Principles and brief history of Cavity QED 1st
S.Haroche, specialized Solvay Lecture, June 4 2010
From the Bohr-Einstein photon box thought experiment…
…to the super-high Q cavities of today’s real experiments…. …exploring the quantum dynamics of atoms and photons in a confined space has progressed a lot…
Bohr’s draft of a box storing and releasing a photon to test quantum laws…
…and its «realization» by Gamov…
1 Early History of Cavity QED: controlling spontaneous emission
Early History: Tailoring spontaneous emission in a confined space Spontaneous processes are random. Only their rate can be predicted and, in the case of photon emission, estimated by classical arguments based on Maxwell’s equations The spontaneous emission rate of an excited state depends on the atom’s state, but also on the structure of the surrounding vacuum, which determines the density of modes into which photons are emitted: an atom within boundaries does not radiate as in free space. Similar effects in beta decay: a neutron lives longer in a nucleus than in free space!
Spontaneous emission enhancement predicted by E.Purcell in 1945… …and the possibility to inhibit spontaneous emission in atoms suggested by D.Kleppner in 1981
Antennae radiating near reflecting surfaces Dipole \\ to mirror and image cancell
Emission inhibited
Dipole # to mirror and image add up
When mirrors are curved, focusing effects enhance the resonances, leading to huge emission enhancement factors
In close-spaced gap, field modes with polarization \\ to mirrors are suppressed
Emission enhanced
"c/"
Dispersive effects: cavity Lamb shifts, Casimir effect
"c/" 100-1000 or larger
2l/! When gap is increased to l = l/2, mode density jumps and undergoes resonances for larger l values
Mode density in cavity (peak increases with Q)
Freespace mode density
l/!
First demonstration of Purcell effect on atoms Ionization of 23S Ionization of 22P
Ionization signal in a ramped electric field applied to the atoms after they leave the cavity. The 22P state ionizes in a larger field, thus at a later time than 23S.
Rydberg atoms prepared in state 23S in a cavity (V=70mm3) resonant with transition 23S22P (!=340 GHz).
Enhancement factor: ! = " Cat / " (23S #22 P ) = 530
Signals corresponding to an average of N atoms crossing together the cavity: N = 3.5, 2 and 1.3 for traces a,b,c respectively. Cavity on resonance (solid line) or offresonance (dashed line).
P.Goy, J-M.Raimond, M.Gross et S.Haroche, PRL 50, 1903 (1983)
Inhibiting the spontaneous emission of circular Rydberg atoms microwave transition
Atom prepared in circular Rydberg state n=22 (orbit parallel to metal plates)
Ionisation detector
Inhibited transition n=22 # n’=21 at " = 0.45mm
R.G.Hulet, E.S.Hilfer et D.Kleppner, PRL 55, 2137 (1985).
! / 2L
Atomic transmission versus "/2L: " is swept by Stark effect, L being kept constant. The sharp signal increase for "/2L=1 demonstrates the inhibition of s.e. of the Rydberg atom which survives longer in its initial state.
Many enhancement and inhibition experiments in microwave, infrared and optical part of spectrum realized since these pionneering studies…
Collective emission in cavity: from Purcell to Dicke Atoms located at equivalent nodal positions in cavity are symetrically coupled to field: they evolve during emission in a subspace invariant by atomic permutation. There is no way to know which atom has emitted when a photon is lost… Two atoms:
e, e !
"S =
1 ( e, g + ge 2
)
! g, g
Strong correlations with entanglement spontaneously build up between atoms, making collective dipole larger than when atoms radiate independently Due to this correlation, the spontaneous emission occurs faster than for single atom: this is Dicke superradiance
Superradiance rate proportional to number N of atoms A double enhancement effect: ! C (N ) = " N! 0 Purcell factor: ~ number of images in cavity wall collectively emitting with one atom
Number of atoms radiating collectively together
Observation of Dicke superradiance in a cavity Sample of N=3200 Sodium atoms prepared in Rydberg state 29S, emitting collectively in a cavity resonant with 29S-28P transition at != 162 GHz. The single atom spontaneous emission rate in free space on this transition is $0 =43s-1. Purcell factor: % = $atC / $0 ~ 70.
The atom-cavity coupling is switched-off after variable time by applying an electric field in cavity (Stark effect). For each interaction time t , we measure the number of atoms in states 29S and 28P after cavity exit. From an ensemble of 900 realizations of experiment, we reconstruct the histograms of the number Ne of excited atoms as a function of t (in units of tD ~ %N/$0 = 460 ns).
