Abstract
Dissipative adaptation is a general thermodynamic mechanism that explains selforganization in a broad class of driven classical manybody systems. It establishes how the most likely (adapted) states of a system subjected to a given drive tend to be those following trajectories of highest work absorption, followed by dissipated heat to the reservoir. Here, we extend the dissipative adaptation phenomenon to the quantum realm. We employ a fullyquantized exactly solvable model, where the source of work on a threelevel system is a singlephoton pulse added to a zerotemperature infinite environment, a scenario that cannot be treated by the classical framework. We find a set of equalities relating adaptation likelihood, absorbed work, heat dissipation and variation of the informational entropy of the environment. Our proof of principle provides the starting point towards a quantum thermodynamics of driven selforganization.
Introduction
When a physical system is simultaneously subjected to both predictable and random energy exchanges, what dictates the likelihood of a given state to be found? From a classical thermodynamic perspective, where energy exchanges are classified as work or heat, the concept of dissipative adaptation has recently been put forward by J. England as the expected answer, at least in the context of driven selfassembly^{1}. Qualitatively speaking, dissipative adaptation establishes that, given a certain drive (an external work source), the most adapted (most likely and lasting) states of a fluctuating system tend to be those with a history of exceptionally high work absorption followed by heat dissipation to the environment. Because heat dissipation is an irreversible process, the higher the dissipation, the less likely the reverse trajectory is. In the long run, the system may appear to us as selforganized in this drivedependent state of highest energyconsuming history. Dissipative adaptation is the recent theoretical development of a long search for the emergence of order from disorder, as inspired by lifelike behavior^{2,3}. Examples revealing this general mechanism of energyconsuming irreversible selforganization span diverse systems, environments, lengths and timescales, as shown both theoretically^{4,5,6} and experimentally^{7,8,9,10,11,12}.
The dissipative adaptation phenomenon has been originally formulated in terms of fluctuation theorems. Fluctuation theorems are equalities relating outofequilibrium processes with thermalequilibrium variables, giving evidence that the fluctuations present in many realizations of a process can provide useful knowledge when summed up. The pioneering example is the socalled Jarzynski equality^{13}, \(\left\langle \exp (\beta {W}_{{\rm{abs}}})\right\rangle =\exp (\beta {{\Delta }}F)\), where β = 1/(k_{B}T) is the inverse temperature, W_{abs} is the work absorbed by the system as described by a timedependent Hamiltonian, ΔF is the variation in the Helmholtz free energy and the brackets is the ensemble average over realizations of the process, initially departing from thermal equilibrium. The Jarzynski equality has been lately derived by Crooks from what he called a microscopically reversible condition^{14},
where the forward \({p}_{i\to j}(t)\) and backward \({p}_{j\to i}^{* }(\tau t)\) probabilities for the system to follow paths linking states i and j are related with the amount of heat stochastically dissipated to the environment, Q_{diss} (a functional of the phasespace trajectory and of the driving protocol performing work on the system). \({p}_{j\to i}^{* }(\tau t)\) is computed with the reversed timedependent protocol. βQ_{diss} relates the statistical irreversibility with the thermodynamic entropy production. Let us call E_{ij} = E_{j} − E_{i} the energy difference between the specific final and initial states. Energy conservation during each stochastic realization,
gives the hint on how the work source fuels the dissipative adaptation. The higher the absorbed work, the more heat can be released, hence the more irreversible the path can become. To emphasize this adaptationwork relation, England rearranges Eq. (1) as^{1}
where the angle brackets denote a weighted average over all microtrajectories with fixed start (i) and end (j, k) points (the coarse graining over microtrajectories describes experimentally accessible states of outofequilibrium selforganizing classical manybody systems; see Perunov et al.^{15} for further details). Equation (3) establishes the classical theoretical framework behind dissipative adaptation by showing how a given final state can be statistically privileged by work consumption.
Here, we extend the concept of dissipative adaptation to the quantum realm. Our main goal is to test the robustness of the dissipative adaptation concept beyond its original theoretical framework discussed above. From a technical viewpoint, at vanishing temperatures (β → ∞), where quantum fluctuations and correlations usually prevail, Eqs. (1) and (3) are ill defined. We employ a systemplusreservoir approach to derive the exact equations of motion of a threelevel lambda (Λ) system driven by a singlephoton pulse added to a zerotemperature environment. We find that the most adapted (selforganized) quantum state of the lambda system is indeed that with a history of highest work absorption followed by maximal heat dissipation, thus characterizing a dissipative adaptation. As a consequence of our work, we establish the starting point of a quantum thermodynamics of driven selforganization, so far unexplored to the best of our knowledge. We hope that the notion of a quantum dissipative adaptation may also provide fresh insights to quantum biology^{16}, not only because adaptation and selforganization are concepts inspired in lifelike behavior, but also because our results may find applications to discussions on quantum signatures in photosynthesis^{17,18,19,20}.
