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

PRX QUANTUM 2, 030302 (2021)

Amro Dodin 1,* and Paul Brumer 2,† 1 Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2 Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 3H6, Canada

(Received 9 March 2021; revised 25 May 2021; accepted 4 June 2021; published 2 July 2021)

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 traditional adiabatic theorems, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can display only coherences that survive in the steady state.

DOI: 10.1103/PRXQuantum.2.030302

I. INTRODUCTION eigenstates, precisely the opposite of what is required in traditional adiabatic theorems. Hence traditional ATs can Quantum systems driven by slowly varying external not, in general, be applied to light-induced processes in fields play an important role in atomic, molecular, and either isolated or open quantum systems. optical physics. The ubiquity of these processes has led Adiabatic dynamics have been successfully deployed to sustained interest in powerful adiabatic theorems (ATs) in experimental settings through adiabatic rapid passage that characterize their dynamics [1–4]. The best-known (ARP) [18] and stimulated Raman adiabatic passage (STI- AT states that a system initialized in an energy eigenstate RAP) [19,20] techniques. In these experiments, the adi- will remain in that state when driven by a slowly varying abatic theorem provides a robust mechanism for high- field. Adiabatic processes have found far reaching utility fidelity state preparation that is not sensitive to slight in quantum dynamics [5,6] and in the development of adi- perturbations in the system properties and has been used, abatic quantum computing (AQC), where they are used to for example, to study molecular reaction dynamics [21] reliably realize quantum state transformations [7,8]. More- and to prepare ultracold molecules [22,23]. However, the over, adiabaticity conditions bound the allowable speed of domain of these techniques has been limited by the appli- state transformations, setting the time cost of AQC algo- cability of the AT, limiting the forms of driving fields and rithms and spurring efforts to find shortcuts to adiabaticity transformation times that can be realized. By relaxing key [9–12]. However, ATs fail for oscillating fields [13–17], restrictions imposed by the traditional AT, the adiabatic since these fields can induce transitions between energy modulation theorem (AMT) introduced below significantly broadens the range of experimental techniques, unlocking *[email protected] the potential for faster, more flexible state preparation and †[email protected] system dynamics. Numerous scenarios do not admit traditional adiabatic Published by the American Physical Society under the terms of approaches. Natural light-induced excitation of biological the Creative Commons Attribution 4.0 International license. Fur- molecules (e.g., in photosynthesis or vision) provides a ther distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and particularly important example of a challenge to standard DOI. ATs insofar as it involves incoherent excitation of open

2691-3399/21/2(3)/030302(12) 030302-1 Published by the American Physical Society AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021) quantum systems with turn-on times that are exceedingly II. THE ADIABATIC MODULATION THEOREMS long compared to the time scale of molecular dynamics. A. Isolated systems For example, human blinking, the turn-on time of light in vision, occurs over 0.2 s as compared to the timescale Consider a family of time-dependent reference Hamil- ˆ ( λ) λ λ of molecular isomerization induced by incident light in tonians H0 t; indexed by the parameter [e.g., may ˆ ˆ ˆ vision, which can, under pulsed laser excitation, occur in be an amplitude H0(t; λ) = λh0(t) or frequency H0(t; λ) = ˆ 50 fs or less. Thus, challenging questions related to the cre- exp(−iλt)h0 + h.c.]. For each value of λ, setting = 1, we ation of molecular coherences in such natural processes define the instantaneous normal modes as requires an AT adapted to oscillating incoherent fields in −iθn(t;λ) open quantum systems [24–27]. |ψn(t; λ)≡e |n(t; λ),(1) In this paper, we derive generalized ATs, termed adia- θ ( λ) ≡ t ( λ) batic modulation theorems, for open and closed quantum where n t; 0 ds n s; is the dynamical phase, systems that apply to oscillatory and other rapidly varying n(t; λ) is an instantaneous quasienergy and the |n(t; λ) ˆ fields, extending adiabaticity conditions to optically driven is the instantaneous normal mode of H0(t; λ). processes and providing faster pathways to adiabaticity, These normal modes and corresponding quasienergies with the potential to accelerate AQC. Moreover, we show are found by solving the eigenproblem that the adiabatic transformation time is limited not by the ˆ energy gaps in the systems but rather by the frequency HF (t; λ)|n(t; λ)=n(t; λ)|n(t; λ),(2) differences of the instantaneous normal modes, which are ˆ ˆ often easier to manipulate. These results exploit the fact where HF (t; λ) ≡ H0(t; λ) − i∂/∂t. That is, the normal that many complicated fields of interest are time-dependent modes are particular solutions to the time-dependent modulations (e.g., frequency- and amplitude-modulated Schrödinger equation (TDSE), oscillations) of simpler fields that have well-understood ∂ dynamics. i |(t)=Hˆ (t)|(t),(3) The AMTs derived below show that a system subjected ∂t to a sufficiently slow modulation of a time-dependent field and are well understood for many driving fields. For exam- conserves the dynamical modes of time-dependent ref- ple, for time-independent Hamiltonians, the normal modes erence Hamiltonians. These theorems contain traditional of static Hamiltonians are energy eigenstates, while for ATs in the limit of modulations of static fields, and gen- periodic Hamiltonians they are Floquet states. eralize easily to modulations of periodic fields, which A modulation of a time-dependent Hamiltonian, are shown to preserve Floquet states rather than energy Hˆ (t; λ), is a transformation that varies λ → λ over a time eigenstates. Moreover, we show that the adiabatic transfor- 0 t interval [0, τ] through a modulation protocol ≡{λ |t ∈ mation time is limited not by the energy gaps in the system t τ } ˆ ( ) ≡ ˆ ( λ ) but rather by the frequency differences of the instanta- [0, ] . The resultant modulated field H t H0 t; t neous normal modes, that are often easier to manipulate. sweeps through the Hamiltonian family defined above ˆ = Remarkably, these constructions allow for the design of [e.g., a field with a time-dependent amplitude for H0 λ ( ) experimental techniques that go beyond the preparation h0 t ]. Note, as an aside, that this is an operator gener- of time-independent states and allow for the preparation alization of modulations in signal processing and encoding of specific dynamics using the well-developed intuition of theory [28]. Generally, the modulated Hamiltonian can be adiabatic theory. significantly more complicated than H0 since the modula- The AMT theorems are widely applicable and two tion function may be highly nonlinear and the modulation significant examples are provided below. In the iso- parameter can vary nontrivially with time. ˆ lated system example, we show how the theorem allows The dynamics induced by H (t) can now be expressed ˆ for control of dynamics on the entire Bloch sphere in in terms of instantaneous normal modes of H0,Eq.(1),to a two-level qubit system. In applications to open sys- give tems, the theorem resolves a long-standing controversy −iθn(t; ) regarding the role of light-induced coherent oscillations |(t)≡ cn(t)e |n(t; λt),(4) in biophysical processes. The generality of the theo- n rems ensure applications to a wide variety of alternative θ ( ) ≡ t ( λ )/ systems. where n t; 0 ds n s; s . The instantaneous state This paper is organized as follows: Sec. II proves the of the system can always be expressed in this form ˆ adiabatic modulation theorems for both isolated and open since HF (t; λt) is always Hermitian and therefore has systems. Sample applications, designed to demonstrate the a basis of eigenstates {|n(t; t)} that can be used to theorems and their applications are provided in Sec. III. expand |(t) with coefficient an(t) ≡n(t; λt)|(t)= Section IV provides a conclusion. cn(t) exp[−iθn(t)][29].

