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Analytical results for the dynamics of parabolic level-crossing modelTo cite this article: Chon-Fai Kam and Yang Chen 2020 New J. Phys. 22 023021

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https://doi.org/10.1088/1367-2630/ab6e4a

New J. Phys. 22 (2020) 023021 https://doi.org/10.1088/1367-2630/ab6e4a

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Analytical results for the dynamics of parabolic level-crossing model

Chon-Fai Kam andYangChenDepartment ofMathematics, Faculty of Science andTechnology, University ofMacau, Avenida daUniversidade, Taipa,Macau, People’sRepublic of China

E-mail: [email protected]

Keywords:non-adiabatic transitions, parabolicmodel, two-levelmodel

AbstractWe study the dynamics of a two-level crossingmodel with a parabolic separation of the diabaticenergies. The solutions are expressed in terms of the tri-confluentHeun equations—the general-ization of the confluent hypergeometric equations.We obtain analytical approximations for the statepopulations in terms of Airy andBessel functions. Applicable expressions are derived for a large part ofthe parameter space.We also provide simple formulas which connect local solution in different timeregimes. The validity of the analytical approximations is shownby comparing them to numericalsimulations.

1. introduction

Level crossingmodels are crucial for the understanding of non-adiabatic transitions in physics, chemistry andbiology [1]. The best known and probablymost widely studied level-crossingmodel is the Landau–Zenermodel,whichwas formulated by Landau in 1932 for analyzing atomic collisions in both near-sudden [2] and near-adiabatic limits [3], andwas subsequently solved by Zener using parabolic cylinder functions [4]. At around thesame time, Stückelberg derived a sophisticated tunneling formula based on analytical continuation of the semi-classicalWKB solutions across the Stokes lines [5], andMajorana derived the transition probability formulaindependently using integral representation of the survival amplitude in connectionwith the dynamics of a spin-1/2 in a time-varyingmagnetic field [6]. The Landau–Zenermodel assumes a constant coupling between barestates in the diabatic basis and a linearly varying separation of diabatic energies [7]. The tunneling probability

z-exp( ) in the Landau–Zenermodel only depends on a single dimensionless parameterz pº -f F F V2 2 1 2( ∣( ) ∣), where f is the couplingmatrix element in the diabatic basis, F1 and F2 are the slopesof the intersecting diabatic potential curves, andV is the velocity of the perturbation variable, e.g. the relativecollision velocity [8]. Since Landau, Zener, Stückelberg andMajorana’s pioneeringworks, the linear two-statemodel has been applied to atomic [9, 10] andmolecular collisions [11], atoms in intense laser fields [12, 13], spintunneling inmolecular nano-magnets [14], tunneling of Bose–Einstein condensates in accelerated opticallattices [15], and optical tunneling inwaveguide arrays [16].

Although the Landau–Zenermodel has achieved great success over the last century, there are indeed caseswhere the assumption of linear crossing between the diabatic states breaks down. To bemore precise, onemayemploy the Stückelberg tunneling formula to the individual tunneling events [5, 17], as long as any pair ofLandau–Zener crossings arewell-separated from each other. In otherwords, for cases inwhich the crossingpointsmerge together as a result of external electric ormagnetic fields, the Landau–Zener linearization fails, andthe linear-dependence of diabatic energies should be replaced by a parabolic one [18]. The parabolicmodel wasfirst introduced by Bikhovskii, Nikitin andOvchinnikova in 1965 in the context of slow atomic collisions [19]and later re-evaluated byDelos andThorson [20, 21] andCrothers [22–24] in various limits. Two decades later,Shimshoni andGefen incorporated environment-induced dissipation and dephasing into the parabolicmodel[25]. Suominen derived analytical approximations for thefinal state populations [26], and applied the results tothe dynamics of cold atoms inmagnetic traps [27]. In the same period, Zhu andNakamura derived an exactformula for the scatteringmatrices in terms of a convergent infinite series, inwhich the coefficients satisfy afive-term recursion relation [28–32]. Nakamura and co-workers applied the results to laser assisted surface ion

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https://doi.org/10.1088/1367-2630/ab6e4ahttps://orcid.org/0000-0002-0012-691Xhttps://orcid.org/0000-0002-0012-691Xhttps://orcid.org/0000-0003-2762-7543https://orcid.org/0000-0003-2762-7543mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/ab6e4a&domain=pdf&date_stamp=2020-02-11https://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/ab6e4a&domain=pdf&date_stamp=2020-02-11http://creativecommons.org/licenses/by/4.0http://creativecommons.org/licenses/by/4.0http://creativecommons.org/licenses/by/4.0

neutralization [33] and the laser-controlled photochromism in functionalmolecules [34]. At the end of thetwentieth century, Vitanov and Suominen studied both super-linear and sub-linear level-crossingmodels basedon theDykhne–Davis–Pechukans formula, which provided good approximation for the transition probabilityin the adiabatic limit [35]. Over the last decade, there is a growing interest in non-adiabatic transitions. Lehtoincorporated super-parabolic level-glancing effects into the parabolicmodel [36, 37], and studied completepopulation inversion due to phase-jump couplings [38]. Zhang and co-workers described the populationdynamics of driven dipolarmolecules in the parabolic level-glancingmodel in terms of the confluentHeunfunctions [39]. Recently, the parabolicmodel has found applications in topological systems, including theinterband tunneling of fermionic atoms near themerging transition ofDirac cones in tunable honeycomboptical lattices [40], and the interband tunneling of two-dimensional electrons near the topological transition intype IIWeyl semimetals [41].

The dynamics of the parabolic level crossingmodel, probably unknown tomost physicists, may bewritten interms of the tri-confluentHeun function [46], which is derived from the general Heun function [47] by thecoalescence of threefinite regular singularities with infinity. In fact, there are in total 61 different classes of two-state level crossingmodels solvable in terms of the general Heun and confluentHeun functions [42–45]. Unlikethe parabolic cylinder function appearing in the Landau–Zenermodel, comprehensive information on theasymptotic behavior of confluentHeun functions, in general, does not exist. Themain difficulty is due to the factthat theHeun functions do not possess any integral representations in terms of simpler special functions. Hence,Majorana’s integral representationmethod is not applicable for determining the transition probability atinfinity. Nevertheless, in the subsequent sections, wewill show that valuable analytical approximations whichonly involves ordinary special functions can still be obtained in a large part of parameter space.

The paper is organized as follows: in section 2, we introduce the parabolic level-crossingmodel in thecontext of a two-level atomdriven by a classical laserfield, inwhich the laser detuning varies quadratically withtime, and theRabi frequency for laser driven atomic transitions is time-independent.We express the final statepopulation in terms of a single Stokesmultiplier. In section 3, we derive analytical approximations for thetransition amplitudes in both short- and long-time regimes. The accuracy of the analytical approximations istested via comparisonwith results obtained fromnumerical integration.We also discuss away to connectanalytical solutions in different time regimes. Finally, in section 4, we conclude our studies, and discussextensions of our results to other level-crossingmodels.

