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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 • 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) • 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. 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 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. 0 c0 i 0 c1 i 1 0 1 0 c0 0 c1 c f c0 0 c1 0 cont dG ,Tbˆ 0 cont The spectral envelope of the single-photon wave packet is given by G ,T kc T dt 0 g (t) it e exp t i t 2kc 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 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], it Bˆ† t j ,T de j G ,T bˆ satisfies bosonic commutation rules if | tj-tk | » T, B tk ,T, B tk ,T jk † • Repeating the process n times, the final state is n 0 1 c0 0 c1 c 1 0 2 n f † where j Bˆ tj ,T 0 cont • 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 Possible experimental limitations and decoherence sources • Lasers’ phase and intensity fluctuations • Spontaneous emission from excited levels |r> • Systematic and random errors in the pulses used to recycle the process • Photon losses due to absorption or scattering • Effects of atomic motion • 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 Effect of laser intensity fluctuations • Fidelity of generation of n entangled photons, P(n) P(n) c 1 e 2 2 T n g2 t T 2 ds 2 s 4 kc 0 t with 2 2 t D t • Laser intensity fluctuations s t with (t) = zero-mean white gaussian noise (T) becomes a Gaussian stochastic variable with variance g4DT/164kc2 • 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 n 2 4 g IT g DT Pn 1 exp 2 4 2 2 kc 8 kc Three different values of the relative fluctuation Fr Fr = 0, 0.1, 0.2 DI 2 T Other parameter values are: g = √I = 60 Mhz, = 1500 Mhz, kc = 25 Mhz, T = 30µsec 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 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 ka • It is evident that the probability to produce the desired wave packet in each cycle is now corrected by a factor kc/(kc+ka) for each polarization • The fidelity in the case of square laser pulses and equal parameter for the two polarizations becomes n 2 k n g IT c Pn 1 exp 2 k k c 2 kc k a a 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 • 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 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 2 2m 0 1 • 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) • 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. rtot 0 0 0 r viN b • Numerical calculation of the fidelity Pn Trvi b 1 rtot T 1 n (the temporal separation guarantees the independence of each generation cycle) 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) 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