030302-2 GENERALIZED ADIABATIC THEOREMS... PRX QUANTUM 2, 030302 (2021)

This approach mirrors that used to derive traditional AMT is that its speed limits are set by the differences in ATs, but replacing the eigenstates with normal modes and normal-mode frequencies, which are often easier to manip- energies with quasienergies. We therefore proceed through ulate than energy gaps between eigenstates, allowing for a similar path to derive adiabaticity conditions under which faster adiabatic transformations. A numerical example is ˆ a system initialized in a normal mode of H0(0; λ0) will discussed in Sec. III A1. remain in that normal mode at all times. Mathematically, ( ) this occurs when cn t are decoupled in the TDSE, Eq. (3). B. Open systems Substituting Eq. (4) into Eq. (3) we find Consider now an adiabatic theorem for open quan- ∂ tum systems. To prove this theorem we take an approach ˙ ( )| ( λ )+ ( )λ˙ | ( λ ) −iθn(t; ) = cn t n t; t cn t t ∂λ n t; t e 0, inspired by Ref. [3]. The dynamics of an open quan- n t tum system are governed by the generalized Liouville-von (5) Neumann (LvN) equation θ ( )/ = ( λ ) where we use the identity d n t; dt n t; t and d ˆ ( λ ) |ρ = L(t)|ρ,(8) Eq. (2) to cancel terms proportional to n t; t . Project- dt ing onto an |m(t; λt), yields equations of motion for the coefficients, in the time-convolutionless form [30–32]. The Liouville ˆ ∂ super operator L(t) in Eq. (8) describes both unitary ˙ ( ) =− ( )λ˙ | | dynamics and nonunitary relaxation processes such as cm t cm t t m ∂λ n t dephasing and dissipation. For example, in the Markovian ˆ λ˙ e−iθnm m|∂H/∂λ|n limit, Eq. (8) can be written in the Lindblad form [30], con- + ( ) t − / ˆ ρˆ cn t − ,(6)sisting of a unitary component i [H, ] and a sum of n=m m n nonunitary Lindblad dissipators. The double-ket notation |· indicates an operator Liouville space with a trace inner λ where we suppress the t and t dependence for brevity, product A|B ≡ Tr{Aˆ †Bˆ }. By analogy with the isolated θ ≡ θ − θ and nm n m. To obtain Eq. (6),Eq.(2) is dif- system, we define modulations by starting with a fam- ferentiated with respect to λt to obtain m|∂/∂λt|n= ˆ ˆ ily of time-dependent Liouvillians Lˆ (t; λ) and allow the m|∂H/∂λ |n/( − ). 0 t n m ˆ Equation (6) takes the same form as the original adia- modulation parameter to vary over time to give L (t) ≡ batic theorem with cross-coupling between normal modes ˆ L0(t; λt). A modified Liouvillian can then be defined by scaled by a rapidly oscillating term [3]. As a result nor- ˆ ˆ Lˆ ( λ) ≡ Lˆ( λ) − ∂/∂ mal modes evolve independently of one another when the F t; t; t. following condition is satisfied: The LvN equation appears similar to the TDSE [Eq. (3)], suggesting that we may be able to apply the same analysis, ˙ ˆ but replacing Hamiltonians with Liouvillians. However, λtm|∂H(t; λt)/∂λt|n max min |nm(t; λt)| ,(7) the Liouvillian and its corresponding modified operator are ≤ ≤τ ( λ ) ≤ ≤τ 0 t nm t; t 0 t completely positive but not necessarily Hermitian [31]and therefore require more care in defining their normal modes. ( λ ) ≡ ( λ ) − ( λ ) λ ˆ where nm t; t n t; t m t; t . Consequently, if t × Lˆ changes much more slowly than the difference in dynami- A given M M modified Liouvillian, F , has an incom- plete set of N ≤ M left and right eigenvectors Qα(t)| cal mode frequencies, the slow modulation will not cross- |P ( ) couple the normal modes of the modulated Hamiltonian. and α t . Associated with each eigenvector is a Jor- Equation (7) has several important features. First, if dan block comprised of nα generalized eigenvectors that ˆ combine to give an orthonormal basis of Liouville space H0(t; λ) are t independent, then their normal modes are energy eigenstates and all Hamiltonian time dependence defined by the generalized eigenproblem is contained in the modulation. In this case, the AMT (j ) ˆ (j ) (j +1) reduces to the traditional AT, simply relabeling the time Qα |LF =Qα |χα +Qα |,(9a) variable as λ. The AMT then extends previous ATs by ˆ (j ) (j ) (j −1) recognizing them as statements about how time-dependent LF |Pα = χα|Pα + |Pα , (9b) transformations of Hamiltonian fields impact dynamical (nα−1) (nα) (0) modes of the TDSE. These coincide with eigenstates for where Qα |≡Qα|, Qα |≡0|, |Pα ≡ |Pα, (−1) static Hamiltonians but the same insight can be applied to and |Pα ≡ |0, where |0 is the zero operator. any field with well-understood dynamics. For example, if Dynamically, the LvN equation does not cross-couple ˆ H0(t; λ) are periodic, Eq. (7) states that slow modulations the Jordan blocks to one another, but can lead to cross- do not couple Floquet states. One key distinction of the coupling within states of the same Jordan block, thereby

