Under the Influence of Spectral Entanglement:Polarization-Entanglement Swapping and Fusion Gates

Travis Humble* and Warren Grice, Oak Ridge National Laboratory, Oak Ridge, TN*Intelligence Community Postdoctoral Research Fellow

Under the Influence of Spectral Entanglement:Polarization-Entanglement Swapping and Fusion Gates

Travis Humble* and Warren Grice, Oak Ridge National Laboratory, Oak Ridge, TN*Intelligence Community Postdoctoral Research Fellow

Introduction Broad Bandwidth SPDC

Quantum Repeater

Polarization-entangled photons are a key resource for implementing quantum information protocols, in part because of the relative ease with which polarization-entangled photon pairs are generated by spontaneous parametric down-conversion (SPDC). In SPDC, a nonlinear optical crystal mediates the down-conversion of a high-energy pump photon into a pair of lower energy photons. Depending on the type of SPDC, the down-converted photons can be either similarly (type-I) or orthogonally (type-II) polarized.

Several schemes use SPDC to prepare polarization-entangled photon pairs. Generically, these schemes superimpose the outcome of down-conversion events, e.g., a type-II cross-ringed configuration, to prepare a polarization-entangled state. Ideally the biphoton state resembles a Bell state:

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ϕ12 =1

2h1,v2 + v1,h2( )

with h and v the horizontal and vertical polarizations, respectively, of photons 1 and 2.

With a broad bandwidth pump pulse, polarization entangled photons can be generated “on demand”.

p, kp2, k2

1, k1

Pump Laser

crystal

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ϕ12 =1

2dω1 dω2 f ω1,ω2( ) h1 ω1( ),v2 ω2( )[∫∫

+ g ω1,ω2( ) v1 ω1( ),h2 ω2( ) ]

Gaussian Joint Spectral Amplitude

has joint spectral amplitudes f(1,2) and g(1,2).

When the Schmidt decomposition

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f ω1,ω2( ) = λ n1/ 2un ω1( )vn ω2( )

n= 0

∞

∑

Polarization Entanglement

The concurrence of polarization density matrix

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C12 = dω1∫ dω2∫ f ω1,ω2( )g ω1,ω2( )*

Spectral Entanglement

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f path ω1,ω2( ) = gpath ω1,ω2( )

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f pol. ω1,ω2( ) = gpol. ω2,ω1( )

Spectral differences correlate with the paths of the photon, not their polarizations, e.g., twin type-I crystal configuration. For this case, the polarization-entanglement is maximal, i.e., C12 = 1.

has a greater-than-unity Schmidt number

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K =1 λ n2

n= 0

∞

∑

Spectral-Polarization Entanglement

But the other photonic degrees of freedom, i.e., spatial and spectral, complicate this description of polarization-encoded qubits.

Spectral differences correlate with the polarizations of the photon, not their paths, e.g., cross-ringed type-II configuration. For this case, polarization-entanglement depends on spectral overlap, i.e., spectral distinguishability.

Contact Information

Travis HumbleComplex Systems GroupOak Ridge National Laboratory(865)-574-6162humblets@ornl.govwww.ornl.gov/~hqt

then the photons are spectrally entangled.

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λn = sech2ϕ tanh2n ϕSchmidt coefficients

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μ =tanh2ϕ

Gaussian linear correlation

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K = cosh2ϕ

How does spectral entanglement affect concurrence obtainable with entanglement swapping?

What is the concurrence after n calls of entanglement swapping with N/2 photon pairs?

Warren GriceComplex Systems GroupOak Ridge National Laboratory(865)-241-2061gricew@ornl.gov

Entanglement following either sequential entanglement swapping or type-I fusion operations as a function of the initial spectral entanglement when =. We assume these protocols are deterministic and we do not account for any entanglement distillation. Note that the interpretation of the ‘concurrence’ is different for these two cases, but that both quantities exhibit identical behaviors with respect to spectral entanglement.

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C1,N = dx dy dzn f12 x,z1( ) f34 z1,z2( )*... fN−1,N zn ,y( )

*∫( )∫∫

× dzn f12 x,z1( ) f34 z1,z2( )*...∫ fN−1,N zn, y( )

*

( )

The multimode polarization-entangled state

Entanglement Swapping

/4

PBS

PBS

2

3

v

h

4

1

BS

PBS

2

3

v

h

4

1

PBS

h

v

How does spectral entanglement affect residual tangle (concurrence squared) obtainable with type-I fusion?

Type-I Fusion

Spectral-Path Entanglement

Types of Entanglement

Four Source Configurations

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C14 = dω d ′ ω ∫ dω f12 ω,ω ( )g34 ω , ′ ω ( )*∫( )

× dω g12 ω,ω ( )*

f34 ω , ′ ω ( )∫( )

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C134 = dω d ′ ω ∫ dω f12 ω,ω ( ) f34 ω , ′ ω ( )*∫( )

× dω g12 ω,ω ( )*g34 ω , ′ ω ( )∫( )

What is the residual tangle after n calls of type-I fusion with N/2 photon pairs?

Joint spectral probability (calculated) for type-II SPDC in a phase-matched 2 mm BBO crystal pumped by a 35 fs Gaussian pulse centered at 405 nm. Difference frequencies are measured with respect to half the pump energy (0) and darker regions indicate regions of higher probability density. While the joint spectra need not be identical, we do assume they are equally normalized. We also assume that any temporal walk-off arising from source configuration is compensated.

Schmidt number

Marginal Bandwidths

Schmidt modes are scaled Hermite functions

and

When spectral differences correlate with polarization, swapping and fusion are optimized by different source configurations.

When spectral differences correlate with path, both swapping and fusion are optimized by the same source configuration.

Cluster-State Preparation

depends on the type of spectral entanglement

The distinction between using sources (a) and (b) is due to the different underlying measurements. While swapping requires interference at the BS, the fusion gate depends on erasing which-path information. Both protocols work best with source (d) as there are no spectral differences or distinguishing information present.

Four source configurations arise from the two types of spectral entanglement and the orientation of the sources relative to one another. Colors red and blue denote spectral differences between the photons.

Applications

For the optimal source configuration source (d) and =, the answer to both questions is

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C1,N =sechNϕ

1− tanhNϕ

Using Schmidt coefficients