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)
Classical optics-Schrödinger electrons analogies
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-Schrödinger electrons analogies
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
D. Dragoman, M. Dragoman, Opt. Commun. 150, 331 (1998)
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
D. Dragoman, M. Dragoman, Opt. Commun. 150, 331 (1998)
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
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
Classical-quantum optics in phase space
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
Classical-quantum optics 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!