Theory: Enrico Brambilla , Alessandra Gatti, Lucia Caspani, Luigi Lugiato

1. Central European Workshop on Quantum Optics 2007 14th Edition, 1-5 June 2007, Palermo, Italy Quantum spatial correlations in parametric down-conversion and detection of faint images Università dell’Insubria, Dipartimento di Fisica, Como Theory: Enrico Brambilla,Alessandra Gatti, Lucia Caspani, Luigi Lugiato Exp. in Como: Ottavia Jedrkiewicz, Paolo Di Trapani (Insubria Univ.) Exp. in Torino: Marco Genovesi et al. (Ist. Galileo Ferrari, Torino)

2. Information on is retrieved from The source may have some excess noise , but for small obj: standard quantum limit Cancellation of excess noise - measurement limited only by shot-noise. INTRODUCTION:Differential measurement of a weak absorbtion coefficient (a) classical source WEAK OBJECT N1’ test beam obj N1 N2 LASER SOURCE (not shot-noise limited) reference beam 50/50 BS obj<< 1 weak absorbtion coefficient

3. signal LASER SOURCE (2) idler 50/50 BS OPO twin beams  The use of quantum correlated twin beams (with <1) improves the signal-to-noise ratio with respect to the SQL. Differential measurement of weak absorbtion coefficient (b) quantum source WEAK OBJECT N1’ test beam obj N1 N2 reference beam Sub-shot noise signal-idler correlation:

4. The technique has been used with single-mode twin beams generated by an OPO: Souto Ribeiro, Schwob, Maitre, Fabre, Opt. Lett. 22, 1893 (1997) (2dB noise reduction, two-photon transition) Jiangrui Gao et al., Opt.Lett. 23, 870 (1998) (4dB noise reduction) Question: can we use the same technique to detect the spatial distribution of the absorbtion coefficient obj(x), i.e the image of a weak object ? - No if the twin beams are single mode. • Yes, if we use multi-mode twin beams, as those generated by single-pass spontaneous parametric down-conversion (PDC) • - we need sub-shotnoise correlation between small regions of the signal and idler cross-section, i.e. quantum correlation in the spatial domain.

5. Microscopic process: q = photon transverse momentum signal (q, p/2+Ω) (q=0, p) k2 k1 q idler pump photon (-q, p/2-Ω) kp Close to degeneracy (<< ωp) the signal and idler photons are emitted symmetically Process of spontaneous parametric down-conversion In the high gain regime (experiment in Como by Ottavia Jedrkiewicz et al.) BBO crystal - type II phase-matching signal - idler emission cones O. Jedrkiewicz et al., PRL 93, 243601(2004) 1 ps pump pulse of frequency p Far field pattern from a single pump pulse at ωp/2 with a 10nm IF (2) crystal

6. evident strong spatial correlation between the two symmetrical images Experimental evidence of twin beam effect in the spatial domain of high gain regime of PDC (close to collinear phase-matching) zoomed signal zoomed idler O. Jedrkiewicz, Y..K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, Phys. Rev. Lett. 93, 243601 (2004). O. Jedrkiewicz, E. Brambilla, M. Bache, A. Gatti, L. A. Lugiato, and P. Di Trapani, J. of Mod. Opt. 53, 575 (2006). • pump beam waist ~ 1 mm • - gain 10-1000 photon/pixel

7. measurement of sub-shot noise correlation spatial ensemble statistics performed over 100 x 40 pixels 3.0 2.5 2.0 i 1.5 Shot-noise level 1.0 Spread of data due to CCD noise (7ph./pixel). 0.5 noise reduction limit =1-=0.25 0.0 10 100 Sub-shot-noise correlation up to 20 photons corresponding to 100 ph./mode

8. Presence of sub-shot noise correlation between symmetrical positions in the PDC far field can be used to enhance the sensitivity of imaging. CCD signal idler ÷obj(x) far-field plane Problem of the Como experiment (regime of high gain-ps pulses): the correlation remains sub-shot noise (σ<1) only for relatively low photon numbers (<20 ph/pixel) would not allow to achieve SNR > 1 (unless σ 0) the CCD noise ( 7 ph./pixels) is not negligible and would be detrimental Can we identify regimes where the quantum character of correlation is maintained at higher photon numbers?

9. z classical, undepleted pump (parametric regime) signal idler q,  1 q+q’,  +’ signal - idler emission cones (2) 2 pump q’, ’ PUMP z lc 0 Modelling parametric down-conversion (at any gain regime) 1 2 Propagation equation along the crystal for signal and idler envelope operators: Microscopic process Phase mismatch function: includes the effects of spatial/temporal walk-off,diffraction, dispersion Numerics:equivalent 3D stochastic model based on the Wigner representation

10. Limit of a plane-wave pump N1 q,  1 f qp =0, p - pump 2 -q, - N2 Perfect intensity correlation betwen any two symmetric far field portions of the two beams N1(q)=N2(-q) Finite size of the pump waist wP uncertainty in the relative propagation directions of twin photons wP FAR FIELD Signal photon pump (2  angular spread of the pump Idler photon Perfect intensity correlation are recovered only for detection areas broader than a coherence area Brambilla, Gatti, Bache, Lugiato, Phys Rev A 69, 023802 (2004)