Agreement between experimental histograms and theory (solid lines in black) J-M Raimond, P.Goy, M.Gross, C.Fabre et S.Haroche, Phys.Rev.Lett. 49, 1924 (1982)
2. The strong coupling regime of CQED in time-domain: Rydberg-atom microwave experiments
From Purcell to Rabi: the strong coupling regime of Cavity QED $c
(
&c 2
" !c = << # c #c
Spontaneous emission in a continuum of cavity modes of width &c = '/Q imparts to atomic excited state a width $c inversely proportional to &c.
From Purcell to Rabi: the strong coupling regime of Cavity QED (
$c
&c
"2 !c = $ #c #c
As cavity Q factor increases, the cavity spectral width &c='/Q decreases and the rate of emission $c shoots up. The perturbative treatment of the Purcell effect breaks down when these two widths become equal.
Atomic dipole
da .E0 " != = da Vacuum h 2h# 0Vc
Rabi frequency
Vacuum fluctuations in Cavity
" $ %c = Q
Strong coupling regime: large dipole, small cavity volume and very large Q factor
From Purcell to Rabi: the strong coupling regime of Cavity QED is a story about a spin and a spring (Jaynes Cummings Hamitonian)
h!eg
h' # † † # % H= $ e e " g g & + h!a a " i $a e g " a g e %& 2 2 e g
The spin: 2-level atom
(
2 1 0 The spring: Cavity mode
$ #t ' $ #t ' e, 0 !! " cos & ) e, 0 + sin & ) g,1 % 2 ( % 2 ( $ # n +1t ' $ # n +1t ' e, n !! " cos & ) e, n + sin & ) g, n +1 2 2 % ( % (
Vacuum Rabi oscillation: (reversible spontaneous emission)
Rabi oscillation sped-up in n photons (stimulated emission)
Rabi oscillation in vacuum or in small coherent field: direct test of photon graininess p(n) n " ! n + 1t % Pe (t) = ( p(n)cos $ ; ' 2 # & n 2
p(n) = e) n
n n!
0
n=0
1 2 3
(n th = 0.06)
n = 0.40 (±0.02)
n = 0.85 (±0.04) n = 1.77 (±0.15) Pe(t) signal
Fourier transform
Inferred p(n)
Brune et al, PRL,76,1800,1996.
n
First strong coupling experiment in CQED: the micromaser (1985)
Rydberg atoms cross one at a time a high Q cavity and build up a manyphoton field in it by cumulative Rabi oscillations: the ultimate maser-laser Meschede et al, PRL 54, 551 (1985)
Herbert Walther 1935-2006
The ideal micromaser: a quantum machine to deliver photons in a box t=l/v
!n = 0
!n = +1 l
Probabilities given by Rabi: Simulations: # & n undergoes 2 ! n +1t + j " Pj (n) = cos % ( staircase-like 2 $ ' evolution, varying randomly j =1 Solid line: j = 0 between ensemble average different Trapping states realizations If trapping ! n0 +1t = 2 p" condition fulfilled, all # Pj =1 (n0 ) = 0 trajectories Photon number converge to n0 converges to n0 (here n0=10) Photon nber histograms at increasing times
The two-photon micromaser:
cavity tuned at half-frequency of transition between same parity levels M.Brune et al, PRL 59, 1899 (1987)
!n = 0
!n = +2
Emits photons by pairs
Single photon emission towards intermediate level is inhibited by CQED
Microlasers in optical CQED The optical version of the micromaser: field builds up from « kicks » produced by atoms crossing one by one the cavity K.An et al, PRL, 73, 3375 (1994).
Lasing of a single atom trapped in a cavity (Caltech group)
J.McKeever et al, Nature, 425, 268 (2003).
3. The strong coupling regime of atomicCQED in optical experiments H.J.Kimble (Caltech), G.Rempe (Garching), T.Esslinger (ETH-Zurich) Chapman (Georgia Tech), Vuletic (MIT), Orozco (Maryland), Blatt (Insbruck), Meschede (Bonn), Lange (Sussex)…
Cavity QED in optical domain: the atom-cavity «!molecule!»
The transmission spectrum of the cavity is split into two components when cavity contains a single atom (from atomic beam or dropped from a MOT). Fourier transform of timedependent Rabi oscillation
Thompson et al, PRL, 68, 1132 (1992)
Single atom detection by cavity field transmission a
b
a b
Depending on laser frequency, a single atom transit across cavity is signaled by a dip or a peak. A 100% efficient atom detector which can count one by one atoms in the cavity J.McKeever et al, PRL 93, 143601 (2004)
Using CQED as single atom counter to study atom-laser statistics similarity with optical laser (Glauber theory)
A.Öttl et al, PRL, 95, 090404 (2005)
Second order atom correlation (BEC atom-laser)
Histogram of number of atoms in time-bin (Poisson)
Second order atom correlation for thermal atom beam
Trapping force of a single photon in a cavity
The atom-cavity dressed energies are atomic position dependent. Their spatial derivative corresponds to a light-force exerted on average by one photon! This force can attract atom inside the cavity. Conversely, the atom modifies the cavity frequency, which changes its response to the field. From an analysis of this field transmission, the position of the atom inside the cavity can be obtained in real time: an atom-cavity microscope tracking atoms. Feed back procedures to improve the trapping have been implemented.
