Full distribution of work done on a quantum system for arbitrary initial states

We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical detector arises naturally and work and its statistics can be investigated in the most general case. In particular, we show that the quantum coherence of the initial state can lead to measurable effects on the moments of the work done on the system. At the same time, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our method can also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling the system to a detector, we can track the energy dissipated in the environment while accessing only the system degrees of freedom.

Figure 1

Quantum work and path integral. Pictorial representation of the unitary evolution of a quantum system from the initial state ${\psi }_0$ (in this case an eigenstate for the initial Hamiltonian) to the generic final state ${\psi }_{\mathcal{T}}$, described in terms of paths in energy space. The time-dependent energy spectrum ${\varepsilon }_i\left(t\right)$ of the system Hamiltonian is plotted in black (dotted lines). Quantum trajectories (dashed blue and solid red) consists of a sequence of jumps between different eigenstates. The solid red trajectory satisfies the constraint (in this case $\mathrm{\Delta }U=0$) while the blue ones do not.

Figure 2

Measuring work and dissipation in open quantum systems. Schematic representation of the sequence of driven evolutions and interactions with the detector with the open-system protocol in Eq. (8). The evolution steps $exp(-i\mathrm{\Delta }t{\widehat{H}}^k)$ are represented by the flat line and characterized by the Hamiltonian $H^k$. Each coupling with the detector is represented by a circle or square. The coupling is either of the form $exp(-i\lambda {\widehat{H}}_S^k)$ (red circle), or $exp\left(i\lambda {\widehat{H}}_S^k\right)$ (blue square). In the blue-dashed region, the evolution is frozen $(\mathrm{\Delta }\rho =0)$ and the Hamiltonian changes by $\mathrm{\Delta }H$. In the red-shadowed region, the Hamiltonian is constant $(\mathrm{\Delta }H=0)$ while the density operator changes by $\mathrm{\Delta }\rho$.