030302-3 AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021) defining decoupled dynamical subspace. (For a concise gives the following equation of motion for the expansion summary of Jordan canonical form in the context of coefficients: quantum adiabatic theorems, see Ref. [3].) Motivated by the isolated systems’ derivation, we con- N nβ −1 ( ) ∂ ( ) sider the modified Liouvillian and aim to expand the ˙(i) = χ (i) + (i+1) − λ˙ j Q(i)| |P j rα αrα rα trβ α ∂λ β , dynamics of the open system in terms of its instantaneous β=1 j =0 t generalized eigenstates. These are given by the (right) (12) generalized equation where we use the Floquet Eq. (10) and the orthonormal- ˆ ˆ (j ) (j ) (j −1) ity condition of generalized eigenstates. By convention, we LF (t; λt)|Pα = χα|Pα + |Pα . (10) (nα) take rα = 0. (j ) The left generalized eigenstates can be similarly defined as Differentiating Eq. (10) for some right eigenstate |Pβ ˆ λ the right eigenstates of Lˆ† . with respect to t at fixed t and projecting onto left F Q(i)| α = β Allowing the modulation parameter to change with time, eigenstate α with gives the instantaneous state of the system can be expanded as a ˆ ∂Lˆ ∂ superposition over the generalized eigenstates in the form (i) (j ) (i) ˆ (j ) Qα | |Pβ + Qα |L |Pβ ∂λt ∂λt N nβ −1 ( ) ( ) ∂χ |ρ(t) ≡ r j (t)|P j (t; λ ), (11) = β Q(i)|P(j ) + χ Q(i)|P˙ (j ) β β t ∂λ α β β α β β=1 j =0 t (i) ˙ (j −1) +Qα |Pβ . (13) (j ) (j ) where rβ (t) ≡Qβ (t; λt)|ρ(t) is a complex-valued expansion coefficient. Similarly, to Eq. (4), any density This expression can be simplified by first noting that operator |ρ can be expressed in this form since the gen- for α = β orthonormality of the basis eliminates the (i) (j ) eralized eigenbasis that generates Jordan canonical form is Qα |Pβ term on the right-hand side of Eq. (13). complete and orthonormal. Equation (9a) can then be used to expand the second term, Substituting the trial form of Eq. (11) into the time- ˆ ( ) Q(i)|Lˆ(∂/∂λ )|P j yielding a recursive expression for convolutionless Liouville-von Neumann equation [Eq. (8)] α t β (i) the projected change of the normal modes. and projecting onto the left generalized eigenstate Qα |,

where χβα ≡ χβ − χα. Iterating recursively through Eq. slowly modulated open quantum systems: (14), the change in the normal modes can be related to the ( − ) nα−i j −S j −Sp (i+p−1) ˆ j Sp change in the modulated Liouvillian by 0 Qα |∂L/∂λ|P ··· β max + 0≤t≤T S p Sp ∂ = = = (−1) p χβα (i) (j ) p 1 k1 0 kp 0 Qα | |Pβ ∂λ |λ˙ −1| min t . (16) ( − ) 0≤t≤T nα−i j −S j −Sp (i+p−1) ˆ j Sp 0 Qα |∂L/∂λ|P = ·· · β + , (15) S p Sp This expression is admittedly unwieldy but provides a = = = (−1) p χ p 1 k1 0 kp 0 βα completely general criteria for open system adiabaticity. While the Jordan form treatment is required for open ≡ q where Sq s=1 ks,andks are the summation indexes systems dynamics in full generality, in many cases the over states in the Jordan block. Liouvillian can be diagonalized, i.e., making all Jordan Finally, by substituting Eq. (15) into Eq. (12) and con- blocks one dimensional. In this case, the recursive itera- sidering only the terms that couple nondegenerate Jordan tion in Eq. (14) and subsequent summations in Eqs. (15) blocks α = β, we obtain an adiabaticity condition for and (16) are not required and a treatment analogous to the

030302-4 GENERALIZED ADIABATIC THEOREMS... PRX QUANTUM 2, 030302 (2021) closed systems case can be used. The resulting expression For the Hamiltonian in Eq. (17), these Floquet states and then takes the same form as Eq. (7) with the slight mod- quasienergies are given by ification that the Liouvillian eigenvalues χα are generally θ i(ω/2)t complex. |+( ω)= sin 2 e t; θ −iφ −i(ω/2)t , (19a) A number of simpler bounds, albeit less tight than cos 2 e e Eq. (16), can be obtained for the adiabaticity condition by cos θ ei(ω/2)t extending the approach discussed in Ref. [3] that treated |−(t; ω)= θ 2 , (19b) sin e−iφe−i(ω/2)t constant Liouvillian. The simplest of these results states 2 λ˙ → that if t 0, e.g., in Eq. (12), then the open quantum ±=± ≡± 2 +|V|2, (19c) system undergoes adiabatic dynamics. Moreover, similarly to the closed system adiabatic theorem, Eq. (16) can be where ≡ 0 − ω/2 is the detuning, φ = arg V is the used to derive the earlier nonmodulated adiabatic theorem coupling phase, θ ≡ arccos(/) = arcsin(|V|/) is the ˆ of Ref. [3] by assuming a constant L0(t). mixing angle, and is the generalized Rabi frequency. A significant example of the open system AMT is pro- Consider then driving this system by a frequency- vided in Sec. III B where the theorem is applied to the modulated field with time varying ω → ωt. The resulting slowly turned-on incoherent (e.g., solar radiation) excita- Hamiltonian can be expressed as a modulation of the tion of molecular systems. ˆ ˆ form discussed above with H(t) = H0(t; ωt), that sweeps through Hamiltonians in the family described by Eq. (17). In general, this problem does not admit a closed-form solu- III. COMPUTATIONAL EXAMPLES tion, but is tractable through direct numerical propagation of the TDSE. However, in the limit where the modu- A. Isolated systems lation changes sufficiently slowly the dynamics can be 1. Rabi-type system solved using the generalized adiabatic theorem for isolated In this section, we consider a two-level system (TLS) systems [Eq. (7)]. driven by a frequency-modulated oscillatory field. We To define the domain in which this theorem applies begin by defining an extension of the Rabi model, the consider the modulation derivative, family of Hamiltonians, ∂ iωt ˆ 0 itVe H(t; ω) = ∗ − ω , (20) ∂ω −itV e i t 0 − iωt ˆ ( ω) = 0 Ve which characterizes the effect of the modulation on the H0 t; ∗ −iωt , (17) V e 0 driving field. Given Eq. (19), we have

