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.

• 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

Main idea: transfer an initial coherent superposition of theatomic levels into a superposition of e.m. continuum excitations,by applying suitable laser pulses with duration T, realizing the Raman transition.

ψ 0( ) = c0 i 0 +c1 i 1[ ]⊗ 0 c0 ⊗ 0 c1 ⊗ 0 cont⇒

ψ1 = 0 c0 ⊗ 0 c1 ⊗ cαα∑ f α ⊗ dωGα ω,T( )∫ ˆ b α ω( )0 cont

Gα ω,T( )=kcα

πdt

gαΩα(t)2δkcα0

T

∫ eiωt exp−μα t( )−iθα t( )[ ]

The spectral envelope of the single-photon wave packetis given by

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],

ˆ B α† tj,T( )= dωe

iωtj Gα∫ ω,T( )ˆ b α ω( )

satisfies bosonic commutation rules if | tj-tk | » T,

Bα tk,T( ),Bα† tk,T( )[ ]≈δαβδ jk

• Repeating the process n times, the final state is

ψ n = 0 c0 0 c1 ⊗ cα

α =0

1

∑ α 1 α 2K α n f α

where α j = ˆ B α† 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>

• Photon losses due to absorption or scattering

• Effects of atomic motion

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

• 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)= cαα∑ 2

1−e−2μα T( )[ ]

nμα T( )=

gα2 t( )

4δ2kcα

dsΩα2 s( )

0

t

∫

• Laser intensity fluctuations Ωα2 t( )=Ωsα

2 t( )+ Dαξ t( )with (t) = zero-mean white gaussian noise (T) becomes a Gaussian stochastic variable with variance g

4DT/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

with

P n( ) = 1−exp−g2IT2δ2kc

−g4DT8δ4kc

2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

n

Three different values of the relative fluctuationFr = 0, 0.1, 0.2

Fr =DI2

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

P n( ) =kc

kc +ka

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

n

1−exp−g2IT

2δ2 kc +ka( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

n

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

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 coincidewith an antinode of both the cavity mode and the laser

fields (which have to be in standing wave configuration)

η =

2πλ

h2mω0

⟨⟨1

• 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.

ρtot 0( )=ψ 0( ) ψ 0( ) ⊗ρNvib

• Numerical calculation of the fidelity

P n( ) = Trvib ψ1 ρtot T( )ψ1{ }[ ]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 sourceonly if the atom is cooled sufficiently close to the motionalground 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

• 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