Continuous-time quantum search and time-dependent two-level quantum systems
Carlo Cafaro Paul M. Alsing



^1SUNY Polytechnic Institute, 12203 Albany, New York, USA
^2Air Force Research Laboratory, Information Directorate, 13441 Rome, New
York, USA


It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett.
120, 060501 (2018)] that the continuous-time formulation of Grover's quantum
search algorithm can be intuitively understood in terms of Rabi oscillations
between the source and the target subspaces.

In this work, motivated by this insightful remark and starting from the
consideration of a time-independent generalized quantum search Hamiltonian as
originally introduced by Bae and Kwon [Phys. Rev. A 66, 012314 (2002)], we
present a detailed investigation concerning the physical connection between
quantum search Hamiltonians and exactly solvable time-dependent two-level
quantum systems. Specifically, we compute in an exact analytical manner the
transition probabilities from a source state to a target state in a number of
physical scenarios specified by a spin-1/2 particle immersed in an external
time-dependent magnetic field. In particular, we analyze both the periodic
oscillatory as well as the monotonic temporal behaviors of such transition
probabilities and, moreover, explore their analogy with characteristic
features of Grover-like and fixed-point quantum search algorithms,
respectively. Finally, we discuss from a physics standpoint the connection
between the schedule of a search algorithm, in both adiabatic and nonadiabatic
quantum mechanical evolutions, and the control fields in a time-dependent
driving Hamiltonian.



Quantum computation (03.67.Lx), Quantum information (03.67.Ac).
Continuous-time quantum search and time-dependent two-level quantum
systems
    Carlo Cafaro^1 and Paul M. Alsing^2
    December 30, 2023
===========================================================================








§ INTRODUCTION


The first continuous-time version of Grover's original discrete quantum search
algorithm was proposed by Farhi and Gutmann in Ref. <cit.>. In
particular, a generalization of the Farhi-Gutmann analog quantum search
algorithm was considered by Bae and Kwon in Ref. <cit.>. Both quantum
search Hamiltonians in Refs. <cit.> are time-independent. The
transition from time-independence to time-dependent in the framework of analog
quantum search algorithms appears originally under the working assumption of
global adiabatic evolution in Ref. <cit.> and later, in the more
advantageous setting of local adiabaticity in Ref. <cit.>. In the
local adiabatic evolution approach, the time-dependent search Hamiltonian is
the linear interpolation between two time-independent Hamiltonians.
Specifically, the system is prepared in the ground state of one of the two
Hamiltonians. Then, this Hamiltonian is adiabatically changed into the other
one whose ground state is assumed to encode the unknown solution of the search
problem. The time-dependence of the Hamiltonian is encoded in the
time-dependent interpolation function. In particular, the precise form of such
a function can be determined by imposing the local adiabaticity condition at
each instant of time during the quantum mechanical evolution of the system.
Roughly speaking, the adiabatic approximation states that a system prepared in
an instantaneous eigenstate of the Hamiltonian will stay close to this
prepared state if the Hamiltonian changes sufficiently slowly
<cit.>. Therefore, any functional form of the interpolation
schedule that satisfies the necessary degree of slowness for the adiabatic
condition to be fulfilled can be formally considered. We emphasize
that there is no fundamental reason to exclude a nonlinear
interpolation of the two static Hamiltonians <cit.>. Indeed, Perez and
Romanelli considered nonadiabatic quantum search Hamiltonians specified in
terms of a nonlinear interpolation of the two time-independent Hamiltonians in
Ref. <cit.>. Depending on both the parametric and temporal form of
the two interpolation functions used to define the time-dependent Hamiltonian,
they presented two search algorithms exhibiting different features. In
particular, while both of them exhibited the typical 𝒪(
√(N)) Grover-like scaling behavior in finding the target state in
an N-dimensional search space, one of the algorithms required a specific
time to make the measurement and find the target state (periodic oscillatory
behavior and original Grover search algorithm,
<cit.>). The other one, instead, caused the
source state to evolve asymptotically into a quantum state that overlapped
with the target state with high probability (monotonic behavior and
fixed-point Grover search algorithm, <cit.>).

Recently, Dolzell, Yoder, and Chuang investigated in detail the possibility
for adiabatic quantum search algorithms to exhibit the fixed-point property in
addition to quadratic quantum speedup <cit.>. From their theoretical
analysis, supported by a number of illustrative examples, they concluded that
depending on the choice of the interpolation schedule of the algorithm, it is
possible to construct quantum search Hamiltonians with a variety of features:
Grover-like scaling and fixed-point property; Grover-like scaling but no
fixed-point property; fixed-point property but no Grover-like scaling. It is
especially relevant to our present work to recognize that their
theoretical investigation was deeply influenced by the intuition that a
two-dimensional system whose dynamics is governed by a time-dependent
Hamiltonian is essentially equivalent to a spin-1/2 particle subject to an
external time-dependent magnetic field. This link between the quantum dynamics
of a spin-1/2 particle in an external magnetic field and quantum search
algorithms was further discussed by Byrnes, Forster, and Tessler in Ref.
<cit.>. In this work, the authors point out that Grover's original
quantum search algorithm occurs in the setting of discrete variable quantum
computing and, in particular, the Grover operator only inverts the sign to one
state. In the framework of quantum computing with continuous variables where
the units of quantum information are called qunats <cit.>,
changing a phase to simply a single quantum subspace of an
infinite-dimensional Hilbert space implies that the Grover operator would have
to invert the phase of an infinitely squeezed momentum state <cit.>.
This task presents two drawbacks: first, it is difficult to prepare such a
state from an experimental standpoint; second, this infinitely squeezed
momentum state would have no quantum mechanical overlap with the solution
states encoded in the position eigenstates. Motivated by these considerations,
Byrnes, Forster, and Tessler present in Ref. <cit.> a continuous-time
generalization of Grover's original quantum search algorithm where not only
the Oracle operator inverts the phase of an arbitrary number of target states
but the Grover operator also inverts the sign to an arbitrary number of
quantum states. In particular, they pointed out that Grover's search
Hamiltonian can be intuitively understood in terms of Rabi oscillations
between the source and the target subspaces.

In this paper, inspired by the analogy discussed in Ref. <cit.>, we
seek to provide a unifying perspective on the above mentioned different types
of analog quantum search algorithms in terms of the quantum mechanics of
two-level quantum systems. In particular, beginning from the analysis of a
time-independent generalized quantum search Hamiltonian as originally
introduced in Ref. <cit.>, we present a detailed investigation
concerning the physical connection between quantum search Hamiltonians and
exactly solvable time-dependent, two-level quantum systems. Specifically, we
analytically calculate the transition probabilities from a source state to a
target state in a number of physical scenarios specified by a spin-1/2
particle immersed in an external time-dependent magnetic field. In particular,
we analyze the observed periodic oscillatory and monotonic temporal behaviors
of such transition probabilities and investigate their analogy with
characterizing features of Grover-like and fixed-point quantum search
algorithms, respectively. Finally, we discuss the physical link between the
schedule of a search algorithm, in both adiabatic and nonadiabatic quantum
mechanical evolutions, and the control fields in a driving time-dependent Hamiltonian.

The layout of the remainder of the paper is as follows. In Section II, we
examine the transition probability from a source state to a target state for a
two-level quantum system whose time-independent Hamiltonian evolution was
originally considered by Bae and Kwon in Ref. <cit.>. In Section III, we
transition from time-independent to time-dependent Hamiltonians. Specifically,
we discuss the temporal behavior of the transition probability from a source
state to a target state for a two-level quantum system immersed in a
time-dependent external magnetic field configuration that specifies the
original scenario considered by Rabi in Ref. <cit.>. The explicit
computations leading to uncover the exact temporal behavior of both quantum
mechanical probability amplitudes and transition probabilities as presented in
Sections II and III (and, in a more detailed fashion, in Appendices A and C,
respectively) allow us to strengthen the original intuitions reported in Refs.
<cit.>. These results also enable a transparent comparison
between the dynamics generated by a time-independent Grover-like quantum
search Hamiltonian and a time-dependent Rabi-like two-level quantum
Hamiltonian. In Section IV, we present the basic notions of adiabatic and
nonadiabatic quantum search Hamiltonians. In the adiabatic case, we emphasize
the intuitive link between the dynamics generated by quantum search
Hamiltonians and the non-relativistic quantum mechanical dynamics of an
electron in an external time-dependent magnetic field. In the nonadiabatic
case, instead, we focus on the temporal behavior of the transition
probabilities from a source state to a target state under suitable working
assumptions on the schedules of the nonadiabatic search Hamiltonian. In
Section V, upon relaxing the working assumption of knowing a priori the
exact time-dependent external magnetic field configuration, we study two
quantum dynamical evolution scenarios specified by suitable time-dependent
magnetic field configurations (determined a posteriori) that produce
transition probabilities showing a temporal monotonic convergence from the
source to the target states. In the former scenario we assume that the phase
and the magnitude of the transverse field are interconnected. In the latter
scenario, while preserving the same formal expression of the complex
transversal field, we do not assume any connection between the phase and the
intensity of the field itself. Finally, exploiting the analogies between the
dynamics generated by a time-independent Grover-like quantum search
Hamiltonian and a time-dependent Rabi-like two-level quantum Hamiltonian
presented in Sections II and III, a comparison between the dynamics of the
newly selected time-dependent Hamiltonians and that of nonadiabatic quantum
search Hamiltonians with a fixed-point property is presented. Our final
remarks appear in Section VI. Several technical details are presented in the
Appendices A, B, C, D, and E.



§ A TIME-INDEPENDENT QUANTUM SEARCH HAMILTONIAN


Consider the time-independent generalized quantum search (GQS) Hamiltonian
ℋ_GQS defined as <cit.>,

    ℋ_GQSdef=E[  α|
    w⟩⟨ w| +β| w⟩⟨ s| +γ| s⟩⟨
    w| +δ| s⟩⟨ s|]  ,

where α, β, γ, δ are complex expansion
coefficients. Furthermore, assume that the quantum state |
w⟩ is the normalized target state while |
s⟩ is the normalized initial state with real quantum
overlap ⟨ w|s⟩ =⟨ s|w⟩ =x that
evolves in an unitary fashion according to Schrödinger's quantum
mechanical evolution law,

    | s⟩↦ e^-i/ℏℋ
    _GQSt| s⟩,

with ℏ≃0.66×10^-15(  eV·)
denoting the reduced Planck constant. Our goal is to compute the time
t^∗ such that 𝒫_| s⟩→| w⟩(  t^∗)  =𝒫_max where
𝒫_| s⟩→| w⟩(  t) is the transition probability defined as,

    𝒫_| s⟩→| w⟩(  t)  def=|⟨
    w|e^-i/ℏℋ_GQSt|s⟩|
    ^2.

Employing the Gram-Schmidt orthonormalization technique, we can construct an
orthonormal set of state vectors starting from the set {|
w⟩, | s⟩}. We obtain,

    {| w⟩, | s⟩}→{| w⟩, |
    s⟩ -⟨ s|w⟩| w⟩}→{| w⟩/‖| w⟩‖, |
    s⟩ -⟨ s|w⟩| w⟩/‖| s⟩ -⟨ s|w⟩| w⟩‖}.

Let us define the quantum state vector | r⟩ as,

    | r⟩def=|
    s⟩ -⟨ s|w⟩| w⟩/‖| s⟩ -⟨ s|w⟩| w⟩‖.

Recalling that ⟨ s|w⟩ =x, Eq. (<ref>) becomes

    | r⟩ =| s⟩ -⟨
    s|w⟩| w⟩/√(⟨
    s|s⟩ -⟨ s|w⟩ ^2)=1/√(1-x^2
    )(  | s⟩ -x| w⟩)
    .

In terms of the orthonormal basis {| w⟩,
| r⟩}, the state | s⟩
becomes

    | s⟩ =| s⟩(  |
    w⟩⟨ w| +| r⟩⟨ r|)  =⟨ w|s⟩|
    w⟩ +⟨ r|s⟩| r⟩.

Note that the quantum overlap ⟨ r|s⟩ is given by,

    ⟨ r|s⟩ =1/√(1-x^2)(  ⟨
    s| -x⟨ w|)  (  |
    s⟩)  =1/√(1-x^2)(  1-x^2)
    =√(1-x^2).

Therefore, by employing Eq. (<ref>), the state |
s⟩ in Eq. (<ref>) can be written as

    | s⟩ =x| w⟩ +√(1-x^2
    )| r⟩.

After a straightforward but tedious computation, we find that the transition
probability P_| s⟩→| w⟩(  t) in Eq. (<ref>) becomes

    𝒫_| s⟩→| w⟩(  t)  =x^2cos^2(  √(h_12h_21/ℏ
    ^2+(  h_11-h_22)  ^2/4ℏ^2)t)
    +|1/2h_11-h_22/ℏx+h_12
    /ℏ√(1-x^2)| ^2/h_12h_21/ℏ^2
    +(  h_11-h_22)  ^2/4ℏ^2sin^2(
    √(h_12h_21/ℏ^2+(  h_11-h_22)
    ^2/4ℏ^2)t)  ,

where h_11, h_12, h_21, and h_22 are defined as,

    h_11def=E[  α+(  β+γ)
    x+δ x^2]  , h_12def=E√(1-x^2
    )(  β+δ x)  ,
       
         h_21def=E√(1-x^2)(  γ+δ x)
    , h_22def=Eδ(  1-x^2)  .

