The Extraction of Higher Order Field Correlations from a First Order Interferometer

1. The Extraction of Higher Order Field Correlations from a First Order Interferometer Scott Shepard Louisiana Tech University

2. Beyond the Diffraction LimitQuantum Phase & Interferometers“H-limits” in the Phase Rep.Phase Rep. of Interferometer Stat.sPhase Function Fitting AlgorithmSuggestions for Future Work

3. Quantum Interferences • “quantum lithography” • resolves beyond the classical diffraction limit • “super-resolving phase measurements” • yield accuracies below the shot noise limit • In essence, N photons of a field at wavelength λ can perform with an effective resolution wavelength of λ/N. • while using lasers, detectors, and the propagation properties associated with wavelength λ

4. Quantum Interferences • “quantum lithography” • resolves beyond the classical diffraction limit • “super-resolving phase measurements” • yield accuracies below the shot noise limit • In essence, N photons of a field at wavelength λ can perform with an effective resolution wavelength of λ/N. • while using lasers, detectors, and the propagation properties associated with wavelength λ • It was thought that these “higher-order fringes” could only be observed in the higher-order correlation functions of the field. • e.g., coincidence detection… • That would severely limit the practicality of such schemes to small values of N • because the apparatus increases in complexity with N • indeed, experimental observations to date are limited to N = 2,3, or 4

5. Quantum Interferences • “quantum lithography” • resolves beyond the classical diffraction limit • “super-resolving phase measurements” • yield accuracies below the shot noise limit • In essence, N photons of a field at wavelength λ can perform with an effective resolution wavelength of λ/N. • while using lasers, detectors, and the propagation properties associated with wavelength λ • It was thought that these “higher-order fringes” could only be observed in the higher-order correlation functions of the field. • e.g., coincidence detection… • That would severely limit the practicality of such schemes to small values of N • because the apparatus increases in complexity with N • indeed, experimental observations to date are limited to N = 2,3, or 4 Fortunately however, we have found a means of extracting this higher-order phase information from a first-order (i.e., standard) interferometer.

6. What an Interferometer Measures Jz _ Jz Φ +2 +1 0 -1 -2 = Φ Harmonic Oscillator Model of Angular Momentum (Schwinger 1952) Interferometer = Rotate by Φ; project onto Jz(Yurke, et. al. 1986) then inferΦ via <Jz> ~ cosΦ(coherent state)

7. Whereas the Phase Representation Estimates Φ Directly • Example: rotated by Φ = π/3 Jz Φ statistics – continuous Jz statistics – discrete Φ +2 +1 0 -1 -2 <Jz> still ~ cosΦ here

8. Whereas the Phase Representation Estimates Φ Directly • Example: rotated by Φ = π/3 Jz Φ statistics – continuous Jz statistics – discrete Φ +2 +1 0 -1 -2 <m> still ~ cos Φ here But <Jz> = 0 for noon states, etc. so they use higher-order (coincidence) detection schemes

9. “H-limits” in Phase Rep • Correct number/phase uncertainty principles exist • (Pegg and Barnett, 1990)(Shapiro, Shepard and Wong, 1991) • but variance has problems anyway, so pedantic but not practical

10. “H-limits” in Phase Rep • Correct number/phase uncertainty principles exist • (Pegg and Barnett, 1990)(Shapiro, Shepard and Wong, 1991) • but variance has problems anyway, so pedantic but not practical • Don’t need Schwarz inequality… • just calculate the performance measures directly/exactly

11. “H-limits” in Phase Rep • Correct number/phase uncertainty principles exist • (Pegg and Barnett, 1990)(Shapiro, Shepard and Wong, 1991) • but variance has problems anyway, so pedantic but not practical • Don’t need Schwarz inequality… • just calculate the performance measures directly/exactly • Get the dependence on N (the ave) directly • not via the variance of n and back again to N

12. “H-limits” in Phase Rep • Correct number/phase uncertainty principles exist • (Pegg and Barnett, 1990)(Shapiro, Shepard and Wong, 1991) • but variance has problems anyway, so pedantic but not practical • Don’t need Schwarz inequality… • just calculate the performance measures directly/exactly • Get the dependence on N (the ave) directly • not via the variance of n and back again to N • We are using complementarity (because the phase rep is) • but we get “H limits” with equality, not as a bound

13. “H-limits” in Phase Rep • Correct number/phase uncertainty principles exist • (Pegg and Barnett, 1990)(Shapiro, Shepard and Wong, 1991) • but variance has problems anyway, so pedantic but not practical • Don’t need Schwarz inequality… • just calculate the performance measures directly/exactly • Get the dependence on N (the ave) directly • not via the variance of n and back again to N • We are using complementarity (because the phase rep is) • but we get “H limits” with equality, not as a bound • variance on finite interval depends on origin • and isn’t useful for multiple-bumped pdfs • define local performance measures: FWHM; and bin.var. (var. given bin) • must also consider “bin errors”

14. “H-limits” for 3 Classes of States noonnooosub n u n u n u +2 +2 +2 +1 +1 +1 2 2 2 0 0 0 - 1 - 1 - 1 1 1 1 - 2 - 2 - 2 0 0 0 n d n d n d 1 2 1 2 1 2 0 0 0 • equivalent performance:

15. “H-limits” for 3 Classes of States noonnooosub n u n u n u +2 +2 +2 +1 +1 +1 2 2 2 0 0 0 - 1 - 1 - 1 1 1 1 - 2 - 2 - 2 0 0 0 n d n d n d 1 2 1 2 1 2 0 0 0 • equivalent performance: • would lead to general limit if sub-harmonics not useful

16. “H-limits” for 3 Classes of States noonnooosub n u n u n u +2 +2 +2 +1 +1 +1 2 2 2 0 0 0 - 1 - 1 - 1 1 1 1 - 2 - 2 - 2 0 0 0 n d n d n d 1 2 1 2 1 2 0 0 0 • equivalent performance: • would lead to general limit if sub-harmonics not useful • but sub can beat that So state optimization remains an open field !!

17. Quantum Phase Representation of Interferometer Statistics Quantum phase representation (simplest case) insert resolution of the identity operator Quantum phase representation of interferometer statistics = convolution of the phase rep. (state) with F xfrm of Wigner d’s (apparatus)

18. Quantum Phase Representation of an Interferometer • Computational advantages • unknown Φ only in phi rep displacement • not in the WD matricies(functions reduced to numbers) • rotation is simple shift in phi rep • Conceptual aspects • the angle rep of the state is convolved with functions that project these angles onto the measurement apparatus. • the Fourier coefficients of these functions are the well known “little d’s” (modeling the fixed beam splitters and photodetectors)

19. Phase Function Fitting Algorithm • Measure stats, the 2J+1 #s: (at some fixed but unknownΦ) • Calculate stats: (for some dummy variable x) • LMS fit the 2j+1 functions to the 2j+1 measured stats(optimal parameter estimation of x) • 9 to 16 digits of accuracy (for “four-photon” state) • Results independent of Φ(no need to null the interf.)

20. Phase Function Fitting Results Not Impossible • H-limits are for widths of PDFs (that we further process) • Really an infinite # of photons (collecting perfect stats) so error 0 is ok. • That’s applicable for “static limit” (slow signals: g-wv.s…) collect over long T (while Φ still ~ constant) so make E=TP big (infinite photons = perfect stats) but P fixed (e.g., N = 4 photon state…) (main constraint) Suggestions for Future Work • Simulate the tracking of a dynamic signal while acquiring and processing the histograms