Generalized Adiabatic Theorems: Quantum Systems Driven by Modulated Time-Varying Fields

Consider a family of time-dependent reference Hamiltonians $\hat{H}_{0}(t;\lambda)$ indexed by the parameter $\lambda$ (e.g. $\lambda$ may be an amplitude $\hat{H}_{0}(t;\lambda)=\lambda\hat{h}_{0}(t)$ or frequency $\hat{H}_{0}(t;\lambda)=\exp(-i\lambda t)\hat{h}_{0}+h.c.$). For each value of $\lambda$, setting $\hbar=1$, we define the instantaneous normal modes as

where $\theta_{n}(t;\lambda)\equiv\int^{t}_{0}ds\epsilon_{n}(s;\lambda)$ is the dynamical phase, $\epsilon_{n}(t;\lambda)$ is an instantaneous quasienergy and the $\ket{n(t;\lambda)}$ is the instantaneous normal mode of $\hat{H}_{0}(t;\lambda)$.

These normal modes and corresponding quasienergies are found by solving the eigenproblem

where $\hat{H}_{F}(t;\lambda)\equiv\hat{H}_{0}(t;\lambda)-i\partial/\partial t$. That is, the normal modes are particular solutions to the Time-Dependent Schrödinger Equation (TDSE),

and are well understood for many driving fields. For example, for time-independent Hamiltonians, the normal modes of static Hamiltonians are energy eigenstates, while for periodic Hamiltonians they are Floquet states.

A modulation of a time-dependent Hamiltonian, $\hat{H}_{0}(t;\lambda)$, is a transformation that varies $\lambda\to\lambda_{t}$ over a time interval $[0,\tau]$ through a modulation protocol $\Lambda\equiv\{\lambda_{t}|t\in[0,\tau]\}$. The resultant modulated field $\hat{\widetilde{H}}_{\Lambda}(t)\equiv\hat{H}_{0}(t;\lambda_{t})$ sweeps through the Hamiltonian family defined above (e.g. a field with a time-dependent amplitude for $\hat{H}_{0}=\lambda h_{0}(t)$). Note, as an aside, that this is an operator generalization of modulations in signal processing and encoding theory Byrne (1993). Generally, the modulated Hamiltonian can be significantly more complicated than $H_{0}$ since the modulation function may be highly nonlinear and the modulation parameter can vary non-trivially with time.

The dynamics induced by $\hat{\widetilde{H}}_{\Lambda}(t)$ can now be expressed in terms of instantaneous normal modes of $\hat{H}_{0}$, Eq. (1), to give

where $\theta_{n}(t;\Lambda)\equiv\int^{t}_{0}ds\epsilon_{n}(s;\lambda_{s})/\hbar$. The instantaneous state of the system can always be expressed in this form since $\hat{H}_{F}(t;\lambda_{t})$ is always Hermitian and therefore has a basis of eigenstates $\{\ket{n(t;\Lambda_{t})}\}$ that can be used to expand $\ket{\Psi(t)}$ with coefficient $a_{n}(t)\equiv\braket{n(t;\lambda_{t})|\Psi(t)}=c_{n}(t)\exp(-i\theta_{n}(t))$ ¹¹1The modified Hamiltonian is defined using a partial time derivative and therefore neglects the implicit time-dependence of $\lambda_{t}$..

This approach mirrors that used to derive traditional ATs, but replacing the eigenstates with normal modes and energies with quasienergies. We therefore proceed through a similar path to derive adiabaticity conditions under which a system initialized in a normal mode of $\hat{H}_{0}(0;\lambda_{0})$ will remain in that normal mode at all times. Mathematically, this occurs when $c_{n}(t)$ are decoupled in the TDSE, Eq. (3). Substituting Eq. (4) into Eq. (3) we find

