Multimode Entanglement in. Continuous Variables

Transcription

1 Multimode Entanglement in Continuous Variables

2 Entanglement with continuous variables What are we measuring? How are we measuring it? Why are we using the Optical Parametric Oscillator? What do we learn?

3 EPR and Entanglement Anybody who is not shocked by quantum theory has not understood it. Niels Bohr

4 EPR s example W δ(x 1 x 2 L)δ(p 1 + p 2 ) (localized in x 1 x 2 e p 1 + p 2 ) A measurement of x 1 yields x 2, as well as a measurement of p 1 gives p 2. But x 2 and p 2 don t commute! [x, p] = i ħ

5 Bohr s reply

6 Entanglement with continuous variables What are we measuring? How are we measuring it? Why are we using the Optical Parametric Oscillator? What do we learn?

7 Classical Description of the Electromagnectic Field Fresnel Representation of a single mode Y E(t)=α exp(iωt) E(t)=E(t)+E * (t) α = X + i Y α E(t)=X cos(ωt)+ Y sen(ωt) φ X

8 Field Quadratures Quantum Optics Ê(t) =â exp(iωt) Ê = Ê(t)+Ê (t) Y α φ X

9 Field Quadratures Quantum Optics Y Uncertainty relation implies in a probability distribution for a given pair of quadrature measurements (a nice description at the W. Schleich book) α φ X Field quadratures behave just as position and momentum operators!

10 Entanglement with continuous variables What are we measuring? How are we measuring it? Why are we using the Optical Parametric Oscillator? What do we learn?

11 Quantum Optics Measurement of the Intense Field We can easily measure photon flux: field intensity (or more appropriate, optical power) Ê

12 Quantum Optics Measurement of the Field Intense Field OK, we have now the amplitude measurement, but that is only part of the history! Amplitude is directly related to the measurement of the number of photon, (or the photon counting rate, if you wish). This leaves an unmeasured quadrature, that can be related to the phase of the field. But there is not such an evident phase operator! Still, there is a way to convert phase into amplitude: interference and interferometers.

13 Building an Interferometer The Beam Splitter b a c d

14 Building an Interferometer The Beam Splitter ± bˆ D2 ĉ BS dˆ D1 Â Homodyning if < > << < > Vacuum Homodyning Quadrature Operator! CalibraCon of the Standard Quantum Level

15 Measurement of the Field in the time domain Y p α q φ X

16 Measurement of the Field in the frequency domain Amplitude 0,1 0,01 0,0 0,1 0,2 0,3 0,4 0,5 Frequency (Hz)

17 A classical field Modulation Coherent state Squeezed state ω 0 Ω ω 0 ω 0 + Ω ω Amplitude Phase

18 Measurement of the field HF Input BPF Peak Detector Spectrum Analyser LPF Ramp Video HF Input Filter LPF Demodulation Chain ADC Beatnote between carrier and sidebands

19 Measurement of the field: sideband picture cos(ω 0 t + θ e ) We have always a combined measurement of quadratures from both sidebands!

20 Phase Rotation of Noise Ellipse Y ± Reflected Beam Amplitude Noise X bˆ D2 ĉ BS dˆ D1 Â

21 For a fixed phase of the electronic oscillator, we are restricted to a linear combination of sidebands. Homodyne measurement will reconstruct the state of the combination of sidebands, but never from the complete sidebands. We need to produced discriminated, asymmetric phase shift for the sidebands.

22 Phase Rotation of Noise Ellipse b in (Vacuum) b out (Transmission) Optical Cavity a in (Input) a out (Reflection) Y X 22

23 Phase Rotation of Noise Ellipse Y Reflected Beam Amplitude Noise X Cavity detuning 23

24 Measurement of the field electronic quadratures

25 Measurement of the field electronic quadratures

26 Homodyne technique can give the complete temporal evolution of a single mode (in time domain). Homodyne technique allows the reconstruction of the covariance matrix for a mode that is a fixed combination of sideband modes (in frequency domain). In the case of laser phase diffusion, it will give us the average over different combinations a mixed state. Self-homodyne technique can give a complete reconstruction of the covariance matrix for sidebands of a given mode in the frequency domain. For two beams reconstruction, phase-shifted electronic oscillators can give the complete reconstruction on the sidebands correlations. Finally, remember that we are measuring noise. It has zero mean value, the first relevant terms are the second order momenta, and if the state is Gaussian, it is the ONLY relevant term.

27 Entanglement with continuous variables What are we measuring? How are we measuring it? Why are we using the Optical Parametric Oscillator? What do we learn?

28 Usual treatment of the OPO: Master Equation

29 χ(2)

30 Usual treatment of the OPO: Master Equation Quasi-probability representation Langevin Equation

31 Usual treatment of the OPO: Langevin Equation Linearization Input Output Formalism Frequency Domain Covariance Matrix X Spectral Matrix Hermitian

32 Tripartite entanglement Continuous Variable Multicolor Gaussian Energy conservation ω 1 + ω 2 = ω 0 Generation of pairs of photons, with pump depletion: ΔN 0 = - (ΔN 1 + ΔN 2 )/2 ΔN 1 = ΔN 2 +

33 Entanglement Test - Simon Positivity under Partial Transposition (discrete variables) non-negative eigenvalues -> Separability Continuous variables: PRL 77, 1413 (1996) PRL 84, 2726 (2000)

34 Entanglement Test - Simon Positivity under Partial Transposition (discrete variables) non-negative eigenvalues -> Separability Continuous variables: PRL 77, 1413 (1996) PRL 84, 2726 (2000)

35 Entanglement Test - Simon q p q q p p q p

36 Entanglement Test - Simon

37 Entanglement Test - Simon

38 Full characterization of the OPO

39 Spectral Covariance Matrix 21 independent terms!

40 Spectral Covariance Matrix 12 independent terms!

41 Twin beams (P 0 > P th )-[Fabre-1987] Squeezed vacuum (P 0 < P th, degenerate) [Kimble 1986] Correlation among pump, signal and idler for P 0 > P th [Cassemiro -2007] [Grosse -2008] Entangled fields - Vacuum (P 0 < P th ) [Kimble 1992] - Intense beams (P 0 > P th ) [Villar 2005] Pump Squeezing (P 0 > P th ) [Fabre 1997] Tripartite Entanglement [Coelho 2009] Changing the basis:

42 Entanglement with continuous variables What are we measuring? How are we measuring it? Why are we using the Optical Parametric Oscillator? What do we learn?

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44 The problem of decoherence Is the main problem for an eventual quantum computer, operating over many entangled qubits. What is the limit for this entanglement? Interaction with the environment! Why producing and keeping them is a hard task? Decoherence: as if the environment where continuously measuring the system! Famous example: Schrödinger Cat Paradox (1935). Also an entangled state

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48 Can there be No such more an ESD surprises? for bipartite Gaussian states? Scenario (1): robust entanglement Scenario (2): disentanglement

49 Tighter conditions for transmission of quantum entanglement!

50 Separable p Robust Fragile q

51 Tripartite entanglement Energy conservation ω 1 + ω 2 = ω 0 Generation of pairs of photons, with pump depletion: ΔN 0 = - (ΔN 1 + ΔN 2 )/2 ΔN 1 = ΔN 2 +

52 The REALLY complete reconstruction of the OPO state

53 ... in a single pass + 12 channels!

54 Real Imaginary

55 What about entanglement? Some combinations of sidebands can have stronger entanglement Laboratoire Kastler-Brossel ENS

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57 Multicolor Quantum Networks

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