Results
Selforganized quantum state
To achieve our main goal, we look for the most elementary scenario where the quantum state of a certain physical system irreversibly selforganizes, as induced by the absorption of energy from an external drive, the excess of which is dissipated to the environment. We choose a threelevel system in Λ configuration, labeled as \(\lefta\right\rangle\), \(\leftb\right\rangle\), and \(\lefte\right\rangle\) (being \(\lefte\right\rangle\) the most excited state, respectively, with transition frequencies ω_{a,b}) (see Fig. 1). To keep the model exactly solvable for both the system and the environment, we assume that the drive source is a singlephoton pulse added to the vacuum state of an infinite environment at zero temperature, T = 0. The environment induces spontaneous emission rates, Γ_{a} and Γ_{b}. Our main results are that
showing that adaptation likelihood at long times, p_{a→b}(t → ∞), is linearly proportional to the average absorbed work \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}\) from a singlephoton pulse of arbitrary shape, resonant with ω_{a}. We call attention to the fact that this result is not immediately expected, since (i) the excitation probability is minimized (p_{a→e}(t) ≪ 1, at all times) when the work absorption is maximized (p_{a→b}(∞) → 1), so the process cannot be thought of as a simple absorptionplusemission picture, and (ii) the final selforganized state is not restrained to be the ground state, which amounts to say that \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}\) does not depend on ω_{b}, so the work is not related to ℏ(ω_{a}−ω_{b}), in particular. The second line in Eqs. (4) characterizes the irreversibility of the process. The absorbed work is partly dissipated to the environment in the form of heat, \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\), satisfying energy conservation
H_{S} here is the Hamiltonian of the system. Finally, we find an exact expression for the informational entropy of the environment at long times, S_{E}(∞), as a function of the average dissipated heat, \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\). The entropy analysis here provides us with both a clearer physical picture of the process and an additional signature of the dissipative adaptation.
Let us suppose that our generic threelevel system in Λ configuration, with lowestenergy states \(\lefta\right\rangle\) and \(\leftb\right\rangle\) and excited state \(\lefte\right\rangle\), is initially in a nondriven steady state, in contact with the environment at temperature T = 0. In this case, at precisely T = 0, the steady state is not uniquely defined, even for nondegenerated cases. We can choose its initial quantum state to be a mixture of the lowestenergy states, as described by the density operator
where \({p}_{a,b}^{(0)}\) depend on the preparation scheme. To give a concrete example, in the preparation by means of a spontaneous emission starting at \(\lefte\right\rangle\), one has \({p}_{a,b}^{(0)}={{{\Gamma }}}_{a,b}/({{{\Gamma }}}_{a}+{{{\Gamma }}}_{b})\). Now we look for the most elementary outofequilibrium stochastic process that drives the system from this (generally mixed) initial state into a final (ideally pure) target state, let us say into state \(\leftb\right\rangle\),
In order to guarantee that the process is irreversible, in the light of Crooks condition, we should also apply the timereversed drive on the system departing from state \({\rho }_{S}(\infty )=\leftb\right\rangle \left\langle b\right\) and find that \({\rho }_{S}^{* }(\infty t)\,\ne\, {\rho }_{S}(0)\) for t→∞.
We employ a systemplusreservoir approach, where we assume a global timeindependent Hamiltonian of the system and its environment,
As we show in what follows, we find a selforganized quantum state in the wellknown dipolar model of lightmatter interaction in the rotatingwave approximation^{21,22},
Here, \({\sigma }_{k}=\leftk\right\rangle \left\langle e\right\) (for k = a, b) and H.c. is the Hermitian conjugate. We consider a continuum of frequencies, ∑_{ω}→∫dωϱ_{ω}, with density of modes ϱ_{ω}. Modes \(\left\{{a}_{\omega }\right\}\) and \(\left\{{b}_{\omega }\right\}\) are the quantized field modes interacting with the transitions \(\lefta\right\rangle\) to \(\lefte\right\rangle\) and \(\leftb\right\rangle\) to \(\lefte\right\rangle\), respectively. The continuum of frequencies allows us to employ a Wigner–Weisskopf approximation to obtain dissipation rates \({{{\Gamma }}}_{a}=2\pi {g}_{a}^{2}{\varrho }_{{\omega }_{a}}\) and \({{{\Gamma }}}_{b}=2\pi {g}_{b}^{2}{\varrho }_{{\omega }_{b}}\). Finally, \({H}_{E}={\sum }_{\omega }\hslash \omega ({a}_{\omega }^{\dagger }{a}_{\omega }+{b}_{\omega }^{\dagger }{b}_{\omega })\) and
where ℏδ_{ab} = ℏ(ω_{a}−ω_{b}) is the energy difference between states \(\leftb\right\rangle\) and \(\lefta\right\rangle\). It is worth emphasizing that our main results in this paper are independent of δ_{ab}.
To keep the model exactly solvable for both the system and the environment, we choose the drive as provided by a propagating pulse containing a single photon. We choose the photon to initially populate only the continuum of modes \(\left\{{a}_{\omega }\right\}\), so the vacuum state of \(\left\{{b}_{\omega }\right\}\) allows us to avoid depleting our target state \(\leftb\right\rangle\). The initial state of the field is
\(\left0\right\rangle ={\prod }_{\omega }\left{0}_{\omega }^{a}\right\rangle \otimes \left{0}_{\omega }^{b}\right\rangle\) is the vacuum state of all the field modes. The initial state of the Λ system is the mixed ρ_{S}(0) given in Eq. (6). The global quantum state is given by
where \(U=\exp (iHt/\hslash )\) for the (timeindependent) global Hamiltonian H [Eq. (8)]. The quantum states of the system and the environment are obtained by the partial traces \({\rho }_{S}(t)={{\rm{tr}}}_{E}[\rho (t)]\) and \({\rho }_{E}(t)={{\rm{tr}}}_{S}[\rho (t)]\).
