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Coherent state

A coherent state is a translated base state of the harmonic oscillator. Coherent states are interesting because they behave classically in some ways.

∣αˉ⟩=Tα∣0⟩=e−iαp^/ℏ∣0⟩. \begin{align*} \ket{\bar \alpha} = T_{\alpha} \ket{0} = e^{-i \alpha \hat p /\hbar} \ket 0. \end{align*}

Consider the expectation values.

⟨αˉ∣x∣αˉ⟩=α⟨αˉ∣p∣αˉ⟩=⟨0∣Tα†pTα∣0⟩=⟨0∣pTα†Tα∣0⟩=⟨0∣p∣0⟩=0⟨αˉ∣H∣αˉ⟩=⟨0∣Tα†(12mp2+12mω2x2)Tα∣0⟩=⟨0∣12mp2+mω22(Tα†xTαTα†xTα)∣0⟩=⟨0∣12mp2+mω22(x+α)2∣0⟩=⟨0∣H∣0⟩+mω2α⟨0∣x∣0⟩+mω22x02⟨0∣0⟩=12ℏω+12mω2α2. \begin{align*} \braket{\bar\alpha | x | \bar\alpha} &= \alpha \\ \braket{\bar\alpha | p | \bar\alpha} &= \braket{0 | T_\alpha\adj p T_\alpha |0} = \braket{0 | p T_\alpha\adj T_\alpha |0} = \braket{0|p|0} = 0 \\ \braket{\bar\alpha | H | \bar\alpha} &= \braket{0 | T_\alpha\adj \left( \frac{1}{2m} p^2 + \frac{1}{2} m\omega^2 x^2 \right) T_\alpha |0} \\ &= \braket{0| \frac{1}{2m} p^2 + \frac{m\omega^2}{2} \left( T_\alpha\adj x T_\alpha T_\alpha\adj x T_\alpha \right) | 0}\\ &= \braket{0 | \frac{1}{2m} p^2 + \frac{m\omega^2}{2} (x + \alpha)^2 |0} \\ &= \braket{0|H|0} + m \omega^2 \alpha \braket{0 | x | 0} + \frac{m\omega^2}{2} x_0^2 \braket{0|0} \\ &= \frac{1}{2} \hbar\omega + \frac{1}{2} m\omega^2 \alpha^2. \end{align*}

This looks like the classical result, with the addition of the zero-point energy term! We also get an interesting result from the Heisenberg time evolution.

⟨αˉ∣xH∣αˉ⟩=⟨αˉ+xScos⁡ωt+1mωpSsin⁡ωt∣αˉ⟩=αcos⁡ωt⟨αˉ∣pH∣αˉ⟩=⟨αˉ∣pScos⁡ωt−mωxSsin⁡ωt∣αˉ⟩=−mωαsin⁡ωt. \begin{align*} \braket{\bar\alpha | x_H | \bar\alpha} &= \braket{\bar\alpha + x_S \cos \omega t + \frac{1}{m\omega} p_S \sin\omega t|\bar\alpha} \\ &= \alpha \cos\omega t \\ \braket{\bar\alpha | p_H | \bar\alpha} &= \braket{\bar\alpha | p_S \cos\omega t - m\omega x_S \sin\omega t | \bar\alpha} \\ &= -m\omega \alpha \sin\omega t. \end{align*}

We see the expectation values oscillate and ⟨pH⟩=ddt⟨xH⟩ \braket{p_H} = \frac{d }{d t} \braket{x_H}. Now consider the uncertainties

⟨αˉ∣x2∣αˉ⟩=⟨0∣(x+α)2∣0⟩=⟨0∣x2∣0⟩+α2⟨0∣0⟩=ℏ2mω+α2⟨αˉ∣p2∣αˉ⟩=⟨0∣p2∣0⟩=mℏω2ΔxΔp=ℏ2. \begin{align*} \braket{\bar \alpha | x^2 |\bar\alpha} &= \braket{0 | (x + \alpha)^2 | 0} \\ &= \braket{0 | x^2 | 0} + \alpha^2 \braket{0|0} \\ &= \frac{\hbar}{2m\omega} + \alpha^2 \\ \braket{\bar \alpha | p^2 | \bar \alpha} &= \braket{0 | p^2 | 0} = \frac{m\hbar\omega}{2} \\ \Delta x \Delta p &= \frac{\hbar}{2}. \end{align*}

We can repeat the exercise for xHx_H and pHp_H and see that the uncertainty bound is saturated for all time. This is interesting, it means that the wavepacket remains coherent as it oscillates over time. Again, this looks classical.

