Optical Parametric Oscillator (OPO) - Axpfep1.if.usp.br

Multipartite entanglement andsudden death in Quantum Optics:continuous variables domainPart II – The OPOMarcelo Martinelli

Question:Dividing the incident beam in two “equal” parts, what will be the result?WPBSi 1?i 2-i sub

Answer:WPBSi 1?i 2 i-subMedidaClassically: i sub= 0Quantically: “photons are clicks on photodetectors” (A. Zeilinger)= 0, ∆ 2 i sub > 0 !

Quantum Optics – Measurement of the Intense FieldWe can easily measure photon flux: optical powerÊbacd

Building an Interferometer – The Beam Splitter±bˆD2ĉBSdˆD1Homodyning if

“Classical” VarianceShot noise !Vacuum Homodyning allows the calibration of the detection, producing a Poissoniandistribution in the output (just like a coherent state).

Parametric Down ConversionEnergy and momentum conservationω 0 = ω 1 + ω 2k 0 =k 1 +k 2Polarization and transverse momentum correlations

Optical Parametric OscillatorPDC + CavitySignalPumpk 0 ,ω 0α 0in(t)χ (2)α 1out(t)α 2out(t)Idlerk 1 ,ω 1k 2 ,ω 2

Optical Parametric Oscillator (OPO)

Optical Parametric OscillatorPDC + CavityPumpk 0 ,ω 0α 1out(t)Signalk 1 ,ω 1α 0in(t)χ (2)α 2out(t)Idlerk 2 ,ω 2Twin photons + phase correlation- Sub-thresholdsqueezed vacuum (degenerate case) - OPAentangled fields (non-degenerate case)- Above threshold: intense entangled fields

Optical Parametric Oscillator (OPO) - ClassicalLet us describe classical properties of the system before weanalyze quantum properties. We’ll consider a Triply ResonantOPO (TR-OPO) in a ring cavity (for simplicity).α 0inα 1outα 0outα 2outr 0r 1r 2R=1lR=1If we consider that the single pass gain issmall, we can approximate the equations forthe amplification inside the crystal

Optical Parametric Oscillator (OPO) - Classicalα 0inα 1outα 0outα 2outr 0r 1r 2R=1lR=1Consistency of the field for a round trip gives us

Optical Parametric Oscillator (OPO) - ClassicalIf δϕ j is small, we can write:where the total loss foreach mode is definedNormalizing the detuning, wehave

Optical Parametric Oscillator (OPO) - ClassicalA first solution of these equations is α 1 = α 2 = 0,corresponding to operation below threshold. We are moreinterested in above-threshold operation. Multiplying thecomplex conjugate of the third equation by the second, wehave:The intracavity pump power is easily obtained and we see itis “clipped”: above-threshold it is always the sameBesides, for , we also haveThe classical equations are already signaling that the intensitiesof signal and idler beams should be strongly correlated and thatthe pump must be depleted.

Optical Parametric Oscillator (OPO) - Classical1,00,80,60,40,20,0-2 0 2

Optical Parametric Oscillator (OPO) - Classical

Optical Parametric Oscillator (OPO) - Classical1,00,80,60,40,20,0-2 0 2

Optical Parametric Oscillator (OPO) - ClassicalFrom 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 tothreshold power on resonance:Substituting α 2 in the first equation, we have

Optical Parametric Oscillator (OPO) - ClassicalSinceandWe getSolving for α j

Optical Parametric Oscillator (OPO) - ClassicalThis gives the photon flux. Considering, for the sake of theargument, the frequency-degenerate case (ω 1 =ω 2 =ω 0 /2), wecan obtain the total output power and the efficiencyWhere η max is the maximum efficiency leading toWe will see that the parameter ξ determines the maximumsqueezing in the above-threshold OPO.

Optical Parametric Oscillator (OPO) - QuantumRest of the Universeα 0inα 1outα 0outα 2outr 0r 1r 2R=1lR=1

Optical Parametric Oscillator (OPO) – Master EquationEvolution of the density operatorSystem + Reservoir + InteractionEvolution of an operator acting only on the system:Master Equation: Evolution of ρ s

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

Quasi-Probability RepresentationsP‐ Glauber ‐ SudarshanWigner

Wigner RepresentationEvident quantum/ classical frontierSqueezed statesYStates with W

Quantum Properties of the OPOThe operators are replaced by amplitudesand the density operator is replaced byUsing the rules

Quantum Properties of the OPOWe obtainFokker-Planck equation

Quantum Properties of the OPOWhich 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 classicaltreatment.Since we will (typically) deal with intense fields, we proceed bylinearizing the fluctuations, neglecting products of fluctuating terms:

Quantum Properties of the OPO

Quantum Properties of the OPODefiningWe getwith

Quantum Properties of the OPOThe subspace related to the subtraction of the fields decouples fromthe sum and the pump fluctuations. However, q - does not have anydecay term, thus the solutions are not strictly stable. As a matter offact, there is phase diffusion and the subtraction of the phases isunbounded. Nevertheless, this is a slow process and we will beinterested in measuring phases with respect to the phase of themean field (in other words, we will follow “adiabatically” the diffusion).Instead of solving these equations in the time domain, we look in thefrequency domain.