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

We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light induced processes. The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time dependent reference Hamiltonians. In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered. Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates. More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically. As examples, we apply the generalized theorem to (a) predict the dynamics of a two level system driven by a frequency modulated resonant oscillation, a pathological situation beyond the applicability of earlier results, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can only display coherences that survive in the steady state.


I. INTRODUCTION
Quantum systems driven by slowly varying external fields play an important role in atomic, molecular and optical physics. The ubiquity of these processes has led to sustained interest in powerful adiabatic theorems (AT's) that characterize their dynamics [1][2][3][4]. The best known AT states that a system initialized in an energy eigenstate will remain in that state when driven by a slowly varying field. Adiabatic processes have found far reaching utility in quantum dynamics [5,6] and in the development of Adiabatic Quantum Computing (AQC) where they are used to reliably realize quantum state transformations [7,8]. Moreover, adiabaticity conditions bound the allowable speed of state transformations, setting the time cost of AQC algorithms and spurring efforts to find shortcuts to adiabaticity [9][10][11][12].
However, ATs fail for oscillating fields [13][14][15][16], since these fields can induce transitions between energy eigenstates, precisely the opposite of what is required in traditional adiabatic theorems. Hence traditional ATs can not, in general, be applied to light-induced processes in either isolated or open quantum systems.
Adiabatic dynamics have been successfully deployed in experimental settings through Adiabatic Rapid Passage (ARP) [17] and Stimulated Raman Adiabatic Passage (STIRAP) [18,19] techniques. In these experiments, the adiabatic theorem provides a robust mechanism for high fidelity state preparation that is not sensitive to slight perturbations in the system properties and has been used, for example, to study molecular reaction dynamics [20] and in the preparation of ultra-cold molecules [21,22]. However, the domain of these techniques has been limited by the applicability of the AT, limiting the forms of driving fields and transformation times that can be realized. By relaxing key restrictions imposed by the traditional AT, the AMT introduced below significantly broadens the range of experimental techniques, unlocking the potential for faster, more flexible state preparation and system dynamics. Thus, challenging questions related to the creation of molecular coherences in such natural processes requires an AT adapted to oscillating incoherent fields in open quantum systems [23][24][25][26].
In this paper we derive generalized AT's, termed Adiabatic Modulation Theorems (AMTs), for open and closed quantum systems that apply to oscillatory and other rapidly varying fields, extending adiabaticity conditions to optically driven processes and providing faster pathways to adiabaticity, with the potential to accelerate AQC. Moreover, we show that the adiabatic transformation time is limited not by the energy gaps in the systems but rather by the frequency differences of the instantaneous normal modes, that are often easier to manipulate. These results exploit the fact that many complicated fields of interest are time-dependent modulations (e.g., frequency and amplitude modulated oscillations) of simpler fields that have well understood dynamics.
The AMT's derived below show that a system subjected to a sufficiently slow modulation of a time-dependent field conserves the dynamical modes of time-dependent reference Hamiltonians. These theorems contain traditional ATs in the limit of modulations of static fields, and generalize easily to modulations of periodic fields, which are shown to preserve Floquet states rather than energy eigenstates. Moreover, we show that the adiabatic transformation time is limited not by the energy gaps in the system but rather by the frequency differences of the instantaneous normal modes, that are often easier to manipulate. Remarkably, these constructions allow for the design of experimental techniques that go beyond the preparation of time-independent states and allow for the preparation of specific dynamics using the well developed intuition of adiabatic theory.
The AMT theorems are widely applicable and two significant examples are provided below. In the isolated system example, we show how the theorem allows for control of dynamics on the entire Bloch sphere in a two level qubit system. In applications to open systems, the theorem resolves a longstanding controversy regarding the role of light-induced coherent oscillations in biophysical processes. The generality of the theorems ensure applications to a wide variety of alternative systems. This paper is organized as follows: Sect. II proves the adiabatic modulation theorems for both isolated and open systems. Sample applications, designed to demonstrate the theorems and their applications are provided in Sect. III.
These normal modes and corresponding quasienergies are found by solving the eigenprob- whereĤ F (t; λ) ≡Ĥ 0 (t; λ) − i∂/∂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,Ĥ 0 (t; λ), is a transformation that varies λ → λ t over a time interval [0, τ ] through a modulation protocol Λ ≡ {λ t |t ∈ [0, τ ]}. The resultant modulated fieldˆ H Λ (t) ≡Ĥ 0 (t; λ t ) sweeps through the Hamiltonian family defined above (e.g. a field with a time-dependent amplitude forĤ 0 = λh 0 (t)). Note, as an aside, that this is an operator generalization of modulations in signal processing and encoding theory [27]. 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 byˆ H Λ (t) can now be expressed in terms of instantaneous normal modes ofĤ 0 , Eq. (1), to give where θ n (t; Λ) ≡ t 0 dsǫ n (s; λ s )/ . The instantaneous state of the system can always be expressed in this form sinceĤ F (t; λ t ) is always Hermitian and therefore has a basis of eigenstates {|n(t; Λ t ) } that can be used to expand |Ψ(t) with coefficient a n (t) ≡ n(t; λ t )|Ψ(t) = c n (t) exp(−iθ n (t)) [28].
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 H 0 (0; λ 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 dθ n (t; Λ)/dt = ǫ n (t; λ t ) and Eq. (2) to cancel terms proportional to ǫ n (t; λ t ). Projecting onto an |m(t; λ t ) , yields equations of motion for the coefficients,ċ where we have suppressed the t and λ t dependence for brevity, and θ nm ≡ θ n − θ m . To obtain Eq. (6), Eq. (2) was differentiated with respect to λ t to obtain m|∂/∂λ t |n = m|∂ˆ H/∂λ t |n /(ǫ n − ǫ m ).
Equation (6) takes the same form as the original adiabatic theorem with cross-coupling between normal modes scaled by a rapidly oscillating term [3]. As a result normal modes evolve independently of one another when the following condition is satisfied: where ǫ nm (t; λ t ) ≡ ǫ n (t; λ t ) − ǫ m (t; λ t ). Consequently, if λ 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, ifĤ 0 (t; λ) 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 λ. 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, ifĤ 0 (t; λ) 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 A 1.

