# Solution for the "Input and Output theory" for circuit/cavity-QED Quantum Architecture [closed]

This question consists of the basic calculations which are often referred to as "trivial" in most references in circuit-QED. However, are not so trivial for anyone who starts fresh into the quantum optics involved for studying the input/output (I/O), and therefore the measurements in actual architecture of QC. (With some quality answer this can become a resource for starters to refer to). The starting point is to study the Quantum Langevin Equations from some 'rare' quantum optics texts. The notation is standard:

$$\Omega:$$ Cavity Frequency

$$\Gamma$$: Dissipation/decay rate

$$b_{\text{in/out}}$$: Input/Output field operators

$$a$$ & $$a^\dagger$$: Cavity modes

The solution to the Langevin equation reads:

$$a(t)=a(0)\text{e}^{-(i\Omega +\Gamma/2)}-\sqrt{\Gamma}\int_0^tds$$ $$\text{e}^{-(i\Omega+\Gamma/2)(t-s)}b_{\text{in}}(s)$$.

(1) Prove the commutators:

$$[a^\dagger(t),b_{\text{in}}(t')]=0 \quad$$ for $$t

$$[a^\dagger(t),b_{\text{in}}(t)]=-\frac{1}{2}\sqrt{\Gamma} \quad$$

$$[a^\dagger(t'),b_{\text{in}}(t')]=-\sqrt{\Gamma}\text{e}^{-(i\Omega+\Gamma/2)(t-t')} \quad$$ for $$t>t'$$

$$[a^\dagger(t),a^\dagger(s)]=\text{e}^{-\frac{1}{2}\Gamma|t-s|}$$

(2) Given the time reversed Langevin equation

$$\dot{a}(t)=(-i\Omega+\Gamma/2)a(t)-\sqrt{\Gamma}b_{\text{out}}(t)$$

find the solution for $$a(t)$$ in terms of the output field at a future time $$T$$.

(3) Prove the relation

$$\tilde{a}(\omega)=\frac{\sqrt{\Gamma}}{i(\omega-\Omega)-\Gamma/2}\tilde{b}_{\text{in}}(\omega)$$

where the standard Fourier transform is defined as: $$\tilde{f}(\omega)=\frac{1}{2\pi} \int_{-\infty}^{\infty}\text{e}^{i\omega t}f(t)dt$$

(4) Given the I/O relation and boundary condition:

$$b_{\text{out}}(t)=b_{\text{in}}(t)+\sqrt{\Gamma}a(t)$$

Prove the simpler relation:

$$\tilde{b}_{\text{out}}(\omega)=\text{e}^{i\delta(\omega)}\tilde{b}_{\text{in}}(\omega)$$

where $$\delta(\omega)$$ is a phase shift which preserves the commutation relations for the I/O operators under the Fourier transform.

• each question should contain a single, laser-focused question. Feel free to edit this question to focus on a single aspect of it, providing as many details about what you understand/know about it etc
– glS
Feb 2, 2021 at 10:47