quantom optics by tooota

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									Tutorial: quantum information Tutorial: quantum information & quantum optics // atomic physics & quantum optics atomic physics

Peter Zoller University of Innsbruck, Institute for Theoretical Physics
SFB Coherent Control €U TMR 1

Road Map
• our goal is to implement quantum networks

•
channel

•

nodes: local quantum computing – quantum processors with atoms channels: communication – transmission of photons

node

• quantum optics toolbox
– photons – atoms
2

Quantum information processing
• quantum computing
ouput quantum weirdness:

| out  | out   Û| in  | in 

superposition entanglement interference nonclonability and uncertainty no decoherence!

quantum processor
input

• quantum communications
| | | | |
teleportation crytography

transmission of a quantum state

3

Qubits, Quantum Gates etc.
• quantum bits or qubits
example: two qubits

n spin1/2

|1i |0i

|  c 00|00  c 01|01  c 10 |10  c 11|11

• single qubit gate
|   Û1|

U1

• two qubit gate
|a |b
control target

|a |a  b
Controlled-NOT 4

Examples from quantum optics
•
ion traps '95

•

cavity QED
cavity

fiber

•

neutral atoms:

laser

µ1
V dip r

µ2

laser

interacting Rydberg dipoles

optical lattice 5

These systems realize manipulation on the single quantum level.

Quantum computing: check list
• quantum memory • single qubit gate • two-qubit gate
– quantum control

• issues
– zero vs. finite temperature – single system vs. ensemble – – – – – scalability robustness speed simplicity ...

• preparation • read out
– "controlled decoherence"

• decoherence

6

Atoms as quantum Memory

single trapped atom:

|0
trap

|1

qubit in longlived internal states

or: atomic ensembles:

+

7

• trapping and cooling of atoms and ions
linear ion trap atom "chips" traps: conservative potential load the trap laser cooling: prepare motional ground state Remarks: long coherence times demonstrated.
Schmiedmayer

Blatt

laser traps FORT
laser

arrays of microtraps: optical lattice
laser

8

Single qubit gates

Raman process laser

Requirements:

Ω1

Ω2

addressing single qubit laser

|0
Rabi frequency spontaneous emission

|1
Ωeff
Γeff

=
∼

1 Ω1 Ω2 4 ∆ 1 Ω2 1,2 Γ 4 ∆2

Remarks: state preparation by optical pumping decoherence due to spontaneous emission (and collisions)

9

Two-qubit gates
•
implement entanglement of two qubits
example: phase gate

|00  |01  |10 

|00 |01 |10

U2

|11  e i |11

•

How? auxiliary collective mode as data bus controllable two body interactions dynamical phases geometric phases (holonomic quantum computing)
10

Two-qubit gates via quantum databus
- Entanglement via collective auxiliary quantum degree of freedom
Collective mode
1 2

quantum data bus switch

Examples: Ion traps (Cirac and Zoller PRL '95) Cavity QED (Pellizzari et at PRL '96)

qubits

state vector:

|   x c x |x N1x N2 x 0  |collective mode
quantum register data bus

gate:
swap

requirement: cooling of the collective mode (= prepare a pure state)
11

Two-qubit gates via two-body interactions

- Controlled two-body interaction
V(R)
1 2

We must design a Hamiltonian H  Et |1 11|  |1 21| so that |1 1  |1 2  ei |1 1  |1 2

qubits

Examples: Cold collisions (Jaksch et al PRL ' 99) Ion trap 2000 (Cirac and Zoller, Nature 2000) Rydberg gate (Jaksch et al. PRL 2000)

12

Ion Trap Quantum Computer '95
J. I. Cirac, P. Zoller PRL '95

• Cold ions in a linear trap
Qubits: internal atomic states Quantum gates: entanglement via exchange of phonons of quantized center-of-mass mode
laser pulses entangle ion pairs

• State vector
|Ψi =

X

cx |xN −1 , . . . , x0 iat om |0iphonon
quantum register databus

• QC as a time sequence of laser pulses • Read out by quantum jumps (100 % efficiency)
13