Garching group (similar films by Caltech group)
C.J.Hood et al, Science, 287, 1457 (2000) . T.Fischer et al, PRL 88, 163002 (2002)
The CQED photon pistol: releasing photons one by one on demand A Raman process on a single atom converts triggering pulses into photons escaping one by one from the cavity
f repumping
Experiment performed with flying atoms, then with trapped atoms in cavity
Kuhn et al, PRL, 89, 067901 (2002) J.McKeever et al, Science, 303, 1992 (2004) M. Hijlkema et al, arXiv:quant-ph/0702034 (2007)
i
Analyzing the photon pistol output The escaping light is analysed after beam splitting by correlation. Coincidence rate measured as a function of delay between the two output channels. Missing central peak is evidence of single photon emission (a photon cannot be «!split!»)
Single atom-non linear optics: the photon blockade effect in optical CQED Once a photon is resonant with the atom-cavity system, a second photon is off-resonant: only one photon at a time can be transmitted by cavity at resonance!
The transmitted light is antibunched Birnbaum et al, Nature 436, 87 (2005).
4. Entanglement experiments in microwave CQED
Two essential ingredients Circular Rydberg atoms Large circular orbit Strong coupling to microwaves
n = 51
e
n = 50
g
Long radiative lifetimes (30ms)
The spin
Level tunability by Stark effect Easy state selective detection Quasi two-level systems
Superconducting microwave cavity Gaussian field mode with 6mm waist Large field per photon Long photon life time improved by ring around mirrors (1ms) Easy tunability
The spring
Possibility to prepare Fock or coherent states with controlled mean photon number
Artist’s view of the Paris microwave CQED set-up (2001-2005 version) Microwave source Oven
Circular Rydberg state preparation
(coherent state)
Cavity State selective detection
Auxiliary microwave (atom manipulation) Raimond, Brune and Haroche RMP, 73, 565 (2001)
The route to circular states: a 53 photon adiabatic process
Controlling the atom-cavity interaction time: atomic velocity selection by optical pumping
Rubidium level scheme with transitions implied in the selective depumping and repumping of one velocity class in the F=3 hyperfine state
In green, velocity distribution before pumping, in red velocity distribution of atoms pumped in F=3, before being excited in circular Rydberg state
Useful Rabi pulses ( quantum knitting) Initial state
|e,0> % |e,0> + |g,1> $ / 2 pulse Creates atom-cavity entanglement P e (t)
|e,0>
0.8
51 (level e) 0.6 51.1 GHz 50 (level g)
0.4
# "t & # "t & e, 0 ! cos % ( e, 0 + sin % ( g,1 $ 2' $ 2'
0.2
time ( ?µ s)
0.0
Hagley et al, PRL 79, 1 (97)
0
30
EPR pairs in CQED
60
90
|e,0> % |g,1> |g,1>% |e,0>
P e (t)
|g,0> % |g,0> (|e> +|g>)|0> % |g> (|1> +|0>)
$ pulse maps atomic state on field and back
0.8
51 (level e) 0.6 51.1 GHz 50 (level g)
0.4
# "t & # "t & e, 0 ! cos % ( e, 0 + sin % ( g,1 $ 2' $ 2'
0.2
time ( ?µ s)
0.0
Maître et al, PRL 79, 769 (97)
0
30
60
90
|e,0> % - |e,0> |g,1> % - |g,1>
P e (t)
|g,0> % |g,0>
2$ pulse: phase gate and quantum logic operations
0.8
51 (level e) 0.6 51.1 GHz 50 (level g)
0.4
# "t & # "t & e, 0 ! cos % ( e, 0 + sin % ( g,1 $ 2' $ 2'
0.2
time ( ?µ s)
0.0 0
30
60
Nogues et al, Nature, 400, 239 (1999); Rauschenbeutel et al, PRL, 83, 5166 (1999)
90
Entangled atom-atom pair mediated by real photon exchange V(t)
g2
e1
Electric field F(t) used to tune atoms #1 and #2 in resonance with C for a determined time t realizing )/2 or ) Rabi pulse conditions
Hagley et al, P.R.L. 79,1 (1997)
+
+
Direct entanglement of two atoms via virtual photon exchange: a cavity-assisted controlled collision (after S.B.Zeng and G.C.Guo, PRL 85, 2392 (2000)).
