1. A deterministic source of entangled photons David Vitali, Giacomo Ciaramicoli, and Paolo Tombesi Dip. di Matematica e Fisica and Unità INFM, Università di Camerino, Italy

2. The efficient implementation of quantum communication protocols needs a controlled source of entangled photons The most common choice is using polarization-entangled photons produced by spontaneous parametric down-conversion, which however has the following limitations: Photons produced at random times and with low efficiency Photon properties are largely untailorable Number of entangled qubits is intrinsically limited (needs high order nonlinear processes)

3. For this reason, the search for new, deterministic, photonic sources, able to produce single photons, either entangled or not, on demand, is very active Proposals involve single quantum dots (Yamamoto, Imamoglu,….) color centers (Grangier,…) coherent control in cavity QED systems (photon gun, by Kimble, Law and Eberly) The cavity QED photon gun proposal has been recently generalized by Gheri et al. [PRA 58, R2627 (1998)], for the generation of polarization-entangled states of spatially separated single-photon wave packets.

4. Relevant level structure: double three-level  scheme, each coupled to one of the two orthogonal polarizations of the relevant cavity mode Single atom trapped within an optical cavity

5. Main idea: transfer an initial coherent superposition of the atomic levels into a superposition of e.m. continuum excitations, by applying suitable laser pulses with duration T, realizing the Raman transition. The spectral envelope of the single-photon wave packet is given by

6. Excitation transfer (when T » 1/k c ): atom  cavity modes  continuum of e.m. modes A second wave packet can be generated if the system is recycled, by applying two  pulses |f> 0  |i> 0 and |f> 1  |i> 1, and repeating the process The two wave packets are independent qubits if they are spatially well separated. In fact, the creation operator for the wave packet generated in the time window [t j,t j +T], satisfies bosonic commutation rules if | t j -t k | » T,

7. Repeating the process n times, the final state is where The residual entanglement with the atom can eventually be broken up by making a measurement of the internal atomic state in an appropriate basis involving |f> 0 and |f> 1. Bell states, GHZ states and their n-dimensional generalization can be generated. Partial entanglement engineering can be realized using appropriate microwave pulses in between the generation sequence

8. Possible experimental limitations and decoherence sources Lasers’ phase and intensity fluctuations Spontaneous emission from excited levels |r>  Photon losses due to absorption or scattering Effects of atomic motion Systematic and random errors in the  pulses used to recycle the process

9. Laser’s phase fluctuations are not a problem because the generated state depends only on the phase difference between the two laser fields  it is sufficient to derive the two beams from the same source Effects of spontaneous emission can be avoided by choosing a sufficiently large detuning   the excited levels are practically never populated Effect of imperfect timing and dephasing of the recycling pulses studied in detail by Gheri et al. The process is robust against dephasing, but the timing of the pulses is a critical parameter

10. Effect of laser intensity fluctuations Fidelity of generation of n entangled photons, P(n) Laser intensity fluctuations  with  (t) = zero-mean white gaussian noise    (T) becomes a Gaussian stochastic variable with variance g  4 D  T/16  4 k c  2 The fidelity P(n), averaged over intensity fluctuations, in the case of square laser pulses with mean intensity I and exact duration T, and with identical parameters for each polarization, becomes with

11. Three different values of the relative fluctuation F r = 0, 0.1, 0.2 Other parameter values are: g = √I = 60 Mhz,  = 1500 Mhz, k c = 25 Mhz, T = 30µsec

12. Three different values of the number of entangled photons, n = 3, 5, 10 Laser intensity fluctuations do not significantly affect the performance of the scheme

13. Effect of photon losses The photon can be absorbed by the cavity mirrors, or it can be scattered into “undesired” modes of the continuum These loss mechanisms represent a supplementary decay channel for the cavity mode, with decay rate k a  It is evident that the probability to produce the desired wave packet in each cycle is now corrected by a factor k c  /(k c  +k a  ) for each polarization  The fidelity in the case of square laser pulses and equal parameter for the two polarizations becomes

14. From the upper to the lower curve, k a /k c = 0, 0.001, 0.005, 0.01 From the upper to the lower curve, n = 3, 5, 10

15. Photon losses can seriously limit the efficiency of the scheme; the fidelity rapidly decays for increasing losses In principle, the effect of photon losses can be avoided using post-selection, i.e. discarding all the cases with less than n photons However, with post-selection the scheme is no more deterministic, and the photons are no more available after detection

16. Effect of atomic motion Effect minimized by trapping the atom and cooling it, possibly to the motional ground state  Lamb-Dicke regime is required Atomic motional degrees of freedom get entangled with the internal levels (space-dependent Rabi frequencies)  decoherence and quantum information loss making the minimum of the trapping potential to coincide with an antinode of both the cavity mode and the laser fields (which have to be in standing wave configuration)

17. Atomic motion is also affected by heating effects due to the recoil of the spontaneous emission and to the fluctuations of the trapping potential However, laser cooling can be turned on whenever needed  heating processes can be neglected. The motional state at the beginning of every cycle will be an effective thermal state  N vib with a small mean vibrational number N. Numerical calculation of the fidelity (the temporal separation guarantees the independence of each generation cycle)

18. From the upper to the lower curve: N = 0.01,0.1, 0.5, 1 Atomic motion do not seriously effect the photonic source only if the atom is cooled sufficiently close to the motional ground state (N < 0.1)

19. Conclusions Cavity QED scheme for the generation, on demand, of n spatially separated, entangled, single-photon wave packets Detailed analysis of all the possible sources of decoherence. Critical phenomena which has to be carefully controlled : imperfect timings of the recycling pulses cooling of the motional state photon losses The scheme is particularly suited for the implementation of multi-party quantum communication schemes based on quantum information sharing