Quantum Coherence via Smooth Optimal Control orn Bartels 1 , Florian - PowerPoint PPT Presentation

Why do we need it? How does it work? An example: control of spin ensembles Quantum Coherence via Smooth Optimal Control orn Bartels 1 , Florian Mintert 1 , Bj¨ obauer 2 , Johannes Majer 2 Tobias N¨ 1 Freiburg Institute for Advanced Studies 2 Atominstitut Vienna 14 September 2012

Why do we need it? How does it work? An example: control of spin ensembles Smooth optimal control: Why do we need it? 1 How does it work? 2 An example: control of spin ensembles 3

Why do we need it? How does it work? An example: control of spin ensembles Why do we need it? 1 How does it work? 2 An example: control of spin ensembles 3

Why do we need it? How does it work? An example: control of spin ensembles How to rotate a qubit?

Why do we need it? How does it work? An example: control of spin ensembles Apply a control pulse! Why is it not that simple?

Why do we need it? How does it work? An example: control of spin ensembles Nitrogen-Vacancy Centers in Diamond Pham et al. , New J. Phys. 13 , 045021 (2011) Davies, Hamer, Proc. R. Soc. London A 348 , 285 (1967) very long coherence times (up to ms): quantum memory

Why do we need it? How does it work? An example: control of spin ensembles Quantum Computers Kubo et al. , PRL 107 , 220501 (2011)

Why do we need it? How does it work? An example: control of spin ensembles Quantum Computers

Why do we need it? How does it work? An example: control of spin ensembles The Problem of Spin Ensembles experiments in Vienna (Majer) and Paris (Est` eve): Ams¨ uss et al., PRL 107 , 060502 (2011) + Kubo et al., PRL 107 , 220501 (2011) problems: inhomogeneous broadening of NV ensemble ( → NMR), inhomogeneous field of antenna (new!)

Why do we need it? How does it work? An example: control of spin ensembles Control of Spin Ensembles optimal control copes with different spin frequencies ω 0 different amplitudes α of the control field

Why do we need it? How does it work? An example: control of spin ensembles Why do we need it? 1 How does it work? 2 An example: control of spin ensembles 3

Why do we need it? How does it work? An example: control of spin ensembles Optimal Control objective of optimal control: maximize fidelity �� � 2 � � max � � ψ f | U f ( t ) ( T ) | ψ i � ω 0 , α � �� � F [ f ( t )] δ F [ f ( t )] = 0 ? δ f ( t )

Why do we need it? How does it work? An example: control of spin ensembles Pulse Shaping with GRAPE/Krotov piecewise constant functions: a i ' a i iterative improvement: F a + δ a ≈ F a + δ a · ∇ a F a + 1 � δ a i δ a j ∂ a i ∂ a j F a 2 i , j Glaser et al. , J. Magn. Reson. 172 , 296 (2005) Krotov, Global Methods in Optimal Control Theory , Dekker (1995)

Why do we need it? How does it work? An example: control of spin ensembles Optimization Algorithm

Why do we need it? How does it work? An example: control of spin ensembles Typical GRAPE Pulse: High Frequency Components Kobzar et al., J. Magn. Reson. 173 , 229 (2005)

Why do we need it? How does it work? An example: control of spin ensembles Our Approach: Smooth Control alternative solution: instead of Caneva, Calarco, Montangero, PRA 84 , 022326 (2011) Romero Isart, Garc´ ıa Ripoll, PRA 76 , 052304 (2007) ⇒ use only a few frequency components: n � H c ( t ) = f ( t ) h f ( t ) = a k sin( k Ω t ) k =1

Why do we need it? How does it work? An example: control of spin ensembles Techniques I: What is U(t)? dynamics periodic in time: H ( t + 2 π i ∂ t | ψ k ( t ) � = H ( t ) | ψ k ( t ) � Ω ) = H ( t ) Floquet: | ψ k ( t ) � = e i ǫ k t | φ k ( t ) � with | φ k ( t + 2 π Ω ) � = | φ k ( t ) � ⇒ eigenvalue problem in Fourier space: � �� � K

Why do we need it? How does it work? An example: control of spin ensembles Techniques II: What is ∇ a F a ?

Why do we need it? How does it work? An example: control of spin ensembles Why do we need it? 1 How does it work? 2 An example: control of spin ensembles 3

Why do we need it? How does it work? An example: control of spin ensembles Robustness against Different Spin Frequencies broad frequency interval: here ∆ ω · T = π

Why do we need it? How does it work? An example: control of spin ensembles Typical Control Pulse: Only 4 Frequency Components achieved fidelities: > 99 . 99 % for the previous example

Why do we need it? How does it work? An example: control of spin ensembles Robustness against Inhomogeneity in the Control Field 1.00 0.95 0.90 fidelity 0.85 4 2 0.80 0 � 2 0.75 � 4 0 T � 2 T 0.70 0 5 10 15 20 25 30 effective inhomogeneity � � �

Why do we need it? How does it work? An example: control of spin ensembles Robustness against Inhomogeneity in the Control Field 1.00 0.95 0.90 fidelity 0.85 4 2 0.80 0 � 2 0.75 � 4 0 T � 2 T 0.70 0 5 10 15 20 25 30 effective inhomogeneity � � �

Why do we need it? How does it work? An example: control of spin ensembles Robustness against Inhomogeneity in the Control Field 1.00 � � � 0.95 0.90 fidelity 0.85 4 2 0.80 0 � 2 0.75 � 4 0 T � 2 T 0.70 0 5 10 15 20 25 30 effective inhomogeneity � � �

Why do we need it? How does it work? An example: control of spin ensembles The more frequency components, the higher the fidelity 1 ∆ ω = 200 ns , T = 3 . 14 µ s, Ω = 1 MHz, 20 % of inhomogeneity (only π/ 2-pulse) 0 � � � � � � 1 log 10 � 1 � Fidelity � � � � � 2 � 3 � � 4 � � 0 2 4 6 8 10 12 14 number of frequency components

Why do we need it? How does it work? An example: control of spin ensembles Generating π -pulses for the experiment ∆ ω : 8 MHz Gaussian FWHM, 25 % of inhomogeneity, n = 15 frequency components, control amplitude < 1.5 MHz � F � > 99 % 0 0.2 0.4 0.6 0.8 1 fidelity

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 100 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 112.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 125 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 137.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 150 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 162.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 175 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 187.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 200 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 212.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 212.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 225 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles First experimental data 1 0.75 P �� 0 � � 0.5 237.5 � 0.25 0 0 100 200 300 400 500 600 t � ns experiments by Tobias N¨ obauer from Atominstitut Vienna