Dynamics of Rydberg atom lattices in the presence of noise and dissipation D zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) vorgelegt der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden von Wildan Abdussalam M.Sc. Eingereicht am 19.1.2017 Verteidigt am 07.08.2017 Eingereicht am 19.1.2017 Verteidigt am 07.08.2017 1. Gutachter: Prof. Dr. Jan Michael Rost 2. Gutachter: Prof. Dr. Walter Strunz To my family and teachers Abstract The work presented in this dissertation concerns dynamics of Rydberg atom lattices in the presence of noise and dissipation. Rydberg atoms possess a number of exaggerated properties, such as a strong van der Waals interaction. The interplay of that interaction, coherent driving and decoherence leads to intriguing non-equilibrium phenomena. Here, we study the non-equilibrium physics of driven atom lattices in the pres- ence of decoherence caused by either laser phase noise or strong decay. In the Vrst case, we compare between global and local noise and explore their eUect on the number of ex- citations and the full counting statistics. We Vnd that both types of noise give rise to a characteristic distribution of the Rydberg excitation number. The main method employed is the Langevin equation but for the sake of eXciency in certain regimes, we use a Marko- vian master equation and Monte Carlo rate equations, respectively. In the second case, we consider dissipative systems with more general power- law interactions. We determine the phase diagram in the steady state and analyse its generation dynamics using Monte Carlo rate equations. In contrast to nearest-neighbour models, there is no transition to long-range-ordered phases for realistic interactions and resonant driving. Yet, for Vnite laser detunings, we show that Rydberg atom lattices can undergo a dissipative phase transition to a long-range-ordered antiferromagnetic phase. We identify the advantages of Monte Carlo rate equations over mean Veld predictions. Having studied the dynamics of Rydberg atom lattices, we study an application of the strong interactions in such systems for quantum information processing. We inves- tigate the coherent exchange of a single photon between a superconducting microwave cavity and a lattice of strongly interacting Rydberg atoms in the presence of local electric Veld Wuctuations plaguing the cavity surface. We show that despite the increased sensi- tivity of Rydberg states to electric Velds, as compared to ground state atoms, the Rydberg dipole-dipole interaction can be used to protect the system against the dephasing induced by the local noise. Using 1{f and laser phase noise models, we show that compared to the case with non-interacting atoms, our system exhibits longer coherence lifetimes and larger retrieval eXciency of the photon after storing into the atoms. v Contents 1 Introduction 1 1.1 Outline . 2 2 Rydberg atoms 5 2.1 Basic properties . 6 2.1.1 The hydrogen atom . 6 2.1.2 Alkali atoms . 8 2.1.3 Dipole moments . 9 2.1.4 Lifetimes . 11 2.2 Rydberg-Rydberg interactions . 12 3 Rydberg ensembles in the presence of noise 17 3.1 Basic Setup . 18 3.2 1D lattice of two-level atoms . 18 3.2.1 Light-atom coupling . 19 3.2.2 Langevin equation . 21 3.2.3 Master equation . 22 3.2.4 Classical rate equation . 24 3.3 Two interacting atoms . 25 3.4 Few-body simulations . 26 3.5 Atom counting statistics . 28 3.6 Implications for experiments . 31 4 Rydberg ensembles in the presence of dissipation 33 4.1 Basic Setup . 35 4.2 Rydberg lattices with two-level atoms . 35 4.3 Rydberg lattices with three-level atoms . 38 4.4 Emergence of long-range order . 43 4.5 Implications for experiments . 45 5 Interfacing Rydberg atoms and solid-state qubits 47 5.1 The Jaynes-Cummings Model . 49 vii 5.2 1/f noise . 51 5.3 Noise eUects on single-atom dynamics . 52 6 Noise-resistant quantum interface 57 6.1 Noise eUects on non-interacting ensembles . 58 6.1.1 The Tavis-Cummings Model . 59 6.1.2 The Langevin equation . 61 6.2 EUects of dipole-dipole interactions . 62 6.3 EUects of dissipative processes . 67 6.4 Transfer Vdelitites . 69 6.5 Robustness against 1{f noise . 72 Conclusions and Outlook 77 6.6 Future perspective . 78 Appendices 81 Appendix 81 Appendix A The multipole expansion of dipole-dipole interactions 83 A.1 Multipole expansion of dipole-dipole interactions . 83 Appendix B Simulation Methods 85 B.1 Quantum simulations of smaller lattice . 85 B.2 Dynamic and Steady-state Monte Carlo . 