Atomic clocks and their applications20.01.2015

• Preliminary concepts

• Cryogenic sapphire oscillator

• Manipulation of atomic state

• Cesium atomic fountain

• Al+ quantum logic clocks

Mechanical clocks: Galileo Galilei

(1564-1642)

• Law of pendulum: period of swing oscillations

does not depend on its amplitude

• Designed a first mechanical clock

• However later he started to play with telescopes

Mechanical clocks: Christian Huygens

• Mathematical derivation of pendulum Law

• Improved mechanical clocks with time error of

1 minute per day (1629-1695)

Longitude prize, England 1714

• October 22, 1707. Lost of navigation due to inaccurate calculation of

position of ships results in sinking of British naval fleet off Isles of Scilly

• Fatalities: 1400-2000. Survivors: 13

• Establishing of Board of Longitude 1714

• Longitude prize:

20.000 £ for method that could determine longitude within 30 NM (56 km)

• John Harrison and George Graham improved the pendulum clock's accuracy to 1 second a

day. Temperature compensation. 1761 JH built marine chronometer and won the price.

What is the clock?

• Clock is pendulum

• We need:

1. Stable period

2. High Q-factor

• Solid state systems: quartz

• dT/T~10-8

• Period fluctuates

Precise clocks: atomic clock

dT/T ~ 10-7

dT/T ~ 3%

Quartz: dT/T ~ 10-8 Caesium: dT/T ~ 10-15

Clock precision isassociated to the qualityfactor of an oscillator

… But quality factor is not everything:

Discuss stabitly versus accuracy

Frequency standard• Faster the pendulum – better the time resolution

• Stable resonance frequency

• High Q-factor

• Feedback control or Referencing

0ω

T/1=Γ1.Measurement2.Corrections

Oscillator~Manipulation

Cryogenic sapphire oscillator

• Whyspering gallery mode

fo =12 GHz, Q ~ 5 x 109

• Cooled at sweet spot of sapphire T=6K

• Excellent short term stability

σy(t) ~ 10-16 (2-100 sec)

Stability and accuracy of the clocks

a) Stable and accurateb) Stable but not accuratec) Not stable but accurated) Not stable and not accurate

e) Allan variance for characterization of different noise sources

Atomic clocks• Faster the pendulum – better the time resolution

• Intrinsic stability of the energy levels

• Long lived atomic transitions: M1 and E2

• Ramsey method to detect a central frequency

1

2

012 ω=− EE

0ω

T/1=Γ1.Measurement2.Corrections

Oscillator~Manipulation

Dipole approximation• Next step: atom is at point R and electron wiggles around with amplitude r

• Bounded electron, or dipole in the Coulomb potential and Transversal field

• We expand A(r,t) over kδr around R

• We write Schrödinger Equation for atomic electron with the following Ansatz

)()),((21 2 reUtrAepm

H Atom +−=

Optical wavelength >> size of atom

( ) ikRikRrRik etAriketAetAtrRA )(...1)()(),( )( ≈++==+ + δδ

),(),(exp),(

),(),()(),(2

22

trrtRAietr

ttritrreUtRAie

m

φψ

ψψ

⋅=

∂∂

=

+

−∇−

M.O.Scully and M.S Zubairy, “Quantum Optics”

Dipole approximation

),(),()(2

),(2

trtRErereUm

ptri φφ

⋅−+=

Interaction Hamiltonian between atom and electric field in dipole approximation:

),(int tREreH

⋅−=

Interaction Hamiltonian between atom and electric field in quadrupole approximation:

),(2int tRErreH

∇⋅⋅−=

Interaction Hamiltonian between atom and magnetic field in dipole approximation:

),(int tRBH

⋅−= µ

Example of Transitions: Barium ion

• The interaction between EM-field is associated with certain Hamiltonian type andmultipole expansion. Different transition types. E1, E2, E3… (electric), M1, M2,…(magnetic)

• Transition selection rules. E1: ΔL = ±1, E2: ΔL = 0, ±2

• E1 and E2 transitions of 138Ba+ ion

• … Just a nice experiment

6S 1/2

6P 1/2

E1:493.4 nm

5D 5/2

5D 3/2

E1:649.6 nm

E2:1761.7 nm

Example of Transitions: Cesium

62S 1/2

62P 3/2

F=4

F=3

E1:852,355 nm

M1:9192631770 Hz

• M1 and E2 Tranistions possess long decay time

• M1 Transition in 133Cs is used to define the second:

The second is time interval comprising of 9192631770 radiation periods betweenhyprfine states of cesium

Primary Time standard NIST http://nist.time.gov

Two-level atom approximation

62S 1/2

62P 3/2

F=4

F=3

E1:852,355 nm

M1:9192631770 Hz M1:

9.19 GHz62S 1/2

62P 3/2

E1:852,355 nm

F=4

F=3

• Two-level atom approximation: transition between two levels is considered

• Radiation is treated to be classical and couples two energy levels

• Atom is quantum system

Optical 2-level system

µ-wave 2-level system

The Bloch vector and Bloch sphere

( )( )

( )( )

1

Im2Re2

2/cos2/sin

1

21

222

21

22

21

21

2

1

22

21

21

=++

−=

==

=

=

=+

+=Ψ

wvu

ccw

ccvccu

ecc

cc

cc

i θ

θϕ

1

2

u

v

w

Felix Bloch (1905-1983)

u and v: in-phase and quadrature components of the dipole momentw : population difference

