Ermakov–Lewis invariant

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

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

The Ermakov–Lewis invariant for this type of interaction is

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

where ${\displaystyle \rho$ :\mathbb {R} \to \mathbb {R} } is a solution to the Ermakov equation^[1]

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

${\displaystyle {\hat {I}}}$ is a unitary transformation of the time independent harmonic oscillator Hamiltonian:^[2]$${\displaystyle {\frac {1}{2}}\left[{\hat {p}}^{2}+{\hat {q}}^{2}\right]={\hat {T}}{\hat {I}}{\hat {T}}^{\dagger },\quad {\hat {T}}=e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}e^{-i{\frac {\dot {\rho }}{2\rho }}{\hat {q}}^{2}}=e^{i{\frac {\ln \rho }{2}}{\frac {d{\hat {q}}^{2}}{dt}}}e^{-i{\frac {{\hat {q}}^{2}}{2}}{\frac {d\ln \rho }{dt}}},}$$This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian. The exponential term ${\displaystyle e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}}$ is a squeeze operator. The other exponential term ${\displaystyle e^{-i{\frac {\dot {\rho }}{2\rho }}{\hat {q}}^{2}}}$ 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.^[3]

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

${\displaystyle \mathbf {x} (t)=S(t)\,\mathbf {x} (0),\qquad S(t)\in \mathrm {Sp} (2,\mathbb {R} ).}$

Introduce the Ermakov–Lewis scaling

${\displaystyle {\binom {Q}{P}}=L(t){\binom {q}{p}},\qquad L(t)={\begin{pmatrix}\rho ^{-1}(t)&0\\[2pt]-{\dot {\rho }}(t)&\rho (t)\end{pmatrix}},}$

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

${\displaystyle {\binom {Q(t)}{P(t)}}=R(\theta (t)){\binom {Q(0)}{P(0)}},\quad R(\theta )={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}},\quad \theta (t)=\int _{0}^{t}\!{\frac {d\tau }{\rho ^{2}(\tau )}},}$

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

${\displaystyle S(t)=L(t)^{-1}R(\theta (t))L(0)={\begin{pmatrix}\rho &0\\{\dot {\rho }}&\rho ^{-1}\end{pmatrix}}R(\theta (t)){\begin{pmatrix}\rho (0)^{-1}&0\\-{\dot {\rho }}(0)&\rho (0)\end{pmatrix}}.}$

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

${\displaystyle W(\mathbf {x} ,t)=W_{0}\!\left(S(t)^{-1}\mathbf {x} \right),}$

any initial Gaussian remains Gaussian. For an initial coherent state of the unit oscillator with mean ${\displaystyle {\boldsymbol {\mu }}_{0}}$ and covariance ${\displaystyle \Sigma _{0}={\tfrac {1}{2}}I}$,

${\displaystyle {\boldsymbol {\mu }}(t)=S(t){\boldsymbol {\mu }}_{0},\qquad \Sigma (t)=S(t)\Sigma _{0}S(t)^{\mathsf {T}}={\tfrac {1}{2}}\,S(t)S(t)^{\mathsf {T}}.}$

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

History

It was proposed in 1880 by Vasilij Petrovich Ermakov (1845-1922).^[4] The paper is translated in.^[5]

In 1966, Ralph Lewis rediscovered the invariant using Kruskal's asymptotic method.^[6] He published the solution in 1967.^[1]