Strongly interacting systems in AMO physics

215 Pages·2011·3.72 MB·English

Strongly interacting systems in AMO physics
A dissertation presented
by
Mohammad Hafezi
to
The Department of Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Physics
Harvard University
Cambridge, Massachusetts
August 2009
c 2009 - Mohammad Hafezi
(cid:13)
All rights reserved.
Thesis advisor Author
Mikhail D. Lukin Mohammad Hafezi
Strongly interacting systems in AMO physics
Abstract
Strong interactions can dramatically change the essence of a physical system. The
behavior of strongly interacting systems can be fundamentally different than those
where the interaction is absent or treated perturbatively. Many examples are known
in solid-state physics including the superconductivity and the Fractional Quantum
Hall Effect. At the same time, tremendous developments have been made in manipu-
lating the interaction between light and matter. These advances have paved the way
to explore the strongly interacting many-body physics in new regimes. This thesis
explores two novel avenues to study strongly interacting systems. First, we investi-
gatethe effect of strong interaction between bosons subjected to an effective magnetic
field. We show that how a Fractional Quantum Hall state of bosons in an optical lat-
tice can be created, characterized and detected in a realistic experiment. Moreover,
we demonstrate that Chern numbers can unambiguously characterize the topological
order of such systems. Second, we investigate the effect of strong interaction between
photons on their transport properties. We theoretically study the transmission of
few-photon quantum fields through a strongly nonlinear optical medium. We develop
a general approach to investigate non-equilibrium quantum transport of bosonic fields
through a finite-size nonlinear medium and apply it to a recently demonstrated ex-
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Abstract iv
perimental system where cold atoms are loaded in a hollow-core optical fiber. We
show that the photonic field can exhibit either anti-bunching or bunching, associated
with the resonant excitation of bound states of photons by the input field. These
effects can be observed by probing statistics of photons transmitted through the non-
linear fiber. As an application, we propose a scheme to realize a single-photon gate,
where the presence or absence of a single “control” photon regulates the propagation
of a “target” photon. Finally, we study optical nonlinearities due to the interaction
of weak optical fields with the collective motion of a strongly dispersive ultracold
gas. We present a theoretical model that is in good agreement with our experimental
observations.
Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Citations to Previously Published Work . . . . . . . . . . . . . . . . . . . vii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bosonic Fractional Quantum Hall states . . . . . . . . . . . . . . . . 3
1.2.1 Introduction to Fractional Quantum Hall Physics . . . . . . . 3
1.2.2 Fractional Quantum Hall and Bose-Einstein Condensates . . . 5
1.3 Strongly interacting photons . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Fractional quantum Hall state in optical lattices 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Quantum Hall state of bosons on a lattice . . . . . . . . . . . . . . . 18
2.2.1 Single particle on a magnetic lattice . . . . . . . . . . . . . . . 18
2.2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Energy spectrum and overlap calculations . . . . . . . . . . . 22
2.2.4 Results with the finite onsite interaction . . . . . . . . . . . . 25
2.3 Chern number and topological invariance . . . . . . . . . . . . . . . . 31
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Chern number and FQHE . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Resolving the degeneracy by adding impurities . . . . . . . . . 39
2.3.4 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Extension of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.1 Effect of the long-range interaction . . . . . . . . . . . . . . . 51
2.4.2 Case of ν = 1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . 54
v
Contents vi
2.5 Detection of the Quantum Hall state . . . . . . . . . . . . . . . . . . 57
2.6 Generating Magnetic Hamiltonian for neutral atoms on a lattice . . . 62
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Photonic quantum transport in a nonlinear optical fiber 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Model: Photonic NLSE in 1D waveguide . . . . . . . . . . . . . . . . 70
3.3 Linear case: Stationary light enhancement . . . . . . . . . . . . . . . 77
3.4 Semi-classical nonlinear case . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.1 Dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.2 Dissipative Regime . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5 Quantum nonlinear formalism: Few-photon limit . . . . . . . . . . . 89
3.6 Analytical solution for NLSE with open boundaries . . . . . . . . . . 95
3.6.1 One-particle problem . . . . . . . . . . . . . . . . . . . . . . . 95
3.6.2 Two-particle problem . . . . . . . . . . . . . . . . . . . . . . 97
3.6.3 Solutions close to non-interacting case . . . . . . . . . . . . . 101
3.6.4 Bound States Solution . . . . . . . . . . . . . . . . . . . . . . 103
3.6.5 Many-body problem . . . . . . . . . . . . . . . . . . . . . . . 106
3.7 Quantum transport properties . . . . . . . . . . . . . . . . . . . . . 110
3.7.1 Repulsive Interaction (κ > 0) . . . . . . . . . . . . . . . . . . 110
3.7.2 Attractive Interaction (κ < 0) . . . . . . . . . . . . . . . . . . 115
3.7.3 Dissipative Regime (κ = i κ ) . . . . . . . . . . . . . . . . . . 117
| |
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4 Single photon switch in a nonlinear optical fiber 122
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2 Description of the scheme . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5 Optical bistability at low light level due to collective atomic recoil 133
5.1 Intoduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Experimental observation . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 Comparison between the model and the experiment . . . . . . . . . . 140
