classical optics – quantum physics analogies daniela dragoman univ. bucharest

CLASSICAL OPTICS – QUANTUM PHYSICS ANALOGIES

Daniela Dragoman

Univ. Bucharest

Classical optics-ballistic electrons analogiesClassical-quantum optics

Classical optics-electrons analogies

electrons photons

wavefunction scalar vector

energy distribution fermions bosons

charge e 0

spin 1/2 1

effective mass m 0

dispersion parabolic linear

exclusion principle yes no

022 AA k

Helmholtz

0)(2

22

EVm

Electron microscope, 1933 Z. Phys. 87, 580 (1934)

Electron opticsErnst Ruska, Nobel Prize 1986

/)](2[ 2/1VEm

A k

2

22

t A

A

MaxwellSchrödinger

tiV

m

2

2

2

Y. Ito, A.L. Bleloch, L.M. Brown, Nature 394, 49 (1998)

f = 0.25 mm f = 1 mm

Fresnel electron lenses

Classical optics-electrons analogies

Electron holography

Other analogies: D. Dragoman, M. Dragoman, Quantum-Classical Analogies, Springer (2004)

Photonic crystals

S. Matthias, F. Müller, U. Gösele, J. Appl. Phys. 98, 023524 (2005)

t

x

Talbot effect

M. Berry, I. Marzoli, W. Schleich, Physics World, June 2001

W.D. Rau et al., Phys. Rev. Lett. 82, 2614 (1999)

A.C. Twitchett et al., Phys. Rev. Lett. 88, 238302 (2002)

Ballistic electrons

visible light

tiVm

)]/([2

2

Schrödinger

Dirac

),( yx

Fti

vi

σ

σ

K.S. Novoselov et al., Science 306, 666 (2004)

E1 E E2 E3

2D

z

x y

EF

3D quantum well

22

2

zp

L

p

mE

E E11 E12 E13

1D

x y

z

quantum wire

222

2 zxpq

L

q

L

p

mE

E E111 E112 E113

0D

x

y

z

quantum dot

2222

2 zyxpqr

L

r

L

q

L

p

mE

mesoscopic structures

NNEEnn /'/'/''sin/sin

Classical optics-Schrödinger electrons analogies

022 AA k

Helmholtz

EVm)]/([2

2Schrödinger

/)](2[ 2/1VEm A cnk /

n1

n2

n3

n4

0

V1

V2

V3

V4

E

J.H. Smet et al., Phys. Rev. Lett. 77, 2272 (1996)

magnetic lens

J. Spector et al., Appl. Phys. Lett. 56, 1290 (1990)

electrostatic lens

J. Spector et al., Appl. Phys. Lett. 56, 2433 (1990)

electron prism

Y. Ji et al., Nature 422, 415 (2003)

electron Mach-Zehnder

B

electron trajectory

S detector array

GRIN axis

2DEG

y

GRIN waveguide/FRFT

D. Dragoman, M. Dragoman, J. Appl. Phys. 94, 4131 (2003)


quantum TE: y component of E

TM: y component of H

A (/)1/2A

2(E‒V)/ ħ

m

1/[2(E‒V)]

quantum TEM

E

k

/m k/

vg vgo

/)( AHA iE

/)](2[ 2/1VEm A

k

z

t r

x

i

medium 2 medium 1

r t

i

boundary conditions

quantum: ,

TE: A,

TM: ,

m/z

/zA

/)( A /ˆ)( zA

dx vg/

EkH ˆ/

2*2 ||/]/Re[||/ miJvg

)2/||/(]2/)/Re[(|//1 22* AAA iWSvgo

group velocities

)]sincos(exp[ˆ),( iii xzikAzx yA

)]sincos(exp[),( iii xzizx

)]sincos(exp[)cosˆsinˆ(),( iiiii xzikAzx xzA

wavefunctions quantum:

TE:

TM:

D. Dragoman, M. Dragoman, Prog. Quantum Electron. 23, 131 (1999)

in addition: condition of same phase across one layer i i zi ioL k L

n

n

N n

n N

m

m

V E

E Vw

b

b

w

w

b

b

w

2

2

2 2

2 2

)()(

)()( 222

EVaVE

EVanVEnN

bw

bwwb

a n m n mb w w b 4 4/

)()()(

2222

22

0NnanN

nnN

wb

bw

wVE

0)( wV

211

22

1 ||

||||

A

AtT NNN

matrix method same field, same T, same

Classical optics-quantum well analogy

V E Vw b wxb nckn /

k c N n nx b b w w/ sin sin

)exp()exp( ziBziA iiiii

• TM excitation• normally incident electrons

0.5

10

5

log s

E (eV)