where we have used the identity $d\theta_{n}(t;\Lambda)/dt=\epsilon_{n}(t;\lambda_{t})$ and Eq. (2) to cancel terms proportional to $\epsilon_{n}(t;\lambda_{t})$. Projecting onto an $\ket{m(t;\lambda_{t})}$, yields equations of motion for the coefficients,

where we have suppressed the $t$ and $\lambda_{t}$ dependence for brevity, and $\theta_{nm}\equiv\theta_{n}-\theta_{m}$. To obtain Eq. (6), Eq. (2) was differentiated with respect to $\lambda_{t}$ to obtain $\braket{m|\partial/\partial\lambda_{t}|n}=\braket{m|\partial\hat{\widetilde{H}}/\partial\lambda_{t}|n}/(\epsilon_{n}-\epsilon_{m})$.

Equation (6) takes the same form as the original adiabatic theorem with cross-coupling between normal modes scaled by a rapidly oscillating term Sarandy and Lidar (2005). As a result normal modes evolve independently of one another when the following condition is satisfied:

where $\epsilon_{nm}(t;\lambda_{t})\equiv\epsilon_{n}(t;\lambda_{t})-\epsilon_{m}(t;\lambda_{t})$. Consequently, if $\lambda_{t}$ changes much more slowly than the difference in dynamical mode frequencies, the slow modulation will not cross couple the normal modes of the modulated Hamiltonian.

Equation (7) has several important features. First, if $\hat{H}_{0}(t;\lambda)$ are $t$-independent, then their normal modes are energy eigenstates and all Hamiltonian time-dependence is contained in the modulation. In this case, the AMT reduces to the traditional AT, simply relabeling the time variable as $\lambda$. The AMT then extends previous AT’s by recognizing them as statements about how time-dependent transformations of Hamiltonian fields impact dynamical modes of the TDSE. These coincide with eigenstates for static Hamiltonians but the same insight can be applied to any field with well-understood dynamics. For example, if $\hat{H}_{0}(t;\lambda)$ are periodic, Eq. (7) states that slow modulations do not couple Floquet states. One key distinction of the AMT is that its speed limits are set by the differences in normal mode frequencies which are often easier to manipulate than energy gaps between eigenstates, allowing for faster adiabatic transformations. A numerical example is discussed in Sect. III.1.1.

Consider now an adiabatic theorem for open quantum systems. To prove this theorem we take an approach inspired by Ref. Sarandy and Lidar (2005). The dynamics of an open quantum system are governed by the Liouville von-Neumann (LvN) equation

in the time-convolutionless form Breuer and Petruccione (2007); Alicki and Lendi (2007); Blum (2012). The double-ket notation $|\cdot\rangle\rangle$ is indicates an operator Liouville space with a trace inner product $\langle\langle A|B\rangle\rangle\equiv\operatorname{Tr}\{\hat{A}^{\dagger}\hat{B}\}$. By analogy with the isolated system we define modulations by starting with a family of time-dependent Liouvillians $\hat{\hat{\mathcal{L}}}_{0}(t;\lambda)$ and allowing the modulation parameter to vary over time to give $\hat{\hat{\mathcal{L}}}_{\Lambda}(t)\equiv\hat{\hat{\mathcal{L}}}_{0}(t;\lambda_{t})$. A modified Liouvillian can then be defined by $\hat{\hat{\mathcal{L}}}_{F}(t;\lambda)\equiv\hat{\hat{\mathcal{L}}}(t;\lambda)-\partial/\partial t$.

