Quantum Memories for Continuous Variable States of Light in Atomic Ensembles Gabriel H´etet A thesis submitted for the degree of Doctor of Philosophy at The Australian National University October, 2008 ii Declaration This thesis is an account of research undertaken between March 2005 and June 2008 at the Department of Physics, Faculty of Science, Australian National University (ANU), Can- berra, Australia. This research was supervised by Prof. Ping Koy Lam (ANU), Prof. Hans- Albert Bachor (ANU), Dr. Ben Buchler (ANU) and Dr. Joseph Hope (ANU). Unless otherwise indicated, the work present herein is my own. None of the work presented here has ever been submitted for any degree at this or any other institution of learning. Gabriel H´etet October 2008 iii iv Acknowledgments I would naturally like to start by thanking my principal supervisors. Ping Koy Lam is the person I owe the most. I really appreciated his approachability, tolerance, intuition, knowledge, enthusiasm and many other qualities that made these three years at the ANU a great time. Hans Bachor, for helping making this PhD in Canberra possible and giving me the opportunity to work in this dynamic research environment. I always enjoyed his friendliness and general views and interpretations of physics. Thanks also to both of you for the opportunity you gave me to travel and present my work to international conferences. This thesis has been a rich experience thanks to my other two supervisors : Joseph Hope and Ben Buchler. I would like to thank Joe for invaluable help on the theory and computer side and Ben for the efficient moments he spent improving the set-up and other innumerable tips that polished up the work presented here, including proof reading this thesis. When I arrived here, the ALE project had just started. I had the pleasure to share a few months in the lab with Katie Pylipas and Charles Harb. I then had the luck to learn how to handle lasers with Oliver Gloeckl. I am really grateful to Oliver for his deep involvement in the project and his rigor all the way through. I gained a great deal from his expertise. Thanks also goes to Magnus Hsu for the knowledge and experimental skills he transmitted to me on the atom side of the experiment. I am also indebted to Amy Peng, Mattias Johnson, Alexander Akulshin, Kirk McKenzie, and also the LKB team in Paris for invaluable help and discussions all along this project. These were really fruitful and enjoyable years, thanks again to all of you. I would like to wish luck to Mahdi and Ben for the rest of their PhD/Honours and thank them for their help during my last months here. I had the opportunity to work with two great researchers, Matt Sellars and Jevon Longdell, from the Research School of Physical Sciences, during my PhD. I want to thank them deeply for allowing me and the group to partake in this awesome work they are doing. This was a truly interesting and really pleasant collaboration. Good continuation with your work. I would certainly not forget to thank Paul Tant, Paul MacNamarra, Neil Devlin, Shane Grieves, and Neil Hinchey for their help with the mechanics and electronics involved in the experimental work. Merci aussi `aRobin Kaiser et les gens du INLN pour leur accueil super sympa `aNice. J’y ai pass´eun mois tr`es agr´eableet tr`esenrichissant. A big collective thanks goes to all the people that have contributed to the work pre- sented here, from the office at the faculties, to my family and my friends in France and here in Australia. Last but certainly not least, to Violaine, for giving me support and happiness during these years. v vi Abstract The field of quantum information science holds great promise for enhancing computing speed and secure communication. In this thesis, we present theoretical studies and ex- periments that aim to develop a key component of quantum communication systems : quantum memories. Specifically, we examine the potential of continuous variable quan- tum memories for light using large ensemble of atoms. It is important to compare the performance of light storage experiments against rele- vant benchmarks. We propose to benchmark efficacies using the classical teleporter. We in- vestigate theoretically the propagation of quantum states through lossy and noisy quantum channels and use the T-V diagram to quantify quantum memory performances. As most experimental demonstrations are performed with rubidium atoms, we built a squeezed light source operating at rubidium wavelengths. We achieved 5.2 dB of squeezing using PPKTP in an optical parametric oscillator and pushed the bandwidth suppression down to 200 Hz, well within the transmission of Electromagnetically Induced Transparency (EIT) based