Quantum state tomography of slow and stored light

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A seminar presented at Reed College February 27, 2013. The talk is an extended version of a talk written for Photonics West 2013.

A seminar presented at Reed College February 27, 2013. The talk is an extended version of a talk written for Photonics West 2013.

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  • 1. 22 Trustees for action. The Program Director may All letters of inquiry and completed formal applications should be mailed inn, an interview with the applicant, or a visit to hard copy to: he full proposal, including staff summary and John Van Zytveld, Ph.D. QUANTUM STATEhe Trustees for their consideration and decision. Senior Program Director ptly when a decision has been reached. While M. J. Murdock Charitable Trustn nearly every proposal received by the Trust, only P. O. Box 1618wed can result in awards. When an application has Vancouver, WA 98668 TOMOGRAPHY OF SLOW ried over for future consideration. Under normal For More Help f a proposal that was declined is not encouraged. If your questions have not been answered by this document or you need some perty of the Trust and will not be returned. It will additional information, please call us at 360.694.8415.munication with the understanding, however, that AND STORED LIGHT Mailing Address: M. J. Murdock Charitable Trust PO Box 1618 Vancouver, Washington 98668 Office Location: Andrew M. C. Dawes, Noah T. Holte, Hunter A. Dassonville M. J. Murdock Executive Plaza 703 Broadway, Suite 710 Pacific University Vancouver, Washington 98660 Reed College Physics Seminar 360.694.8415 Contact: Phone WA: February 27, 2013 Fax: 360.694.1819 Phone OR: 503.285.4086 Website: www.murdock-trust.org Friday, March 29, 13
  • 2. Quantum State of Light All the “knowable” information about an optical signal. frequency* { amplitude For a plane wave: phase propagation direction polarization* * we’ll ignore these for todayFriday, March 29, 13
  • 3. Preserving the Quantum State Storing information in the quantum state is delicate Fidelity: how well does a stored light system preserve the quantum state? Efficiency: how well does a stored light system preserve the signal amplitude?Friday, March 29, 13
  • 4. Quantum State of Slow Light TextFriday, March 29, 13
  • 5. and Stopped LightFriday, March 29, 13
  • 6. Slow & Stopped Light 2.5 (a) Control field (b) Polariton 2 11 Ψ(z,t) θ 1.5 150 0.8 π/2 0.6 100 0.5 Ω (t) 1 t Ω (0) 0.4 0.5 50 0.2 a 00 0 0 0 25 50 75 100 125 150 0 50 100 150 0 50 100 150 0 40 80 120 t z 2.5 2.5 (c) Photon (d) Spin Coherence σcb(z,t) 2 2 E(z,t) 1.5 150 1.5 150 1 100 1 100 t t 0.5 50 0.5 50 0 0 0 50 100 150 c 0 0 0 50 100 150 0 40 80 120 0 40 80 120 z z Figure 3.A dark-state polariton can be stopped and re-accelerated by ramping the contro ty as shown in (a).The broken line shows the mixing angle between photonic and spin stat herent amplitudes of the polariton ␺, the electric field E of the photon, and the spin coheren ted in (b-d). medium. The width of the transparency transition that maps the signal window, and thus vg , is a function of the coherent superposition of t atomic density and the control beam in- states, |g1ʹ and |g 2ʹ . In so doing tensity, and is therefore under experimen- energy of the signal photons i tal control. In particular, vg decreases near- in the creation of new control ly linearly with both quantities. tons. The resulting atomic spinFriday, March 29, 13
  • 7. Storing InformationFriday, March 29, 13