88 B.3 Mean Veld .................................... 89 Appendix C The population inversion due to the presence of dark state 93 Appendix D State truncation 95 References 97 Nomenclature 111 List of Figures 2.1 The radial charge and a comparison of eUective potentials for hydrogen and alkaline atoms. 8 2.2 Radial wave function of Rubidium . 10 2.3 Scheme of two interacting atoms . 13 2.4 Scheme of two three-states atoms . 14 2.5 Comparison of eigenenergy E` (a, b) and state E´ (c, d) as the function the interaction strength V0{∆ (a, c) and R{Rvdw. 15 3.1 Schematics of a one-dimensional lattice in the presence of noise . 18 3.2 Time variations of the random force F(t) and phase φ ptq of the electric Veld of the laser wave. 22 3.3 The averaged fraction of Rydberg atoms as a function of time in the pres- ence of noise for the case of Markovian and strong decoherence limit . 24 3.4 The steady state of excitation distribution in the presence of global and local noise for a two-atom scenario . 26 3.5 Relaxation of the fraction of Rydberg atoms for a few atoms system . 27 3.6 Variance as a function of time for a few atom system . 28 3.7 Dynamics of the excitation distribution in the presence of global and local noise for a few atom system . 30 3.8 Steady-state variance in the presence of global and uncorrealated noise . 31 4.1 Schematics of a two-dimensional lattice with a three-level ladder-type sys- tem in the presence of dissipative processes . 34 4.2 Fraction of Rydberg atoms in a two-dimensional lattice as a function of p0 for a Monte Carlo wave function and a classical rate equation . 37 4.3 q parameter as a function of excitation probability p0 for the case 1D and 2D 38 4.4 Ground and excited population of single-atom case for three level scheme 39 4.5 Comparison between a rate equation and optical bloch equation . 41 4.6 (a) q parameter as a function of p0 for the case of 1D system and (b) 2D system. The q parameter as a function of ∆ for the case of 2D system is shown in panel (c) . 43 4.7 Order parameter q as a function the power-law exponent . 44 ix 4.8 Order parameter q as a function laser detunings ∆ . 44 4.9 Comparison of order parameters q via a Monte carlo rate equation and mean Veld method . 45 5.1 Schematics of a Rydberg atom coupled to SC cavity . 48 5.2 Schematics of eigen states and energies for atom-cavity interactions. 50 5.3 Rabi oscilllation of single photon between cavity and single atom. 50 5.4 1/f power spectral density . 53 5.5 Atoms-cavity coupling as the function of atoms-cavity surface distance . 53 5.6 Rabi oscilllation of single photon between cavity and single atom in the presence of 1{f noise. 54 6.1 Schematics of Rydberg atoms coupled to superconducting cavity . 58 6.2 Rabi oscilllation of single photon between cavity and N=16 non-interacting atom. 60 6.3 Rabi oscilllation of single photon between cavity and N=100 non-interacting atom. 62 6.4 Coupling strengths between SC cavity and the symmetric state of 10 ˆ 10 interacting atoms subjected to Wuctuating noise from the cavity surface . 65 6.5 Cavity population for the case of 1D and 2D lattices . 66 6.6 Collective coupling of atoms as the function of eigen energies for for (a) 1D lattice and (b) 2D lattice. The energy of symmetric state and ∆E is shown in (c) and (d), respectively. The rest parameters are N “ 100 and V0 “ 25g...................................... 66 6.7 Cavity population in the presence of decay for γs “ 0.1g and N “ 4 . 67 6.8 The dynamics of cavity population and the shift of eigen energy of sym- metric state . 68 6.9 Storing and retrieval of microwave photon between SC cavity and atoms . 70 6.10 Panel (a) and (c) shows the maximum retrieval η0 and the storage lifetime τstor as the function of interaction strength in the absence (triangles) γs “ 0 and the presence of spontaneous decay (crosses) γs “ 0.01g, γp “ 0.01g for a 2D case . (b) and (d) shows of η0 and τstor as the function of atom number in the absence (stars) and the presence of interactions (1D-squares, 2D-triangles) for V0 = 25g. The rest of parameters are Γ = 5g and γ = 14g . 71 6.11 Cavity population as a function of time for realistic parameters . 73 6.12 Storage lifetime and eXciency oUset as the function of lattice spacing and atom number . 75 B.1 Comparison of simulation results between monte carlo wave function and density matrix master equation . 88 D.1 The truncation of state during atom loss processes . 95 List of Tables 2.1 The properties of Rydberg states .