Probing nonclassical light fields with energetic witnesses in waveguide quantum electrodynamics

We propose an operational scenario to characterize the nature of energy exchanges between two coupled but otherwise isolated quantum systems. Defining work as the component stemming from effective unitary interactions performed by each system on one another, the remnant is stored in the correlations and generally prevents full energy extraction by local operations. Focusing on the case of a qubit coupled to the light field of a waveguide, we establish a bound relating work exchange and local energy extraction when the light statistics is coherent, and that gets violated in the presence of a quantum light pulse. Our results provide operational, energy-based witnesses to probe nonclassical resources.

Figure 1

(a)–(c) Operational scenario. (a) Alice prepares the initial state of the bipartite system $qf$ at time $t=t_0$, spending the amount of work $-W_{\text{A}}$ (blue). (b) At time $t=0$ the two subsystems are interacting, exchanging work (blue) and correlation energy (blue and red). (c) At time $t$ Bob locally extracts work from both subsytems, $W_{\text{B}}$. Due to the correlations, some locally unextractable energy may remain (red). (d)–(f) Waveguide setup. A qubit is located at the position $x=0$ of a 1DWG. (d) At $t=t_0=-\left|x\right|/v$, Alice prepares a pulse of light and the qubit in a pure state. The field propagates without dispersion in the region $x<0$ (resp. $x>0$) at the velocity $v$. (e) At $t=0$ the pulse impinges the qubit. (f) At time $t$ the qubit and the pulse are still entangled, preventing full energy extraction by local unitary operations.

Figure 2

Classical ergotropy bound and correlation energy. At $t=t_0$ the qubit is prepared in $|g\rangle$ and a light pulse at the position $x=-v{t}_0$. The pulse envelope reads $\alpha \left(t\right)=\sqrt{\gamma }{e}^{\gamma t/2-i{\omega }_{0}t}$ for $t\in [-\infty ,0]$ and $\alpha \left(t\right)=0$ for $t\in (0,\infty ]$ (see Ref. [33]). The extra amount of work $\mathrm{\Delta }{W}_{\text{B}}\left(t\right)$ (see text) is plotted for the coherent field case (solid purple) and the single-photon case (solid blue). The correlation energy $\mathcal{Q}\left(t\right)$ is plotted for the coherent (dashed red) and single-photon (dashed orange) cases. We set $t_0=-5/\gamma$.