Input-Output Formalism for Few-Photon Transport

Abstract

We extend the input-output formalism of quantum optics to analyze few-photon transport in waveguide quantum electrodynamics (QED) systems. We provide explicit analytical derivations for one- and two-photon scattering matrix elements based on the quantum causality relation. The computation scheme can be generalized to N-photon scattering systematically.

Appendix

1.1.1 Derivation of Input-Output Formalism

The Heisenberg equations associated with the Hamiltonian (1.2)–(1.4) are

$$\begin{aligned} \frac{d}{dt}c_{k}= & {} -i\,k\,c_{k}-i\,\frac{V}{\sqrt{v_g}}\,a,\end{aligned}$$

(1.49)

$$\begin{aligned} \frac{d}{dt}a= & {} -i\,\left[ a,\,H_{\text {c}}\right] -i\,\frac{V}{\sqrt{v_g}}\int dk\,c_k\,. \end{aligned}$$

(1.50)

We let

$$\begin{aligned} \varPhi (t)\equiv \int \,\frac{dk}{\sqrt{2\pi }}\, c_{k}(t),\nonumber \end{aligned}$$

and define the input and output operators

$$\begin{aligned} c_{\text {in}}(t)= & {} \int \,\frac{dk}{\sqrt{2\pi }}\, c_{k}(t_0)\,e^{-ik(t-t_0)},\nonumber \\ c_{\text {out}}(t)= & {} \int \,\frac{dk}{\sqrt{2\pi }}\, c_{k}(t_1)\,e^{-ik(t-t_1)},\nonumber \end{aligned}$$

with $t_0\rightarrow -\infty ,t_1\rightarrow +\infty$.

After multiplying (1.49) by the factor $\exp (ikt)$, we integrate it from an initial time $t_0$ to get

$$\begin{aligned} c_{k}(t)=c_{k}(t_0)e^{-ik(t-t_0)}-i\,\frac{V}{\sqrt{v_g}}\int _{t_0}^td\tau \, a(\tau )e^{-ik(t-\tau )},\nonumber \end{aligned}$$

and then integrate it with respect to k to get

$$\begin{aligned} \varPhi (t)=c_{\text {in}}(t)-i\frac{V}{\sqrt{v_g}}\frac{1}{2}\sqrt{2\pi }\,a(t)=c_{\text {in}}(t)-i\frac{\sqrt{\gamma }}{2}\,a(t). \end{aligned}$$

(1.51)

Here, notice that we integrate over half the delta function, which results in a factor of 1 / 2 and $\gamma$ is defined as $\gamma \equiv 2\pi V^2/v_g$.

Furthermore, plugging (1.51) into (1.50) results in

$$\begin{aligned} \frac{d}{dt}a =-i\,\left[ a,\,H_{\text {c}}\right] -\frac{\gamma }{2}a-i\,\sqrt{\gamma }\,c_{\text {in}}(t).\nonumber \end{aligned}$$

Similarly, we integrate (1.49) up to a final time $t_1>t$ and obtain

$$\begin{aligned} \varPhi (t)=c_{\text {out}}(t)+i\frac{\sqrt{\gamma }}{2}\,a(t). \end{aligned}$$

(1.52)

Combining (1.51) and (1.52), we finally obtain

$$\begin{aligned} c_{\text {out}}(t)=c_{\text {in}}(t)-i\sqrt{\gamma }\,a(t). \end{aligned}$$

(1.53)

1.1.2 Derivation of Effective Hamiltonian

We first prove that the propagator of the cavity can be computed using the effective Hamiltonian (1.32). That is, when $H_{\text {c}}=H_{\text {c}}^{(0)}\equiv \omega _c\, a^{\dag }a$,

$$\begin{aligned} G^{(0)}({t', t})=\widetilde{G}^{(0)}({t', t}),\nonumber \end{aligned}$$

where

$$\begin{aligned} G^{(0)}({t', t})\equiv & {} \langle 0| {\mathscr {T}}\,{a}(t') {a^{\dag }}(t) |0\rangle ,\nonumber \\ \widetilde{G}^{(0)}({t', t})\equiv & {} \langle 0| {\mathscr {T}}\,\widetilde{a}(t') \widetilde{a^{\dag }}(t) |0\rangle . \nonumber \end{aligned}$$

a(t) and $a^{\dag }(t)$ are Heisenberg operators in the input-output formalism (1.8)–(1.10). $\widetilde{a}(t)$ and $\widetilde{a^{\dag }}(t)$ are defined as

$$\begin{aligned} \widetilde{a}(t)=e^{i H_{\text {eff}} t}\, a\, e^{-i H_{\text {eff}} t},\,\,\,\,\,\,\,\,\widetilde{a^{\dag }}(t)=e^{i H_{\text {eff}} t} \,a^{\dag }\, e^{-i H_{\text {eff}} t}, \end{aligned}$$

(1.54)

whose evolution is controlled by the effective Hamiltonian (1.32 ) .