2. The parabolicmodel

Within the rotating-wave approximation, thewave amplitudes of a two-level atomdipole-interacting with aclassical electricfield are governed by the following coupled equations (details are provided in appendix A)

d d= - + = +

a

ta fa

a

tf a ai

d

d 2, i

d

d 2, 11 1 2

21 2* ( )

where the detuning d w wº - 0 is the difference between the laser carrier frequencyω and the Bohr transitionfrequency w0, and º - f E d ege p

1

2 0* *· is the Rabi frequency for laser driven atomic transitions, where E0 is the

envelope of the electric field, dge is the transition dipolemoment, and ep is the polarization vector of the incident

electric field. After applying the gauge changes = ò da C e s1 1 dti

2 0 and = ò d-a C e s2 2 dti

2 0 , the coupled equationsbecomes

ò ò= =d d-Ct

f CC

tf Ci

d

de , i

d

de , 2s s1 i d 2

2 i d1

t t

0 0* ( )

or equivalently

d+ - + =C

t

f

f

C

tf C a

d

di

d

d0, 3

21

21 2

1

⎛⎝⎜

⎞⎠⎟ ∣ ∣ ( )

d- + + =C

t

f

f

C

tf C b

d

di

d

d0, 3

22

22 2

2

*

*

⎛⎝⎜⎜

⎞⎠⎟⎟ ∣ ∣ ( )

whereC1 andC2 satisfy the normalization condition + =C C 11 2 2 2∣ ∣ ∣ ∣ . The problemof non-adiabatic transitionis to determine the transition probability ¥C1 2∣ ( )∣ , subjected to the conditions -¥ =C 01∣ ( )∣ and

-¥ =C 12∣ ( )∣ . Using the change of variable ò= -C U p sexp dt

1 11

2 0{ }, where dº -p f fi , we obtain the

following differential equation

+ =U

tJ t U a

d

d0, 4

21

2 1( ) ( )

2

New J. Phys. 22 (2020) 023021 C-FKamandYChen

d d= - - - -J t ft

f

f

f

fb

1

2

d

di

1

4i . 42

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟( ) ∣ ∣ ( )

In the conventional Landau–Zenermodel, the laser detuning varies linearly with time, d aºt t( ) , and the theRabi frequency is time-independent, which yields

a a+ - + =

U

tf

tU

d

d

i

2 40. 5

21

22

2 2

1

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

Equation (5) becomes the parabolic cylinder equation + + - =U n z U 011

2

1

42

1( ) via the change ofvariables aº p-z te i 4 1 2 and aºn i f 2∣ ∣ . The transition probability ¥C1 2∣ ( )∣ can be obtained from theasymptotic expansions of the parabolic cylinder function (details are provided in appendix B).

In contrast to the conventional Landau–Zenermodel, the laser detuning in the parabolicmodel variesquadratically with time, and theRabi frequency is time-independent

d a bº + ºt t t f t1

2, const, 62( ) ( ) ( )

whereα andβ are assumed to be real parameters. Before going into themathematical details, onemay analyzethe physicalmeaning of themodel by rewriting equation (1) in the formof a Schrödinger equation for a two-level system

yy y

ñ= ñ º ñ

t

t

E t f

f E tt H t ti

d

d, 7

1

2*

⎛⎝⎜

⎞⎠⎟

∣ ( ) ( )( )

∣ ( ) ( )∣ ( ) ( )

where d= - º -E t E t t 21 2( ) ( ) ( ) are the diabatic energy levels, f is the diabatic coupling,y ñ ºt a t a t, T1 2∣ ( ) ( ( ) ( )) , and a t1( ) and a t2( ) are the probability amplitudes of the diabatic basis states. As showninfigure 1, the energy levels E t1( ) and E t2( ) cross twice at times determined by d =t 0( ) , i.e. t=0 and

a b= -t 2 . In otherwords, the crossing of diabatic energy levels happenswhen the laser field is in resonancewith the atomic transition (w w= 0). But in the adiabatic basis formed by the instantaneous eigenstates ofH(t),the crossing of diabatic energy levels changes into an avoided crossing of adiabatic energy levels, where theadiabatic energy levels are given by d= + t f t 22 2( ) ( ( ) ) , and the non-adiabatic coupling between theadiabatic states has the form

f f

f f f fdº

á ñ

á ñ á ñ=

-D

- +

- - + +

g tt t

t t t t

f

tt , 8

2( )

( )∣ ( )( )∣ ( ) ( )∣ ( ) ( )

( ) ( )

where f ñ t∣ ( ) are the instantaneous eigenstates ofH(t), andD º -+ - t t t( ) ( ) ( ) is the gap between theadiabatic levels.

We now analyze the parabolicmodel based on equations (4a) and (4b), a direct computation yields

a b a ab b+ - - + + + =

U

tf

t t t tU

d

d

i

2

i

2 4 4 160. 9

21

22

2 2 3 2 4

1

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

Equation (9)may be transformed into the canonical formof tri-confluentHeun equation. Let us perform thetransformation a bº +-z h t1( )with l= - -h6 9

42 and l bº 4, then equation (9) becomes the second

canonical formof the tri-confluentHeun equation (THE2 equation) [46]

mx

n x+ - + - - =U

zz z z U a

d

d 4

3

2

9

40, 10

21

2

22 4

1

⎛⎝⎜

⎞⎠⎟ ( )

mab

xn x

ab

º + + = º -f hh

b16 4

, 3, 3 , 1024

22

2 2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

whichmay be transformed into the first canonical formof the tri-confluentHeun equation (THE1 equation)[46]

x m n+ + + + - =V

zz

V

zz V

d

d3

d

d3 0, 11

21

22 1

1( ) ( ( ) ) ( )

via the transformation = x- +U Ve z z1 112

3( ) . To be precise and for later convenience, we define lº ph e 3 2i 6 1 3( )for l > 0, and define m n x w m w n w x w=U z U z, , ; , , ;1 1 4 3 2( ) ( ) for w = pei 3.

Similar to Zener’s approach to the linear level-crossingmodel [4], the transition probabilitymay be derivedfrom the asymptotic expansions of the tri-confluentHeun function at different sectors in the complex plane.The THE2 equation has two independent solutions m n xT z, , ;1( ) and m n xT z, , ;2( ), where m n xT z, , ;1( ) hasthe following asymptotic expansion in the sector < pzarg

2∣ ∣ [46]

3


åm n x m n x= x- + - -n

T z z a z, , ; e , , , 12z z

kk

k1

1

0

12

33( ) ( ) ( )( )

and m n xT z, , ;2( ) has the following asymptotic expansion in the sector <

Using the relation ò a= - +bC U s s sexp d

t1 1

i

2 0 22( ){ }, we obtain

b -¥ »p

- - + - - -b a a

bC t A t a6 e , 16t t

1 1i

3 223 6

32

2 16

32( )( ) ( ) ( )

b -¥ » -p- + - -b a a

bC t A b3i 6 e . 16t t

1 1i

313 6

32

2 16

32( )( ) ( ) ( )

The constantA1 is determined by = =C fC f1 2∣ ∣ ∣ ∣ ∣ ∣ , which yields =b -A

f1 3 6

13( )∣ ∣ . The large t∣ ∣ solutions

equations (16a) and (16b) can also be obtained from themethod of direct integration (details are provided inappendix C). For ¥t , we have a b= +-z h t1( ) and p= -zarg 6. Hence, equation (13), the asymptoticexpansion for m n xT z, , ;2( )may not be used. In order to evaluate m n xT z, , ;2( ) for ¥t , we have to use theconnection formula [46]

m n x m n x m n x m n x= - T z T z C T z, , ; , , ; , , , , ; , 172 1 1( ) ( ) ( ) ( ) ( )

where m n x w m n w xº -C C, , , ,4 2( ) ( ), m n x w m n w x wº -T z T z, , ; , , ;1 1 4 2( ) ( ), and m n xC , ,( ) is theStokesmultiplier which connects the asymptotic expansions ofU z1( ) at different sectors, and is an entirefunction ofμ, ν and ξ [48]. For ¥t , we have a b+ = - p-h targ 1

6( ( )) and w a b+ = p-h targ 1

6( ( )) .

Hence, wemay use the asymptotic expansion of m n xT z, , ;1( ) and obtain

b

w m n w x

¥ »

- -

p- - + - + -

+ -

b a ab

b a ab

U t A t

C

6 e

, , e . 18

t t

t t

1 1i

3 2

4 2 i

23 12

34

2 16

32

123

42 1

632

( )( ) [( )( ) ] ( )( )

Using the relation = - +b a

C U e t t1 1i 12

34

2( ) , we obtainw m n w x ¥ » - - -

abC t A C , , e , 191 1 4 2

i3

6 2( ) ( ) ( )

which yields w m n w x¥ = -C A C , ,1 2 1 4 2 2∣ ( )∣ ∣ ( )∣ . Hence, thefinal transition probability ¥C1 2∣ ( )∣ depends onlyon the Stokesmultiplier m n xC , ,( ). However, it is in general not an easy task to obtain exact formulas for theStokesmultipliers. AlthoughZhu andNakamura derived an exact formula for the Stokesmultipliers in terms ofa sophisticated infinite series generated by afive-term recursion relation [49], a compact formula for the finaltransition probability similar to the Landau–Zener formula has not been derived. In the following section, wederive concise and explicit expressions for the transition dynamics in the parabolicmodel where d t( ) is given byequation (6), and provide an analytical approximation for connecting solutions of the transition amplitude indifferent time regimes.

