C. Hunt A. Adiabatic perturbation theory

Recent interest in adiabatic transitions has been spurred by the prospects for adiabatic quantum computing:^1-21 Adiabatic quantum computing is potentially applicable when the solution of a computational problem can be mapped to the ground state | 0_P  of a Hamiltonian H_P. If the ground state | 0  of a Hamiltonian H₀ can be prepared, and if the Hamiltonian H₀ can be converted gradually to H_P over time, then ground state will evolve adiabatically from | 0  to the solution state | 0_P . A time dependence such as H(t) = H₀ (1 – t/T) + (t/T) H_P is used, so that H₀ converts to H_P as t goes from 0 to T, and then | 0  → | 0_P  in the limit as t → T and T → ∞.¹

In simulations run on standard computers, adiabatic quantum computing has been applied to solve problems where the states can be represented as strings of digits, and energy “costs” can be assigned to specific features of the digit string.^{1-5,7,15,18,21} These features permit the con-struction of the Hamiltonian H_P. Typically the state | 0  is the uniformly weighted superposition of all possible digit strings in the space defined by the problem.^1-5 Examples of problems approached in this way include the “exact cover” problem,¹ which involves finding ways to satisfy long lists of conditions on multiple elements of the data strings, the search for marked vertices in a graph,¹⁸ and Ramsey-type problems,²¹ exemplified by the determination of the minimum value of N(m, n) such that in every graph with N points there are at least m points that are completely connected to each other pairwise, or at least n points that are connected to no others in the set of n.²¹

The simulations suggest that the time required to solve such problems can be reduced from a super-exponential dependence on the problem size in classical computing to a polynomial dependence in quantum computing (see, e.g., Ref. 1). The speed-up achievable with adiabatic quantum computing is equivalent (up to polynomial factors) to the speed-up from other forms of quantum computing.^14,22 Although adiabatic quantum algorithms appear to require exponential time in the worst cases for NP-complete problems,²² quantum computing has the potential to significantly expand the set of numerically tractable problems. Physical systems that have been used in quantum computers include nuclear spins,^6,23-27 cold trapped ions,^28-32 photons,^33-35 and superconducting loops;^36,37 applications have been suggested—and in some cases, implemented experimentally—in studying conformational energetics of proteins,¹² chemical dynamics,³⁸ electronic structure,^{24,27,32,33,39} and spin systems,^17,23,25,30 in addition to problems in graph theory and number theory.^{1,11,18,21,40} Reviews have been provided by Childs and van Dam,¹⁵ by Kassal, Whitfield, Perdomo-Ortiz and Aspuru-Guzik²⁰ and by Buluta and Nori.⁴¹

Applications in quantum computing have also prompted re-investigation of the conditions needed for the validity of the adiabatic theorem, including new experimental work.⁴² One standard criterion for the validity of the adiabatic approximation has been stated in terms of the instantaneous eigenstates | n  of the full Hamiltonian at time t, the instantaneous transition energies E_n(t) – E_m(t), and the time-derivative of the perturbation ∂H(t)/∂t, in this form: For all distinct states m and n, it is required that |  m | ∂H(t)/∂t | n  | / [E_n(t) – E_m(t)]² << 1 at all times t in the interval [0, T], assuming that the perturbation is first applied at t = 0.⁴³ However, counterexamples have been identified for which this criterion is met, yet the adiabatic theorem does not hold.⁴⁴ Extensive analysis and commentary have ensued.^45-58 In particular, the standard criterion is not sufficient when resonant transitions occur.^47,52,59 To remove this limitation, a relatively simple alternate condition has been suggested in terms of the scaled time s, defined by s = t/T: it is required that |  m | ∂H(s)/∂s | n  | / [E_n(s) – E_m(s)] |² << T for all distinct states m and n, at all scaled times s in the interval [0, 1].⁵⁹ Other forms of rigorous conditions for adiabatic time evolution have been derived.^{10,47,52,55,60-64}

