Download A deterministic source of entangled photons

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

  dG  ,Tbˆ   0
cont
The spectral envelope of the single-photon wave packet
is given by
G  ,T  
kc 

T
 dt
0
g   (t) it
e exp  t   i  t 
2kc 
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],
it
Bˆ† t j ,T   de j G  ,T bˆ  
satisfies bosonic commutation rules if | tj-tk | » T,
†
B

t
,T

,
B
  k  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 t j ,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
g2 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 g4DT/164kc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
n

2
4




g IT g DT 

Pn   1 exp  2  4 2 


 2 kc 8 kc 



Three different values of the relative fluctuation
Fr = 0, 0.1, 0.2
Fr 
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




Pn  
1  exp  2

 

kc  ka  
 2 kc  k 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 vib
N
• Numerical calculation of the fidelity

Pn  Trvib  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