1. Introduction to
quantumoptics with ions
PD Dr. Kilian Singer
Universität Mainz
www.quantenbit.de/#/teach/Public%20Outreach/
start

2. Ion Gallery
Boulder, USA: Hg+
arhus, Denmark: 40Ca+ (red) and 24Mg+ (blue)
Oxford, England: 40Ca+ coherent breathing motion of a 7-ion linear
Innsbruck, Austria: 40Ca+

3. Paul trap

4. fluorescence
detection
ring
electrode
endcap
electrode
endcap
electrode
cooling
beam
z
lens
y
x
Paul trap

5. Binding in three dimensions
Electrical quadrupole potential
Binding force for charge Q
leads to a harmonic binding:
no static trapping in 3 dimensions
Laplace equation requires
Ion confinement requires a focusing force in 3 dimensions, but
such that at least one of the coefficients is negative,
e.g. binding in x- and y-direction but anti-binding in z-direction !
trap size:

6. Dynamical trapping: Paul‘s idea
time depending potential
with
leads to the equation of motion for a particle with charge Q and mass m
takes the standard form of the Mathieu equation
(linear differential equ. with time depending cofficients)
with substitutions
radial and axial trap radius

7. Regions of stability
time-periodic diff. equation leads to Floquet Ansatz
If the exponent µ is purely real, the motion is bound,
if µ has some imaginary part x is exponantially growing and the motion is unstable.
The parameters a and q determine if the motion is stable or not.
Find solution analytically (complicated) or numerically:
a=0, q =0.1 a=0, q =0.2
time time
excursion
excursion
a=0, q =0.3 a=0, q =0.4
time time
excursion
excursion
a=0, q =0.5 a=0, q =0.6
time time
excursion
excursion
a=0, q =0.7 a=0, q =0.8
time time
excursion
excursion
a=0, q =0.9 a=0, q =1.0
time time
excursion
excursion
6 1019
-3 1019
unstable

8. time
positionintrap
micromotion
1D-solution of Mathieu equation single Aluminium dust particle in trap
Two oscillation frequencies
slow frequency: Harmonic secular motion, frequency w increases with increasing q
fast frequency: Micromotion with frequency W
Ion is shaken with the RF drive frequency (disappears at trap center)
Lissajous figure

9. 3-Dim. Paul trap stability diagram
for a << q << 1 exist approximate solutions
The 3D harmonic motion with frequency wi can
be interpreted as being caused by
a pseudo-potential Y
leads to a quantized harmonic oscillator

10. Real 3-Dim. Paul traps
ideal 3-Dim. Paul trap with equi-potental
surfaces formed by copper electrodes
endcap electrodes at distance
ideal surfaces:
but non-ideal surfaces do trap also well:
rring ~ 1.2mm
A. Mundt, Innsbr

11. ideal 3 dim. Paul trap with equi-potental
surfaces formed by copper electrodes
non-ideal surfaces
rring ~ 1.2mm
numerical calculation
of equipotental lines
similar potential
near the center
Real 3-Dim. Paul traps

12. determine the quadrupole part of the potential:
ideal:
real:
rring ~ 1.2mm
Paul trap: wire ring + endcaps
Paul Straubel trap: ring only
inverted Paul Straubel trap: endcaps only
Paul trap
main advantages:
good optical access
large observation angle
fully harmonic
potential
Loss factor
Real 3-Dim. Paul traps

13. x
y
2-Dim. Paul mass filter stability diagram
time depending potential
with
dynamical confinement in the x- y-plane
with substitutions
radial trap radius

14. 2-Dim. Paul mass filter stability diagram

15. x
y
A Linear Paul trap
plug the ends of a mass filter by positive electrodes:
mass filter blade design side view
RF
RF 0V
0V Uend Uend
numerically calculate the axial electric potential,
fit parabula into the potential
and get the axial trap frequency
with k geometry factor
z0

16. Innsbruck linear ion trap
1.0mm
5mm
MHz5radialwMHz27.0 axialw
Blade design
eVdepthtrap F. Schmidt-Kaler, et al.,
Appl. Phys. B 77, 789 (2003).

17. MACOR frame
compensation
electrodes
Electrodes
Innsbruck linear ion trap
F. Schmidt-Kaler, et al.,
Appl. Phys. B 77, 789 (2003).