A detailed derivation of Eq. (<ref>) appears in Appendix A. From Eq.
(<ref>), it follows that the maximum 𝒫_max=𝒫
_| s⟩→| w⟩(
t^∗) of 𝒫_| s⟩→| w⟩(  t) occurs at the instant t^∗,

    t^∗def=πℏ/√((  h_11
    -h_22)  ^2+4h_12h_21),

and, in addition, is equal to

    𝒫_max=|1/2h_11-h_22/ℏx+h_12/ℏ√(1-x^2)| ^2/h_12h_21
    /ℏ^2+(  h_11-h_22)  ^2/4ℏ^2
    .

Finally, using Eq.(<ref>) and observing that α and δ must
be real coefficients while β=γ^∗, t^∗ in Eq.
(<ref>) and 𝒫_max in Eq. (<ref>) become

    t^∗(  α, β, δ, x)
    def=2/√(4[  αδ+Re
    ^2(  β)  -|β| ^2]
    x^2+4Re(  β)  (  α+δ)
    x+[  (  α-δ)  ^2+4|β|
    ^2]  )πℏ/2E,

and,

    𝒫_max(  α, β, δ,
    x)  =4[  |β| ^2-Re
    ^2(  β)  ]  x^4+[  (  α+δ)
    ^2-8(  |β| ^2-Re^2(
    β)  )  ]  x^2+4Re(  β)
    (  α+δ)  x+4|β| ^2/4[
    αδ+Re^2(  β)  -|β| ^2]  x^2+4Re(  β)
    (  α+δ)  x+[  (  α-δ)
    ^2+4|β| ^2]  ,

respectively. For the sake of completeness, we point out that in the limiting
case of α=δ=1 and γ=β^∗=0, we recover the original
Farhi-Gutmann (FG) results,

    t_FG^∗=πℏ/2E·1/x, and
    𝒫_max^(  FG)  =1.

Moreover, we also recover the transition probability 𝒫_|
s⟩→| w⟩^(  GF
)  (  t),

    𝒫_| s⟩→| w⟩^(  FG)  (  t)  def=
    x^2cos^2(  Ex/ℏt)  +sin^2(  Ex/ℏt)  .

It is insightful to point out that the minimum search time t^∗ in Eq.
(<ref>) needed to achieve the maximal success probability 𝒫
_max in Eq. (<ref>) is inversely proportional to the energy level
separation Δλdef=λ_+-λ_- where
λ_± denote the two real eigenvalues of the Hermitian matrix
[  ℋ_GQS]  _{| w⟩, | r⟩},

    λ_±def=1/2[  (  h_11
    +h_22)  ±√((  h_11-h_22)  ^2+4h_12h_21
    )]  .

Therefore, it appears reasonable from our analysis that the finiteness of the
minimum search time t^∗ can be explained in terms of the familiar
phenomenon of avoided level crossing in quantum perturbation theory
<cit.>. Such a phenomenon is caused by the effect of an external
perturbation on a two-level quantum system. Specifically, the two energy
levels move apart in opposite direction after applying the perturbation. The
higher level increases while the lower level decreases. Furthermore, the
energy splitting is more prominent when the unperturbed Hamiltonian is
degenerate. This link between analog quantum search algorithms and
perturbations in quantum theory will be further explored in the next sections.



§ PERIODIC OSCILLATORY BEHAVIOR AND THE RABI TIME-DEPENDENT
HAMILTONIAN


In this section, we begin by computing the transition probability from a
source state | s⟩ to a target state |
w⟩ for a two-level quantum system immersed in a time-dependent
external magnetic field configuration that characterizes the original scenario
considered by Rabi in Ref. <cit.>. For a brief review on the concept of
interaction representation in quantum mechanics in the context of
time-dependent perturbations, we refer to Appendix B.



 §.§ Transition probability: General setting


The general expression of the time-dependent Hamiltonian of the system that we
are investigating is given by,

    ℋ^(  full)  (  t)  def=ℋ_0^(  free)  +𝒱
    ^(  interaction)  (  t)  .

In particular, to study the original scenario considered by Rabi in Ref.
<cit.>, assume that the time-independent free Hamiltonian
ℋ_0^(  free) and the time-dependent
interaction potential 𝒱^(  interaction)
(  t) are defined as <cit.>,

    ℋ_0^(  free)  def=
    E_1| E_1⟩⟨ E_1|
    +E_2| E_2⟩⟨ E_2|, and
    𝒱^(  interaction)  (  t)
    def=Γ e^iω t| E_1⟩⟨ E_2| +Γ e^-iω t| E_2
    ⟩⟨ E_1|,

respectively. The two parameters ω and Γ are positive and
real. Furthermore, we assume that E_2>E_1 and ⟨
E_i|E_j⟩ =δ_ij for any 1≤ i, j≤2. Using Eqs.
(<ref>) and (<ref>), we obtain

    iℏ(
    [ ċ_1; ċ_2 ])  =(
    [           𝒱_11(  t)  𝒱_12(  t)
    e^iω_12t; 𝒱_21(  t)  e^iω_21t            𝒱_22(
    t) ])  (
    [ c_1; c_2 ])  ,

that is,

    iℏ(
    [ ċ_1; ċ_2 ])  =(
    [                   0  Γ e^i(  ω-ω_21)  t; Γ e^-i(  ω-ω_21)  t                   0 ])  (
    [ c_1; c_2 ])  ,

where ċ_idef=dc_i/dt for any 1≤ i,
j≤2 and ω_21def=(  E_2-E_1)
/ℏ denotes the characteristic frequency of the two-level system. The
matrix equality in Eq. (<ref>) yields a system of two coupled first
order ordinary differential equations,

    iℏċ_1=Γ e^i(  ω-ω_21)  t
    c_2, and iℏċ_2=Γ e^-i(  ω-ω
    _21)  tc_1.



At this point, we recall that the transition probabilities
𝒫_| s⟩→| E_1
⟩(  t) and 𝒫_| s⟩→| E_2⟩(  t) are defined as
𝒫_| s⟩→| E_1
⟩(  t)  def=|
c_1(  t)  | ^2 and 𝒫_|
s⟩→| E_2⟩(  t)
def=| c_2(  t)  | ^2,
respectively. After some algebra (for more details, we refer to Appendix C),
we find that the transition probabilities 𝒫_|
s⟩→| E_1⟩(  t)
and 𝒫_| s⟩→|
E_2⟩(  t) are given by

    𝒫_| s⟩→| E_1
    ⟩(  t)  =(  1-x^2)  cos^2(
    √(Γ^2/ℏ^2+(  ω-ω_21)
    ^2/4)t)  +{  1-[  (  ω-ω
    _21)  /2x-Γ/ℏ√(1-x^2)/√(Γ^2/ℏ^2+(  ω-ω_21)  ^2/4)
    ]  ^2}sin^2(  √(Γ^2/ℏ^2
    +(  ω-ω_21)  ^2/4)t)  ,

and,

    𝒫_| s⟩→| E_2
    ⟩(  t)  =x^2cos^2(  √(Γ^2
    /ℏ^2+(  ω-ω_21)  ^2/4)t)
    +[  (  ω-ω_21)  /2x-Γ/ℏ√(1-x^2)/√(Γ^2/ℏ^2+(
    ω-ω_21)  ^2/4)]  ^2sin^2(  √(Γ^2/ℏ^2+(  ω-ω_21)  ^2
    /4)t)  ,

respectively. In Fig. 1, we report the oscillatory and periodic temporal
behavior of the transition probabilities 𝒫_|
s⟩→| E_2⟩(  t)
and 𝒫_| s⟩→|
E_1⟩(  t) at the original Rabi resonance where
ω=ω_21 with x=0, Γ=1, and ℏ=1. For the sake of
completeness, observe that

    _I⟨ s(  t)  |s(  t)  ⟩
    _I=𝒫_| s⟩→|
    E_1⟩(  t)  +𝒫_| s⟩→| E_2⟩(  t)  =1,

where the quantum state | s(  t)  ⟩
_I is such that,

    | s(  t)  ⟩ _Idef
    =⟨ E_1|s(  t)  ⟩ _I|
    E_1⟩ +⟨ E_2|s(  t)  ⟩
    _I| E_2⟩,

with | s(  0)  ⟩ _Idef=√(1-x^2)| E_1⟩ +x|
E_2⟩.



 §.§ Time-dependent magnetic field in the Rabi Hamiltonian


At this point, in an effort to emphasize several physics insights between
analog quantum search algorithms and time-dependent Hamiltonians, we recall
that the most general time-dependent Hermitian Hamiltonian operator
ℋ(  t) with a (  2×2)-matrix
representation can be recast as,

    ℋ(  t)  def=a(  t)
    ℐ+b⃗(  t)  ·σ⃗,

where σ⃗ denotes the vector of Pauli matrices, ℐ is
the identity operator, a(  t) is a time-dependent function, and
b⃗(  t) is the vector given by

    b⃗(  t)  def=‖b⃗(
    t)  ‖b̂(  t)  =√([  b_x(
    t)  ]  ^2+[  b_y(  t)  ]  ^2+[
    b_z(  t)  ]  ^2)b̂(  t)  .

Recalling that the Pauli vector operator σ⃗ is defined as
<cit.>,

    σ⃗def=[  σ_x, σ
    _y, σ_z]  =[  (
    [ 0 1; 1 0 ])  ,(
    [  0 -i;  i  0 ])  , (
    [  1  0;  0 -1 ])  ]  ,

we note that the matrix representation [  ℋ(  t)
]  _ℬ_canonical of the Hamiltonian ℋ
(  t) in Eq. (<ref>) with respect to the canonical
basis ℬ_canonical is given by,

    [  ℋ(  t)  ]  _ℬ_canonical
    =(
    [    a+b_z b_x-ib_y; b_x+ib_y    a-b_z ])  .

For the sake of notational simplicity, we have omitted to make explicit the
time-dependence of the scalar function a=a(  t) and the vector
b⃗=b⃗(  t) in Eq. (<ref>). The Rabi-like
Hamiltonian ℋ^(  full)  (  t) in
Eq. (<ref>) can be rewritten as,

    ℋ^(  full)  (  t)  =-μ⃗
    ·B⃗,

 where the vector B⃗ denotes the external time-dependent magnetic
field with constant magnitude,

    B⃗(  t)  def=(  B_1cos(
    ω t)  , B_1sin(  ω t)  , 
    B_0)  ,

with B_1, B_0, and the angular frequency ω belonging to ℝ
_+\{  0}. The vector operator μ⃗ is the
magnetic moment of the electron,

    μ⃗def=eℏ/2mcσ⃗,

with e being the electric charge of the electron, m is its mass, and c
denotes the speed of light. Finally, the quantity eℏ/(2mc)≃
5.8×10^-9(  eV/gauss) denotes the Bohr magneton.

We point out that ℋ^(  full)
(  t) in Eq. (<ref>) is not only time-dependent but,
in general, the Hamiltonian at two different instances does not commute,

    [  ℋ^(  full)  (  t^')
    , ℋ^(  full)  (  t^'')  ]  ≠0,

where t^'≠ t^''. For instance, taking t^'=0
and t^''=π/(  2ω) and recalling the
commutation relations between the Pauli matrices <cit.>,

    [  σ_i, σ_j]  def=
    σ_iσ_j-σ_jσ_i=2iε_ijkσ
    _k,

we obtain,

    [  ℋ^(  full)  (  0)  ,
    ℋ^(  full)  (  π/2ω)
    ]  =(  eℏ/2mc)  ^22i[  B_1^2
    σ_z-B_0B_1(  σ_x+σ_y)  ]
    ≠0.

Fundamentally, Eq. (<ref>) is a consequence of the fact that in
the original Rabi scenario, although the amplitude of the magnetic field is
stationary, the direction of this field does change in time.



 §.§ Comparison between the Rabi and the Farhi-Gutmann Hamiltonians


Having introduced the magnetic field B⃗ and the magnetic moment
μ⃗, the connection between ℋ^(  full)
(  t) in Eq. (<ref>) and ℋ^(
full)  (  t) in Eq. (<ref>) becomes clear
once one considers the following correspondences,

    ω=ω, E_2=-E_1def=-eℏ/2mcB_0, and Γdef=-eℏ/2mc
    B_1.

A schematic correspondence between selected features that characterize the
quantum dynamics arising from the Rabi and the Farhi-Gutmann Hamiltonians
appear in Table I. In particular, equating the energy gaps and the interaction
strengths in the two frameworks, it follows that

    x=x(  B_0, B_1)  =B_0/B_11/√(1+(  B_0/B_1)  ^2).