The LvN equation appears similar to the TDSE [Eq. (3)], suggesting that we may be able to apply the same analysis, but replacing Hamiltonians with Liouvillians. However, the Liouvillian and it’s corresponding modified operator are completely positive but not necessarily Hermitian Alicki and Lendi (2007) and therefore require more care in defining their normal modes. A given $M\times M$ modified Liouvillian, $\hat{\hat{\mathcal{L}}}_{F}$, has an incomplete set of $N\leq M$ left and right eigenvectors $\langle\langle\mathcal{Q}_{\alpha}(t)|$ and $|\mathcal{P}_{\alpha}(t)\rangle\rangle$. Associated with each eigenvector is a Jordan Block comprised of $n_{\alpha}$ generalized eigenvectors that combine to give an orthonormal basis of Liouville space defined by the generalized eigenproblem

where $\langle\langle\mathcal{Q}_{\alpha}^{(n_{\alpha}-1)}|\equiv\langle\langle\mathcal{Q}_{\alpha}|$,$\langle\langle\mathcal{Q}_{\alpha}^{(n_{\alpha})}|\equiv\langle\langle 0|$, $|\mathcal{P}_{\alpha}^{(0)}\rangle\rangle\equiv|\mathcal{P}_{\alpha}\rangle\rangle$, and $|\mathcal{P}_{\alpha}^{(-1)}\rangle\rangle\equiv|0\rangle\rangle$, where $|0\rangle\rangle$ is the zero operator. Dynamically, the LvN Equation does not cross couple the Jordan blocks to one another, but can lead to cross coupling within states of the same Jordan block, thereby defining decoupled dynamical subspace. (For a concise summary of Jordan canonical form in the context of quantum adiabatic theorems, see ref. Sarandy and Lidar (2005).)

Motivated by the isolated systems derivation, we consider the modified Liouvillian and aim to expand the dynamics of the open system in terms of its instantaneous generalized eigenstates. These are given by the (right) generalized equation

The left generalized eigenstates can be similarly defined as the right eigenstates of $\hat{\hat{\mathcal{L}}}_{F}^{\dagger}$.

Allowing the modulation parameter to change with time, the instantaneous state of the system can be expanded as a superposition over the generalized eigenstates in the form

where $r_{\beta}^{(j)}(t)\equiv\langle\langle\mathcal{Q}_{\beta}^{(j)}(t;\lambda_{t})|\rho(t)\rangle\rangle$ is a complex valued expansion coefficient. Similarly, to Eq. (4), any density operator $|\rho\rangle\rangle$ can be expressed in this form since the generalized eigenbasis that generates Jordan canonical form is complete and orthonormal.

Substituting the trial form of Eq. (11) into the time convolutionless Liouville-von Neumann equation [Eq. (8)] and projecting onto the left generalized eigenstate $\langle\langle\mathcal{Q}_{\alpha}^{(i)}|$, gives the following equation of motion for the expansion coefficients:

where we have used the Floquet Equation (10) and the orthonormality condition of generalized eigenstates. By convention, we take $r_{\alpha}^{(n_{\alpha})}=0$.

Differentiating Eq. (10) for some right eigenstate $|\mathcal{P}_{\beta}^{(j)}\rangle\rangle$ with respect to $\lambda_{t}$ at fixed $t$ and projecting onto left eigenstate $\langle\langle\mathcal{Q}_{\alpha}^{(i)}|$ with $\alpha\neq\beta$ gives

This expression can be simplified by first noting that for $\alpha\neq\beta$ orthonormality of the basis eliminates the $\langle\langle\mathcal{Q}_{\alpha}^{(i)}|\mathcal{P}_{\beta}^{(j)}\rangle\rangle$ term on the right hand side of Eq. (13). Equation (9a) can then be used to expand the second term, $\langle\langle\mathcal{Q}_{\alpha}^{(i)}|\hat{\hat{\mathcal{L}}}({\partial}/{\partial\lambda_{t}})|\mathcal{P}_{\beta}^{(j)}\rangle\rangle$ yielding a recursive expression for the projected change of the normal modes.

where $\chi_{\beta\alpha}\equiv\chi_{\beta}-\chi_{\alpha}$. Iterating recursively through Eq. (14), the change in the normal modes can be related to the change in the modulated Liouvillian by

where $S_{q}\equiv\sum_{s=1}^{q}k_{s}$, and $k_{s}$ are the summation indexes over states in the Jordan Block.

