coherent population trapping (cpt) electromagnetically induced transparency … · 2020. 1. 6. ·...

Light interaction with three-level systems:

Coherent Population Trapping (CPT)

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

Stimulated Raman Adiabatic Passage (STIRAP)

Jordi MompartDepartment of Physics, Universitat Autònoma de Barcelona, Catalonia

From July 7th to July 9th (10:00 to 11:30) 2015,

C700, Level C, Lab 3, OIST

MOTIVATION:MOTIVATION:

3 IS MUCH MORE THAN 23 IS MUCH MORE THAN 2

TWO-LEVEL ATOM:

MOTIVATION: 3 IS MUCH MORE THAN 2 (3)

ω|2>

ω

AC-Stark splitting

Dressed atom

Light shifts

Dipole force

ω=|2>, E2

|1>, E1

ω0 ω0

|2 >, E2+ +

|1 >, E1+ +

|2 >, E2- -

|1 >, E1- -

ω

|1>

ω0Dipole force

Rabi oscillations

Qubit gates

Optical nutation

RAP

Radiation pressure

….

ω

|3> Assume the atom initially in:

21)0( 21 cct +==ψ

THREE-LEVEL ATOM:


,0E

|2>|1>

21)0( 21

)(·· 1221221120

2

2120

∗∗∗∗ +++=+= ccccccccEctccEctPabs

)Re2(·

)(·

12221120

2112221120

ρρρρρρρ

++=

+++=

Ect

Ect

populations atomic coherence(quantum interference)

Example: for 0)Re2(· 12221120 =++=⇒ ρρρEctPabs( )21

21

)0( −==tψ

Dark state

ATOMIC COHERENCE EFFECTS

Three-level optical techniques:

• Coherent Population Trapping (CPT)


• Coherent Population Trapping (CPT)

• Electromagnetically Induced Transparency (EIT)

• Stimulated Raman Adiabatic Passage (STIRAP)

Some applications:

• Laser cooling based on CPT

• High precission magnetometry (atomic clocks) based on CPT

• Lasing without inversion based on EIT or CPT

DARK STATE PHYSICS

• Lasing without inversion based on EIT or CPT

• Slow/super luminal light based on EIT

• Quantum memories based on EIT

• Robust and efficient population transfer based on STIRAP

• Subwavelength lithography/microscopy based on STIRAP

• Single photon gun based on STIRAP

• …

recent application: light harvesinglight harvesinglight harvesinglight harvesing


Coherent excitation transfer via the dark-state channel in a bionic systemHui Dong, Da-Zhi Xu, Jin-Feng Huang and Chang-Pu Sun

Light: Science & Applications (2012) 1, e2; doi:10.1038/lsa.2012.2

Light absorption and energy transfer were studied in a bionic system with donors and an acceptor. In

the optimal case of uniform couplings, this seemingly complicated system was reduced to a three-level

Λ-type system. With this observation, we showed that the efficiency of energy transfer through a dark-state channel, which is free of the spontaneous decay of the donors, was dramatically improved…

donors

acceptor

1 REVIEW OF TWO-LEVEL OPTICAL SYSTEMS

OUTLINE OF THE LECTURE


1 REVIEW OF TWO-LEVEL OPTICAL SYSTEMS

4 APPLICATIONS OF THREE-LEVEL TECHNIQUES

2 THREE-LEVEL OPTICAL SYSTEMS

Rabi Oscillations3 LA ‘RIGA NERA’

5 THREE-LEVEL ATOM OPTICS

6 CONCLUSIONS

ω|2>

|1>

ω0ω=|2>, E2

|1>, E1

ω0 ω0

|2 >, E2+ +

|1 >, E1+ +

|2 >, E2- -

|1 >, E1- -|1 >, E1

REVIEW OF TWOREVIEW OF TWO--LEVEL LEVEL

OPTICAL SYSTEMSOPTICAL SYSTEMS11

1.1 TWO-LEVEL ATOM: SEMICLASSICAL THEORY

1. REVIEW OF TWO-LEVEL OPTICAL SYSTEMS (9)

OUTLINE

1.4 RAPID ADIABATIC PASSAGE

1.2 AC-STARK SPLITTING AND DRESSED ATOM

Rabi Oscillations1.3 RABI OSCILLATIONS

1.5 RADIATION PRESSURE

1.1 TWO-LEVEL ATOM: SEMICLASSICAL THEORY

Introduction

Semiclassical TheoryLight classically described: Maxwell Eqs.