A plot of Eq. (<ref>) appears in Fig. 2. Having pointed out these
correspondences, we also remark that ℋ^(  full)
(  t) in Eq. (<ref>) can be written as

    ℋ^(  full)  (  t)  =1/2
    Ω_ℋ(  t)  n̂(  t)  ·σ⃗,

where Ω_ℋ(  t) and n̂(  t)
are given by,

    Ω_ℋ(  t)  def=-eℏ/mc√((  B_0)  ^2+(  B_1)  ^2),

and,

    n̂(  t)  def=1/√((
    B_0)  ^2+(  B_1)  ^2)(  B_1cos(  ω
    t)  , B_1sin(  ω t)  , B_0)
    ,

respectively. Note that Ω_ℋ(  t) in Eq.
(<ref>) does not depend on time while the unit vector n̂(  t) becomes time-independent only if ω is set equal
to zero. In the remainder of this section, in order to facilitate discussion
of the formal comparison between Grover-like time-independent quantum search
Hamiltonians and Rabi-like time-dependent two-levels Hamiltonians, we focus
without any loss of generality on the original Farhi-Gutmann Hamiltonian. In
particular, before comparing the transitions probabilities in Eqs. (<ref>)
and (<ref>), inspired by the Hamiltonian decomposition as presented in Eq.
(<ref>), we note that the original Farhi-Gutmann Hamiltonian
ℋ_FG^(  full) can be formally
written as

    ℋ_FG^(  full)  def
    =E| w⟩⟨ w| +E|
    s⟩⟨ s| =ℋ_FG^(
    free)  +ℋ_FG^(  interaction
    )  ,

where | s⟩def=√(1-x^2
)| r⟩ +x| w⟩. In addition, the
states | r⟩ and | w⟩ replace
the states | E_1⟩ and | E_2⟩, respectively. The Hamiltonians ℋ_FG^(
free) and ℋ_FG^(
interaction) in Eq. (<ref>) are given by,

    ℋ_FG^(  free)  def
    =E(  1-x^2)  | r⟩⟨
    r| +E(  1+x^2)  | w⟩⟨
    w|

and,

    ℋ_FG^(  interaction)  def=Ex√(1-x^2)| r⟩⟨
    w| +Ex√(1-x^2)| w⟩⟨
    r|,

respectively. From Eqs. (<ref>), (<ref>), and (<ref>), we note
that

    E_1def=E(  1-x^2)  , E_2
    def=E(  1+x^2)  >E_1, Γdef=Ex√(1-x^2), and ωdef
    =0.

From Eq. (<ref>), the energy gap Δ Edef=
E_2-E_1=2Ex^2 and, furthermore, the characteristic frequency
ω_21 of the two-level system becomes

    ω_21def=Δ E/ℏ=2Ex^2
    /ℏ.

In particular, using Eqs. (<ref>) and (<ref>) it can be
shown that at resonance, that is when ω=ω_21, the analogue of
Eq. (<ref>) obtained in the original Farhi-Gutmann framework becomes

    t_Rabi^∗=πℏ/2·1/Γ=πℏ/2·(  2mc/| e|ℏ)  ·1/B_1, and 𝒫_max^(
    Rabi)  =1.

Having pointed out these formal correspondences between the two quantum
dynamical scenarios in Eqs. (<ref>) and (<ref>), we conclude by
emphasizing the formally identical periodic oscillatory temporal behavior in
the transition probabilities obtained in Eqs. (<ref>) and (<ref>).



Despite these formal analogies between the Grover-like and the Rabi-like
scenarios, the spin-1/2 particle can be regarded as immersed in a stationary
magnetic field in the former case while it is subject to a time-dependent
magnetic field configuration in the latter one. For this reason, we observe
distinctive features in the two scenarios, both in the mathematical methods
employed (mathematical physics) and in the type of perturbation of the quantum
system (quantum physics). From a mathematical physics standpoint, instead of
using familiar matrix algebra methods and finding an explicit expression for
the unitary temporal evolution operator as we did in the first scenario, we
were compelled to employ the integration of ODEs approach to determine the
probability amplitudes in the second quantum dynamical setting specified by a
non-stationary magnetic field configuration (with constant amplitude but
time-dependent direction) assumed to be known a priori
. Unfortunately, the possibility of solving in an exact analytical
fashion these systems of coupled linear ODEs with time-dependent coefficients
is rather unusual. From a quantum physics standpoint, when transitioning from
the first to the second scenario, we go from a time-independent to a
time-dependent perturbation. In the latter scenario, recalling the
considerations on the link between anticrossing and minimum search time
pointed out at the end of Section I, one can think of changing the
perturbation by suitably varying the magnetic field amplitude and/or its
frequency in order to minimize the search time. Interestingly, the passage
through resonance, either by adiabatically varying the magnetic field
or the frequency, occurs in magnetic resonance phenomena. Thus, motivated by
the remarkable insights presented in Ref. <cit.>, we proposed in
Sections II and III a link between the physics of two-level quantum systems
and analog quantum search algorithms. This link, emphasizing the role played
by perturbations, leads naturally to the consideration of the concept of
adiabaticity in Section IV.

The next section serves as a bridge between Section III and Section
V. In the latter section, we shall investigate the possibility of finding
suitable magnetic field configurations yielding transition probabilities with
a monotonic temporal behavior that one would observe in a quantum dynamical
evolution under a fixed-point quantum search Hamiltonian. In Section IV, we
shall revisit the concepts of adiabatic and nonadiabatic quantum search algorithms.



§ ADIABATIC AND NONADIABATIC QUANTUM SEARCH


The intuitive analogy between the quantum evolution of a spin-1/2 particle
in an external magnetic field and the dynamics arising from a time-dependent
Hamiltonian of a two-level quantum system was originally exploited in the
context of fixed-point adiabatic quantum search by Dalzell, Yoder, and Chuang
in Ref. <cit.>. For the sake of transparency, we point out that no
explicit temporal behavior of transition probabilities was considered by these
authors. On the contrary, such a temporal dependence was investigated in two
dynamical scenarios proposed by Perez and Romanelli in the framework of
nonadiabatic quantum search in Ref. <cit.>. It is worth noting
however, no physical connection between the quantum search Hamiltonians and
the driven time-dependent two-level quantum systems was addressed by these
latter authors.

In what follows, motivated by our physical insights presented in Sections II
and III, we revisit the basic concepts of adiabatic and nonadiabatic quantum
search algorithms. The review of both adiabatic and nonadiabatic quantum
search algorithms together with the material that will be introduced in
Section V will help us establish a physical link between the schedule of a
search algorithm and the control fields that specify a time-dependent driving Hamiltonian.



 §.§ The adiabatic framework


The quantum mechanical evolution in the case of adiabatic quantum search can
be described in terms of a time-dependent Hamiltonian ℋ
_adiabatic(  t) specified by means of a linear
interpolation of two time-independent Hamiltonians <cit.>,

    ℋ_adiabatic(  t)  def=[
    1-s(  t)  ]  ℋ_0+s(  t)
    ℋ_1,

where ℋ_0def=ℐ-|
s⟩⟨ s|, ℋ_1def=ℐ-| w⟩⟨ w|, and ⟨ w|s⟩def=x∈(
0, 1). For the sake of completeness, we remark that there is
no fundamental reason why one could not consider a nonlinear interpolation of
Hamiltonians in Eq. (<ref>). For more details on the possibility of
considering a nonlinear interpolation when ⟨ w|s⟩
=π/2, we refer to Ref. <cit.>. The time-dependent function s(
t) that defines the linear interpolation is a smooth function such
that s(  0)  =0 and s(  T)  =1, with T being the
run time of the search algorithm. Stated otherwise, the functions s(
t) and 1-s(  t) denote a turn-on and a turn-off
function, respectively. The basic idea in the adiabatic quantum search
algorithm can be expressed as follows. Assuming that the ground state of the
Hamiltonian ℋ_1 encodes the solution of the search problem, to
find the target state with large probability, we prepare the system in the
ground state of the Hamiltonian ℋ_0. Then, we change
ℋ_0 adiabatically to ℋ_1 from t=0 to t=T. The
fulfillment of the adiabaticy condition determines the possible temporal
expressions of the function s(  t). In particular, adapting the
evolution rate ds/dt to the local adiabaticity condition

    |ds/dt|≤ε[  λ
    _1(  t)  -λ_0(  t)  ]  ^2
    /⟨λ_1(  t)  |dℋ
    _adiabatic/dt|λ_0(  t)  ⟩,

Roland and Cerf obtained the following optimum expression of s(
t) in Ref. <cit.>,

    s(  t)  =1/2+1/2x/√(1-x^2)tan[
    2ε x√(1-x^2)t-tan^-1(  √(1-x^2)/x)  ]  .

The quantity [  λ_1(  t)  -λ_0(
t)  ] in Eq. (<ref>) is the energy gap between the ground
state |λ_0(  t)  ⟩ and the first
excited state |λ_1(  t)  ⟩ of the
Hamiltonian ℋ_adiabatic(  t) while the
real parameter ε is such that 0<ε≪1. We
remark that the reduced Planck constant ℏ is set equal to one in Eq.
(<ref>) so that ε has the physical dimensions of an energy
with joules equal to seconds^-1 since ℏ=1. Note that imposing
s(  T)  =1 in Eq.(<ref>), in the limit of N approaching
infinity with x=1/√(N) and N being the dimensionality of the search
space, we can obtain the typical Grover-like scaling behavior (for instance,
see the first relation in Eq. (<ref>)),

    Tx≪1≈π/2ε x.

The method used by Dalzell, Yoder, and Chuang to establish whether or not a
given adiabatic quantum search Hamiltonian possesses the fixed-point property
was developed by exploiting a physics intuition. Specifically, they noted that
the quantum search Hamiltonian in Eq. (<ref>) can be formally
viewed as a Hamiltonian describing the quantum evolution of a spin-1/2
particle (an electron, for instance) in an external magnetic field. Then,
using a Bloch sphere geometric description, the evolution of the quantum state
of the system can be described as a spin precessing on the sphere around the
magnetic field vector at a temporal rate proportional to the magnitude of the
field itself. At the beginning, the source state | s⟩
(that is, the spin of the particle at t=0) and the magnetic field vectors
are parallel. At the end of the evolution, while the magnetic field vector is
parallel to the target state | w⟩, the final state
|ψ(  T)  ⟩ at t=T (that is, the spin
of the particle at t=T) is only approximately parallel to |
w⟩. If the search algorithm has a fixed-point property, then
there must exist an upper bound on the angle between |ψ(
T)  ⟩ and | w⟩. The existence of
such an upper bound was estimated, in certain scenarios, by Dalzell, Yoder,
and Chuang in Ref. <cit.> thanks to the above mentioned geometric
description of physical origin together with clever sequences of
time-dependent change of coordinates to study the quantum mechanical
Schrödinger evolution.

We briefly recall that in Grover's original search scheme, the algorithm must
be stopped at a precise instant in order to have a large success probability.
Instead, a search algorithm shows a fixed-point property when the transition
probability from the source state to the target state remains high even when
the algorithm runs longer than necessary. In Ref. <cit.>, the
authors consider algorithms specified by schedules s(  t;
ε, w) parametrized in terms of two real
parameters ε and w. The former parameter quantifies how fast the
interpolation between the two static Hamiltonians occurs, while the
latter is a lower bound on the fraction of target states M/N with w≤
M/N. Observe that M denotes the number of target states while N is the
dimensionality of the Hilbert search space. The algorithm has a Grover-like
scaling if the run time T is T(  w)  =𝒪(
1/√(w)), that is, there exists positive constants c and w_0
such that

    | T(  w)  |≤ c/√(w),

for any w≥ w_0. Furthermore, the algorithm has a fixed-point property
if there exists a function δ(  ε) such that,

    𝒫_| s⟩→| w⟩(  T; ε, w)  ≥1-[  δ(
    ε)  ]  ^2,

for all w≤ M/N and δ(  ε) approaches zero
as ε goes to zero.

Depending on the choice of the schedule s(  t;
ε, w), Dalzell, Yoder, and Chuang were able to find
a variety of types of search algorithms: some algorithms have both Grover-like
scaling and fixed-point property, some have Grover-like scaling but are not
fixed-point, and some are fixed-point but lack Grover-like scaling. For
example, algorithms defined by means of a schedule s(  t;
ε, w) with a temporal rate of change ds/dt
proportional to the second power of the energy gap Δ_w(
s)  def=√(1-4s(  1-s)  (
1-w)  ) like the one in Ref. <cit.> exhibit both a Grover-like
scaling and a fixed-point property. For an overview of the various scenarios
covered in Ref. <cit.>, we refer to Table II.

Interestingly, the imposition of the local adiabaticity condition by Roland
and Cerf together with the idea of using a sequence of time-dependent change
of coordinates to study the Schrödinger evolution as proposed by Dalzell,
Yoder, and Chuang find their inspiration into the physics of magnetic
resonance phenomena. In the former case, the authors were inspired by the
so-called adiabatic first passage method employed in nuclear magnetic
resonance experiments <cit.>. In the latter case, instead, the
authors were influenced by the theoretical methods used to explain the process
of passing through resonance by studying the perturbation in a rotating
coordinate system where it becomes non-resonant <cit.>.



