Coherent state (Quantum mechanics)
| John R. Klauder (2009), Scholarpedia, 4(9):8686. | doi:10.4249/scholarpedia.8686 | revision #121886 [link to/cite this article] |
The phrase coherent states refers to a set of vectors in Hilbert space that enjoy several properties. The most widely accepted definition involves a continuous parametrization and a resolution of unity that involves a weighted integral over one dimensional projection operators onto the set of coherent states.
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Generic Formulation
As a generic example, let \( L \) be a label space locally equivalent to the space \( \textbf{R}^n \) and \( l=(l^1,l^2,l^3,\ldots,l^n)\in L\). Hilbert space vectors are represented by \( \textstyle |\psi\rangle\in\mathfrak H\), and the inner product of two such vectors is denoted \( \langle\phi|\psi \rangle \), a complex number linear in the second element and anti-linear in the first element. The set of coherent states \( \{|l\rangle\}\subset \mathfrak{H} \) is determined by a continuous map \( l\rightarrow |l \rangle\)such that if \( l_n\rightarrow l \) then \( \|\,|l_n\rangle-|l\rangle\,\|\rightarrow 0\) where \( \|\,|\psi\rangle\|=+\sqrt{\langle\psi|\psi\rangle} \ .\) Finally, the unit operator \( I \) must be represented by \(I=\int |l\rangle\langle l|\,d\mu(l)\), where \( d\mu(l) \) is a positive, absolutely continuous measure, i.e., \( d\mu(l) =\rho(l)\,d^nl\ ,\) with \(\rho(l)>0\) almost everywhere. The coherent states induce a functional representation of the Hilbert space \(\mathfrak{H}\) by continuous functions \(\psi(l)=\langle l|\psi\rangle\) with the inner product \((\psi,\psi)=\int |\psi(l)|^2\,d\mu(l)\ .\)
Without loss of generality, it is common to assume the coherent states are normalized so that \( \|\,|l\rangle\|=1 ; \) however, the coherent state overlap function \( \langle l'|l\rangle \) is generally nonzero. The overlap function is a continuous function of positive type and thus can serve as the reproducing kernel of a reproducing kernel Hilbert space, which thus admits two distinct expressions with which to compute the Hilbert space inner product. The coherent states are often defined with the aid of a group such as \( |l\rangle=U(l)\, |\eta\rangle \ ,\) where the unit vector \(|\eta\rangle\) is called the fiducial vector.
Canonical Coherent States -- Limited Version
The canonical coherent states are defined as \( |z\rangle=e^{za^\dagger-z^*a}\,|0\rangle\), where \(a\) and \(a^\dagger \) are traditional annihilation and creation operators which obey \( [a,a^\dagger]=I\), and the fiducial vector \(|0\rangle\) satisfies \(a\,|0\rangle=0\ .\) These vectors are also given by
\[\tag{1} |z\rangle=e^{\textstyle -|z|^2/2}\sum_{n=0}^\infty (z^n/\sqrt{n!}\,)\,|n\rangle\;,\]
where the orthonormal vectors \(\{|n\rangle\}_{n=0}^\infty\) satisfy \(a^\dagger a\,|n\rangle=n|n\rangle\ .\)
The coherent state overlap function is given by \(\langle z|z'\rangle=
\exp[-|z|^2/2-|z'|^2/2+z^*z']\ ,\) the integration measure for the resolution of unity is
\(d\mu(z)=dx\,dy/\pi\ ,\) where \(z=x+iy\ ,\) and the domain of integration is the entire
complex plane. The canonical coherent states enjoy the property that \( a\,|z\rangle=z\,|z\rangle\ ,\) and its adjoint \(\langle z|\,a^\dagger=z^*\,\langle z|\ ,\) which implies that the diagonal matrix elements of a normal ordered operator are given by
\[\tag{one:label exists!} \langle z|\Sigma_{m,n} c_{m,n}a^{m\,\dagger}\,a^n\,|z\rangle=\Sigma_{m,n}c_{m,n} \,z^{*\,m}\,z^n\;.\]
This relation implies that if \(\langle z|B|z\rangle=0\) for all \(z\ ,\) then \(B=0\ ,\) and conversely. In addition, the anti-normal ordered operator enjoys a diagonal representation given by
\[\tag{2} \Sigma_{m,n}d_{m,n}a^m\,a^{\dagger \,n}=\int \Sigma_{m,n}d_{m,n}a^m\,|z\rangle\langle z|a^{\dagger \,n}\,d\mu(z) =\int[\Sigma_{m,n}d_{m,n} z^m\,z^{*\,n}]\,|z\rangle\langle z|\,d\mu(z)\;.\]
Equation (2) implies that effectively every operator admits a unique diagonal representation as a weighted integral of coherent state projection operators. This property has been of use in quantum optics.