The proof is as follows. When $t'>t$,

$$\begin{aligned} \frac{\partial }{\partial t'} G^{(0)}({t', t})= & {} \langle 0|\,\frac{d{a}(t')}{dt'}\, {a^{\dag }}(t)\, |0\rangle \nonumber \\= & {} -i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \langle 0|{a}(t')\, {a^{\dag }}(t)|0\rangle -i\sqrt{\gamma }\,\langle 0|{ c}_{\text {in}}(t')a^{\dag }(t)|0\rangle \nonumber \\= & {} -i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, G^{(0)}({t', t})\,,\nonumber \\ \frac{\partial }{\partial t} G^{(0)}({t', t})= & {} \langle 0|\,{a}(t')\frac{d{a}^{\dag }(t)}{dt} \,|0\rangle \nonumber \\= & {} i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \langle 0|{a}(t')\, {a^{\dag }}(t)|0\rangle +i\sqrt{\gamma }\,\langle 0|a(t')\,{c}^{\dag }_{\text {out}}(t)|0\rangle \nonumber \\= & {} i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, G^{(0)}({t', t}),\nonumber \end{aligned}$$

where we plug in the input-output formalism (1.9) and (1.10) and use the respective quantum casuality (1.15) and (1.16) so that $\langle 0|{ c}_{\text {in}}(t')a^{\dag }(t)|0\rangle =\langle 0|a^{\dag }(t)\,{ c}_{\text {in}}(t')|0\rangle =0$ and $\langle 0|a(t')\,{c}^{\dag }_{\text {out}}(t)|0\rangle =\langle 0|{c}^{\dag }_{\text {out}}(t)\,a(t')|0\rangle =0$ when $t'>t$. On the other hand, by (1.54), one can compute

$$\begin{aligned} \frac{\partial }{\partial t'} \widetilde{G}^{(0)}({t', t})= & {} \langle 0|\,\frac{d\widetilde{a}(t')}{dt'}\, \widetilde{a^{\dag }}(t)\, |0\rangle =-i\,\langle 0|\,[\widetilde{a}, H_{\text {eff}}](t')\, \widetilde{a^{\dag }}(t)\, |0\rangle \nonumber \\= & {} -i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \langle 0|\widetilde{a}(t')\, \widetilde{a^{\dag }}(t)|0\rangle =-i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \widetilde{G}^{(0)}({t', t}),\nonumber \\ \frac{\partial }{\partial t} \widetilde{G}^{(0)}({t', t})= & {} \langle 0|\widetilde{a}(t')\frac{d\widetilde{a}^{\dag }(t)}{dt} \,|0\rangle =-i\,\langle 0|\,\widetilde{a}(t')\,[\widetilde{a^{\dag }}, H_{\text {eff}}](t)\, |0\rangle \nonumber \\= & {} i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \langle 0|\widetilde{a}(t')\, \widetilde{a^{\dag }}(t)|0\rangle =i\,\left( \omega _c-i\,\frac{\gamma }{2}\right) \, \widetilde{G}^{(0)}({t', t}).\nonumber \end{aligned}$$

So $G^{(0)}({t', t})$ and $\widetilde{G}^{(0)}({t', t})$ satisfy exactly the same differential equations when $t'>t$. They also have the same initial values at $t'= t$. Therefore, by the uniqueness theorem for differential equations, we can conclude that $G^{(0)}({t', t})=\widetilde{G}^{(0)}({t', t})$.

Now for a general Hamiltonian $H_{\text {c}}=H_{\text {c}}^{(0)}+V$, according to the perturbation theory in quantum field theory, all Green functions in principle are completely determined by the propagator $G^{(0)}$and the interaction vertices. The vertices only rely on the form of the interaction term V and is independent of the waveguide photons. As a result, all Green functions, including higher-order ones, can be computed by the effective Hamiltonian (1.32).

In [18, 35], the effective Hamiltonian is obtained in the path integral formalism by integrating out the waveguide degrees of freedom in the full Hamiltonian. The derivation here only relies on the input-output formalism (1.8)–(1.10) and the resulting quantum causality (1.15) and (1.16).