Performance of shortcut-to-adiabaticity quantum engines Obinna Abah Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast, United Kingdom Workshop on Quantum Science and Quantum Technologies ICTP, Trieste 2017 Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 1 / 21
News.. Sales of inefficient vacuum cleaners banned By Euronews · last updated: 01/09/2017 Powerful vacuum cleaners are to be banned from today after the European Union introduced new rules which aim to improve energy efficiency across the continent. 'Widespread misconception’ The European Environment Bureau (EEB) said: "Power doesn't always equal performance, though the misconception has become widespread.” "Some efficient models maintained high standards of dust pick-up while using significantly less energy - due to design innovation." From BBC News 01/09/2017 Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 2 / 21
Outline Motivation 1 Introduction Downscaling engines Four-stroke Otto engine 2 Shortcut-to-adiabaticity engine 3 Fast and efficient engines Generic bounds on quantum machines Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 3 / 21
Introduction Miniaturization: is about building smaller devices Drexler 1981 fundamental limit = atomic structure of matter � Transistor: 1947 Today
Introduction Mobile phone: 1973/1983 Today weight 1.1kg, 30min talk time, 10h charge time, price 4000$ �
Introduction ”There is plenty of room at the bottom”: Feynman 1959 ”Consider any machine – for example, an automobile – and ask about the problems of making an infinitesimal machine like it” Two basic strategies: Follow engineers Follow nature Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 6 / 21
Macroscopic heat engine → convert thermal energy into mechanical work = motion Carnot efficiency: η = Work produced Heat absorbed ≤ 1 − β h = 1 − T c β c T h (James Watt 1783: η ∼ 5 − 7 % ) → maximum efficiency Today’s gasoline engines: η ∼ 25 − 30 % Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 7 / 21
Downscaling of heat engines Car engine Piezoresistive engine Steeneken et al., Nat. Phys. 7 (2011) Size m mm µm nm Nano heat engine (Classical or quantum) Mini engine Colloidal engine Blickle-Bechinger, Nature Phys. (2011) Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 8 / 21
Single atom heat engine Rossnagel et al., Science 352, 325 (2016) Reservoir engineering: • Cold reservoir: laser (Doppler) cooling (always on) • Hot reservoir: electrode noise (switched on/off) Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 9 / 21
Classical four-stroke heat engine ( ) ( ) 1 2 Work B A B C ( ) 3 ( ) 4 D D C A Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 10 / 21
Quantum Otto heat engine (3) Isentropic expansion Work done W 3 D C (4) Cold isochore (2) Hot isochore Heat removed Heat added Q Q 4 2 (1) Isentropic compression Work done W 1 A B Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 11 / 21
Quantum Otto heat engine: theory ℏ ℏ ℏ ℏ ω β ω ω β ω ∗ = 1 Q 2 2 = 2 2 2 H 2 coth H coth D C 2 2 2 2 D C A B ℏ ℏ ℏ ℏ ω β ω ω β ω ∗ = 2 Q 1 1 = 1 1 1 H 1 coth H coth B A 2 2 2 2 Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 12 / 21
Quantum Otto heat engine Abah et al, PRL 112 , 030602 (2012) w b w w æ b w ö h æ h ö h h * = Q = H 1 2 coth ç 2 2 ÷ H 2 coth ç 2 2 ÷ è ø è ø D 2 2 C 2 2 D C A B w æ b w ö æ ö h h h w b w h * = Q ç ÷ = ç ÷ H 2 1 coth 1 1 H 1 coth 1 1 è ø è ø B 2 2 A 2 2 Question How can we speed up the heat engine? Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 13 / 21
National waiters day gadgets – shoes, tray, ... optimal protocol Scientific Report 4 : 6208 (2014) Waiters race, fast service is a priority! Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 14 / 21