Quantum matter: Schrödinger eq. + Bohr’s atomic model


Quantum matter: Schrödinger eq. + Bohr’s atomic model

Radiation pressure Laser cooling of atoms and ions

Dipole force Optical traps for neutral atoms

Some examples of phenomena described by the Semiclassical Theory:

Dipole force Optical traps for neutral atoms

Rabi oscillations Qubit manipulation

Optical nutation Decoherence

Perturbation of a quantum system:

Temporal evolution of the probability amplitudes

Evolution of a free atom

with a Hamiltonian 0H

∂General solution:

n ti− ω∑=

=n

ia 2 1


)()( 0 tHtt

i ψψ =∂∂

h ieatn

i

tii i∑

=

−=1

)( ωψ∑=i

i1

iEiH i=0

22)( iati =ψPerturbation: 0 0H H V with V H≡ + <<

)(taa ii →nn

Then: ( ) ( ) ieaVieaiain

i

tiii

n

i

tiiii ii ∑∑

=

−

=

− +=−11

ωω ωω hh⋅

( )∑=

−−−=n

i

tiik kieaiVk

ia

1

ωωh

⋅

(Electric dipole) interaction of a plane wave light field with

a two-level atom:

ω|2>

ω

Model:


ω

|1>

ω0

)cos(),( 0 kztEtzE −= ωrr

)cos()( 0 tEtE ωrr

=

o

A53.02

0 =>>= aπλ

( ) )(·)()(· tEretEV ADE rrrr −−=−= µElectric dipole interaction:

EDA (Electric Dipole approximation)

A53.00 =>>= ak

λ

∫−=−=−= rdrurrueirjere ijijij3* )()(

rrrrrrrrµ

0)( 32 =−= ∫ rdrurerrrrµ)(rui

rrhas, in general, a well defined parity

Electric dipole moment operator:


0)( =−= ∫ rdrure iiiµ2)(rui

rr is a symmetric function in rr

)(ruirr )(ru j

rrif and have same parity

)(ruirr )(ru j

rrif and have oposite parity

0=ijµr

0≠= jiij µµ rr

=

0 0µµ r

rr

For the two-level atom, we end up with:

=00

0µ

µ rr

For the two-level atom, we end up with:

( )( )

ΩΩ

=0cos

cos0

t

tV

ωω

h

h

h

rr00·Eµ−≡Ωwhere is the Rabi frequency

2)(1)()( 21 21titi etaetat ωωψ −− +=

Atomic evolution

with:

( ) 00cos21 aetiaeV

ia tii t ωω

ω −− Ω=−=⋅


( ) 221 00cos21 aetiaeV

ia tii t ωω

ω −− Ω=−=h

( ) 112 00cos12 aetiaeV

ia tii t ωω

ωΩ=−=

h

⋅

⋅

( ) ( )[ ]00 aeeia titi ωωωω +−−− +Ω=⋅

“Rotating Wave Aproximation” (RWA):

RWA aeia ti∆Ω=⋅( ) ( )[ ] 21 002

aeeia titi ωωωω +−−− +Ω=⋅

⋅ ( ) ( )[ ] 12 002

aeeia titi ωωωω +− +Ω=

RWA

0ωω −≡∆with

being the detuning

21 2aeia ti∆Ω=

12 2aeia ti∆−Ω=

⋅

⋅

Exact solution within the RWA

∆Ω 02

=Ω+∆− aaia

Two-level atom in the electric dipole interaction with a plane wave (EDA+RWA):


21 2aeia ti∆Ω=

12 2aeia ti∆−Ω=

⋅

⋅

02 111 =

Ω+∆− aaia

02 2

2

22 =

Ω+∆+ aaia

⋅

⋅⋅⋅

⋅⋅

General solution:( ) ( ) 2/2/

1 )( titi BeAeta Ω′+∆Ω′−∆ +=1( ) ( ) 2/2/

2 )( titi DeCeta Ω′+∆−Ω′−∆− +=

22 ∆+Ω≡Ω′with being

the Generalized Rabi Frequency

( ) ( )( )( ) ( )( ) 2

1)(

/2/2/

/2/2/

2

1

h

h

tiEtiti

tiEtiti

eDeCe

eBeAet

−Ω′+∆−Ω′−∆−

−Ω′+∆Ω′−∆

++

+=Ψ


1.2 AC-STARK SPLITTING AND DRESSED ATOM

( ) 2eDeCe ++

00 =−=∆ ωω

On resonance light field

|2 >, E+ +

00 >−=∆ ωω

Blue detuned light field

2211Ω′

±∆−=±hhEE

2222Ω′

±∆+=±hhEE

ω=|2>, E2

|1>, E1

ω0 ω0

|2 >, E2+ +

|1 >, E1+ +

|2 >, E2- -

|1 >, E1- -

Ω

Ω

ω >|2>, E2

|1>, E1

ω0 ω0

|2 >, E2+ +

|1 >, E1+ +

|2 >, E2- -

|1 >, E1- -

ω

Ω′

Ω′

“Light Shifts”

Ω>>∆

+

∆Ω+∆≈

∆Ω+∆=Ω′ ...