 §.§ The nonadiabatic framework


The quantum mechanical evolution in the case of nonadiabatic quantum search
can be described in terms of a time-dependent Hamiltonian ℋ
_nonadiabatic(  t) as <cit.>,

    ℋ_nonadiabatic(  t)  def
    =f(  t)  ℋ_0+g(  t)  ℋ
    _1,

where ℋ_0def=ℐ-|
s⟩⟨ s|, ℋ_1def=ℐ-| w⟩⟨ w|, and ⟨ w|s⟩def=x∈(
0, 1). Unlike the adiabatic scenario, it is not required in
this case to adiabatically drive the source state (that is, the known ground
state of ℋ_0) into the target state (that is, the unknown ground
state of ℋ_1). For this reason, there is a certain freedom in
choosing the expressions of f(  t) and g(  t)
in Eq. (<ref>). In Ref. <cit.>, Perez and Romanelli
investigated the dynamics generated by two nonadiabatic quantum search
Hamiltonians as in Eq. (<ref>). In the first case, the functions
f(  t) and g(  t) were chosen to be equal to

    f(  t)  def=-1/21/x√(1-x^2
    )|α t+γ|sin(  θ_0+2Ω
    _0t)  ,

and,

    g(  t)  def=-1/21/x√(1-x^2
    )|α t+γ|[  2x√(1-x^2)cos(
    θ_0+2Ω_0t)  +(  1-2x^2)  sin(
    θ_0+2Ω_0t)  ]  ,

respectively. Note that θ(  t)  def=
θ_0+2Ω_0t denotes the mixing angle, γdef
=Ω_ℋ(  0), and αdef
=(  dΩ_ℋ/dt)  _t=0. In this first case, it
was shown in Ref. <cit.> that the transition probability
𝒫_| s⟩→| w⟩(  t) exhibits an oscillatory and periodic temporal behavior
where the amplitude of these oscillations remains constant in time.
Furthermore, the target state is found at a specific instant in time that
shows the typical Grover-like scaling behavior since it is proportional to
√(N)=1/x. In the second algorithm, the functions f(  t)
and g(  t) were imposed to be equal to

    f(  t)  def=1,

and,

    g(  t)  def=1-2x^2+x√(1-x^2)(
    b+ct)  ,

respectively. The two quantities b and c in Eq. (<ref>) are two
parameters that characterize the function g(  t). In this
second case, the transition probability 𝒫_|
s⟩→| w⟩(  t)
exhibits an oscillatory but non-periodic behavior where the amplitude of the
oscillations decreases in time. In other words, although in a
non-monotonic fashion, the search Hamiltonian nevertheless drives the system
towards a fixed point and the asymptotic state at time T∝√(N)=1/x
overlaps with a large probability (although not exactly equal to one) with the
target state. In both scenarios, the precise expressions of the functions
f(  t) and g(  t) were determined by imposing
that the coupled system of linear ODEs with time-dependent coefficients that
arise from the time-dependent Schrödinger equation iℏ∂
_t𝒰(  t)  =ℋ(  t)  𝒰
(  t) leads to analytical solutions. Indeed, the integration
problem reduces in both cases to a Weber-like differential equation whose
general solution is a superposition of parabolic cylinder functions
<cit.>. This type of mathematical scheme was originally employed
by Zener in Ref. <cit.>. For further details on this specific
technical aspect, we refer to Appendix D.

In the next section, in view of the results achieved thus far, we investigate
the possibility of finding suitable magnetic field configurations yielding
transition probabilities with a monotonic temporal behavior that one would
observe in a quantum dynamical evolution under a fixed-point quantum search
Hamiltonian. In addition, we shall discuss the connection between the schedule
of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical
evolutions, and the control fields in a time-dependent driving Hamiltonian.



§ MONOTONIC BEHAVIOR AND TIME-DEPENDENT HAMILTONIANS


So far, we have assumed to know a priori the configuration of the
external time-dependent magnetic field that specifies the external
perturbation of the quantum system. In addition, we have considered quantum
dynamical evolutions causing a periodic oscillatory behavior of the transition
probabilities which was required to single out a specific instant in time in
order to find the target state with certainty. In this section, we depart from
these working conditions in a number of ways. Firstly, since it is challenging
to find the exact analytical solutions to the Schrödinger equation, the
exact configuration of the magnetic field will only be determined a
posteriori. Secondly, we shall be considering quantum dynamical evolutions
yielding transition probabilities that present a temporal monotonic
convergence towards the target state. Specifically, the asymptotic quantum
state can overlap with the target state with a large probability and no
specific instant in time need to be selected in order to find the desired
state with almost certainty. Thus, thanks to the analogies between the
dynamics generated by a time-independent Grover-like quantum search
Hamiltonian and a time-dependent Rabi-like two-level quantum Hamiltonian
presented in Sections II and III, a comparison between the dynamics of the
newly selected time-dependent Hamiltonians and that of quantum search
Hamiltonians with a fixed-point property is brought forward in this section.
Inspired by Messina and collaborators, we propose two novel applications of
the techniques proposed in Refs. <cit.> in order to study
two-level systems in an analytical fashion. This way, we aim to uncover
additional relevant physical insights about quantum search and two-level
quantum systems relying on the work presented in Sections II, III, and IV.

 Assume that the matrix representation with respect to the
computational basis ℬ_canonicaldef
={| w⟩def=1
0, | r⟩def=0
1} of a time-dependent su(  2)-Hamiltonian
ℋ(  t) is given by,

    [  ℋ(  t)  ]  _ℬ_canonical
    def=(
    [   Ω(  t)   ω(  t); ω^∗(  t)  -Ω(  t) ])  ,

where ω(  t) and Ω(  t) in Eq.
(<ref>) denote the so-called complex transverse field and
real longitudinal field, respectively. We convey that longitudinal
fields are oriented along the z-direction while transversal fields
lie in the xy-plane. Observe that, up to a term proportional to the identity
(see Eqs. (<ref>) and (<ref>)), [
ℋ(  t)  ]  _ℬ_canonical
parametrizes the most general form of a (  2×2)-Hermitian
matrix. Moreover, recall that we are interested in computing transition
probabilities which, in turn, are given in terms of the squared modulus of
complex probability amplitudes. Therefore, since the term proportional
to the identity would only generate a time-dependent phase factor, it can
simply be ignored. For the sake of completeness, we also point out that the
reason why we used the su(  2)-terminology is because
the matrix representation of the Hamiltonian in Eq. (<ref>) can be
rewritten in terms of a linear superposition of the three generators {
iσ_x, -iσ_y, iσ_z} of
su(  2), the Lie algebra of the special unitary group
SU(  2). Furthermore, the elements of su(
2) are anti-Hermitian (  2×2)- matrices with
trace zero. In particular, in analogy to Eqs. (<ref>), (<ref>),
and (<ref>), we point out that ℋ(  t) with its
matrix representation in Eq. (<ref>) can be recast as,

    ℋ(  t)  =1/2Ω_ℋ(  t)
    n̂(  t)  ·σ⃗,

with Ω_ℋ(  t) and n̂(  t)
given by,

    Ω_ℋ(  t)  def=2√(|ω(  t)  | ^2+[  Ω(  t)
    ]  ^2),

and,

    n̂(  t)  def=1/Ω_ℋ
    (  t)  (  2|ω(  t)  |cos[  ϕ_ω(  t)  ]  , -2|ω(  t)  |sin[  ϕ_ω(  t)
    ]  , 2Ω(  t)  )  ,

respectively. For further technical details that can be helpful to explain in
more detail what follows in this section, we refer to Appendix E.



 §.§ Applications


In what follows, we consider two quantum dynamical evolution scenarios
specified by suitable time-dependent magnetic field configurations that
generate transition probabilities exhibiting a temporal monotonic convergence
from the source to the target states. In the first scenario, we assume that
the phase ϕ_ω(  t) and the magnitude |ω(  t)  | of the transverse field ω(
t) are interconnected. In the second scenario, although keeping the
same formal expression of ω(  t), we do not assume any
connection between the phase and the intensity of the complex
transverse field.



  §.§.§ Interconnection between phase and magnitude of the transverse
field


In this first scenario, assume that the complex auxiliary function
X(  t) is given by

    X(  t)  def=csin[  ϕ(  t)
    ]  e^iϕ(  t)  ,

with ϕ(  t)  ∈


ℝ, c∈


ℝ, and ϕ(  0)  =0. Furthermore, assume that the real
longitudinal field Ω(  t) is fixed and the complex
transverse field ω(  t) is given by,

    ω(  t)  def=ω_0e^-ξ t
    e^iϕ_ω(  t)  ,

with known ω_0>0 and ξ>0 (that is, only |ω(
t)  | is fixed a priori). Use of the condition
ω(  t)  =α^2(  t)  Ẋ(  t) and Eq. (<ref>) imply that if one imposes the working assumption that
the a priori unknown phase ϕ_ω(  t) in Eq.
(<ref>) is such that <cit.>,

    |ω(  t)  |/c=Ω(
    t)  /ℏ+ϕ̇_ω(  t)  /2,

an exact analytical solution to the Schrödinger evolution can be found.
Equation (<ref>) is the so-called generalized time-dependent out of
resonance condition in a generalized Rabi-like scenario
<cit.>. Furthermore, having fixed |ω(  t)  |, Ω(  t), and
X(  t), the phase ϕ_ω(  t) that
appears in Eq. (<ref>) can be obtained by integrating with respect to
time Eq. (<ref>). Taking into account these working conditions and
following the analysis reported in the previous subsection, it can readily be
shown that the complex probability amplitudes α(  t) and β(  t) are given by

    α(  t)  =√(ℏ^2+c^2cos^2[
    Φ(  t)  ]  /ℏ^2+c^2)exp{  i[
    ϕ_ω(  t)  /2-tan^-1(  ℏ/√(ℏ^2+c^2)tan[  Φ(  t)  ]  )
    ]  },

and,

    β(  t)  =c/√(ℏ^2+c^2)sin[
    Φ(  t)  ]  exp{  i[  ϕ_ω(  t)  /2-π/2]  },

respectively. The real function Φ(  t) in Eqs.
(<ref>) and (<ref>) is defined as,

    Φ(  t)  def=√(ℏ^2+c^2)
    /ℏ c∫_0^t|ω(  t^')
    | dt^'.

Using Eqs. (<ref>), (<ref>), (<ref>) and, in addition, noting
that

    α(  t)  β^∗(  t)  +α^∗(
    t)  β(  t)  =2|α(  t)
    | ^2/ℏcsin[  ϕ(  t)  ]  ,

the transition probability 𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>)
becomes,

    𝒫_| s⟩→| w⟩(  t)  =|α(  t)  | ^2
    x^2+|β(  t)  | ^2(  1-x^2
    )  +2|α(  t)  | ^2/ℏcsin[  ϕ(  t)  ]  x√(1-x^2).

The real phase ϕ(  t) in Eq. (<ref>) is defined
as,

    ϕ(  t)  def=tan^-1{ℏ/√(ℏ^2+c^2)tan[  Φ(  t)  ]
    }.

Using Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), the
squared modulus quantities |α(  t)  |
^2 and |β(  t)  | ^2 become

    |α(  t)  | ^2=1/ℏ^2
    +c^2{ℏ^2+c^2cos^2[  √(ℏ^2
    +c^2)/ℏ cω_0/ξ(  1-e^-ξ t)  ]
    },

and,

    |β(  t)  | ^2=c^2/ℏ
    ^2+c^2sin^2[  √(ℏ^2+c^2)/ℏ c
    ω_0/ξ(  1-e^-ξ t)  ]  ,

respectively. Observe that at the resonance (that is, when c approaches
infinity in Eq. (<ref>)) and in the asymptotic limit of time approaching
infinity, |α(  t)  | ^2 and
|β(  t)  | ^2 approach zero and one,
respectively, provided that the coefficient ξ is defined as

    ξdef=2/(  2n+1)  π√(ℏ^2+c^2)/cω_0/ℏ,

where n∈


ℤ. To obtain a monotonic convergent behavior without any transient
oscillation, we impose n=0 in Eq. (<ref>). For n∈


ℤ
\{  0}, the convergence is non-monotonic since it
appears only after some transient oscillations. Furthermore, note that
csin[  ϕ(  t)  ] in the expression of
𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>) is given by

    csin[  ϕ(  t)  ]  =ctan[  ϕ(
    t)  ]  /√(1+tan^2[  ϕ(  t)  ]  )
    ,

where,

    tan[  ϕ(  t)  ]  =ℏ/√(ℏ
    ^2+c^2)tan[  √(ℏ^2+c^2)/ℏ cω_0/ξ(  1-e^-ξ t)  ]  .

Therefore, using Eqs. (<ref>), (<ref>) and (<ref>), in the
asymptotic limit of time approaching infinity at the resonance, we find that
csin[  ϕ(  t)  ] approaches zero. 