We can write a coherent state in terms of the energy eigenstates it’s composed of. First write TαT_\alpha in terms of the raising and lowering operators. We apply the Baker-Campbell-Hausdorff formula and use the fact that [a†,a][a\adj,a] is a constant.

Tα=e−iαp/ℏ=eα(a†−a)/2d where d=ℏmω=eαa†/2de−αa/2de−α2/4d2. \begin{align*} T_\alpha &= e^{-i \alpha p/\hbar} = e^{\alpha (a\adj - a) / \sqrt 2 d} \text{ where } d = \sqrt{\frac{\hbar}{m\omega}} \\ &= e^{\alpha a\adj/\sqrt 2 d} e^{-\alpha a / \sqrt 2 d} e^{-\alpha^2 / 4d^2}. \end{align*}

Then expand a coherent state in terms of this operator. Looking at the last exponential, we notice that a∣0⟩=0a \ket 0 = 0, so all but the first term of the Taylor expansion will be annihilated.

∣αˉ⟩=e−α2/4d2eαa†/2de−αa/2d∣0⟩=e−α2/4d2∑n1n!(α2d)n(a†)n∣0⟩=e−α2/4d2∑n1n!(α2d)n∣n⟩=∑ncn∣n⟩. \begin{align*} \ket{\bar \alpha} &= e^{-\alpha^2 / 4d^2} e^{\alpha a\adj / \sqrt 2d} e^{-\alpha a / \sqrt 2d} \ket 0 \\ &= e^{-\alpha^2/4d^2} \sum_n \frac{1}{n!} \left( \frac{\alpha}{\sqrt 2d} \right) ^n \left( a\adj \right)^n \ket 0 \\ &= e^{-\alpha^2 / 4d^2} \sum_n \frac{1}{\sqrt{n!}} \left( \frac{\alpha}{\sqrt 2d} \right) ^n \ket n \\ &= \sum_n c_n \ket n. \end{align*}

The terms ∣cn∣2|c_n|^2 form a Poisson distribution with mean λ=α2/2d2\lambda = \alpha^2 / 2d^2. Immediately we see ⟨N⟩=λ \braket{N} = \lambda and ΔN=λ\Delta N = \sqrt \lambda. Since H=(N+12)ℏωH = (N + \frac 12) \hbar\omega, we know ΔH=ℏωλ\Delta H = \hbar \omega \sqrt \lambda.

So far in this derivation we have set α∈R\alpha \in \mathbb R and used the translation operator. We can generate a coherent state to allow α∈C\alpha \in \mathbb C using the displacement operator to define

∣αˉ⟩=D(α)∣0⟩=eαa†−α∗a∣0⟩. \begin{align*} \coh \alpha = D(\alpha) \ket 0 = e^{\alpha a\adj - \alpha\conj a} \ket 0. \end{align*}

Another way to define coherent states is as eigenstates of the lowering operator

a∣αˉ⟩=aeαa†−α∗a∣0⟩=[a,eαa†−α∗a]∣0⟩,since a∣0⟩=0=[a,αa†−α∗a]eαa†−α∗a∣0⟩,since [a,a†] constant=eαa†−α∗a∣0⟩a∣αˉ⟩=α∣αˉ⟩. \begin{align*} a\coh\alpha &= a e^{\alpha a\adj - \alpha\conj a} \ket 0 \\ &= [a,e^{\alpha a\adj - \alpha\conj a}] \ket 0, && \text{since } a \ket 0 = 0 \\ &= [a, \alpha a\adj - \alpha\conj a] e^{\alpha a\adj - \alpha\conj a} \ket 0, && \text{since } [a,a\adj] \text{ constant} \\ &= e^{\alpha a\adj - \alpha\conj a} \ket 0 \\ a\coh \alpha &= \alpha \coh \alpha. \end{align*}

We see complex α\alpha is allowed since aa is not hermitian. We can interpret the real and imaginary parts physically by considering the position and momentum of a generalized coherent state.