3. Analytical approximations for the transition amplitude

In the previous section, we have shown that the dynamics of a two-level atomdipole-interacting with an off-resonant classical electric fieldwith constant amplitude and parabolic detuning can be solved in terms of the tri-confluentHeun functions.We discussed the relationship between thefinal transition probability and the Stokemultipliers which connect asymptotic expansions of the tri-confluentHeun functions. However, for practicalpurposes, we develop analytical approximations to the transition amplitudes, rather than to solve theconnection problem rigorously.

To beginwith, let us rewrite equation (9), the differential equationwhich governsU1, as

tbt b

tab

+ - + - =U

f Ud

d

i

2 160, 20

21

22

22

2

2

2

1

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥∣ ∣ ( )

where t a bº +t . From equation (20), we see that the sign ofα/β does not alter the nature of the equation.Hence, without loss of generality, wemay assume thatα/β is a positive number. Equation (20) can also berewritten as

tab

bt a t b t+ + - - + =

Uf U

d

d 16

i

2 8 160. 21

21

22

4

2

2 2 2 4

1

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

For largeα/β, we compare the last two terms in equation (21). The short time regime is defined byb t a t16 82 4 2 2⪅ , i.e. a b»t t ;*∣ ∣ ⪅ whereas the long time regime is defined by b t a t16 82 4 2 2⪆ , i.e.t t*∣ ∣ ⪆ . In the short time regime, the quadratic term is dominant in equation (21). Hence, the transitiondynamics can be described by the Landau–Zener formula, and solved via the parabolic cylinder functions. Butfor smallα/β, we should compare the third and the last terms in equation (21). Hence, the short time regime isdefined by b t bt16 22 4 ⪅ ∣ ∣ , i.e. t b -2 ;1 3∣ ∣ ⪅ ∣ ∣ whereas the long time regime is defined by t b -2 1 3∣ ∣ ⪆ ∣ ∣ . Inthe short time regime, the linear term is dominant in equation (21). Hence, the transition dynamics is solved viathe Airy functions, which is different from that obtained from the Landau–Zener formula. In the long time

5


regimewhere the quartic term is dominant in equation(21), the transition dynamics can be solved via the Besselfunctions.

In the following subsections, we discuss in detail the analytical approximations for the transition amplitudein different time regimes.

3.1.Dynamics in the short-time regime for smallα/βIn this subsection, we analyze the dynamics for the transition amplitude in the time regime t t*∣ ∣ ⪅ , that isb t a t16 82 4 2 2⪅ .We require a b2 2 3∣ ∣ ⪅ , so that a t bt8 22 2 ⪅ ∣ ∣ is satisfied.Wemay rewriteequation (21) as

tab

bt a t b t+ + - = -

Uf U U

d

d 16

i

2 8 16, 22

21

22

4

2 1

2 2 2 4

1

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

where the right-hand side of equation (22) is regarded as a known function of τ. Let us denote

t= + +bb

ab

z fi2

2i 216

13 4

2( )( ) [ ∣ ∣ ], thenwe obtain t = - +b b ab- z fi2 2i 2 1613 4

2( )( ) ∣ ∣ andb a

bp

- = - - +

º

+ -U

zzU z z z z U

z U

d

d

1

4

i

2 16

, 23

21

2 1

2 32

4

2 1

4 1

⎜ ⎟⎧⎨⎩

⎛⎝

⎞⎠

⎫⎬⎭[( )( )]( ) ( )

where gº b ab

z 2i 1i2

1 3( ) ( ) and g º +a abf1 2 16 4 2( )∣ ∣ . From equation (23), we see that the homogeneousequation =U zU1 1¯ ¯ is solved by the Airy functions zAi( ) and zBi( ). To be specific, we define the cubic root of

bi2

as bp

e i2

6

13( )∣ ∣ for b b= ∣ ∣, so that Î p pzarg ,6 76( ) for b > 0 and Î - -p pzarg ,76 6( ) for b < 0. The

particular solution of equation (23) is given by

ò

ò

z p z zz z

z

z p z zz z

z

=-

+

U z zU

zU

AiBi

Ai , Bid

BiAi

Ai , Bid , 24

pz

z

z

z

14 1

4 1

0

0

W

W

( ) ( ) ( ) ( )¯ ( )

{ ( ) ( )}

( ) ( ) ( )¯ ( )

{ ( ) ( )}( )

where z z p=Ai , Bi 1W{ ( ) ( )} is theWronskian of zAi( ) and zBi( ). As the homogeneous solutionU z1¯ ( ) is alinearly combination of zAi( ) and zBi( ), we need to evaluate integrals of the form ò z y zdn 2 and ò z y y zdn 1 2 ,where y refers to zAi( ) or zBi( ) respectively. After integration, equation (23) is solved by (details can be found inappendixD)

» - + ¢U zG z

zU z G z U z1

1

2

d

d, 251 1 1⎜ ⎟

⎛⎝

⎞⎠( )

( ) ¯ ( ) ( ) ¯ ( ) ( )

where º +U z c z c zAi Bi1 1 1¯ ( ) ( ) ( )with c1 and c2 being arbitrary constants, and

b

ab

= - ++

+ - + - +

G zz az a b

z

abz b

a

1

4

i

2 9

2

7

2

5

4

9

2

3

6

7 16, 26

2 3 4 3 22

24

2⎜ ⎟

⎜ ⎟⎛⎝⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

( )

( )

where º ++ -a z z and º + -b z z . Infigure 2, we compare the analytical approximation of the transitionprobability with the numerical solution. The result shows that the transition probability is well-described byequations (25) and (26) in the short-time regime for smallα/β.

3.2.Dynamics in the short-time regime for largeα/βIn this subsection, we analyze the dynamics for the transition amplitude in the time regime t t*∣ ∣ ⪅ , so thatb t a t16 82 4 2 2⪅ .We require a b2 2 3∣ ∣ ⪆ , so that a t bt8 22 2 ⪆ ∣ ∣ is satisfied. In this regime, it is better touse the original equationwhich governsU1. From equation (9), we obtain

a a b ab+ - + » -

U

tf

tU

t tU

d

d

i

2 4

i

2 4, 27

21

22

2 2

1

3

1

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟∣ ∣ ( )

where the right-hand side of equation (27) is regarded as a known function of t. Let us denote a= p-z te i 4 1 2 ,thenwe obtain

p- + » - + ºU

za

zU

z

z

z

zU z U

d

d 4 2 4, 28

21

2

2

1

3

1 3 1* *

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ( ) ( )

6


where aº - -a fi 1 22∣ ∣ and a bº p-z e i 4 3 2* . The homogeneous equation = +U a z U11

42

1( )¯ ¯ is solvedby the parabolic cylinder functions U a z,( ) and - U a z, i( ). As a remark,U a z,( ) can also bewritten asDn(z), whereDn(z) satisfies + + - =D n z D 0n n

1

2

1

42( ) .We choose the pair of linearly independent solutions

as -U a z U a z, , , i{ ( ) ( } , where the plus andminus signs are for a > 0 and a < 0 respectively. The particularsolution of equation (28) is given by

ò

ò

z p z zz z

z

z p z zz z

z

=--

-

+ --

U z U a zU a U

U a U a

U a zU a U

U a U a

,, i

, , , id

, i,

, , , id , 29

pz

z

z

z

13 1

3 1

0

0

W

W

( ) ( ) ( ) ( )¯ ( )