B. Open Systems
Consider now an adiabatic theorem for open quantum systems. To prove this theorem we take an approach inspired by Ref. [3]. The dynamics of an open quantum system are governed by the Liouville von-Neumann (LvN) equation in the time-convolutionless form [29][30][31]. The double-ket notation |· is indicates an operator Liouville space with a trace inner product A|B ≡ Tr{Â †B }. By analogy with the isolated system we define modulations by starting with a family of time-dependent Liouvillianŝ L 0 (t; λ) and allowing the modulation parameter to vary over time to giveL Λ (t) ≡L 0 (t; λ 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 [30] and therefore require more care in defining their normal modes.
A given M × M modified Liouvillian,L F , has an incomplete set of N ≤ M left and right eigenvectors Q α (t)| and |P α (t) . Associated with each eigenvector is a Jordan Block comprised of n α generalized eigenvectors that combine to give an orthonormal basis of Liouville space defined by the generalized eigenproblem in the context of quantum adiabatic theorems, see ref. [3].) 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 ofL † F . 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 (j) where we have used the Floquet Equation (10) and the orthonormality condition of generalized eigenstates. By convention, we take r Differentiating Eq. (10) for some right eigenstate |P (j) β with respect to λ t at fixed t and projecting onto left eigenstate Q This expression can be simplified by first noting that for α = β orthonormality of the basis term on the right hand side of Eq. (13). Equation (9a) can then be used to expand the second term, Q yielding a recursive expression for the projected change of the normal modes.
where χ βα ≡ χ β − χ α . 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 ≡ q s=1 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 α = β, 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 [3] that treated constant Liouvillian. The simplest of these results states that ifλ t → 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. [3] by assuming a constantL 0 (t).
A significant example of the open system AMT is provided in Sect. III B where the theorem is applied to the slowly turned-on incoherent (e.g., solar radiation) excitation of molecular systems.