Physics behind the quantum gate
• Swap the qubit (internal state) to phonon data bus
|0 A  |1 A |0 phonon
laser

|0 A |0 phonon  |1 phonon 

• Two qubit gate
1 step 1: 2 swap qubit 1 to phononic bus CNOT qubit 2 + phonon mode

laser

|0 A |0 phonon  |0 A |0 phonon
step 2:

|0 A |1 phonon  |0 A |1 phonon
laser

|1 A |0 phonon  |1 A |1 phonon |1 A |1 phonon  |1 A|0 phonon

step 3:

laser

14

The Innsbruck Linear Ion Trap
R. Blatt et al. 1997 - 2001

Achievements & expectations:

• • • • • •

storing ~ 10 qubits ground state cooling addressing single ions two qubit gate ... soon!? few qubits within the next years? decoherence times: ~ 40 gates

15

Boulder Linear Ion Trap
C. Monroe, D. Wineland et al., Nature 2000

• Lithographic trap

• Maximally entangled state of N=4
ions

|0000  |1111
Mølmer - Sørensen protocol (no indivdual addressing)

ions
200 µm

• Bell state measurements • Two atom entangled state
interferometry
16

A fast 2-qubit gate with Rydberg atoms
D. Jaksch, J.I. Cirac, P.Z., S. Rolston, M. Lukin and R. Cote, PRL 2000

• Rydberg atom in constant electric field
• setup • linear Stark effect
energy atom

• permanent dipole moment

|r

E

n ~ 15 ...60

  n2
laser

huge!

|g

E
E  1kV/cm R  opt /2  300 nm E  60 GHz V dip  4 GHz for n  15
17

• Large dipole-dipole interaction
µ1
V dip r

µ2

Two-qubit gate: internal dynamics

• atomic configuration
Atom 1
V dip  u

• two qubit gate
Atom 2
|r 2
laser

|r 1
laser  1

|11i → |10i → |01i →

|11i |10i |01i

2

|1 1
qubit

|0 1

|1 2
qubit

|0 2

|00i → eiφ |00i
• dipole – dipole interaction = phase shift • force !?
18

Scheme:
•

 1,2  u

gate time t  2/ 1,2  1/ trap

Energy levels of two atom 1 + atom 2 atom 1 R atom 2

| rr 〉
V dip  u
2

| r 1〉
1

| 1r 〉
2

| r 0〉
1

| 0r〉
Large dipole-dipole interaction shifts level off resonance:

| 11〉

| 01〉

| 10 〉

| 00 〉

No double excitation no force!

•

Laser pulse sequence

"dipole blockade"

π

1

2π

2

π

1

time
19

• Laser pulse 1
atom 1 R atom 2

| rr 〉
V dip  u

| r 1〉
1

| 1r 〉

| r 0〉
1

| 0r〉

| 11〉

| 01〉

| 10 〉

| 00 〉

π

1

2π

2

π

1

time
20

• Laser pulse 2
atom 1 R atom 2

| rr 〉
V dip  u
2

| r 1〉
2

| 1r 〉

| r 0〉

| 0r〉

No excitation!

"dipole blockade"

| 11〉

| 01〉

| 10 〉

| 00 〉

minus sign!

π

1

2π

2

π

1

time
21

• Laser pulse 3
atom 1 R atom 2

| rr 〉
V dip  u

| r 1〉
1

| 1r 〉

| r 0〉
1

| 0r〉

Fast phase gate

|00   |00 |01   |01 |10   |10 |11   |11
no force

| 11〉

| 01〉

| 10 〉

| 00 〉

π

1

2π

2

π

1

time
22

Atomic ensembles
M Lukin, L.M. Duan et al., PRL 2001

• mesoscopic atomic ensembles (instead of microscopic quantum
objects)
- coherent manipulation of collective excitations of atomic ensembles

|r
laser laser

N  100 atoms

|g
 10 m
ground state

|q

-underlying physics:
dipole blockade
23

Manipulating collective excitations
|r

•

ground state

|g
laser

|q

|g N 
•

 |g 1 |g 2  |g N 

one excitation (Fock state)

|g N1 q   i |g 1  |q i  |g N 
•
two excitations

laser

blockade

|g N2 q   i,j |g 1  |q i  |q j  |g N
etc.
24

cont.

