Anomalous dynamical evolution and nonadiabatic level crossing in exactly solvable time-dependent quantum systems
Hong Cao Shao-Wu Yao Li-Xiang Cen
quant-ph









Center of Theoretical Physics, College of Physics, Sichuan University, Chengdu 610065, China


School of Material Science and Engineering, Chongqing Jiaotong
University, Chongqing 400074, China


Center of Theoretical Physics, College of Physics, Sichuan University, Chengdu 610065, China


lixiangcen@scu.edu.cn
Center of Theoretical Physics, College of Physics, Sichuan University, Chengdu 610065, China




The anomalous dynamical evolution and the crossing of nonadiabatic
energy levels are investigated for exactly solvable time-dependent
quantum systems through a reverse-engineering scheme. By exploiting a typical driven model,
we elucidate the peculiarities of its dynamics with anomalous
behavior: the evolution of the adiabatic states and of the nonadiabatic ones exhibits opposite
behavior with their representative vectors evolving from a
parallel state to an antiparallel state;
the nonadiabatic level crossing is
identified as a necessary consequence since the crossing point corresponds
exactly to the perpendicular point of the two vectors in the parametric space.
In the light of these results, we show that
various driven models with anomalous dynamical evolution can be designed
and they offer alternative protocols for the quantum state control.




Anomalous dynamical evolution and nonadiabatic level crossing in exactly
solvable time-dependent quantum systems
    Li-Xiang Cen
    December 30, 2023
================================================================================================================







§ INTRODUCTION


Nonadiabatic dynamics generated by quantum systems with explicitly
time-dependent Hamiltonians plays an important role in various branches of
quantum physics. Especially, when the diabatic energies of two interacting
quantum states are forced to cross, e.g., by a linearly driving external
field, the nonadiabatic transition between adiabatic levels is known as the
Landau-Zener (LZ) tunneling <cit.>. The latter offers a simple way
to understand the wave phenomenon of the quantum system and the resulting
quantitative formula of the transition probability has widespread
applications ranging from quantum optics and atomic physics <cit.> to chemistry and biophysics <cit.>. Recently, application of the LZ driving and its
generalized protocols to the quantum state manipulation has attracted much
attention in the context of quantum information processing <cit.>. Experimental
demonstration of the corresponding coherent dynamical transition has been
reported in a variety of physical systems, e.g., the Rydberg atoms <cit.>,
the superconducting quantum interference device <cit.>,
and the nitrogen-vacancy center in diamond <cit.>.

The evolution of the energy level and the wavefunction of a driven quantum
system can be distinctly different under the adiabatic or the nonadiabatic
driving. In the case of the adiabatic evolution, the LZ-type driven model
will exhibit the avoided level crossing which is
well understood. Consider as an example the dynamics generated by a
two-level system with the following Hamiltonian

    H(t)=1/2(
    [  Ω _z(t)  Ω _x(t);  Ω _x(t) -Ω _z(t) ]),

in which the bare energies ±1/2Ω_z (t) change from Ω_z<0 to Ω_z >0 at some time instant, e.g., at t=0. The occurrence of a
perturbative Ω_x(t) around t=0 will lift the degeneracy of the
energies as E_±^ad(t)= ±1/2√(Ω_x^2(t)+Ω_z^2(t)),
which results in the emergence of the avoided level crossing (the
particular case with both Ω_x(t) and Ω_z(t) vanishing at t=0
was investigated in Ref. <cit.>). The adiabatic following of the
corresponding instantaneous eigenstate of the Hamiltonian then leads to the
population transfer from one bare state to the other. On the other hand, as
the nonadiabatic driving is concerned, the dynamics generated by this kind
of models depends heavily on the pulse shape of the driving field.
For example, in the standard LZ protocol with linearly driving field,
the nonadiabaticity-induced transition will destroy the desired population
transfer. Notwithstanding, recent studies on the variants of the LZ model
display that the complete population transfer could be achieved through the
avoided level crossing under a tangent-shape driving <cit.> even
when the corresponding evolution is in a nonadiabatic manner.