We obtain analytical expressions for the probabilities \({p}_{k}(t)=\left\langle k\right{\rho }_{S}(t)\leftk\right\rangle\), for k = a, b, e. Equation (6) allows us to write the reduced state of the system as \({\rho }_{S}(t)={p}_{a}^{(0)}{{\rm{tr}}}_{E}\left[U\lefta,{1}_{a}\right\rangle \left\langle a,{1}_{a}\right{U}^{\dagger }\right]+{p}_{b}^{(0)}{{\rm{tr}}}_{E}\left[U\leftb,{1}_{a}\right\rangle \left\langle b,{1}_{a}\right{U}^{\dagger }\right]\). It proves useful to write p_{k}(t) in terms of transition probabilities, p_{n→k}(t), as
where n = a, b, and
Equations (13) and (14), which follow directly from the mixedstate structure of the initial state, unravel a close analogy with the notation used in Eq. (1) for the microscopically reversible condition. Equation (13) has no relation with the degree of Markovianity in the dynamics of the reduced state ρ_{S}(t), though. On the contrary, Valente et al.^{23} characterize nonMarkovianity in a quite similar scenario. The photon initially at modes \(\left\{{a}_{\omega }\right\}\) does not interact with the threelevel system initially at \(\leftb\right\rangle\) [as derived from Eqs. (6), (9), and (12)], so we find that p_{b→a}(t) = p_{b→e}(t) = 0 and p_{b→b}(t) = 1. Because the transition probabilities vanish identically, regardless of the initial pulse shape, the backwards probability also vanishes,
We have defined the reverse drive protocol here as the mirrored shape of the initial pulse (further discussed below). Put simply, we revert only the drive, not the entire universe. The final global state of the system plus the environment is obviously reversible in our model since it is governed by the global unitary operator U in Eq. (12). Equation (15) explains the second line in Eqs. (4). We now compute
where \(\left\xi (t)\right\rangle \equiv U\lefta,{1}_{a}\right\rangle\). Since our H conserves the total number of excitations, we can restrict our model to the oneexcitation subspace, \(\left\xi (t)\right\rangle =\psi (t)\lefte,0\right\rangle +{\sum }_{\omega }{\phi }_{\omega }^{a}(t){a}_{\omega }^{\dagger }\lefta,0\right\rangle +{\phi }_{\omega }^{b}(t){b}_{\omega }^{\dagger }\leftb,0\right\rangle .\) We find that
and similarly with p_{a→a}(t). Without loss of generality, we have defined a onedimensional realspace representation for the amplitudes, \({\phi }_{n}(z,t)\equiv {\sum }_{\omega }{\phi }_{\omega }^{n}(t)\exp (i{k}_{\omega }z)\), which characterizes the pulse shape. We have also employed a linear dispersion relation, ω = ck_{ω}, and approximated the density of modes by a constant, ϱ_{ω} ≈ ϱ. Note that, in Eq. (9), we have implicitly assumed the threelevel system to be positioned at z_{S} = 0; otherwise, a phase term such as \(\exp (i{k}_{\omega }{z}_{S})\) should have been included in the sum over modes.
To go a step further, as to obtain explicit expressions for the transition probabilities, we solve the Schrödinger equation for \(\left\xi (t)\right\rangle\) (see Methods). Our intention with keeping modes \(\left\{{b}_{\omega }\right\}\) initially in the vacuum state, as we did in Eq. (11), was to minimize excitations promoting the unwanted backward (\(\leftb\right\rangle \to \lefta\right\rangle\)) transitions. Now that we have defined the amplitudes ϕ_{n}(z, t), we see that Eq. (11) implies ϕ_{b}(z, 0) = 0, which we combine with (34) and substitute in (17). After a change of variables, we find that
Although the main results in this paper do not depend on the initial pulse shape in modes \(\left\{{a}_{\omega }\right\}\) (i.e., on the choice of ∣ϕ_{a}(z, 0)∣), it is worth working out an explicit example. To that end, we now set ϕ_{a}(z, 0) to have an exponential envelope profile of linewidth Δ and a central frequency ω_{L} (see Eqs. (36) and (37) in the Methods), as typical in spontaneous emission. We are particularly interested in the resonant condition ω_{L} = ω_{a} (see the Methods for the more general solution). We find, in the monochromatic limit, Δ ≪ Γ_{a} + Γ_{b}, and at long times t ≫ Δ^{−1}, that
When Γ_{a} = Γ_{b}, we have that p_{a→b}(∞) = 1. Equation (19) reveals the ideal driven selforganization we have been looking for. As stated before, the selforganization in our model does not depend on δ_{ab}. As a final step, we combine Eq. (19) and p_{b→b}(t) = 1 to see that, in the ideal (monochromatic and resonant) regime,
that is, \({\rho }_{S}(\infty )=\leftb\right\rangle \left\langle b\right\) and \({\rho }_{S}^{* }(\infty t)=\leftb\right\rangle \left\langle b\right\) for all times t. Hence, in the t → ∞ limit, \({\rho }_{S}^{* }(\infty t)\,\ne\, {\rho }_{S}(0)\), as was to be shown.