• qubits
|  |g N   |g N1 q
+
superposition

• entanglement of ensembles

25

Atomic ensembles: quantum memory for light
•
purpose

[
incoming light pulse unknown (arbitrary) state known shape of wave packet

Atomic ensemble

]
outgoing light pulse same state reshaping

storage medium

•

how? example ...
write

cavity

a1
atoms

read

theory: Lu Ming Duan; M. Lukin and M. Fleischhauer PRL '00 exp: R. Walsworth et al, PRL '01 L. Hau et al., Nature '01

cavity

laser 26

Teleportation with coherent light + atomic ensembles
• theory: Lu Ming Duan, J.I. Cirac, P.Z. and E. Polzik, PRL Dec 2000 • experiment: E. Polzik et al., Nature 2001 • features
– so far: quantum computing and communications requires single atoms and single photons cavity high-Q cavities atom
laser

– now: can we get away with ... atomic ensembles? free space?

laser

ensemble of atoms

simpler!

– continuous variable quantum communications – we use measurements to generate EPR states

27

Quantum Communication
• the goal of quantum communication is to transmit a quantum state
reliably Alice

|φi |φi

|φi |φi

Bob

|φi

transmission of a quantum state

• The quantum state can be a qubit
|  |0  |1
or a continuous variable quantum state

|  

 

dx |x x

 x  position   x, p  i  p  momentum
28

Teleportation of Continuous Quantum Variables
• continuous variable quantum states
|   dx |x x
 x  position  p  momentum

  x, p  i

• continuous variable protocol
prepare an EPR state as a resource
T A B
Vaidman Braunstein Kimble (exp)

|EPR   dP|P A |  P B   dX |X A |X B

eigenstate of commuting operators:
  X A  X B |EPR  X 1|EPR   P A  P B |EPR  P 1|EPR

   XA  XB    PA  PB

2 2

 0  0
29

cv teleportation cont. the initial state is
T A B

| in   | T  |EPR AB

we make a Bell measurement

  X2 XT  XA  projective measurement on |X 2, P 2 TA   T  PA  P2 P

result of measurement:

|out   |X 2, P 2 TA  TA X 2 ,P 2|in 
   |X 2,P 2 TA  e iX2 P B e iP 2X B | B

apart from a displacement B is in |

classical communication and displacement

X2 P2 |out   |X 2, P 2 TA  | B
30

Physical realization with squeezed light
Braunstein and Kimble '96, Caltech exp '98

• squeezed vacuum as a cv EPR state
pump a

|EPR  e ra b ab |0 A  |0 B



 

b

1   2  n0  n |n A  |n B   tanhr
1 2

with quadrature components a 

  X A  iP A  with a, a    1

– position representation: Gaussian

X A , X B |E  expX A  X B  2 e 2r  X A  X B  2 e 2r 
– variances

  X A  X B  2  e 2r   P A  P B  2  e 2r
31

Implementation of cv teleportation
Caltech present

1.

EPR: two mode squeezed light
1 2

1.

atomic / spin squeezing

parametric downconversion

EPR source

coherent light atoms 1

atoms 2 measure EPR

quantum variables - quadrature components EPR correlations - light squeezing

- atomic collective "spin" - spin squeezing

•

Questions: interactions? noise?

32

teleportation cont. Caltech present

2.

Bell measurement
Bell

classical communications out Bell in 1 EPR source 2

atoms 1 EPR

atoms 2

atoms 3

teleport 3 to 2

coherent light

• Questions:
- efficiency of Bell measurement 33 - can we do better than with light?

