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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

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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

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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

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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

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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

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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

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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

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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

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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

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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)

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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)

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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.

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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.

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Supplementary : Preparation Fidelity