Finally, by substituting Eq. (15) into Eq. (12) and considering only the terms that couple non-degenerate Jordan blocks $\alpha\neq\beta$, we obtain an adiabaticity condition for slowly modulated open quantum systems:

This expression is admittedly unwieldy but provides a completely general criteria for open system adiabaticity. A number of simpler but less tight bounds can be obtained for the adiabaticity condition by extending the approach discussed in Sarandy and Lidar (2005) that treated constant Liouvillian. The simplest of these results states that if $\dot{\lambda}_{t}\to 0$, e.g., in Eq. (12), then the open quantum system undergoes adiabatic dynamics. Moreover, similarly to the closed system adiabatic theorem, Eq. (16) can be used to derive the earlier non-modulated adiabatic theorem of Ref. Sarandy and Lidar (2005) by assuming a constant $\hat{\hat{\mathcal{L}}}_{0}(t)$.

A significant example of the open system AMT is provided in Sect. III.2 where the theorem is applied to the slowly turned-on incoherent (e.g., solar radiation) excitation of molecular systems.

A particularly important case in open system quantum mechanics involves the adiabatic turn-on of incoherent radiation that is incident on a molecule and the role of quantum coherences in biological processes (e.g photosynthesis and vision). In particular, oscillatory coherences have been observed experimentally in the excitation of biological molecules with pulsed lasers Engel et al. (2007); Zhang et al. (2015); Ishizaki et al. (2010); Johnson et al. (2015). By contrast, we have argued, supported by numerical studies, that natural processes rely upon slowly-turned-on incoherent light, which leads to steady-state transport Jiang and Brumer (1991); Brumer (2018); Dodin and Brumer (2019) with no oscillating coherences. As shown below, the application of the open system adiabatic theorem proves that if a system is driven by very-slowly turned on light, then the only coherences that will be observed are those that survive to the steady state. In particular, the pulsed laser generated coherences noted above do not survive and are irrelevant under natural biological conditions.

Specifically, consider a molecular system initially prepared in the absence of a driving light field. In this case, the system is initially in a simple equilibrium steady state (i.e., with vanishing dynamical frequency). As such, before the radiation field is turned on, the system is found in a Jordan block with zero eigenvalue. At $t=0$ an incoherent light field is turned on on a time scale much slower than the dynamics of the molecule, a typical circumstance in light induced biophysical processes. This corresponds to a modulation in the limit of $\dot{\lambda}_{t}\to 0$, ensuring adiabaticity [see Eq. (16] of the underlying dynamics. As a result, at all times, the system is found in a Jordan block with vanishing eigenvalue, that is, in a steady state. This then indicates that the only coherences observed in the slow turn-on limit are those that survive in the steady-state, such as those seen in previous theoretical studies Koyu et al. (2020); Dodin and Brumer (2019); Reppert et al. (2019); Axelrod and Brumer (2018, 2019).

In many cases, consistency with thermodynamics requires the system to have one steady state given by the Gibbs state $\rho\propto\exp(-\hat{H}/k_{B}T)$ which shows no coherences between non-degenerate energy eigenstates. This result suggests that the non-steady state coherent dynamics observed under finite turn-on times are a consequence of the rapid turn on of the incoherent light field. Figure 2 discussed below provides a numerical example of these predictions for a popular generic V-system model Dodin and Brumer (2019); Dodin et al. (2016).

The ability to show that no coherent dynamics survive the slow turn-on limit of incoherent light without requiring any calculation highlights the intuitive power of the adiabatic theorems derived in this letter. Notably, this situation is far beyond the bounds of applicability of previous adiabaticity condition as the system dynamics are driven by a noisy incoherent light field. This produces a driving Hamiltonian that fluctuates extremely rapidly, on the order of 1 fs, much faster than the energy gap between states and consequently the underlying molecular dynamics. A numerical example follows below.