Scheme
ex
atoms coherent light

• atomic level scheme
3 4

• incident light:
- coherent - linear light polarization

ex

= σ+ + σ

−

σ+
1
M=− 1 2

σ2
M =+ 1 2

• atoms initially:
- prepared in superposition

1

2

34

Scheme
ex
atoms coherent light

• atomic level scheme
3 4

• incident light:
- coherent - linear light polarization

ex

= σ+ + σ

−

σ+
1
M=− 1 2

σ2
M =+ 1 2

• atoms initially:
- prepared in superposition

• Remarks:
light propagation problem spontaneous emission noise

1

2

35

1. Atoms as a continuous variable quantum system
• continuous atomic operators
atoms z

number of atoms 2N a  AL  1

  z, t   lim z0

1 Az

 izzi zz |  i  |

• collective spin operators for the atomic ground states
a Sx a Sy a Sz

= = =

ρA 2 ρA 2i ρA 2

Z Z

L 0 L 0 L 0

Z

³ ³

σ 12 + σ 12 −

† σ 12 † σ 12

´

dz dz

´

angular momentum: Bloch vector

(σ 11 − σ 22) dz
36

atoms cont.

•

superposition of the two ground states: coherent spin state
1 2

1
Bloch vector

2

|1  |2 

N

Bloch picture

 S a   S a , S a , S a    N a , 0, 0 x y z 2
• quantum fluctuations
S a , S a   iS a y z x
S a S a  y z
1 2

z
Pa

|S a | x

y
X
a

we treat S a classically and rescale x

X a , P a   i

X a P a 

1 2

x

coherent spin state =vacuum state there are many cv quantum states around it:

canonical communation relations

| a    dX a |X a X a 37 

2. Light (polarization) as cv quantum system
• one dimensional light propagation problem: polarization
ex
z

E  z, t  

 0 40 A

 a i z, t e ik 0 z 0 t
i  ,

a i z, t , a  z  , t  j

  ij z  z  

• input
strong coherent input with linear polarization photon number

• Stokes parameter
p Sx p Sy p Sz

= = =

hai (0, t)i = αt
R
T 0

c 2 c

Z

T 0

2Np = 2c

|α t|2 dt À 1

2i Z c 2

Z

T 0 T

³ ³

† a1 a2

+

† a2a1

³

´

dτ dτ

a† a2 1

−

a† a1 2

a† a1 1

0

−

a†a2 2

´

´

dτ

38

light cont.

• input and output
ex
atoms z

Rem.: free field a i z, t  a i t  z/c commutation relations for a free field:
p p p S y , S z   iS x

Sp x,y,z

input

output

S p x,y,z

• for linear light polarization the expectation value of the Stokes vector is
N  S p   S p , S p , S p    2p , 0, 0 x y z

Stokes picture

z
Pp

we treat S p  S p   N p classically x x

X , P   i
| p    dX p |X p X p 

p

p

y
Xp
39

x

3. Dynamics: quantum noise propagation problem
coherent light atoms 1 in

3
atoms 2 out

4

σ+
1

σ2

• Maxwell-Bloch equations
i |g| ρAσii |g| ρAγσii ∂ ai (z, τ) = ai (z, τ ) − ai (z, τ) + noise ∂z ∆c 2∆2³ c ³ ´ ´ 2 2 0 † † † † i |g| a2 a2 − a1 a1 |g | γ a2a 2 + a1 a1 ∂ σ12 +noise σ12 = σ12 − ∂τ ∆ 2∆2
approx. by a spatial mean field
a p H ∼ Sz Sz ∼ Pp Pa
Kuzmich Polzik 40
2 2

effective Hamiltonian:

Dynamics: effective Hamiltonian
• we model the atom-light interaction as effective time evolution
atoms input output

U eff  expiP p P a 
phase shift quadratic

Kuzmich Polzik noise !?

X p  X p  U  X p U eff eff P p  P p  U  P p U eff eff

• whole system + process
coherent light input atoms 1 atoms 2 output measurement

• we can describe dynamics in Heisenberg or Schrödinger picture

41

3.1 Dynamics in the Heisenberg picture
• effective dynamics • Heisenberg equation:
light:
output

atoms input

X p  X p  P a P p  P p
atoms:

Xp 

X p



U  X p U eff eff

P p  P p  U  P p U eff eff

X a  X a  P p P a  P a

U eff  expiP p P a 

42

Teleportation Step 1: atomic EPR correlations
• first round
coherent light atoms 1 in measure atoms 2 out

X p1

X p1

X p1  Xp1  P a1  Pa2 
measure out in: vacuum noise to obtain

Measurement of X p1 projects the two atomic ensembles into an (approximate) eigenstate of P a1  P a2

P a1  P a2  2 

1 12 2

 e 2r

43

cont.