∂ ˆ V |−(t; ω) H(t; ω)+(t; ω)|=i t, (21) that characterize driving by a field with frequency ω. Here ∂ω 0 is the energy difference between the states |0 and |1, = · µ required for the generalized adiabatic theorem. the coupling coefficient is given by V E 01, E is the µ Substituting Eqs. (21) and (20) into Eq. (7) shows that electric field vector driving the system, 01 is the transition dipole moment between the two states and we set = 1. the generalized adiabatic theorem applies when This family of Hamiltonians comprises the standard Rabi |˙ω || | model with well-understood dynamics for all values of ω. t V t max t min {2|t|}, (22) ≤ ≤τ 2 ≤ ≤τ It is useful to briefly review the dynamics induced by 0 t 2 t 0 t Eq. (17) from the perspective of Floquet’s theorem to highlight the normal modes that play an important role where we indicate the parameters that change over time ˆ upon frequency modulation using the subscript t. Here in the generalized adiabatic theorem. Since H0(t) is peri- | |= 2 +| |2 odic with period T = 2π/ω, its dynamical normal modes t t V . are two T-periodic Floquet states, |±(t; ω) with time- When Eq. (22) is well satisfied, then a system initialized |± independent quasienergies ±. These states are eigenstates in one of the generalized adiabats will remain in that of the Floquet Hamiltonian, satisfying state at all times. To demonstrate this, we consider a simple linear modulation ωt = ω0 +˙ωt, where ω˙ is the constant frequency-sweep velocity, and where the field, initially in ˆ resonance = = 0, is swept linearly to =τ = 4 .In H (t; ω)|±(t; ω)=±|±(t; ω), (18a) t 0 t 0 F Fig. 1, numerically exact dynamics are obtained for a range ∂ Hˆ (t; ω) ≡ Hˆ (t; ω) − i . (18b) of sweep velocities through Runge-Kutta integration of the F 0 ∂t TDSE. These dynamics are then compared to the adiabatic

030302-5 AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021)

2. 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, experimental adiabatic rapid passage and stimulated Raman adiabatic passage techniques have com- bined the rotating-wave approximation with a rotating reference frame in order to apply the traditional AT to optically driven systems. While this approach can be effective for treating systems driven by near-resonant monochro- matic lasers or transform-limited pulses, it imposes several restrictions on the types of experiments that can be realized. The AMT, however, removes many of these limitations, allowing for driving beyond the rotating-wave approximation, driving of a transition by multichromatic fields, and driving of more simultaneous transitions than is allowed under standard conditions. The removal of these limitations can have significant effects on possible experimental techniques. (For example, a recently derived adiabaticity condition [33] allowed for the use of specifically FIG. 1. Dynamics of a two-level system driven by a designed pulse sequences to realize adiabatic transforma- frequency-modulated field. The frequency of the driving field is tions in finite time [34].) Moreover, the AMT allows for the swept linearly from a resonant frequency ω0 = 0 to ωτ = adiabatic treatment of new phenomena that lie far outside 50 with sweep velocity ω˙ (i.e., ωt = 0 +˙ωt). This transfor- of the rotating-wave regime, such as Sisyphus cooling, and mation takes place over a time interval τ = 40/ω˙ . The yellow may address experimental challenges that are intractable to blue traces show the dynamics under a range of sweep veloc- using simpler driving protocols, such implementing stim- ities while the red dashed line shows the adiabatic prediction ulated hyper-Raman adiabatic passage in the presence of expected for infinitely slow-sweep velocities. Autler-Townes shifts from spectator states [35,36]. To appreciate the advantages afforded by the AMT, consider issues in the application of the traditional AT to theorem predictions, showing excellent agreement for slow oscillatory Hamiltonians. This has been carried out in the modulations. (We note that the time taken to perform this interaction picture by moving to a rotating reference frame modulation varies significantly for different sweep veloc- where the oscillatory Hamiltonian time dependence can be ities, and we plot the dynamics in Fig. 1 on a normalized removed. If this can not be done, then the Marzlin-Sanders /τ time axis t .) inconsistency [16] prevents application of the traditional This example is helpful in highlighting the construction adiabatic theorem. Hence, applications of the traditional and flexibility of the AMT. The adiabatic speed limit is AT are limited to systems where an appropriate choice of not set by the energy gap but rather by the Rabi frequency basis and reference Hamiltonian can be found to remove on the right-hand side of Eq. (22). This feature, typical all oscillatory time dependence. of the AMT, is remarkable since it allows for manipula- Consider then the limits of the transformations pos- tion of the adiabatic time scales simply by increasing the sible by an appropriate choice of reference Hamiltonian | | intensity of the driving field V, and hence t , and allows in the interaction picture. Let H(t) be an arbitrary time- for adiabatic transformations of near degenerate systems dependent Hamiltonian on manageable timescales. Moreover, the normal modes, given by the Floquet states |±, follow the dynamics of ( ) = ( )| |+ ( )| | the system under fixed frequency driving. By following H t Ei t i i Vij t i j , (23) these oscillatory dynamics the oscillations of the driving i ij field that typically lead to the failure of traditional ATs are removed from consideration [13–16]. In this particularly where Ei(t) are the time-dependent diagonal energies and simple case where only one frequency of light drives the Vij (t) the off-diagonal couplings. system, this is equivalent to moving to a rotating reference An interaction picture, such as the commonly used frame using the interaction picture. The instantaneous nor- rotating reference frame, is defined by a choice of a ref- mal modes can therefore be thought of as a more general erence Hamiltonian H0. For simplicity, we can work in the method for following the dynamics of the system when an eigenbasis of the reference Hamiltonian to give the rep- = (0)| | interacting reference frame cannot be defined. resentation H0 i Ei i i . Equation (23) can then be

