Lindblad Equation & IWOP

Author: Xu-Dong Liu | Date: April 20, 2026

1. Operator Representation of the Vacuum Projector

We begin by defining the falling factorial for a variable $x$ :

$(x)_{l}=x(x-1)\cdots(x-l+1)$

For an integer $n \geq 0$ , this is related to the binomial coefficient by:

$(n)_{l}=\frac{n!}{(n-l)!}=l!\binom{n}{l}$

In the context of quantum mechanics, the number operator $N=a^{\dagger}a$ acts on a Fock state $|n\rangle$ such that $N|n\rangle=n|n\rangle$ . Consequently, the normally ordered operator $a^{\dagger l}a^{l}$ satisfies:

$a^{\dagger l}a^{l}|n\rangle=(n)_{l}|n\rangle=l!\binom{n}{l}|n\rangle$

Summing over the spectrum, we can write the identity $a^{\dagger l}a^{l}=l!\binom{N}{l}$ . Consider the vacuum projector $|0\rangle\langle 0|$ . Its eigenvalue is 1 for the vacuum state and 0 for all other Fock states. This behaves like the delta function $\delta_{n,0}$ , which can be represented by the binomial expansion of $(1-1)^{n}$ :

$\delta_{n,0}=(1-1)^{n}=\sum_{l=0}^{n}(-1)^{l}\binom{n}{l}$

Replacing the eigenvalue $n$ with the operator $N$ , we obtain:

$|0\rangle\langle 0|=\sum_{l=0}^{\infty}\frac{(-1)^{l}}{l!}l!\binom{N}{l}=\sum_{l=0}^{\infty}\frac{(-1)^{l}}{l!}a^{\dagger l}a^{l}$

By the definition of normal ordering (denoted by $::$ ), this power series is equivalent to the normally ordered exponential:

$|0\rangle\langle 0|=:e^{-a^{\dagger}a}:$

2. Lindblad Equation

Consider a coherent state $|\alpha(t)\rangle$ evolving in a lossy environment. Its complex amplitude decays according to the semi-classical equations:

$\dot{\alpha}=-\frac{\gamma}{2}\alpha, \quad \dot{\alpha}^{*}=-\frac{\gamma}{2}\alpha^{*}$

where $\gamma$ is the decay rate. The density matrix for this state is defined as the projector $\rho(t)=|\alpha(t)\rangle\langle\alpha(t)|$ . Using the properties of the IWOP technique, this projector can be expressed in its normally ordered form:

$\rho(t)=:e^{-(a^{\dagger}-\alpha^{*})(a-\alpha)}:$

Taking the time derivative of $\rho(t)$ , we note that the operators $a$ and $a^{\dagger}$ are time-independent in the Schrödinger picture. The derivative acts only on the complex amplitudes within the normal ordering:

$\frac{d\rho}{dt}=:\frac{\partial}{\partial t}[-(a^{\dagger}-\alpha^{*})(a-\alpha)]\rho(t):$

Applying the chain rule to the product in the exponent:

$\begin{aligned} \frac{d\rho}{dt} &= :[-\frac{\partial(a^{\dagger}-\alpha^{*})}{\partial t}(a-\alpha)-(a^{\dagger}-\alpha^{*})\frac{\partial(a-\alpha)}{\partial t}]\rho(t): \\ &= :[\dot{\alpha}^{*}(a-\alpha)+\dot{\alpha}(a^{\dagger}-\alpha^{*})]\rho(t): \end{aligned}$

Substituting the decay rates:

$\begin{aligned} \frac{d\rho}{dt} &= -\frac{\gamma}{2}:[\alpha^{*}(a-\alpha)+\alpha(a^{\dagger}-\alpha^{*})]\rho(t): \\ &= -\frac{\gamma}{2}:[\alpha^{*}a+\alpha a^{\dagger}-2|\alpha|^{2}]\rho(t): \end{aligned}$

To reconstruct the standard Lindblad form, we rearrange the terms within the normal ordering symbols to group the “jump” and “dissipation” components:

$\frac{d\rho}{dt}=\gamma:(|\alpha|^{2}-\frac{1}{2}\alpha^{*}a-\frac{1}{2}\alpha a^{\dagger})\rho:$

Finally, we utilize the fundamental mapping between the eigenvalues $\alpha, \alpha^{*}$ and the operators acting on the normally ordered density matrix $\rho = |\alpha\rangle\langle \alpha|$ :

$|\alpha|^{2}\rho=\alpha\rho\alpha^{*}\leftrightarrow a\rho a^{\dagger}$ (The Quantum Jump term)
$\alpha^{*}a\rho=a\rho\alpha^{*}\leftrightarrow a^{\dagger}a\rho$ (Acting on the left)
$\alpha a^{\dagger}\rho=\alpha\rho a^{\dagger}\leftrightarrow\rho a^{\dagger}a$ (Acting on the right)

Substituting these operator identifications and using the anti-commutator notation $\{A,B\}=AB+BA$ , we arrive at the standard Lindblad Master Equation for a damped harmonic oscillator:

$\frac{d\rho}{dt}=\gamma(a\rho a^{\dagger}-\frac{1}{2}\{a^{\dagger}a,\rho\})$

Appendix A: Derivation of the Projector using IWOP Technique

The Integration Within an Ordered Product (IWOP) technique allows us to perform integration directly on normally ordered operators. We start with the identity for a coherent state $|\alpha\rangle$ generated from the vacuum $|0\rangle$ via the displacement operator $D(\alpha)=e^{\alpha a^{\dagger}-\alpha^{*}a}$ :

$|\alpha\rangle\langle\alpha|=D(\alpha)|0\rangle\langle0|D^{\dagger}(\alpha)$

Using the result $|0\rangle\langle 0|=:e^{-a^{\dagger}a}:$ and the property that displacement operators can shift the operators $a\rightarrow a-\alpha$ and $a^{\dagger}\rightarrow a^{\dagger}-\alpha^{*}$ within normal ordering:

$|\alpha\rangle\langle\alpha|=:D(\alpha)e^{-a^{\dagger}a}D^{\dagger}(\alpha):=:e^{-(a^{\dagger}-\alpha^{*})(a-\alpha)}:$

Appendix B: Non-orthogonality of Coherent States

The overlap (fidelity) between two distinct coherent states $|\alpha\rangle$ and $|\beta\rangle$ can be calculated using the normally ordered projector. The probability $|\langle\alpha|\beta\rangle|^{2}$ is:

$|\langle\alpha|\beta\rangle|^{2}=\langle\beta|(|\alpha\rangle\langle\alpha|)|\beta\rangle=\langle\beta|:e^{-(a^{\dagger}-\alpha^{*})(a-\alpha)}:|\beta\rangle$

By the property of normal ordering, we replace the operators with their respective eigenvalues $\beta^{*}$ and $\beta$ :

$|\langle\alpha|\beta\rangle|^{2}=e^{-(\beta^{*}-\alpha^{*})(\beta-\alpha)}\langle\beta|\beta\rangle=e^{-|\beta-\alpha|^{2}}$

Since $e^{-|\beta-\alpha|^{2}}>0$ for all finite $\alpha, \beta$ , coherent states are never strictly orthogonal.

Appendix C: Amplitude Decay

In non-Hermitian quantum mechanics, decay can be described by an effective Hamiltonian. For a damped harmonic oscillator:

$H_{eff}=H_{0}-i\frac{\gamma}{2}a^{\dagger}a$

In the Heisenberg picture (ignoring $\omega$ for simplicity), the equation of motion for the annihilation operator $a$ is:

$\dot{a}=i[H_{eff},a]=i[-i\frac{\gamma}{2}a^{\dagger}a,a]=\frac{\gamma}{2}[a^{\dagger}a,a]$

Using $[a,a^{\dagger}]=1$ , the commutator is $[a^{\dagger}a, a] = -a$ . Substituting this back:

$\dot{a}=\frac{\gamma}{2}(-a)=-\frac{\gamma}{2}a$

Taking the expectation value with respect to $|\alpha\rangle$ , where $\langle a\rangle=\alpha$ , we obtain the classical decay equation $\dot{\alpha}=-\frac{\gamma}{2}\alpha$ .

Appendix D: Derivation of Poisson Distribution via Fock State Projector

We demonstrate how Poissonian photon statistics emerge naturally from the normally ordered vacuum projector. A Fock state $|n\rangle$ is defined as:

$|n\rangle=\frac{a^{\dagger n}}{\sqrt{n!}}|0\rangle$

The projector onto the n-photon state is $|n\rangle\langle n|=\frac{1}{n!}a^{\dagger n}|0\rangle\langle0|a^{n}$ . Substituting $|0\rangle\langle 0|=:e^{-a^{\dagger}a}:$ :

$|n\rangle\langle n|=\frac{1}{n!}a^{\dagger n}(:e^{-a^{\dagger}a}:)a^{n}=:\frac{(a^{\dagger}a)^{n}}{n!}e^{-a^{\dagger}a}:$

The probability $P(n)$ in a coherent state $|\alpha\rangle$ is:

$P(n)=\langle\alpha|:\frac{(a^{\dagger}a)^{n}}{n!}e^{-a^{\dagger}a}:|\alpha\rangle=\frac{(\alpha^{*}\alpha)^{n}}{n!}e^{-\alpha^{*}\alpha}\langle\alpha|\alpha\rangle$

Since $\alpha^{*}\alpha=|\alpha|^{2}=\bar{n}$ , we arrive at the Poisson distribution $P(n)=\frac{\bar{n}^{n}e^{-\bar{n}}}{n!}$ .

Appendix E: Direct IWOP Derivation of the Coherent State Vector

To derive $|\alpha\rangle$ with maximum efficiency, we use the projector $|\alpha\rangle\langle\alpha|=:e^{-(a^{\dagger}-\alpha^{*})(a-\alpha)}:$ .

Step 1: Determining the Vacuum Overlap

The vacuum expectation value is $|\langle 0|\alpha\rangle|^{2} = \langle 0|:e^{-(a^{\dagger}-\alpha^{*})(a-\alpha)}:|0\rangle = e^{-|\alpha|^{2}}$ . Choosing a real phase convention, $\langle 0|\alpha\rangle=e^{-|\alpha|^{2}/2}$ .