{ ( ) ( )}

( ) ( ) ( )¯ ( )

{ ( ) ( )}( )

where z z- = p +U a U a, , , i ie i a2 14W{ ( ) ( )} ( ) is theWronskian of zU a,( ) and z-U a, i( ) . As thehomogeneous solutionU z1¯ ( ) is a linearly combination ofU a z,( ) and -U a z, i( ) , we need to evaluate integralsof the form ò z y zdn 2 and ò z y y zdn 1 2 , where y refers to zU a,( ) or z-U a, i( ) respectively. After integration,equation (28) can still be solved by (details can be found in appendixD)

» - + ¢U zG z

zU z G z U z1

1

2

d

d, 301 1 1⎜ ⎟

⎛⎝

⎞⎠( )

( ) ¯ ( ) ( ) ¯ ( ) ( )

where º + -U z c U a z c U a z, , i1 1 2¯ ( ) ( ) ( ) with c1 and c2 being arbitrary constants, and= - + -G z a z z6 8 62 *( ) ( ) ( ). Aswe can see from figure 3(a), the Landau–Zener solution

= + -U z c U a z c U a z, , i1 2 1 2 2∣ ¯ ( )∣ ∣ ( ) ( )∣ coincides with the exact result of the transition probability obtainedfrom equation (20), but the analytical approximation equation (30) does not provide better result than theLandau–Zener solution. Infigure 3(b), we compare the transition probability obtained from U z1 2∣ ¯ ( )∣ with theexact solution, and the result shows that the final transition probability is well-described by the Landau–Zenerformula in thewhole-time range for largeα/β.

3.3.Dynamics in the long-time regimeIn this subsection, we analyze the dynamics for the transition amplitude in the time regime t t*∣ ∣ ⪆ , so thatb t a t16 82 4 2 2⪆ .Wemay rewrite equation (21) as

tb t a

bbt a t

p t

+ = - - + +

º

UU f U

U

d

d 16 16

i

2 8

, 31

21

2

2 4

12

4

2

2 2

1

2 1

⎛⎝⎜

⎞⎠⎟∣ ∣

( ) ( )

where the right-hand side of equation (31) is regarded as a known function of τ. The homogeneous equationb t = -U U1

1

162 4

1¯ ¯ is solved by the Bessel functions of order 1/6, that is, t btºw J 121 1 6 3( ) andt btº -w J 122 1 6 3( ). The particular solution of equation (31) is given by

Figure 2.The transition probability U t1 2∣ ( )∣ subjected to the initial conditions =U t 0i1∣ ( )∣ and =U t fi1∣ ( )∣ ∣ ∣ . Numerical solution ofequation (3a)with d t( ) is given by equation (6) is depicted in blue, approximate solution of the homogeneous equation =U zU1 1¯ ¯ isdepicted in red solid line, and the analytical approximation to equation (23), which is given by equation (25), is depicted in red dashedline. Here, = -t 1i , tf=2, =f 1∣ ∣ , a = 0.2 and b = 2.

7


ò

ò

t tt p t t

t tt

tt p t t

t tt

=-¢ ¢ ¢

¢ ¢¢

+¢ ¢ ¢

¢ ¢¢

t

t

t

t

U ww U

w w

ww U

w w

,d

,d , 32

p1 12 2 1

1 2

21 2 1

1 2

0

0

W

W

( ) ( ) ( ) ( )¯ ( )

{ ( ) ( )}

( ) ( ) ( )¯ ( )

{ ( ) ( )}( )

where t t p= -w w, 31 2W{ ( ) ( )} is theWronskian of tw1( ) and tw2( ). As the homogeneous solutionU1¯ is alinearly combination ofw1 andw2, we need to evaluate integrals of the form ò t tºI w dn n 2 and

ò t tºJ w w dn n 1 2 , wherew refers tow1 orw2 respectively. After integration, equation (32) is solved by (detailscan be found in appendixD)

ttt

t t t» - + ¢UG

U G U11

2

d

d, 331 1 1⎜ ⎟

⎛⎝

⎞⎠( )

( ) ¯ ( ) ( ) ¯ ( ) ( )

Figure 3.The transition probability U t1 2∣ ( )∣ subjected to the initial conditions =U t 0i1∣ ( )∣ and =U t fi1∣ ( )∣ ∣ ∣ . Numerical solution ofequation (3a) is depicted in blue, approximate solution of the homogeneous equation = +U a z U1

1

42

1( )¯ ¯ is depicted in red solid line,and the analytical approximation to equation (28), which is given by equation (30), is depicted in red dashed line. Here, =f 1∣ ∣ ,a = 2, and b = 0.2.

8


where t t bt t bt= + -U c J c J12 121 1 1 6 3 2 1 6 3¯ ( ) ( ) ( )with c1 and c2 being arbitrary constants, and

tab

t bt

b t bt

a t bt

= + -

- -

- -

G f F

F

F

16 31,

5

6;

7

6,

8

6,

9

6;

12

i

2 121, 1;

8

6,

9

6,

10

6;

12

8 301,

7

6;

9

6,

10

6,

11

6;

12. 34

24

2

3

2 3

3 2

4

2 3

3 2

2 5

2 3

3 2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

( ) ∣ ∣

( )

Here F a a b b z, , ; , , ;p q p q1 1( ) is the generalized hypergeometric function of order p, q, and tG ( ) has thefollowing asymptotic expansion in the limit t ¥

tab t

»G . 352

2( ) ( )

Aswe can see from figure 4, the analytical approximation of the transition probability U1 2∣ ∣ , which is determinedby equation (33)with t a b t» -G 2 2 1( ) , agrees with the exact result in the long-time regime.

3.4. Analytical approximations to the connection problemIn the last subsections, we have studied the dynamics for the transition amplitude in both short- and long-timeregimes. In this subsection, we focus on the problemof connecting different local solutions of the transitiondynamics. Aswe showed infigure 3(b), the transition dynamics for largeα/β can bewell-described by theLandau–Zener formula. Hence, we concentrate on the transition dynamics for smallα/β.Without loss ofgenerality, we only consider the parabolic glancing case with a = 0. Then equation (20)may bewritten as

l l+ - + =U

tf t t U

d

d2i 0, 36

21

22 2 4

1(∣ ∣ ) ( )

where l bº 4. Substitution of rºU e S1 i into equation (36) yields the following set of differential equations

rr

l- + + =t

S

tf t a

1 d

d

d

d0, 37

2

2

22 2 4⎜ ⎟⎛⎝

⎞⎠ ∣ ∣ ( )

r l r- =t

S

tt b

d

d

d

d2 0, 372 2⎜ ⎟

⎛⎝

⎞⎠ ( )

where the initial conditions are r -¥ = 0( ) and r -¥ = 1( ) . In the long time limit, we expect that thetransition probability r = U2 1 2∣ ∣ converges toward a stationary value.Hence, we assume r » 0̈ and

l= +S f t2 2 2 4∣ ∣ in equation (37a), which yields l» +S t t S33 0( ) and r l l r+ =f t t2td

d2 2 2 4 2( ∣ ∣ ) . A direct

computation yields

Figure 4.The transition probability U t1 2∣ ( )∣ subjected to the initial conditions =U t 0i1∣ ( )∣ and =U t fi1∣ ( )∣ ∣ ∣ . Numerical solution ofequation (20) is depicted in blue, approximate solution of the homogeneous equation b t = -U U1

1

162 4

1¯ ¯ is depicted in red solid line,and the analytical approximation to equation (31), which is given by equations (33) and (35), is depicted in red dashed line. Here,= -t 5i , = -t 1f , =f 1∣ ∣ , a = 1, and b = 2.

9


rr l

lr» +

+ºt

t

f tF t

21 , 38

ff

2

2 2 4( )

∣ ∣( ) ( )

where r r» -l

-t1ff

24

2

2( )∣ ∣ and r r» - l -tf f2 522∣ ∣ in the long-time limit.Wewill use this solution in laterdiscussions.