21

1122

If

Cohen-Tannoudji, Ann. Phys. (Paris) 7 (1962) 423 and 469

( )ΩΩ′∆+22/


( )∆

Ω+≈Ω′+∆−=+

2

1112/

22hhh EEE

( )∆

Ω−≈Ω′−∆+=−

2

2222/

22hhh EEE

0>>∆If

(blue detuned light)

( )∆

Ω+≈−2

112/

hEE

( )Ω+22/

0<<∆If(red detuned light)

Dipole Force

( )2

2

12/

EE δδ −=∆

Ω= h )(1

)(4

)( 21 rIrErF rrrrhrrr

∇∆

−∝Ω∇∆

−=∇−=

( )∆

Ω−≈+2

222/

hEE(red detuned light)

Dipole Force.

Example 1: Optical Lattices

Stationary wave: interference of two counter-propagating lasers

If Atoms move towards the anti-nodes0<<∆


http://www.mpq.mpg.de/~haensch/bec/experiments/lattice.htmlImmanuel Bloch in Ted Hänsch's group at the LMU Munich and the MPQ in Garching

0>>∆If Atoms move towards the anti-nodes

Atoms move towards the nodes

0<<∆If

Wavelength: 850 nm

(approx. 60 nm detuning)

Lattice Spacing: 425 nm

Lattice type: simple cubic

Beam waist: 120 µm

Dipole Force

Example 2: Optical Microtraps

G. Birkl and W. Ertmer, Universität Hannover

Trapping at the focii of a microlenses array


- P = 1 mW per trap

- Trap width = 7 µm (1µm)

- 85Rb

- Diode laser of 1-10 mW

- 100 µm lense separation

154 1010 −−= sω- Trapping frequency:

Fluorescence reported in the experimentRubidium atoms

G. Birkl, F.B.J. Buchkremer, R. Dumke, M. Volk, W. Ertmer, Optics Comm. 191, 67 (2001)

R. Dumke, M. Volk, T. Müther, F.B.J. Buchkremer, G. Birkl, W. Ertmer,

Phys. Rev. Lett. 89, 097903 (2002)

- Trap depth = 1 mK

- Atoms per trap: 100 (1)

154 1010 −−= sxωxωω 10≈⊥

Microlenses

Principal laser, red-detuned

Rabi Oscillations

ω|2>

|1>

ω0( ) ( )( )

( ) ( )( ) 2

1)(

/2/2/

/2/2/

2

1

h

h

tiEtiti

tiEtiti

eDeCe

eBeAet

−Ω′+∆−Ω′−∆−

−Ω′+∆Ω′−∆

++

+=Ψ


1.3 RABI OSCILLATIONS

Ω′Ω=

Ω′Ω−=

Ω′∆−Ω′

=Ω′

∆+Ω′=

2222DCBA

====

0)0(

1)0(

2

1ta

taSi

1p2

2sin2

2t

Ω′

Ω′Ω=)( *

222 cctp ≡

|1>

1/41/9

t2πΩ

4πΩ

6πΩ

2sin t

Ω′=)( 222 cctp ≡

Rabi Oscillations

Example: Single 40Ca+ Ion in a Paul Trap


F. Schmidt-Kaler et. al. arXiv:quant-ph/0003096 (21/March/2000)

JOURNAL OF MODERN OPTICS, 2000, VOL. 47, NO. 14/15, 2573 -2582

Light pulses of well defined area

2~2

sin1~2

cos

22

sin12

cos)( // 21

Ω+

Ω=

Ω+

Ω= −−

tit

etiett tiEtiE hhψ0=∆Si


ππ

ππ

2/3

2/

0

=Ω=Ω=Ω=Ω=Ω

t

t

t

t

( )( )

( )( )~

2~1~2/2)(

2)(

2~1~2/2)(

1~)(

−=Ψ⇒

+−=Ψ⇒

=Ψ⇒

+=Ψ⇒

=Ψ⇒

it

it

it

t

22

sin12

cos

+

= tit

Hadamard gate

Phase flip

Bit flip

Singel qubit operations

πππππ

4

2/7

3

2/5

2

=Ω=Ω=Ω=Ω=Ω

t

t

t

t

t

( )( )