Shortcut-to-adiabaticity (STA) ... inducing a ”fast motion video of the adiabatic dynamics.” Effective Hamiltonian: H eff ( t ) = H 0 ( t ) + H i STA ( t ) H i STA ( t ) - STA driving Hamiltonian and i = (1 , 3) - compression/expansion steps � fast and reduces irreversible losses ω (0) = ω i , ω (0) = 0 , ˙ ω (0) = 0 , ¨ ω ( τ ) = ω f , ω ( τ ) = 0 , ˙ ω ( τ ) = 0 , ¨ For harmonic oscillator: LCD technique m Ω 2 t − ω 2 x 2 � � H STA = t 2 ω 2 � − 3 ˙ + ¨ ω t � m Demirplak and Rice, JPC A 107 , 9937 (2003) t x 2 = Berry, JPA 42 , 365303 (2009) 4 ω 2 2 2 ω t Chen et al, PRL 109 , 100403 (2010) t del Campo, PRL 111 , 100502 (2013) Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 15 / 21
Energetic cost of the shortcut driving 10 Elementary power analysis 8 Energetic cost P inst = 2 + 2 cos ω t 6 P inst 4 4 3 2 Power 2 0 P avg 5 10 15 20 1 Time τ Cost of the driving: 0 0 2 4 6 8 10 12 � τ Time H i H i � � � � τ = (1 /τ ) dt SA ( t ) STA 0 � T P avg = (1 / T ) 0 P inst dt Nonadiabatic work (friction): � W i � NA = � W i � − � W i � AD Abah and Lutz, EPL 118 , 40005 (2017) - the actual and the adiabatic work Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 16 / 21
Performance of STA quantum engines Efficiency: − ( � W 1 � STA + � W 3 � STA ) η STA = energy output energy input = H 1 H 3 � � � � � Q 2 � + τ + STA STA τ Power: P STA = energy output = −� W 1 � STA + � W 3 � STA . Cycle time τ cycle 2.0 0.70 0.65 1.5 STA 0.68 Efficiency NA Power 0.60 1.0 QSL 0.67 0.67 0.55 0.5 16 20 24 STA NA QSL 0.50 0.0 0 5 10 15 20 25 5 10 15 20 25 Time τ Time, τ
Generic bounds: quantum speed limit (QSL) Quantum: limits the speed of evolution of a system Anandan and Aharonov, PRL (1990) QSL time: τ QSL = � L ( ρ i ,ρ f ) � H STA � τ ≤ τ L ( ρ i , ρ f ) - the Bures angle between density operators STA = − � W 1 � AD + � W 3 � AD Efficiency: η STA ≤ η QSL � Q 2 � + � ( L 1 + L 3 ) /τ STA = − � W 1 � AD + � W 3 � AD Power: P STA ≤ P QSL Abah and Lutz, EPL 118, 40005 (2017) τ 1 QSL + τ 3 QSL 2.0 0.70 0.65 1.5 STA 0.68 Efficiency NA Power 0.60 1.0 QSL 0.67 0.67 0.55 0.5 16 20 24 STA NA QSL 0.50 0.0 0 5 10 15 20 25 5 10 15 20 25 Time τ Time, τ Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 18 / 21
Fast and efficient quantum engines Question: Is it true for every shortcut-to-adiabaticity protocol? 3.5 Abah and Lutz, arXiv. 1707.09963 (2017) 3.0 LCD Q ∗ - Adiabaticity parameter 2.5 CD Q * IE 2.0 LCD - local counterdiabatic driving 1.5 CD - counterdiabatic driving IE - inverse engineering 1.0 0.0 0.2 0.4 0.6 0.8 1.0 NA - nonadiabatic driving t / τ 0.70 3.0 LCD 2.5 0.65 0.68 CD 2.0 Efficiency Power IE 0.60 1.5 NA 0.67 16 20 24 1.0 0.55 IE CD 0.5 LCD NA 0.50 0.0 0.50 0.55 0.60 0.65 0.70 0 5 10 15 20 Time τ Efficiency � simultaneous increase of efficiency and power for fast cycles Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 19 / 21
Take-Home message STA engine are energy efficient machines outperform their convention counterpart � Quantum speed limit impose bounds to performance fundamental limit for quantum machines � tighter than the second law of thermodynamics � Power doesn’t always equal performance overall efficiency is important quantity � Obinna Abah (QUB) Energy efficient nanoscale machines September, 2017 20 / 21
References ⋆ Energy efficient quantum machines O. Abah and E. Lutz EPL (Europhys. Lett.) 118 , 40005 (2017) (3) Isentropic expansion Work done W 3 D C (4) Cold isochore (2) Hot isochore Heat removed Q 4 Heat added Q 2 (1) Isentropic compression Work done W 1 A B ⋆ Performance of shortcut-to-adiabaticity quantum engines O. Abah and E. Lutz arxiv: 1707.09963 (2017) ⋆ Shortcut-to-adiabaticity quantum refrigerator O. Abah, M. Paternostro, and E. Lutz ( to appear soon ) Thanksforyourattention !!
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