Interaction-Free Ion-Photon Gates

Interaction-Free Ion-Photon Gates (Milan, May 17, 2007) Mladen Pavičić pavicic@grad.hr ; Web: http://m3k.grad.hr/pavicic University of Zagreb Interaction-Free Ion-Photon Gates p. 1/1

Early Interaction-Free Experiments Ancient Ideas: Renninger (1960), Dicke (1981), Pavičić (1986): Whenever we fail to detect interference we know that something is there. Interaction-Free Ion-Photon Gates p. 2/1

Early Interaction-Free Experiments Ancient Ideas: Renninger (1960), Dicke (1981), Pavičić (1986): Whenever we fail to detect interference we know that something is there. E.g., Pavičić (1986): Interaction-Free Ion-Photon Gates p. 2/1

Ring Resonator Harry Paul & Pavičić (Berlin, 1996) Interaction-Free Ion-Photon Gates p. 3/1

Ring Resonator Harry Paul & Pavičić (Berlin, 1996) Matteo was also there at the time Interaction-Free Ion-Photon Gates p. 3/1

Ring Resonator Harry Paul & Pavičić (Berlin, 1996) Matteo was also there at the time Let us calculate what we get at D r : Interaction-Free Ion-Photon Gates p. 3/1

Interference Reflected portion of the incoming beam: B 0 = A R Interaction-Free Ion-Photon Gates p. 4/1

Interference Reflected portion of the incoming beam: B 0 = A R After a full round trip this joins it: B 1 = A 1 R R 1 R e iψ Interaction-Free Ion-Photon Gates p. 4/1

Interference Reflected portion of the incoming beam: B 0 = A R After a full round trip this joins it: B 1 = A 1 R R 1 R e iψ All round trips: interference (a geometric progression) the total amplitude (D r ): B = i=0 B i = A R 1 eiψ 1 R e iψ Interaction-Free Ion-Photon Gates p. 4/1

Resonator Int.-Free Experiments ψ = (ω ω res )T phase per round-trip (r-t); ω incoming frequency; T r-t time; ω res resonance frequency (λ/2 = L/k, L r-t length) Interaction-Free Ion-Photon Gates p. 5/1

Resonator Int.-Free Experiments ψ = (ω ω res )T phase per round-trip (r-t); ω incoming frequency; T r-t time; ω res resonance frequency (λ/2 = L/k, L r-t length) So, ω = ω res B = 0 Interaction-Free Ion-Photon Gates p. 5/1

Resonator Int.-Free Experiments ψ = (ω ωres)t phase per round-trip (r-t); ω incoming frequency; T r-t time; ωres resonance frequency (λ/2 = L/k, L r-t length) So, ω = ωres B = 0 ABS (a) M M ABS ABS ABS (b) M M O O D r t D D t D r Interaction-Free Ion-Photon Gates p. 5/1

Classical Efficiency The efficiency of the suppression of the reflection into D r when there is no object in the resonator; ρ is the measure of losses Interaction-Free Ion-Photon Gates p. 6/1

Let object be atom 87 Rb has closed shells up to 4p and an electron in ground state 5s (J = L + S); We consider only one excited state: 5p 1/2. Interaction-Free Ion-Photon Gates p. 7/1

Let object be atom 87 Rb has closed shells up to 4p and an electron in ground state 5s (J = L + S); We consider only one excited state: 5p 1/2. Total nuclear ang. mom. K and J give the total ang. mom. of the atom: F = J + K. 87 Rb has K = 3/2, and its j = 1/2 ground states are split by hyperfine interaction into doublets with F = K ± j = 3/2 ± 1/2 = 2, 1. Interaction-Free Ion-Photon Gates p. 7/1

Atom vs. photon To excite and deexcite electrons between m = ±1 and m = 0 we must use circularly polarized photons with j p = 1 and m jp = ±1 Interaction-Free Ion-Photon Gates p. 8/1

Atom vs. photon To excite and deexcite electrons between m = ±1 and m = 0 we must use circularly polarized photons with j p = 1 and m jp = ±1 When an atom receives angular momentum of a photon, the following selection rules must be met: l = ±1, m = m jp = ±1. Interaction-Free Ion-Photon Gates p. 8/1