( )( )1~)(

2~1~2/2)(

2~)(

2~1~2/2)(

1~)(

=Ψ⇒

−=Ψ⇒

−=Ψ⇒

+−=Ψ⇒

−=Ψ⇒

t

it

it

it

t Phase flip

Light pulses of well defined area

Example: Cavity Quantum ElectroDynamics (CQED)

Rydberg states (n=50,51)

Two-level atom Superconducting Fabry-Perot

resonator

S. Haroche Group. ENS & College de France


|e>

|g>

micrones

Rydberg states (n=50,51)

Quantum Rabi oscillations

ep

resonator

5 cm

Atom-photon entanglement

e 0⊗=ψ

1+=Ω ng

ep

)( sti µ

campatomin e 0⊗=ψ

( )102

1 ⊗+⊗= gieoutψ

Hagley et al., Phys. Rev. Lett. 79, 1 (1997)

2/π

Optical Nutation

ω|2>

ω0

Two-level atom + spontaneous emission

γ=1/τWhich is the probability dp that an atom

suffers a spontaneous emission in a dt?


|1>

ω0 γ=1/τ. 2· ( )·spon emissiondp p t dtγ=

)(2 tp

Collection of two-level atoms + spontaneous emission

stationaryp2


1.4 RAPID ADIABATIC PASSAGE

ω|2>

ω0

21 2aeia ti∆Ω=

ti∆−Ω

⋅

ωω −≡∆h

rr00·Eµ−≡Ω

N. V. Vitanov, T. Halfmann, B. W. Shore, K. Bergmann, Annual Review of Physical Chemistry, 52, 763 (2001)

|1>

ω012 2

aeia ti∆−Ω=⋅

∆−Ω−Ω−

=

2

1

2

1

2/

2/0

a

a

a

ai&

&

Taking out ~ for simplicity

0ωω −≡∆

tieaa

aa∆↔

↔

22

11

~

~ 21~

2~ aia

Ω=&

212~~

2~ aiaia ∆+Ω=&

Change of variables:

Two state Hamiltonian

2

2sin1cos

2cos1sin

ϑϑϑϑ

−=−

−−=+ ( )22

2

1 Ω+∆±∆−=±E

Eigenstates and eigenvalues:

( )∆Ω=ϑ2tanwith and


ω|2>

|1>

ω02sin1cos

2cos1sin

ϑϑϑϑ

−=−

−−=+

( )221 Ω+∆±∆−=±E

Eigenstates and eigenvalues:

( )∆Ω=ϑ2tanwith and

0ωω −≡∆h

rr00·Eµ−≡Ω

( )2

Ω+∆±∆−=±E( )∆

=ϑ2tanwith and

Adiabatic theorem by Max Born and Vladimir Fock:M. Born and V. Fock. Zeitschrift für Physik, 51, 165-180 (1928)

“A physical system remains in its instantaneous eigenstate if a given

perturbation is acting on it slowly enough and if there is a gap between the

eigenvalue and the rest of the Hamiltonian's spectrum”

Rapid Adiabatic Passage (RAP). Assume: )();( tt Ω=Ω∆=∆

Example: the atom initially at

1)()(0)(0 0000 =−=Ψ⇒=⇒Ω>>∆>∆ tttt ϑ1)( 0 =Ψ t

At : with

At : with 2)()(2

)(0 −=−=Ψ⇒=⇒Ω>>∆<∆ ffff ttttπϑ

Consists in adiabatically following or from to

Rapid Adiabatic Passage (RAP). Assume:

−+ 21

)();( tt Ω=Ω∆=∆

On resonance force (nothing to do with the dipole force that works

far from resonance):

1. Light absorption 2. Spontaneous emission


1.5 RADIATION PRESSURE

ivr ω fvr

kr

hrr −= if vv

Doppler effect: ·vkω ω′ = −r r

Laser cooling:

ω ω0ω

( )ωω >0(preferential absorption)

“Optical Molasses”

T. Hänsch, A. Schawlow, Opt. Commun 13 (1975) 68.

Example:

Magneto Optical Trap (MOT)

W. B. Phillips Group, “Laser Cooling and Trapping”, NIST


http://www.colorado.edu/physics/2000/index.pl

(The Atomic Lab / Bose Einstein Condensate)

http://physics.nist.gov/Divisions/Div842/Gp4/group4.html

Na atoms

T < 1 mK

coherent population trapping (cpt) electromagnetically induced transparency … · 2020. 1. 6. ·...

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