⟨αˉ∣x∣αˉ⟩=d2⟨αˉ∣a†+a∣αˉ⟩=d2(α∗+α)⟨αˉ∣αˉ⟩=2d Re⁡[α],⟨αˉ∣p∣αˉ⟩=iℏ2d⟨αˉ∣a†−a∣αˉ⟩=2ℏdIm⁡[α]. \begin{align*} \braket{\bar\alpha | x | \bar\alpha} &= \frac{d}{\sqrt 2} \braket{\bar \alpha | a\adj + a | \bar\alpha} \\ &= \frac{d}{\sqrt 2} (\alpha\conj + \alpha) \braket{\bar \alpha | \bar\alpha} \\ &= \sqrt 2 d\, \re[\alpha], \\ \braket{\bar\alpha | p | \bar\alpha} &= \frac{i\hbar}{\sqrt 2d} \braket{\bar\alpha|a\adj-a|\bar\alpha} = \sqrt 2\frac{\hbar}{d} \im[\alpha]. \end{align*}

The real component of α\alpha corresponds to a translation in xx, and the complex component to a translation in pp. In position space, we see (where d=h/mωd = \sqrt{h/m\omega})

αa†−α∗a=(x02d+ip0d2ℏ)12d(x−ipmω)−(x02d−ip0d2ℏ)12d(x+ipmω)=idℏ(p0dx−x0pdmω)=iℏ(x0p−p0x). \begin{align*} \alpha a\adj - \alpha\conj a &= \left( \frac{x_0}{\sqrt 2d} + i \frac{p_0d}{\sqrt 2\hbar} \right) \frac{1}{\sqrt 2d} \left( x-i \frac{p}{m\omega} \right) - \left( \frac{x_0}{\sqrt 2d} - i \frac{p_0d}{\sqrt 2\hbar} \right) \frac{1}{\sqrt 2d} \left( x + i \frac{p}{m\omega} \right) \\ &= \frac{i}{d\hbar} \left( p_0 d x - \frac{x_0 p}{dm\omega} \right) \\ &= \frac{i}{\hbar} (x_0 p - p_0 x). \end{align*}

Now consider the Heisenberg time evolution.

DH(α,−t)=e−iHt/ℏDS(α)eiHt/ℏ=eαaH†(−t)−α∗aH(−t)=exp⁡(αe−iωtaS†−α∗eiωtaS)=DS(αe−iωt). \begin{align*} D_H(\alpha,-t) &= e^{-i Ht/\hbar} D_S(\alpha) e^{iHt/\hbar} \\ &= e^{\alpha a_H\adj(-t) - \alpha\conj a_H(-t)} \\ &= \exp \left( \alpha e^{-i\omega t} a_S\adj - \alpha\conj e^{i\omega_t} a_S \right) \\ &= D_S(\alpha e^{-i\omega t}). \end{align*}

Then apply this to the Schrödinger time evolution of the coherent state

∣αˉ,t⟩=e−iHt/ℏDS(α)∣0⟩=e−iHt/ℏDS(α)eiHt/ℏe−iHt/ℏ∣0⟩=DH(α,−t)e−iωt/2∣0⟩=e−iωt/2DS(αe−iωt)∣0⟩=e−iωt/2∣αe−iωt‾⟩. \begin{align*} \ket{\bar\alpha,t} &= e^{-iHt/\hbar} D_S(\alpha) \ket 0 \\ &= e^{-iHt/\hbar} D_S(\alpha) e^{iHt/\hbar} e^{-iHt/\hbar} \ket 0 \\ &= D_H(\alpha,-t) e^{-i\omega t/2} \ket 0\\ &= e^{-i\omega t/2} D_S(\alpha e^{-i\omega t}) \ket 0 \\ &= e^{-i\omega t/2} \ket{\overline{\alpha e^{-i \omega t}}}. \end{align*}

We can also show that coherent states form an (overcomplete) basis. Each coherent state is a linear combination of every other other coherent state – no two states are orthogonal. This makes sense intuitively, every gaussian contains some component of every other gaussian. We can thus write a number state in terms of coherent states.

The overcompleteness is interesting. Whereas there are countably infinite number states, there are uncountable coherent states. This is a possible consequence of coherent states being eigenstates of a non-hermitian operator.