  • 8. Temporal Optimization (Novikova et al.) Novikova et al. “Optimal control of light pulse storage and retrieval,” PRL 98, 243602 (2007).Friday, March 29, 13
  • 9. Next step: Spatial mode optimization? In Out* State Rb vapor Detector *Compare to recent full 3D theory Zeuthen et al. “Three-dimensional theory of quantum memories based on lambda-type atomic ensembles,” PRA 84, 043838 (2011).Friday, March 29, 13
  • 10. Problems! - PD In Out* Rb vapor PD Local Oscillator LO and signal aren’t mode-matched!Friday, March 29, 13
  • 11. Problems! - PD In Out* Rb vapor PD Local Oscillator LO and signal aren’t mode-matched! A new approach needs to keep mode informationFriday, March 29, 13
  • 12. Quantum OpticsFriday, March 29, 13
  • 13. Quantum Optics i(k · x !t) u(x, t) = u0 e Mode function (plane wave) ˆ E = u⇤ (x, t)ˆ† + u(x, t)ˆ a a Electric field operator 1 † xp = p a + a ˆ ˆ ˆ 2 Quadrature operators i yp = p a † ˆ ˆ a ˆ 2Friday, March 29, 13
  • 14. Quantum Optics i(k · x !t) u(x, t) = u0 e Mode function (plane wave) ˆ E = u⇤ (x, t)ˆ† + u(x, t)ˆ a a Electric field operator 1 † xp = p a + a of E -field ents ˆ ˆ ˆ 2 compon Quadrature operators os and sin ~c i yp = p a † ˆ ˆ a ˆ 2Friday, March 29, 13
  • 15. Optical Phase Space Quadratures are the axes in phase space Classical Optics Quantum Optics Uncertainty yp yp xp xpFriday, March 29, 13
  • 16. Wigner and Q functions • Quasi-probability distributions • Representation of the quantum state • 3D look at phase space (Wigner ✽ Gaussian)Friday, March 29, 13
  • 17. Example of Wigner and Q-functions Schrödinger cat state | i / |↵i + | ↵i Wigner Q-functionFriday, March 29, 13
  • 18. Example of Wigner and Q-functions Schrödinger cat state | i / |↵i + | ↵i Wigner Q-function e nt gl em nt an t e esen epr ’t r C anFriday, March 29, 13
  • 19. Measuring the Quantum State of Light Balanced Homodyne Tomography Balanced Array Detection Smithey et al. PRL 70, 1244 (1993) Unbalanced Array Detection Beck PRL 84, 5748 (2000) Beck et al. PRL 87, 253601 (2000)84, 5748 29, 13 Friday, March (2000)
  • 20. Quantum State Tomography Constructing the quantum state of light from measurements of the quadrature componentsFriday, March 29, 13
  • 21. A New ApproachFriday, March 29, 13
  • 22. Unbalanced Array Detection of Spatial Modes Local Oscillator q CCD Array Signal x q ~ 5 mradFriday, March 29, 13
  • 23. Unbalanced Array Detection - Theory Local Oscillator k CCD Array Signal k kS xS(x) = |ELO (x) + ES (x) exp(ikS · x)|2 2 2 ⇤ = |ELO (x)| + |ES (x)| + [ELO (x)ES (x) exp(ikS · x) + c.c.]Friday, March 29, 13
  • 24. Unbalanced Array Detection - Theory Fourier Transform of detected intensity: 0 e e⇤ e e⇤ e S(k) = ELO ( k) ⌦ ELO (k) + ES ( k) ⌦ ES (k) ⇤ + f (k kS ) + f ( k kS ) where 2nd order classical LO noise eLO ( k) ⌦ ES (k) f (k) = E ⇤ eFriday, March 29, 13
  • 25. Unbalanced Array Detection - Theory † Each detector pixel nj = ˆ aj aj ˆ ˆ X measures all modes aj = ˆ exp [ i2⇡jk/N ] ˆk b k 8 (vac) >ˆk <b N/2  k < M, ˆk = ˆ(lo) (Signal + LO + vacuum) b bk M  k  M, > (s) :ˆ bk M < k < N/2. Fourier transform CCD ˆ 1 X Kp = p exp [i2⇡pj/N ] nj ˆ output to measure: N j (p is the index of the measured mode)Friday, March 29, 13
  • 26. Unbalanced Array Detection - Theory Assume LO is strong X ⇣ M ⌘ ˆ Kp = ⇤ˆ(s) ˆ†(vac) (i.e. classical) field: k bk+p + k bk p k= M Assume LO is in a single ˆ ⇤ˆ(s) ˆ†(vac) . plane-wave mode: Kp = 0 bp + 0b p ˆ(s) 1 bp = p (ˆp + iˆp ) x y 2 Each entry in the FFT output is Kp (for mode p) A measurement of the signal quadratures + a vacuum componentFriday, March 29, 13
  • 27. A Toy ModelFriday, March 29, 13
