1. LPHYS’07 – Leon – August 22 nd 2007 Alessandro Zavatta, Valentina Parigi, Myungshik Kim, and Marco Bellini Istituto Nazionale di Ottica Applicata (INOA) - CNR, Florence, Italy LENS and Physics Dept., University of Florence, Italy School of Mathematics and Physics, The Queen’s University, Belfast, UK Alessandro Zavatta, Valentina Parigi, Myungshik Kim, and Marco Bellini Istituto Nazionale di Ottica Applicata (INOA) - CNR, Florence, Italy LENS and Physics Dept., University of Florence, Italy School of Mathematics and Physics, The Queen’s University, Belfast, UK Probing quantum commutation rules by adding and subtracting single photons to/from a light field Bled 26-28 March 2008 Dipartimento di Fisica INOA-CNR

2. LPHYS’07 – Leon – August 22 nd 2007 Outline Experimental implementation of photon creation and annihilation operators Application to a completely classical light field Nonclassical and counterintuitive features Direct probing of fundamental quantum rules Bled 26-28 March 2008

3. LPHYS’07 – Leon – August 22 nd 2007 Implementing single-photon addition and subtraction Use photon creation and annihilation operators “Photon-added” state “Photon-subtracted” state Conditional generation schemes Bled 26-28 March2008 Bled 26-28 March 2008

4. LPHYS’07 – Leon – August 22 nd 2007 Use parametric down-conversion in a nonlinear crystal Stay in the low-gain regime (eliminates higher-order excitations) … whenever an idler photon is detected, the signal is prepared in a single-photon-excited version of the initial state Stimulated emission regime Emission probability increases with Adding a single photon to a light field Inject a seed pure state in the signal mode Bled 26-28 March2008 Bled 26-28 March 2008 A. Zavatta, S. Viciani, M. Bellini, Science 306, 660 (2004); Phys. Rev. A, 72, 023820 (2005)

5. LPHYS’07 – Leon – August 22 nd 2007 (Pseudo-)thermal light source Rotating ground glass disk Diffuses the input coherent beam and adds random amplitude and phase fluctuations Bled 26-28 March2008 Bled 26-28 March 2008 Single-mode fiber

6. LPHYS’07 – Leon – August 22 nd 2007 For a generic mixed input state and upon conditioning on single idler photon detection Mixed state input The addition of a single photon turns any state into a nonclassical one due to the lack of the vacuum terms C. T. Lee, Phys. Rev. A 52, 3374 (1995) Thermal input state Bled 26-28 March2008 Bled 26-28 March 2008

7. LPHYS’07 – Leon – August 22 nd 2007 Glauber-Sudarshan P distribution Wigner distribution Both distributions are always negative at the origin Nonclassical states independent of Nonclassical states independent of Nonclassicality of photon-added thermal states Bled 26-28 March2008 Bled 26-28 March 2008

8. LPHYS’07 – Leon – August 22 nd 2007 Experimental single-photon addition Bled 26-28 March2008 Bled 26-28 March 2008

9. LPHYS’07 – Leon – August 22 nd 2007 P θ (x θ ) xθxθ Complete set of quadrature distributions Gated time-domain acquisition Ultra-high bandwidth (~100 MHz) Low electronic noise (S/N >12 dB) High subtraction efficiency (~60 dB @ 82 MHz) Ultrafast homodyne detection Density matrix elements Density matrix elements Wigner function Maximum likelihood estimation A. I. Lvovsky, J. Opt. B: Quantum Semiclassical Opt. 6, 556 (2004) Z. Hradil, D. Mogilevtsev, J. Rehacek, Phys. Rev. Lett. 96, 230401 (2006) Bled 26-28 March2008 Bled 26-28 March 2008 A. Zavatta, M. Bellini, P.L. Ramazza, F. Marin, F.T. Arecchi, J. Opt. Soc. Am. B 19, 1189 (2001).