0 1

0

logT

1.2 1.4

10

5

logT

log s

N

1

0

GaAs/AlAsLb =25 Å, Lw = 45 Å, mw = 0.067m0, mb = 0.15m0, Vb = 1 eV, Vw = 0

nb =1, nw =1.5 0=2c/0 = 1 m

Lbo = 0.44 m

Lwo = 0.16 m

D. Dragoman, M. Dragoman, Opt. Commun. 133, 129 (1997)


Classical optics-quantum wire analogy

z

x

L Lw Lb

)()(

)]()/([)]()/([ 22222

bw

bowwob

EEaEE

EELcpnaEELcpnN

a m n m nw b b w 4 4/

2)2/( www mLpVE

2)2/( bbb mLpVE

)()(

)/)(1()(2222

22min

222

NnaNn

Lcpann

wb

owb

max min / a

0.2 0.4 0.6

10

5

0

-12

-2

-7

logT

log s

E (eV)

3.4 3.5

10

0

log s

logT

3.6

-2

-12

N

GaAs/AlAs L = 100 Å, Lw = Lb = 100 Å

nw = 3.6 (GaAs), nb = 3.47 (Al0.2Ga0.8As) 2c/max = 1 m

p = 1 Lwo = 1.02 m, Lbo =1.34 m p = 2 Lwo = 0.94 m, Lbo = 1.25 m

Lo – not determined directly; chosen to match the optical beam width

D. Dragoman, M. Dragoman, IEEE J. Quantum Electron. 33, 375 (1997)

dzlxqdzit

dzdLxjIztzdrl

zlxqzirlxpzi

zx

qqq

jqjqjjqjqpq

qqqp

p

,)/sin()](exp[

0,)sin(/)/sin()}sin()](sin[){()/2(

0,)/sin()exp()/sin()exp(

),(,

)(

)1)(1(222

21

2

222

nnK

KamE

l

lc

o

22 )/(/2 lpmEp 22 )/(/2 LjmEj

)/(

)]/(sin[

)/(

)]/(sin[

2 Ljlq

Ljlq

Ljlq

Ljlql

lqj dxLxjlxqI

0)/sin()/sin(

z l l

L

d x

)1)(1(

)1()( 22

21

222

aK

annanKN

a n n 14

24/

22 )/()/2( lpmEK

21

22

22

min)1(

2

1

nan

an

l

p

mE

l L l Lo o/ /

d dl l dL Lo o o / /

)/(/ 21

22 zjzpjp nnk

q

qqq

qq intT 22 ||||

Classical optics-quantum wire analogy

lo = 5 m Lo = 10 m, do = 5 m, independent of p

D. Dragoman, M. Dragoman, IEEE J. Quantum Electron. 33, 375 (1997)

0.1 0.2 0.30

1

2

E (eV)

T

log s

14.5

13.5

12.5

0

0.5

1

1 20

1

2

-14.5

-13.5

-12.5

1.5 0.5 0

0.5

1

N

T

log s

GaAs/AlAsl = d = 100 Å, L = 200 Å

nw = 3.6 (GaAs)nb = 3.47 (Al0.2Ga0.8As)

Classical optics-quantum dot analogy

x

z

Lw Lb Lx

Ly

Lb

)(')(

)('

)(1

2222

wb

wb

bw

nmoEEaEE

EEn

aEE

nc

22 )/()/( yxnm LmLn 22 )/()/( yoxonmo LmLn

)/(' 22wwbb mnmna

E V mb b nm b 2 2 /

wnmww mVE 2/2

max /c nnmo b min /c nnmo w i i zi io iL k L

2min

2max

2max

22min

2

2min

2 1

)(')(

1))((2)(

aEEm

Lwbw

wwbw EE 0

wb EE 0

2max

2min

2min

22max

2

2max

2 1

)()('

