Quantum Damped Harmonic Oscillator

Density Operator

The central object in computing statistical and quantum expectation values is the density operator, which at a specific initial time $t_0$ , is given as the following sum over all relevant quantum states $\{ | \psi_\ell \rangle | \ell = 1,2,3,\dots,N \}$ :

$\rho[t_0] \equiv \sum_{\ell=1}^N p_\ell | \psi_\ell \rangle \langle \psi_\ell | .$

These $N$ linearly independent states need not be orthogonal; whereas the $p_\ell$ describes the statistical probability that the $\ell$ -th state be `measured’; in the sense that — for some arbitrary operator $O$ — the quantum statistical expectation value is given by

$\langle\langle O \rangle\rangle[t_0] = \sum_\ell p_\ell \langle \psi_\ell | O | \psi_\ell \rangle .$

These probabilities $\{ p_\ell \}$ may parametrize partial knowledge of the quantum system at hand; or may result from `coarse graining’ the system from a more fundamental one — below, we will exploit the latter perspective. In any case, the reason why we use the density operator to compute the above expectation value, is because the latter can be expressed as a trace involving the former:

$\langle\langle O \rangle\rangle[t_0] = \text{Tr}\left[ O \cdot \rho[t_0] \right]$ .

For example, the trace of the density operator itself is

$\langle\langle \mathbb{I} \rangle\rangle[t_0] = \text{Tr}\left[ \rho[t_0] \right] = \sum_\ell p_\ell = 1 = \sum_\lambda \lambda$ ;

and the trace of the square of the density operator — aka purity — is

$\text{Pu}[t_0] \equiv \text{Tr}\left[ \rho[t_0]^2 \right] = \langle\langle \rho \rangle\rangle[t_0] = \sum_\lambda \lambda^2$ ;

where I have exploited the Hermitian character of $\rho$ to phrase its purity in terms of its eigensystem, namely, $\rho[t_0] | \lambda \rangle = \lambda | \lambda \rangle$ . Thus, we have not only shown that its eigenvalues must be bounded between 0 and 1, their sum of the squares — which yields purity itself — must also be similarly bounded:

$0 \leq \left( \text{Pu}\left[ t_0 \right] = \sum_{\lambda} \lambda^2 \right) \leq 1$ .

Time Evolution

If our quantum system were self-contained, i.e., `closed’, the time evolution of each and every $| \psi_\ell \rangle$ is simply governed by

$| \psi_\ell[t \geq t_0] \rangle = K[t,t_0] | \psi_\ell \rangle ;$

where $K$ the time evolution operator itself obeys the Schrodinger equation. With $H$ denoting its total Hamiltonian,

$i \dot{K} = H K, \qquad K[t = t_0] = \mathbb{I} .$

Its solution in the position representation is often phrased as a path integral:

$\langle \vec{x} | K[t,t'] | \vec{x}' \rangle = \mathcal{N} \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q} \exp\left[ i \int_{t'}^t L[\vec{q},\dot{\vec{q}}] dt'' \right] .$

This in turn tells us, the density operator evolves in time via a pair of path integrals — one for the ket and one for the bra:

$\langle \vec{x} | \rho[t] | \vec{y} \rangle = \int_{\mathbb{R}^{2D}} d^D\vec{x}' d^D\vec{y}' \langle \vec{x} | K[t,t_0] | \vec{x}' \rangle \langle \vec{x}' | \rho[t_0] | \vec{y}' \rangle \langle \vec{y}' | K[t,t_0] | \vec{y} \rangle .$

For notational convenience, we re-phrase it as

(Time.Evol) $\langle \vec{x} | \rho[t] | \vec{y} \rangle \equiv \int_{\mathbb{R}^{2D}} d^D\vec{x}' d^D\vec{y}' K\overline{K}[t,t_0;\vec{x}',\vec{y}'] \langle \vec{x}' | \rho[t_0] | \vec{y}' \rangle ,$

where we may write the double path integrals as

$K\overline{K}[t,\vec{x},\vec{y};t',\vec{x}',\vec{y}'] = |\mathcal{N}|^2 \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \exp\left[ i \int_{t'}^t \left( L[\vec{q}_1,\dot{\vec{q}}_1] - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} \right) dt'' \right] .$