Energetics of the selforganization
We now need to verify that the selforganized quantum state we have found can indeed be classified as dissipative adaptation. We shall find that ideal selforganization costs maximal work absorption, followed by maximal dissipation. This is not an obvious relation because in the ideal selforganization (which takes place in the monochromatic limit, as we have shown above), the excitation probability is minimized (rather than maximized),
For instance, in the case of an incoming pulse with exponential profile (as used in Eq. (19)), \( \psi (t){ }^{2}\le 4{{\Delta }}{{{\Gamma }}}_{a}/{({{{\Gamma }}}_{a}+{{{\Gamma }}}_{b})}^{2}\ll 1\), for Δ ≪ Γ_{a} + Γ_{b}. Extremely lowexcitation probabilities may leave the false impression that no energy is absorbed neither dissipated at all. To address this issue, we need to resolve energy transfer into work and heat in our model. As mentioned earlier, work and heat can be regarded as the predictable versus the random energy exchanges, respectively. We restrict ourselves once again to the resonant case, ω_{L} = ω_{a}, keeping in mind however an arbitrary pulse shape. The average energy of the Λ system driven by this resonant photon is given by
The resonant condition here avoids dynamic Stark shifts that would otherwise bring extra (dispersive) energetic contributions from timedependent frequencies, as shown by Valente et al.^{24}. To be more precise, dispersive (refractive) energetic contributions depend on the average interaction energy^{24}, which vanishes at resonance (ω_{L} = ω_{a} implies that \(\left\langle {H}_{I}(t)\right\rangle \equiv {\rm{tr}}[\rho (t){H}_{I}]=0\); see details in the Methods). That justifies why we have used only H_{S} in Eq. (22). As inspired by Eq. (3), we are interested in the work performed on the system during the dynamical transition starting from \(\lefta\right\rangle\) and arriving at \(\leftb\right\rangle\). Therefore, we now consider \({\rho }_{S}(0)=\lefta\right\rangle \left\langle a\right\), for which p_{e}(t) = ∣ψ(t)∣^{2}. From the Schrödinger equation (see Eq. (32) in the Methods), we find that
where \({\rm{Re}}\left[\bullet \right]\) stands for the real part. Equation (23) clearly shows us that the excited state of the Λ system is governed by a (predictable) drivedependent term, related to ϕ_{a}(−ct, 0), and a (random) spontaneous emission term, proportional to Γ_{a} + Γ_{b}. This motivates us to define the total average absorbed work \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}\) and the total average absorbed heat \({\left\langle {Q}_{{\rm{abs}}}\right\rangle }_{a}\) in the dynamics starting at \({\rho }_{S}(0)=\lefta\right\rangle \left\langle a\right\) as
where
and
Our definition of work, Eq. (25), can also be written as \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}=\mathop{\int}\nolimits_{0}^{\infty }\left\langle \left({\partial }_{t}d(t)\right){E}_{{\rm{in}}}(t)\right\rangle dt\) (in the Heisenberg picture and in the rotatingwave approximation), revealing a more clear link to our classical notion of work. Here, d(t) = U^{†}dU is the dipole operator, where d = ∑_{k}d_{ek}σ_{k} + H.c., the driving field operator at the system’s position is \({E}_{{\rm{in}}}(t)={\sum }_{\omega }i{\epsilon }_{\omega }\left({a}_{\omega }+{b}_{\omega }\right)\exp (i\omega t)+{\rm{H.c.}}\), the coupling is \({g}_{a}={d}_{ea}{\epsilon }_{{\omega }_{a}}/\hslash\), and the average is calculated at the initial state \(\lefta,{1}_{a}\right\rangle\). The Heisenberg picture also explains why work can be finite even though the average driving field is precisely zero at the singlephoton state (i.e., \(\left\langle {1}_{a}\right{E}_{{\rm{in}}}(t)\left{1}_{a}\right\rangle =0\)). This shows how the dissipative adaptation can be robust to a phaseincoherent work source, in contrast to the classical forces of welldefined phases used in ref. ^{4} (in our model, a semiclassical driving field with a welldefined phase would correspond to an initial coherent, or Glauber, state \(\left\alpha \right\rangle ={\prod }_{\omega }\left{\alpha }_{\omega }\right\rangle\), where \({a}_{\omega }\left{\alpha }_{\omega }\right\rangle ={\alpha }_{\omega }\left{\alpha }_{\omega }\right\rangle\)). In Eq. (26), \({{{\Gamma }}}_{a,b}\propto {g}_{a,b}^{2}\) reveals that the heat is related with the variance of the interaction energy, explaining why the vanishing average interaction energy does not hinder energy exchange in the form of heat and confirming its stochastic nature. The energy conservation in Eq. (5) results, of course, from defining \({\left\langle {Q}_{{\rm{abs}}}\right\rangle }_{a}\equiv {\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\) in Eq. (24). Finally, \(\left\langle {H}_{S}(0)\right\rangle =0\) and \(\left\langle {H}_{S}(\infty )\right\rangle ={p}_{a\to b}(\infty )\hslash {\delta }_{ab}\). We remind that, during the dynamical transition from \(\lefta\right\rangle\) to \(\leftb\right\rangle\), we have \(p_e(t) = \psi(t)^2\), allowing us to establish an exact adaptationenergy relation between Eqs. (18) and (26). We finally find our quantum adaptationwork relation,