In the general case of the nonadiabatic evolution, the Hamiltonian itself is
no longer an invariant of the system. The energy levels are then defined by
the expectation values of the Hamiltonian over the nonadiabatic bases, e.g.,
the eigenvectors of the Lewis-Riesenfeld (LR) invariant <cit.>.
The phenomenon of the nonadiabatic level crossing (NLC), if it does occur,
would have nothing to do with the degeneracy or the symmetry of the system. So the
question arises: what is the dynamical implication with respect to the
real NLC? This issue had hardly been investigated, probably because of
the fact that the case of driven quantum systems that could exhibit the
NLC phenomenon is very rare. In this paper we shall explore this issue
in virtue of exactly solvable time-dependent quantum systems that are constructed
through a reverse-engineering scheme. We will first describe the anomalous dynamical
behavior in a nonadiabatically driven system, that is, the nonadiabatic state of the system
exhibits opposite behavior from its adiabatic state. We then demonstrate that the occurrence
of the NLC constitutes a necessity condition for such particular dynamical behavior.
The general description of the driving protocol and illustration of
various examples with distinct features will be presented within the
reverse-engineering scheme.




§ A TYPICAL DRIVEN MODEL WITH ANOMALOUS DYNAMICAL EVOLUTION



Let us firstly consider a driven quantum system described by the following
Hamiltonian

    H(t)    =   Ω_x(t)J_x+Ω_z(t)J_z  
       =   ϵ/√(ϵ ^2t^2+1)J_x+ϵ (ϵ
    ^2t^2-1)/ϵ ^2t^2+1J_z,

where J_i (i=x,y,z) denote the angular momentum operators satisfying
[J_i,J_j]=iε _ijkJ_k and the driving field has two components
along the x and z axes, respectively. The orientation of the driving
field, specified by α⃗_h(t)=Ω⃗(t)/|Ω⃗
(t)|, tends to +z axis at t→±∞. It indicates that
the instantaneous adiabatic eigenstates of H(t), expressed as |ψ
_m^ad(t)⟩ =e^-iθ _h(t)J_y|m⟩ with |m⟩ denoting
the eigenstate of J_z  and
θ_h(t)=arccosΩ _z(t)/√(Ω _x^2(t)+Ω _z^2(t)),
will return to its initial state
|ψ _m^ad(-∞ )⟩ =|m⟩ at t→ +∞. This
can also be understood from the fact that the adiabatic energy levels E_m^ad(t) undergo the avoided crossing twice during the evolution [see
Fig. <ref> (a)].




To resolve the nonadiabatic evolution of the system governed by the
Schrödinger equation (setting ħ =1)

    i∂/∂ t|ψ (t)⟩ =H(t)|ψ (t)⟩ ,

it is direct to verify that the system possesses a dynamical invariant

    I(t)=-ϵ t/ϵ ^2t^2+1J_x-1/ϵ ^2t^2+1J_y-
    ϵ t/√(ϵ ^2t^2+1)J_z

which satisfies i∂ _tI(t)=[H(t),I(t)]. According to the LR theory <cit.>,
the solution to the Schrödinger equation can be achieved by the
instantaneous eigenstate of I(t) equipped with a phase factor.
Let us express I(t) as I(t)=α⃗_0(t)·J⃗ in which
α⃗_0(t)=(sinθ _0cosφ _0,sinθ _0sinφ
_0,cosθ _0) and θ _0(t) and φ _0(t) are given by

    θ _0(t)=π/2+arctan (ϵ t),  φ _0(t)=3π/2-arctan (ϵ t).

The eigenstate of I(t) can then be obtained explicitly as |ϕ _m(t)⟩
=e^i(π -φ _0)J_ze^iθ _0J_y|m⟩. So the dynamical basis
of the system |ψ _m(t)⟩ is formulated as |ψ _m(t)⟩
=e^iΦ _m(t,t_0)|ϕ _m(t)⟩, where
Φ _m(t,t_0) is the so-called LR total phase given by

    Φ _m(t,t_0)    =   ∫_t_0^t⟨ϕ _m(t^')|i∂
    _t^'-H(t^')|ϕ _m(t^')⟩ dt^'
       =   -m∫_t_0^tϵ ^2τ/√(ϵ ^2τ ^2+1)dτ .