030302-6 GENERALIZED ADIABATIC THEOREMS... PRX QUANTUM 2, 030302 (2021) rewritten in the interaction picture to give By contrast, we have argued, supported by numerical studies, that natural processes rely upon slowly turned i[H0(t)/]t −i[H0(t)/]t HI (t) = e [H(t) − H0]e on incoherent light, which leads to steady-state transport ( ) [24,41,42] with no oscillating coherences. These two sets ( ) − 0 / = ( ) − 0 | |+ ( ) i[Eij ]t| | [Ei t Ei ] i i Vij t e i j , of studies differ primarily in the nature of the optical fields’ i ij driving dynamics. In the laser experiments, the dynam- (24) ics are induced by sequences of intense femtosecond laser pulses that prepare specific excited state superpositions at (0) = (0) − (0) a specific time. In contrast, the steady-state studies treat where Eij Ei Ej . Considering the conditions under which oscillatory time exciting fields that are stochastic white noises (or equiv- dependence can be removed from Eq. (24) provides a alently broad thermalized photon baths at approximately set of conditions for when AT-based approaches can be 5800 K) with slowly increasing intensity that continuously applied. This requires a basis in which the diagonal ele- excite statistical mixtures of excited states with stochas- ments of H(t) have no oscillatory time dependence, defin- tic phase. The differences in the resulting excitations lead to different predictions of the role of coherences in nat- ing the eigenbasis of the reference Hamiltonian H0.The off-diagonal elements show that the only way to eliminate ural settings and even in the rate of these photoinduced iωij t processes in nature [42]. As shown below, the application oscillatory time dependence in Vij (t) is if Vij ∝ e for ( ) a frequency ω = E 0 /. This condition that prescribes a of the open-system adiabatic theorem proves that if a sys- ij ij tem is driven by very-slowly turned-on light, then the only specific functional form for the time dependence of Vij is very restrictive and suggests that the strategy of remov- coherences that will be observed are those that survive to ing oscillatory Hamiltonian time dependence will only be the steady state. In particular, the pulsed-laser-generated effective in specific situations. However, this approach has coherences noted above do not survive and are irrelevant under natural biological conditions. been successful since the oscillatory functional form Vij ∝ − ω Specifically, consider a molecular system initially pre- e i ij t is precisely what is seen when modeling optically driven transitions in the rotating-wave approximation. pared in the absence of a driving light field. In this case, This condition, however, does limit the application of the system is initially in a simple equilibrium steady state the traditional adiabatic theorem to specific types of optical (i.e., with vanishing dynamical frequency). As such, before the radiation field is turned on, the system is found in a excitation. First, the excitation must be well modeled using = the rotating-wave approximation. If this is not the case, Jordan block with zero eigenvalue. At t 0 an incoherent then the off-diagonal matrix elements would be real val- light field is turned on on a time scale much slower than ued [e.g., of the form V ∝ sin(ω t)] and the interaction the dynamics of the molecule, a typical circumstance in ij ij light-induced biophysical processes. This corresponds to picture transformation would leave a residual oscillatory ˙ ω a modulation in the limit of λ → 0, ensuring adiabaticity component (e.g., of the form e2i ij t). Moreover, it pre- t cludes the driving of one transition by more than one [see Eq. (16)] of the underlying dynamics. As a result, at all frequency of light, preventing the use of multicolor exci- times, the system is found in a Jordan block with vanish- tation. Finally, for a discrete N-dimensional system, the ing eigenvalue, that is, in a steady state. This then indicates N(N − 1) off-diagonal elements impose up to N(N − 1) that the only coherences observed in the slow turn-on limit ( ) are those that survive in the steady state, such as those seen conditions ω = E 0 conditions on the N diagonal ele- ij ij in previous theoretical studies [24,43–46]. Crucially, this ments of H . Beyond two-dimensional spin systems, it is 0 conclusion does not depend on any particular realization not guaranteed that it will be possible to simultaneously of the incoherent light or on any approximation scheme satisfy all of these conditions. As a result, removal of oscil- for the dynamics. latory time dependence can only be guaranteed when a In many cases, consistency with thermodynamics total of N or fewer transitions are driven by an oscilla- requires the system to have one steady state given by the tory external field unless additional resonance conditions Gibbs state ρ ∝ exp(−Hˆ /k T), which shows no coher- are met. The AMT shares none of these limitations. B ences between nondegenerate energy eigenstates. This result suggests that the nonsteady-state coherent dynam- B. Open-system adiabatic turn on of incoherent light ics observed under finite turn-on times are a consequence A particularly important case in open-system quantum of the rapid turn on of the incoherent light field. Figure 2 mechanics involves the adiabatic turn on of incoherent discussed below provides a numerical example of these radiation that is incident on a molecule and the role of predictions for a popular generic V-system model [24,25]. quantum coherences in biological processes (e.g., pho- The ability to show that no coherent dynamics survive tosynthesis and vision). In particular, oscillatory coher- the slow turn-on limit of incoherent light without requir- ences have been observed experimentally in the excita- ing any calculation highlights the intuitive power of the tion of biological molecules with pulsed lasers [37–40]. adiabatic theorems derived in this paper. Notably, this

030302-7 AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021)

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 ρˆ:

ρ˙gg =−[r1(t) + r2(t)]ρgg + [r1(t) + γ1]ρe e 1 1 + [r2(t) + γ2]ρe e + 2p[ r1(t)r2(t) √ 2 2 + γ γ ]ρR , (25a) 1 2 e1e2 ρ˙ = ( )ρ − ( ) + γ ρ eiei ri t gg [ri t i] eiei √ − p[ r (t)r (t) + γ γ ]ρR , (25b) 1 2 1 2 e1e2 1 ρ˙ =− [r (t) + r (t) + γ + γ + 2i]ρ e1e2 2 1 2 1 2 e1e2 p + r (t)r (t)(2ρ − ρ − ρ ) 2 1 2 gg e1e1 e2e2 p √ − γ γ (ρ + ρ ), (25c) 2 1 2 e1e1 e2e2 γ ( ) FIG. 2. Dynamics of a V system driven by slowly turned on where i is the spontaneous emission rate and ri t is the incoherent light. The intensity of the exciting field, i.e., the mean rate of excitation to (and stimulated emission from) state photon number, turns on with a rate α following an exponential |ei. The alignment parameter p characterizes the proba- ( ) ∝ − (−α ) activation function n t [1 exp t ]. We consider a system bility of simultaneous excitation to states |e1 and |e2 and driven by a blackbody source with temperature T = 5800 K with is defined as ω = ground to excited state excitation energy of ge 1.98 eV. Both µ · µ excited states have a natural line width of γ = 1 GHz with a p ≡ 1 2 , (26) splitting of = 0.001γ . The yellow to blue traces show the μ1μ2 dynamics under a range of turn-on rates while the dashed red where µ is the transition dipole moment between the trace shows the adiabatic turn-on prediction. i ground state |g and excited state |ei, assumed real for situation is far beyond the bounds of applicability of pre- simplicity. The spontaneous emission and excitation rates ( ) = γ ( ) vious adiabaticity condition as the system dynamics are are related by detailed balance to give ri t in t where ( ) driven by a noisy incoherent light field. This produces n t is the mean occupation number of the resonant ther- a driving Hamiltonian that fluctuates extremely rapidly, mal field mode. It is the slow turn on of the incoherent ( ) on the order of 1 fs, much faster than the energy gap light that leads to a time-dependent occupation number n t between states and consequently the underlying molecular of the thermal field reflecting its time-dependent intensity. dynamics. A numerical example follows below. That is, the slow modulation of the Liouvillian arises due Consider then the basic minimal model: an open three- to slow changes in the statistics of the exciting field, in this level system under irradiation by slowly turned-on inco- case in the mean number of photons in the thermal field herent light. This model has been previously treated where modes. numerical results showed [25] that coherent dynamics, i.e., In order to apply the open-system AMT, it is useful |ρ = coherences in the excited state, vanished under slow turn to write Eq. (25c) in matrix form. By writing [ρ , ρ , ρ , ρ , ρ ]T as a column vector of the on of the exciting field. The system has a ground-energy gg e1e1 e2e2 e1e2 e2e1 | | | density-matrix elements, Eq. (25c) can be written in the eigenstate, g , and two excited states, e1 and e2 that are ˆ separated by an energy and is excited by an incoherent LvN form of Eq. (8), where Lˆ(t) is a matrix given by