valid for a photon of arbitrary pulse shape and resonant with ω_{a}, as well as for arbitrary δ_{ab}, as stated in Eqs. (4). In the case of the exponential pulse used in Eq. (19), for instance, we find that \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}=\hslash {\omega }_{a}\,4{{{\Gamma }}}_{a}/({{{\Gamma }}}_{a}+{{{\Gamma }}}_{b}+{{\Delta }})\), maximized in the monochromatic regime (Δ ≪ Γ_{a} + Γ_{b}). If in addition Γ_{a} = Γ_{b}, we get \({\left\langle {W}_{{\rm{abs}}}\right\rangle }_{a}=2\hslash {\omega }_{a}\), which is twice the initial average energy contained in the singlephoton pulse. This counterintuitive result reinforces the notion that work is the amount of energy transferred during a process, rather than the average energy stored in a system at a given time (after all, unitary evolutions with timeindependent Hamiltonians conserve energy; here, \({\partial }_{t}\left\langle H(t)\right\rangle =0\) implies that \(\left\langle {H}_{S}(t)\right\rangle +\left\langle {H}_{I}(t)\right\rangle +\left\langle {H}_{E}(t)\right\rangle ={p}_{b}(0)\hslash {\delta }_{ab}+\hslash {\omega }_{L}\), so at resonance and for p_{b}(0) = 0 we find \(\left\langle {H}_{S}(t)\right\rangle \le \hslash {\omega }_{L}\)). Equation (27) is our key signature of a quantum dissipative adaptation. The seeming lowexcitation issue (∣ψ(t)∣^{2} ≪ 1, in the monochromatic limit) has finally been clarified, given that the time integral of ∣ψ(t)∣^{2} (in Eq. (18)) is not only finite, but also linearly proportional to the work required for the quantum dissipative adaptation to take place.
Entropy in the selforganization
In the classical formulation of dissipative adaptation, entropy production has a key role in establishing the connection between energy transfer and statistical irreversibility, requiring no detailed knowledge on the state of the environment. Here, irreversibility is readily characterized by the asymmetry between the forward, p_{a→b}(t), and the backward, \({p}_{b\to a}^{* }(t)\), processes. Nevertheless, we have the advantage that we keep the full description of the quantum state of the system plus the environment, ρ(t) (at the expense of a greater degree of generality in our global Hamiltonian H). With \({\rho }_{E}(t)={{\rm{tr}}}_{S}[\rho (t)]\) at hands (see Methods), we now seek to describe what happens to the environment during and after the drive interacts with the threelevel system. We calculate the exact expression for the von Neumann entropy,
See Eqs. (45) and (46) in the Methods for the analytic expression. We have found that our classical intuition, namely, that better adaptation produces more entropy in the environment, can be found by an appropriate distinction between the classical and the quantum contributions to S_{E}.
The idea behind this distinction is the following. Let us first suppose that the system is initially at \(\lefta\right\rangle\). A highly monochromatic incoming photon will fully induce transition from state \(\lefta\right\rangle\) to \(\leftb\right\rangle\). Hence, the outgoing photon will be detected at modes \(\left\{{b}_{\omega }\right\}\), as well. Now, by considering an initially mixed state of the threelevel system as given by \((1/2)\lefta\right\rangle \left\langle a\right+(1/2)\leftb\right\rangle \left\langle b\right\), a highly monochromatic incoming photon will have 1/2 probability of leaving at \(\left\{{a}_{\omega }\right\}\) (in the cases where it encounters the system at \(\leftb\right\rangle\)) and 1/2 of leaving at \(\left\{{b}_{\omega }\right\}\) (in the cases where it encounters the system at \(\lefta\right\rangle\)), so the final global state would be mixed and separable,
\(\left{\tilde{1}}_{b}\right\rangle\) and \(\left{1}_{a}^{{\rm{free}}}\right\rangle\) have been defined in Eqs. (43) and (44) below (see Methods). This is what we call the classical contribution to the final mixed state of the field: the initial mixture of the system is fully transferred to the environment.
Let us now consider again that the system is initially at \(\lefta\right\rangle\) (i.e., \({p}_{a}^{(0)}=1\)). However, let us assume that the linewidth of the pulse is of the order of the dissipation rates of the threelevel system. In that case, the final state of the global system becomes entangled,
N_{k} being the average number of photons at modes k (see Eq. (42) in the Methods). Therefore, the quantum state of the environment in this case, \({\rho }_{E}(\infty )={{\rm{tr}}}_{S}[\left\xi (\infty )\right\rangle \left\langle \xi (\infty )\right]\), is also mixed between modes \(\left\{{a}_{\omega }\right\}\) and \(\left\{{b}_{\omega }\right\}\). However, the mixture in this case arises from a sustained system–environment quantum entanglement rather than from the statistical mixture in the initial state of the system.
