Catalytic Coherence

Because of conservation of energy we cannot directly turn a quantum system with a definite energy into a superposition of different energies. However, if we have access to an additional resource in terms of a system with a high degree of coherence, as for standard models of laser light, we can overcome this limitation. The question is to what extent coherence gets degraded when utilized. Here it is shown that coherence can be turned into a catalyst, meaning that we can use it repeatedly without ever diminishing its power to enable coherent operations. This finding stands in contrast to the degradation of other quantum resources and has direct consequences for quantum thermodynamics, as it shows that latent energy that may be locked into superpositions of energy eigenstates can be released catalytically.

Figure 1

Rigid translation property. A two-level system interacts with a reservoir whose energy levels can be described as a ladder. By design, the interaction shifts the whole state of the reservoir “rigidly” down or up along this ladder, depending on whether the two-level system absorbs (left part) or donates (right part) energy. For example, by removal of one quantum, the uniform superposition $|{\eta }_{L,l_0}⟩=\sum _{l=0}^{L-1}|l_0+l⟩/\sqrt{L}$ is shifted to $|{\eta }_{L,l_0-1}⟩$. If $L$ is large, the difference between $|{\eta }_{L,l_0-1}⟩$ and $|{\eta }_{L,l_0}⟩$ is small. Hence, the state of the reservoir does not change much by the loss or gain of a quantum, which enables coherent operations on the two-level system. In the limit of large $L$, these implementations can be made perfect. This is analogous to how coherent states on a bosonic mode can implement coherent operations on an atom via the Jaynes-Cummings model.

Figure 2

Regenerative catalytic cycles. When the energy reservoir interacts with the two-level system (left part) the projection of its state onto the number states can increase with at most one level up and down in energy. As long as the projection onto the ground state of the reservoir remains zero, the set of channels that the reservoir can induce on the system stays intact from one interaction to the next. By using another two-level system in a pure excited state (right part) to inject energy into the reservoir, the state is translated “rigidly” up along the energy ladder by one step. By alternating every use of the reservoir with such a pumping, the state of the reservoir can be kept away from the ground state, thus maintaining the coherence properties indefinitely. Note that the state of the energy reservoir does change from one cycle to the next; e.g., the range of number states onto which it projects may become broader for each step. However, the relevant aspects of the state, which determine its capacity to induce channels, remain constant.

Figure 3

Decay of coherence in the Jaynes-Cummings model. The JC model exhibits a decay of coherence over repeated use. However, there is a remnant of the catalytic property, in the sense that the “lifetime” of the coherence (counted in number of iterations) appears to increase indefinitely merely by an increase of the average energy of the initial state. These graphs depict the decay of the fidelity by which the superposition $|\varphi ⟩=(|{\psi }_0⟩-i|{\psi }_1⟩)/\sqrt{2}$ can be created from the ground state $|{\psi }_0⟩$, in a collection of two-level systems that sequentially interact with one single reservoir, according to the JC model for a fixed time step. Each curve corresponds to a different initial state of the reservoir and shows $F_k^{(m)}=⟨\varphi |\mathrm{\Phi }({\sigma }_m^{(k)})|\varphi ⟩$, against the number of times $k$ the reservoir has been used. Each curve corresponds to an initial state ${\sigma }_m^{(0)}=|{\eta }_{L,l_0(m)}⟩⟨{\eta }_{L,l_0(m)}|$ for $|{\eta }_{L,l_0(m)}⟩$ with $l_0(m)=(4m+1{)}^2-24$ and $L=50$, for $m=5$, 7, 9, 11, 14, 18, 24, 31, 40, 52, 67, 87, 113, 147, 191. Note that the width of the superposition does not change with $m$. The dotted line is the value of $(1+|⟨{\eta }_{L,l_0(m)}|\mathrm{\Delta }|{\eta }_{L,l_0(m)}⟩|)/2=0.99$, which would be the fidelity reached in the doubly infinite ladder model for these initial states.