Two modes: 1/*#1/* 1 +1/*2 %1 1 ( ! = 10 "6 # eg ' + * & $1 $ 2 )
Relatively insensitive to cavity Q and thermal photons
A thought experiment about complementarity EinsteinBohr discussion at Solvay 1927 Particle/slit entanglement
– Microscopic slit: set in motion when deflecting particle. Which path information and no fringes – Macroscopic slit: impervious to interfering particle. No which path information and fringes – Wave and particle are complementary aspects of the quantum object.
A “modern” version of Bohr’s proposal with a Mach-Zehnder interferometer
_
_ &
D
•Interference between two well-separated paths. • Getting a which-path information?
A “modern” version of Bohr’s proposal: Mach-Zehnder with a moving beam-splitter
_
&_
D
• Massive beam splitter: negligible motion, no which- path information, fringes • Microscopic beam splitter: which path information and no fringes
Complementarity and entanglement P
a
M
B1
b
•
O _
A more general analyzis of Bohr’s experiment –
Initial beam-splitter state
0
–
Final state for path b
!
–
Particle/beam-splitter state
&
_
Final fringes signal
" a "b
• Small mass, large kick NO FRINGES • Large mass, small kick FRINGES
D
M'
– Particle/beam-splitter entanglement – (an EPR pair if states orthogonal) –
B2
0!
0 ! =0 0 ! =1
" = "a 0 + "b !
Complementarity and decoherence Entanglement with another system destroys interference • explicit detector (beam-splitter/ external) • uncontrolled measurement by the environment (decoherence)
_
& _
D
Complementarity, decoherence and entanglement intimately linked
A more realistic system: Ramsey interferometry •
Two resonant $/2 classical pulses on an atomic transition e/g M
a B1
R1
b
R2
_ &
D B2
M' 1.0
0.8
0.6
Pg
Which path information? Atom emits one photon in R1 or R2
0.4
0.2
Ordinary macroscopic fields (heavy beam-splitter) Field state not appreciably affected. No "which path" information FRINGES Mesoscopic Ramsey field (light beam-splitter) Addition of one photon changes the field. "which path" info NO FRINGES 0.0
0
10
20
30
40
Fréquence relative (kHz)
50
60
An object at the quantum/classical boundary Coherent field in a cavity
From quantum to classical
•
•
Vacuum or small field:
•
Large quantum fluctuations. A field at the single-photon level is a quantum object
•
Large field
State produced by a classical source coupled for finite time to the cavity mode: field defined by complex amplitude '
a
!
'
•
A picture in phase space (Fresnel plane)
•
Small quantum fluctuations. A field with more than 10 photons is almost a classical object.
Bohr’s experiment with a Ramsey interferometer Store one Ramsey field in a high Q cavity
S e
Atom-cavity interaction time Tuned for $/2 pulse Possible even if C empty Initial cavity state
– Small field
_
R2 g
!
D
C
• Ramsey fringes contrast
•
& _
R1
– Intermediate atom-cavity state
– Large field
From quantum to classical classical
" =
!e ! g
!e " ! g " ! FRINGES
!e = 0 , !g = 1 NO FRINGE
1 e, ! e + g, ! g 2
(
)
Quantum/classical limit for an interferometer Fringes contrast versus photon number N in first Ramsey field Fringes vanish for quantum field 0.8
photon number plays the role of the beamsplitter's "mass"
0.7
Fringes contrast
0.6
An illustration of the (N() uncertainty relation :
0.5 0.4
• Ramsey fringes reveal field pulses phase correlations.
0.3 0.2 0.1 0.0 0
2
4
Nature, 411, 166 (2001)
6
N
8
10
12
14
16
• Small quantum field: large phase uncertainty and low fringe contrast Not a trivial blurring of the fringes by a classical noise: atom/cavity entanglement can be erased
An elementary quantum eraser • Another thought experiment
_
_ &
D
Two interactions with the same beamsplitter assembly erase the which path information and restore the interference fringes
Ramsey “quantum eraser” •
A second interaction with the mode erases the which path info 1.0 0.9
e,0
_
e,0
0.8 0.7
&
1 ( e,0 + g,1 2
0.6
|g,1>
)
0.5
Pe 0.4 0.3 0.2 0.1 0.0
10
12
14
Ramsey fringes without fields ! – Quantum interference fringes without external field – A good tool for quantum manipulations
Atom found in g: one photon in C whatever the path:no info and fringes
16
18
20
22
24
A conditional quantum eraser: new perspective on EPR