III. COMPUTATIONAL EXAMPLES
A. Isolated Systems

Rabi type system
In this section, we consider a two-level system (TLS) driven by a frequency modulated oscillatory field. We begin by defining an extension of the Rabi model, the family of Hamil- that characterize driving by a field with frequency ω. Here ∆ 0 is the energy difference between the states |0 and |1 , the coupling coefficient is given by These states are eigenstates of the Floquet Hamiltonian, satisfyinĝ . For the Hamiltonian in Eq. (17), these Floquet states and quasi energies are given by where ∆ ≡ ∆ 0 − ω/2 is the detuning, φ = arg V is the coupling phase, θ ≡ arccos(∆/Ω) = arcsin(|V |/Ω) is the mixing angle and Ω is the generalized Rabi frequency.
Consider then driving this system by a frequency modulated field with time varying ω → ω t . The resulting Hamiltonian can be expressed as a modulation of the form discussed above withˆ H(t) =Ĥ 0 (t; ω t ), that sweeps through Hamiltonians in the family described by Eq. (17). In general, this problem does not admit a closed form solution, but is tractable through direct numerical propagation of the TDSE. However, in the limit where the modulation changes sufficiently slowly the dynamics can be solved using the generalized adiabatic theorem for isolated systems [Eq. (7)].
To define the domain in which this theorem applies consider the modulation derivative, which characterizes the effect of the modulation on the driving field. Given Eq.
required for the generalized adiabatic theorem.
Substituting Eqs. (21) and (20) into Eq. (7) shows that the generalized adiabatic theorem applies when where we have indicated the parameters that change over time upon frequency modulation using the subscript t. Here |Ω t | = ∆ 2 t + |V | 2 . When Eq. (22) is well satisfied, then a system initialized in one of the generalized adiabats |± will remain in that state at all times. To demonstrate this, we consider a simple linear modulation ω t = ω 0 +ωt, whereω is the constant frequency sweep velocity, and where the field, initially in resonance ∆ t=0 = 0, is swept linearly to ∆ t=τ = 4∆ 0 . In Fig. 1 allows for manipulation of the adiabatic time scales simply by increasing the intensity of the driving field V , and hence |Ω t |, and allows for adiabatic transformations of near degenerate systems on manageable timescales. Moreover, the normal modes, given by the Floquet states |± , follow the dynamics of the system under fixed frequency driving. By following these oscillatory dynamics the oscillations of the driving field that typically lead to the failure of traditional AT's are removed from consideration [13][14][15][16]. In this particularly simple case where only one frequency of light drives the system, this is equivalent to moving to a rotating reference frame using the interaction picture. The instantaneous normal modes can therefore be thought of as a more general method for following the dynamics of the system when an interacting reference frame cannot be defined.

Generalizing Adiabatic Experiments
As an indication of the additional role of the AMT in isolated systems, note that it allows a wide range of experimental applications beyond that of the traditional AT. Currently, (For example, a recently derived adiabaticity condition [32] allowed for the use of specifically designed pulse sequences to realize adiabatic transformations in finite time. [33]) Moreover, the AMT allows for the adiabatic treatment of new phenomena that lie far outside of the rotating wave regime, such as Sisyphus cooling, and can address experimental challenges that are intractable using simpler driving protocols, such implementing Stimulated Hyper-Raman Adiabatic Passage in the presence of Autler-Townes shifts from spectator states [34,35].
To appreciate the advantages afforded by the AMT, consider issues in the application of the traditional AT to oscillatory Hamiltonians. This has been carried out in the Interaction Picture by moving to a rotating reference frame where the oscillatory Hamiltonian timedependence can be removed. If this can not be done, then the Marzlin-Sanders inconsistency [16] where E i (t) are the time-dependent diagonal energies and V ij (t) the off-diagonal couplings.
An interaction picture, such as the commonly used rotating reference frame, is defined by a choice of a reference Hamiltonian H 0 . For simplicity, we can work in the eigenbasis of the reference Hamiltonian to give the representation H 0 = i E (0) i |i i|. Equation (23) can then be rewritten in the interaction picture to give ij / . This condition that prescribes a specific functional form for the timedependence of V ij is very restrictive and suggests that the strategy of removing oscillatory Hamiltonian time-dependence will only be effective in specific situations. However, this approach has been successful since the oscillatory functional form V ij ∝ e −iω ij t is precisely what is seen when modelling optically driven transitions in the rotating wave approximation. This condition, however, does limit the application of the traditional adiabatic theorem to specific types of optical excitation. First, the excitation must be well modelled using the rotating wave approximation. If this is not the case, then the off-diagonal matrix elements would be real valued (e.g. of the form V ij ∝ sin(ω ij t)) and the interaction picture transformation would leave a residual oscillatory component (e.g. of the form e 2iω ij t ). Moreover, it precludes the driving of one transition by more than one frequency of light, preventing the use of multi-color excitation. Finally, for a discrete N dimensional system, the N(N − 1) off diagonal elements impose up to N(N − 1) conditions ω ij = E (0) ij conditions on the N diagonal elements of H 0 . Beyond two-dimensional spin systems, it is not guaranteed that it will be possible to simultaneously satisfy all of these conditions. As a result, removal of oscillatory time-dependence can only be guaranteed when a total of N or fewer transitions are driven by an oscillatory external field unless additional resonance conditions are met.
The AMT shares none of these limitations.