To reiterate: for a closed system, the $\vec{q}_{1,2}$ path integrals factorize because the bra and the ket evolve independently. However, if the system arose out of coarse graining or `tracing out’ some other degrees of freedom that is otherwise irrelevant to the problem at hand, it is then entirely possible that the $\vec{q}_{1,2}$ s become coupled. Hence, eq. (Time.Evol) still represents the time evolution of an initial density operator, but the time-evolution operator itself now reads

(In.In) $K\overline{K}[t,\vec{x},\vec{y};t',\vec{x}',\vec{y}'] = |\mathcal{N}|^2 \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \exp\left[ i \int_{t'}^t \left( L[\vec{q}_1,\dot{\vec{q}}_1] - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} + L_\text{IF}[\vec{q}_1,\vec{q}_2] \right) dt'' \right] .$

This doubled-path integral is usually considered part of the Schwinger-Keldysh formalism, though it should really be known as the Feynman-Vernon path integral.

As an example how such $\vec{q}_{1,2}$ coupling may arise, if we had started with an additional degree of freedom $\vec{Q}$ , where $L$ is the Lagrangian involving only the $\vec{q}$ s and $L_0$ only the $\vec{Q}$ s, while $L_1$ couples the $\vec{q}$ s and $\vec{Q}$ s; the total time evolution operator would be

$|\mathcal{N}|^2 \int_{\mathbb{R}^D} d^D\vec{X} \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{X}'}^{\vec{X}} \mathcal{D}\vec{Q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \int_{\vec{Y}'}^{\vec{X}} \mathcal{D}\vec{Q}_2 \\ \times \exp\Bigg[ i \int_{t'}^t \Bigg( L[\vec{q}_1,\dot{\vec{q}}_1] + L_0[\vec{Q}_1,\dot{\vec{Q}}_1] + L_1[\vec{q}_1,\dot{\vec{q}}_1,\vec{Q}_1,\dot{\vec{Q}}_1] \\ \qquad\qquad - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} - \overline{L_0[\vec{Q}_2,\dot{\vec{Q}}_2]} - \overline{L_1[\vec{q}_2,\dot{\vec{q}}_2,\vec{Q}_2,\dot{\vec{Q}}_2]} \Bigg) dt'' \Bigg] .$

We see the influence action arises from

$\exp\left[ i \int_{t'}^t L_\text{IF} dt'' \right] \delta^{(D)}[\vec{X}'-\vec{Y}'] \\ = \int_{\mathbb{R}^D} d^D\vec{X}\int_{\vec{X}'}^{\vec{X}} \mathcal{D}\vec{Q}_1 \int_{\vec{Y}'}^{\vec{X}} \mathcal{D}\vec{Q}_2 \\ \times \exp\Bigg[ i \int_{t'}^t \Bigg( L_0[\vec{Q}_1,\dot{\vec{Q}}_1] + L_1[\vec{q}_1,\dot{\vec{q}}_1,\vec{Q}_1,\dot{\vec{Q}}_1] - \overline{L_0[\vec{Q}_2,\dot{\vec{Q}}_2]} - \overline{L_1[\vec{q}_2,\dot{\vec{q}}_2,\vec{Q}_2,\dot{\vec{Q}}_2]} \Bigg) dt'' \Bigg] .$

The presence of the $\delta^{(D)}[\vec{X}'-\vec{Y}']$ on the LHS is due to the fact that, if we further traced over the $\vec{q}$ degrees of freedom, the result of the time-evolution $K\overline{K}$ must be an identity, so that total probability is always preserved to be unity. This argument is likely not a proof, but I believe ought to be satisfied by a non-trivial fraction of closed many- or few-body quantum systems. Note, too, that this integration over the $\vec{Q}$ may be viewed as `coarse graining’, by `averaging’ the $\vec{q}-$ dynamics over its interaction with the $\vec{Q}$ s.

To preserve probability $\text{Tr}[\rho[t \geq t_0]]=1$ for any initial density operator in eq. (Time.Evol), we therefore need

(Prob.Conserv) $\int_{\mathbb{R}^D} d^D\vec{x} K\overline{K}[t,\vec{x}=\vec{x}';t',\vec{y},\vec{y}'] = \delta^{(D)}[\vec{y}-\vec{y}'] .$

We also expect the purity to remain bounded between 0 and 1:

(Pu.Bound) $0 \leq \text{Pu}[t \geq t_0] \leq 1$ .

