Probing quantum interference effects in the work distribution

What is the role of coherence in determining the distribution of work done on a quantum system? We approach this question from an operational perspective and consider a setup in which the internal energy of a closed system is recorded by a quantum detector before and after the system is acted upon by an external drive. We find that the resulting work distribution depends on the initial state of the detector as well as on the choice of the final measurement. We consider two complementary measurement schemes, both of which show clear signatures of quantum interference. We specifically discuss how to implement these schemes in the circuit QED architecture, using an artificial atom as the system and a quantized mode of the electromagnetic field as the detector. Different measurement schemes can be realized by preparing the field either in a superposition of Fock states or in a coherent state and exploiting state-of-the art techniques for the characterization of microwave radiation at the quantum level. More generally, the single bosonic mode we utilize is arguably the minimal quantum detector capable of capturing the complementary aspects of the work distribution discussed here.

Figure 1

Measuring the work distribution in a circuit-QED architecture. (a) Illustration of the setup. An artificial atom (the system) is dispersively coupled to a quantized field (the detector). The atom and the field can be independently addressed by control fields. The system is measured by characterizing the field leaking out of the cavity. (b) Proposed measurement scheme, applying to both schemes A and B.

Figure 2

Results for scheme A. (a),(b) Husimi-$Q$ function $Q_{{\rho }_a}\left(\alpha \right)=\langle \alpha |{\rho }_a|\alpha \rangle$ in the $X\text{−}P$ plane for the detector in the state superposition $(|0\rangle +e^{i\phi }|1\rangle )/\sqrt{2}$, with $\phi =0$ (a) and $\phi =0.76$ (b). (c) Real (solid) and imaginary part (dashed, dot-dashed) of the characteristic function ${\mathcal{G}}_{\lambda }$ for an initial state with and without coherence. (d),(e) Corresponding quasiprobability distributions $\mathcal{P}\left(W\right)$, as obtained from the Fourier transform of ${\mathcal{G}}_{\lambda }$.

Figure 3

Results for scheme B. (a),(b) Initial (a) and final (b) Husimi-$Q$ function in the $X\text{−}P$ plane for a detector prepared in a coherent state with $\alpha =5$. (c) Corresponding probability distribution, obtained by integrating the distribution (b) over the radial coordinate. The distribution is the same regardless of whether there is some initial coherence. (d),(e) Same as in (a) and (b) for a coherent state with $\alpha =2.5$. (f) Corresponding probability distribution for an initial state with (blue, solid) and without initial coherence (red, dashed). The difference between the two distributions is purely due to quantum interference effects.