Finally, we can conclude that 𝒫_|
s⟩→| w⟩(  t) in
Eq. (<ref>) asymptotically approaches the limiting value of 1-x^2
under the above mentioned working conditions. In the additional working
assumption of very small x, one can achieve a large numerical value of the
transition probability 𝒫_| s⟩→| w⟩ that, ideally, can approach unity. A plot that
exhibits the monotonic temporal behavior of the transition probabilities
𝒫_| s⟩→| w⟩(  t) and 𝒫_| s⟩→| r⟩(  t) at the generalized
Rabi resonance where c approaches infinity with x=0 and ξ=1 appears in
Fig. 3.



  §.§.§ Disconnection between phase and magnitude of the transverse
field


In this second illustrative example, we assume that the complex
auxiliary function is given by

    X(  t)  def=A(  t)  e^iϕ(
    t)  ,

with A(  t)  ∈


ℝ, ϕ(  t)  ∈


ℝ, and A(  0)  =0. Furthermore, assume that the complex
transverse field ω(  t) is fixed and defined as,

    ω(  t)  =ω_0e^-ξ te^iϕ_ω(
    t)  ,

with known ω_0>0 and ξ>0 (that is, both |ω(
t)  | and ϕ_ω(  t) are fixed
a priori). Unlike the previous application, the phase ϕ_ω(  t) can be arbitrary now. In this case, using ω(
t)  =α^2(  t)  Ẋ(  t) and Eq.
(<ref>) together with defining a real function Θ(
t) with Θ(  0)  =0 as

    Θ(  t)  def=tan^-1(  Ȧ
    /Aϕ̇)  ,

it happens to be possible to compute exactly the probability amplitudes
α(  t) and β(  t) provided that the
real longitudinal field Ω(  t) satisfies the
following relation <cit.>,

    Ω(  t)  =ℏ/2[  Θ̇(
    t)  -ϕ̇_ω(  t)  ]  +|ω(  t)  |sin[  Θ(  t)
    ]  [  2/ℏ∫_0^t|ω(
    t^')  |cos[  Θ(  t^')
    ]  dt^']  .

In particular, following the line of reasoning reported in the previous
subsection, the complex probability amplitudes α(  t) and β(  t) become,

    α(  t)  =cos[  1/ℏ∫_0^t|ω(  t^')  |cos[  Θ(
    t^')  ]  dt^']  ·exp[  -i∫
    _0^t(  Θ̇(  t^')  -ϕ̇_ω(  t^')  /2+|ω(
    t^')  |sin[  Θ(  t^')
    ]  /ℏsin[  2/ℏ∫_0^t^'|ω(  t^'')  |cos[  Θ(
    t^'')  ]  dt^'']  )
    dt^']  ,

and

    β(  t)  =1/iℏα(  t)  A(
    t)  e^iϕ(  t)  ,

respectively. Furthermore, the quantities A(  t) and
ϕ(  t) in Eq. (<ref>) are given by,

    A(  t)  def=ℏtan[  1/ℏ∫_0^t|ω(  t^')  |cos[  Θ(  t^')  ]  dt^']
    ,

and

    ϕ(  t)  def=∫_0^t2|ω(  t^')  |/ℏsin[
    Θ(  t^')  ]  /sin[  2/ℏ
    ∫_0^t^'|ω(  t^'')
    |cos[  Θ(  t^'')  ]
    dt^'']  dt^',

respectively. In what follows, we assume that Θ(  t) is
defined as

    cos[  Θ(  t)  ]  def=cosh
    ^-2(  ξ t)  ,

where the coefficient ξ in Eq. (<ref>) plays the same role
covered by ξ that was originally introduced in Eq. (<ref>).
Using Eqs. (<ref>), (<ref>), and (<ref>), after
some straightforward but tedious algebra, we determine that the squared
modulus quantities |α(  t)  | ^2 and
|β(  t)  | ^2 are given by

    |α(  t)  | ^2=cos^2{2ω_0/ℏξ[  e^ξ t/1+e^2ξ t-tan
    ^-1(  e^-ξ t)  +(  π/4-1/2)
    ]  },

and

    |β(  t)  | ^2=sin^2{2ω_0/ℏξ[  e^ξ t/1+e^2ξ t-tan
    ^-1(  e^-ξ t)  +(  π/4-1/2)
    ]  },

respectively. Observe that in the asymptotic limit of time approaching
infinity, |α(  t)  | ^2 and
|β(  t)  | ^2 approach zero and one in
a strictly monotonic fashion, respectively, provided that the coefficient
ξ in Eq. (<ref>) is defined as

    ξdef=4/πω_0/ℏ(
    π/4-1/2)  .

Using Eqs. (<ref>), (<ref>), (<ref>) and, in
addition, noting that

    α(  t)  β^∗(  t)  +α^∗(
    t)  β(  t)  =2|α(  t)
    | ^2/ℏA(  t)  sin[  ϕ(  t)
    ]  ,

the transition probability 𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>)
becomes,

    𝒫_| s⟩→| w⟩(  t)  =|α(  t)  | ^2
    x^2+|β(  t)  | ^2(  1-x^2
    )  +2|α(  t)  | ^2/ℏA(  t)  sin[  ϕ(  t)  ]  x√(1-x^2
    ).

Noting that |α(  t)  | ^2A(
t) approaches zero as time approaches infinity, we conclude that
𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>) asymptotically approaches the limiting
value of 1-x^2 under the above mentioned working conditions.

In analogy to the previous application, under the additional
working assumption of very small x, one can achieve a large numerical value
of the transition probability 𝒫_| s⟩→| w⟩ that, ideally, can approach unity. A
plot that illustrates the monotonic temporal behavior of the transition
probabilities 𝒫_| s⟩→|
w⟩(  t) and 𝒫_|
s⟩→| r⟩(  t) with
x=0 and ξ=1 appears in Fig. 4. For the sake of completeness, we finally
point out that the explicit expression for the longitudinal field
Ω(  t) in Eq. (<ref>) can be computed by
means of Eqs. (<ref>) and (<ref>) where the phase
ϕ_ω(  t) can be chosen arbitrarily.



 §.§ Link with adiabatic and nonadiabatic search Hamiltonians


In the adiabatic case, it is possible to express the schedule s(
t) of a search algorithm in terms of the ratio between the transverse
and longitudinal fields ω(  t) and Ω(
t), respectively. Indeed, using Eqs. (<ref>) and
(<ref>), it follows that

    s(  t)  =s(  ω(  t)  , Ω(
    t)  )  =2x√(1-x^2)-ω(  t)
    /Ω(  t)  (  1-2x^2)  /2x√(1-x^2)
    -2ω(  t)  /Ω(  t)  (
    1-x^2)  .

From Eq. (<ref>), it is evident that the schedule function s(
t) that appears in the Hamiltonian ℋ_adiabatic
(  t) depends on the ratio between the fields ω(
t) and Ω(  t) when the adiabatic search evolution
is regarded as the quantum mechanical evolution of an electron immersed in a
time-dependent external magnetic field. In particular, the sign of the speed
ds/dt of the schedule s(  t) is determined by the rate of
change in time of the ratio ω(  t)  /Ω(  t) since

    ds/dt=2x√(1-x^2)/[  2x√(1-x^2)-2ω(  t)  /Ω(  t)  (  1-x^2)
    ]  ^2·d/dt(  ω(  t)  /Ω(  t)  )  .

We point out that in order to satisfy the boundary conditions
s(  0)  =0 and s(  T)  =1, it is necessary that
ω(  0)  =-x√(1-x^2), ω(  T)  =0,
Ω(  0)  =-(  1/2)  (  1-2x^2), and
Ω(  T)  =1/2. For example, in the case of the Roland-Cerf
schedule in Eq. (<ref>), the transverse field ω(  t)
can be written as,

    ω(  t)  =1/2{  x^2tan[  2ε
    x√(1-x^2)t-tan^-1(  √(1-x^2)/x)  ]
    -x√(1-x^2)},

where the run time T equals

    T=1/ε1/x√(1-x^2)tan^-1(  √(1-x^2)/x)  .

In the nonadiabatic case, using Eqs. (<ref>) and (<ref>),
it follows that the analogue of Eq. (<ref>) becomes

    ω(  t)  /Ω(  t)  =2f(
    t)  x√(1-x^2)/f(  t)  (  1-2x^2)
    -g(  t)  ,

that is, after some algebra,

    g(  t)  =[  (  1-2x^2)  -2Ω(
    t)  /ω(  t)  x√(1-x^2)]  f(
    t)  .

Interestingly, setting f(  t)  =1, x=1/√(N), and
Ω(  t)  /ω(  t)  =at+b with a and b in ℝ, Eq. (<ref>) reduces to Eq.(<ref>), the latter being
introduced without any particular physical motivation by Perez and Romanelli
in Ref. <cit.>. From Eq. (<ref>), it becomes clear that
once the function f(  t) is chosen to be constant, in order to
have the simplest time-dependent nonadiabatic search Hamiltonian
ℋ_nonadiabatic(  t), the function g(
t) must depend linearly on time. For the sake of completeness, we
emphasize that the expressions of f(  t) and g(
t) in Eqs. (<ref>) and (<ref>), respectively, were chosen by
Perez and Romanelli by imposing that Ω_ℋ(  t)
in Eq.(<ref>) was linear in time. As a final remark, we point out that
if we set f(  t)  +g(  t) equal to zero, from Eq.
(<ref>) it is evident that the ratio ω(  t)
/Ω(  t) becomes constant and equal to

    ω(  t)  /Ω(  t)  =x/√(1-x^2).

Furthermore, requiring that the field ω(  t) be
real so that its phase ϕ_ω(  t) is equal to
zero in Eq. (<ref>), the generalized out of resonance condition becomes

    ω(  t)  /Ω(  t)  =c/ℏ.

Finally, being within the framework of nonadiabatic quantum search, we are
able to find an interpretation of the parameter c that originally appears in
Eq. (<ref>) in terms of the quantum mechanical overlap xdef=⟨ w|x⟩ as c(  x)  =ℏ
x/√(1-x^2) by making use of Eqs. (<ref>) and (<ref>). In
conclusion, our application of the mathematical methods introduced in Refs.
<cit.> to quantum search problems is useful for a number
of reasons. First, it helps in finding search Hamiltonians yielding
exact analytical expressions of transition probabilities from a source state
to a target state with strictly monotonic behavior as evident from Eqs.
(<ref>), (<ref>), (<ref>), and (<ref>). Second, it enhances the
physical interpretation of the schedule functions as is evident from Eqs.
(<ref>) and (<ref>). Finally, it helps in clarifying the
meaning of mathematically introduced quantities, such as the parameter c, by
linking them to geometrical quantities with a clear physical significance such
as the quantum mechanical overlap x.



§ FINAL REMARKS


In this paper, we investigated the connection between analog quantum search
and two-level quantum systems. A number of quantitative results together with
some insightful remarks were found. Our main findings can be outlined in some
detail as follows.



  * First, we characterized in a quantitative manner the intuitive analogy
between a Grover-like quantum search dynamics defined in terms of the
time-independent Hamiltonian in Eq. (<ref>) and a Rabi-like quantum
mechanical evolution of a spin-1/2 particle in an external time-dependent
magnetic field specified by the time-dependent Hamiltonian in Eq.
(<ref>). This task was achieved in two steps. In the first step, we
presented a detailed and exact analytical computation of the transition
probabilities from a source state to a target state in both scenarios as
reported in Eqs. (<ref>) (for details, see Appendix A) and (<ref>) (for
details, see Appendix C). A plot exhibiting the characteristic periodic
oscillatory behavior of the transition probabilities in the Rabi-like case
appears in Fig. 1. In the second step, we provided a schematic correspondence
between relevant physical quantities in the two scenarios, including the
energy gaps and the interaction strengths. Such a correspondence appears in
Table I and, in particular, enabled us to physically interpret the quantum
mechanical overlap between the source state and the target state in the
Grover-like scenario in terms of the ratio of magnetic field intensities in
the Rabi-like framework as evident from Eq. (<ref>) and Fig. 2. This
first set of results, motivated by the insightful remarks appearing in Refs.
<cit.>, are especially interesting since they quantify for
the first time in a detailed and exact analytical fashion the physical link
between Grover-like quantum algorithms and Rabi oscillations originating from
a time-dependent two-level quantum system Hamiltonian.


  * Second, to further advance our understanding of this physical link
between quantum search problems and two-level quantum systems dynamics
yielding transition probabilities with periodic oscillatory temporal behavior,
we explored the possibility of extending such a physical connection to
fixed-point quantum algorithms specified by a monotonic convergence towards
the target state. To accomplish this task, we first revisited the concepts of
both adiabatic and nonadiabatic quantum search algorithms characterized,
unlike the original time-independent Grover-like search Hamiltonians, by
time-dependent search Hamiltonians. In the adiabatic case, we focused
primarily on the role played by the rate of change in time of the
schedule s(  t) of the algorithm in determining its
Grover-like scaling behavior and/or its fixed-point property feature. A
summary of this specific review appears in Table II. In the nonadiabatic case,
instead, we attempted to determine the motivation behind the choice of
particular temporal expressions of the search Hamiltonians as in Eq.
(<ref>) in terms of their schedule functions that appear in Eqs.
(<ref>), (<ref>), (<ref>), and (<ref>). Our Section IV
constitutes an original and unifying critical reconsideration of both
adiabatic and nonadiabatic quantum search Hamiltonians as originally presented
in Refs. <cit.> in terms of the schedule of the algorithm.