To unravel these two entropy contributions, we define the classical contribution to the environment entropy as
where \(S(\bullet )\equiv {\rm{tr}}[\bullet \mathrm{ln}\,\bullet ]\) is the von Neumann entropy (see Eq. (51) in the Methods). In Eq. (31), we have taken into account that only the term proportional to \({p}_{a}^{(0)}\) in ρ_{S}(0) generates the entanglement discussed in Eq. (30). Now we focus on the long time limit, t→∞. Most interestingly, we have analytically expressed \({S}_{E}^{c}(\infty )\) as a function of p_{a→b}(∞) (see Eqs. (45)–(49) in the Methods). Figure 2 illustrates that \({S}_{E}^{c}(\infty )\) vs. p_{a→b}(∞) (solid black line) is a monotonic function, in contrast with the nonmonotonic S_{E}(∞) vs. p_{a→b}(∞) (dashed blue line). This monotonic behavior shows that the more organized is the threelevel system, the higher is the classical contribution to the entropy of the environment at long times t→∞. Besides having derived \({S}_{E}^{c}(\infty )\) as a function of p_{a→b}(∞), we have also expressed p_{a→b}(∞) as a function of the average dissipated heat in the \(\lefta\right\rangle\) to \(\leftb\right\rangle\) transition, \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\), with the help of Eq. (27). This establishes the function S_{E}(∞) vs. \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\), as mentioned earlier. In addition, we have derived \({S}_{E}^{c}(\infty )\) as a function of \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\). This is even more meaningful, because we have found that the function \({S}_{E}^{c}(\infty )\) vs. \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\) is monotonic. To provide an example illustrating this monotonicity, let us take the degenerate case (δ_{ab} = 0) with equal dissipation rates (Γ_{a} = Γ_{b}). In that case, we have that \({p}_{a\to b}(\infty )={\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}/(2\hslash {\omega }_{a})\), showing that \({S}_{E}^{c}(\infty )\) vs. \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\) is monotonic, since \({S}_{E}^{c}(\infty )\) vs. p_{a→b}(∞) is monotonic as well (Eq. (50) in the Methods provides the more general relation). The monotonicity here strengthens the signature of dissipative adaptation. Namely, maximal adaptation (irreversible selforganization) not only costs maximal work absorption (as we have shown in the energetics analysis), but also maximizes the dissipated heat which, in turn, maximizes the classical contribution to the environment entropy.
Discussion
Our results establish the quantum dissipative adaptation underlying the driven selforganization of a quantum state, going beyond the classical formulation. We have explored an elementary fully quantized model, exactly solvable for both the system and the environment, where the irreversibly selforganized quantum state of a threelevel system is owing to the work absorption from a singlephoton pulse, part of which is dissipated to the environment (as shown in Eqs. (4) and (5)). The irreversibility of this selforganization became clear from the asymmetry in the transition probabilities in the forward and backward processes. Finally, with the purpose of providing an additional signature of dissipative adaptation, we have analytically investigated the environment entropy. We have found that the classical contribution to the environment entropy variation is a monotonic function of p_{a→b}(∞) (as illustrated in Fig. 2), showing that maximizing adaptation not only requires maximal work absorption, but also leads to maximal increase in the classical contribution to the environment entropy, due to maximal dissipated heat.
We remind that the meaning of this increase in the environment entropy is that of a statistical mixing between the field modes \(\left\{{a}_{\omega }\right\}\) and \(\left\{{b}_{\omega }\right\}\), as we have analyzed in Eq. (29). Our model’s dynamics do not lead to the thermalization of the environment, which always remains out of equilibrium. If we wish to take the analogy further, so as to give a thermodynamic meaning to the entropy increase, we will have to add extra ingredients. In practice, there could be for instance a slight increase in the local temperature of the environment in the vicinity of the Λ system. To describe that kind of effect, the model should consider a finite heat capacity environment (in contrast to our infinitesize environment) and some auxiliary lightmatter coupling mechanism that could effectively create an interaction between the many frequency modes of the light field. Such features provide means for the environment state to eventually approximate a Gibbs state at some new finite temperature \(T^{\prime} \, > \,T=0\) (producing even more entropy than that we have calculated from our model), in the spirit of Timpanaro et al.^{25}.
Before concluding, we would like to point out how our model can be significant to quantum manybody systems. Our intention is to indicate how broadly applicable the concept of quantum dissipative adaptation may become. First, we notice that the Λ structure of energy levels can arise from the quantization of collective degrees of freedom describing many interacting atoms and electrons, as happens in the socalled artificial atoms (e.g., electronhole pairs in semiconductor quantum dots^{26} and the quantized magnetic flux in superconducting rings^{27}). Lodahl et al.^{26} and Gu et al.^{27} also discuss how these artificial atoms can be driven by singlephoton pulses propagating in onedimensional waveguides, building the closest possible scenario to that in our model.
