EE-LE E OPTI T C A L S Y TE

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1 1> p p γ 1 γ > 3 c 3> p p +> > 1> THREE-LEVEL OPTICAL SYSTEMS

2 . THREE-LEVEL OPTICAL SYSTEMS () OUTLINE.1 BASIC THEORY.1 STIRAP: stimulated raman adiabatic passage. EIT: electromagnetically induced transparency.3 CPT: coherent population trapping.4 RAP: rapid adiabatic passage, revisited

3 . THREE-LEVEL OPTICAL SYSTEMS (3).1 BASIC THEORY Physical Model 1> Ε α, α α β > Ε β, β 3> r Eα r Eβ α β r Eα cos r Eβ cos 1 3 α β αt βt Electric dipole interaction: r ˆ r µ µ 1 r µ 1 r µ 3 r µ 3 V ( ) V ADE h α cosα t r r r r µ E( t) ( e) E( t) α cosα t β cosβ t β cosβ t r r r r µ 1 Eα µ 3 Eβ α β h h

4 3 1 1 cos cos cos cos β β β β α α α α t t t t H h Total Hamiltonian:. THREE-LEVEL OPTICAL SYSTEMS (4) V H H + Interaction Picture: 1 U UH H β α α h H IP β β α α h H h t H i IP U e ˆ Interaction Picture: with U UH H RWA

5 . THREE-LEVEL OPTICAL SYSTEMS (5) Density Matrix: coherent dynamics i & ρ, h i h [ H ρ] In the interaction picture: & ρ [ H, ρ ] H IP With: H H IP h α α α β β ( ) α β Density Matrix: adding the incoherent dynamics & i ρ, h Lρ [ H H ρ ] Lρ IP + where accounts for: α spontaneous emission incoherent pumping 1> elastic and inelastic collisions fluctuations in the phase and/or amplitude of the laser fields α γ 1 γ 3 β > β 3>

6 . THREE-LEVEL OPTICAL SYSTEMS (6) & i ρ, h ( α ρ1 ) ( ) + ( α ρ1 Im β ρ3 ) ( ρ ) & ρ11 Im + γ 1 ρ & ρ Im γ 1ρ γ 3ρ & ρ33 Im β 3 + γ 3 ρ β 13 ( ) ρ13 ρ1 + α Γ & i ρ3 13 ρ 13 with Γ 13 β α ρ [ H H ρ ] Lρ IP + & β ρ 1 i ( ρ ρ11) α α ρ1 ρ13 & β ρ3 i ( ρ ρ33) ρ3 α β ρ13 ( γ )/ Γ1 ρ 1 with Γ1 1 + γ 3 ( γ )/ Γ3 ρ 3 with Γ3 1 + γ 3 & α κ α + ig ρ 1 α R Im ρ 1 Re ρ1 κ: losses and g : gain constant For accounts for light absorption/amplification yields the index of refraction

7 . THREE-LEVEL OPTICAL SYSTEMS (7). STIRAP: stimulated raman adiabatic passage "Coherent population transfer among quantum states of atoms and molecules" K. Bergmann, H. Theuer, and B. W. Shore Rev. Mod. Phys. 7, 13 (1998) P Pump 1 S Stokes 3 H ( t) h P ( t) P ( t) P S ( t) S ( t) ( ) P S Energy eigenstates: P S 1 + sin Θ cos Θ 3 D cos Θ 1 sin Θ 3 1 sin Θ 1 + cos Θ 3 tan Θ P ( t) / S ( t) λ D λ ± ± P + S ± P S For (Raman resonance condition)we still have D cosθ 1 sin Θ 3 For, there is no dark state P S STIRAP: adiabatically following the dark state from 1 to 3, i.e., Θ( t ) º Θ( t ) 9º f

8 . THREE-LEVEL OPTICAL SYSTEMS (8) T Adiabaticity condition: ± T > 1

9 . THREE-LEVEL OPTICAL SYSTEMS (9).3 EIT: electromagnetically induced transparency For S, the probe absorption profile is Lorentzian p > γ γ 3 p 1 3> Im{χ} 1> p For S, S and P << S K. J. Boller et al., Phys. Rev. Lett. 66, 593 (1991) J. E. Field et al., Phys. Rev. Lett. 67, 36 (1991) J. R. Boon et al., Phys. Rev. A 57, 133 (1998) 1> p p γ 1 γ > 3 S 3> p p +> > 1> Im{χ} s - c / s c / p

10 . THREE-LEVEL OPTICAL SYSTEMS (1) p p +> > { A } + A1 + A1 + A A1 + P abs α A1 Re 1> Interference term due to quantum interference Steady-state solution for << R with s p s Im ρ p 1 Two Lorentzian resonances steady state K1 K 1 ( ρ ρ ) 11 + K ( ) ( ) s / Γ13 + p + Γ 13 Γ1 ( ) ) / + Γ Γ + ( Γ + Γ ) s p 1 13 p 1 13 Two dispersive resonances quantum interefernce contribution K s Im ρ3 ( s / ) p ) ( / ) + Γ Γ + ( Γ + Γ ) s p Γ1 Γ13 p 1 13

