1. Quantum information in bright colors Instituto de Física - USP Paulo A. Nussenzveig Paraty – 2009

2. Antônio Sales Felippe Barbosa Jonatas César Luciano Cruz Paulo Valente The Team Katiúscia Cassemiro Alessandro Villar Marcelo Martinelli Paulo Nussenzveig

3. Lectures 1 st Lecture – Continuous variables (CV): entanglement, squeezing, OPO basics. 2 nd Lecture – More on OPOs, bipartite entanglement below and above threshold. 3 rd Lecture – Direct generation of tripartite three-color entanglement; “Entanglement Sudden Death” in a CV system.

4. Einstein, Podolsky & Rosen’s paper

6. EPR’s example |  (x 1 – x 2 – L)  (p 1 + p 2 )(localized in x 1 – x 2 and p 1 + p 2 ) A measurement of x 1 yields x 2, just as a measurement of p 1 gives p 2. But x 2 and p 2 don’t commute! ↔ [x, p] = i ħ

7. EPR’s conclusion If (1) is false, then (2) is also false! Hence, (1) should be true: quantum theory, although it allows for correct predictions, must be incomplete. Measurements should just reveal pre-existing states, which are not described by this incomplete theory.

8. Bohr’s reply Bohr discusses complementarity, but his paper does not give sufficient arguments to rule out the EPR program. (This story goes on with the theorems by John Bell and experiments to violate Bell’s inequalities, and GHZ-states etc.)

9. Quantum Optics It is our purpose here to investigate these “spooky” correlations by using electromagnetic fields. Our bright beams of light have properties that are described just as position and momentum observables. We begin by describing these properties and then we will study them in a specific system, the optical parametric oscillator (OPO). Borrowing a line from an anonymous reviewer, these systems are of “great interest to the quantum information community, but also to a broad audience interested in the latest progress on sophisticated optical systems designed for quantum information applications”.

12. Field Quantization Each mode is a harmonic oscillator, with the Hamiltonian given the commutation relations The electric field is:

13. Field Quadratures And also as X and Y are the field quadrature operators, satisfying The electric field can be decomposed as Thus,

14. Quantum Optics Field quadratures behave just as position and momentum operators. Thus, we can expect to observe phenomena such as EPR-type correlations among optical fields. Before we proceed, we notice that the uncertainty relation sets a minimum bound on the product of the variances of orthogonal quadratures. For coherent states the variances are both equal to 1 (the so-called Standard Quantum Limit – SQL). For squeezed states one variance is smaller than 1, while the orthogonal quadrature necessarily has excess noise.

15. Coherent x Squeezed X Y Y X

16. Noise Measurements ± BS D2 D1 S.A. (Balanced) Homodyne Detection If field b is strong, we can replace the operator by its mean value If field b is the vacuum, we can obtain A’s intensity noise by measuring n +

17. The Optical Parametric Oscillator (OPO) New quantum light from a classic system

18. Optical Parametric Oscillator (OPO) Crystal OPO Signal Idler Pump 532 nm ~1064 nm

19. Optical Parametric Oscillator (OPO)

20. 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). ℓ   in   out   out   out r0r0 r1r1 r2r2 R=1 The hamiltonian has three terms:

21. Optical Parametric Oscillator (OPO) The amplitudes will be given in photon flux (photons per second). If we consider that the single pass gain is small, we can approximate the equations for the amplitudes, for propagation inside the crystal as: And, for a round trip:

22. Optical Parametric Oscillator (OPO) If  j is small, we can write: where the total loss for each mode is defined Normalizing the detuning, we have

23. Optical Parametric Oscillator (OPO) A first solution of these equations is  1 =  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.

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

25. Optical Parametric Oscillator (OPO) Since We get and Solving for  j 

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

27. Quantum properties of the OPO We resume from the Hamiltonian and the master equation: We will sketch the general method, which consists of choosing a representation (quasi-probability distribution) and converting the master equation into a set of stochastic differential equations. Our choice is to use the Wigner function, in spite of the fact that higher-order derivatives appear, which we simply neglect…

28. Quantum properties of the OPO The operators are replaced by amplitudes and the density operator is replaced by Using the rules

29. Quantum properties of the OPO We obtain Neglecting higher-order derivatives, we have a Fokker-Planck equation

30. Quantum properties of the OPO Which is equivalent to a set of Langevin equations 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:

31. Quantum properties of the OPO Defining with We get

32. 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 instead in the frequency domain.

33. Wiener-Khintchine theorem From Optical Coherence and Quantum Optics, L. Mandel e E. Wolf We wish to study a stationary random process z(t) We can try where S(  ) would represent the strength of fluctuations associated to a Fourier component of z(t). However, z(t) is nonzero for t  , so the above definition is not mathematically sound. is well defined for a large number of functions z(t), approaching zero when    if  z(t)  = 0. This problem was removed by Wiener and Khintchine who noticed that the auto-correlation function

34. Wiener-Khintchine Theorem The auto-correlation function and the spectral density (or power spectrum) are related by Fourier transforms. Time series of photocurrent measurements Power spectrum Auto-correlation function Wiener- Khintchine theorem Spectrum Analyzer Experimentally

35. Singularities in the power spectrum If  z   0, then  (  )  |  z  | 2 for   . This causes singularities to appear in S(  ). A well-behaved function can be obtained again by writing z(t) =  z  + z 0 (t), where z 0 (t) has a zero mean. For z 0 (t), we have  0 (0) gives the variance of z(t) Singularity removal in S(  ) Experimentally Oscilloscope DC output light DC Filter Our Detector z(t)z(t) z(t)z(t) HF output z0(t)z0(t) Spectrum Analyzer

36. Wait: not so fast! What do we really measure? How about the commutation relations? How do we define the SQL? i(t)i(t) Field operators depend on time, leading to slight changes in the commutation relations:

37. Wait: not so fast! In the frequency domain, In every detection system, we have a finite bandwidth, so we measure

38. OK, back to the OPO The Fourier transform is We concentrate on the subtraction subspace: giving

39. OK, back to the OPO The output field fluctuations are Finally,

40. OK, back to the OPO The subtraction subspace gives a minimum uncertainty product, for  = 0. Twin beams!