Atomic clocks and their applications...Atomic clocks • Faster the pendulum – better the time...
Transcript of Atomic clocks and their applications...Atomic clocks • Faster the pendulum – better the time...
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