  * Third, we exploited recent mathematical techniques developed within the
framework of exactly solvable two-level time-dependent su(
2)-Hamiltonian models in order to provide two exact analytical
expressions of transition probabilities exhibiting monotonic convergence from
the source state to the target state. The outcomes of our first application
appear in Eqs. (<ref>) and (<ref>) and are displayed in Fig. 3.
Furthermore, the findings of our second application are reported in Eqs.
(<ref>) and (<ref>) and are illustrated in Fig. 4. More interestingly,
thanks to the reformulation of the adiabatic and nonadiabatic search
Hamiltonians in Eqs. (<ref>) and (<ref>),
respectively, in terms of the su(  2)-Hamiltonian model
in Eq. (<ref>), we uncovered some relevant insights. In the adiabatic
case, we were able to express the schedule and its speed in terms of the ratio
of the transverse and longitudinal fields that define the particular
time-dependent magnetic field configuration in which the spin-1/2 particle
is immersed. These results appear in Eqs. (<ref>) and (<ref>).
In the nonadiabatic case, we found a compact general relation between the two
schedule functions f(  t) and g(  t) where,
again, the ratio between the longitudinal and the transverse fields emerges.
This relation appears in Eq. (<ref>) and, interestingly, coincides
with the physically unmotivated choice that appears in Eq. (<ref>).
Finally, we were able to assign some clear interpretation in terms of
geometric quantities with physical meaning to certain auxiliary mathematical
parameters that enter the above mentioned mathematical techniques. A
particular case of such an assignment appears by combining Eqs. (<ref>)
and (<ref>). The importance of our third set of results is threefold.
First, we extend the application of recently introduced mathematical
techniques in Refs. <cit.> to quantum search
problems. Second, we were able to give a physical interpretation of
the schedule function in terms of control magnetic fields. Finally, we
extended the physical interpretation of fixed-point quantum search algorithms
in terms of the non-relativistic quantum mechanical evolution of an electron
immersed in specific time-dependent magnetic field configuration.


The main physical insights that emerge from our investigation can be
summarized as follows. A quantum search Hamiltonian can be regarded as a
superposition of an unperturbed Hamiltonian and a perturbation (see Section
III). In particular, the minimum search time appears to be inversely
proportional the energy level separation defined in terms of the eigenvalues
of the (total) Hamiltonian. In principle, by modifying the perturbation, one
can change the minimum search time. The perturbation can be time-independent
or time-dependent. In the case of a time-independent perturbation (see Section
II), the constant perturbation can be characterized by a number of parameters.
By suitably tuning these parameters, one can minimize the minimum search time
or, alternatively, minimize the minimum time to achieve a nearly optimal
success probability value. The same reasoning can be extended to the
time-dependent scenario where, however, a richer realm of possibilities can be
explored. For instance, one can observe relevant dynamical features of a
quantum system by varying the external perturbation in an adiabatic or
nonadiabatic fashion (see Section IV). The role played by the time-dependent
schedules of the search algorithms in both adiabatic and nonadiabatic settings
can be described in terms of a time-dependent perturbation. Finally,
expressing the perturbation in terms of time-dependent transverse and
longitudinal fields, one can suitably choose one of these two fields
(and, unfortunately, determine the other one only a posteriori) in such
a manner that a desired temporal behavior of the transition probability from
the source state to the target state is recovered (see Section V).

Despite our serious effort to propose a unifying perspective on quantum search
algorithms in terms of the physics of two-state quantum systems, our work is
not free of limitations. For instance, a clear limitation of our analysis is
that we were unable to predict the exact or, for that matter, guess an
approximate analytical temporal behavior of the transition probability from
the source state to the target state for an a priori
chosen time-dependent magnetic field configuration. However, the insights
that emerge from our work open up the possibility of exploring a number of
intriguing questions. For instance, given our enhanced understanding of the
role played by the schedule of a search algorithm, we have reason to believe
that the work presented in this paper will help further enhance the
comprehension of our recent information geometric analysis of quantum
algorithms viewed from a statistical thermodynamics standpoint as proposed in
Ref. <cit.>. More generally, given that the phenomenon of anticrossing
is controlled by the parameters behind the perturbation and results into
lowering of energy, it would be worthwhile exploring in a more detailed manner
the link between minimum search time and avoided crossing in the hope of
finding out an optimal tradeoff between speed (minimum search time), fidelity
(success probability), and energy minimization requirements in actual physical
implementations of quantum search algorithms <cit.>. We leave these
intriguing investigations to future efforts.


C. C. is grateful to the United States Air Force Research Laboratory (AFRL)
Summer Faculty Fellowship Program for providing support for this work. Any
opinions, findings and conclusions or recommendations expressed in this
manuscript are those of the authors and do not necessarily reflect the views
of AFRL. Finally, constructive criticism from an anonymous referee
leading to an improved version of this manuscript are sincerely acknowledged
by the authors.


99                                                                                               


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§ DERIVATION OF EQ. (<REF>)


In this Appendix, we derive Eq. (<ref>) that appears in Section II. Note
that the matrix representation of the Hamiltonian ℋ_GQS
in Eq. (<ref>) with respect to the orthonormal basis {| w⟩, | r⟩} where
⟨ w|r⟩ =δ_wr, with δ_wr denoting the
Kronecker delta, is given by

    [  ℋ_GQS]  _{| w⟩, | r⟩}def=(
    [ ⟨ w|ℋ_GQS|w⟩ ⟨
    w|ℋ_GQS|r⟩; ⟨ r|ℋ_GQS|w⟩ ⟨
    r|ℋ_GQS|r⟩ ])  .

Using the above mentioned orthonormality conditions ⟨
w|r⟩ =δ_wr together with Eqs. (<ref>) and
(<ref>), we obtain

    [  ℋ_GQS]  _{| w⟩, | r⟩}=(
    [ h_11 h_12; h_21 h_22 ])  ,

where,

    h_11def=E[  α+(  β+γ)
    x+δ x^2]  , h_12def=E√(1-x^2
    )(  β+δ x)  ,
       
         h_21def=E√(1-x^2)(  γ+δ x)
    , h_22def=Eδ(  1-x^2)  .

Observe that ℋ_GQS is an Hermitian operator. Therefore,
imposing the condition that ℋ_GQS=ℋ_GQS
^† where the dagger symbol “†
” denotes the usual Hermitian conjugation operation,

    (
    [ h_11 h_12; h_21 h_22 ])  =(
    [ h_11^∗ h_21^∗; h_12^∗ h_22^∗ ])  ,

it follows that α and δ must be real coefficients while
β=γ^∗. The symbol “∗”
 in Eq. (<ref>) denotes the usual complex conjugation operation.
Let us proceed to diagonalize the Hermitian matrix [  ℋ
_GQS]  _{| w⟩, |
r⟩} in Eq. (<ref>). The two real
eigenvalues λ_± of the matrix are given by,

    λ_±def=1/2[  (  h_11
    +h_22)  ±√((  h_11-h_22)  ^2+4h_12h_21
    )]  .

The eigenspaces ℰ_λ_- and ℰ_λ_+
that correspond to the eigenvalues λ_- and λ_+ are given
by

    ℰ_λ_-def=Span{|
    v_λ_-⟩}, and ℰ_λ_+
    def=Span{| v_λ_+
    ⟩},

respectively. The two eigenvectors | v_λ_-⟩
and | v_λ_+⟩ corresponding to λ_+
and λ_- can be written as

    | v_λ_+⟩def=(
    [ 1/2h_21[  (  h_11-h_22)  +√((
    h_11-h_22)  ^2+4h_12h_21)];                                                        1 ])  ,

and,

    | v_λ_-⟩def=(
    [ 1/2h_21[  (  h_11-h_22)  -√((
    h_11-h_22)  ^2+4h_12h_21)];                                                        1 ])  ,

respectively. For ease of notational simplicity, let us define two
complex quantities A and B as follows

    Adef=1/2h_21[  (  h_11-h_22)
    -√((  h_11-h_22)  ^2+4h_12h_21)]  ,

and,

    Bdef=1/2h_21[  (  h_11-h_22)
    +√((  h_11-h_22)  ^2+4h_12h_21)]  .

Employing Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), the
eigenvector matrix ℳ_ℋ_GQS and its inverse
ℳ_ℋ_GQS^-1 corresponding to the Hamiltonian
matrix [  ℋ_GQS]  _{|
w⟩, | r⟩} are given by,

    ℳ_ℋ_GQSdef=(
    [ A B; 1 1 ])  ,

and,

    ℳ_ℋ_GQS^-1def=1/A-B(
    [  1 -B; -1  A ])  =ℳ_ℋ_GQS^†,

respectively. In terms of the matrices ℳ_ℋ_GQS, ℳ_ℋ_GQS^-1, and a diagonal matrix
H_GQS-diagonal, the matrix [  ℋ_GQS
]  _{| w⟩, |
r⟩} in Eq. (<ref>) can be expressed as

    [  ℋ_GQS]  _{| w⟩, | r⟩}=ℳ_ℋ
    _GQSH_GQS-diagonalℳ_ℋ_GQS
    ^-1=(
    [ A B; 1 1 ])  (
    [ λ_-   0;   0 λ_+ ])  (
    [  1/A-B -B/A-B; -1/A-B  A/A-B ])  ,

where H_GQS-diagonal is given by,

    H_GQS-diagonaldef=[  ℋ_GQS
    ]  _{| v_λ_-⟩,
    | v_λ_+⟩}=(
    [    ⟨ v_λ-|ℋ_GQS|v_λ-⟩  ⟨ v_λ-|ℋ_GQS|v_λ_+
    ⟩;  ⟨ v_λ_+|ℋ_GQS|v_λ
    -⟩ ⟨ v_λ_+|ℋ_GQS
    |v_λ_+⟩ ])  =(
    [ λ_-   0;   0 λ_+ ])  .

The eigenvalues in Eq. (<ref>) are defined in Eq. (<ref>)
while A and B are given in Eqs. (<ref>) and (<ref>),
respectively. At this juncture, we recall that our goal is to compute the time
t^∗ such that 𝒫_| s⟩→| w⟩(  t^∗)  =𝒫_max where
the transition probability 𝒫_| s⟩→| w⟩(  t) is defined in Eq.
(<ref>). By means of standard matrix algebra methods, 𝒫
_| s⟩→| w⟩(
t) can be recast as

    𝒫_| s⟩→| w⟩(  t)  def=|⟨
    w|e^-i/ℏℋ_GQSt|s⟩|
    ^2=|⟨ w|ℳ_ℋ_GQS
    e^-i/ℏH_GQS-diagonalℳ_ℋ
    _GQS^†|s⟩| ^2.

Using the matrix notation with components expressed relative to the
orthonormal basis {| w⟩, |
r⟩}, states | w⟩ and
| s⟩ are given by

    | w⟩def=(
    [ 1; 0 ])  , and def| s⟩ =(
    
    [        x; √(1-x^2) ])  ,

respectively. Using Eqs. (<ref>), (<ref>), and
(<ref>), the quantum state amplitude ⟨ w|e^-i/ℏℋ_GQSt|s⟩ that appears in the
expression of the fidelity 𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>)
becomes

    ⟨ w|e^-i/ℏℋ_GQSt|s⟩
    =1/A-B[  Ae^-i/ℏλ_-t(
    x-B√(1-x^2))  -Be^-i/ℏλ_+t(
    x-A√(1-x^2))  ]  ,

and, consequently, its complex conjugate ⟨ w|e^-i/ℏℋ_GQSt|s⟩ ^∗ is given by,

    ⟨ w|e^-i/ℏℋ_GQSt|s⟩
    ^∗=1/A-B[  Ae^i/ℏλ_-t(
    x-B√(1-x^2))  -Be^i/ℏλ_+t(
    x-A√(1-x^2))  ]  .

Note that,

    e^-i/ℏλ_-t=e^-i/ℏh_11+h_22
    /2te^ia/ℏt and, e^-i/ℏλ_+
    t=e^-i/ℏh_11+h_22/2te^-ia/ℏ
    t.

Recalling Eq. (<ref>), the real quantity a in Eq. (<ref>)
is defined as

    adef=1/2√((  h_11-h_22)
    ^2+4h_12h_21).