11. Explanation:Unbalance in the test and reference arms due to imperfect detection Example: different losses 1 2,with1 2 M = degeneracy factor ~ number of modes collected by the detector ~ 2.5 in the experiment of Como (few temporal mode since coh ~pump ~ 1ps) (1- 2)2 <<1 in the experiment  small contribution of the term Enoise Deterioration of signal-idler correlation with increasing gain Important: it is not predicted by the basic PDC theory: if size of detectors > xcoh

12. Inacuracy in the determination of the symmetry center Unbalance due to an error xshiftin determining the center of symmetry of the far field pattern (xshift ~ size of CCD pixel = 20m, xcoh ~40 m) σ1-2 numerical simulation sub-shot noise Correlation 1-2=1-  shot-noise level sub-shot noise correlation are extremely sensitive to xshift when Enoise is large

13. Transition from quantum to classical correlation (raw estimate): more tolerance on the error in the symmetry center xshift if Enoise is small.

14. Comparison between experimental data and numerical simulation (xshift=3m in the numerical) ●Experimental data ▲ Numerical simulation standard quantum limit

15. Solution II: decrease the excess noise! Regime of low-gain (small number of photon per mode) - long pulse duration. It allows to increase the degeneracy factor M : signal and idler statistics become Poissonian-like  Excess noise makes the quantum character of correlation very fragile with respect to unavoidable experimental imperfections Solution I:careful balance of the twin beams, use of algorithms to retrieve the symmetry center with sub-pixel precision (L. Caspani et al., preprint): it works, but not so efficiently.

16. M~15 M~1000 Numerical simulation for increasing pump pulse duration assuming xshift=6 m By increasing pump M, the slope of σs-i decreases  sub-shot noise correlation also for large photon number

17. Numerical simulation of the detection of a weak object (x) weak object (x) =0.04 CCD signal (2) reference idler Gaussian pump pulse wpump=1500 μm pumpup to 8ns flens FAR FIELD Relevant quantities evaluated from the simulations:

18. SNR as a function of the pump pulse time pump (g=0.9 is kept constant). =1 PDC source 1-2=0.04, F=4 =0.9 1-2=0.115, F=2.7 =0.8 1-2=0.24, F=1.95 classical source (SQL) object PDC SQL

19. low excess-noise low excess-noise high excess noise high excess noise Role of the excess noise SNR, Fvs error in the symmetry center xshift <N1> fixed =3500 ph./pixel More tolerance to experimental imperfections for low excess-noise

20. Conclusions • Imaging scheme that exploits the spatial quantum correlation of twin-beams, for high-sensitivity imaging. • Quantum nature of spatial correlation is more robust towards unavoidable experimental errors for a low excess noise  regime of low-gain, long-pulses. • In this condition  substancial enhancement of the SNR with respect to the SQL in a domain of parameters accessible to experimental implementations. • Resolution limit for high-sensitivity imaging: the coherence length characterizing the spatial correlation.

21. 1.2   0.99 1.0 0.8 FWHM 0.6 xcoh2 pixels 0.4 G(x) 0.2 0.0 -0.2 -0.4 -0.6 -20 -10 0 10 20 x(pixels) Local character of the signal-idler correlation Correlation of signal-idler intensity fluctuations: Correlation function profile

22. ordinary - extraordinary emission cones (2) PUMP z lc 0 3D modelling of parametric down-conversion (at any gain regime) microscopic process 1 q,  1 q+q’,  +’ 2 2 pump q’, ’ FAR FIELD Numerics: equivalent stochastic model with classical-looking equations in the Wigner representation Starting point: propagation equation along the crystal for signal and idler envelope operators: signal undepleted pump idler Ap(q,) = spatio-temporal Fourier transform of a classical, undepleted, pump (parametric regime) g ÷ (2) parametric gain Phase mismatch function: includes effects such as GVM, spatial walk-off, diffraction, dispersion, etc.

23. The improvement in SNR using the PDC source (with respect to the SQL) is (general formula) (weak absorbtion limit) F Conditions for high-sensitivity detection: 1) Gain in SNR is substantial only if correlations are well below shot-noise 2) Large photon number operation is needed to have SNR>1 (for realistic values of 1-2)  1-2

24. Outline • Introduction • Measurement for the detection of weak amounts of absorbtion • - Principle of operation of differential measurements • - Beating the SQL by using sub-shot noise correlation of twin beams • Question: can we also image a weak object with the same technique ? • I. Demonstration of quantum correlation in the spatial domain in the high gain • regime of spontaneous parametric down-conversion (Exp. of Como). • I.High-sensitivity Imaging, simulation of the experiment with realistic parameters . • - concluding remarks

25. vs xshiftfor increasing pump pulse duration pump <N1>=3500 ph./pixel Signal-idler correlation high excess noise Enoise ~ sat~400 g=4.5, xcoh30m g=2.5, xcoh15m low excess noise sat ~ 1+Enoise ~ 1.25 Correlation is more robust for long pump pulses with a low excess noise