'))((2)(

a

aEEm

Lbwb

bb

0.5 1

5

0

-4

-5 -9

0

0.25 0.5 1 0.75

logT

log s

E (eV)

nw = 3.6 (GaAs)nb = 3.41 (Al0.3Ga0.7As) Lxo = Lyo = 0.5 m

Lwo = 2.16 m, Lbo = 0.68 m2c/min = 1.27 m2c/max = 1.2 m


1.02 1.04

5

0

-4

-9

1 1.02 1.04

0

-5

w

logT

log s

w / min

GaAs/AlAs

Lx = Ly = 100 Å, Lw = 100 Å, Lb = 20Å

x

z

Lx

Ly

d

lx

ly

1 2

dzlyqlxpzit

dzLyqLxpzBzA

zlyqlxpzirlymlxnzi

yx

qpyxpqpq

qpyxpqpqpqpq

qpyxpqpqyxnm

,)/cos()/cos()exp(

0,)/cos()/cos()]cos()sin([

0,)/cos()/cos()exp()/cos()/cos()exp(

),(

,1

,22

,11

222111 )/()/(/)(2 yxnm lmlnVEm 222

222 )/()/(/)(2 yxpq LqLpVEm

nmqp

pqpqtT 1,

12 /||

l L l Lx x xo xo/ /

yoyoyy LlLl //

d dl l dl lo xo x yo y / /

222

121

1

1)(2

yxxo

x

l

m

l

n

L

lVE

m

n

c

22

21 )/()( mlml xxo

222

2

21

2

1

2212

22 1)(

2)(

yxxo

x

xo

x

l

m

l

n

l

l

c

n

m

mVV

mc

l

ln

Classical optics-quantum dot analogy

0.2 0.25 0.36

4

2

0

-8

-10

-12

-14

logT

log s

E (eV)

n1=3.41 (Al0.3Ga0.7As) n2=3.6 (GaAs) lxo= 0.5 m

lyo = 0.5 m Lxo = Lyo = 1 m do = 0.5 m


1 1.2 1.46

4

2

logT

log s

-8

-14

-12

-10 -2

-4

-6

0

1 1.2 1.4

w w / min

Al0.12Ga0.88As/GaAs

m1 = 0.076m0

m2 = 0.067m0 V1 = 0.1 eV, V2 = 0 lx = ly = 100 Å Lx = Ly = 200 Å d = 100 Å

type I

Eg1 Eg2

Ec

Ev

type II

Eg1

Ev Eg2

Ec

type III

Eg1

Eg2 Ev

Ec

Classical optics-ballistic electrons analogy: type II and III heterostructures

PEEC

PEEC

b

b

cj

vj

v

c

/)(

/)(

21

12

2

1

)()(

)()(20

20

20

20

20

20

220

20

22

NKNK

NKNKN

wwbb

wwbbbw

2112 CC

02/)( 21

)/())((

)/())((20

2000

22

jjcjvj

jjcjvjj

EEEE

EEEEK

w

cwwvw

r

ErEE

1

)()(

)()(

cbcwwcbvw

vbcwwvbvwb

EErEE

EErEEr

AlGaAs: d1 = 6 m, d2 = 0.5 m, D = 4 mncl = 3, nco2 = 3.1, ncl1 = 3 Lwo = 2 mm, Lbo = 0.32 mm

E (eV) rw

rb

www CCr 2112 /

bbb CCr 2112 / nco1

ncl2

InAs/AlSb: Lb = 10 Å, Lw = 15 Å

D. Dragoman, J. Appl. Phys. 88, 1 (2000)

v

c

c

v

v

c

P

EEP

EE

dz

di

0

0

1210 )/2( gEPmm

quantum

jcjvjj PEEEE /))((

22/ NLL jjjjojj

jjvjcj CCEEEE 1221 /)/()(

jjj CC 2112

jjN sin

conditions

2d2

2D

2d1

n

y

ncl1

nco1

nco2

ncl

x z

b b w w

optics: coupled modes

)exp()exp( 222111 ziazia eeE

]2/)(exp[)()( 21 zizazb ii

2

1

21

12

2

1

b

b

C

C

b

b

dz

di

A

mnnnm dAnnk

C ee*22

0

0 )(4

Classical optics-Dirac electrons analogy

2

1

2

1

2

1 )(0

0VE

ikk

ikkvv

yx

yxFF kσ

Vk

ktz

y

x

2

21 yx EE

)](exp[)/sgn(0

01

1212 issM

)exp(0

01

iM ret

)exp()exp(

1

2

1

2

1 yikxikis

yx

)sgn( VEs )/(tan 1xy kk2222 /)( yFx kvVEk

quantum states

)exp(

1

2

1

iE

EJ

y

xpolarization states

0)()())exp(1(2))exp(1(2 **2/12/1

TTTT ii orthogonality

y

x

y

xr

y

xopt

y

x

E

E

i

i

E

En

E

EH

E

E

dz

di

)ˆ( 20

evolution law in gyrotropic and electro-optic media (AgGaSe2)