18. Aarhus, Denmark
Linear ion traps in Rood design
trap electrodes are
nearly in ideal geometry
R= 4mm
r0= 3.5mm
harmonic trapping
in a large region, and
even outside the center

19. Rood design
Boulder, USA
Innsbruck, Austria

20. Leibfried, Schätz
Physik Journal 3 (2004) 23
Schulz, NJP (2008)
d ~ 250µm
Stick et al., Nature
Physics 2, 36 (2006)
d ~ 60µm
Seidelin et al.,
PRL 96, 253003 (2006)
d ~ 40µm
Hensinger et al.,
Appl. Phys. Lett.,
034101 (2006)
d ~ 200µm
Segmented micro traps overview
Labaziewicz et al,
PRL 100, 013001 (2008)
Kielpinski et al,
Nature 417, 709 (2002)

21. RF
DC
RF
DC
• Micro structuring - high number and density of ions
- high trap frequency and gate speed
• Segmentation - Processor and memory section
- Transport of Ion crystals
- Separation of ions
- Cavity QED region
500µm
250µm250µm
100µm
Schulz et al., Fortschr. Phys. 54, 648 (2006)
Ulm segmented micro trap

22. ceramics,
fs-Laser stuctured,
Gold-coated
base
spacer
top
Ulm segmented micro trap: Fabrication
Schulz et al.,
New Journal of Physics 2008

23. Loading zone / Transfer zone Experiment zoneSandwich construction
• Ti/Au on Al2O3-Wafer
(10nm/400nm)
• fs-Laser cut in Au/Ti and Al2O3
• adjusting and
• gluing
into Chip Carrier
• bonding
Ulm segmented micro trap: Fabrication
Schulz et al.,
New Journal of Physics 2008

24. … assembled and connected
in the UHV recipient
Schulz et al.,
New Journal of Physics 2008

25. Planar micro traps
RF
DC
DC
RF
DC
RF DCDC RF
DC RFRF DC DC
Trapping potential
Pros:
Easy fabrication
High precision
Small sizes possible (few µm)
high surface quality
Cons:
shallow trap potential

26. Planar micro traps S. Seidelin, et. al.,
PRL 96 253003, (2006).
J. Labaziewicz, et. al.,
PRL 100, 013001 (2008).

27. Equilibrium positions in the axial potential
z-axis
mutual ion repulsiontrap potential
find equilibrium positions x0: ions oscillate with q(t) arround
condition for equilibrium:
dimensionless positions with length scale
D. James, Appl. Phys.
B 66, 181 (1998)

28. Equilibrium positions in the axial potential
numerical solution (Mathematica),
e.g. N=5 ions
equilibrium positions
set of N equations for um
-1.74 -0.82 0 +0.82 +1.74
force of the
trap potential
Coulomb force
of all ions from left side
Coulomb force
of all ions from left side

29. -40 -30 -20 -10 0 10 20 30 40
0
1
2
3
4
5
6
7
8
9
10
NumberofIons
z-position (µm)
Experiment: equilibrium positions
equilibrium positions
are not equally spaced
H. C. Nägerl et al.,
Appl. Phys. B 66, 603 (1998)
theory
experiment
minimum inter-ion distance:

30. Eigenmodes and Eigenfrequencies
Lagrangian of the axial ion motion:
m,n=1m=1
N N
describes small excursions
arround equilibrium positions
with
and
N
m,n=1m=1
N N
D. James, Appl. Phys.
B 66, 181 (1998)
linearized Coulomb interaction leads to Eigenmodes, but the
next term in Tailor expansion leads to mode coupling, which is
however very small.
C. Marquet, et al.,
Appl. Phys. B 76, 199
(2003)

31. Eigenmodes and Eigenfrequencies
Matrix, to diagonize
numerical solution (Mathematica),
e.g. N=4 ions
Eigenvectors
Eigenvalues
for the radial modes:
Market et al., Appl. Phys.
B76, (2003) 199depends on N
pictorial
does not

32. Common mode excitation: Experiment
H. C. Nägerl, Optics
Express / Vol. 3, No. 2 /
89 (1998).

33. time
position
Center of
mass mode
breathing
mode
Common mode excitations
H. C. Nägerl, Optics
Express / Vol. 3, No. 2 /
89 (1998).

34. Breathing mode excitation
H. C. Nägerl, Optics
Express / Vol. 3, No. 2 /
89 (1998).

35. Basics: Harmonic oscillator
Why? The trap confinement is leads to three
independend harmonic oscillators !
here only for the linear direction
of the linear trap no micro-motion
treat the oscillator quantum mechanically and
introduce a+ and a
and get Hamiltonian
Eigenstates |n> with:

36. Harmonic oscillator wavefunctions
Eigen functions
with orthonormal Hermite polynoms
and energies:

37. Two – level atom
Why? Is an idealization which is a good approximation to real
physical system in many cases
two level system is connected
with spin ½ algebra using the
Pauli matrices
D. Leibfried, C. Monroe,
R. Blatt, D. Wineland,
Rev. Mod. Phys. 75, 281 (2003)