With the above analytical solution, we are able to describe the anomaly of
the dynamical evolution of the model. It is seen that as t goes from
-∞ to +∞, the invariant I(t) will go from +J_z to -J_z.
Indeed, the two representative vectors α⃗_h(t) and α⃗_0(t),
accounting respectively for the
orientation of H(t) and I(t) in the parametric space, will evolve from
the parallel state with θ _h=θ _0=0 to the antiparallel state with
θ _h=0 and θ _0=π.
Therefore, the nonadiabatic basis state |ψ _m(t)⟩ exhibits the
opposite behavior from the adiabatic state |ψ _m^ad(t)⟩ as it
leads to a complete population inversion |m⟩↔
|-m⟩ during the evolution.
Furthermore, the peculiarity of the dynamical behavior can also be recognized from
the different crossing phenomena of the corresponding adiabatic
and nonadiabatic energy levels. As the adiabatic levels exhibit only avoided crossings, the
nonadiabatic levels of the model, formulated as

    E_m(t)    ≡   ⟨ψ _m(t)|H(t)|ψ _m(t)⟩
       =   -mϵ ^4t^3/(1+ϵ ^2t^2)^3/2,

exhibit a real crossing at t_c=0. For the two-level case with
the azimuthal quantum number j=1/2, these level crossing phenomena are
illustrated in Fig. <ref>(a).




§ GENERAL DESCRIPTION OF THE PROTOCOL






 §.§ Condition associated with the nonadiabatic level crossing


To reveal the connection between the above
described anomalous dynamical evolution and the NLC phenomenon,
it is worthy to note that at the crossing point t_c=0 the orientation of
α⃗_h is vertical to α⃗_0
as α⃗_h is fixed in the x-z plane and
α⃗_0(t_c)=(0,-1,0) is exactly along the y axis.
In fact, this is not accidental in view that the nonadiabatic level
E_m(t) can be expressed as

    E_m(t)   =   ⟨ϕ _m(t)|H(t)|ϕ _m(t)⟩
       =   ⟨ m|e^-iθ_0J_ye^-i(π-φ_0)J_zH(t)e^i(π-φ_0)J_ze^iθ_0J_y|m⟩
       =   mΩ⃗(t)·α⃗_0(t).

That is to say, there exists a general correspondence between the orthogonal
relation α⃗_0·α⃗_h=0 and the NLC
with all E_m=0. The evolution of the inner product
f(α⃗_0,α⃗_h)≡α⃗_0(t)·α⃗_h(t) of the model
(<ref>) is illustrated in Fig. <ref>(b). Mathematically, the
orthogonality here can also be understood as that the two operators, I(t)
and H(t), have vanishing Frobenius inner product <cit.>: tr
[I(t_c)H(t_c)]=0. As a consequence, the occurring
of the NLC can be identified as a necessity
condition of the anomalous dynamical evolution: as the two vectors α⃗_0(t) and α⃗_h(t) change continuously from a parallel
state to an anti-parallel state in the parametric space, it is unavoidable
that they should go through a perpendicular position during the evolution.



 §.§ The reverse-engineering scheme


In the light of the above result, we now address the question how to achieve
such kind of driven quantum systems with the similar anomalous dynamical
behavior. To this end, let us consider the dynamics generated by the driven
model of Eq. (<ref>) with general field pulses Ω _x(t) and Ω _z(t).
We suppose that the system possesses a dynamical invariant
I(t)≡α⃗(t)·J⃗ with
its components α _i(t) fulfilling

    α̇_x(t)    =   -Ω _z(t)α _y(t),  
    α̇_y(t)    =   Ω _z(t)α _x(t)-Ω _x(t)α _z(t),
    
    α̇_z(t)    =   Ω _x(t)α _y(t).