26. SNR, Fvsxshiftas the pump pulse duration pump is increased and the parametric gain is decreased ( so that )

27. CONCLUSIONS • The detection of weak images beating the SQL can be achieved • using multi-mode twin beams generated by spontaneous parametric • down-conversion. • A substantial gain in SNR can be obtained if: • 1) A careful balance of the test and the reference beam is achieved • 2) The excess noise is as low as possible so that correlation • are more robust (use of long pulses or cw pump)

28. xcoh=15 μm, pixel side =20μm xcoh=30μm, pixel side =20μm Role of the coherence area Only when the detection area is close to the coherence area, the degree of correlation σ approaches its maximum value 1-significant improvement of SNR  the coherence area sets a limit for the spatial resolution of high-sensitivity imaging

29. vsshot-noisefor increasing values of xshift Numerical simulations: by increasing the error xshift in the determination of the center of symmetry, the slope of the curves increase. A detailed numerical analysis (see e.g. O. Jedrkiewicz et al. J. of Mod. Opt. 53, 575 (2006).)

30. Transition from quantum to classical correlation: more tolerance on the error in the symmetry center xshift if Enoise is small. We need CCD noise in the weak absorbtion experiment, Solution: we can make M large by increasing the pump pulse time signal and idler statistics become Poissonian

31. SNR as a function of the pump pulse time pump (the gain/mode is kept constant). PDC source σs-i=0.4 classical source (SQL)

32. Effect of shift on correlation center Comparison between experimental and numerical results No scattering contribution No background noise correction taken into account in numerics Shot-noise limit

33. BBO type I Shutter time 40ms Pumped at 400nm by a continuous diode laser

35. Far field image of the selected portion of PDC fluorescence No temporal statistics is made. Statistical ensemble constituted only by spatial replicas. Averages are only SPATIALperformed inside box (4000 pix) for each single laser pulse Inserting the 10nm IF allows to locate on the CCD the collinear degeneracy point The generated pairs of signal and idler phase-conjugate modes propagate at symmetrical angles with respect to the pump direction in order to fulfill the phase-matching constraints, and each pair of symmetrical spots charac- terizing the far field represents a spatial replica.

36. ordinary - extraordinary emission cones (2) PUMP z z lc 0 q,  1 q+q’,  +’ 2 p q’, ’ Modelling parametric down-conversion (any gain regime) 1 2 Starting point:propagation equation along the crystal for signal and idler envelope operators: Ap(q,) spatio-temporal Fourier transform of a classical, undepleted, pump (parametric regime) Phase mismatch function: includes the effects of temporal and spatial walk-off, diffraction, dispersion

37. Spatial entanglement in the far field of PDC (high gain regime) NEAR FIELD (2) PUMP NEAR FIELD FAR FIELD For symmetry reason we have In an ideal experiment (plane-wave pump, no detection losses), the theory predicts In a real experiment more precisely shot-noise level  = detection quantum efficiency

40. (2) SIGNAL IDLER FAR FIELD NEAR FIELD pump lc=5mm SIGNAL IDLER • Finite size of the pump waist wP --> uncertainty in the propagation directions of twin photons • uncertainty in the transverse momentum of photon 1 from a measurement of the momentum of photon 2 • Perfect intensity correlation recovered for detection areas larger than • Finite crystal length--> uncertainty in the twin photon position due to diffraction spread • uncertainty in the position of photon 1 from a measurement of the position of photon 2 • Perfect spatial intensity correlation for detection areas larger than Brambilla, Gatti, Bache and Lugiato, Phys. Rev. A 69, 023802 (2004)

41. Numerical simulation for the detection of a weak object obj(x) weak object obj(x) CCD test signal (2) reference idler Gaussian pump pulse flens flens NEAR FIELD FAR FIELD Parameters of the simulation: g=0.9, flens=5cm wpump=1500m pump up to 8 ns obj=0.04

42. χ(2) signal idler OPO twin beams Note: for σ1-2=1 we recover the SQL result Differential measurement of weak absorbtion coefficient (b) quantum source WEAK OBJECT N1’ test beam  N1 N2 reference beam Sub-shot noise signal-idler correlation:

43. Experimental set-up to measure PUMP PULSE p= 352nm, 1ps O. Jedrkiewicz, Y..K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, Phys. Rev. Lett. 93, 243601 (2004). O. Jedrkiewicz, E. Brambilla, M. Bache, A. Gatti, L. A. Lugiato, and P. Di Trapani, J. of Mod. Opt. 53, 575 (2006). spatial filter +200 mm Selection of a portion of PDC fluorescence around collinear direction type II BBO rectangular aperture (4mm) High quantum efficiency CCD (=89%) signal polarizing beamsplitter CCD idler 1 ~ 2 704nm low-band pass filter

44. Spatially entangled (multi-mode)beams: photon numbers correlated in time and in the beam cross sections sub-shot noise Correlation with local detection Ordinary (single-mode)twin beams: photon number correlated in time, but uncorrelated in space sub-shot noise correlation only with bucket detector [courtesy of Claude Fabre]