38. Two – level atom
Why? Is an idealization which is a good approximation to real
pyhsical system in many cases
gn ,1
en ,1
en,
en ,1
gn ,1
gn,
together with the harmonic oscillator leading to the ladder of
eigenstates |g,n>, |e,n>:
levels not coupled

39. Laser coupling
dipole interaction, Laser radiation with frequency wl, and intensity |E|2
Rabi frequency:
the laser interaction (running laser wave) has a spatial dependence:
Laser
with
momentum kick, recoil:

40. Laser coupling
in the rotating wave approximation
using
and defining the Lamb
Dicke parameter h:
Raman transition: projection of Dk=k1-k2
x-axis
if the laser direction is at an angle f to the vibration mode direction:
x-axis
single photon transition

41. Interaction picture
In the interaction picture defined by
we obtain for the Hamiltonian
with
coupling states with vibration quantum numbers
laser detuning D

42. 2-level-atom harmonic trap
Laser coupling
dressed system
„molecular
Franck Condo
picture
dressed system
gn ,1
en ,1
en,
en ,1
gn ,1
gn,
„energy
ladder“
picture

43. 2-level-atom harmonic trap
Laser coupling
dressed system
„molecular
Franck Condon“
picture
„energy
ladder“
picture
S
D
D
S
Sn ,1
Dn ,1
Dn,
Dn ,1
Sn ,1
Sn,

44. Lamb Dicke Regime
carrier:
red sideband:
blue sideband:
laser is tuned to
the resonances:

45. kicked wave function is non-orthogonal to the other
wave functions
Wavefunctions in momentum space
kick by the laser:

46. 0,S
0,D
Prob.forD
electronic
excitation
laser pulse length in µs
electronic
excitation
Coherent qubit rotation
Carrier flops

47. 0,S
0,D
1,D
1,S
internal
electronic
state
Vibrational quanta
laser pulse length in µs
Coherent qubit rotation
Carrier flops
carrier and sideband
Rabi oscillations
with Rabi frequencies
and

48. What is a quantum gate?
• QBit ist kleinste quantenmechanische
Informationseiheit:
-Verknüpfung von QBits mittels
-1Qbit Gatter
-Und 2 Qbit-Gatter=> Quanten-Computer
Damit kann man universellen Quantencomputer bauen!

49. 1 Qubit Gatter – Licht
Wechselwirkung
1D-WW
Atom-Licht-WW

50. What is a Superposition?
two different quantum states
are simultaneous existing
at the very same place

51. two different quantum states
are simultaneous existing
at the very same place
What is a Superposition?

52. Vector on the Bloch-sphere
two different quantum states
are simultaneous existing
at the very same place
What is a Superposition?

53. 10  
in superposition of Zero and One
A single quantum-bit
two different quantum states
are simultaneous existing
at the very same place
What is a Superposition?

54. P
S
t = 7 ns
397 nm
D
729 nm
„qubit“
Energy
|1>
|0>
A single Ion
in superposition
of Zero and One
two different quantum states
are simultaneous existing
at the very same place
What is a Superposition?

55. What is entanglement?
Solvay Konferenz 1927
Entangled particles show correlations and can only
propery discribed by a common quantum state,
regardless how far appart they are.
Still open
Problem:
Entanglement
measures and
characterization

56. Co-Carrier
Raman
|1>
|0>
PopulationPopulation
Raman Pulsdauer [µs]
Poschinger et al,
arXive 0902.2826
Bei Messung
50% der Fälle

57. 2 Qubit gate
CNOT-GATE
Particle A Particle B
CNOT
Particle A Particle B

58. 2 Qubit gate
CNOT-GATE
Particle A Particle B
CNOT
Particle A Particle B

59. Generation of Entanglement
Particle A Particle B Particle A Particle
Particle A Particle B
CNOT
1 Qbit gate on
each A and B
does not generate
entanglement:
Measurement would result in uncorrelated results
Measurement results in always the same outcomes at A and B !!!

60. Properties of the
complete system are fully determined,
but properties of sub-systems
completely undeterined
Erwin Schrödinger 1935:
"I would call entanglement not one but rather
the characteristic trait of quantum mechanics,
the one that enforces its entire departure from
classical lines of thought."
What is Entanglement?