Since the eigenvalues of I(t) are independent of time, we can set |α⃗(t)|=1 and express it as α⃗(t)=(sinθcosφ
,sinθsinφ ,cosθ ). We then regard Eqs. (<ref>
)-(<ref>) as algebraic equations of Ω _x,z(t). Explicitly,
Eq. (<ref>) indicates

    Ω _x(t)=-θ̇(t)/sinφ (t),

and Eq. (<ref>) [or Eq. (<ref>) equivalently] indicates

    Ω _z(t)    =   φ̇(t)+Ω _x(t)θ (t)cosφ (t)
    
       =   φ̇(t)-θ̇(t)θ (t)φ (t).

That is to say, for any given I(t) with specified θ (t) and φ (t), one can construct the field components Ω _x,z(t) in
virtue of Eqs. (<ref>) and (<ref>). Note that similar
reverse-engineering schemes have ever been proposed, e.g., in Refs. <cit.>
and <cit.>. In comparison, there is no redundant
parameter in the present scheme and the driving field of the target Hamiltonian
here is uniquely determined. On other words, it indicates
that the driving protocol with two field components is sufficient to generate
any desired SU(2) rotation to the wavefunction.

If we set θ (t)=θ _0(t) and φ (t)=φ _0(t) specified in Eq. (<ref>)
and substitute them into Eqs. (<ref>) and (<ref>), one
promptly obtains the previous model of Eq. (<ref>). To illustrate
the selection of θ(t) and φ(t) and the resulting anomalous
dynamical behavior, one is led to notice that as t→±∞, there is
φ̇_0(t)→ 0 and Ω _z is then dominated by
the term Ω _xθ _0cosφ_0 [cf. Eq. (<ref>)].
So the relation between the two angles θ _0 and θ _h at
t→±∞ is characterized by

    θ _h=cosφ_0 θ _0.

The change of the relative orientation of α⃗_h(t) and α⃗ _0(t), i.e., from the parallel state to the antiparallel state, is
then well understood because cosφ _0, according to Eq. (<ref>),
alters its sign (from +1 to -1) during the evolution.
This illuminates a way to contrive similar driving protocols with the
anomalous dynamical behavior.



 §.§ Further examples of the driving protocol



For the second example let us consider a single-axis driven model described
by

    H(t)=ϵ J_x+Ω _z(t)J_z,

in which Ω_x=ϵ is time independent and the driving component Ω _z(t) is
given by

    Ω _z(t)=5ϵ e^2ϵ t-ϵ e^6ϵ t/
    1+e^4ϵ t.

This model can be constructed from Eqs. (<ref>) and (<ref>) by
setting the two angles as

    θ (t)=π/2-arctan e^2ϵ t,  φ (t)=2arctan
    e^2ϵ t.

As t goes from -∞ to +∞, these angles will go from (θ
,φ )=(π/2,0) to (θ ,φ )=(0,π).
Differing from the former model, the representative vectors
α⃗_h(t) and α⃗(t) of the Hamiltonian and of
the dynamical invariant here,
are along the +x axis initially at t→ -∞.
They arrive eventually at an anti-parallel state,
α⃗_h(t)→ -z and α⃗(t)→ +z
as t→ +∞.
The anomalous dynamical behavior can be specified explicitly
by the evolution of the wavefunction. The instantaneous adiabatic states,
say, for the two-level case with azimuthal quantum number j=1/2,
will evolve as

    |ψ _±^ad(-∞ )⟩ =√(2)/2(|+⟩±
    |-⟩ )t→ +∞⟶|∓⟩ ,

in which we have used the notation “±" for “±1/2".
In comparison, the eigenstate of I(t), accounting for the
the nonadiabatic evolution of the system, will evolve as

    |ϕ _±(-∞ )⟩ =√(2)/2(|+⟩± |-⟩ )
    t→ +∞⟶|±⟩ .

That is, the nonadiabaticity of the evolution induces a complete exchange between the two
adiabatic basis states as t→ +∞.




On the other hand, the nonadiabatic energy level of the system is worked out to be

    E_m(t)=mϵ[ 4e^4ϵ t/(1+e^4ϵ t)^3/2-
    e^4ϵ t-1/(1+e^4ϵ t)^1/2] .