quantum memories. A study of EIT is presented theoretically and experimentally. We discovered theoretically that population exchange between the ground states is respon- sible for excess noise on the output field due to phase-insensitive amplification of the probe in the medium. Phase space numerical models were developed to calculate the efficiency of information storage and retrieval. We use this code and our benchmarks to calculate the parameter regimes that would enable EIT to act as a quantum memory. Experiments performed in a warm rubidium vapour cell showed transmission of 2 dB of squeezed light through an EIT feature. Using the same squeezed source, biased entanglement was used to show quantum state delay of 2 µs while also preserving state inseparability. We show that the presence of buffer gas eliminates the excess noise observed in samples containing natural mixtures of rubidium. Using a similar set-up, we demonstrate the writing and read-out of a classical pulse to and from the ground states of rubidium atoms using EIT and examine the limitations of the technique. Alternative quantum memories for light using photon echo techniques are also pro- posed. After analyzing the standard photon echoes techniques and controlled reversible inhomogeneous broadening (CRIB) for quantum state storage, we discovered an efficient quantum memory protocol. The Gradient Echo Memory (GEM) uses two-level atoms and a reversible detuning that varies linearly with propagation distance across the atomic sample. Using numerical simulations, we show that GEM is perfectly efficient in the limit of a large number of atoms. We compare our results with experimental demonstrations performed in solid state systems. The memory dynamics are interpreted as the formation of polaritonic modes that can be controlled by the detuning slope. We found that, as GEM stores information in the Fourier domain, it can store signals that have a large time- bandwidth product, an important feature for quantum memories. We propose a direct extension of the GEM protocol using three-level atoms in an off-resonant Raman config- uration and present proof of principle demonstrations of this Λ-GEM memory scheme in a rubidium vapour by controlling a linear Zeeman shift. vii viii Contents Declaration iii Acknowledgments v Abstract vii Introduction 1 I Tools for quantum memory characterisation 5 1 Elements of quantum-atom-optics theory 7 1.1 Quantum states of light for storage . 7 1.1.1 Heisenberg uncertainty principle . 7 1.1.2 Experimental observables . 8 1.1.3 The density matrix . 9 1.1.4 Wigner function . 11 1.1.5 Quantum states of light . 11 1.1.6 Other quasi-probability distributions . 14 1.2 Interaction between a single mode light field and a single atom . 15 1.2.1 Description of the model . 15 1.2.2 Reversible evolution of the density matrix . 16 1.2.3 Damping of the coherences and populations . 17 1.2.4 Heisenberg-Langevin equations . 19 1.3 Interaction of quantum states of light with atomic ensembles . 20 1.3.1 The Maxwell equation . 20 1.3.2 Local averaging . 21 1.4 Detection and production of Gaussian states . 22 1.4.1 Phase-insensitive photodetection . 22 1.4.2 Balanced detection . 25 1.4.3 Fabry-Perot cavity . 26 1.4.4 The parametric amplifier . 29 1.4.5 Classical and quantum correlations . 33 2 Benchmarks for continuous variable quantum memories 35 2.1 Why use quantum memories ? . 35 2.1.1 Quantum key distribution . 35 2.1.2 Quantum computing . 36 2.1.3 Current quantum memories . 37 2.2 Channel transmission and noise . 38 2.2.1 Modelling transmission and noise . 39 2.2.2 Displaced coherent states . 40 ix x Contents 2.2.3 Criteria for continuous variable entanglement . 41 2.3 Quantum information benchmarks . 45 2.3.1 Fidelity measure . 46 2.3.2 T-V diagram . 47 2.3.3 Noise-loss diagram . 50 2.3.4 Single and multi-mode light storage . 51 2.3.5 Time-bandwidth product . 51 2.4 Conclusion . 53 3 Squeezed light for quantum memories in rubidium 55 3.1 Introduction . 56 3.2 Non-linear conversion . 56 3.2.1 Birefringence . 58 3.2.2 Quasi-phase matching . 58 3.2.3 Periodically poled KTP . 59 3.3 Second harmonic generation and down-conversion . 59 3.3.1 The laser . 59 3.3.2 Generation of ultra-violet light. 60 3.3.3 Mode cleaning cavity . 62 3.3.4 Optical parametric oscillation. 63 3.4 Squeezing at rubidium wavelengths. 64 3.4.1 Quantum noise detection . 64 3.4.2 Laser noise . 64 3.4.3 Vacuum squeezing . 65 3.4.4 Locked squeezing . 65 3.5 Low frequency squeezing . 68 3.5.1 FM to AM conversion . 68 3.5.2 Noise locking . 68 3.5.3 Orthogonal mode for the cavity locking . 68 3.6 Digital analysis of squeezed states . 70 3.6.1 Time domain analysis . 70 3.6.2 Sub-kHz squeezing .