We can also envision the quantum dissipative adaptation in ensembles of noninteracting atoms or spins. That is more easily seen when we realize that our idea of a driven selforganized quantum state is notably similar to the dynamics induced by stimulated Raman adiabatic passage (STIRAP)^{28}, a versatile and robust technique that has been performed, e.g., in ultracold gases, in doped crystals and in nitrogenvacancy centers. STIRAP consists of an efficient population transfer between two discrete quantum states of an ensemble of emitters (usually the lowest levels of Λ systems) by coherently coupling them with two radiation fields (wellcontrolled classical pulses) through an unpopulated intermediate state. The connection we have in mind between the driven selforganization provided by STIRAP and dissipative adaptation becomes more evident in light of the recent proposal for using STIRAP as a tool for spectral hole burning (SHB) in inhomogeneously broadened systems^{29}. The reason is that the mechanism behind standard SHB^{30} turns out to be precisely that of classical dissipative adaptation, as discussed by Kedia et al.^{31}. Namely, those dipoles that get excited by the resonant drive (whose frequency can be swept on demand) can become irreversibly trapped in dark states (at sufficiently low temperatures). The difference concerning the newly proposed STIRAPbased SHB^{29}, or the singlephoton pulse that we have studied here, is the quantum coherent nature of the process. Understanding how STIRAPbased SHB depends on temperature seems a valuable opportunity for widening the concept of quantum dissipative adaptation (similarly, Ropp et al.^{10} show a groundbreaking experiment on temperaturedependent dissipative selforganization in optical space).
Optomechanical nanoresonators^{32} also hold the promise of displaying some kind of quantum dissipative adaptation in the lines of our results. We glean this notion from Kedia et al.^{31}, where signatures of dissipative adaptation are shown in disordered networks of classical bistable springs. In the mechanical nanoresonators by Yeo et al.^{32}, bistability can arise from a strainmediated coupling between the center of mass of an oscillating nanowire and the quantum state of a single semiconductor quantum dot embedded therein. This kind of coupling between an optically controllable microscopic degree of freedom (within the quantum dot) and a mesoscopic degree of freedom (the nanoresonator center of mass) opens appealing perspectives for studying dissipative adaptation at a quantumclassical boundary of a manybody system. As a last word on resonators, it seems relevant investigating whether the vibrationassisted exciton transport found in photosynthetic lightharvesting antennae^{17} could be related with a quantum dissipative adaptation.
To conclude, the quantum dissipative adaptation we have found can be regarded as a proof of principle, in need of generalizations towards many directions. As a first example, it would be worth investigating dissipative adaptation in larger Hilbert spaces (i.e., in a vast energy landscape, in the words of ref. ^{1,4}). Multistability makes selforganization in classical manybody systems a fascinating problem^{1,4,31}. Curiously, the multistability from complex classical systems is reminiscent of our zerotemperature model. Among the huge amount of available “nonorganized” stable states that our Λ system can occupy, only a single (exceptional) state is populated when the work source is optimal. In other words, although we find an infinite number of (pure or mixed) combinations of states \(\lefta\right\rangle\) and \(\leftb\right\rangle\) that are equally stationary in the nondriven zerotemperature environment, the system always ends up in the rare state \(\lefta\right\rangle\) (irrespective of the infinitely many possible initial states) once it has been driven by the suitable pulse (which maximizes the absorbed work in the \(\lefta\right\rangle\) to \(\leftb\right\rangle\) transition).
It also remains to be investigated how other sources of work and finite temperatures affect dissipative adaptation in quantum systems. In our Λ system, for instance, we expect that the forward (\(\lefta\right\rangle\) to \(\leftb\right\rangle\)) transition will keep being favored with respect to the backward (\(\leftb\right\rangle\) to \(\lefta\right\rangle\)) transition at finite temperatures, as long as the work source is strong enough. Also, the time dependence of this asymmetry should be transient or stationary, depending on the work source type (pulsed or continuous). In quantum thermodynamics, the influence of the environment temperature on entropy production at the quantum regime, on the one hand, has been the subject of many recent studies^{25,33,34,35,36}. The focus on modelindependent aspects is justified not only due to the interest in generality, but also due to the closest possible analogies with classical fluctuation theorems and with the Landauer erasure principle (that establishes how information erasure is connected to thermodynamic entropy production due to heat dissipation). On the other hand, Manzano et al.^{37}, for instance, provide an alternative formalism for quantum fluctuation theorems that go beyond thermalequilibrium states of the environment. It may be the case that the quantum fluctuation theorems developed in the papers above (and in the references therein) present fruitful methods in the search for a general quantum theory of dissipative adaptation. In summary, we provide the starting point towards a quantum thermodynamics of driven selforganization.
Methods
Global systemplusreservoir quantum dynamics
To obtain explicit expressions for the transition probabilities in Eq. (17), we need to solve the Schrödinger equation \(i\hslash {\partial }_{t}\left\xi (t)\right\rangle =H\left\xi (t)\right\rangle\), where \(\left\xi (t)\right\rangle =\psi (t)\lefte,0\right\rangle +{\sum }_{\omega }{\phi }_{\omega }^{a}(t){a}_{\omega }^{\dagger }\lefta,0\right\rangle +{\phi }_{\omega }^{b}(t){b}_{\omega }^{\dagger }\leftb,0\right\rangle\). After a Wigner–Weisskopf approximation and using that \({\phi }_{n}(z,t)\equiv {\sum }_{\omega }{\phi }_{\omega }^{n}(t)\exp (i{k}_{\omega }z)\), with a linear dispersion relation, ω = ck_{ω}, and ∑_{ω} → ∫dωϱ_{ω} ≈ ϱ∫dω, we find that
with
and
where \({{{\Gamma }}}_{k}=2\pi {g}_{k}^{2}\varrho\).