It is straightforward to verify that as ϵ t_c=1/4ln (2+√(5)
) there is E_m(t_c)=0, that is, all the nonadiabatic energy levels should
intersect at the particular point t=t_c. Specifically, at the point of
ϵ t_c=1/4ln (2+√(5)), there
are

    α⃗(t_c)=(√(10)-3√(2)/4,√(2√(5)-4)/2,
    √(√(5)+1)/2),

and

    Ω⃗(t_c)/ϵ =(1,0,√(10√(5)-22)/2).

So the orthogonality of these two vectors at the crossing point t=t_c
can be verified straightforwardly. In Fig. <ref> we plot the schematic of the evolving
trajectories of α⃗(t) and α⃗_h(t) for the model in the parametric space.


To obtain the third protocol, we set θ (t) and φ (t) as

    θ (t) =arccos [tanh (2ϵ t)],   φ (t) =π/2-arctan [tanh (ϵ t)],

via which the driving field components are achieved as

    Ω _x(t) =2ϵ(ϵ t)^1/2(2ϵ t),   Ω _z(t) =ϵ [3(2ϵ t)-2].

The curves of the field pulses Ω _x(t) and Ω _z(t) are plotted
in Fig. <ref>(a). According to Eq. (<ref>), cosφ will evolve
from -√(2)/2 to √(2)/2 as the time goes from
-∞ to +∞. So the ratio θ _h/θ should
alter its sign at t→±∞ according to Eq. (<ref>).
Specifically, the orientation of α⃗(t) will go from -z to
+z, so the complete population transfer |m⟩↔ |-m⟩
 can be realized through the nonadiabtic evolution. On the other hand,
as α⃗_h(t) tends to the -z axis as t→±∞, the adiabatic state will return to its initial state.
At t=0, there are θ (0)=φ (0)=π/2,
which indicates that α⃗(0) is along the y axis, hence is
vertical with α⃗_h(t) at that time instant. In Fig. <ref>(b) we depict
the corresponding NLC for the model with j=1/2.









§ DISCUSSION AND CONCLUSION


The anomalous dynamical evolution described above represents the
extreme behavior associated with the nonadiabatic driving.
It is worthy to mention that
such an extreme manifestation of the nonadiabtic effect could also be found in
the standard LZ model. According to the LZ formula, the nonadiabatic transition
between the two adiabatic states over the whole evolution
is specified by P=e^-πΔ^2/2ν in which Δ describes
the constant coupling between the two bare states and ν stands for
the scanning rate of the linearly driving field.
If the scanning rate satisfies ν/Δ^2→ +∞,
the nonadiabtic evolution will lead to completely opposite behavior with
that of the instantaneous adiabtic state. In comparison,
the driving fields in all our proposed protocols have finite scanning rate
hence are more realistic physically.

The driven model with the anomalous dynamical evolution offers an alternative
scenario for the state transfer that is distinctly different from the existing schemes,
e.g., the transitionless quantum driving <cit.>, the so-called shortcut-to-adiabaticity
protocol <cit.> and their recent development <cit.>.
Given a general time-dependent Hamiltonian H(t),
the transitionless algorithm
cancels out the nonadiabatic effect by introducing an auxiliary counter-diabatic field
and ensures that the dynamical evolution of the system follows a normal trajectory,
i.e., remains in the instantaneous eigenstate of H(t). In this sense, the strategies
of the anomalous dynamical evolution and of those existing protocols are complementary
and may have different applications to the design of quantum state control in experimental systems.


To summarize, we have investigated the anomalous dynamical evolution for the
time-dependent quantum systems by virtue of a reverse-engineering method.
Our study demonstrates that the occurring of the anomaly, i.e., the evolution
of the nonadiabatic states exhibits opposite behavior with that of the adiabatic ones,
is always concomitant with the NLC in these nonadiabtically driven systems.
In particular, we have proposed three cases of
such kind of driving protocols. In all these protocols the relative
orientation of the representative vectors of the adiabatic and nonadiabatic
bases undergoes a change from the
parallel state to the antiparallel state, and the nonadiabatic levels
of each system exhibit the anticipated crossing phenomenon.





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