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Bibliography 145
A EIT and band gap 160
B Numerical methods 166
Contents vii
C Effect of noise 170
D Photon-Photon interaction in Double-V system 174
E Strategy for single photon gate 180
F Vacuum Rabi splitting in electromagnetically induced photonic crystal186
G Theoretical model for collective atomic recoil 193
G.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
G.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
G.3 Limit of large decoherence: Population diffusion . . . . . . . . . . . . 200
G.4 Subrecoil Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
List of Figures
1.1 Optical nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Atom-photon interaction . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Hollow-core photonic band gap fiber . . . . . . . . . . . . . . . . . . 11
2.1 Hofstadter’s Butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Laughlin wave function overlap ν = 1/2 . . . . . . . . . . . . . . 26
2.3 Energy spectrum and gap . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The Laughlin wave function overlap for different α and U . . . . . . . 30
2.5 Twist angles and the toroidal boundary condition . . . . . . . . . . . 36
2.6 Chern number in the presence of impurity . . . . . . . . . . . . . . . 43
2.7 Chern number (degenerate case) for three atoms . . . . . . . . . . . . 46
2.8 Chern number (degenerate case) for four atoms . . . . . . . . . . . . 47
2.9 Energy level crossing for high magnetic field . . . . . . . . . . . . . . 49
2.10 Effect of dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . 53
2.11 The Laughlin wave function overlap: ν = 1/4 . . . . . . . . . . . . . 57
2.12 Structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.13 Rotating an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Four-level atomic system for creating strong nonlinearity . . . . . . . 74
3.2 Linear transmission spectrum . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Optimization of the transmission for different optical densities . . . . 80
3.4 Shifted resonances due to nonlinearity . . . . . . . . . . . . . . . . . 85
3.5 Positive and negative nonlinearity in the semi-classical approximation 86
3.6 Transmission versus photon number: dispersive case . . . . . . . . . . 87
3.7 Transmission versus photon number: absorptive case . . . . . . . . . 88
3.8 Two-photon wave function with different modes . . . . . . . . . . . . 103
3.9 Energy of two-photon states occupying different modes . . . . . . . . 104
3.10 Energy of bound states versus strength of nonlinearity (1) . . . . . . 105
3.11 Two-photon wave function of a bound state (1) . . . . . . . . . . . . 106
3.12 Energy of bound states versus strength of nonlinearity(2) . . . . . . . 107
3.13 Two-photon wave function of a bound state (2) . . . . . . . . . . . . 108
viii
List of Figures ix
3.14 Reaching the steady-state: dispersive case . . . . . . . . . . . . . . . 111
3.15 Two-photon wave function: Repulsive case . . . . . . . . . . . . . . . 112
3.16 Scaling of g with nonlinearity strength: repulsive case . . . . . . . . 113
2
3.17 Scaling of the anti-bunching with optical density and cooperativity . 114
3.18 Resonances in presence of attractive interaction . . . . . . . . . . . . 117
3.19 Reaching the steady-state: absorptive case . . . . . . . . . . . . . . . 118
3.20 g versus optical density and cooperativity: absorptive case . . . . . . 119
2
3.21 Two-photon wave function suppression due to nonlinear absorption . 120
4.1 Single-photon gate scheme . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Fidelity of single-photon gate . . . . . . . . . . . . . . . . . . . . . . 127
4.3 Reflection dependence on the position of the control photon . . . . . 129
4.4 Reflection spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.1 Schematic indicating the pump and probe beams used for the RIR . . 135
5.2 Optical absorption bistability due to the collective motion of atoms . 137
5.3 Comparison between the experiment and the theoretical model . . . . 141
5.4 Scaling of the decoherence with the detuning . . . . . . . . . . . . . 142
A.1 Schematic of a three-level system and standing control field . . . . . 161
A.2 Band gap formation for different optical densities . . . . . . . . . . . 164
A.3 Band gap edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.4 Band gap structure for strong control fields . . . . . . . . . . . . . . . 165
C.1 Level diagram corresponding to the noise effect . . . . . . . . . . . . 173
D.1 Double-V system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
D.2 Energy spectrum of a double-V system . . . . . . . . . . . . . . . . . 177
E.1 Single-photon gate scheme: realistic atomic diagram . . . . . . . . . . 181
E.2 Fidelity of the single-photon gate . . . . . . . . . . . . . . . . . . . . 183
F.1 Vacuum Rabi splitting and the transfer matrix . . . . . . . . . . . . . 189
F.2 Shifting the cavity resonance and the transfer matrix . . . . . . . . . 191
F.3 Rabi splitting for low optical density . . . . . . . . . . . . . . . . . . 192
Citations to Previously Published Work
Parts of Chapter 2 have appeared in the following papers:
“Fractional quantum Hall effect in optical lattices”, Mohammad Hafezi,
Anders S. Sørensen, Eugene A. Demler and Mikhail D. Lukin, Phys. Rev.
A 76, 023613 (2007);
“Characterization of topological states on a lattice with Chern number”,
Mohammad Hafezi, Anders S. Sørensen, Mikhail D. Lukin and Eugene A.
Demler, Euro. Phys. Lett. 81, 1 (2008).
Parts of Chapters 3 and 4 have appeared in the following papers:
“Photonic quantum transport in a nonlinear optical fiber”, Mohammad
Hafezi,DarrickE.Chang,VladimirGritsev, EugeneA.DemlerandMikhail
D. Lukin submitted to Phys. Rev. Lett., (e-print: arXiv:0907.5206).
“Efficient All-Optical Switching Using Slow Light within a Hollow Fiber”,
Michal Bajcsy, Sebastian Hofferberth, Vladko Balic, Thibault Peyronel,
Mohammad Hafezi, Alexander S Zibrov, Vladan Vuletic, Mikhail D Lukin
Phys. Rev. Lett. 102, 203902 (2009).
Parts of Chapter 5 have appeared in the following paper:
“Optical bistability at low light level due to collective atomic recoil”,
Mukund Vengalattore, Mohammad Hafezi, Mikhail D. Lukin, Mara Pren-
tiss, Phys. Rev. Lett. 101, 063901 (2008).
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