From cavity optomechanics to the Dicke quantum phase transition

Transcription

1 From cavity optomechanics to the Dicke quantum phase transition (~k; ~k)! p Rafael Mottl Esslinger Group, ETH Zurich Cavity Optomechanics Conference 2013, Innsbruck

2 Motivation & Overview Engineer optomechanical interactions between ultracold atoms and the light of an optical cavity generic optomechanical coupling H int = Ga y ax Dicke-type optomechanical coupling H int = g(a y + a)x mechanical mode softening leading to the Dicke phase transition influence of dissipation on the fluctuations driving the phase transition

3 1cm The Experiment high-finesse optical cavity length = 178 m Finesse = Bose-Einstein condensate harmonic trap 10 5 Rb atoms T 100 nk 150 μm Related Experiments: A. Hemmerich, J. Reichel, D. Stamper-Kurn, V. Vuletic, C. Zimmermann

4 Optomechanically coupling atoms and light generic optomechanical coupling H int = Ga y ax Nature of the drive field: dispersive limit:! p! a À close to the cavity resonance:! p! c» Dicke-type optomechanical coupling H int = g(a y + a)x Dispersively coupling the cavity light field and collective atomic momentum states special starting point in the field of optomechanics: - mechanical oscillator is naturally prepared in its ground state

5 Generic optomechanical coupling Atoms -> Light: dispersive cavity resonance shift Light -> Atoms: AC-Stark shift per photon as a dynamical dipole potential ^H a c / cos 2 (x)^a y^a 4! rec j 2~ki ^ª = p Nj0i + ^c j 2~ki j0i creates a collective momentum excitation ^H int = Z ^ª y (~r) ^H a c ^ª(~r)d 3 r ^H int / ^a y^a(^c y + ^c) / ^a y^a ^X Gupta et al. PRL 99 (21), (2007), Brennecke et al. Science 322, 235 (2008), Murch et al. Nat. Phys. 4 (7), 561 (2008)

6 Effects of the generic optomechanical coupling Optomechanical bistability H int = Ga y ax Squeezed light generation Single photon optical bistability Quantum measurement induced backaction Gupta et al. PRL 99 (21), (2007), S. Ritter et al. Appl. Phys. B 95, 213 (2009), Murch et al. Nat. Phys. 4 (7), 561 (2008)

7 Dicke-type optomechanical coupling (~k; ~k) Dynamical dipole potential of the interference field between scattered light and transverse pump ^H int / cos(kx) cos(ky)(^a + ^a y )! p j ~k; ~ki 2! rec J + = c y 1 c 0 j0; 0i collective atomic operators ^H int = Z ^ª y (~r) ^H a c ^ª(~r)d 3 r ^H int / (^a + ^a y )( ^J + + ^J ) K. Baumann et al., Nature 464, 1301 (2010), D. Nagy et al. Phys. Rev. Lett. 104, (2010)

8 Dicke-type coupling in the thermodynamic limit (~k; ~k) ^H int / (^a + ^a y )( ^J + + ^J )! p Almost all atoms in the ground state Thermodynamic limit: N À 1 describe collective atomic operators in a single bosonic mode around the ground state ^J+ + ; ^J ; ^J z! ^b;^b y ^H int / (^a + ^a y )(^b + ^b y ) / (^a + ^a y ) ^X Coupling strength / pump power position operator of the bosonic excitation mode K. Baumann et al., Nature 464, 1301 (2010), D. Nagy et al. Phys. Rev. Lett. 104, (2010)

9 Consequence of the Dicke-type OM interaction mode softening of the momentum mode H int = g(a y + a)x Critical behavior of the drivendissipative system Dicke quantum phase transition

10 energy Effect of the Dicke-type OM coupling! p ^H = ~!^a y^a + ~! 0^by^b + ~ (^a + ^a y )(^b + ^b y )! c! =! p! c! À! 0! 0 = 2! rec j ~k; ~ki Energy spectrum: ² ph =! ² at =! 0 p 1 2= 2cr!

11 energy Detection: Bragg spectroscopy! 0 probe beam! p + ± ! p Ballistic expansion J. Stenger et al. PRL 82, 4569 (1999), J. Steinhauer et al. PRL 88, (2002), R. Mottl et al. Science 336, 1570 (2012)

12 Mode softening of the excited momentum state shading: ab-initio calculation including collisions and trapping R. Mottl et al. Science 336, 1570 (2012)

13 Dicke quantum phase transition normal phase superradiant phase Transverse pump power hai = 0 ha y ai 6= 0 hai 6= 0 Theory: H. Ritsch, P. Domokos (2002); observed with thermal atoms: V. Vuletic (2003)

14 Observing the Dicke quantum phase transition occupation of momentum states K. Baumann et al., Nature 464, 1301 (2010)

15 Consequence of the Dicke-type OM interaction mode softening of the momentum mode H int = g(a y + a)x Critical behavior of a drivendissipative system Dicke quantum phase transition

16 Fluctuations driving the phase transition?? mode softening of the mechanical oscillator adiabatic elimination of the cavity field: ^a / (^b + ^b y ) Contains the information about the fluctuations of the mechanical mode!

