Adiabatic perturbation theory: a new tool in quantum mechanics

Adiabatic perturbation theory: a new tool in
quantum mechanics
Gustavo Garcia Rigolin

Structure of the talk
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The adiabatic approximation (non degenerate);
The adiabatic approximation (degenerate);
Standard strategies to correct the adiabatic
approximation (in a nutshell);
The adiabatic perturbation theory (APT);
– Exactly solvable model justifying APT;
– Numerical examples (in a nutshell);
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The degenerate adiabatic perturbation theory (DAPT);
– Exactly solvable model justifying DAPT;
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Corrections to Berry phase and Wilczek-Zee phases.

References
Beyond the Quantum Adiabatic Approximation:
Adiabatic Perturbation Theory
G.R., Gerardo Ortiz, Victor Hugo Ponce
Physical Review A 78, 052508 (2008);
(Arxiv: 0807.1363v1 [quant-ph]).
Adiabatic Perturbation Theory and Geometric Phases for
Degenerate Systems
G.R., Gerardo Ortiz
Physical Review Letters 104, 170406 (2010);
(Arxiv: 1001.5255v2 [quant-ph]).

The adiabatic approximation
If a system's Hamiltonian H(t) slowly changes
slowly changes with time,
say, from t=0 to t = T, and the system starts in one of the
eigenvectors of H(t), then during the whole evolution it will
stay in the corresponding snapshot/instantaneous
eigenvector of H(t). In particular, for a cyclic evolution
of H(t),

The adiabatic approximation

The adiabatic approximation for
degenerate systems
If a system's Hamiltonian H(t) slowly changes
slowly changes with time,
say, from t=0 to t = T, and the system starts in one of the
eigenvectors of H(t), then during the whole evolution it will
stay in the corresponding snapshot/instantaneous
eigenspace of H(t).

Formalizing the Adiab. Approx.:
Rescaled Schrödinger Equation
where

The “ Berry ansatz”: no-degeneracies
where
snapshot
eigenvector
and

Geometric and dynamical phases

No degeneracies
Eigen-energies
Eigen-energies
time
time

Degeneracies
Eigen-energies
time

What are the b n
(s) coefficients?
Coupled Eqs.
Comes from
Schroedinger Equation

The “strong” adiabaticity condition

Starting at the ground state

Another way of expressing the strong
adiabaticity condition

Usual ways of correcting the adiabatic
approximation: standard approach
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1) We formally integrate the Schroedinger Equation
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wich results in

Standard approach
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2) We use the following identity (chain rule)

Standard approach
● Which gives (up to first order)
Time dependence at the r.h.s

Standard approach
● 3) Solve it iteratively up to order 1:

The wave function to order one

The wave function to order one: starting
at the ground state |0(0)>
No term with n=0

Other methods
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“Iterative rotating-basis method”

Problems with this method
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It is not a perturbative method in the the small
parameter v=1/T.
Only “asymptotically convergent”; After a few
iterations the corrections get worse and worse.

Dyson series
Good perturbative corrections if H(t) = H(0) + λV(t),
with λ

Dyson series: drawbacks to correct the
adiabatic approximation
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Not a perturbative expansion about the adiabatic
approximation. Snapshot eigenstates not used as the
basis for the perturbative correction;
Not suitable as a perturbation series if H(t) cannot
be put as H(0) + λV(t).
Example:

Dyson series: drawbacks to correct the
adiabatic approximation

Dyson series: drawbacks to correct the
adiabatic approximation
For w

Adiabatic perturbation theory
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Relevant properties:
– “Easily” generalizable to degenerate Hamiltonians;
– It has physical appeal; it is intuitive; true perturbative
expansion in v=1/T;
– No differential equations; all correcting terms come from
algebraic recursive relations;
– It is operational; can be applied to real problems; not
only a mathematical “beauty” (or “beast”); it is
numerically stable;
– Transforms the time dependent Schrödinger equation
into a set of eigenvalue/eigenvector problems.

Insights leading to APT
● Come up with the right ansatz for ;
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What sort of properties should the ansatz have?
– It should make it explicit the dependence of
with respect to terms of order
and lower ;
and lower

The Ansatz

The Ansatz

Initial conditions

Adiabatic approximation must be the
zeroth order

Inserting the ansatz into the
Schroedinger Equation
Problem for v → 0? No!

The recursive relations

The order 1 correction
History (integration) of
evolution is important

Order 1 correction starting at |0(0)>

The wave function up to order 1

Sufficient conditions to the validity of
the adiabatic approximation

The example justifying the APT
z
B
θ
φ
y
x

The model's time dependent
Hamiltonian

Snapshot eigenvectors and eigenvalues

Exact solution with initial state |0(0)>

Expanding the exact solution
where

Expanding the exact solution

Expanding the exact solution

Corrections up to order 2 via APT

Corrections up to order 2 via APT

Corrections up to order 2 via APT

Corrections up to order 2 via APT

Corrections up to order 2 via APT
Putting everything together and comparing with the expansions coming
from the exact result we realize that we get the same expressions!

Numerical examples
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Three cases whose Hamiltonian looks like
● All three cases we start at the ground state |0(0)>

Infidelity between the exact solution
and what we get from APT
Exact solution
Normalized state coming from APT

Numerical results
t t²
order 0
order 1
t³
order 2

Numerical results: only order 2
t¹
t²
t³

Degenerate adiabatic perturbation
theory: a bit of notation
The “vector of vectors” notation:

The degenerate adiabatic approximation
in this notation

The DAPT ansatz

The DAPT ansatz

Properties of the ansatz
Initial conditions:
Starting at the ground state :

Schrödinger equation in matrix form
and the recursive relation
Insert the ansatz into SE and
get recursive relations by left
multiplying to

What comes out of the recursive
relations?
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For p = 0 and n=m we get the Wilczek-Zee phase
and the zeroth order as the adiabatic approximation

What comes out of the recursive
relations?
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For p = 0 and n m and p=1 with n=m we get the
first order correction:

What comes out of the recursive
relations?

Dirac matrices
Exactly solvable problem

Pictorial view of the exactly solvable
model
Eigen-energy
Time

Eigenvectors and eigenvalues

Exact solution starting at |0 0 (0)>

Expansion of the exact solution up to
first order
Matches exactly what comes from DAPT.

Expansion of the exact solution up to
first order
Matches exactly what comes from DAPT!

Corrections to the Berry phase
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Key concept: Aharonov-Anandan phase (state
returns to itself)

Corrections to the Berry phase

Berry phase corrected up to order 1

How to measure the correction
B 2
is such that the upper
beam acquires the proper
dynamical phase
B 1
is rotated back and
forth not too slowly

Corrections to the Wilczek-Zee phase
Since we have the corrected state up to first order we can get
the corrected WZ phase too. To first order we have,
Adiabatic state
What is actually seen

Corrections to the Wilczek-Zee phase
To order zero we had,

Corrections to the Wilczek-Zee phase
Keeping terms up to first order,
Giving the following non-abelian phase,

Final considerations
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We are looking for quantum phenomena thought to
be adiabatic, but whose experimental results do not
agree with theoretical predictions relying solely on
this assumption;
We are looking forward generalizations of APT to
the degenerate case (done: DAPT); mixed states;
open systems; relativistic quantum mechanics.