• second round: we first rotate the atomic spins
X a1  P a1 P a1  X a1
and then we measure

X a2  P a2 P a2  X a2

X p2  X p2  X a1  X a2 
measure out in: vacuum noise

to obtain

• result: we have generated a cv EPR state
P a1  P a2  2 
X a1  X a2  
2

1 12 2 1 12 2

 e 2r
e
2r

for   5 we have r  2

44

Teleporation Step 2: Bell measurement
Bell
0

Xp1 = X p1 − κ(Xa1 − Xa3) Xp2 = Xp2 − κ(Pa1 + Pa3 )
atoms 1 EPR atoms 2
0

measure out

in: vacuum noise

to obtain

atoms 3

efficiency

  1

1 12 2

• combining 1 and 2: teleporation
fidelity

F

1 1
1 12

 2

1 2 2

for   5 we have F  96%
45

Noise & Imperfections
• noise, and modelling of the noise as "beam splitters"
vacuum coherent atoms 1 light vacuum atoms 2 vacuum

t

spontaneous emission

propagation loss as a "beam splitter"

spontaneous emission

• example: propagation losses
teleporation fidelity

F   1/ 1  t

Even with a notable transmission loss rate t  0. 2, quantum teleportation with a remarkable high fidelity F  0. 7 is still achievable.

46

3.2 Wave function
• initial state
|ψi = |ψip |ψia1 |ψia2

light atoms 1 atoms 2

∼
vacuum noise

Z

∞

dp|pie

− 1 p2 2

−∞

Z

∞

−∞

dpa1 |pa1 ie

− 1 p2 2 a1

Z

∞

−∞

dpa2 |pa2 ie− 2 pa2

1

2

light

z
Xp

atoms 1 and 2

z

y
P
p

Xa

y
Pa
47

x

x

light atoms 1 atoms 2

• interaction
– time evolution

U
– wave function

= exp(iκˆpa2 ) exp(iκˆpa1 ) pˆ pˆ

|ψi ∼

Z

∞

dp

−∞

Z

∞

dpa1

−∞

Z

∞

−∞

dpa2 |pi|pa1 i|pa2 i eiκp(pa1 +pa2 ) e− 2 (p

1

2

+p2 +p2 ) a1 a2

light + atom entangled by interaction

initial vacuum noise

48

light atoms 1 atoms 2

measure

• measurement
– the atomic wave function after the measurement is

|ψia = h¯|ψi ∼ x
∼ Z
∞

dp

−∞

Z

∞

dpa1

−∞

Z

∞

−∞

dpa2 |pa1 i|pa2 ie

ip[κ(pa1 +pa2 )+¯] − 1 (p2 +p2 +p2 x a1 a 2

e

49

light atoms 1 atoms 2

measure

• measurement
– the atomic wave function after the measurement is

|ψia = h¯|ψi ∼ x
∼ Z
∞

dpa1

−∞

Z

∞

−∞

dpa2 |pa1 i|pa2 ie

− 1 (κ(pa1 +pa2 )+¯)2 − 1 (p2 +p2 ) x a1 a2 2 2

e

initial: uncorrelated

pa 2

interaction + measurement

pa1
1 ∆p = √ 2

50

light atoms 1 atoms 2

measure

• measurement
– the atomic wave function after the measurement is

|ψia = h¯|ψi ∼ x
∼ Z
∞

dpa1

−∞

Z

∞

−∞

dpa2 |pa1 i|pa2 ie

− 1 (κ(pa1 +pa2 )+¯)2 − 1 (p2 +p2 ) x a1 a2 2 2

e

initial: uncorrelated

pa 2

interaction + measurement

pa 2

EPR

pa1
1 ∆p = √ 2
∆(pa1 + pa2 )2 =

pa1
1 ≡ e−2r 1 + 2κ2 51


								
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