17 Real-time observation of leaking photons Single-photon counter transition point F. Brennecke et al. PNAS 110 (29) (2013)

18 Fluctuations around the ground state normal-phase Hamiltonian ^H = ~!^ay^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) diagonalize the Hamiltonian fluctuations around the ground state F. Brennecke et al. PNAS 110 (29) (2013)

19 Fluctuations around the ground state normal-phase Hamiltonian ^H = ~!^ay^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) diagonalize the Hamiltonian fluctuations around the ground state F. Brennecke et al. PNAS 110 (29) (2013)

20 Openness of the cavity Vacuum input noise Cavity dissipation induces measurement backaction Vacuum input noise drives system into a steady state with enhanced fluctuations F. Brennecke et al. PNAS 110 (29) (2013)

21 Fluctuations of the steady state normal-phase Hamiltonian ^H = ~!^ay^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) Quantum Langevin equations _^a = i[^a; ^H] ^a + p 2 ^a in _^b = i[^b; ^H] ^b + p 2 ^b in F. Brennecke et al. PNAS 110 (29) (2013)

22 Fluctuations of the steady state Damping of the mechanical mode? ^a / (^b + ^b y ) normal-phase Hamiltonian ^H = ~!^ay^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) Quantum Langevin equations _^a = i[^a; ^H] ^a + p 2 ^a in _^b = i[^b; ^H] ^b + p 2 ^b in F. Brennecke et al. PNAS 110 (29) (2013)

23 Correlation analysis of the cavity output field g (2) ( ) / h^a y ( )^a y (0)^a(0)^a( )i i ^a / (^b + ^b y ) ( i) F. Brennecke et al. PNAS 110 (29) (2013)

24 Determination of the atomic damping rate! s =2¼ B. Öztop et al. NJP 14, (2012), F. Brennecke et al. PNAS 110 (29) (2013)

25 Effective quantum Langevin description Contributions to intracavity field: coherent cavity component cavity backaction fluctuations thermal fluctuations normal-phase Hamiltonian ^H = ~!^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) Quantum Langevin equations _^a = i[^a; ^H] ^a + p 2 ^a in _^b = i[^b; ^H] ^b + p 2 ^b in D. Nagy et al, PRA 84, (2011), B. Öztop et al, NJP 14, (2012), D. Torre et al, PRA 87, (2013), F. Brennecke et al. PNAS 110 (29) (2013)

26 Effective quantum Langevin description normal-phase Hamiltonian ^H = ~!^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) Quantum Langevin equations _^a = i[^a; ^H] ^a + p 2 ^a in _^b = i[^b; ^H] ^b + p 2 ^b in D. Nagy et al, PRA 84, (2011), B. Öztop et al, NJP 14, (2012), D. Torre et al, PRA 87, (2013), F. Brennecke et al. PNAS 110 (29) (2013)

27 Effective quantum Langevin description Scaling of the fluctuations h(^b + ^b y ) 2 i Steady state: exponent -0.9 Closed system: exponent -0.5 normal-phase Hamiltonian ^H = ~!^a y^a + ~! 0^by^b + ~ (^a y + ^a)(^b + ^b y ) Quantum Langevin equations _^a = i[^a; ^H] ^a + p 2 ^a in _^b = i[^b; ^H] ^b + p 2 ^b in D. Nagy et al, PRA 84, (2011), B. Öztop et al, NJP 14, (2012), D. Torre et al, PRA 87, (2013), F. Brennecke et al. PNAS 110 (29) (2013)

28 Cavity team Tilman Esslinger Renate Landig Lorenz Hruby Tobias Donner Ferdinand Brennecke Rafael Mottl Kristian Baumann (Stanford) Thank you very much for your attention!

29 Summary realizing a Dicke-type optomechanical interaction with ultracold atoms in an optical cavity H int = g(a y + a)x mode softening induced by Dicketype optomechanical interactions changed fluctuations at a drivendissipative phase transition R. Mottl et al. Science 336, 1570 (2012) F. Brennecke et al. PNAS 110 (29) (2013)