x = 0

x

x

y

V = V1 V = V2

in r t

gate

t

in

r

1 2

2211 sinsin nn 22

11 sinsin

FF v

VE

v

VE

Snell law

refraction at an interface

Stokes parameters

sin

cos

0

1||

3

2

1

20

sjS

sjS

S

S

y

x

yx

Tyxj ,

*, )( 2

023

22

21 SSSS

FvVEn /)(

x = 0 x = 0- 0 0+

2

1

-1

in

t r

in

r t

)exp()exp(

)exp()exp(

2211

221112

isis

isisr

)exp()exp(

cos2

2211

1112

isis

st

21

2112

nn

nnrem

21

112

2

nn

nt em

D. Dragoman, J. Opt. Soc. Am. B, in press

400

300

200

100

0

-100

-200

-300 0 200 400 600 800 -200 -400 -600 -800

I (A)

V (mV)

M. Dragoman et al., J. Appl. Phys. 106, 044312 (2009)

Metamaterials for ballistic electrons

x

y

x

y

iH

E

iH

E

dz

d

0

0

harmonic plane waves ballistic electrons

0/)(2

/0

EV

m

dz

d dzdm /)/(

VE

m

0

0

0

(left-handed) metamaterial barrier in a

semiconductor with negative effective mass

GaN, AlN, In0.53Ga0.47As, InAs, InP

dzzikA

dzzikBzikA

zzikBzikA

),exp(

0),exp()exp(

0),exp()exp(

33

2222

1111

m1 > 0, V1 m2 < 0, V2 m3 > 0, V3

k1 k2 k3

/)](2[ 2/13,13,13,1 VEmk

/)](2[ 2/1222 VEmk

d

g zvdz0

)(/

m1 = m3 = 0.4m0, m2 = –0.02m0, V1 = V3 = 0, V2 = 0.5 eV

d = 30 nm

d = 34 nm

d = 30 nmslabhomogeneous m1

homogeneous m2

D. Dragoman, M. Dragoman, J. Appl. Phys. 101, 104316 (2007)

'2

'

2

')/'exp(

2

1),( * dq

qq

qqipqpqW

Classical-quantum optics in phase space

pure quantum state

)|(|)]2exp(1[2

1|

2

)Im(),Re( 21

)}4cos()22exp(2]2)(2exp[]2)(2{exp[

)]2exp(1[

1),(

222

21

22

21

22

21

221

W

Wint Wmix

dyikyp

yx

yx

kpxW )exp(

222),( *

20

2

20

2 )(exp

)(exp)()()(

x

dx

x

dxAdxEdxExE GG

),(),(),()2cos(2exp2

exp2

)(2exp

)(2exp

2exp

2),(

int20

220

22

20

2

20

220

2202

pxWpxWpxWpkdx

xxpk

x

dx

x

dxxpkkxApxW

coherent optical field

opaque region

opaque region

opaque region

Gaussian slits

2d

x0

)||2exp(2

)( 2

Wcoherent quantum state ||a

0

02

01

/

2/

/

xd

kpx

xx

mask

plane wave f0

f0 f0

f

x

y

f

output plane

yout

xout

x

y


0 50 1000

100

200

300224

4

( )x1i

259

801 i

I (a.u.)

yout (a.u.)

mimicking quantum decoherence in phase space

S. Deléglise et al., Nature 455, 510 (2008)

D. Dragoman, M. Dragoman, Opt. Quantum Electron. 33, 239 (2001)

optical incoherent field in phase space


nnnaann ||ˆˆ|ˆFock states

2exp

!2

1)(

24/1q

qHn

q nn

n

22

22 exp2

)1(),(

pq

pqLpqW n

n

n

)(2

1)(

222

2

2

22

qnqqdq

dnn

0)()12(2

22

2

22

xE

r

xVVn

dx

dr n

2

24/1

2 2exp

!2

1)(

r

Vx

r

xVH

nr

VxE n

nn

2

222

22

222

2exp2

)1(),( p

V

rkx

r

Vp

V

rkx

r

VL

kpxW n

n

n

0krnV

k

rV

/1

// 2

p

qS

p

q

p

q

s

s )()2/exp(0

0)2/exp(

),(),( 22211211 psqspsqsWpqW ss

p

qS

p

q

p

q

s

s ),()2/sinh(cos)2/cosh()2/sinh(sin

)2/sinh(sin)2/sinh(cos)2/cosh(

n = 0

n = 1

n = 5

zx

2

220

2 1)(r

xnxn

D. Dragoman, Optik 112, 497 (2001)

qtan(/2) qtan(/2)