⎡ √ √ ⎤ −[r1(t) + r2(t)] r1(t) + γ1 r2(t) + γ2 2p[ r1(t)r2(t) + γ1γ2]2p[r12(t) + γ12] ⎢ p p ⎥ ⎢ r1(t) −[r1(t) + γ1]0 − [r12(t) + γ12] − [r12(t) + γ12]⎥ ˆ ⎢ 2 2 ⎥ Lˆ(t) = ⎢ r (t) 0 −[r (t) + γ ] − p [r (t) + γ ] − p [r (t) + γ ]⎥ , ⎢ 2 2 2 2 12 12 2 12 12 ⎥ ⎣ ( ) − p ( ) + γ − p ( ) + γ − − − γ ⎦ pr12 t 2 [r12 t 12] 2 [r12 t 12] i r 0 ( ) − p ( ) + γ − p ( ) + γ − − − γ pr12 t 2 [r12 t 12] 2 [r12 t 12] i r 0 (27)

030302-8 GENERALIZED ADIABATIC THEOREMS... PRX QUANTUM 2, 030302 (2021) √ √ where r12(t) = r1(t)r2(2) and γ12 = γ1γ2 are the geo- the limit of an infinitely slow turn-on time. This requires metric mean pumping and decay rates and r(t) = [r1(t) + only evaluating the steady state of the Liouvillian (i.e., the ( ) / γ = (γ + γ )/ r2 t ] 2and 1 2 2 are the corresponding arith- χα = 0) eigenspace at various fixed values of ri, since the metic means. Equation (27) is analogous to writing the system is initialized in the ground state—the steady state Hamiltonian as a matrix when expressing the TDSE in at the initial ri(0) = 0 field. matrix form with the exception that the Liouvillian is not These equations of motion can be analytically solved Hermitian. In this form, the open-system AMT is applied by diagonalizing the matrix in Eq. (27) giving several analogously to the isolated-system AMT as was done in distinct dynamical regimes [25]. In particular, in the 2 2 Sec. III A1, using the Liouvillian in place of the Hamilto- limit where γ/P 1, with p = + (1 − p )γ1γ2, nian. The only added complication that can arise is that the a regime examined below, the system shows long-lived operator may not be diagonalizable since it is not Hermi- quasistationary coherences that eventually decay to give tian, in which case it must be placed into Jordan normal an incoherent thermal state [48]. Under excitation by a form using standard linear algebra techniques [47]. In this field with time-dependent intensity of the form n(t) = section, we pursue an alternative route, since we are inter- n[1 − exp(−αt)], the dynamics are given by ested in the simpler question of estimating the dynamics in

2 2 1 rj p −α − p t ri −α − γ ρ (t) = 1 − e t − α 1 − e 2γ + 2γ 1 − e t − α 1 − e 2 t (28a) eiei γ (2 / γ)− α γ γ − α 2 p 2 2 2

√ 2 2 p r1r2 1 p −α − p t 1 −α − γ ρ (t) = 1 − e t − α 1 − e 2γ + 2γ 1 − e t − α 1 − e 2 t , e1e2 γ (2 / γ)− α γ γ − α 2 p 2 2 2 (28b)

with γ and p defined above. A detailed derivation of Eq. show coherences that survive in the steady state. In the (28) along with an in-depth discussion of the underlying case of two baths, steady-state coherences associated with physics are given in Ref. [25]. transport will persist [24,43]. While Eq. (25) is complicated, the dynamics are quali- −1 tatively simple. When the turn-on time τr = α is shorter γ/2 IV. CONCLUSION than the coherence lifetime 2 p , the quasistationary coherences lead to a potentially long-lived superposition We present a new set of generalized adiabatic theorems of excited states, which rises on a timescale τr. The coher- that apply to rapidly fluctuating fields with slowly vary- γ/2 ences then decay on a timescale 2 p . As the turn-on ing properties. These results express the dynamics of the time becomes comparable to or longer than the coher- modulated field in terms of the dynamical normal modes of ence lifetime, the quasistationary superposition is not fully time-dependent reference Hamiltonians. The resulting adi- excited, leading to a decrease in the generated coherences abaticity conditions take a form that is conceptually similar that disappear entirely in the τr →∞limit. These dynam- to pre-existing theorems but bounds adiabaticity based on ics are shown for a variety of turn-on rates, α, in the solid the rate of a slowly changing transformation rather than the traces of Fig. 2 for the case of p = 1, γ1 = γ2. rapid underlying field. These results significantly extend These traces can be compared to the dashed red trace, the applicability of the adiabatic theorem and its under- which shows the predictions of the open-system adiabatic lying intuition to, e.g., the domain of optical excitations theorem. As discussed above, since the system is initial- that play a central role in atomic, molecular, and optical ized in a steady state of the zero-field case, a sufficiently physics. slow turn on leads to dynamics that remain in the instanta- The AMTs presented in this work provide fundamen- neous steady state at n(t). In this example involving only tal insights into quantum dynamics and can be applied to a single bath, the steady state is the thermal Gibbs state develop new techniques in quantum control, sensing, and with a temperature determined by n(t), and therefore shows information processing. no coherences. Consequently, as illustrated in this exam- Fundamentally, these theorems show that the notions of ple, the open-system adiabatic theorem guarantees that a adiabaticity first formulated for eigenstates of slowly vary- sufficiently slow turn on of an incoherent field will only ing Hamiltonians can be extended to fields that oscillate

