A deterministic source of entangled photons

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

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/kc ): atom  cavity modes  continuum of e.m. modes • A second wave packet can be generated if the system • is recycled, by applying two p 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 [tj,tj+T], satisfies bosonic commutation rules if | tj-tk | » 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>a • Systematic and random errors in the p pulses • used to recycle the process • Photon losses due to absorption or scattering • Effects of atomic motion

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) with • Laser intensity fluctuations  with x(t) = zero-mean white gaussian noise  ma (T) becomes a Gaussian stochastic variable with variance ga4DaT/16d4kca2 • 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

11. Three different values of the relative fluctuation Fr = 0, 0.1, 0.2 Other parameter values are: g = √I = 60 Mhz, d = 1500 Mhz, kc = 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 kaa • It is evident that the probability to produce the desired • wave packet in each cycle is now corrected by a factor • kca/(kca+kaa) for each polarization a • 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, ka/kc = 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 • Atomic motional degrees of freedom get entangled with the • internal levels (space-dependent Rabi frequencies) • decoherence and quantum information loss • Effect minimized by • trapping the atom and cooling it, possibly to the motional • ground state  Lamb-Dicke regime is required • 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 rNvib 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 • photon losses • cooling of the motional state • The scheme is particularly suited for the implementation • of multi-party quantum communication schemes based • on quantum information sharing