Aswe have discussed in the last subsections, the transition dynamics for the a = 0 case can bewell-approximated by the Airy andBessel functions in the short- and long-time regimes, respectively. Hence, we onlyneed to connect the local solutions at two critical times t1 and t2—for Î -¥t t, 1( ], we have

= +U c w t c w t ;1I

1I

1 2I

2( ) ( )( ) ( ) ( ) for Ît t t,1 2[ ], we have = +U c z c zAi Bi1II 1II 2II( ) ( )( ) ( ) ( ) withº +b

bz t f ;i

2

1 3 2i 2( )( ) ∣ ∣ and for Î ¥t t ,2[ ), we have = +U c w t c w t1III 1III 1 2III 2( ) ( )( ) ( ) ( ) , wherebºw t J t 121 1 6 3( ) and bº -w t J t 122 1 6 3( ). Then the connection formulas for the transition amplitudes

are

= = =U U U U t t a, , for ; 391I

1II

1I

1II

1 ( )( ) ( )( ) ( )

= = =U U U U t t b, , for . 391II

1III

1II

1III

2 ( )( ) ( )( ) ( )

Adirect computation yields

= - =cf w t

w t w tc

f w t

w t w ta

,,

,, 40i

i i

i

i i1

I 2

1 22I 1

1 2W W

∣ ∣ ( ){ ( ) ( )}

∣ ∣ ( ){ ( ) ( )}

( )( ) ( )

=-

cU t z t U t z t

z t z tb

Bi Bi

Ai , Bi, 40

It

I

1II 1 1

d

d 1 1 1 1

1 1W

( ) ( ( )) ( ) ( ( ))

{ ( ( )) ( ( ))}( )( )

( ) ( )

=-

cU t z t U t z t

z t z tc

Ai Ai

Ai , Bi, 40

I It

2II 1 1 1 1 1

d

d 1

1 1W

( ) ( ( )) ( ) ( ( ))

{ ( ( )) ( ( ))}( )( )

( ) ( )

=-

cU t w t U t w t

w t w td

,, 401

III 1II

2 2 2 1II

2 2 2

1 2 2 2W

( ) ( ) ( ) ( ){ ( ) ( )}

( )( )( ) ( )

=-

cU t w t U t w t

w t w te

,, 402

III 1II

2 1 2 1II

2 1 2

1 2 2 2W

( ) ( ) ( ) ( ){ ( ) ( )}

( )( )( ) ( )

where b p=z t z tAi , Bi i 2 1 3W{ ( ( )) ( ( ))} ( ) is theWronskian of z tAi( ( )) and z tBi( ( )),l l= -w t t J t 31 5 2 5 6 3( ) ( ) , l l=w t t J t 32 5 2 5 6 3( ) ( ) , and p= -w t w t, 31 2W{ ( ) ( )} is theWronskian of

w t1( ) and w t2( ). Hence, the dynamics of the transition amplitude is determined up to two critical times t1 and t2.Aswe can see from equation (36), the long-time limit is determined by l lt t22 4 , which yields lt 2 ;1 3( )whereas the short-time limit is determined by l lt t22 4 , which yields lt 2 1 3( ) . Hence, wemayapproximate the critical times t1 and t2 by l- 2 1 3( ) and l2 1 3( ) respectively.

Infigure 5(a), we compare the analytical approximation to the transition probability with the exact solutionsof equation (36) obtained fromnumerical integration. The result shows that the transition probability is well-described by ÈU U1I 2 1II 2∣ ∣ ∣ ∣( ) ( ) for Î -¥t t, 2( ). But for the long-time regime, the deviations from the analyticexpressions and the numerical results are prominent, as the analytic expressions involving Bessel function

quickly decrease to zero at ¥t , e.g. = » -bpl

b p-w t t J t t tcos1 1 6 123 6 1

123

3( ) ( ) ( ).Motivated by the

physical fact that the transition probability approaches a stationary value at infinity, wemodify the transitionamplitudeU1

III( ) by replacing +c w c w1III

1 2III

2( ) ( ) by r + +c w c we Si 1

III1 2

III2

( ) ( ) , where l» +S t S33 0 and ρ isdetermined by equation (38). A direct computation yields

r = + -U t

F t S t

U t

U t

F t

F ta1

1, 41f

1II

2

22

2

1II

2

1II

2

2

2

2⎛⎝⎜⎜

⎞⎠⎟⎟( )( ) ( )

( )( )

( )( )

( )( ) ( )

( )I I

I

l= - --S

S t

U t

U t

F t

F t

tbcot

1

3, 410 1

2

1II

2

1II

2

2

2

23⎡

⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥( )

( )( )

( )( )

( )( )

( )I

I

=-

cU t w t U t w t

w t w tc

,, 411

III 1II

2 2 2 1

II2 2 2

1 2 2 2W

˜ ( ) ( ) ˜ ( ) ( ){ ( ) ( )}

( )( )( ) ( )

=-

cU t w t U t w t

w t w td

,, 412

III 1

II2 1 2 1

II2 1 2

1 2 2 2W

˜ ( ) ( ) ˜ ( ) ( ){ ( ) ( )}

( )( )( ) ( )

where l= +S f t2 2 2 4∣ ∣ and r lº - +U U F t Scos 3f1II

1II 3

0˜ ( )( ) ( ) . For ¥t , asw1 andw2 decreases as -t 1,

thefinal transition probability ¥U1 2∣ ( )∣ is given by r f2 , which is determined by equation (41a). Infigure 5(b), we

10


compare themodified analytical approximation of the transition probability with the numerical solution ofequation (36). The result shows that the transition probability U1 2∣ ∣ is well-described by

È ÈU U UI1 2 1II 2 1III 2∣ ∣ ∣ ∣ ∣ ∣( ) ( ) ( ) in thewhole time range after the replacement+c w c w1

III1 2

III2

( ) ( ) r + +c w c we Si 1III

1 2III

2( ) ( ) .

4. Conclusion

To summarize, we studied the transition dynamics of the parabolicmodel—a two-state system subject to aquadratically detuning over an infinite time interval. The solutions are expressed in terms of the tri-confluentHeun functions, which are the generalizations of the conventional confluent hypergeometric functions. Insteadof rigorously solving the Stokesmultipliers which connect the asymptotic solutions of the transition amplitudes,we derived concise analytical approximations to the transition amplitudes in both short- and long-time regimes,and provided practical formulas for connecting local solutions in different regimes.We gave applicableestimation of the critical times that separate different time regimes. The transition dynamics is shown to bewell-described by the analytical formulas in thewhole-time range by comparisonwith exact results.

One limitation of our current work is that, as the constant coupling between bare states is assumed to lastfrom = -¥t to = ¥t , and the detuning as a quadratic function of time goes to infinity when ¥t , thephase of the propagator parametrized in the Stückelberg variablesmay diverge as a result of this unphysicalassumption, as shown inVitanov and co-workers’works on the Landau–Zenermodel [51–53].

In future works, wewould like to extend our study to both super-linear and sub-linearmodels, inwhich thedetuning is a polynomial in timewith cubic or higher degrees, or a rational function in time.We alsowant toderive analytical formulas for the propagator and the associated Stokes constants, and determine the criticaltimeswhich separate the short- and long-time regimes rigorously, instead of giving a rough estimate of themagnitude.

Acknowledgments

This research is funded by The Science andTechnologyDevelopment Fund,Macau SAR (File no. FDCT 023/2017/A1).