Using Eq. (<ref>), the complex probability amplitudes in Eqs.
(<ref>) and (<ref>) become

    ⟨ w|e^-i/ℏℋ_GQSt|s⟩
    =e^-i/ℏh_11+h_22/2t[  A(
    x-B√(1-x^2))  /A-Be^ia/ℏt-B(
    x-A√(1-x^2))  /A-Be^-ia/ℏt]  ,

and,

    ⟨ w|e^-i/ℏℋ_GQSt|s⟩
    ^∗=e^i/ℏh_11+h_22/2t[  A^∗(  x-B^∗√(1-x^2))  /A^∗-B^∗e^-ia/ℏt-B^∗(  x-A^∗√(1-x^2))
    /A^∗-B^∗e^ia/ℏt]  ,

respectively. Using Eqs. (<ref>) and (<ref>) and introducing the
following three quantities

    Ãdef=A(  x-B√(1-x^2))
    /A-B, B̃def=-B(  x-A√(1-x^2))  /A-B, and α̃=a/ℏt,

the transition probability 𝒫_| s⟩→| w⟩(  t) in Eq.
(<ref>) becomes

    𝒫_| s⟩→| w⟩(  α̃)  =[  Ãe^iα̃
    +B̃e^-iα̃]  [  Ã^∗e^-iα̃+B̃^∗e^iα̃]  =|Ã| ^2+|B̃| ^2+2ÃB̃^∗cos(  2α̃)  ,

where we point out that ÃB̃^∗ is real since
h_12=h_21^∗. By exploiting standard trigonometric relations in a
clever sequential order, we obtain

    𝒫_| s⟩→| w⟩(  α̃)        =|Ã|
    ^2+|B̃| ^2+2ÃB̃^∗
    cos(  2α̃) 
         =|Ã| ^2+|B̃|
    ^2+2ÃB̃^∗[  cos^2(  α̃)
    -sin^2(  α̃)  ] 
         =|Ã| ^2+|B̃|
    ^2+2ÃB̃^∗cos^2(  α̃)
    -2ÃB̃^∗sin^2(  α̃) 
         =|Ã| ^2sin^2(  α̃)
    +|Ã| ^2cos^2(  α̃)
    +|B̃| ^2sin^2(  α̃)
    +|B̃| ^2cos^2(  α̃)
    +2ÃB̃^∗cos^2(  α̃)  -2ÃB̃^∗sin^2(  α̃) 
         =(  |Ã| ^2+|B̃| ^2-2ÃB̃^∗)  sin^2(
    α̃)  +(  Ã^2+B̃^2+2Ã
    B̃^∗)  cos^2(  α̃) 
         =|Ã-B̃| ^2sin^2(
    α̃)  +|Ã+B̃| ^2cos
    ^2(  α̃)  ,

that is,

    𝒫_| s⟩→| w⟩(  α̃)  =|Ã-B̃|
    ^2sin^2(  α̃)  +|Ã+B̃| ^2cos^2(  α̃)  .

Finally, using Eqs. (<ref>), (<ref>), (<ref>), and
(<ref>), 𝒫_| s⟩→|
w⟩(  α̃) in Eq. (<ref>) becomes

    𝒫_| s⟩→| w⟩(  t)  =x^2cos^2(  √(h_12h_21/ℏ
    ^2+(  h_11-h_22)  ^2/4ℏ^2)t)
    +|1/2h_11-h_22/ℏx+h_12
    /ℏ√(1-x^2)| ^2/h_12h_21/ℏ^2
    +(  h_11-h_22)  ^2/4ℏ^2sin^2(
    √(h_12h_21/ℏ^2+(  h_11-h_22)
    ^2/4ℏ^2)t)  .

Appendix A ends with the derivation of the exact temporal behavior of
𝒫_| s⟩→| w⟩(  t) in Eq. (<ref>).



§ TIME-DEPENDENT PERTURBATION


In this Appendix, we briefly review the concept of interaction representation
in the context of time-dependent perturbations. Furthermore, we comment on the
complexity of finding quantum mechanical probability amplitudes in an exact
analytical manner when the perturbation depends on time. The material
presented in this Appendix can be helpful to further clarify our work in
Section III.

Assume that a two-state quantum system is subject to a time-dependent
Hamiltonian evolution specified by ℋ^(  full)
(  t),

    ℋ^(  full)  (  t)  def=ℋ_0^(  free)  +𝒱
    ^(  interaction)  (  t)  ,

where ℋ_0^(  free) denotes the
time-independent free Hamiltonian while 𝒱^(
interaction)  (  t) is the time-dependent
interaction potential. Furthermore, let ℋ_0^(
free)  | E_k⟩ =E_k|
E_k⟩ for any 1≤ k≤2. The energy eigenstates {| E_k⟩} are such that ⟨
E_n|E_m⟩ =δ_nm for any 1≤ n, m≤2. Then, the
temporal evolution of a quantum state | s⟩def=| s(  0)  ⟩,

    | s(  0)  ⟩def=
    
    
    ∑_k=1^2
    c_k(  0)  | E_k⟩,

can be described as,

    | s(  t)  ⟩def=
    
    
    ∑_k=1^2
    c_k(  t)  e^-i/ℏE_kt| E_k
    ⟩.

From Eq. (<ref>), observe that the presence of a nonzero interaction
potential implies that the coefficients c_k(  t) are
time-dependent. Instead, if 𝒱^(  interaction)
(  t)  =0 then c_k(  t)  =c_k(  0).
To find an equation satisfied by the coefficients c_k(  t),
we proceed as follows <cit.>. The interaction representation
of a quantum state that at time t=0 is given by | s(
0)  ⟩ is defined as,

    | s(  t)  ⟩ _Idef
    =e^i/ℏℋ_0^(  free)  
    t| s(  0)  ⟩ _S.

The symbols “I” and “S” in Eq. (<ref>) denote the interaction and the
Schrödinger representations, respectively. Note that | s(
0)  ⟩ _S=| s(  0)
⟩ _I=| s(  0)  ⟩.
Multiplying both sides of Eq. (<ref>) by iℏ, differentiating with
respect to t, and recalling that iℏ∂_t| s(
0)  ⟩ _S=ℋ_0^(  free
)  | s(  0)  ⟩ _S, we obtain

    iℏ∂_t| s(  t)  ⟩ _I
    =𝒱_I| s(  t)  ⟩
    _I

The operator 𝒱_I in Eq. (<ref>) is the interaction
representation of the interaction potential 𝒱^(
interaction) and is defined as,

    𝒱_Idef=e^i/ℏ
    ℋ_0^(  free)  t𝒱^(
    interaction)  e^-i/ℏℋ_0^(
    free)  t.

Multiplying both sides of Eq. (<ref>) for ⟨ E_k|
from the left and using the completeness relation,

    ∑_n=1^2
    | E_n⟩⟨ E_n| =ℐ
    ,

with ℐ denoting the identity operator, we obtain

    iℏ∂_t⟨ E_k|s(  t)  ⟩
    _I=
    
    
    ∑_n=1^2
    ⟨ E_k|𝒱_I|E_n⟩⟨
    E_n|s(  t)  ⟩ _I.

Observe that c_k(  t)  def=⟨
E_k|s(  t)  ⟩ _I, and

    ⟨ E_k|𝒱_I|E_n⟩ =⟨
    E_k|𝒱^(  interaction)  (  t)
    |E_n⟩ e^i/ℏ(  E_k-E_n)
    t=𝒱_kn(  t)  e^iω_knt,

where 𝒱_kn(  t)  def=⟨
E_k|𝒱^(  interaction)  (  t)
|E_n⟩ and ω_kndef=(
E_k-E_n)  /ℏ. Finally, using Eq. (<ref>), Eq.
(<ref>) becomes

    iℏ∂_tc_k(  t)  =
    
    
    ∑_n=1^2
    𝒱_kn(  t)  e^iω_kntc_n(  t)
    .

Equation (<ref>) is a system of coupled first order ordinary
differential equations with time-dependent coefficients that need to be
integrated in order to find the coefficients c_k(  t). We
remark that once the time-dependent quantum mechanical probability amplitudes
{  c_k(  t)  } are obtained, the transition
probabilities 𝒫_| s⟩→|
E_1⟩(  t) and 𝒫_|
s⟩→| E_1⟩(  t)
are given by

    𝒫_| s⟩→| E_1
    ⟩(  t)  def=[  c_1(
    t)  ]  [  c_1(  t)  ]  ^∗=|
    c_1(  t)  | ^2, and 𝒫_|
    s⟩→| E_2⟩(  t)
    def=[  c_2(  t)  ]  [
    c_2(  t)  ]  ^∗=| c_2(  t)
    | ^2,

respectively.

In the remainder of this Appendix, we provide further technical details on the
structure of the ODEs that arise from Eq. (<ref>). Recall that the
probability amplitudes c_1(  t) and c_2(  t)
satisfy the following differential equations,

    iℏċ_1=𝒱_11c_1+𝒱_12e^iω_12
    tc_2,

and,

    iℏċ_2=𝒱_21e^iω_21tc_1+𝒱
    _22c_2,

respectively, where ω_12=-ω_21 since ω_21
def=(E_2-E_1)/ℏ and ċ_idef=dc_i/dt with i∈{  1, 2}.
Differentiating both sides of Eq. (<ref>) with respect to time, we obtain

    iℏc̈_1-𝒱̇_11c_1-𝒱_11ċ
    _1=(  𝒱̇_12+iω_12𝒱_12)
    e^iω_12tc_2+𝒱_12e^iω_12tċ_2

From Eq. (<ref>), it is found that c_2 equals

    c_2=iℏ/𝒱_12e^-iω_12tċ_1
    -𝒱_11/𝒱_12e^-iω_12tc_1.

Using Eqs. (<ref>) and (<ref>), ċ_2 can be rewritten as

    ċ_2=𝒱_22/𝒱_12e^-iω_12tċ_1-𝒱_11𝒱_22/iℏ𝒱_12
    e^-iω_12tc_1+𝒱_21/iℏe^-iω_12
    tc_1.

Finally, substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>),
we arrive at

    c̈_1-(  𝒱̇_12+iω_12𝒱
    _12/𝒱_12+𝒱_11+𝒱_22/iℏ)  ċ_1+[  𝒱_11(  𝒱̇
    _12+iω_12𝒱_12)  /iℏ𝒱_12
    -𝒱̇_11/iℏ-(  𝒱
    _11𝒱_22-𝒱_21𝒱_12)  /ℏ^2
    ]  c_1=0.

Proceeding along this same line of reasoning, we readily obtain

    c̈_2-(  𝒱̇_21+iω_21𝒱
    _21/𝒱_21+𝒱_11+𝒱_22/iℏ)  ċ_2+[  𝒱_22(  𝒱̇
    _21+iω_21𝒱_21)  /iℏ𝒱_21
    -𝒱̇_22/iℏ-(  𝒱
    _11𝒱_22-𝒱_21𝒱_12)  /ℏ^2
    ]  c_2=0.

In general, Eqs. (<ref>) and (<ref>) are second-order linear ordinary
differential equations with non-constant coefficients. Observe that in the
special case in which 𝒱_11=0, 𝒱_22=0,
𝒱_12=Γ e^iω t, and 𝒱_21=Γ
e^-iω t, Eqs. (<ref>) and (<ref>) become

    c̈_1-i(  ω-ω_21)  ċ_1+Γ^2
    /ℏ^2c_1=0,

and,

    c̈_2+i(  ω-ω_21)  ċ_2+Γ^2
    /ℏ^2c_2=0,

respectively. Unlike Eqs. (<ref>) and (<ref>), Eqs. (<ref>) and
(<ref>) are second-order linear ordinary differential equations with
constant coefficients. While this set of two ODEs will be explicitly
integrated in Appendix C, the exact analytical integration of Eqs.
(<ref>) and (<ref>) can be generally quite challenging.



§ INTEGRATION OF EQ. (<REF>)


In this Appendix, we integrate the system of coupled ODEs in Eq.
(<ref>) that appear in Section III.

After some algebraic manipulations, the two relations in Eq. (<ref>
) lead to the following second order ordinary differential equation with
time-independent coefficients for c_2,

    c̈_2+i(  ω-ω_21)  ċ_2+Γ^2
    /ℏ^2c_2=0.

We note that a particular solution of Eq. (<ref>) can be written as
c_2(  t)  =Ae^α t with A and α denoting two
complex constants. In particular, the constant α can be found
by substituting the condition c_2(  t)  =Ae^α t into Eq.
(<ref>). We obtain the following constraint equation,

    α^2+α i(  ω-ω_21)  +Γ^2
    /ℏ^2=0,

which, in turn, leads to the two complex roots

    α_±def=-i/2(  ω-ω
    _21)  ± i√(Γ^2/ℏ^2+(
    ω-ω_21)  ^2/4).