The adiabatic theorem has been generalized to open systems,^65-69 adiabatic evolution with feedback,⁷⁰ and systems without energy gaps between the ground and excited states.^71-73 In the context of molecular quantum mechanics, Liu and Hunt⁷⁴ have used nonlocal susceptibility densities to prove a “force relay” theorem that holds when the electrons respond adiabatically to a perturbation: The expectation value of the force on the electrons at order n in the perturbation is equal to the force on the nuclei from the change in the electronic charge density at the next higher order, n+1.⁷⁴ This result stems from a hypervirial theorem applied to the momentum operator for the electrons,^75-83 combined with Ehrenfest’s theorem.⁸⁴

Equilibrium statistical mechanics establishes a connection between quantum mechanical and thermodynamic adiabaticity.⁸⁵ A differential change in the energy can be separated into differentials (which are in general inexact) representing heat and work. The reversible work is determined by differential changes in the energy levels, with no change in population; while the reversible heat is determined by differential changes in the state populations, with no change in the energy values themselves.⁸⁵ Processes that are adiabatic quantum mechanically correspond to processes that involve only reversible work, and not heat. The extension of this result to systems in non-equilibrium steady states has been considered by Keizer.^86,87

B. Nonadiabatic transitions

In the Landau-Zener theory of electron transfer^88,89 and in the Tully-Preston surface-hopping model,^90,91 the transition probability is expressed in terms of the nuclear velocity in the vicinity of an avoided crossing, where transitions are expected to be most probable. More generally, in the “fewest switches” algorithm for molecular dynamics with nonadiabatic electronic transitions developed subsequently by Tully,⁹² transitions may occur anywhere the electronic coupling between states is significant, and among any number of electronic states.

In reaction dynamics several variants of the surface-hopping model have been developed: Miller and George⁹³ employed a generalized Stückelberg method⁹⁴ to determine the transition probabilities. Blais, Truhlar, and Mead⁹⁵ introduced algorithmic improvements; also, as in the work of Stine and Muckerman,⁹⁶ they ensured energy conservation by adjusting the component of the momentum after the “hop” along the direction determined by the matrix element of the gradient operator, evaluated between adiabatic states in the appropriate multi-dimensional coordinate space (when the matrix element is non-zero). A closely related version of the surface-hopping model has been developed by Parlant and Gislason.⁹⁷ Cline and Wolynes⁹⁸ have used a phenomenological collision operator to treat curve-crossing processes in solution; and the nonadiabatic relaxation of excess electrons in fluids has been analyzed by Space and Coker.⁹⁹

The Landau-Zener model^88,89 and related approaches have been very widely employed. To cite just a few examples, such approaches have been used in recent studies of dissociative recombination of DCO⁺ and DOC⁺ by Larson et al.;¹⁰⁰ within the ab initio multiple-spawning approach used by Kim et al.,¹⁰¹ in their work on using laser-induced conical intersections (LICIs) in order to control the photoisomerization of 1,3-cyclo-hexadience; and in the theoretical work of Palii et al.¹⁰² on manipulating the polarization of mixed-valence inorganic compounds using electric pulses. The Landau-Zener model has been incorporated into the theory of proton transfer by Cukier and Zhu¹⁰³ and the theory of hydrogen-atom transfer by Cukier;¹⁰⁴ the latter theory is related to work on proton-coupled electron transfer reviewed by Cukier and Nocera.¹⁰⁵ The Zener formalism has been modified by Nakamura and Zhu^106,107 to include tunneling, and the modified approach has been used by Zimmerman et al.¹⁰⁸ in calculations on the production of two triplet excitons following the absorption of a single photon by pentacene or tetracene. Perturbative and other corrections of the Stueckelberg-Landau-Zener formula have been dis-cussed by Child,¹⁰⁹ with applications to the theory of molecular predissociation, collision cross sections and rate constants. Recent applications of the Landau-Zener theory in physics include the work of Gaudreau et al.¹¹⁰ on coherent control of three-spin states in triple quantum dots and the work of Miladinovic et al.¹¹¹ on the motion of photons in a double optical cavity.