11 . THREE-LEVEL OPTICAL SYSTEMS (11) K. J. Boller, A. Imamoglu, S. Harris, Phys. Rev. Lett. 66, 593 (1991) VOLUME 66, NUMBER PHYSICAL REVIEW LETTERS MAY 1991 Observation of Electromagnetically Induced Transparency K.-J. Boller, A. Imamoğlu, and S. E. Harris Edward L. Ginzton Laboratory, Stanford University, Stanford, California 9435 (Received 1 December 199) We report the first demonstration of a technique by which an optically thick medium may rendered transparent. The transparency results from a destructive interference of two dressed states which are created by applying a temporally smooth coupling laser between a bound state of an atom and the upper state of the transition which is to be made transparent. The transmitance of an autoionizing (ultraviolet) transition in Sr is changed from exp(-) without a coupling laser to exp(-1) in the presence of a coupling laser.

12 . THREE-LEVEL OPTICAL SYSTEMS (1) Slow light. Reducing group s velocity via EIT L. H. Hau et al., Nature 397, 594 (1999). M. M. Kash et al., Phys. Rev. Lett. 8, 59 (1999). Im{χ} Re{χ} s - c / s c / p s - c / s c / p Phase velocity: Group velocity: v c / n c v g dn n( ν ) + ν dν dn > dν v g << c

13 . THREE-LEVEL OPTICAL SYSTEMS (13) L. H. Hau et al., Nature 397, 594 (1999).

14 . THREE-LEVEL OPTICAL SYSTEMS (14) Superluminical light c v g dn n( ν ) + ν dν Λ p p γ 1 γ 3 > s p p +> > dn < dν 1> 3> 1> v g >> c Im {χ } R e{χ } s - c / s c / p s - c / s c / p D. Mugnai et al., Phys. Rev. Lett. 84, 483 (). L. J. Wang et al., Nature 46, 77 (). microwaves visible

15 . THREE-LEVEL OPTICAL SYSTEMS (15) L. J. Wang et al., Nature 46, 77 ()

16 . THREE-LEVEL OPTICAL SYSTEMS (16) other non-linear optics induced by atomic coherences VOLUME 8, NUMBER 3 PHYSICAL R EVIEW L ETTERS 7 JUNE 1999 Nonlinear Optics at Low Light Levels S. E. Harris Edward L. Ginzton Laboratory, Stanford University, Stanford, California 9435 Lene Vestergaard Hau Rowland Institute for Science, 1 Edwin H. Land Boulevard, Cambridge, Massachusetts 14 and Department of Physics, Harvard University, Cambridge, Massachusetts 138 (Received 1 December 1998) We show how the combination of electromagnetically induced transparency based nonlinear optics and cold atom technology, under conditions of ultraslow light propagation, allows nonlinear processes at energies of a few photons per atomic cross section. VOLUME 84, NUMBER 7 PHYSICAL REVIEW LETTERS 14 FEBRUARY Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons M. D. Lukin 1 and A. Imamoğlu 1 ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 138 Department of Electrical and Computer Engineering, and Department of Physics, University of California, Santa Barbara, California 9316 (Received 19 October 1999) Two light pulses propagating with slow group velocities in a coherently prepared atomic gas exhibit dissipation-free nonlinear coupling of an unprecedented strength. This enables a single-photon pulse to coherently control or manipulate the quantum state of the other. Processes of this kind result in generation of entangled states of radiation field and open up new prospectives for quantum information processing energies of a few photons per atomic cross section.

17 . THREE-LEVEL OPTICAL SYSTEMS (17) EIT from an adiabatic point of view. Let us assume that the Stokes laser is a continuous wave (cw) laser: P Pump 1 S Stokes 3 P S D cos Θ 1 sin Θ 3 In this case: Θ º Θ Xº Θ º 1 a 1 + b 3 1 Xº tan 1 ( Pump ) MAX Stokes The atom starts in state 1>, interacts with the laser fields and, eventually, it ends up again in state 1>, without absorbing any photon. ELECTROMAGNETICALLY INDUCED TRANSPARENCY

18 . THREE-LEVEL OPTICAL SYSTEMS (18).4 CPT: coherent population trapping 1> α α γ 1 γ 3 β > β 3> Let us assume now: -, are continuous wave α β - The initial state of the atom is not known γ - 1 γ 3 for simplicity In the interaction picture, the Hamiltonian of the three-level system is: H H 3L H IP D B N N h α α α β [ 1 3 ] 1 β α [ ] 1 α β N ( α β β ) Let s define the dark and bright states as: α + β H H 3L 3L D B h ( α β ) α 3 N h ( α + β ) N h + ( α β ) β 3 N h α + β

19 . THREE-LEVEL OPTICAL SYSTEMS (19) α β H H 3L 3L D B h ( N h ( N α α + β ) β α ) 3 h + N ( α β ) β 3 h α + β Let us assume, for simplicity: R; ; γ 1 γ3 α β α β D 1 ( 1 3 ) B + ( 1 3 ) D B After several cycles of absorption and spontaneous emission, the atom is eventually trapped in the dark state

