Controlling Quantum Systems Controlling Quantum Systems with Spatial Adiabatic Passage Thomas Busch
Motivation: Complex Quantum System Dynamics Step 1 : learn how to control small quantum systems (one or two particles) i ~ ∂ ∂ t | ψ ( x 1 , x 2 ; t ) i = H ( t ) | ψ ( x 1 , x 2 ; t ) i Step 2 : learn how to control large quantum systems (more than two particles) i ~ ∂ ∂ t | ψ ( x 1 , x 2 , x 3 , . . . ; t ) i = H ( t ) | ψ ( x 1 , x 2 , x 3 , . . . ; t ) i bottom-up: following experimental progress, clean and controllable systems
Guiding Principles Why Find Understand Develop QM in real life Fundamental QM Applications of QM | i Entanglement Quantum Computing Energy Transport Non-locality Quantum Simulators Charge Transfer Decoherence Quantum Metrology Quantum Phase … … Transitions …
Challenge quantum systems are very fragile and have a massive Hilbert space identify systems which can be engineered develop techniques for quantum engineering find techniques for scaling quantum systems up do this in close collaboration with experimentalists System of Choice: ultracold atoms
Examples of Projects @ OIST superfluid vortices as creation of quantum walks as nano-sensors topological matter non-classical states quantum memories for single atom states spin-orbit coupled multicomponent atom-ion a diabatic engineering superfluids superfluids hybrid systems techniques & shortcuts
Spatial Adiabatic Passage How to move an atom? ? ! Solution : Tunneling (increase and decrease the distance between traps) Problem : Fragile Process (Rabi Oscillations) success depends on good control of three parameters interaction time t i approach time t a minimum distance between traps a min precise experimental control necessary for high fidelities ( > 99.9999%)
Spatial Adiabatic Passage How to move an atom? Solution : Tunneling (increase and decrease the distance between traps) Problem : Fragile Process (Rabi Oscillations) success depends on good control of three parameters interaction time t i approach time t a minimum distance between traps a min precise experimental control necessary for high fidelities ( > 99.9999%)
Spatial Adiabatic Passage Sequential Tunneling 1 time 2 same problem, only twice
Quantum Systems Unit: Spatial Adiabatic Passage Counterintuitive Tunneling 2 time 1 ( STIRAP) 100% transfer! K Eckert, M Lewenstein, R Corbalán, G Birkl, W Ertmer, J Mompart Physical Review A 70 , 023606 (2004)
Why does this work? | 3 i | 1 i | 2 i Ω 12 Ω 23 ✏ | 1 i | 2 i | 3 i eigenstate connects 1 and 3 Ω 12 0 | 1 i ✏ Ω 12 Ω 23 | 2 i | Ψ i = cos θ | 1 i � sin θ | 3 i ✏ Ω 23 0 | 3 i ✏ tan θ = Ω 12 Ω 23 cos θ : 1 → 0 θ : 0 → π TRANSFER: tan θ : 0 → ∞ 2 sin θ : 0 → 1 10
All good. Now what? generalise beyond 1D 1 find shortcuts to avoid adiabatic restrictions find shortcuts to avoid adiabatic restrictions identify suitable experimental settings for observation 2 generalise to many particle systems generalise to many particle systems identify non-classical correlations develop into other engineering tools: quantum state preparation, deterministic single atom source ….