AppendixA. Equations ofmotion for a two-level atomdipole-interactingwith a classicaldrivingfield

In this appendix, we provide the background knowledge of the two-level atomdescription of light–matterintegrations for readers.When the two target atomic levels are nearly resonant with the driving field, while onthe same time the other atomic levels are detuned far off resonance, wemay regard the system as two discretenon-degenerate states, e.g. the ground state ñg∣ and thefirst excited state ñe∣ , then theHamiltonian of the atom

Figure 5.The transition probability U t1 2∣ ( )∣ subjected to the initial conditions =U t 0i1( ) and =U t fi1( ) ∣ ∣ . Numerical solution ofequation (36) is depicted in blue, and the analytical approximations of the transition probability are depicted in red solid lines. Infigure 5(a), b b= + -U c t J t c t J t12 12 ;1

III1

III1 6

32

III1 6

3( ) ( )( ) ( ) ( ) whereas infigure 5(b),r b b= + + -U c t J t c t J te 12 12S1

III i1

III1 6

32

III1 6

3( ) ( )( ) ( ) ( ) . Here, =f 1∣ ∣ , a = 0, b = 2, = -t 3.20971 , t2 = 2.0335, andl= =t 2 1.5874

13* ( ) .

11


may bewritten as = ñá + ñáH E e e E g ge gatomˆ ∣ ∣ ∣ ∣. For the case when the atom interacts with an external electricfield under dipole approximation, the interactionHamiltonian becomes = -H d E rint cmˆ ˆ · (ˆ ), where rcmˆ is theposition operator for the center ofmass.When theDe Broglie wavelength of the atom is small compared to theinteratomic spacing, the center ofmass positionmay be treated classically [50].

In the following, we discuss the coherent excitation of the two-level atomunder the driving of an externalelectric field. Let us denote the state of the two-level atom as y ñ = ñ + ñt c t e c t ge g∣ ( ) ( )∣ ( )∣ , the Schrödingerequations for the twowave amplitudes have the form

= - á ñ - á ñ ct

E t e e c e g c ad E d Eid

d, A1e e e g( ( ) ∣ ˆ · ∣ ) ∣ ˆ · ∣ ( )

= -á ñ + - á ñc

tg e c E t g g c bd E d Ei

d

d, A1

ge g g∣ ˆ · ∣ ( ( ) ∣ ˆ · ∣ ) ( )

where º qd xˆ ˆ is the transition electric dipolemoment of the atom. For atom that possesses inversion symmetry,the energy eigenstates are symmetric or anti-symmetric, and hence the expectation values of the dipolemomentvanishes, á ñ = á ñ =e e g gd d 0∣ ˆ ∣ ∣ ˆ ∣ . Let us denote the energies of the two atomic levels as w= Ee 2 0 and

w= -Eg 2 0, where w º - E Ee g0 is the energy difference between the two atomic levels. For amonochromatic

wave, the electric fieldmay bewritten as ò w t t= +EE eexp i d h.c.t

p1

2 0 0{ [ ( ) ] }, where E0 is the complex

electric field amplitude at the position rcm, and ep is the polarization vector of the incident electric field. If we

denote the transition dipolemoment á ñe gd∣ ˆ ∣ as dge , the Schrödinger equations for the twowave amplitudesmaybe simplified as

òw= - +w t t

c

tc E e c a

did

d 2 2e h.c. , A2e e

gep g

00

i dt

0

⎡⎣⎢

⎤⎦⎥· ( )

( )

ò w= - + -w t t

c

tE c c b

dei

d

d 2e h.c.

2. A2

g gep e g0

i d 0t

0

* ⎡⎣⎢

⎤⎦⎥· ( )

( )

In the reference frame rotating about the z-axis with frequency w t( ), the state of the two-level atom isòy w t y¢ ñ = ñ

t

t tS texp d z

i

0∣ ( ) ( ( ) )∣ ( ) . Then the relationships between thewave amplitudes in the rotated andunrotated frame are

ò ò= =w t t w t t-a c a ce , e . A3e e g gd dt t

i2

0

i2

0 ( )( ) ( )

Substitution of equation (A3) into equations (A2a)–(A2b) yields

d

d

=- - +

=- + +

ò

ò

w t t

w t t-

a

t

ta

E Ea

a

t

E Ea

ta

d e d e

d e d e

id

d 2 2e

2,

id

d 2 2e

2,

ee ge p ge p g

gge p ge p e g

0 2i d 0

0 0 2i d

t

t

0

0

**

*** *

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( ) · ·

· · ( )

( )

( )

where d w wº -t 0( ) is the laser detuning. Applying the rotatingwave approximation, we neglect the fast-oscillating terms like ò w t texp 2i d

t

0{ ( ) } in the Schrödinger equations, and obtain

d d= - + = +

a

t

ta fa

a

tf a

tai

d

d 2, i

d

d 2, A4e e g

ge g*

( ) ( ) ( )

where º -

f d eE ge p20 ** · is the Rabi frequency for the transition dipolemoment. For the special casewhen the

laser detuning vanishes, the coupled-mode equations become =a fai e g and =a f ai g e* , which are equivalent to

- + = - + =af

fa f a a

f

fa f a0, 0. A5e e e g g g2 2*

*̈ ∣ ∣ ̈ ∣ ∣ ( )

When the atom is initially at the ground state, we have =c 0 1g ( ) and =c 0 0e ( ) , or equivalently =a 0 1g ( ) and=a 0 0e ( ) . For the special case that f is a constant, equation (A5) is solved by =a ftsine and =a ftcosg , or

equivalently = ò w t t-c fte sine dti

2 0( ) and = ò w t tc fte cosg d

ti2 0

( ) in the lab frame, which results in the occupationprobabilities =c ftsine 2 2∣ ∣ and =c ftcosg 2 2∣ ∣ .

Appendix B. The Landau–Zenermodel

In the Landau–Zenermodel, thewave amplitude after the change of variable ò=aU C s sexp d

t1 1

i

2 0{ } is

governed by the Schrödinger equation + =U J t U 01 1̈ ( ) , where = - +a aJ t f t2 i2 4

2 2

( ) ∣ ∣ . After the change of

12


variable a= p-z te i 4 1 2 and a=n fi 2∣ ∣ , the Schrödinger equation becomes the parabolic cylinder equation + + - =U n z U 01

1

2

1

42

1( ) , and is solved by the parabolic cylinder functionDn(z), which has the followingasymptotic expansion for large z∣ ∣

p

p

p

» å--

<

» å--

-G -

å+

» --

+ -G -

p

p

-=

¥

-=

¥

+ =¥

-

+

D z zn

k zz

D z zn

k z n z

n

k z

zn n

z n za

e2

, arg3

4,

e2

2 e e 1

2

e 11

2

2 e eB1

nz n

kk

k

nz n

kk

k

n z

n kk

k

z nn z

n

022

022

i

1 02

2

2

i

1

14

2

14

214

2

14

214

2{ }

( ) ( )!( )

∣ ∣

( ) ( )!( ) ( )

( )!( )

( )( )

( )

p p+

+ ++ Î

n n

zz b1

1 2

2, arg

4,

5

4. B1

2⎜ ⎟⎛⎝

⎞⎠{ }( )( ) ∣ ∣ ( )

For a > 0, we have a= p-z te i 4 ∣ ∣ , whereas for a < 0, we have a= pz tei 4 ∣ ∣ . For -¥t , we have= pz Re i3 4 with aºR t∣ ∣ ∣ ∣, which yields

g

-¥ = »

-¥ = »-

ag

p pg g

a a

-

C t A D R A R

C tf

Cft

C

e e e ,

ie e,

t

t t

1i 4

i3i 4 3 4 i

2 1 1

2

i2

2 i2

2

( ) ( )

( )

where g aº f 2∣ ∣ . Hence, -¥ = pgC A e1 3 2∣ ( )∣ ∣ ∣ and -¥ =C 02∣ ( )∣ , which violates the initial conditions-¥ =C 01( ) and -¥ =C 12∣ ( )∣ .We now check the linearly independent solutions

= p- - - - D z D Ri en n1 1 i 4( ) ( ) . From equation (B1a), we obtain

-¥ » p g a g - - - -C t A R ae e , B3t1 i 4 i 2 i 12( ) ( )( )

a -¥ »

aC t

t

fC b

e, B3

t

2

i 2

1

2

( ) ( )

which yields -¥ =C 01∣ ( )∣ and g-¥ = g p -C A e2 1 2 4∣ ( )∣ ∣ ∣∣ ∣ ∣ ∣ . Hence, the choice of parametersg= g p -A e1 2 4∣ ∣ ∣ ∣ satisfies the initial conditions -¥ =C 01( ) and -¥ =C 12∣ ( )∣ . For +¥t , we have

= pz Ri e 3i 4 , which yields

p gg

¥ »G +

p gg

-C t R a

2 e

i 1, B41

1 2 2i( ) ∣ ∣

( )( )

∣ ∣

g a ¥ »a

p g p g- -C tf

R bie

e e . B4t

2

i 21 2 3 i 4 i

2

( ) (∣ ∣ ∣ ∣ ) ( )∣ ∣

Adirect computation yields ¥ = p g-C e2 2 2∣ ( )∣ ∣ ∣ and ¥ = - p g-C 1 e1 2 2∣ ( )∣ ∣ ∣.