030302-9 AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021) on arbitrary timescales similar to or even faster than the ACKNOWLEDGMENTS system dynamics. As illustrated by the example of incoher- This work is supported by the US Air Force Office of ent excitation of biomolecular systems in Sec. III B, these Scientific Research Under Contracts No. FA9550-17-1- concepts can significantly simplify our understanding of 0310 and No. FA9550-20-1-0354. highly complex dynamical problems involving rapidly oscillating and stochastic fields by revealing that, in some cases, their behavior is primarily determined by a much simpler set of slowly varying properties. These general- [1] M. Born and V. Fock, Beweis des adiabatensatzes, Z. Phys. ized adiabaticity concepts can enable the use of much 51, 165 (1928). simpler models or numerical techniques to describe previ- [2] T. Kato, On the adiabatic theorem of quantum Mechanics, ously intractable or challenging problems in these regimes, J. Phys. Soc. Jpn. 5, 435 (1950). similarly to the application of the adiabatic theorem to jus- [3] M. S. Sarandy and D. A. Lidar, Adiabatic approximation in tify existing adiabatic quantum dynamics method. As a open quantum systems, Phys.Rev.A71, 012331 (2005). [4] A. Messiah, Quantum Mechanics: Two Volumes Bound as concrete example, we show that incoherent excitation by One (Dover Publications, Mineola, NY, 2017). adiabatically turned-on light can be understood by simply [5] L. D. Landau, Zur theorie der energieubertragung. ii. Phys. calculating the steady states at different intensities rather Z. Sowjetunion 2, 1 (1932). than undertaking the generally far more complicated task [6] C. Zener, Non-adiabatic crossing of energy levels, Proc. R. of full dynamic propagation. Soc. London A 137, 696 (1932). Practically, the AMTs significantly expand the design [7] T. Albash and D. A. Lidar, Adiabatic quantum computation, space of quantum adiabatic transformations such as ARP Rev. Mod. Phys. 90, 015002 (2018). and STIRAP by enabling the use of rapidly varying time- [8] R. Barends, et al., Digitized adiabatic quantum computing dependent fields, which can allow for the design of pro- with a superconducting circuit, Nature 534, 222 (2016). [9] E. Torrontegui, I. Lizuain, S. González-Resines, A. tocols that are faster and more robust to environmental Tobalina, A. Ruschhaupt, R. Kosloff, and J. G. Muga, noise and system heterogeneity (e.g., by exploiting com- Energy consumption for shortcuts to adiabaticity, Phys. binations of resonant fields to induce rapid dynamics). Rev. A 96, 022133 (2017). These methods have found wide-spread applicability in [10] A. del Campo, Shortcuts to Adiabaticity by Counterdiabatic a variety of fields due to their ability to robustly trans- Driving, Phys.Rev.Lett.111, 100502 (2013). form one quantum state into another, including in adiabatic [11] E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Mod- [7] and circuit-based quantum computing [49–52], gas- ugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. phase spectroscopy [19,20], single-atom cavity quantum Chen, and J. G. Muga, in Advances in Atomic, Molecular, electrodynamics [53–57] and the preparation of ultracold and Optical Physics, edited by E. Arimondo, P. R. Berman, and C. C. Lin (Academic Press, San Diego, California, molecules [58–61]. The ability to accelerate adiabatic pro- USA, 2013), Vol. 62, p. 117. tocols can present important new opportunities in these [12] A. D. Campo and M. G. Boshier, Shortcuts to adiabaticity fields as the success of adiabatic transformations in real- in a time-dependent box, Sci. Rep. 2, 648 (2012). istic settings is often limited by their ability to out-race [13] M. H. S. Amin, Consistency of the Adiabatic Theorem, external relaxation channels. Phys.Rev.Lett.102, 220401 (2009). In addition to accelerating adiabatic protocols in exist- [14] J. Du, L. Hu, Y. Wang, J. Wu, M. Zhao, and D. Suter, Exper- ing applications, the AMTs allow for a fundamentally imental Study of the Validity of Quantitative Conditions new type of adiabatic transformation. While previously in the Quantum Adiabatic Theorem, Phys. Rev. Lett. 101, reported ATs are restricted to transformations from one 060403 (2008). [15] D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Quan- static eigenstate to another, the AMTs allow for trans- titative Conditions Do Not Guarantee the Validity of the formations from one type of dynamics to another. For Adiabatic Approximation, Phys.Rev.Lett.95, 110407 example, given a three-level quantum system with states (2005). |0, |1,and|2, previous ATs could realize transforma- [16] K.-P. Marzlin and B. C. Sanders, Inconsistency in the tions from one of these states to another (e.g., from |0 to Application of the Adiabatic Theorem, Phys.Rev.Lett.93, |1). The AMTs also allow for transformations that start 160408 (2004). with a wave function that dynamically cycles between the [17] R. Dann and R. Kosloff, Inertial theorem: Overcoming states in one direction (e.g., |0→|1→|2) and ends the quantum adiabatic limit, Phys.Rev.Res.3, 013064 with a wave function that cycles in the opposite direction (2021). [18] V. S. Malinovsky and J. L. Krause, General theory of pop- (|0→|2→|1). This new ability to control dynamics ulation transfer by adiabatic rapid passage with intense, and not just state preparation may open new applications chirped laser pulses, Eur. Phys. J. D 14, 147 (2001). for adiabatic control protocols that would not be possible [19] N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. with previous results (e.g., in quantum thermodynamics Bergmann, Stimulated raman adiabatic passage in physics, where controlling the direction of flux around a cycle may chemistry, and beyond, Rev. Mod. Phys. 89, 015006 be useful). (2017).

030302-10 GENERALIZED ADIABATIC THEOREMS... PRX QUANTUM 2, 030302 (2021)

