Quantum Optics exam. M2 LOM and Nanophysique. 28 November 2017

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1 Quantum Optics exam M LOM and Nanophysique 8 November 017 Allowed documents : lecture notes and problem sets. Calculators allowed. Aux francophones (et francographes) : vous pouvez répondre en français. 1 Spontaneous emission in strontium 1 = 30 MHz 1 P i 1 = 461 nm 3 = 7 Hz 3 P i 3 = 689 nm 1 Si Figure 1 Atomic structure of strontium. The atomic structure of strontium shown in figure 1. The ground state 1 S has spin S = 0 and orbital angular momentum L = 0. The state 1 P has S = 0 and L = 1 and 3 P has S = 1 and L = 1. The decay rates and wavelengths are indicated in the figure. 1. Recall the relation between the decay rate Γ of a state, the dipole matrix element and the wavelength of a transition.. Calculate the ratio of the dipole matrix elements d 3 and d 1 associated with the transition ( 3 P 1 S) and ( 1 P 1 S) respectively. 3. If S is an exactly conserved quantum number, d 3 should vanish. Why? 4. Because of relativistic effects, the state 3 P is not a pure S = 1 state, but has some 1 P character (S = 0). The true energy eigenstate associated with the transition 3 P to 1 S is therefore 3 P 3 P + ɛ 1 P. Calculate ɛ. 1

2 Single-photon source from an atomic ensemble E (a) (b) (c) (d) 1 1, 1, 0 g i g i g i Detector Figure (a) Atomic states involved in the single-photon source. (b) Scattering of a first photon following the action of the first laser. (c) Scattering of a second photon following the action of the second laser. (d) Experimental arrangement, with the detector placed in the direction 0. A narrow band single-photon source can be realized using a cloud of cold atoms. We model each atom as a three level system, as shown in Figure (a). There are two stable ground states g 1 and g separated by a small energy difference and an excited state e connected to each ground state by an electric dipole transition. An example is Rb in which the ground states are hyperfine levels and the transition to the excited state corresponds to λ = 780 nm for both transitions. We will ignore polarizations. The protocol operates in three steps. First we send in a pulse of light with wavevector 1 and Rabi frequency Ω 1, detuned from the transition g 1 to e by a frequency (Fig. b). Sometimes an atom scatters a photon into a mode with wavevector 0 (Fig. d). For an atom at position r j this scattering process is described by the Hamiltonian (h.c. means hermitian conjugate) : 1 = η 1 e i 1 r j (â e i r j g j g 1 j + h.c.), (1) Here η 1 is a quantity depending on Ω 1 and. In writing the Hamiltonian in this way we are treating the laser as a classical field. Next, the photon is detected by a detector in a narrow solid angle. Finally, we send in a second laser with wave-vector 1 and Rabi frequency Ω, coupling the states g and e (Fig. c). A photon with wave-vector is scattered by atom j. This process is described by the Hamiltonian : () = η e i r j (â e i r j g 1 j g j + h.c.), () with η defined in analogy to η 1. We will see in the problem that this protocol results in a heralded single photon source, with the photon emitted in a well defined direction, conditioned on the detection of the first photon on the detector.

3 1. Without trying to derive the Hamiltonians (1) and (), explain briefly why they describe the scattering processes represented in Figs. (b-c).. Consider first the case of a single atom, located at position r j. The atom and field are prepared in state g 1, 0 where 0 denotes the electromagnetic vacuum. After the application of 1 for a duration t the state of the system is ψ(t) = e i H(j) 1 t g 1, 0. (3) Assuming η 1 t 1, calculate ψ(t) to first order in η 1 t. 3. We now consider an ensemble of N atoms randomly positioned in the cloud. The total Hamiltonian describing the situation is : H 1 = 1. (4) The initial state of the atomic ensemble and field is g 1, g 1..., g 1 0. Using the same method as in the previous question, show that the quantum state after the scattering is, to first order in η 1 t : ψ(t) g 1, g 1..., g 1 0 i A j g 1, g 1,...g,j, g 1,...g 1 1, (5) where g 1, g 1,...g,j, g 1,...g 1 describes a state in which atom j is in state g and all the others are in g 1, and 1 denotes a single photon in mode Give the expression of the constant A j. 4. When the detector registers a photon, what is the normalized state ψ at of the atomic ensemble? Is it an entangled state? 5. Following the detection of a photon in mode 0, we apply the second laser pulse for a duration t. Justify that the rate of scattering a photon in a solid angle dω in the direction is proportional to (do not attempt to derive this expression from first principles) : dω () g 1, g 1..., g 1 1 H () ψ at 0 (6) with H () = N H(j) (), and 1 the state of the field corresponding to one photon with a wavevector. 6. Show that : and give the expression of. dω () 3 e i r j, (7)

4 7. Remembering that the atoms are placed randomly in the cloud, find the direction of scattering of the last photon as a function of 0, 1 and. Reproduce figure (d) and draw the wave-vector of the last photon. 8. Explain why this protocol implements an heralded single photon source. 9. Propose an experiment to verify that this source emits single photons. 10. Discuss the advantages and disadvantages of this method compared to generating single photons with a single Rb atom excited by a short laser pulse? 11. The two laser pulses 1 and are separated in time by t. During this delay, the atoms move : an atom moving with a velocity v j and located at position r j after the first pulse will be at r j + v j t just before the application of the second laser. Show that in this case dω () e i( r j+b v j ), (8) and give the expression of B as a function of 0, 1 and t. 1. Consider the case = 0. Why does the motion of the atoms degrade the performance of the source? 13. Using the formula B T m e ik v j t e 1 K v t (9) with v = the width of the Doppler distribution of the atoms at temperature T, calculate an order of magnitude of the highest temperature one can tolerate if t = 10 µs (impose the geometry you want for 0 with respect to 1 ). Tae m = g and the Boltzman constant B = J/K. 4

5 3 Article Faonas et al., Two-plasmon quantum interference 1. A photon propagating in the silicon nitride waveguide is converted in a plasmon when it reaches the border of the PMMA on Au region. Compare the size of the plasmonic region with the decay length of plasmonic mode and justify the choice of the size.. Why does the HOM (TPQI) visibility decrease with the length of the directional coupler? 3. What would one observe if one did the experiment at 700 nm instead of at 814 as was actually done. 4. The authors give an attenuation length (6.8 microns) and a coupling efficiency (0.66) at each dielectric-plasmon interface. Does this efficiency have an effect in the result in Fig Why was it important for the authors to stress that the contrast for a classical light source is less that 50%? 5