Quantum Optomechanical Heat Engine

Transcription

1 Quantum Optomechanical Heat Engine ~Ando-Lab Seminar on Sep. 2nd~ Kentaro Komori 1

2 Optomechanics Thermodynamics 2

3 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 3

4 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 4

5 What is the heat engine? ü The machine producing work from heat For example ² Car engine (a kind of otto cycle) 1. Intake, 2. Compression, 3. Power, 4. Exhaust ² Steam turbine Fuel Gas Work Heat 5

6 Cycles at thermodynamics Ø Carnot cycle p iq = isothermal quasistatic aq = adiabatic quasistatic aq iq T H T L iq Ø Giving the maximum efficient aq ε 0 =1 T L T H V 6

7 Cycles at thermodynamics Ø Otto cycle (considered at this seminar) p i = isochoric T H i T H ' aq aq T L ' i T L Ø The efficient is smaller than one of carnot ε =1 T L T ' H < ε 0 V 7

8 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 8

9 Hamiltonian Ø Linearized optomechanical coupling Η =!Δâ â +!ω m ˆb ˆb +!G( ˆb + ˆb )(â + â ) optical mechanical optomechanical α + â : average + photon annihilation ˆb Δ : phonon annihilation : cavity detuning ω m : mechanical resonant frequency 9

10 Hamiltonian Ø Linearized optomechanical coupling Η =!Δâ â +!ω m ˆb ˆb +!G( ˆb + ˆb )(â + â ) =!Δ X 2 c + P2 c 2 X c = â + â 2, P c = â â 2i X m = ˆb + ˆb 2, P m = ˆb ˆb 2i +!ω m X 2 m + P2 m 2 + 2!GX m X c : optical quadrature : mechanical quadrature 10

11 Hamiltonian Ø Linearized optomechanical coupling Η =!Δâ â +!ω m ˆb ˆb +!G( ˆb + ˆb )(â + â ) =!Δ X 2 c + P2 c 2 +!ω m X 2 m + P2 m 2 + 2!GX m X c X+ 2 =!ω + P2 + X 2 + +!ω + P2 2 2 diagonalization =!ω + Â Â +!ω ˆB ˆB new bosonic annihilation operators (linear combinations of â, â, ˆb, ˆb ) 11

12 Hamiltonian Ø the frequency of polaritons ω ± = Δ2 +ω 2 m ± (Δ2 ω 2 m ) 2 16G 2 Δω m 2 Focusing on the normal-mode ˆB ω + ω ~ ω m (Δ < ω m ) phononlike ω ω ~ Δ( ω m < Δ < 0) photonlike 12

13 Situation Excitation ˆB Excitation ˆB T a almost 0K T b finite ω ~ ω m (Δ < ω m ) phononlike T a almost 0K T b finite ω ~ Δ( ω m < Δ < 0) photonlike 13

14 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 14

15 Otto cycle p T H (1) isentropic expansion (4) hot isochoriclike transition T H ' T L ' (2) cold isochoriclike transition T L (3) isentropic compression V (1),(3) changing detuning (laser frequency) 15

16 Situation Excitation ˆB (4) (1) (3) (2) Excitation ˆB T a almost 0K T b finite ω ~ ω m (Δ < ω m ) phononlike T a almost 0K T b finite ω ~ Δ( ω m < Δ < 0) photonlike 16

17 (1) Isentropic expansion ω i ~ ω m (Δ < ω m ) phononlike N i = ˆB ˆB remains unchanged ωi,t i optical energy damping ω f ~ Δ( ω m < Δ < 0) photonlike increased radiation pressure = work 17

18 (2) Cold Isochoriclike transition N i Excitation ˆB N f κ Ø Thermal mean particle number decreases. T a almost 0K T b finite ω f ~ Δ( ω m < Δ < 0) photonlike 18

19 (3) Isentropic compression ω f ~ Δ( ω m < Δ < 0) photonlike N f = ˆB ˆB ω f,t f remains unchanged ω i ~ ω m (Δ < ω m ) phononlike 19

20 (4) Hot Isochoriclike transition Excitation ˆB N f T a almost 0K T b finite ω i ~ ω m (Δ < ω m ) phononlike N i Ø Thermal mean particle number increases. 20 γ

21 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 21

22 Work and efficiency E 1 =!ω i N i E 4 =!ω i N f E 2 =!ω f N i Ø total work W = E 1 E 2 + E 3 E 4 E 3 =!ω f N f Ø received heat Q = E 1 E 4 22

23 Work and efficiency Ø efficiency η = W Q =1 ω f ω i W =!(ω i ω f )(N i N f ) Q =!ω i (N i N f ) Δ i = 3ω m T a = 0,T b =

24 Work and efficiency When G ω m <<1 (weak coupling) and Δ f <<1,the approximate values are ω m ω i = ω m N i = n b " ω f = Δ f 2G2 N f = 1+ 4Δ f G2 % 3 ω $ m # ω ' n a + 2G2 m & ω n 2 b m W " = Δ f + 2G2 2!ω m ω m ω +1 %)" $ ' 1 2G2 % $ 2 'n b G2, +. 2 # m &* + # ω m & ω m -. 24

25 Work and efficiency In the high temperature limit of phonon bath, k B T b!ω m >>1, the efficiency at maximum work is # η W =1 Δ f % $ ω m <1!Δ f 2k B T b +!ω m 4k B T b & ( c.f. Carnot limit ' η <1+!Δ f 2k B T b 25

26 Timescale τ 4 τ 1 τ 2 τ 3 1/ G << τ 1,τ 3 <1/κ preventing transition keeping N i, N f between  and ˆB unchanged 26

27 Timescale τ 4 τ 1 τ 2 τ 3 waiting thermalization 1/κ < τ 2 <<1/γ preventing thermalization of  27

28 Timescale τ 4 τ 1 τ 2 τ 3 1/γ < τ 4 waiting thermalization 28

29 Timescale τ 4 τ 1 τ 2 τ 3 requirement 1/ τ 4 < γ <<1/ τ 2 < κ <1/ τ 1,3 << G << ω m 29

30 Contents Ø What is the heat engine? Ø Hamiltonian and polaritons Ø Otto cycle Ø Work and efficiency Ø Cosmic blackbody radiation 30

31 Cosmic blackbody radiation cold hot photon phonon 0K finite optical regime hot CMB 2.7K cold coldatom pk microwave regime 31

32 Cosmic blackbody radiation Ø Can extract energy from CMB! requirement thermal motion < coherent momentum recoil 2!k hot CMB 2.7K cold coldatom pk microwave regime In a case of lithium atoms, not to exceed a pk for 300 GHz photons 32

33 Future work Ø Evaluating the role of imperfections due to the coupling to the thermal reservoirs. Ø Considering the effects of non-adiabatic transitions between polaritons and nonideal control of maintaining the intracavity power. Ø Method to extract work from the cycle actually 33

34 Summary Ø Heat engine can be realized with optomechanical system. Ø It is possible to produce an Otto cycle by varying the cavity detuning in terms of two effective reservoirs, which consist phonon and photon. Ø In principle, this system can extract work from cosmic microwave background. 34