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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
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• 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)

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• 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.

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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

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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
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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 fgt0?
  igt0 and
• fgt1? igt1 , 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,tjT,

satisfies bosonic commutation rules if tj-tk
T,
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• 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 fgt0
  and fgt1.

• 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

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Possible experimental limitations and
decoherence sources

• Lasers phase and intensity fluctuations

• Spontaneous emission from excited levels rgta

• Systematic and random errors in the p pulses
• used to recycle the process

• Photon losses due to absorption or scattering

• Effects of atomic motion

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• Lasers 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

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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

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Three different values of the relative
fluctuation Fr 0, 0.1, 0.2
Other parameter values are g vI 60 Mhz, d
1500 Mhz, kc 25 Mhz, T 30µsec
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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
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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/(kcakaa) for each polarization a

• The fidelity in the case of square laser pulses
  and equal
• parameter for the two polarizations becomes

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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
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• 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

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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)

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• 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)
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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 lt 0.1)
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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