Reexamination of pure qubit work extraction

Many work extraction or information erasure processes in the literature involve the raising and lowering of energy levels via external fields. But even if the actual system is treated quantum mechanically, the field is assumed to be classical and of infinite strength, hence not developing any correlations with the system or experiencing back-actions. We extend these considerations to a fully quantum mechanical treatment by studying a spin-1/2 particle coupled to a finite-sized directional quantum reference frame, a spin-$l$ system, which models an external field. With this concrete model together with a bosonic thermal bath, we analyze the back-action a finite-size field suffers during a quantum-mechanical work extraction process and the effect this has on the extractable work and highlight a range of assumptions commonly made when considering such processes. The well-known semiclassical treatment of work extraction from a pure qubit predicts a maximum extractable work $W=kTlog2$ for a quasistatic process, which holds as a strict upper bound in the fully quantum mechanical case and is attained only in the classical limit. We also address the problem of emergent local time dependence in a joint system with a globally fixed Hamiltonian.

Figure 1

Top: The qubit starts in a known pure state $|0\rangle$. The unpopulated level $|1\rangle$ is then raised to a high energy without any work cost. Bottom: The qubit is now coupled to a thermal bath and allowed to equilibrate. The raised level is then gradually lowered back to its originally position while keeping the qubit in equilibrium with the bath. Eventually the qubit is in a completely mixed state, with an increase in (von Neumann) entropy of $\Delta S=log2$, and an amount of work $W=kTlog2$ is said to be extracted during the process.

Figure 2

Schematic of the joint system, consisting of a spin-$l$ reference frame and a spin-$1/2$ qubit, associated with the angular-momentum operators $\widehat{L}$ and $\widehat{S}$, respectively, in the particular initial state ${\sigma }_0=|l,m\rangle \langle l,m|\otimes |0\rangle \langle 0|$.

Figure 3

Schematic of the time-dependent Hamiltonian protocol using the example of an $l=1$ reference starting in the fully coherent state $|1,1\rangle$. Initially, without the coupling, $f\left(0\right)=0$, both the qubit's and the reference's states are fully degenerate. When the interaction strength is turned up, the degeneracy is broken as the energy levels shift. Once the maximum desired coupling is reached the qubit is couple to a thermal bath and the qubit-reference interaction $f\left(t\right)$ is slowly tuned back to zero again. As in the semiclassical protocol the qubit is maximally mixed after the protocol is finished. But, in addition, the population of the reference has also shifted and the reference is not in a pure state anymore, having lost some of its polarization and gained in entopy.

Figure 4

Scaled expectation values of the reference's $z$ component of angular momentum as a function of time for the same process as shown in Fig. 5. One unit of time is chosen to correspond to one iteration (ignoring the time required for raising the unoccupied level), i.e., this plot corresponds to the first 10 iterations in Fig. 5. We observe a direct correlation between the value of $\langle {\widehat{L}}_z\rangle$ and the extracted work in the corresponding iteration but also between the rate of polarization loss of the reference and a surplus in the extracted work beyond the semiclassical result.

Figure 5

Work extracted from the joint qubit-reference-system over a single iteration, plotted against the number of iterations. The same reference is used for consecutive iterations but with a new pure qubit. This plot clearly shows the trade-off between single-shot work extraction and reference degradation. For small $l$ a large amount of work is initially extracted, but the reference rapidly loses its ability to induce a level splitting and extract further work (cf. Fig. 4). For large $l$ the reference is nearly unperturbed and we approach the semiclassical $W=kTlog2$ result, repeatable over many iterations with the same reference.

Figure 6

Scaled expectation values of the angular-momentum components of the qubit (dashed blue) and reference (dash-dotted red) as a function of time for $l=10,\theta =\pi /4$, and $dt={10}^{-5}$. Initially at $t=0$ the reference starts fully polarized along $-x$ and the qubit along $-z$. Subsequently, the reference rotates around an axis in the $y\text{−}z$ plane determined by the value of $\theta$ (cf. Fig. 8). At the time where $\langle {\widehat{L}}_z\rangle$ reaches a maximum (vertical dashed line), the level splitting in the qubit is maximal and we begin thermalizing it. This in turn affects the rotation of the reference, leaving it with some finite $\langle {\widehat{L}}_y\rangle$ after completion of the protocol, i.e., when $\langle {\widehat{L}}_z\rangle =0$ again and the qubit is maximally mixed. By this mechanism, energy is transferred from the bath to the reference (cf. Fig. 7).

Figure 7

Energy stored in the reference as a function of time over one iteration of the protocol for $\theta =\pi /4$ and $dt={10}^{-5}$ for different reference sizes. At $t=\pi /2,\langle {\widehat{L}}_z\rangle$ reaches a maximum and the thermalization or lowering process begins. We find that even with a comparatively low value of $l$ such as $l=10$ [green (middle)] we get very close to the semiclassical limit $W=kTlog2$. In this specific case, we find that for $l=10$ an energy $E\approx 0.9998kTlog2$ is transferred from the bath into the reference, accompanied by and entropy gain of only $\Delta S\approx 7.9\times {10}^{-4}log2$, resulting in $\Delta E-kT\Delta S\approx 0.9990kTlog2$. However, if the reference is too small, a considerably lower energy is transferred to the system. The discontinuous gap for $l=2$ is due to the idealized notion of an infinitely strong bath coupling. The inset shows the power as a function of time.

Figure 8

Same plot as Fig. 7 but for fixed $l=10,dt={10}^{-5}$ and different values of $\theta$. We see that for $\theta =\pi /4$ we extract the maximum amount of energy. As $\theta$ tends to either zero or $\pi /2$ a lower amount of energy is transferred to the reference. The maximum level splitting is also considerably smaller for these values, as is evident from the discontinuity at the start of the thermalization process.

Figure 9

Expectation value $\langle {\sigma }_z\rangle$ for the qubit as a function of time, when coupled to a single-mode bosonic bath of frequency $\omega$. From right to left, the dashed vertical lines mark the times at which $\langle L_z\rangle =\frac{\omega }{\sqrt{2}}$ for $\omega =10$, 30, and 60, respectively. We see that these points exactly coincide with the times at which the qubit is most affected by the bath, i.e., they are in resonance. The dash-dotted black line gives $\langle {\sigma }_z\rangle$ as it would be obtained from using the idealized bath and reduced Hamiltonian presented in the previous considerations. The reference used in this plot is of size $l=75$, and the bath mode dimension is $D=7$. Other parameters are $\beta =0.05$ and $\alpha =2$.

Figure 10

Scaled expectation values of the angular-momentum components of the qubit (dashed blue) and reference (dash-dotted red) as a function of time for $l=10,\theta =\pi /4$, and $dt={10}^{-4}$. The same reference is reused for five iterations, each with a new pure qubit. A clear accumulation of the reference's polarization in the $y$ direction is visible. The specific parameters were chosen to show a slow but visible degradation of the reference over the five iterations. Higher (lower) values of $l$ lead to slower (faster) degradation.

Figure 11

Energy stored in the reference as a function of time over five iterations of the protocol for $l=10,\theta =\pi /4$, and $dt={10}^{-4}$ as in Fig. 10. During each consecutive iteration we transfer less and less energy onto the reference and the power drops. In the limit of infinitely many iterations, the reference completely polarizes in the $y$ direction, saturating the amount of energy it can store. The green inset shows the (von Neumann) entropy of the reference, which is monotonically increasing, also hinting at the constant degradation of the reference.