AppendixC. Large t solution of d ¢ + =y t y f yi 02( ) ∣ ∣

In this appendix, we consider a slightly general formof equations (3a) and (3b), and analyze the large variablesolution of d ¢ + =y t y f yi 02( ) ∣ ∣ , where d a bº + +t t t1

22( ) is a polynomial of time. Let us denote

y=UV, whereV obeys d + =t V f Vi 02( ) ∣ ∣ , and is solved by ò d= ¢-V V f t texp i dtt

02 1

0{ ∣ ∣ ( ) } . Substitution

of y=UV into d ¢ + =y t y f yi 02( ) ∣ ∣ yields

dd

dd

d- - =U

f

tt U

f

t

f t

tU

2ii

i0. C1

2 4

2

2

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

̈ ∣ ∣( )

( ) ∣ ∣( )

∣ ∣ ( )( )

( )

For large t∣ ∣, equation (C1) is approximated by n =U Ui 0̈ , which is solved b0y

ò ò d» ¢ +¢

U U t t t Uexp i d dt

t

t

t1 0

0 0{ ( ) } . For the Landau–Zener case that d a=t t( ) , we obtain

= gV V t t0 0 i( ) , òa a» ¢ ¢ + -U U t z Ue i 2 i e d

z

z z1 0

2 20

0

2

, where g aº f 2∣ ∣ and º az ti2

. Substitution

of the result into y=UV yields

pa

a

apa

»

+

=- = -

g g

g g

y c t t c t

c V t U t c V t U c z

2ierfc

i

2,

e i 2,2i

erfc ,

1i

2i

1 0 0i

1 02

2 0 0i

0 1 0

⎛⎝⎜

⎞⎠⎟

( )

13


where the complementary error function zerfc ( ) has the following asymptotic expansion

åp= --

+-

=

-

zz

k

zR z aerfc

e1

2 1

2, C2

z

n

nk

k n0

1

2

2

( ) ( ) ( ) !!( )

( ) ( )

òpº-

-

¥- -R z

n

ns s b

1 2

2e d . C2n

n

n z

n s2 1

2 2( ) ( ) ( )!!

( )

Hence, for large t∣ ∣, we obtain

a»

+a g g-y t

ct c t

ie . C3t1 i 2 i 1 2 i

2( ) ( )

As a result, thewave amplitudesC1 andC2 in equations (3a) and (3b) subjected to the conditions -¥ =C 01( )and -¥ =C 12∣ ( )∣ are asymptotically determined by

a» »

-a g g- - - -C tc

t C tc

ft

ie ,

i. C4t1

1 i 2 i 12

1 i2( ) ( ) ( )

Hence, we obtain ¥ -¥ = pg-C C e2 2∣ ( )∣ ∣ ( )∣ .We now analyze the parabolicmodel with d a b= +t t t122( ) .

Following similar procedures described previously, we obtain

a b=

+

g

V Vt

ta

2, C50

i⎛⎝⎜

⎞⎠⎟ ( )

òa b

» ¢ + ¢ ¢ +U U t t t U bexp i2 6

d , C5t

t

22 3

00

⎜ ⎟⎧⎨⎩

⎛⎝

⎞⎠

⎫⎬⎭ ( )

where º +a bU U t texp i2 1 2 02

6 03{ ( )}.We now analyze the large t∣ ∣behavior of the integral ò ¢p

¥ - ¢ te dt

tn( ) ,

where p tn( ) is an n-order polynomial of t, e.g. p = +a bt i t t3 2

26

3( ) ( ). A direct computation gives

pp

p= - -

pp

p--

-

ta

d

d

ee

e, C6

n

n

n2

nn

n⎛⎝⎜

⎞⎠⎟

̈ ( )

pp

pp

pp

pp

= - - +p p p p- - - -

tb

d

d

e e 3e e, C6n

n

n

n

n

n

n

n3 2

2

4

3

3

n n n n⎛⎝⎜

⎞⎠⎟

̈ ̈ ̈ ( )( )

which yields

ò ppp

¢ » - +pp¥

- ¢-

tt

t

te d

e1 . C7

t

tt

n n2

nn ⎡

⎣⎢⎤⎦⎥( )

̈ ( )( )

( )( )( )

Hence, we obtain

a a b»

++

+b a bg g+

+

a b

y t ct t

ct

t

e

i 2, C8

t tt

t1

i

22 2

i

2

i2

26

3 ⎛⎝⎜

⎞⎠⎟( )( ) ( ) ( )

( )

where = -c V U1 0 2 and ò= + +a b¥c V U U t t texp i d

t2 0 0 2 22

63

0{ [ ( )] } . Hence, we obtain

a a b»

+ +b

g- + -a b

C t ct t

t

ta

e

i 2, C9

t t

1 1

i

22

i2

26

3 ⎛⎝⎜

⎞⎠⎟( ) ( ) ( )

( )

a b»

-+

g-

C tc

f

t

tb

i

2, C92

1i⎛

⎝⎜⎞⎠⎟( ) ( )

where ¥ -¥ =C C 12 2∣ ( )∣ ∣ ( )∣ , which is not consistent with the exact solution of the final transitionprobability. Hence, we conclude that the asymptotic solutions alone cannot guarantee a correctfinal transitionprobability. Amore careful treatment of the connection problem is needed for the parabolicmodel.

AppendixD. Integrals involving products of Airy, parabolic cylinder, andBesselfunctions

Herewe derive some general results regarding integrals of the form òºI z y zdn n 2 and òºJ z y y zdn n 1 2 ,where y1and y2 are solutions of the differential equation =y fy . If wewrite = + ¢ + ¢I Py Qyy Ryn 2 2,we obtain

= + ¢ + ¢ + ¢ ¢J Py y Q y y y y Ry yn 1 21

2 1 2 2 1 1 2( ) ,where P andQ satisfy = -P R fR1

2and = - ¢Q R ,andR is a solution

of the third-order differential equation ¢¢¢ - ¢ - ¢ =R fR f R z4 2 2 n. As a consequence,º + + + - + n n n R z3 2 1 2 n 3( )( )( ) is a solution of the third-order differential equation¢¢¢ - ¢ - ¢ = + + ¢+ f f z n f z f4 2 2 2 3n 3( ( ) ). Let us denote y1 and y2 as two independent solutions of

14


=y fy and y y,1 2W{ }being theWronskian of y1 and y2. A direct computations shows thatW is a constant.Then for any linear combinations of y1 and y2, i.e. a bº +y y y1 2, we have

= ¢ - ¢y R y Ry1

2. D1nL ( )

D.1. Integrals of products of Airy functionsHerewe evaluate indefinite integrals of the form òºI z y zdn n 2 and òºJ z y y zdn n 1 2 in terms of Airy functionsand theirfirst derivatives, where y1 and y2 are solutions of the Airy differential equation =y zy . A directcomputation shows that

=+

--

+ ¢ - ¢

+- -

+

+ - -

-

In

zn n

z y nz yy z y

n n n

nI a

1

2 1

1

2

1 2

2 2 1D2

nn n n n

n

1 2 2 1 2

3

⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎫⎬⎭( )