Therefore, the general solution of Eq. (<ref>) can be formally written as

    c_2(  t)  =𝒜e^α_+t+ℬe^α_-
    t,

where the constants 𝒜 and ℬ belong to ℂ. Substituting Eq. (<ref>) into Eq. (<ref>), c_2(
t) becomes

    c_2(  t)  =e^-i/2(  ω-ω_21)
    t[  𝒜exp(  i√(Γ^2/ℏ^2
    +(  ω-ω_21)  ^2/4)t)  +ℬ
    exp(  -i√(Γ^2/ℏ^2+(  ω
    -ω_21)  ^2/4)t)  ]  ,

where exp(  ·) denotes the exponential function. Recalling
that e^ixdef=cos(  x)  +isin(
x) for any x in ℝ, after some algebraic manipulations, we deduce that c_2(  t) in Eq. (<ref>) can be recast as

    c_2(  t)  =e^-i/2(  ω-ω_21)
    t[  𝒞cos(  √(Γ^2/ℏ^2
    +(  ω-ω_21)  ^2/4)t)  +𝒟
    sin(  √(Γ^2/ℏ^2+(  ω
    -ω_21)  ^2/4)t)  ]  ,

with 𝒞def=𝒜+ℬ and
𝒟def=i(𝒜-ℬ). Imposing that
c_2(  0)  =x, we find that 𝒞=x and, as a
consequence,

    c_2(  t)  =e^-i/2(  ω-ω_21)
    t[  xcos(  √(Γ^2/ℏ^2+(
    ω-ω_21)  ^2/4)t)  +𝒟sin(
    √(Γ^2/ℏ^2+(  ω-ω_21)
    ^2/4)t)  ]  .

For the sake of notational simplicity, let us introduce the following
real quantity

    Ωdef=√(Γ^2/ℏ^2
    +(  ω-ω_21)  ^2/4).

Combining the first relation in Eq. (<ref>) with Eqs. (<ref>)
and (<ref>), it can be verified that c_1(  t) satisfies
the condition

    c_1(  t)  =-iΓ/ℏ∫ e^i/2(
    ω-ω_21)  t[  xcos(  Ω t)
    +𝒟sin(  Ω t)  ]  dt.

Observe that for any complex coefficients a and b, up to an
unimportant constant of integration, we have

    ∫ e^atsin(  bt)  dt=e^at/a^2+b^2[
    asin(  bt)  -bcos(  bt)  ]  ,

and,

    ∫ e^atcos(  bt)  dt=e^at/a^2+b^2[
    bsin(  bt)  +acos(  bt)  ]  .

Using Eqs. (<ref>) and (<ref>), integration of Eq. (<ref>)
yields

    c_1(  t)  =-iℏ/Γe^i(  ω-ω
    _21)  t{[  Ω x+i/2(  ω
    -ω_21)  𝒟]  sin(  Ω t)  -[
    Ω𝒟-i/2(  ω-ω_21)  x]
    cos(  Ω t)  }.

To find the integration constant 𝒟, we impose that c_1(
0)  =√(1-x^2). We therefore obtain,

    𝒟def=i(  ω-ω
    _21)  /2x-Γ/ℏ√(1-x^2)/√(Γ^2/ℏ^2+(  ω-ω_21)  ^2/4)
    .

At this point, we recall that the transition probabilities 𝒫
_| s⟩→| E_1⟩(
t) and 𝒫_| s⟩→|
E_2⟩(  t) are defined as 𝒫
_| s⟩→| E_1⟩(
t)  def=| c_1(  t)  |
^2 and 𝒫_| s⟩→|
E_2⟩(  t)  def=|
c_2(  t)  | ^2, respectively. The quantum mechanical
probability amplitudes c_1(  t)  def
=⟨ E_1|s(  t)  ⟩ _I and
c_2(  t)  def=⟨ E_2|s(
t)  ⟩ _I are given in Eqs. (<ref>) and
(<ref>), respectively. Furthermore, the quantities Ω and
𝒟 are defined in Eqs. (<ref>) and (<ref>), respectively.
After some algebra, we find that the transition probabilities 𝒫
_| s⟩→| E_1⟩(
t) and 𝒫_| s⟩→|
E_2⟩(  t) are given by

    𝒫_| s⟩→| E_1
    ⟩(  t)  =(  1-x^2)  cos^2(
    √(Γ^2/ℏ^2+(  ω-ω_21)
    ^2/4)t)  +{  1-[  (  ω-ω
    _21)  /2x-Γ/ℏ√(1-x^2)/√(Γ^2/ℏ^2+(  ω-ω_21)  ^2/4)
    ]  ^2}sin^2(  √(Γ^2/ℏ^2
    +(  ω-ω_21)  ^2/4)t)  ,

and,

    𝒫_| s⟩→| E_2
    ⟩(  t)  =x^2cos^2(  √(Γ^2
    /ℏ^2+(  ω-ω_21)  ^2/4)t)
    +[  (  ω-ω_21)  /2x-Γ/ℏ√(1-x^2)/√(Γ^2/ℏ^2+(
    ω-ω_21)  ^2/4)]  ^2sin^2(  √(Γ^2/ℏ^2+(  ω-ω_21)  ^2
    /4)t)  ,

respectively. Appendix C ends with the derivations of the exact temporal
behaviors of 𝒫_| s⟩→| E_1⟩(  t) and 𝒫
_| s⟩→| E_2⟩(
t) in Eqs. (<ref>) and (<ref>), respectively.



§ PROBABILITY AMPLITUDES IN THE ZENER CASE


In this Appendix, we present some technical details on the mathematical scheme
employed by Zener in Ref. <cit.> and used by Perez and Romanelli in
Ref. <cit.> as mentioned in Section IV. This approach helps
with solving a special case of Eqs. (<ref>) and (<ref>). In
particular, Zener considered a special case of Eq. (<ref>) given by

    c̈_1+iα tċ_1+f^2c_1=0,

with α and f^2 denoting two real constants. Upon a suitable
change of the dependent variable c_1,

    c_1(  t)  → c_1(  t)  def
    =e^-iα/4t^2u_1(  t)  ,

Eq. (<ref>) becomes

    ü_1+(  f^2-iα/2+α^2/4t^2)
    u_1=0.

Multiplying Eq. (<ref>) by i/α and defining,

    ndef=if^2/α, and zdef=√(α)e^-iπ/4t,

Eq. (<ref>) can be recast as

    i/αü_1(  t)  +(  n+1/2-1/4z^2)  u_1(  t)  =0.

In terms of the independent variable z, Eq. (<ref>) becomes

    u_1^''(  z)  +(  n+1/2-1/4
    z^2)  u_1(  z)  =0,

where u_1^'def=du_1/dz. Finally, we recognize
that Eq. (<ref>) is a Weber differential equation whose general
solution u_1(  z; n) can be written as a superposition
of independent solutions 𝒟_n(  z) and
𝒟_-n-1(  iz) as follows

    u_1(  z; n)  =a𝒟_n(  z)
    +b𝒟_-n-1(  iz)  ,

where a and b are integration constants. The functions
𝒟_n(  z) in Eq. (<ref>) are known as parabolic
cylinder functions <cit.> and are defined as,

    𝒟_n(  z)  def=2^n/2+1/4/√(z)𝒲_n/2+1/4, 
    -1/4(  z^2/2)  .

The functions 𝒲_k, m(  z) in Eq.
(<ref>) are known as the Whittaker functions <cit.> and are defined
as,

    𝒲_k, m(  z)  def=
    z^m+1/2e^-z/2
    
    
    ∑_n=0^∞
    (  m-k+1/2)  _n/n!(  2m+1)  _n
    z^n,

where (  z)  _n denotes the Pochhammer symbol defined as
<cit.>,

    (  z)  _ndef=Γ(  z+n)
    /Γ(  z)  =(  z+n-1)  !/(  z-1)
    !=z(  z+1)  ...(  z+n-1)  .

The quantity Γ in Eq. (<ref>) is the Euler gamma function. For
further details on the Weber equation, the parabolic cylinder functions, and
the Whittaker functions, we refer to Ref. <cit.>.



§ ANALYTICALLY SOLVABLE QUANTUM TWO-LEVEL SYSTEMS


In this Appendix, we present a few preliminary mathematical remarks inspired
by the recent investigations presented by Messina and collaborators in Refs.
<cit.>. These remarks can be helpful to further clarify
our work presented in Section V.

The matrix representation of the unitary evolution operator 𝒰
(  t) generated by the time-dependent Hamiltonian ℋ
(  t) in Eq. (<ref>) with iℏ𝒰̇
(  t)  =ℋ(  t)  𝒰(  t) where 𝒰̇def=∂_t𝒰 can
be described as,

    [  𝒰(  t)  ]  _ℬ_canonical
    def=(
    [    α(  t)    β(  t); -β^∗(  t)  α^∗(  t) ])  .

Note that unitarity requires that the complex probability amplitudes
α(  t) and β(  t) satisfy the
normalization condition,

    |α(  t)  | ^2+|β(
    t)  | ^2=1.

Observe that the temporal evolution of a quantum source state |
s⟩,

    | s⟩def=x| w⟩
    +√(1-x^2)| r⟩,

under the unitary evolution operator 𝒰(  t) in Eq.
(<ref>) can be described in terms of the following mapping,

    x√(1-x^2)→α(  t)
    x+β(  t)  √(1-x^2)-β^∗(  t)
    x+α^∗(  t)  √(1-x^2).

Therefore, the probability that the source state | s⟩
transitions into the target state | w⟩ under
𝒰(  t) is given by,

    𝒫_| s⟩→| w⟩(  t)  def=|⟨
    w|𝒰(  t)  |s⟩| ^2,

that is,

    𝒫_| s⟩→| w⟩(  t)  =|α(  t)  | ^2
    x^2+|β(  t)  | ^2(  1-x^2
    )  +[  α(  t)  β^∗(  t)
    +α^∗(  t)  β(  t)  ]  x√(1-x^2),

with α(  t)  β^∗(  t)  +α^∗(  t)  β(  t)  =2[  α_R(
t)  β_R(  t)  +α_I(
t)  β_I(  t)  ] being a real
quantity. In the last statement, we assumed that α(  t)
and β(  t) could be rewritten as,

    α(  t)  def=Re[
    α(  t)  ]  +iIm[  α(
    t)  ]  =α_R(  t)  +iα_I
    (  t)  ,

and,

    β(  t)  def=Re[
    β(  t)  ]  +iIm[  β(
    t)  ]  =β_R(  t)  +iβ_I
    (  t)  ,

respectively. From Eq. (<ref>), it is evident that in order to
compute transition probabilities, one needs to know the evolution operator
𝒰(  t) in terms of the complex probability amplitudes
α(  t) and β(  t). Ideally, having
fixed the fields ω(  t) and Ω(  t) in
Eq. (<ref>) due to the motivation of physical arguments, one would try to
integrate the coupled system of first order ordinary differential equations
with time-dependent coefficients generated by the relation iℏ𝒰̇(  t)  =ℋ(  t)
𝒰(  t) with 𝒰(  0)
=ℐ,

    iℏα̇(  t)  =Ω(  t)  α(
    t)  -ω(  t)  β^∗(  t)  , and
    iℏβ̇(  t)  =ω(  t)  α^∗(  t)  +Ω(  t)  β(  t)
    ,

where α(  0)  =1 and β(  0)  =0.
Unfortunately, this general approach can rarely be solved in an exact manner.
If one is willing to specify only the functional form of one of the two fields
ω(  t) and Ω(  t) however, it is
possible to introduce clever mathematical schemes that allow to solve
analytically the system of ODEs in Eq. (<ref>), at least for certain
relevant functional forms of the non-fixed field <cit.>.
In what follows, we build a great part of our discussion on the very
interesting works presented in Refs. <cit.>.

For the sake of reasoning, let us assume that we fix the field Ω(
t). After some algebra, it can be shown that the second relation in
Eq. ((<ref>) is solved by β(  t),

    β(  t)  =1/iℏα(  t)  X(
    t)  ,

where X(  t) with X(  0)  =0 is a complex
auxiliary function defined as,

    X(  t)  def=∫_0^tω(
    t^')  /α^2(  t^')  dt^'.

From Eq. (<ref>), it follows that

    ω(  t)  =α^2(  t)  Ẋ(  t)
    .

Using Eqs. (<ref>), (<ref>), and (<ref>), it can be shown
that the first relation in Eq. (<ref>) leads to the following
differential equation for α(  t),

    α̇(  t)  =-[  i/ℏΩ(
    t)  +Ẋ(  t)  X^∗(  t)  /ℏ^2+| X(  t)  | ^2]
    α(  t)  .

Noting that Ẋ(  t)  X^∗(  t)  def=Re[  Ẋ(  t)  X^∗(  t)  ]  +iIm[  Ẋ(
t)  X^∗(  t)  ], integration of Eq.
(<ref>) yields

    α(  t)  =ℏ/√(ℏ^2+| X(
    t)  | ^2)exp[  -i/ℏ∫_0^t
    Ω(  t^')  dt^'-i∫_0^tIm[  Ẋ(  t^')  X^∗(
    t^')  ]  /ℏ^2+| X(  t^')  | ^2dt^']  .

In conclusion, once the auxiliary complex function X(  t) and the real longitudinal field Ω(  t) have been
chosen, we can compute the probability amplitude α(  t)
from Eq. (<ref>), the transversal field ω(  t) from
Eq. (<ref>), and finally, the probability amplitude β(
t) from Eq. (<ref>). Clearly, once we have α(
t) and β(  t), we can compute the transition
probability 𝒫_| s⟩→|
w⟩(  t) in Eq. (<ref>).


