Quântica Oscilador Paramétrico

Transcription

1 Luz e Átomos como ferramentas para Informação Quântica Oscilador Paramétrico Ótico Inst. de Física Marcelo Martinelli Lab. de Manipulação Coerente de Átomos e Luz

2 Parametric Down Conversion Energy and momentum conservation ω 0 = ω 1 + ω 2 k 0 =k 1 +k 2 Polarization and transverse momentum correlations

3 Optical Parametric Oscillator PDC + Cavity Signal Pump k 0,ω 0 α 0 in (t) χ (2) α 1 out (t) α 2 out (t) Idler k 1,ω 1 k 2,ω 2

4 Optical Parametric Oscillator (OPO)

5 Optical Parametric Oscillator PDC + Cavity Pump k 0,ω 0 α 1 out (t) Signal k 1,ω 1 α 0 in (t) χ (2) α 2 out (t) Idler k 2,ω 2 Twin photons + phase correlation - Sub-threshold squeezed vacuum (degenerate case) - OPA entangled fields (non-degenerate case) - Above threshold: intense entangled fields

6 Optical Parametric Oscillator (OPO) - Classical Let us describe classical properties of the system before we analyze quantum properties. We ll consider a Triply Resonant OPO (TR-OPO) in a ring cavity (for simplicity). α 0 in α 1 out α 0 out α 2 out r 0 r 1 r 2 R=1 l R=1 If we consider that the single pass gain is small, we can approximate the equations for the amplification inside the crystal

7 Optical Parametric Oscillator (OPO) - Classical α 0 in α 1 out α 0 out α 2 out r 0 r 1 r 2 R=1 l R=1 Consistency of the field for a round trip gives us

8 Optical Parametric Oscillator (OPO) - Classical If δϕ j is small, we can write: where the total loss for each mode is defined Normalizing the detuning, we have

9 Optical Parametric Oscillator (OPO) - Classical A first solution of these equations is α 1 = α 2 = 0, corresponding to operation below threshold. We are more interested in above-threshold operation. Multiplying the complex conjugate of the third equation by the second, we have: The intracavity pump power is easily obtained and we see it is clipped : above-threshold it is always the same Besides, for, we also have The classical equations are already signaling that the intensities of signal and idler beams should be strongly correlated and that the pump must be depleted.

10 Optical Parametric Oscillator (OPO) - Classical 1,0 0,8 0,6 0,4 0,2 0,

11 Optical Parametric Oscillator (OPO) - Classical

12 Optical Parametric Oscillator (OPO) - Classical 1,0 0,8 0,6 0,4 0,2 0,

13 Optical Parametric Oscillator (OPO) - Classical From the first equation we can derive the threshold power, given the intracavity pump field (α 1 = α 2 = 0) An important parameter will be the ratio of incident power to threshold power on resonance: Substituting α 2 in the first equation, we have

14 Optical Parametric Oscillator (OPO) - Classical Since and We get Solving for α j

15 Optical Parametric Oscillator (OPO) - Classical This gives the photon flux. Considering, for the sake of the argument, the frequency-degenerate case (ω 1 =ω 2 =ω 0 /2), we can obtain the total output power and the efficiency Where η max is the maximum efficiency leading to We will see that the parameter ξ determines the maximum squeezing in the above-threshold OPO.

16 Optical Parametric Oscillator (OPO) - Quantum Rest of the Universe α 0 in α 1 out α 0 out α 2 out r 0 r 1 r 2 R=1 l R=1

17 Optical Parametric Oscillator (OPO) Master Equation Evolution of the density operator System + Reservoir + Interaction Evolution of an operator acting only on the system: Master Equation: Evolution of ρ s

18 Quantum Properties of the OPO Hamiltonian and the master equation: OK, simpler now? We can improve this if we change from the density matrix into an equivalent representation: it will replace (ordering sensitive) operators by c-numbers. But the nonclassicallity makes P representation a tricky choice...

19 Quasi-Probability Representations P Glauber Sudarshan Wigner

20 Wigner Representation Evident quantum/ classical frontier Squeezed states Y States with W<0 α φ X Fock states

21 Quantum Properties of the OPO The operators are replaced by amplitudes and the density operator is replaced by Using the rules

22 Quantum Properties of the OPO We obtain Fokker-Planck equation

23 Quantum Properties of the OPO Which is equivalent to a set of Langevin equations (Do you remember the Brownian Motion?) The mean values in steady state are the same as in the classical treatment. Since we will (typically) deal with intense fields, we proceed by linearizing the fluctuations, neglecting products of fluctuating terms:

24 Quantum Properties of the OPO

25 Quantum Properties of the OPO Defining We get with

26 Quantum Properties of the OPO Defining We get with

27 Quantum Properties of the OPO The subspace related to the subtraction of the fields decouples from the sum and the pump fluctuations. However, q - does not have any decay term, thus the solutions are not strictly stable. As a matter of fact, there is phase diffusion and the subtraction of the phases is unbounded. Nevertheless, this is a slow process and we will be interested in measuring phases with respect to the phase of the mean field (in other words, we will follow adiabatically the diffusion). Instead of solving these equations in the time domain, we look in the frequency domain.

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

29 Usual treatment of the OPO: Langevin Equation Linearization Input Output Formalism Frequency Domain

30 Covariance Matrix X Spectral Matrix Complete description of the state: Wigner function (for a Gaussian State)

31 Covariance Matrix 36 independent terms!

32 Covariance Matrix 18 independent terms!

33 Noise correlations Signal Y p 1 Idler Y p 2 α α q 1 φ Signal - Idler q - X q 2 Y φ Signal + Idler X p + p - α + q + φ + X

34 Energy Conservation ω 1 + ω 2 = ω 0 δi 1 - δi 2 = 0 Intensity Correlation δϕ 1 + δϕ 2 = δϕ 0 Phase Anti correlation A. Heidmann et al., PRL. 59, 2555 (1987) A. S. Villar et al., PRL 95, (2005) Signal - Idler q - Y Signal + Idler p + p - α + q + φ + X

35 EPR s example ψ 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

36 Bohr s reply