Integrating Eq. (32) for ψ(0) = 0 gives
which depends on the initial photon wavepacket,
ω_{L} is the central frequency of the pulse and \({\phi }_{a}^{{\rm{shape}}}(z,0)\) is its spatial shape. For the sake of providing an explicit and physically motivated example, we consider in Eq. (19) the shape of the initial photon wavepacket to be an exponential (raising in space, equivalent to decaying in time for a rightpropagating pulse), as typical from spontaneous emission,
where Δ is the pulse linewidth and \(N=\sqrt{2\pi \varrho {{\Delta }}}\) is a normalization factor (considering ϕ_{b}(z, 0) = 0). Finally, and substituting \({{{\Gamma }}}_{a}=2\pi {g}_{a}^{2}\varrho\), we have that
where
and δ_{L} ≡ ω_{L} − ω_{a}.
The average interaction energy is given by
From Eqs. (6), (12), and the definition \(\left\xi (t)\right\rangle \equiv U\lefta,{1}_{a}\right\rangle\), we have that
Using H_{I} from Eq. (9), we have that \(\left\langle b,{1}_{a}\right{H}_{I}\leftb,{1}_{a}\right\rangle =0\) and \(\left\langle \xi (t)\right{H}_{I}\left\xi (t)\right\rangle =2(\hslash {g}_{a}{\rm{Im}}[{\psi }^{* }(t){\phi }_{a}(ct,0)]+\hslash {g}_{b}{\rm{Im}}[{\psi }^{* }(t){\phi }_{b}(ct,0)])\). Choosing ϕ_{b}(z, 0) = 0 obviously makes \({\rm{Im}}[{\psi }^{* }(t){\phi }_{b}(ct,0)]=0\). By choosing the resonance condition, δ_{L} = 0, we have from Eqs. (35) and (36) that \({\rm{Im}}[{\psi }^{* }(t){\phi }_{a}(ct,0)]=0\). This shows that \(\left\langle {H}_{I}(t)\right\rangle =0\) at resonance.
Entropy of the environment
The quantum state of the environment in our model is obtained from the global initial state \(\rho (0)=({p}_{a}^{(0)}\lefta\right\rangle \left\langle a\right+{p}_{b}^{(0)}\leftb\right\rangle \left\langle b\right)\otimes \left{1}_{a}\right\rangle \left\langle {1}_{a}\right\). We obtain that
We have defined \({N}_{k}(t)\equiv {\sum }_{\omega } {\phi }_{\omega }^{k}(t){ }^{2}\), with
and
To explicitly calculate the von Neumann entropy,
we need to find the eigenvalues λ_{j} of ρ_{E}. The exact diagonalization of ρ_{E} gives us four nonzero eigenvalues, namely, \({\lambda }_{1}={p}_{a}^{(0)} \psi (t){ }^{2}\), \({\lambda }_{2}={p}_{a}^{(0)}{N}_{b}(t)\), and
Terms N_{a} and N_{b} can be interpreted as the average number of photons in each continuum of modes. Their mathematical expressions, however, are invariably connected with the threelevel system probabilities, namely,
We calculate the overlap between the states representing the free propagation and the reemitted photon at mode a in the realspace representation,
where ϕ_{a, free}(z,t) ≡ ϕ_{a}(z − ct, 0).
With the aid of (32) and (33), and using integration by parts, we find that
valid at long times t→∞. Equation (49) provides the core connection between the overlap and the transition probability at long times that we needed. Additionally, N_{a}(∞) = 1 − N_{b}(∞) = 1 − p_{a→b}(∞) and λ_{1}(∞) = 0. Since Eqs. (27) and (24) link p_{a→b}(∞) to \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\),
we have thus analytically established the function S_{E}(∞) vs. \({\left\langle {Q}_{{\rm{diss}}}\right\rangle }_{a}\), valid at long times t → ∞, under the resonant condition ω_{L} = ω_{a}, for any arbitrary pulse shape.
Finally, the classical contribution to the entropy is \({S}_{E}^{c}\equiv {S}_{E}{p}_{a}^{(0)}S({{\rm{tr}}}_{S}[\left\xi (t)\right\rangle \left\langle \xi (t)\right])\), where
for which we also have the analytic solution.
Data availability
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Code availability
The code used to produce the figure in this article is available from the corresponding author upon request.
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Acknowledgements
This work was supported by the Serrapilheira Institute (grant no. Serra191232056) and by Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCTIQ), Brazil.
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Valente, D., Brito, F. & Werlang, T. Quantum dissipative adaptation. Commun Phys 4, 11 (2021). https://doi.org/10.1038/s42005020005120
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DOI: https://doi.org/10.1038/s42005020005120
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