(\theta)\vert J,J\rangle\, \end{align} (see also this post on spin coherent states).

Using the usual tools, this can also be written as \begin{align} \vert\theta,\varphi\rangle &= e^{\zeta' J_-}e^{-\eta J_z}e^{\zeta J_+}\vert L,L\rangle \, ,\\ &=e^{\zeta' J_-}e^{-\eta J_z}\vert L,L\rangle \end{align} where \begin{align} \zeta=\tan\frac{\theta}{2}e^{i\varphi}\, ,\qquad \eta= -2\log(\cos\vert\zeta\vert)\, . \end{align} Because the space spanned by $\{\vert J,M\rangle, M=-J,\ldots,J\}$ is finite-dimensional, it is easily shown that $J_-$ cannot have eigenstates, so this property of harmonic oscillator coherent states does not generalize. Nevertheless, generalized coherent states do share other properties with the harmonic-oscillator coherent states. For instance, \begin{align} \Delta \tilde J_x\Delta\tilde J_y=\frac{1}{4}\langle \tilde J_z\rangle \end{align} where $\tilde J_i= R(\theta,\varphi) J_i R^{-1}(\theta,\varphi)$ and all quantities are evaluated for the state $\vert\theta,\varphi\rangle$. Again, these are not eigenstates of $J_z$ but they have "nice" evolution properties when $H=\omega J_z$.

Connection with classical limit:

Finally, Onofri [in Onofri, Enrico. "A note on coherent state representations of Lie groups." Journal of Mathematical Physics 16.5 (1975): 1087-1089; unfortunately I cannot find an open-access file for this] showed that, with the definition of Perelomov, one could construct a (classical) Poisson bracket using the variable $\zeta$ and its conjugate, or the generalization of this variable when something other than angular momentum is considered. In the case of angular momentum, this bracket is written as \begin{align} \{f,g\}= \frac{\partial f}{\partial \zeta}\frac{\partial f}{\partial \zeta^*} -\frac{\partial f}{\partial \zeta^*}\frac{\partial f}{\partial \zeta} \end{align} which can be expressed, using the explicit expression for $\zeta$ in terms of the angular variables $\theta,\varphi$, up to scaling factors, as \begin{align} \{f,g\}=\frac{1}{r\sin\theta} \left(\frac{\partial f}{\partial \theta}\frac{\partial f}{\partial \varphi} -\frac{\partial f}{\partial \varphi}\frac{\partial f}{\partial \theta}\right) \end{align} thus highlighting the role of coherent states in the transition from quantum to classical mechanics.