( )( )( )

( )

º - - ¢ ¢ + ¢R zR y R yy R y b1

2, D2n n n n

2 2⎜ ⎟⎛⎝⎞⎠ ( )

whereRn satisfies the recursion relation

=+

- - --Rn

n n n R z a1

2 2 11 2 2 , D3n n n3( )

[ ( )( ) ] ( )

= - = - = -R R z R z b1, 3, 5, D30 1 2 2 ( )

which yields = - +R z 3 73 3( ) and = - +R z z4 94 4( ) . Hence, a direct computation yields

= - ¢ = + ¢ - ¢I zy y I z y y y zy a,1

3, D40 2 2 1 2 2 2( ) ( )

= - + ¢ - ¢I z y zy y z y b1

51 2 , D42 3 2 2 2( ) ) ( )

= + ¢ - + ¢I z y z y y z y c1

73 3 , D43 4 2 2 3 2( ( ) ) ( )

= - + + ¢ - + ¢I z z y z y y z z y d1

92 4 1 4 . D44 5 2 2 3 4 2( ) ( ) ( ) ) ( )

Let us denote =y zAi1 ( ), =y zBi2 ( ), and a b= +y y y1 2. From equation (D1), we obtain

= ¢ = ¢ -y y yz

y y a,3

1

6, D50 1L L ( )

= ¢ - =+

¢ -yz

yz

y yz

yz

y b5 5

,3

7

3

14, D52

2

3

3 2

L L ( )

=+

¢ -+

yz z

yz

y c4

9

2 2

9. D54

4 3

L ( )

D.2. Integrals of products of parabolic cylinder functionsIn this sub-appendix, we evaluate indefinite integrals of the form òºI z y zdn n 2 and òºJ z y y zdn n 1 2 in terms ofparabolic cylinder functions and theirfirst derivatives, where y1 and y2 are both solutions of the parabolic

cylinder differential equation = +y a z y14

2( ) . If wewrite º - - ¢ ¢ + ¢I R zR y R yy R yn n n n n1

22 2( ) , we

obtain ¢¢¢ - + ¢ - =R a z R zR z4 2n n n2( ) .Wemay expandRn in a series aså =¥ c zk k

k0 , where the coefficients ck

satisfies the following three-term recursion relation

+ + - - - == +

+ -k k kc akc k ck n

2 1 4 12, 1;0, otherwise.

k k k2 2

⎧⎨⎩( )( ) ( )


= + - ¢I z a y y a1

24 2 , D61 2 2 2( ) ( )

15


= - - -

+ ¢ + - ¢

Iz

az a y

zy y a z y b

1

3 22 16 2

4 16 2 , D6

3

42 2 2

2 2

⎡⎣⎢⎛⎝⎜

⎞⎠⎟

( ) ] ( )

which corresponds to = -R 21 and = -R a z16 2 33 2( ) . Let us denote =y U a z,1 ( ), = -y U a z, i2 ( ) , anda b= +y y y1 2. From equation (D1), we obtain

= ¢ = - --

¢y y yz

ya z

y2 ,2

3

16 2

3. D71 3

2

L L ( )

D.3. Integrals of products of Bessel functionsIn this sub-appendix, we evaluate indefinite integrals of the form ò t tºI y dn n 2 and ò t tºJ y y dn n 1 2 in terms ofBessel functions and their first derivatives, where y1 and y2 are both solutions of the differential equation

l t = -y y2 4 . Using the ansatz l t= + - ¢ ¢ + ¢I R R y R yy R yn n n n n1

22 4 2 2( ) , we obtain

l t l t t¢¢¢ + ¢ + =R R R4 8 2n n n n2 4 2 3 , whereRn satisfies the following recursion relation

t l=

- ++ + +

++R

n R

n n n

2 4 5

1 2 3. D8n

nn

3 26( )

( )( )( )( )


tl

= + ¢I y y a4

1

4, D93

42

22 ( )

tl

tl

= - ¢ + ¢I y yy y b6

1

6 6, D94

52

2 22 ( )

tl

tl

tl

= + - ¢ + ¢I y yy y c8

1

8 4 8, D95

6

22

2

2

22( ) ( )

which corresponds to l=R 1 43 2( ), t l=R 64 2( ) and t l=R 85 2 2( ).We need to evaluate Inwith=n 0, 1, 2.Wemay expandRn in a series as tå =

¥ ck kk

0 , where the coefficients ck satisfy the following two-termrecursion relation

l+ + + + + = = -+k k k c k ck n

6 5 4 4 22, 3;0, otherwise.

k k62

⎧⎨⎩( )( )( ) ( )


t l t

t

=+ + +

-+

+ + ++

=-

+ + +

lt

+

+ + + + +

Rn n n

n

n n n

F

n n n

2

3 2 11

4 5

9 8 7

2 1, ; , , ;

3 2 1, D10

n

n

n n n n n

3 2 6

32 3

5

6

7

6

8

6

9

6 3

23⎜ ⎟

⎡⎣⎢

⎤⎦⎥

⎛⎝

⎞⎠( )

( )( )( )( )

( )( )( )

( )( )( )( )

where F a a b b z, , ; , , ;p q p q1 1( ) is the generalized hypergeometric function of order p, q. To be specific, let usdenote t lt=y J 31 1 6

3( ), t lt= -y J 32 1 6 3( ), and a b= +y y y1 2. From equation (D1), we obtain( =n 0, 1, 2)

t

t

=-

+ +

--

+ + +¢

lt

lt

+ + + + +

+ + + + +

yF

n ny

F

n n ny

1, ; , , ;

2 1

2 1, ; , , ;

3 2 1. D11

n

n n n n n

n n n n n

22 3

5

6

3

6

7

6

8

6 3

2

32 3

5

6

7

6

8

6

9

6 3

2

3

3

L

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

( )( )( )

( )( )( )( )

In particular, for ¥z and = -p q 1, the generalized hypergeometric function -F za b; ;p q ( ) has thefollowing asymptotic expansion in p

where k º - +q p 1, n º å - å + -a b q p 2l l l l ( ) . ForRnwith n=0 or 2, we have k = 2,n = - +n 5 3( ) , and

p

p

»

»G

G G G

G G G

p-

-

- -

+ + +

-

p+

+

+

E zz

H zz z

e4

e ,

2, D13

z

n

n n n

2,3i i 2

2,3

1

61

1

6

2

6

3

6

2

6

3

6

4

6

nn

n

56 5

6

56( )

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( )

( )( )

where the plus andminus signs are for n=0 and n=2 respectively. For n=0, we have - -z z ;1 5 6 whereasfor n=2, we have - -z z7 6 1 . Hence, a direct computation yields t »

b tR2

82( ) and

tp l t

ltp

»G G

+ -R3

6

31

1

3cos

2

3

5

6.0 3

5 31

3

1

6

2

23⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎡⎣ ⎤⎦ ⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

( )( )

( )

For n=1, since -F z1, 1; , , ;2 38

6

9

6

10

6( ) is a linear combination of z−1 and -z zln1 near = ¥z , tR1( ) is alinear combination of t-2 and t t- ln2 . Hence, we have t t tR R R2 1 0( ) ( ) ( ) near t = ¥.

ORCID iDs

Chon-Fai Kam https://orcid.org/0000-0002-0012-691XYangChen https://orcid.org/0000-0003-2762-7543

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1. introduction2. The parabolic model3. Analytical approximations for the transition amplitude3.1. Dynamics in the short-time regime for small α/β3.2. Dynamics in the short-time regime for large α/β3.3. Dynamics in the long-time regime3.4. Analytical approximations to the connection problem

4. ConclusionAcknowledgmentsAppendix A.Appendix B.Appendix C.Appendix D.D.1. Integrals of products of Airy functionsD.2. Integrals of products of parabolic cylinder functionsD.3. Integrals of products of Bessel functions

References

Analytical results for the dynamics of parabolic level-crossing model · 2020. 12. 1. ·...

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