61. Laser pulse
Entanglement by Laser pulses
get two ions

62. Laser-Puls
Entanglement by Laser pulses

63. two ions in entangled state
Entanglement by Laser pulses

64. Measurement
two ions in entangled state
detection
Laser
detection
Laser
|1> |0>

65. .... Repeated measurements
Fully determined
correlation
Measurement results
left
Measurement results
right
Subsystems completly random
Erwin Schrödinger 1935:
"I would call entanglement not one but
rather the characteristic trait of quantum
mechanics, the one that enforces its entire
departure from classical lines of thought."

66. Temporal sequence
of quantum logic
operations
INPUT OUTPUT
Basics of a quantum computer
Single
qubit gate
two-qubit
gatetime

67. applications in physics and informatics
 P. Shor, 1994: factorization of large numbers, L digits, is much more
efficient on a quantum computer than with a classical computer:
classical computer: ~exp(L1/3), quantum computer: ~ L2
 L. Grover, 1997: search data base - quantum computer: ~ L
 simulation of Schrödinger equations or any unitary evolution
 spin interactions, quantum phase transitions
 quantum cryptography / repeaters / quantum links
 improved atomic clocks
 understanding the fundamentals of quantum mechanics /
Gedanken-Experimente
 Experiments with entangled matter
Why?

68. The requirements for experimental qc
• Qubits store superposition information, scalable physical system
• Ability to initialize the state of the qubits
• Universal set of quantum gates: Single bit and two bit gates
• Long coherence times, much longer than gate operation time
• Qubit-specific measurement capability
D. P. DiVincenzo,
Quant. Inf. Comp. 1
(Special), 1 (2001)
Qubit
Transformation
10  

69. Scalable device?
Experimental status
Quantum Information Roadmaps
http://qist.ect.it/
http://qist.lanl.gov/

70. Quantum gate proposal
control bit target bit
• single bit rotations and
quantum gates
• small decoherence
• unity detection efficiency
• scalable
J. I. Cirac P. ZollerW. Paul

71. • single bit rotations and
quantum gates
• small decoherence
• unity detection efficiency
• scalable
J. I. Cirac P. Zoller21121:   NOTControlled
0111
1101
1010
0000




0111
1101
1010
0000




control bit target bit
Quantum gate proposal

72. Cirac & Zoller gate
with two ions

73. 



S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S D
,S D
0 0
control qubit
target qubit
SWAP
1 2
control target
Controlled-NOT operation

74. 



S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S D
,S D
0 0
control qubit
target qubit
|0>, |1>
1 2
Controlled-NOT operationControlled-NOT operation

75. 



S S S S
S D S D
D D DS
D D D S
ion 1
motion
ion 2
,S D
SWAP-1
,S D
0 0
control qubit
target qubit
|0>, |1>
1 2
Controlled-NOT operationControlled-NOT operation

76. SWAP and SWAP-1
,0S
,1S
,0D
,1D

SWAP
,0S
,1S
,0D
,1D

SWAP-1
starting with |n=0> phonons,
write into and read from the common vibrational mode
-pulse on blue SB
control bit control bit

77. Conditional phase gate
Effect:
phase factor of -1
for all, except |D,0 >
target bit
2
2
Composite pulse phase gate
I.Chuang,
MIT Boston
Rabi frequency:
1W nhBlue SB:

78. Composite phase gate (2 rotation)
       1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R R f         

1
2
3
4
,0 ,1S Don2

79. Population of |S,1> - |D,2> remains unaffected
       1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R R f         

4
3
2
1

80. ion 1
motion
ion 2
,S D
SWAP-1
,S D
0 0
SWAP
Ion 1
Ion 2
pulse sequence:
control bit
target bit
laser frequency
pulse duration
optical phase
Controlled-NOT operation

81. input
output
Fidelity of Cirac-Zoller CNOT
|<Yexp| Yideal >|2
F. Schmidt-Kaler et al.,
Nature 422, 408 (2003)
Fidelity : 73%
M. Riebe et al.,
PRL 97, 220407 (2006)
Fidelity : 92,6%

82. GHZ state:
W state:
C. Roos et al., Science
304, 1478 (2004)

83. Ion 3
Ion 2
Ion 1
Deterministic generation of GHZ state

84. Experiment Ideal
Fidelity: 72 %
Tomography of the GHZ state
C. Roos et al., Science
304, 1478 (2004)

85. What is a Schrödinger’s cat ?

87. Kilian Singer - Symposium
Hybride Quantensysteme Ulm
87
Schrödingerkatze im
Bewegungszustand
D. Leibfried et al., Nature 422, 412 (200
P.J. Lee et al., J. Opt. B. 7, 371 (2005)
Bestimmung der Phasenraumtrajektorie eines verschränkten Wellenpackets
U. G. Poschinger, A. Walther, KS, F. Schmidt-Kaler, Physical Review Letters 105,
263602 (2010).