Shortcut To Adiabaticity Idea: add terms to Hamiltonian that compensate for diabatic excitations when driving is non-adiabatic H = H 0 + H 1 0 Ω 12 ( t ) 0 0 0 i Ω 13 ( t ) H 0 ( t ) = ~ H 1 ( t ) = ~ Ω 12 ( t ) 0 Ω 23 ( t ) 0 0 0 2 2 0 Ω 23 ( t ) 0 − i Ω 13 ( t ) 0 0 [ Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105 , 123003 (2010) ] Ω 12 Ω 23 Ω 13 12
Spatial Adiabatic Passage in 2D up to now: Ω 12 Ω 23 1 2 3 Ω 31 symmetry breaking gives additional coupling use this degree of freedom to make new states create angular momentum Tunneling-induced angular momentum for single cold atoms R. Menchon-Enrich, S. McEndoo, J. Mompart, V. Ahufinger and TB, 13 Phys. Rev. A 89, 013626 (2014)
Shortcut To Adiabaticity 2 3 0 Ω 12 ( t ) i Ω 13 ( t ) H ( t ) = ~ Ω 12 ( t ) 0 Ω 23 ( t ) 2 − i Ω 13 ( t ) Ω 23 ( t ) 0 1 but: shortcut Hamiltonian is imaginary! Ω 13 ⇠ h 1 |H| 3 i cannot get this phase dynamically for transition between eigenstates nice idea, but cannot be implemented for SAP…?! use geometric phase!? 14
Geometric Phase Brief Reminder : Aharanov Bohm Effect assume charged particle moving in a magnetic field that is constant everywhere (or localised) phase is added when particle moves from to ~ ~ r j r i Z ~ r j � ij = q A · d ~ ~ are the positions of the wells ~ l r i ~ and is the magnetic vector potential ~ A ~ r i total phase in a closed loop: Φ = � 12 + � 23 + � 31 = q l = q I 2 3 A · d ~ ~ ~ Φ B ~ magnetic flux through 1 the closed path around the triangle 15
Geometric Phase Ω 12 e − i φ 12 Ω 13 e i φ 31 0 H AB = − ~ Ω 12 e i φ 12 Ω 23 e − i φ 23 0 2 Ω 13 e − i φ 31 Ω 23 e i φ 23 0 what we want: 0 Ω 12 − i Ω 13 H = − ~ Ω 12 Ω 23 0 2 i Ω 13 Ω 23 0 φ 31 = − π engineer field such that φ 12 = φ 23 = 0 and 2 requires field with specific spatial profile 16
Geometric Phase change basis using only local phases i 2 ( φ 12 + φ 23 ) e 0 i 2 ( − φ 12 + φ 23 ) U = 0 e 0 e − i 2 ( φ 12 + φ 23 ) 0 0 so that we get Ω 31 e i Φ 0 Ω 12 AB = U H AB U � 1 = − ~ H 0 Ω 12 Ω 23 0 2 Ω 31 e � i Φ Ω 23 0 and therefore only need l = − ~ ⇡ I Φ = − π A · d ~ ~ Φ B = 2 q 2 only relevant value is the total phase (or total flux) can be achieved with homogeneous field distribution 17
Spatial Adiabatic Passage in 2D 2 3 1 with shortcut pulse no shortcut pulse 0.9 full transfer (adiabatic) 0.7 time time 0.5 0.3 0.1 low transfer (fast) phase phase can also be inverted to measure magnetic fields! 18
All good. Next … generalise beyond 1D generalise beyond 1D 1 find shortcuts to avoid adiabatic restrictions find shortcuts to avoid adiabatic restrictions identify suitable experimental settings for observation 2 generalise to many particle systems generalise to many particle systems identify non-classical correlations develop into other engineering tools: deterministic single atom source
Interactions resonance not guaranteed dark state not guaranteed strongly interacting bosons non-interacting bosons (non-interacting fermions) g = 0 g = ∞ 20
Weak Interactions Three-well Bose-Hubbard model: 1 1 diagonalise and find two energy bands 1 1 1 1 particles are in different wells E = 1 2 particles are in same well E = 1 + U 2 2 21
Weak Interactions E = 1 . 05 E = 1 . 25 level crossings make following interaction leads to band separation the dark state effectively impossible co-tunneling restores the dark state 22
Strong Interactions Three-well Fermi-Hubbard model: diagonalise and find two energy bands particles are in different wells E = 2 particles are in same well E = 2 − | U | 23
Strong Interactions | U | = 0 . 3 | U | = 0 . 15 interaction leads to band separation level crossings make following co-tunneling restores the dark state the dark state effectively impossible 24
Exact Diagonalisation interaction band is isolated, but crossings still exist! adiabatic and diabatic dynamics can lead to full transfer 25
Summary Adiabatic techniques are not necessarily slow or limited to single particles. Spatial Adiabatic Passage possess an experimentally implementable Shortcut to Adiabaticity Interactions can lead to band-separation that allow to use single particle ideas for many-particle systems 26
Collaborations Irina Reshodko Lee O’Riordan Tara Hennessy Albert Benseny Yongping Zhang Jeremie Gillet Angela White Rashi Sachdeva Thomas Fogarty James Schloss TB Andreas Ruschhaupt Anthony Kiely
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