M. Fox, “Quantum Optics: An introduction”

The Bloch vector and Bloch sphere

31

321

12

21

2

2

eeW

eweveuR

WRdtRd

vwwuv

vu

ceci

ceci

R

R

R

tiR

tiR

δ

δδ

δ

δ

+Ω=

++=

×=

Ω−=Ω+−=

=

Ω=

Ω=

−

e1

e2

e3Ground state

Excited state

W

Consider also gyroscope weel. Ifyou apply a momentum its startsto precess

M. Fox, “Quantum Optics: An introduction”C.Foot, “Atomic Physics”http://en.wikipedia.org/wiki/Gyroscope

Ramsey fringes

Norman Ramsey(1915-2011)

)2/(cos2/

)2/sin(2

112

)(

1)0(0)0(2

2

2

222

2

0

)()(

0

)(

2

12

1)(

2

2)(

1

0

0

0

0

0

Tc

eeetc

cc

ceci

ceci

p

ppR

iTi

iR

tiR

tiR

pp

δδτδττ

ωωωω

τωωωω

τωω

ωω

ωω

Ω=

−−

+−

−Ω=

==

Ω=

Ω=

−−

−

−−

−

,

T

τp

The experimental sequence is analogous to Young’s double slit experiment!

Ramsey fringes on caesium

62S 1/2

M1:9192631770 Hz

F=4

F=3

• π/2 pulse e3 → e2, accumulated phase due to detuning –e2, next π/2: -e2 → e3

• Period of the fringes: Δf = 1/T. FWHM = 1/2T

F. Riehle, “Frequency standards: basics and applications”

T

Atomic fountain

The principle of fountain

• MOT and cooling to ~1 μK. Moving up.

• Twice interaction with microwave at 9.2 GHz

• State detection by fluorescence on 852 nm

• Repeat

Caesium fountain atomic clocks

Caesium fountain atomic clocksCSF1 and CSF2 in PTB

dt/T=1 sec in 30 million years

Primary defenition of a second!

High precision time standards are usedin Telecom, GPS, Transport, TV, Radio….

http://www.ptb.de

Trapped ions versus trapped atoms

• Q = Δν/ν = 2 ν T

• Increase the frequency and increase T (interrrogation time)

• Trapped ions: T is very long but ions must also be very cold

• No collisions between ions

• Hg+: 40 GHz (Hyperfine splitting of ground state)

• Accuracy is better than for Cs atomic fountain

Accuracy of atomic clocks

Cooling of ion motion

Trapping 138Ba+

Ring diameter 1.2 mmView angle at 45 deg

Doppler cooling. TD ~ 1 mK

Amplitude ~ 40 nm

Averaging over many oscillations period results inmotion of ion in harmonic pseudo potential Ψ(r,z)with 3 oscillation modes

1 MHz 1.2 MHz

2.3 MHz

E

x,y,z

Resolved sideband coolingexcited

ground

λ = 200-500 nmГ ~ 10 MHz

Ion displacement

Ener

gy

Ion’s vibrational states

Ωx/2π~ 1 MHz

v 1+v1−v

Resolved sideband cooling

Ion’s vibrational states

1 20 3

Laser probe frequency

Ω−0ω 0ω Ω+0ω

• The transition linewidth Г is less then trap frequency Ωx

• Absorption spectrum is different when ion is cooled: RedSideBand dissappear

• Minimum Temperature less than recoil of the photon (theory) ~ 10 μK

Abso

rptio

n

Resolved sideband cooling

Ion’s vibrational states

1 20 3

Laser probe frequency

Ω−0ω 0ω Ω+0ω

• The transition linewidth Г is less then trap frequency Ωx

• Absorption spectrum is different when ion is cooled: RedSideBand dissappear

• Minimum Temperature less than recoil of the photon (theory) ~ 10 μK

Abso

rptio

n

Sideband cooling of Hg+ ion

F. Diedrich et al, PRL 62, 403 (1989)

• Doppler cooling on S → P

• Sideband cooling on S→ D

• Electron shelving to detect population of S1/2 state

• Lower sideband is suppressed

• <nv> ≈ 0.05

• T ~ 10 μK

Clock experiments with single ions

2S Microwave:1 - 40 GHz

Hyperfine splitting ofthe ground state

2P

readout

Be+, Mg+, Ca+, Sr+, Ba+

Hg+, Cd+, Zn+, Yb+

∞→τ

2S

2P

2D

2S

2P1

3P0

Ca+, Sr+, Ba+, Hg+, Yb+ Al+, In+

sec 1→τ

Orbital levels and/orforbidden transitions

optical clock

http://www.iontrap.umd.eduhttp://heart.c-704.uibk.ac.at Fritz Riehle, “Frequency standards: Basics and Applications”

h 1→τ

Principle of optical clock

Aluminium as clock reference

1S

1P1

3P0

h 1→τ

X• 27Al+ (I=5/2)

• Narrow optical transition

• No induced frequency shifts

• …But, no accessible cooling transition

nm 167=λ

nm 43.267=λ

Quantum logic spectroscopy

• Use additional ion with accessible optical transition

• … and quantum logic for spectroscopy

• Be+ ion for sympathetic cooling, internal state preparation of Al+ and internal state detection of Al+

Quantum logic spectroscopy