Atom vs. photon (ctnd.) By solving Schrödinger equation for our three-level system Ĥ Ψ = i h Ψ t, Interaction-Free Ion-Photon Gates p. 9/1

Atom vs. photon (ctnd.) By solving Schrödinger equation for our three-level system Ĥ Ψ = i h Ψ t, we arrive at the Hamiltonian Ĥ = h 2 0 Ω 1 (t) 0 Ω 1 (t) 2 Ω 2 (t) 0 Ω 2 (t) 0 Ω 1 and Ω 2 are Rabi frequencies Interaction-Free Ion-Photon Gates p. 9/1

Excited state drops out One of the eigenstates of the Hamiltonian is Ψ 0 = 1 Ω 2 1 (t) + Ω 2 2(t) (Ω 2(t) g 1 Ω 1 (t) g 2 ) Interaction-Free Ion-Photon Gates p. 10/1

Excited state drops out One of the eigenstates of the Hamiltonian is Ψ 0 = 1 Ω 2 1 (t) + Ω 2 2(t) (Ω 2(t) g 1 Ω 1 (t) g 2 ) It depends only on dark states g 1 and g 2 Interaction-Free Ion-Photon Gates p. 10/1

Excited state drops out One of the eigenstates of the Hamiltonian is Ψ 0 = 1 Ω 2 1 (t) + Ω 2 2(t) (Ω 2(t) g 1 Ω 1 (t) g 2 ) It depends only on dark states g 1 and g 2 We can use this to obtain a direct transfer of electrons from g 1 to g 2 without either emitting or absorbing photons on the part of atom in the following way Stimulated Raman adiabatic passage (STIRAP). Interaction-Free Ion-Photon Gates p. 10/1

STIRAP Experimentally, let photons be laser beams. Interaction-Free Ion-Photon Gates p. 11/1

STIRAP Experimentally, let photons be laser beams. Now, let us switch on and off the second laser before switching on and off the first one. Interaction-Free Ion-Photon Gates p. 11/1

STIRAP Experimentally, let photons be laser beams. Now, let us switch on and off the second laser before switching on and off the first one. This can be described by g 1 Ψ 0 2 = 1 for t g 2 Ψ 0 2 = 1 for t + Adiabatic complete population transfer g 1 g 2 is STIRAP: Interaction-Free Ion-Photon Gates p. 11/1

STIRAP g 1 g 2 F=2 h ω 1 σ + m= 0 e σ h ω 2 Ω 2 F=2 F=1 m= 1 g 1 Ω 1 STIRAP g 2 m= 1 Interaction-Free Ion-Photon Gates p. 12/1

Interaction-free excitation A left-hand circularly polarized photon could excite an atom from its ground state g 1 to its excited state e and a right-hand circularly polarized photon could excite the atom from g 2 to e. So an L-photon will see the atom in g 1 but will not see it when it is in g 2. With an R-photon, the opposite is true. Interaction-Free Ion-Photon Gates p. 13/1

State notation We feed our resonator with +45 and 45 linearly polarized photons. In front of an atom we place a quarter-wave plate (QWP) to turn a 45 -photon into an R-photon and a 45 -photon into an L-photon. Behind the atom we place a half-wave plate (HWP) to change the direction of the circular polarization and then another QWP to transform the polarization back into the original linear polarization. Interaction-Free Ion-Photon Gates p. 14/1

State notation (ctnd.) We denote the atom states as follows: 0 = g 1, 1 = g 2 They are control states; atom is control qubit. We denote the photon states as follows: 0 = 45, 1 = 45 They are target states; photons are target qubits. For example, 01 means that the atom is in state g 1 and the photon is polarized along 45. Interaction-Free Ion-Photon Gates p. 15/1

Interaction-free CNOT gate (a) M g1 M (b) M g 2 M QWP HWP QWP QWP HWP QWP 0,1 M ABS 1 ABS 0 PBS M HWP M D 0,1 M ABS 0 ABS 1 PBS M HWP M D M 0,1 SBS M 1,0 SBS (a) The atom is in state g 1 and can absorb 1 ; (b) The atom is in state g 2 and can absorb 0 ; 00 00, 01 01, 10 11, 11 10 Interaction-Free Ion-Photon Gates p. 16/1