Ermakov–Lewis invariant

Template:Short description In quantum mechanics, the Ermakov–Lewis invariant is a conserved quantity used to analyze explicitly time-dependent systems, especially the time-dependent harmonic oscillator. Because many quantum Hamiltonians are time-dependent, methods that identify constants of motion or invariants are central; the Ermakov–Lewis invariant provides such an integral of motion and underpins exact or approximate solutions. It is one of several invariants known for this system.

Let $\mathrm{\Omega }:ℝ\to ℝ$ be a function. It defines a time dependent harmonic oscillator Hamiltonian reads

$\hat{H}=\frac{1}{2}[{\hat{p}}^2+{\mathrm{\Omega }}^2(t){\hat{q}}^2].$

The Ermakov–Lewis invariant for this type of interaction is

$\hat{I}=\frac{1}{2}[{\left(\frac{\hat{q}}{\rho }\right)}^2+(\rho \hat{p}-\dot{\rho }\hat{q}{)}^2],$

where $\rho :ℝ\to ℝ$ is a solution to the Ermakov equation

$\ddot{\rho }+{\mathrm{\Omega }}^2\rho ={\rho }^{-3}.$

$\hat{I}$ is a unitary transformation of the time independent harmonic oscillator Hamiltonian:$$\frac{1}{2}[{\hat{p}}^2+{\hat{q}}^2]=\hat{T}\hat{I}{\hat{T}}^{†},\hat{T}=e^{i\frac{\ln \rho }{2}(\hat{q}\hat{p}+\hat{p}\hat{q})}{e}^{-i\frac{\dot{\rho }}{2\rho }{\hat{q}}^2}=e^{i\frac{\ln \rho }{2}\frac{d{\hat{q}}^2}{dt}}{e}^{-i\frac{{\hat{q}}^2}{2}\frac{d\ln \rho }{dt}},$$This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian. The exponential term $e^{i\frac{\ln \rho }{2}(\hat{q}\hat{p}+\hat{p}\hat{q})}$ is a squeeze operator. The other exponential term $e^{-i\frac{\dot{\rho }}{2\rho }{\hat{q}}^2}$ is a shear operator (momentum-dependent phase shift).

This approach simplifies problems such as the Quadrupole ion trap, where an ion is trapped in a harmonic potential with time dependent frequency.

Phase-space geometry

Wigner function for number states a) n = 0, b) n = 1, and c) n = 19. Marginal distributions for x and p are recovered by integrating over p and x respectively.

This invariant has a geometrically intuitive interpretation in Wigner's form of phase-space quantum mechanics.

For any quadratic Hamiltonian the phase-space flow is linear and symplectic. Writing $𝐱=(q,p{)}^{𝖳}$, the classical evolution is

$𝐱(t)=S(t)𝐱(0),S(t)\in \mathrm{S}\mathrm{p}(2,ℝ).$

Introduce the Ermakov–Lewis scaling

$(\genfrac{}{}{0ex}{}{Q}{P})=L(t)(\genfrac{}{}{0ex}{}{q}{p}),L(t)=\left(\begin{array}{cc}{\rho }^{-1}(t)& 0\\ -\dot{\rho }(t)& \rho (t)\end{array}\right),$

for which the invariant becomes $\frac{1}{2}(Q^2+P^2)$ and the dynamics is a pure rotation. In these variables

$(\genfrac{}{}{0ex}{}{Q(t)}{P(t)})=R(\theta (t))(\genfrac{}{}{0ex}{}{Q(0)}{P(0)}),R(\theta )=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right),\theta (t)={\int }_0^t\frac{d\tau }{{\rho }^2(\tau )},$

with $\rho$ solving the Ermakov equation (see above). Transforming back gives

$S(t)=L(t{)}^{-1}R(\theta (t))L(0)=\left(\begin{array}{cc}\rho & 0\\ \dot{\rho }& {\rho }^{-1}\end{array}\right)R(\theta (t))\left(\begin{array}{cc}\rho (0{)}^{-1}& 0\\ -\dot{\rho }(0)& \rho (0)\end{array}\right).$

Because the Wigner function evolves by the pullback of the classical flow for quadratic Hamiltonians,

$W(𝐱,t)=W_0(S(t{)}^{-1}𝐱),$

any initial Gaussian remains Gaussian. For an initial coherent state of the unit oscillator with mean ${𝝁}_0$ and covariance ${\mathrm{\Sigma }}_0=\frac{1}{2}I$,

$𝝁(t)=S(t){𝝁}_0,\mathrm{\Sigma }(t)=S(t){\mathrm{\Sigma }}_{0}S(t{)}^{𝖳}=\frac{1}{2}S(t)S(t{)}^{𝖳}.$

Geometrically, the contours of $W(\cdot ,t)$ are ellipses whose axes "breathe" via $\rho (t)$ and whose orientation rotates by $\theta (t)$. In the scaled coordinates $(Q,P)$, or equivalently under the unitary $\hat{T}$ given above, the state is a circular Gaussian rotating at constant angular velocity, so all deformation in the laboratory $(q,p)$ plane is captured by the time-dependent squeeze $L(t)$ and variable-velocity rotation $R(\theta )$.

History

It was proposed in 1880 by Vasilij Petrovich Ermakov (1845-1922). The paper is translated in.

In 1966, Ralph Lewis rediscovered the invariant using Kruskal's asymptotic method. He published the solution in 1967.