Consider then the basic minimal model: an open three level system under irradiation by slowly turned-on incoherent light. This model has been previously treated where numerical results showed Dodin et al. (2016) that coherent dynamics, i.e. coherences in the excited state, vanished under slow turn-on of the exciting field. The system has a ground energy eigenstate, $\ket{g}$, and two excited states, $\ket{e_{1}}$ and $\ket{e_{2}}$ that are separated by an energy $\Delta$ and is excited by an incoherent light source with time-dependent intensity. The dynamics of this system, in the weak coupling limit, is characterized by the following Partial Secular Bloch-Redfield Master equations for the matrix elements of the density operator $\hat{\rho}$:

where $\gamma_{i}$ is the spontaneous emission rate and $r_{i}(t)$ is the rate of excitation to (and stimulated emission from) state $\ket{e_{i}}$. The alignment parameter $p$ characterizes the probability of simultaneous excitation to states $\ket{e_{1}}$ and $\ket{e_{2}}$ and is defined as

where $\bm{\mu_{i}}$ is the transition dipole moment between the ground state $\ket{g}$ and excited state $\ket{e_{i}}$, assumed real for simplicity. The spontaneous emission and excitation rates are related by detailed balance to give $r_{i}(t)=\gamma_{i}\overline{n}(t)$ where $\overline{n}(t)$ is the mean occupation number of the resonant thermal field mode. It is the slow turn-on of the incoherent light that leads to a time dependent occupation number $\overline{n}(t)$ of the thermal field reflecting it’s time dependent intensity. That is, the slow modulation of the Liouvillian arises due to slow changes in the statistics of the exciting field, in this case in the mean number of photons in the thermal field modes.

These equations of motion can be analytically solved giving several distinct dynamical regimes. In particular, in the limit where $\overline{\gamma}/\Delta_{P}\gg 1$, with $\overline{\gamma}=(\gamma_{1}+\gamma_{2})/2$ and $\Delta_{p}=\sqrt{\Delta^{2}+(1-p^{2})\gamma_{1}\gamma_{2}}$, a regime examined below, the system shows long-lived quasi-stationary coherences that eventually decay to give an incoherent thermal state Tscherbul and Brumer (2014). Under excitation by a field with time-dependent intensity of the form $\overline{n}(t)=\overline{n}(1-\exp(-\alpha t))$, the dynamics are given by

with $\overline{\gamma}$ and $\Delta_{p}$ defined above. A detailed derivation of Eq. (27) along with an in depth discussion of the underlying physics are given in Ref. Dodin et al. (2016).

While Eq. (25) is complicated, the dynamics are qualitatively simple. When the turn on time $\tau_{r}=\alpha^{-1}$ is shorter than the coherence lifetime $2\overline{\gamma}/\Delta_{p}^{2}$, the quasi-stationary coherences lead to a potentially long-lived superposition of excited states which rises on a timescale $\tau_{r}$. The coherences then decay on a timescale $2\overline{\gamma}/\Delta_{p}^{2}$. As the turn on time becomes comparable to or longer than the coherence lifetime, the quasi-stationary superposition is not fully excited, leading to a decrease in the generated coherences that disappear entirely in the $\tau_{r}\to\infty$ limit. These dynamics are shown for a variety of turn on rates, $\alpha$, in the solid traces of Fig. 2 for the case of $p=1$, $\gamma_{1}=\gamma_{2}$.

These traces can be compared to the dashed red trace which shows the predictions of the open system adiabatic theorem. As discussed above, since the system is initialized in a steady state of the zero field case, a sufficiently slow turn on leads to dynamics that remain in the instantaneous steady state at $\overline{n}(t)$. In this example involving only a single bath, the steady state is the thermal Gibbs state with a temperature determined by $\overline{n}(t)$, and therefore shows no coherences. Consequently, as illustrated in this example, the open systems adiabatic theorem guarantees that a sufficiently slow turn-on of an incoherent field will only show coherences that survive in the steady-state. In the case of two baths, steady state coherences associated with transport will persist Dodin and Brumer (2019); Koyu et al. (2020).