[20] K. Bergmann, N. V. Vitanov, and B. W. Shore, Perspec- [37] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Man- tive: Stimulated raman adiabatic passage: The status after cal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, 25 years, J. Chem. Phys. 142, 170901 (2015). Evidence for wavelike energy transfer through quantum [21] O. Kaufmann, A. Ekers, C. Gebauer-Rochholz, K. U. Met- coherence in photosynthetic systems, Nature (London) 446, tendorf, M. Keil, and K. Bergmann, Dissociative charge 782 (2007). transfer from highly excited Na Rydberg atoms to vibra- [38] Y. Zhang, S. Oh, F. H. Alharbi, G. S. Engel, and S. Kais, tionally excited Na2 molecules, Int. J. Mass Spectrom. Low Delocalized quantum states enhance photocell efficiency, Energy Electron-Mol. Interact. (Stamatovic Honor) 205, Phys. Chem. Chem. Phys. 17, 5743 (2015). 233 (2001). [39] A. Ishizaki, T. R. Calhoun, G. S. Schlau-Cohen, and G. R. [22] M. Külz, M. Keil, A. Kortyna, B. Schellhaaß, J. Hauck, K. Fleming, Quantum coherence and its interplay with protein Bergmann, W. Meyer, and D. Weyh, Dissociative attach- environments in photosynthetic electronic energy transfer, ment of low-energy electrons to state-selected diatomic Phys. Chem. Chem. Phys. 12, 7319 (2010). molecules, Phys. Rev. A 53, 3324 (1996). [40] P. J. M. Johnson, A. Halpin, T. Morizumi, V. I. [23] K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Prokhorenko, O. P. Ernst, and R. J. D. Miller, Local vibra- Kobayashi, P. Naidon, T. Kishimoto, M. Ueda, and S. tional coherences drive the primary photochemistry of Inouye, Coherent Transfer of Photoassociated Molecules vision, Nat. Chem. 7, 980 (2015). Into the Rovibrational Ground State, Phys.Rev.Lett.105, [41] X.-P. Jiang and P. Brumer, Creation and dynamics of molec- 203001 (2010). ular states prepared with coherent vs partially coherent [24] A. Dodin and P. Brumer, Light-induced processes in nature: pulsed light, J. Chem. Phys. 94, 5833 (1991). Coherences in the establishment of the nonequilibrium [42] P. Brumer, Shedding (incoherent) light on quantum effects steady state in model retinal isomerization, J. Chem. Phys. in light-induced biological processes, J. Phys. Chem. Lett. 150, 184304 (2019). 9, 2946 (2018). [25] A. Dodin, T. V. Tscherbul, and P. Brumer, Coherent dynam- [43] S. Koyu, A. Dodin, P. Brumer, and T. V. Tscherbul, ics of V-type systems driven by time-dependent incoherent ArXiv:2001.09230 (2020). radiation, J. Chem. Phys. 145, 244313 (2016). [44] M. Reppert, D. Reppert, L. A. Pachon, and P. Brumer, [26] Y.-C. Cheng and G. R. Fleming, Dynamics of light har- ArXiv:1911.07606 (2019). vesting in photosynthesis, Annu. Rev. Phys. Chem. 60, 241 [45] S. Axelrod and P. Brumer, An efficient approach to the (2009). quantum dynamics and rates of processes induced by [27] A. Chenu and G. D. Scholes, Coherence in energy trans- natural incoherent light, J. Chem. Phys. 149, 114104 fer and photosynthesis, Annu. Rev. Phys. Chem. 66,69 (2018). (2015). [46] S. Axelrod and P. Brumer, Multiple time scale open sys- [28] C. L. Byrne, Signal Processing: A Mathematical Approach tems: Reaction rates and quantum coherence in model (A K Peters/CRC Press, Wellesley, Mass, 1993). retinal photoisomerization under incoherent excitation. J. [29] The modified Hamiltonian is defined using a partial time Chem. Phys. OSQD2019, 014104 (2019). derivative and therefore neglects the implicit time depen- [47] D. B. Damiano and J. B. Little, A Course in Linear Alge- dence of λt. bra (Dover Publications, Mineola, NY, 2011), Illustrated [30] H.-P. Breuer and F. Petruccione, The Theory of Open Quan- edition ed. tum Systems (Oxford University Press, Oxford, United [48] T. V. Tscherbul and P. Brumer, Long-Lived Quasistationary Kingdom, 2007). Coherences in a V-type System Driven by Incoherent Light. [31] R. Alicki and K. Lendi, Quantum Dynamical Semi- Phys.Rev.Lett.113, 113601 (2014). groups and Applications in Lecture Notes in Physics,Vol. [49] D. Møller, J. L. Sørensen, J. B. Thomsen, and M. Drewsen, 286 (Springer-Verlag Berlin Heidelberg, Berlin, Germany, Efficient qubit detection using alkaline-earth-metal ions and 2007). a double stimulated raman adiabatic process, Phys.Rev.A [32] K. Blum, Density Matrix Theory and Applications 76, 062321 (2007). (Springer Science & Business Media, New York, USA, [50] F. Vewinger, J. Appel, E. Figueroa, and A. I. Lvovsky, 2012). Adiabatic frequency conversion of optical information in [33] Z.-Y. Wang and M. B. Plenio, Necessary and sufficient con- atomic vapor, Opt. Lett., OL 32, 2771 (2007). dition for quantum adiabatic evolution by unitary control [51] S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, and fields, Phys. Rev. A 93, 052107 (2016). W. K. Hensinger, Simple Manipulation of a Microwave [34] K. Xu, T. Xie, F. Shi, Z.-Y. Wang, X. Xu, P. Wang, Y. Dressed-State Ion Qubit, Phys. Rev. Lett. 111, 140501 Wang, M. B. Plenio, and J. Du, Breaking the quantum adi- (2013). abatic speed limit by jumping along geodesics, Sci. Adv. 5, [52] N. Timoney, I. Baumgart, M. Johanning, A. F. Varón, M. B. eaax3800 (2019). Plenio, A. Retzker, and C. Wunderlich, Quantum gates and [35] S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and memory using microwave-dressed states, Nature 476, 185 K. Bergmann, Stimulated hyper-raman adiabatic passage. (2011). II. Static compensation of dynamic stark shifts, Phys. Rev. [53] A. S. Parkins, P. Marte, P. Zoller, and H. J. Kimble, Syn- A 58, 4691 (1998). thesis of Arbitrary Quantum States via Adiabatic Trans- [36] L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. fer of Zeeman Coherence, Phys. Rev. Lett. 71, 3095 Shore, and K. Bergmann, Stimulated hyper-raman adiabatic (1993). passage. I. The basic problem and examples, Phys. Rev. A [54] M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, Vacuum- 58, 4683 (1998). Stimulated Raman Scattering Based on Adiabatic Passage

030302-11 AMRO DODIN and PAUL BRUMER PRX QUANTUM 2, 030302 (2021)

in a High-Finesse Optical Cavity, Phys.Rev.Lett.85, 4872 Feshbach Molecules to a Lower Vibrational State, Phys. (2000). Rev. Lett. 98, 043201 (2007). [55] C. Nölleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, [59] J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, and S. Ritter, Eﬃcient Teleportation Between Remote N. Bouloufa, O. Dulieu, H. Ritsch, and H.-C. Nägerl, Quan- Single-Atom Quantum Memories, Phys. Rev. Lett. 110, tum gas of deeply bound ground state molecules, Science 140403 (2013). 321, 1062 (2008). [56] T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, Single- [60] F. Lang, K. Winkler, C. Strauss, R. Grimm, and J. atom single-photon quantum interface, Science 317, 488 H. Denschlag, Ultracold Triplet Molecules in the Rovi- (2007). brational Ground State, Phys.Rev.Lett.101, 133005 [57] T. Wilk, S. C. Webster, H. P. Specht, G. Rempe, and A. (2008). Kuhn, Polarization-Controlled Single Photons, Phys. Rev. [61] K.-K. Ni, S. Ospelkaus, M. H. G. D. Miranda, A. Pe’er, B. Lett. 98, 063601 (2007). Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. [58] K. Winkler, F. Lang, G. Thalhammer, P. V. D. Straten, R. S. Jin, and J. Ye, A high phase-space-density gas of polar Grimm, and J. H. Denschlag, Coherent Optical Transfer of molecules, Science 322, 231 (2008).

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