20 . THREE-LEVEL OPTICAL SYSTEMS () G. Alzetta, A. Gozzini, L. Moi and G. Orriols, Nuovo Cimento B 36 (1976) 5. E. Arimondo, and G. Orriols, Nuovo Cimento Lett. 17 (1976) 333. G. Orriols, Nuovo Cimento B 53 (1979) 1. E. Arimondo, Coherent population trapping in laser spectroscopy, Progress in Optics, vol 35, ed. E. Wolf (Amsterdam: Elseveier) F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, Phys. Rev. A 55, 371 (1997) Probe absorption (a.u.) Fluorescenece (a.u.) Coherent population trapping on the D 1 line of a thermal sodium beam

21 . THREE-LEVEL OPTICAL SYSTEMS (1) In the V-scheme, there is no CPT due to spontaneous emission. 1 3 D B Coherent population trapping in two-electron three-level systems with aligned spins J. Mompart, R. Corbalán, L. Roso, Phys. Rev. Lett. 88, 363 ()..6.5 D c Probe absorption /γ bc α

22 . THREE-LEVEL OPTICAL SYSTEMS ().5 RAP: rapid adiabatic passage, revisited > 1> 1 1 / / a a a a i & & N. V. Vitanov and B. W. Shore. Stimulated Raman adiabatic passage in a two-state system. Phys. Rev. A 73, 534 (6) ( ) ) ( 1 ) ( ) ( t i t i e t a e t a t ψ + + h r r µ E We define new variables: Im } Im{ Re } Re{ ρ ρ ρ ρ a a w a a v a a u w v u w v u w v u dt d. Following: There is one eigenvector with null eigenvalue: sin cos ) ( θ ϑ θ u w d with tanθ

23 . THREE-LEVEL OPTICAL SYSTEMS (3).5 RAP: rapid adiabatic passage, revisited u Re{ a a > r r µ E h 1> 1 v Im{ a a 1 } Re ρ } Im ρ 1 1 ρ ρ11 w a a u d dt + v + w u v w 1. u v w w d( θ ) wcosϑ u sinθ RAP, revisited: v with u tanθ The atom is initially in the ground state with: > ; >> θ ( t ) w 1, u v d( t ) At the end: As: d( t) < ; >> θ ( t ) π t f Then, at the end: w 1, u v ρ ( t ) 1 f

24 3 LA RIGA NERA

25 MAIN ARTICLES FIRST EXPERIMENT G. Alzetta, A. Gozzini, L. Moi and G. Orriols, Nuovo Cimento B 36 (1976) 5. FIRST THEORY E. Arimondo, and G. Orriols, Nuovo Cimento Lett. 17 (1976) 333. G. Orriols, Nuovo Cimento B 53 (1979) 1. MAIN REVIEW ARTICLE E. Arimondo, Coherent population trapping in laser spectroscopy, Progress in Optics, vol 35, ed. E. Wolf (Amsterdam: Elseveier) 1996.

26 Experiments with a sodium vapor cell Na vapor cell laser beam oven

27 Hemholtz coils Uniform magnetic field single coil Variable magnetic field Coil with oscillating current Radiofrequency field λ/4 waveplate Circular polarization

28 resonant fluorescence phenomena Tuning the laser wavelength to the D 1 sodium line

29

30 Sodium atom

31 Energy levels of the Na electron (ground and first excited) 3P 3P 3/ 3P 1/ F 3,, 1, F, 1 D 1 D 4 D 1 589, nm 589,6 nm 6 D 3S 3S 1/. F F GHz m F (from +F to -F) Orbital Electron angular spin momentum L S1/ Fine structure J Nuclei spin I 3/ Hyperpine structure F Magnetic field Zeeman B sublevels

32 Orientation of an atomic vapor by optical pumping Orientation of the atomic angular momenta Linear polarization σ + B Simplified case J 1/ σ + angular momentum +1 J 1/ + - m J +1/ - 1/ λ/4 B J 1/ +1/ - 1/ conservation of angular momentumr selective excitation spontaneous emission to all states atoms acumulate in a particular Zeeman sublevel that corresponds to an orientation of the angular momentum atoms do not absorb light and, eventually, do not produce fluorescence

33 Linearly polarized laser beam tuned to the sodium D 1 line Circularly polarized beam (by means of a λ/4 waveplate)

34 Magnetic resonance induced by the radiofrequency field

35 +1/ - 1/ llumσ + light Resonance condition ν RF ν àtom (E +1/ E -1/ )/h ν RF B +1/ - 1/

36 Modifying the frequency or Modifying the magnetic field

37 Two-photon (RF) resonance

38 Laser induced non-absorbing resonance RF RIGA NERA B

39 RIGA NERA Destructive interference between the two absorbing pathways At the Riga Nera, there is a resonance between the ν hyperfine splitting and the frequency of two consecutive light modes ν ν ν RF B Laser spectrum Frequency Dark state non absobing superposition

40 Frequency standards and atomic clocks based on the dark line

41 Micromagnetometer and atomic clocks based on the D1 line Photodiode NIST, Boulder Rb cell Optical elements VCSEL laser