10. LPHYS’07 – Leon – August 22 nd 2007 n=1.15 _ _ n=0.1 _ _ Thermal states Single-photon-added thermal states Experimental tomographic reconstruction Adding a photon makes the states highly nonclassical. Their Wigner function becomes negative at the origin No correction for the finite homodyne detection efficiency Bled 26-28 March2008 Bled 26-28 March 2008 A. Zavatta, V. Parigi, M. Bellini, Phys. Rev. A 75, 052106 (2007)

11. LPHYS’07 – Leon – August 22 nd 2007 The experimental inefficiency in the preparation and detection can be simply introduced as a loss by a BS of transmittivity . Both the finite efficiency and the statistical noise contribute to reduce the visibility of nonclassical features. Wigner negativity and efficiencies Fock states Bled 26-28 March2008 Bled 26-28 March 2008 A. Zavatta, V. Parigi, M. Bellini, Phys. Rev. A 75, 052106 (2007)

12. LPHYS’07 – Leon – August 22 nd 2007 How to subtract a single photon ? Bled 26-28 March2008 Bled 26-28 March 2008 On/off detection Input state

13. LPHYS’07 – Leon – August 22 nd 2007 Experimental single-photon subtraction Bled 26-28 March2008 Bled 26-28 March 2008 J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004)

14. LPHYS’07 – Leon – August 22 nd 2007 Subtracting a single photon from a thermal light state preserves its classicality but changes the Wigner function to a non-Gaussian one. n=0.5 _ _ n=1.0 _ _ Thermal states Photon-subtracted thermal states Tomographic reconstructions Bled 26-28 March2008 Bled 26-28 March 2008

15. LPHYS’07 – Leon – August 22 nd 2007 n=0.5 _ _ n=1.0 _ _ Thermal states Photon-subtracted thermal states n  sub =0.7 _ _ n  sub =1.3 _ _ n  =0.33 _ _ n  =0.65 _ _ Mean photon numbers The mean number of photons in the state increases after photon subtraction !!! Bled 26-28 March2008 Bled 26-28 March 2008

16. LPHYS’07 – Leon – August 22 nd 2007 for a coherent state for a Fock state for a thermal state Subtracting a single photon from a state may increase the resulting mean photon number if the initial state was super-Poissonian. where Fano parameter !!! The strange case of photon subtraction Photon creation and annihilation operators do not perform deterministic photon addition and subtraction ! M. Ueda, N. Imoto, T. Ogawa, Phys. Rev. A 41, 3891 (1990) Bled 26-28 March2008 Bled 26-28 March 2008

17. LPHYS’07 – Leon – August 22 nd 2007 Combining quantum operators Bled 26-28 March2008 Bled 26-28 March 2008

18. LPHYS’07 – Leon – August 22 nd 2007 Experimental sequences of single-photon addition and subtraction Bled 26-28 March2008 Bled 26-28 March 2008

19. LPHYS’07 – Leon – August 22 nd 2007 Directly probing quantum commutation rules Direct test of quantum commutation rules Contrary to naïve classical expectations, “adding” and then “subtracting” a single photon modifies the state Bled 26-28 March2008 Bled 26-28 March 2008

20. LPHYS’07 – Leon – August 22 nd 2007 Homodyne data and reconstructions Reconstructions corrected for the finite detection efficiency (  =64%) Limited preparation efficiency (  =92%) in the photon-addition stage adds small contaminations Clearly different final states Bled 26-28 March2008 Bled 26-28 March 2008

21. LPHYS’07 – Leon – August 22 nd 2007 Experimental Wigner functions Nonclassical: negative WF Nonclassical but positive WF V. Parigi, A. Zavatta, M. S. Kim, M. Bellini, Science 317, 1890 (2007) Bled 26-28 March2008 Bled 26-28 March 2008

22. LPHYS’07 – Leon – August 22 nd 2007 Conclusions First implementation of sequences of basic quantum operators Direct test of fundamental quantum algebraic rules: noncommutativity of creation and annihilation operators Nonclassical Multi-photon state generation through enhancement cavity Bled 26-28 March 2008 Generation of SPATS and P-function reconstruction

23. LPHYS’07 – Leon – August 22 nd 2007 Florence Quantum optics Group Bled 26-28 March 2008 azavatta@inoa.it www.inoa.it/home/QOG azavatta@inoa.it www.inoa.it/home/QOG