The Exercise

A harmonic oscillator with annihilation operator $a$ , the system, exchanges quanta with a second oscillator, its environment, with annihilation operator $b$ . The coupling $a^\dagger b + b^\dagger a$ moves one quantum at a time between the two, as a beam splitter does with two modes of light; acting for a time, it gives the joint unitary $U(\theta) = \exp[-i\theta(a^\dagger b + b^\dagger a)]$ , the angle $\theta$ being the coupling strength times the time. The environment starts in its vacuum $|0\rangle$ . Tracing it out in its number basis $|k\rangle$ leaves the system with an operation whose elements are $E_k = \langle k|U(\theta)|0\rangle$ , one for each number $k$ of quanta the environment has taken up. The exercise is to compute them and to recognize in them the elements of amplitude damping, with $\gamma = \sin^2\theta$ the probability that a single quantum is lost.

The Coupling in Each Sector

The coupling conserves the total number of quanta, so $U(\theta)$ acts separately on each sector of $n$ quanta in all. The coupling matrix $G_n$ of that sector is the $(n+1)\times(n+1)$ matrix of $a^\dagger b + b^\dagger a$ in the basis $|n-j\rangle|j\rangle$ , $j = 0, \dots, n$ , of states with $j$ quanta in the environment; it is tridiagonal, with $(G_n)_{j,j+1} = (G_n)_{j+1,j} = \sqrt{(j+1)(n-j)}, \qquad j = 0, \dots, n-1,$ (1)^✦ and zeros elsewhere. The sector propagator $U^n(\theta)$ , the evolution of the sector of $n$ quanta, is the matrix exponential of $-i\theta G_n$ , $U^n(\theta) = e^{-i\theta G_n}.$ (2)^✦

The Beam Splitter Elements

The amplitude for $k$ of $n$ quanta to pass into the environment is $c_{n,k}(\theta) = \sqrt{\binom{n}{k}}\,\cos^{n-k}\theta\,(-i\sin\theta)^k,$ (3) and the beam splitter element $E_k(\theta)$ is the operator on the oscillator’s state space that takes $|n\rangle$ to $|n-k\rangle$ with this amplitude, $E_k(\theta) = \sum_{n \ge k} c_{n,k}(\theta)\,|n-k\rangle\langle n|.$ (4)

Amplitude Damping

Under amplitude damping each quantum of the oscillator is lost independently, with probability $\gamma$ , and $k$ of $n$ quanta are lost with the binomial probability $p_{n,k}(\gamma) = \binom{n}{k}(1-\gamma)^{n-k}\gamma^k.$ (5)^✦ The damping element $D_k(\gamma)$ is the operator on the oscillator’s state space whose amplitudes are the square roots of these probabilities, $D_k(\gamma) = \sum_{n \ge k} \sqrt{p_{n,k}(\gamma)}\,|n-k\rangle\langle n|.$ (6)^✦

The Claim

Theorem 1. Let $\theta$ be a real angle. The beam splitter elements satisfy the completeness relation $\sum_k E_k(\theta)^\dagger E_k(\theta) = I.$ (7)^✦ For $k \le n$ , the matrix element of $E_k(\theta)$ from $|n\rangle$ to $|n-k\rangle$ is the entry of the sector propagator in row $k$ and column $0$ , $\langle n-k|E_k(\theta)|n\rangle = U^n(\theta)_{k0}.$ (8)^✦ For every $k$ , each matrix element of $E_k(\theta)$ in the number basis has the modulus of the same matrix element of the damping element at $\gamma = \sin^2\theta$ , $|\langle r|E_k(\theta)|s\rangle| = |\langle r|D_k(\sin^2\theta)|s\rangle| \qquad \text{for all } r, s.$ (9)^✦ For $0 \le \gamma \le 1$ , the damping elements satisfy the completeness relation $\sum_k D_k(\gamma)^\dagger D_k(\gamma) = I.$ (10)^✦

What Is Checked

The claim is stated over two families of operation elements: the beam splitter elements $E_k(\theta)$ of (4), with their amplitudes (3), and the damping elements $D_k(\gamma)$ of (6), with the loss probabilities (5).

Of these constructions the library proves that both families are complete, (7) and (10), so that each is the set of elements of a trace preserving operation on the oscillator.