Recently Domcke and Yarkony¹¹² have reviewed the role of conical intersections in photoinduced molecular dynamics and in spectroscopy, with reference to work by Teller,¹¹³ Nikitin and Umanskii,¹¹⁴ and Desouter-Lecomte et al.,¹¹⁵ as well the one-dimensional Zener model⁸⁹ and Tully’s least-switching algorithm⁹² (see also the review of nonadiabatic chemical dynamics and prospects for the future given in Ref. 116)^. Tully’s fewest switches algorithm⁹² has been used by Tavernelli et al.¹¹⁷ in a study of the ultrafast relaxation of a photoexcited singlet metal-to-ligand charge-transfer state, with the inclusion of solvent effects. The algorithm has been combined with “on the fly” calculations of the electronic energies, gradients, and nonadiabatic coupling vectors by Nelson et al.,¹¹⁸ and used in work by Clark et al.¹¹⁹ on speeding the torsional relaxation of oligofluorenes in solution into the sub-picosecond time regime. Other recent applications of surface-hopping methods include the work of Lan et al.¹²⁰ on the photo-induced nonadiabatic decay of benzylidene malonitrile, which involves twisting around a double bond and the conversion of one of the bond carbon atoms to a pyramidal structure; modeling of photo-induced dynamics in distyrylbenzene, by Nelson et al.;¹²¹ and simulations of excited state decay following photoisomerization of a bacterial photoreceptor, by Boggio-Pasqua et al.¹²² Related work has covered photofragmentation and recombination in cluster ions of I₂ ∙ Ar_n,¹²³ nonadiabatic excited state dynamics in pyrazine,¹²⁴ photodissociation dynamics of F₂ in solid argon,¹²⁵ methane photodissociation dynamics,¹²⁶ exciton dissociation and charge separation in poly-p-phenylenevinylene oligomers,¹²⁷ environmental effects near conical intersections,¹²⁸ electron-phonon relaxation in quantum dots,¹²⁹ and solvent dynamics in proton-coupled electron transfer.¹³⁰ Jasper and Truhlar found that using a stochastic model for decoherence improved agreement between surface-hopping and quantum results for the photodissociation of Na ∙ FH.¹³¹ Dang and Herman have recently discussed hops in classically forbidden regions.¹³²

C. Derivation of intermediate results

1) We show that the terms with j ≠ k in the sum _j b_k^(1)*(t) b_j⁽¹⁾(t)  k | H(t)  H₀₀(t) | j  exp(i_kjt) [appearing in the third group of terms in Eq. (38) for E⁽³⁾(t)] drop out of the final expression for E⁽³⁾(t). These terms cancel with the contributions to _k b_k⁽²⁾(t) b_k^(1)*(t) (E_k  E₀) and its complex conjugate that stem from boundary terms S_k(t) obtained from the integration by parts, in the derivation of Eq. (42) for b_k⁽²⁾(t). The boundary term S_k(t) is defined by

S_k(t) = i/ħ   k | H(t)  H₀₀(t) | j  exp(i_kjt) (i_kj)^1 b_j⁽¹⁾(t) . (C.1)

j ≠ 0

The net contribution from the boundary terms is given by

   k | H(t) | j  exp(i_kjt) b_j⁽¹⁾(t) (ħ_jk)^1 b_k^(1)*(t) (E_k  E₀)

k ≠ 0, j ≠ 0

j ≠ k

+    j | H(t) | k  exp(i_jkt) b_j^(1)*(t) (ħ_jk)^1 b_k⁽¹⁾(t) (E_k  E₀)

j ≠ k

k ≠ 0, j ≠ 0

j ≠ k

+    k | H(t) | j  exp(i_kjt) b_k^(1)*(t) (ħ_kj)^1 b_j⁽¹⁾(t) (E_j  E₀) , (C.2)

j ≠ k

   k | H(t) | j  exp(i_kjt) b_k^(1)*(t) b_j⁽¹⁾(t) [ ħ_k0/(ħ_jk) + ħ_j0/(ħ_kj) ]

k ≠ 0, j ≠ 0

j ≠ k

k ≠ 0, j ≠ 0

j ≠ k

= (1)    k | H(t) | j  exp(i_kjt) b_k^(1)*(t) b_j⁽¹⁾(t) . (C.3)