B. Open System Adiabatic Turn-on of Incoherent Light
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 [36][37][38][39]. 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 [23,40,41] 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 ofλ t → 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 [23,[42][43][44][45].
In many cases, consistency with thermodynamics requires the system to have one steady state given by the Gibbs state ρ ∝ exp(−Ĥ/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 [23,24].
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 [24] 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, |g , and two excited states, |e 1 and |e 2 that are separated by an energy ∆ 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 operatorρ: ρ gg = −(r 1 (t) + r 2 (t))ρ gg + (r 1 (t) + γ 1 )ρ e 1 e 1 + (r 2 (t) + γ 2 )ρ e 2 e 2 + 2p( r 1 (t)r 2 (t) + √ γ 1 γ 2 )ρ R e 1 e 2 (25a) ρ e 1 e 2 = − 1 2 (r 1 (t)+r 2 (t)+γ 1 +γ 2 +2i∆)ρ e 1 e 2 + p 2 r 1 (t)r 2 (t)(2ρ gg −ρ e 1 e 1 −ρ e 2 e 2 )− p 2 √ γ 1 γ 2 (ρ e 1 e 1 +ρ e 2 e 2 ) where γ i is the spontaneous emission rate and r i (t) is the rate of excitation to (and stimulated emission from) state |e i . The alignment parameter p characterizes the probability of simultaneous excitation to states |e 1 and |e 2 and is defined as where µ i is the transition dipole moment between the ground state |g and excited state |e i , assumed real for simplicity. The spontaneous emission and excitation rates are related by detailed balance to give r i (t) = γ i n(t) where 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 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 γ/∆ P ≫ 1, with γ = (γ 1 + γ 2 )/2 and ∆ p = ∆ 2 + (1 − p 2 )γ 1 γ 2 , a regime examined below, the system shows long-lived quasi-stationary coherences that eventually decay to give an incoherent thermal state [46]. Under excitation by a field with time-dependent intensity of the form n(t) = n(1 − exp(−αt)), the dynamics are given by with γ and ∆ p defined above. A detailed derivation of Eq. (27) along with an in depth discussion of the underlying physics are given in Ref. [24].
While Eq. (25) is complicated, the dynamics are qualitatively simple. When the turn on time τ r = α −1 is shorter than the coherence lifetime 2γ/∆ 2 p , the quasi-stationary coherences lead to a potentially long-lived superposition of excited states which rises on a timescale τ r .
The coherences then decay on a timescale 2γ/∆ 2 p . 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 τ r → ∞ limit. These dynamics are shown for a variety of turn on rates, α, in the solid traces of Fig. 2 for the case of p = 1, γ 1 = γ 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 n(t). In this example involving only a single bath, the steady state is the thermal Gibbs state with a temperature determined by 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 [23,42].

IV. SUMMARY
We have presented a new set of generalized adiabatic theorems that apply to rapidly fluctuating fields with slowly varying properties. These results express the dynamics of the modulated field in terms of the dynamical normal modes of time dependent reference Hamiltonians. The resulting adiabaticity conditions take a form that is conceptually similar to preexisting theorems but bounds adiabaticity based on the rate of a slowly changing transformation rather than the rapid underlying field. These results significantly extend the applicability of the adiabatic theorem and its underlying intuition to, e.g., the domain of optical excitations that play a central role in atomic, molecular and optical physics.