Aspects of quantum work

Various approaches of defining and determining work performed on a quantum system are compared. Any operational definition of work, however, must allow for two facts: first, that work characterizes a process rather than an instantaneous state of a system and, second, that quantum systems are sensitive to the interactions with a measurement apparatus. We compare different measurement scenarios on the basis of the resulting postmeasurement states and the according probabilities for finding a particular work value. In particular, we analyze a recently proposed work meter for the case of a Gaussian pointer state and compare it with the results obtained by two projective and, alternatively, two Gaussian measurements. In the limit of a strong effective measurement strength the work distribution of projective two energy measurements can be recovered. In the opposite limit the average of work becomes independent of any measurement. Yet the fluctuations about this value diverge. The performance of the work meter is illustrated by the example of a spin in a suddenly changing magnetic field.

Figure 1

The pdf of work supplied on a two-level system by a sudden change of the Hamiltonian from $H_i$ to $H_f$ given by Eqs. (61) and (62), respectively, is displayed as it is generated by a work meter with a pure Gaussian state in position representation. The level separation of the Hamiltonian after the switch is given by ${\epsilon }_f=2{\epsilon }_i$. Both panels refer to an initial density matrix (64) with ground-state probability $p=0.7$. In panel (a) the work pdf is displayed for vanishing nondiagonal elements of the initial density ($q=0$) and different values of the work variance ${\sigma }_e^2$ which are ${\sigma }_e^2=0.01\times {\epsilon }_i^2$ (red), $0.1\times {\epsilon }_i^2$ (blue), and ${\epsilon }_i^2$ (black). At the smallest variance, the work pdf consists of clearly separated peaks, which overlap at the intermediate variance and are completely washed out at the large variance. In those cases where peaks can be identified they are always exactly located at the positions of the work values (67) obtained by projective measurements. These values are indicated by the four thin vertical black lines. Panel (b) exemplifies the effect of a nondiagonal initial density matrix at the intermediate variance ${\sigma }_e^2=0.1\times {\epsilon }_i^2$. As a reference, the solid blue line refers to the diagonal case ($q=0$) whereas the red dashed curve corresponds to $q=\sqrt{0.21}$ and the black dash-dotted line corresponds to $q=-\sqrt{0.21}$. The nondiagonal elements of the initial density matrix $\rho \left(0\right)$ lead to a substantial distortion of the work pdf. In particular, the positions of the maxima do no longer coincide with those of the projective work values.

Figure 2

The average work approaches asymptotically the difference of untouched energy mean values $\prec w\succ$ (3) as a function of the variance ${\sigma }_e^2$ [see Eq. (44)]. All curves are for a density matrix of the type (64) with $p=0.7$. The average work is displayed for $q=\sqrt{0.21}$ by the upper red curve, for $q=0.5\times \sqrt{0.21}$, 0, $-0.5\times \sqrt{0.21}$, and $-\sqrt{0.21}$ by the blue, green, and lower red curves, respectively. The according asymptotic values given by Eq. (44) are indicated by thin horizontal lines of respective color. The final energy is chosen as ${\epsilon }_f=2\times {\epsilon }_i$. The change in the average work is largest for the two limiting values $q=\pm \sqrt{p(1-p)}=\pm \sqrt{0.21}$ and vanishes for $q=0$.

Figure 3

The exact work pdf for ${\sigma }_e^2={\epsilon }_i^2$ (narrow blue curve) and ${\sigma }_i^2=2\times {\epsilon }_i^2$ (broad blue curve) is compared to the limiting result of the imprecise measurement limit for two values of the variance ${\sigma }_e^2$ (thin red curves). At the smaller value ${\sigma }_w^2={\epsilon }_i^2$ the Gaussian factor of the convolution integral on the right-hand side of Eq. (54) is not yet broad enough to suppress the negative contributions around $w=-2\times {\epsilon }_i$. At the larger variance, the approximate work pdf only insignificantly differs from the exact one. The other parameters, $p=0.7$ and $q=\sqrt{0.21}$, are the same for all curves.