q/sin q/tan(/2) q/tan(/2)

qsin

fractional Fourier transform of order

D. Dragoman, Optik 111, 393 (2000)valid also for superpositions of Fock states

Review on classical optics-quantum phase space analogies: D. Dragoman, Prog. Opt. 42, 433 (2002)

squeezed states

qqs )2/exp( pps )2/exp(

])2/[exp()2/exp()()(ˆ qqS

Classical-quantum optics analogies: the fractional Fourier transform

nmHnm

aaaa nmnm

,2

exp),(!!

110,0

2exp

12

*,

0,212

*1

2

nmiiHnm

iaaaa nm

nm

nm

,2

exp),(!!

10,0

2exp

12

*,

0,212

*1

2

),min(

0

**, )!()!(!

!!)1(),(

nm

l

lnlml

nm lnlml

nmH

))(2/(exp),( 2211q aaaaiK

)(exp'

,,)(exp''''')',(

2211

22110',',,

cl

aaaai

nmnmaaaainmnmxxKnmnm

)(exp'

,,)(exp''''')',(

2211

22110',',,

cl

aaaai

nmnmaaaainmnmxxKnmnm

Kq

R(-/4)

K-cl

R(-/4)

x

y

x’

y’

D. Dragoman, J. Opt. Soc. Am. A 26, 274 (2009)

')'()',()]([ dqqqqKqF

sin

'

tan2

)'(exp

sin2

)2/exp()',(

22cl xxxx

xxiiii

Kclassic )','('),,( 2121 xxxx xx

sin2tan2

expsin2

)2/exp(),(

**22

q iiiK

quantum

'', 2121 ixxixx

||]2/)ˆˆ[( 121 xXX

||]2/)ˆˆ[( 221 xPP

|'|]2/)ˆˆ[( 121 xXX

|'|]2/)ˆˆ[( 221 xPP

Classical-quantum optics analogies: computation

2n states of an n-qubit system can only be realized by 2n distinct optical paths OR NOT ?

0010 0000 1100

0011 0001 1101

1110

1111

…..

…..

0

1

logic gates: image forming devices + phase shifters

fsph

fcyl

fsph + fcyl

D. Dragoman, Optik 113, 425 (2002)

classic computation quantum computation

bit: 0 or 1 qubit: |0 + |1

an n-bit classical register stores only one from 2n possible states

an n-qubit register stores a superposition of 2n states

possible to inquire at any time about the state of any bit in the memory

the post-measurement state of a qubit is either |0 or |1

bits can be copied no-cloning theorem

no entanglement entanglement of qubits is possible

logic gate: Boolean operator + irreversible fanout gates + ancilla

quantum logic gate: unitary and reversible operators

universal logic gates: NAND or NOR

universal logic gates: one-qubit gates + CNOT

sequential computation (common)

parallel processing

possible to read the result of computation in one-step

quantum interference is needed for an efficient reading of the result

f f

2f

0 1

1 0

NOT

f f

2f

00

01

10

11

00 11

01 10 C-NOT

Conclusions

• Analogies between classical and quantum states exist, although they refer to completely different realities (fermions versus bosons) or theoretical approaches (operators versus algebraic functions)

•The quantum-classical analogies offer a means to generate, measure, and design optical structures similar to mesoscopic quantum structures but much easier to fabricate, control and characterize

• These analogies can be used to study phase space distribution functions of quantum optical states without worrying about decoherence and the impossiblity of measuring non-commutative variables

• The analogies between classical optics and quantum physics reveal, if used properly, the (sometimes subtle) differences between these two realms and help understanding the essence of quantum behavior

• For all these reasons it is my belief that quantum-classical analogies are worth pursuing

classic

quantum

Thank you for your attention!

classical optics – quantum physics analogies daniela dragoman univ. bucharest

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