Lecture 18

Lecture 18: Adiabatic Population Transfer in Few-Level Quantum Systems

** Application of adiabatic approximation to few-level quantum systems

Excuse me, there are no true few-level systems, right?

** Example: matter transport “without” transit: Phys. Rev. A, 2008.

Consider a three-level system similar to what we discussed before:

making RWA

A three-level systeminteracting withtwo on-resonancelaser fields

continued on the next slide

pump field

Stokes field

Introducing the dressed-state picture:Introducing the dressed-state picture:

we then have:

oscillating terms no longer there

Using the on-resonance conditionsUsing the on-resonance conditions

Because the reference point of energy levels is arbitrary, for convenience let

us set

Effective Hamiltonian becomes

Three-level atominteracting with twoon-resonance fields

pump field

Stokes field

To apply the adiabatic approximation, find first To apply the adiabatic approximation, find first all the eigenstates of the effective Hamiltonian:all the eigenstates of the effective Hamiltonian:

Two nonzeroeigenvalues:

with eigenvectors:

One zero/null eigenvalue:

with the eigenvector:

no degeneratestates! good for adiabatic approximation !

So all eigenvalues and eigenfunctionsfor arbitrary field parameters are found here

X

STRAP: Stimulated Raman Adiabatic Passage: STRAP: Stimulated Raman Adiabatic Passage:

Counter-intuitive ordering of the pump and the Stokers field

EpEs

Note: we first slowly turn on the coupling between |2> and |3>and then we slowly turn on the coupling between |1> and |2>

pump field

Stokes field

Stokes field

|1>

|2>

|3>

Basic picture of STIRAP in terms of the adiabatic approximation:Basic picture of STIRAP in terms of the adiabatic approximation:

InitiallyInitially

FinallyFinally

= |1>

= |3>

adiabatic state

|3>|3>

|2>|2>

|1>|1>

If we adiabatically turn on the two coupling If we adiabatically turn on the two coupling fields in the counter-intuitive order, the fields in the counter-intuitive order, the system dressed by the field will remain on the system dressed by the field will remain on the null eigenstate. Hence, we will be able to null eigenstate. Hence, we will be able to realize a complete population transfer from realize a complete population transfer from state |1> to state |3>. Because what we rely state |1> to state |3>. Because what we rely on is the null eigenstate that has a node on on is the null eigenstate that has a node on state |2>, during the population transfer state state |2>, during the population transfer state |2> will not be populated (approximately) |2> will not be populated (approximately) !! !!

Understanding STIRAP:

Extensions of STIRAP to many-level systems:Extensions of STIRAP to many-level systems:

N intermediate states

Degeneracy N

Initial state

Gong and Rice, Phys. Rev. A (2005).

What if the Stokes field amplitude is always What if the Stokes field amplitude is always much larger than the pump field amplitude? much larger than the pump field amplitude?

InitiallyInitially

FinallyFinally

= |1>

= |1>

What does this analysis imply?

Electromagnetically Induced Transparency (EIT): Electromagnetically Induced Transparency (EIT):

The effect of EIT on a typical absorption line. A weak probe normally experiences absorption shown in blue. A second coupling beam induces EIT and creates a "window" in the absorption region (red). This plot is a computer simulation of EIT in an InAs/GaAs quantum dot

Matter transport in a triple-well systemMatter transport in a triple-well system

Tunneling rate from well 1 to well 2 is

Tunneling rate fromwell 2 to well 3 is

Assuming the same on-siteenergy for the three wells, the Hamiltonian is

Matter Transport Without Transit (Rab et al, Physical Review A, 2008)Matter Transport Without Transit (Rab et al, Physical Review A, 2008)

Adiabatically tuning the two tunneling rates, to ensure that the system stays on the null eigenstate, and to ensure thatthe tunneling between 2 and 3 is turned on first. Then just as in STIRAP, all matter can be transported from well 1 to well 3, without ever being in well 2 !

Matter transport without transit: Matter transport without transit:

A realistic calculation taking into account A realistic calculation taking into account the weak interactions between the cold atoms:the weak interactions between the cold atoms:

Movie by Rab et al, downloaded from Electronic Physics Auxiliary Publication Movie by Rab et al, downloaded from Electronic Physics Auxiliary Publication Service of Physical Review (EPAPS), American Physical SocietyService of Physical Review (EPAPS), American Physical Society

Tutorial Classroom Discussion: Adiabatic behavior in a Landau-Zener process

In this Hamiltonian, ∆ describes the coupling strength between two levels, and 2γ describes the energy bias between the two levels.

Applying adiabatic approximation, find what happens if we slowly

change the bias parameter γ from to . The initial state of

the system is or .

Concluding remarks:Concluding remarks:

Physical insights based on adiabatic approximation are playing a key Physical insights based on adiabatic approximation are playing a key role in today’s two related physics research frontiers: quantum control role in today’s two related physics research frontiers: quantum control and quantum information science. and quantum information science.

This is largely because (i) adiabatic manipulation of quantum systems is This is largely because (i) adiabatic manipulation of quantum systems is often robust to small fluctuations in the driving field parameters and (ii) often robust to small fluctuations in the driving field parameters and (ii) the physical picture based on the adiabatic approximation is non-the physical picture based on the adiabatic approximation is non-perturbative and easy to understand. perturbative and easy to understand.