j ≠ k

S = _k_j b_k^(1)*(t) b_j⁽¹⁾(t)  k | H(t)  H₀₀(t) | j  exp(i_kjt) . (C.4)

j ≠ 0, k ≠ 0

j ≠ k

2) We obtain the normalized states at second order in the perturbation, from time-independent perturbation theory, for the instantaneous Hamiltonian H(t) = H₀ +  H(t). At second order, the states have the form

| 0(t)  = N ₀ [ | 0  +   j | H(t) | 0  (ħ_0j)^1 | j 

j ≠ 0

j ≠ 0,

for the perturbed ground state and

| k(t)  = N _k [ | k  +   n | H(t) | k  (ħ_kn)^1 | n 

n ≠ k

n ≠ k,

for the perturbed excited state, where N ₀ and N _k are the normalization factors. Then

 0(t) | 0(t)  = N ₀² [ 1 + ²   0 | H(t) | j   j | H(t) | 0  (ħ_0j)^2 + O (³) ] , (C.7)

j ≠ 0

N ₀ = [ 1 + ²   0 | H(t) | j   j | H(t) | 0  (ħ_0j)^2 ]^1/2 (C.8)

j ≠ 0

and similarly for N _k.

 j(t) | k(t)  =  j | k  +   j | H(t) | k  (ħ_jk)^1  k | k  +   j | H(t) | k  (ħ_kj)^1  j | j 

+ ²   j | H(t) | m  (ħ_jm)^1  n | H(t) | k  (ħ_kn)^1  m | n 

m ≠ j n ≠ k

+ ²   j | H(t) | m   m | H(t)  H_jj(t) | k  (ħ_mj)^1 (ħ_kj)^1  k | k 

m ≠ j

n ≠ k

 j(t) | k(t)  =  j | k 

+ ²  j | H(t) | m   m | H(t) | k  (ħ_jm)^1 (ħ_km)^1

m ≠ j,

+ ²   j | H(t) | m   m | H(t)  H_jj(t) | k  (ħ_mj)^1 (ħ_kj)^1

m ≠ j

+ ²   j | H(t)  H_kk(t) | n   n | H(t) | k  (ħ_nk)^1 (ħ_jk)^1 . (C.10)

We split the second summation (the third term on the right-hand side) in Eq. (C.10) into the set of terms with m ≠ j and m ≠ k, plus the term with m = k. In the third summation, we first change the summation index from n to m, and then similarly split the third summation into the set of terms with m ≠ j and m ≠ k, plus the term with m = j. This gives

 j(t) | k(t)  =  j | k 

+ ²  j | H(t) | m   m | H(t) | k  (ħ_jm)^1 (ħ_km)^1

m ≠ j,

m ≠ k

m ≠ j

+ ²  j | H(t) | k   k | H(t)  H_jj(t) | k  (ħ_kj)^1 (ħ_kj)^1

+ ²   j | H(t)  H_kk(t) | m   m | H(t) | k  (ħ_mk)^1 (ħ_jk)^1

m ≠ k

+ ²  j | H(t)  H_kk(t) | j   j | H(t) | k  (ħ_jk)^1 (ħ_jk)^1 . (C.11)

Now

²  j | H(t) | k   k | H(t)  H_jj(t) | k  (ħ_kj)^1 (ħ_kj)^1

= ²  j | H(t) | k  [H_kk(t)  H_jj(t)] (ħ_kj)^1 (ħ_kj)^1

+ ² [H_jj(t)  H_kk(t)]  j | H(t) | k  (ħ_jk)^1 (ħ_jk)^1

= 0 . (C.12)

Also

²  j | H(t) | m   m | H(t) | k  (ħ_jm)^1 (ħ_km)^1

m ≠ k

m ≠ j

+ ²   j | H(t)  H_kk(t) | m   m | H(t) | k  (ħ_mk)^1 (ħ_jk)^1

m ≠ k

m ≠ j

m ≠ j,

 [(ħ_jm)^1 (ħ_km)^1 + (ħ_mj)^1 (ħ_kj)^1 + (ħ_mk)^1 (ħ_jk)^1] . (C.13)

We observe that

[(ħ_jm)^1 (ħ_km)^1 + (ħ_mj)^1 (ħ_kj)^1 + (ħ_mk)^1 (ħ_jk)^1]

= (ħ_jm)^1 (ħ_km)^1 (ħ_kj)^1 [ħ_kj  ħ_km + ħ_jm] = 0 , (C.14)

and finally we obtain

 j | k  = _jk (C.15)

thus establishing that the perturbed states are orthogonal to order ².

