PT-Quantum Mechanics

In 1998, Bender and Boettcher found that a wide classof Hamiltonians, even though non-Hermitian, can stillexhibit entirely real spectra if they obey parity-timerequirements or PT symmetry.

Parity Time (PT)Optics

Demetri Christodoulides,

Kostas Makris, Ramy El-Ganainy, and George Siviloglou,CREOL/ UCF

Roberto Morandotti,

David Duchesne, INRS-Canada

Greg Salamo,Aqiang Guo, K. Yeruva,Department of PhysicsUniversity of Arkansas

PT-

OpticsPT-symmetrysystem

Real part: evenImaginary part: odd Gain/loss

One example a PT Optical Potential is thePT directional coupler

Gain Loss

Gain = Loss

Index

x

Passive PT directional couplerExperiment measurable --- Output Power

TotalOutputPower

Lossless

Loss

PT-

OpticsHow should we make the PTdirectional coupler?

Gain Loss

Gain = Loss

x

We rely on Nanoscience

The effort to understand and designstructures at the nano size and seektheir application

Line up 50 atoms end-to-endand you get one nanometer

Take the diameter of a hair and divideby 100,000 and you have a diameter ofnanometer size

Why Nanoscale Materials

and cut it in half,and then halfagain ….and yetin half again….until you havenanosize

Same element but

totally different

properties

Element in the Periodic Table

Take any element or compound

the same element orcompound will havevery different optical,electrical, ormechanicalproperties dependingon its size!

The Search for Underlying Rulesat the Nanoscale

CdSe CdSe –– but each a different size! but each a different size!

Why this Change in Behavior?New Rules When We Go

Very Small

Easy to Cause Flow

If it’s Small it is Difficultto Cause Flow

?

One of our Arkansas Growth & Fabrication& Imaging Facilities

Molecular Beam Epitaxy (MBE)

Source of Atoms

III

V

substrate

Heater

Beam ofAtoms

substrate

Mono-Layer

13.5

nm

0

nm

15nm

13.5

nm

0

nm

13.5

nm

0

nm

15nm200 nm x 200nm

Strainrelaxation

Surface energy

Stablesurfacesor facets

What Nanoscience can do for you in researchExplore the new rules at the nanoscale

Starting GaAs

InAs

2.2 ML InAs

on AlAs2.1 ML InAs

on AlAs

Scanning Tunneling Microscopy(STM)

A

Piezo

Tip

= 5 Å

Bias+

–

200nm

Even More Than Size = New Behavior!We can form Molecules or Chains or Solids

made of Quantum Dots

Dots in aPeacock Feather

400nm

400nm

Arkansas GrownQuantum Dots

Color in a peacockfeather

PT-

OpticsHow should we make the PTdirectional coupler?

No Loss Loss

x

Design of a PT Coupler

3.5 μm

1.88 μm Al0.20Ga0.80As

Al0.26Ga0.74As

GaAs: substrate

Tuning loss byVarying Cr-Width

8 μμ 2.0 μμ

0.62 μm

index=3.3695

index=3.24956

index=3.27739

62°

The introduction of loss must be done in way that does not perturb the even refractiveprofile. This is physically demanding since The presence of loss is typicallyaccompanied by an index perturbation (because of the Kramers-Kronig relations).

Top-view: Loss = 35 cm-1

Cr width = 4 μμ

2 μμ

ESEM cross view

ESEM image of one PTcoupler

Experimental Set-Up

IPG Fiber Laser IR Camera

X40 Obj X20 Obj

Sample

Detector

CylindricTelescorpe

Aperture

B.S

Ar Laser

Camera

B.S

Sample image (top)

Loss of Isolated waveguide structure as a function of Crwidth

Lossless

Loss

Experimental observation of PT-symmetry breaking

Conclusions

•We show for the first time that passivePT-symmetry breaking can be observedwithin the realm of optics.

•This abrupt phase transition leads to acounterintuitive loss induced opticaltransparency in specially designedpseudo-Hermitian potentials.

for PT

PT symmetry in QM PT symmetry in Optics

Schrödinger-like equations appear in In optics (paraxial equation )

Potential V(x)

Index of refraction n(x)

Passive PT directional couplerExperiment measurable --- Output Power

TotalOutputPower

Lossless

Loss

Real and imaginary parts of the optical dielectric function ofCr

Appl. Opt. 37, 5271-5283 (1998)

Choose Cr:

At 1.55mm, themetal leads to heavylosses while the realpart of indexmismatch atminimum!

At 1.55μμ

Gain LossLoss = Gain

PT directional coupler supermodes below and above phase transition

below phase transitionabove phase transition

Gain Loss

Gain = Loss

Lossless Loss

Passive PT directional coupler: Gain = 0

Lossless (Gain=0) and LossLossless Loss

Passive PT directional coupler: Gain = 0

below phase transition above phase transition

Output Power?

Z

Passive PT directional coupler

Supermode at z (below phase transition)

Lossless

Loss

Input

Output Power?

Z

Passive PT directional couplerExperiment measurable --- Output Power

Supermode at z (above phase transition)

Lossless

Loss

Input