  • 28. Model intensity Interference of two plane waves at θ = 5 mrad Signal = LO / 100 y (pixel #) x (pixel #) (visibility = 0.02, exaggerated by auto-range)Friday, March 29, 13
  • 29. Calculate FFTFriday, March 29, 13
  • 30. Calculate FFTFriday, March 29, 13
  • 31. Calculate FFT Pick one kx & histogram Re and Im valuesFriday, March 29, 13
  • 32. Model histogram (in phase space) A) Signal = LO / 100 B) Signal = LO / 1000 C) Signal = LO / 100 with 1 rad phase shift A) B) 2 X1 400 0 -15 -12 -9 -6 -3 -2 300 -4 X2 200 -6 100 -8 C) -10 0 -12Friday, March 29, 13
  • 33. The ExperimentFriday, March 29, 13
  • 34. Unbalanced Array Detection - Experiment • Two resonant laser fields (Control, Probe) Requirements: • CCD needs to be low noise & high QE • AOM beams need stable phase relationshipFriday, March 29, 13
  • 35. 3 Tunable Laser Systems Commercial optics mount 780 nm laser diode Diffraction grating Piezo stack controls grating angle Δν ~ 6 GHzFriday, March 29, 13
  • 36. High quantum-efficiency CCD installed • 98% Quantum Efficiency • Peak detection λ ~ 780 nm 780 nm • 100% 1340 x 400 pixels 90% 80% 70% • 2 - 3 per pixel per hr e- Quantum Efficiency (%) 60% 50% (LN cooled) 40% 30% 20% 10% 0% 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 With optional UV coating BR_eXcelon B_eXcelon BR B F (for non-eXcelon cameras only) Wavelength (nm)Friday, March 29, 13
  • 37. Acousto-Optic Modulators Reflected Sound Wave ν0 + 80 MHz Light in Light out ν0 ν0 ν0 - 80 MHz Sound Wave (80 MHz)Friday, March 29, 13
  • 38. AOMs installed • 1 x 2 cm • Optical post mount • Deflection ~ 10 mradFriday, March 29, 13
  • 39. AOM DriversFriday, March 29, 13
  • 40. AOM DriversFriday, March 29, 13
  • 41. AOM DriversFriday, March 29, 13
  • 42. AOM Driver (finished)Friday, March 29, 13
  • 43. 80 MHz (1.4 kHz BW)Friday, March 29, 13
  • 44. AOM Beat Signal Stable phase relationship Interference Pattern 50/50 AOM AOMFriday, March 29, 13
  • 45. What’s Next? 1) Proof of principle experiment with plane waves 2) Implement slow light protocol in warm Rubidium vapor 3) Measure state of slow light in Rb vapor Repeat with stopped light in warm Rb, then switch to cold (trapped) Rb vapor.Friday, March 29, 13
  • 46. Trustees for action. The Program Director mayn, an interview with the applicant, or a visit to THANKS: All letters of inquiry and completed formal applications should be mailed in hard copy to: 22 he full proposal, including staff summary and John Van Zytveld, Ph.D.he Trustees for their consideration and decision. Senior Program Director ptly when a decision has been reached. While M. J. Murdock Charitable Trustn nearly every proposal received by the Trust, only P. O. Box 1618 Noah T. Holte For More Helpwed can result in awards. When an application has ried over for future consideration. Under normal Vancouver, WA 98668 Hunter A. It will perty of the Trust and will not be returned. Dassonvillequestions have not beencall us at 360.694.8415. or you need some f a proposal that was declined is not encouraged. If your additional information, please answered by this document Marcus Kienlenmunication with the understanding, however, that NSF Simone Carpenter M. J. MurdockRCSA Mailing Address: Charitable Trust Jennifer Novak PRISM (Pacific U.) PO Box 1618 Vancouver, Washington 98668 Bryson Vivas Murdock Foundation Office Location: M. J. Murdock Executive Plaza 703 Broadway, Suite 710 Vancouver, Washington 98660 Contact: Phone WA: 360.694.8415 Phone OR: 503.285.4086 Fax: 360.694.1819 Website: www.murdock-trust.org Friday, March 29, 13