Canonical Coherent States -- Extended Version
The simple form of the complex version of the canonical coherent states masks some general properties related states enjoy. For this purpose, we pass from a complex notation to a real version where \(z=(q+ip)/\sqrt{2\hbar}\) and \(a=(Q+iP)/\sqrt{2\hbar}\ ,\) where \([Q,P]=i\hbar\,I\ ,\) the usual commutation relation between traditional Heisenberg variables. The coherent states now are chosen, with a different phase convention, as
\[\tag{3} |p,q\rangle=e^{\textstyle-ipQ/\hbar}\,e^{\textstyle iqP/\hbar}\,|\eta\rangle\;, \]
where now the fiducial vector is a general unit vector and the dependence of the
coherent states on the choice of \(|\eta\rangle\) is left implicit. With both \(Q\) and \(P\) chosen self adjoint, the vectors \(|p,q\rangle\) are
continuously labeled, and moreover, for any choice of \(|\eta\rangle \!\ ,\) these states admit a resolution of unity in the form
\[\tag{4} \int |p,q\rangle\langle p,q)\,d\mu(p,q)=I\;, \]
where \(d\mu(p,q)=dp\,dq/2\pi\hbar\ .\) Thus, these states lead to a vast family of distinct coherent state Hilbert space representations by continuous functions based on the relation \(\psi(p,q)=\langle p,q|\psi\rangle\ .\)
Despite the wealth of these representation, the form of Schrödinger's equation is the same for all of them! Given proper domain requirements, this universal form follows, for any choice of \(|\eta\rangle\ ,\) because
\[\tag{5} [-i\hbar(\partial/\partial q)]\psi(p,q)=\langle \eta|\,e^{\textstyle-ipQ/\hbar}\,e^{\textstyle iqP/\hbar}\,P\,|\psi\rangle=\langle p,q|\,P\,|\psi\rangle\;,\]
\[\tag{6} [q+i\hbar(\partial/\partial p)]\psi(p,q)=\langle \eta|\,e^{\textstyle-ipQ/\hbar}\,e^{\textstyle iqP/\hbar}\,Q\,|\psi\rangle=\langle p,q|\,Q\,|\psi\rangle\;. \]
Consequently, Schrödinger's equation becomes
\[\tag{7} i\hbar(\partial/\partial t)\,\psi(p,q,t)=\textbf{H}(-i\hbar(\partial/\partial q), q+i\hbar(\partial/\partial p))\,\psi(p,q,t)\;. \]
The choice of \(|\eta\rangle\) enters through the initial condition. In the coherent state representation, the propagator is a solution to (7) given by the function
\[\tag{8} K(p'',q'',T;p',q',0)=\langle p'',q''|\, e^{\textstyle-iT\textbf{H}/\hbar}\,|p',q'\rangle\;, \]
and the initial condition at \(T=0\ ,\) is given by the coherent state overlap function
\[\tag{9} \lim_{T\rightarrow0} K(p'',q'',T;p',q',0)=\langle p'',q''|p',q'\rangle\;, \]
where the fiducial vector finally enters.
Let us impose the modest conditions \(\langle\eta|P|\eta\rangle=0\) and \(\langle \eta|Q|\eta\rangle=0\ ,\) called physically centered, and which leads to \(\langle p,q|P|p,q\rangle=p\) and
\(\langle p,q|Q|p,q\rangle=q\ ,\) implying that the physical meaning of the labels \((p,q)\) is that of mean values of the operators \((P,Q)\ .\)
For such fiducial vectors, it follows that the coherent state propagator given in (8) admits a formal phase space path integral representation given by
\[\tag{10} K(p'',q'',T;p',q',0) = M\int e^{(i/\hbar)\int[i\hbar\langle p,q|(d/dt)|p,q\rangle-\langle p,q|\textbf{H}|p,q\rangle]dt}\;Dp\,Dq =M\int e^{(i/\hbar)\int[p{\dot q}-H(p,q)]dt}\;Dp\,Dq\;. \]
A proper interpretation of this expression arises from a suitable lattice prescription as can be
seen in the literature. It is quite remarkable that the same formal phase space path integral
admits an alternative interpretation as a typical configuration space propagator as well!
Spin Coherent States
An analogous story applies to other sets of coherent states, such as the spin coherent states, defined for \(\hbar=1\) for a fixed spin value \(s\in\{1/2,1/3/2,2,\ldots\}\) by
\[\tag{11} |\theta,\phi\rangle= e^{\textstyle-i\phi S_3}\,e^{\textstyle-i\theta S_2}\,|s;s\rangle\;, \]
where the fiducial vector \(|s;s\rangle\) is the extremal weight vector of the operator
\(S_3\ ,\) i.e., one of the family of eigenvectors such that \(S_3\,|m;s\rangle=m\,|m;s\rangle\ ,\) \(-s\leq m\leq s\ .\) These coherent states are evidently continuous in their labels, and moreover
\[\tag{12} \int|\theta,\phi\rangle\langle\theta,\phi|\, d\mu(\theta,\phi)=I_s\;,\;\;\;\;\;\;\;\;\;\;\; d\mu(\theta,\phi)=\frac{2s+1}{4\pi}\,\sin(\theta)\,d\theta\,d\phi\;,\]
where \(I_s\) is the \((2s+1)\times(2s+1)\) unit matrix.
