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Quantum Optics Final Project
Many-body Rabi oscillations in Rydberg atomic ensembles
Huy Nguyen
Quantum Optics Final Project
April 17th, 2018
Quantum Optics Final Project
Outline
▪ Applications of Rydberg atoms in quantum information
▪ Many-body Rabi oscillations▪ Excitation dynamics in small lattices
▪ Decoherence mechanisms▪ Multiply excited Rydberg states▪ Intermediate P state excitations
▪ Generation of entanglement
Quantum Optics Final Project
Rydberg Atoms
Tunable Interactions [1]
▪ Interaction strength over 12 orders of magnitude
[1] M. Saffman, T. G. Walker, and K. Molmer, RMP 82, 2313 (2010)
[2] S.-Y. Lan et al., Opt. Exp. 17, 13639 (2009)
Multiplexed Quantum Memory [2]
▪ Many applications in quantum information
Quantum Optics Final Project
Single atom qubits [1]
▪ Pro: Easier implementation
▪ Con: Slow manipulations of quantum state
Rydberg Mediated Quantum Gates
[1] M. Saffman, T. G. Walker, and K. Molmer, RMP 82, 2313 (2010)
Ensemble qubits
▪ Pro: Strong atom-field coupling
▪ Con: Dependent on Rydberg blockade mechanism
Quantum Optics Final Project
Excitation dynamics in small lattices
Excitations driven by coherent laser:
Interactions between excited states:
[3] G. Wu et al. / Physics Letters A 379 (2015) 143-148
Quantum Optics Final Project
Toy Model – 3 Site Lattice
Reflection symmetry imposed by open boundary condition [3]
Symmetric Subspace Reduction
[3] G. Wu et al. / Physics Letters A 379 (2015) 143-148
Quantum Optics Final Project
Excitation dynamics in small lattices
Weak Interaction Strength
▪ Periodic beating
Strong Rydberg Interaction
▪ Coherent oscillations▪ No visible damping
[3] G. Wu et al. / Physics Letters A 379 (2015) 143-148
Quantum Optics Final Project
Decoherence due to neighboring atoms
▪ Damped Rabi oscillations
10 Lattice Site Dynamics
Rich Excitation Dynamics
▪ Collapse and revival of Rydberg polariton
[3] G. Wu et al. / Physics Letters A 379 (2015) 143-148
Quantum Optics Final Project
Many Body Rabi Oscillations
[4] Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Nat. Phys. 8, 790 (2012)
Collective Dicke States
Enhancement of Atom-Field Coupling We wish to model inhomogeneous lightshift caused by doubly excited statesonto singly excited Rydberg states
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Possible dephasing mechanisms
▪ Collisions
▪ Atomic motion
▪ Radiative decay
▪ Atom loss
▪ Stark shifts
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Interaction-induced inhomogeneous lightshifts
Effective Hamiltonian to model decoherence:
Strategy:
▪ Consider uniform excitation Ω𝑖= Ω𝑗 = Ω
▪ Solve low dimensional Hilbert system analytically
▪ Perform spatial average of positiondependent light shifts across sample distribution
Quantum Optics Final Project
Two Dimensional Hilbert Space – Analytic Solutions
Collective states
Analytic expressions for coefficients
[4] Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Nat. Phys. 8, 790 (2012)
Effective Rabi Frequency
Quantum Optics Final Project
Probability Density Function – Uniform vs Gaussian
Gaussian vs Uniform density sphere
Probability density function for n-dimensional sphere with Gaussian density distribution
Probability density function for n-dimensional sphere with uniform density distribution [5]
[5] Shu-Ju Tu and Ephraim Fishbach (2001)
Quantum Optics Final Project
Probability Density Function – Uniform vs Gaussian
Gaussian vs Uniform Density Sphere
Probability density function for 3-dimensional sphere with Gaussian density distribution
Probability density function for 3-dimensional sphere with uniform density distribution
[5] Shu-Ju Tu and Ephraim Fishbach (2001)
Quantum Optics Final Project
Analytic expressions for averaged coefficients
Gaussian density distribution averaged:Airy and Airy prime functions
Uniform density distribution averaged : Gamma and Incomplete Gamma functions
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Estimating blockade parameters
van der Waals coefficient [6]
Bounds for van der Waals shift Ratio characterizing blockade
[4] Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Nat. Phys. 8, 790 (2012)[6] L.Beguin et al. PRL (2013)
Effective Rabi frequency of two-photon transition
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Varying blockade ratio - Dephasing
+
-
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Multi-excitation induced Stark shifts
Atom-Field Hamiltonian
[7] P. Berman
Wish to investigate the effect of multiple atoms in the intermediate 𝑝 state
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3 Atom Collective State Amplitudes
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Collective amplitudes
System of differential equations for collective amplitudes
Multiple p excitations causes effective damping of Rabi oscillation
[7] P. Berman
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Generation of Entanglement – CNOT Gate
Generating Bell State
1. Prepare two qubit input state:
2. Apply CNOT gate:
3. Output state is maximally entangled (ideal scenario)
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Measure of Entanglement
Violation of Bell inequality
Overlap with Bell State
Increase in entanglement with more atoms and stronger Rydberg blockade
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Summary
▪ Rydberg ensemble qubits allow for fast quantum state preparation and manipulation
▪ Several mechanisms lead to damping of Rabi oscillations▪ Doubly excited Rydberg states▪ Multiple intermediate P state excitations
▪ Breakdown of Rydberg blockade leads to reduced fidelity of quantum gate operations
▪ Combine both mechanisms as well as include additional effects such as atom loss and radiative decay.
Quantum Optics Final Project
Questions?
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References
[1] M. Saffman, T. G. Walker, and K. Molmer, RMP 82, 2313 (2010)
[2] S.-Y. Lan et al., Opt. Exp. 17, 13639 (2009)
[3] G. Wu et al. / Physics Letters A 379 (2015) 143-148
[4] Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Nat. Phys. 8, 790 (2012)
[5] Shu-Ju Tu and Ephraim Fishbach (2001)
[6] L.Beguin et al. PRL (2013)
[7] Paul R. Berman, V. S. (2011). Principles of Laser Spectroscopy and Quantum Optics.Princeton: Princeton University Press.
Quantum Optics Final Project
Supplementary : Preparation Fidelity
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