4) We provide intermediate results that are needed to evaluate  k(t) | (t)  to second order in , using Eq. (62). Up to terms of second order in ,

 (t) | (t) ^1/2 = [1 + ²  c_n⁽¹⁾*(t) c_n⁽¹⁾(t)]^1/2

n ≠ 0

= [1 – (1/2) ²  c_n⁽¹⁾*(t) c_n⁽¹⁾(t)] (C.16)

 k(t) | ⁽⁰⁾(t)  =   k | H(t) | 0  (ħ_k0)^1

+ ²   k | H(t) | n  n | H(t)  H_kk(t) | 0  (ħ_nk)^1 (ħ_0k)^1 (C.17)

n ≠ k

+ ^   k | H(t) | n  (ħ_kn)^1 c_n⁽¹⁾(t) exp[i (E_n – E₀) t/ħ] (C.18)

n ≠ k

and

In their proof of the adiabatic theorem, Born and Fock¹³³ derived equations for quantities that are essentially equivalent to  k(t) | (t) . In this section, we note the differences between the two approaches, but also demonstrate that the results are consistent.

Born and Fock¹³³ introduced a matrix Y_mn(t), with the elements

Y_mn(t) =  m(s) | _n(t)  (D.1)

where | m(s)  denotes the mth instantaneous eigenstate of the Hamiltonian at the scaled time s given by s = t/T, and | _n(t)  denotes the full wave function at time t for a system that starts in the eigenstate | n  at t = 0. Thus our quantity  k | (t)  corresponds to Y_k0(t), in their notation. Born and Fock used a frequency variable _k(s) defined by

_k(s) = (1/ħ) ₀^s W_k(s) ds , (D.2)

where W_k(s) is the kth instantaneous eigenvalue of the full Hamiltonian H(s) at the scaled time s, and the spectrum is assumed to be discrete, although degeneracies at intermediate times are not excluded. The quantities Y_mn(t) are related to coefficients c_mn(s) by

c_mn(s) = Y_mn(t) exp(i_mT) , (D.3)

where _m denotes _m(s = 1). From Eqs. (29) and (38) of Ref. 133, Born and Fock defined a matrix P_mn(s) that induces transitions:

P_mn(s) =  i   m(s) | j   j | dH(s)/ds | k   k | n(s) 

j, k

∙ [W_m(s) – W_n(s)]^1 exp[iT(_m – _n)] . (D.4)

c_mn⁽¹⁾(s) = _mn + i ₀^s P_mn(s) ds . (D.5)

From Eqs. (D.2)-(D.5), it is apparent that the result for Y_mn(t) given by Born and Fock depends on the instantaneous eigenstates and instantaneous eigenvalues at all times t between the initial application of the perturbation (t = 0) and time t.

Additionally, we note that the result for c_mn⁽¹⁾(s) in Eq. (D.5) is of first order in the transition matrix P_mn; but by construction, c_mn⁽¹⁾(s) includes terms of all orders in the perturbation parameter . The term of first order in  is readily extracted from Eq. (D.5), however, based on the observation that dH(s)/ds is itself first order in . To leading order, we find that

| Y_k0(t) |² = |  k | (t)  |² = ² | b_k⁽¹⁾(t) |² , (D.6)

which is consistent with Eq. (55) of the current work.

1976), Part B, p. 217.

(2012).

(2011).

Ladrier, Z. R. Wasilewski, and A. S. Sachrajda, Nature Phys. 8, 54 (2012).

Lorquet, J. Phys. Chem. 89, 214 (1985).

(2011).