Quantum DHO

If we start from the outset with the following Lagrangians

$L[\vec{q},\dot{\vec{q}}] = \frac{1}{2} \dot{\vec{q}}^2 - \frac{1}{2} \left( \omega^2 - i \alpha \right) \vec{q}^2$

and

$L_{\text{IF}} = - \gamma (\vec{q}_1 - \vec{q}_2) \cdot (\dot{\vec{q}}_1 + \dot{\vec{q}}_2) - i g \vec{q}_1 \cdot \vec{q}_2$

— where $\omega>0$ and, for now, $\gamma,\alpha,g \in \mathbb{R}$ — then we will discover that this describes a damped harmonic oscillator. In particular, its one-point position expectation value is

$\text{Tr}[X \cdot \rho[t]] = \langle\langle X[t \geq t_0] \rangle\rangle \\ = \langle\langle X[t_0] \rangle\rangle \left( 2 \gamma \mathcal{G}_{\text{DHO}}[t-t_0] - \partial_{t_0} \mathcal{G}_{\text{DHO}}[t-t_0] \right) + \langle\langle P[t_0] \rangle\rangle \mathcal{G}_{\text{DHO}}[t-t_0] ,$

where

$\mathcal{G}_{\text{DHO}}[\tau] = \exp[-\gamma \cdot \tau] \sin\left[ \tau \sqrt{\omega^2 - \gamma^2} \right]/\sqrt{\omega^2 - \gamma^2}$

and, hence, $\langle\langle X[t] \rangle\rangle$ both obey the DHO oscillator equation

$\left( \frac{d^2}{d \tau^2} + 2 \gamma \frac{d}{d \tau} + \omega^2 \right) \mathcal{G}_{\text{DHO}}[\tau] = 0 = \left( \frac{d^2}{dt^2} + 2 \gamma \frac{d}{dt} + \omega^2 \right) \langle\langle X[t] \rangle\rangle$

with initial conditions $\mathcal{G}_{\text{DHO}}[\tau = 0] = 0$ , $\partial_\tau \mathcal{G}_{\text{DHO}}[\tau = 0] = 1$ , $\langle\langle X[t_0] \rangle\rangle$ and $(d/d t) \langle\langle X[t=t_0] \rangle\rangle = \langle\langle P[t_0] \rangle\rangle$ . Furthermore, we may identify the DHO retarded Green’s function as

$G^+_{\text{DHO}}[\tau] = \Theta[\tau] \mathcal{G}_{\text{DHO}}[\tau] .$

The appearance of $\gamma$ in $\mathcal{G}_{\text{DHO}}[\tau]$ , which $\langle\langle X[t] \rangle\rangle$ is built out of, indicates we must impose the non-negativity of $\gamma$ to prevent a runaway solution:

$\gamma \geq 0$ .

Next, if one proceeds to impose probability conservation in eq. (Prob.Conserv), it turns out

$\alpha = g$ ;

for otherwise the ensuing integrals may not yield the Dirac $\delta-$ functions on the RHS.

Moreover, to guarantee that purity be bounded — i.e., eq. (Pu.Bound) holds — even in the asymptotic future ( $t \to \infty$ ), one would find that

$\alpha \geq 2 \gamma \omega .$

In other words, as long as we have a damped harmonic oscillator — namely, $\gamma,\omega > 0$ — its quantization appears to require a non-zero imaginary part of its frequency-squared; and this imaginary part cannot be arbitrarily small, since it must be greater than or equal to $2\gamma \omega$ . In fact, had we set $\alpha=0$ from the start, the purity would blow up exponentially as $e^{2\gamma t}$ in the asymptotic future, violating the requirement that it stays below unity.

Finally, the variance of the position operator $\langle\langle X[t]^2 \rangle\rangle = \text{Tr}[X^2 \cdot \rho[t]]$ , in the $t \to \infty$ limit, can be shown to be completely independent of the initial density operator $\rho[t_0]$ . The infinite time limit of the density operator $\rho[\infty]$ is in fact a thermal one, provided one identifies the inverse temperature as

$\beta \equiv T^{-1} = \frac{2}{\omega} \text{coth}^{-1}\left[ \frac{\alpha}{2 \gamma \omega} \right].$

References

N. Agarwal and Y.Z. Chu, “Initial value formulation of a quantum damped harmonic oscillator,” [arXiv:2303.04829 [hep-th]].

Author: Yi-Zen Chu

I am a theoretical physicist, with research interests spanning gravitation and field theory, particle cosmology and Mathematica software development. View all posts by Yi-Zen Chu

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