These states lead to a coherent state representation of Hilbert space by \(\psi(\theta, \phi)=\langle\theta,\phi|\psi\rangle\ ,\) and the propagator of a dynamical spin Hamiltonian is given by
\[\tag{13} K(\theta'',\phi'',T;\theta',\phi',0)=\int e^{ i\int[i\langle\theta,\phi| (d/dt)|\theta,\phi\rangle-\langle\theta,\phi|\textbf{H}|\theta,\phi\rangle]dt}\;D\mu(\theta.\phi)=\int e^{ i\int[s\cos(\theta){\dot\phi}-H(\theta,\phi)]dt}\,D\mu(\theta,\phi)\;.\]
Generalizations
The two examples offered above are more or less representative of the form of coherent states and one of their applications to dynamics. Many other sets of coherent states are based on different groups -- as well as those not involving groups whatsoever!
An example of the non-group variety is given by the set of states
\[\tag{14} |z\rangle\equiv N(|z|)^{-1/2}\sum_{n=0}^\infty (z^n/\sqrt{\rho_n}\,)\,|n\rangle\;,\]
where the range of the complex variable \(z=x+iy\) is limited to \(N(|z|)=\Sigma_{n=0}^\infty (|z|^{2n}/\rho_n)<\infty\ .\) The weight function \(W(|z|)\) of the measure \( d\mu(z)=W(|z|)\,dx\,dy/\pi\) is determined by
\[\tag{15} \rho_n\equiv\int |z|^{2n}\,N(|z|)^{-1}\,W(|z|)\,d|z|^2\;.\]
in order that
\[\tag{16} \int |z\rangle\langle z|\,d\mu(z)=\sum_{n=0}^\infty \rho_n^{-1}\,|n\rangle\langle n|\,\int|z|^{2n}\, N(|z|)^{-1}\,W(|z|)\,d|z|^2=\sum_{n=0}^\infty|n\rangle\langle n|=I\;.\]
For example, choose \(N^{-1}\,W=e^{-|z|^2}\ ,\) which leads to \(\rho_n=n!\ .\) It follows that \(N=e^{|z|^2}\) and so \(W=1\ ,\) leading to the canonical coherent states of \eqref{zero}. Instead, if \(N^{-1}\,W=e^{-|z|}/2\ ,\) then \(\rho_n
=(2n+1)!\ ,\) \(N(|z|)=\sinh(|z|)/|z|\ ,\) and \(W(|z|)=(1-e^{-2|z|})/4|z|\ ,\)
which provides an example of coherent states that are not connected to a transitive group.
One of the main purposes of coherent states is to offer Hilbert space representations that are not of the conventional orthonormal -- discrete or continuous -- variety. These new representations open up the possibility to look at conventional problems from new perspectives.
A perfect example of how a new view can be useful is offered by the so-called Segal-Bargmann representation composed of analytic functions \(f(z)\) with \(d\mu(z)=e^{-|z|^2}\,dx\,dy/\pi\) (which is closely related to the coherent state representation discussed in Sec. 2). In this formulation, the representation of the creation operator \(a^\dagger\) is multiplication by \(z\) while the representation of the annihilation operator \(a\) is given by \(d/dz\ .\) For the harmonic oscillator with the zero-point energy subtracted, the quantum Hamiltonian is given by \(\hbar\omega\,a^\dagger a\ .\) Thus the time-independent Schrödinger equation reads
\[\tag{17} \hbar\omega\,z\,(d/dz)\,f(z)=E\,f(z)\;, \]
with a solution given (up to a constant factor) by
\[\tag{18} f(z)=z^{E/\hbar\omega}\;. \]
Now we invoke analyticity of the solution, namely, \(E/\hbar\omega=n\ ,\) or specifically
\[\tag{19} E_n=n\,\hbar\,\omega\;,\;\;\;\;\;\;\;\;\;\;\;n\in\{0,1,2,3,\ldots\} \]
which is recognized as the right spectrum for the oscillator! Is there an easier derivation?
References
- Klauder, J R and Sudarshan, E C G (1968). Fundamentals of Quantum Optics, Chapters 7 and 8, W A Benjamin, New York. ISBN-13: 9780486450087.
- Klauder, J R and Skagerstam, B-S (1985). Coherent States: Applications in Physics and Mathematical Physics, World Scientific, Singapore. ISBN 9971-966-52-2.
- Perelomov, A M (1986). Generalized Coherent States and Their Applications, Springer, Berlin. ISBN-13: 9783540159124.
Internal references
- Jean Zinn-Justin (2009) Path integral. Scholarpedia, 4(2):8674.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
See also
Analytic function, Harmonic oscillator, Hilbert space, Measure, Path integral, Reproducing kernel, Schrödinger equation
