Quantum Many Body Physics with Strongly Interacting Matter and Light. Marco Schiro Princeton Center for Theoretical Science

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1 Quantum Many Body Physics with Strongly Interacting Matter and Light Marco Schiro Princeton Center for Theoretical Science Nice, INLN Condensed Matter Seminar July

2 Interacting Light and Matter Far From equilibrium Condensed Matter Light used to probe phases or (more recently) to induce phases of matter Ultra-Cold Atoms Light used to trap (optical lattices) to probe or to excite atoms Quantum Many Body Physics with Light (?) Non-Linear Optics with Single Photons!

3 Outline Route to Non-Linear Optics with Single Photons: Coupling Light and Matter at the Quantum Level Many Body Physics with Atoms and Photons: Perspectives and Challenges Photon Lattice with Rabi Non-Linearity

4 Exploring Light-Matter Interaction at the Quantum Level

5 Cavity Quantum ElectroDynamics Trapping photons and atoms into cavities! J. M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73, 565 (2001) Optical/Microwave 3D cavities supporting discrete modes Alkali or Rydberg Atoms trapped into the cavity Light-Matter coupling (Dipole) H int d E

6 A Solid-State Analogue: Circuit QED Circuit QED A. Blais et al, Phys. Rev. A (2004) Microwave Transmission Line Resonators Superconducting-based to avoid dissipation Artificial Atoms: Superconducting Qubits Mesoscopic SC system with atom-like spectrum Capacitive/Inductive Coupling

7 How Strong Can the Coupling Between Light and Matter Be? Poor-Man Estimate of Light-Matter Coupling g g = d at E 0 /~ d at el ~ r E2 0 V Fine Structure Constant Limit M. Devoret, S. Girvin&R. Schoelkopf, Ann Phys (2007) r r L r 2 g el r 2 r ~ 0 V Circuit QED works in a regime where L~r!! Coupling via current (i.e. magnetic field) can be even larger!

8 Energy Scales in CQED Circuit QED Cavity QED Photon Frequency/Atom splitting r, q Light-Matter Coupling (Vacuum Rabi Splitting) g r, q Photon Losses and Atomic Decay Rates (Dissipation), Strong-Coupling Regime of cqed g, Typical Circuit QED:! r 10 Ghz g 1Ghz, 500 khz

9 Effective Photon-Photon Interactions by Light-Matter Coupling

10 Quantum Models of Light and Matter Cavity QED Circuit QED The Rabi Model H Rabi = r a a + q + + g a + a + + Harmonic Mode (Photon) Anharmonic Mode (Atom) Linear coupling (dipole) I. Rabi (1936) Often in Rotating Wave Approximation (requires g min( r, q) ) Rabi r a H JC = r a + a + q q XModel Jaynes,Cumming (1963) The Jaynes-Cumming + g a + hc + g a + + hc

11 Atom-Induced Photon Non Linearity H JC = r a a + q + + g a + hc Elementary Light-Matter Excitations: Polaritons N pol = a a + + [H JC,N pol ]=0 Weakly Anharmonic Spectrum n, + = sin n n + cos n n 1 n, = cos n n sin n n 1 tan n =2g p n/( + p 2 /4+ng 2 ) =! r! q E n± = r n + 2 ± p 2 /4+ng 2 Exp Realization with Circuit QED under driving L. Bishop et al, Nat. Phys 5, 109 (2009) /2

12 Photon-Blockaded Transport A.J Hoffmann et al, Phys. Rev. Lett 107, (2011) Steps in the Photon Transmission through a driven cavity Strongly Dispersive Regime: g r q Effective Photon Hamiltonian H JC r a a + U eff a a 2 U eff g 4 / r q 3

13 Many Body Physics with Strongly Interacting Light and Matter

14 Single cqed unit coupled to Quantum Baths Experiments in Paris (T. Kontos Lab), Princeton(J. Petta Lab), ETH (K.Ennslin Lab, A. Imamoglu Lab) Quantum Impurity Physics with Light&Matter! M. Schiro&K. Le Hur (in preparation) M. Delbeq et al PRL (2011) Coupled Arrays of Resonators with SC Qubits Experiments at Princeton! A. Houck s Lab) J. Koch and K. Le Hur PRA (2009) J. Koch, A. Houck and H. E. Tureci Nat. Phys (2012)

15 Arrays of Resonators and SC Qubits H i JC = r a i a i + q + i i + g a i i + hc H hopping = X ij J ij a i a j Jaynes-Cumming Lattice Model H JCL = X ij J ij a i a j + X i H i JC Effective Photon-Photon Interactions Hopping between adjacent resonators/cavities Bose-Hubbard Physics(?) Delocalized bosonic particles with an effective onsite interaction...

16 Mott to Superfluid Transition of Polaritons Jaynes-Cumming Lattice Model H JCL = X ij a J ij i a j + h.c. + X i Hartmann et al, Greentree et al, Schmidt et al,rossini et al, Tomadin et al,.. H i JC µ X i N i pol J=0 -- Polaritons [H JC,N pol ]=0 J<Jc -- Mott Insulator of Polaritons J>Jc -- Superfluid ha i i6=0 U(1) Spontaneous Symmetry Breaking J. Koch and K. Le Hur PRA (2009)

17 The importance of being a Photon Unlike massive particles Photons are not conserved in Nature Think of QED! Photons have zero chemical potential! µ ph = E N =0 The density of a Photon Gas is set by drive-dissipation or interaction with matter!

18 Can we drive photons into steady states with non trivial quantum correlations?

19 Open Questions Effective Chemical Potential for Polaritons? Increase Light-Matter Coupling: effective driving Counter-Rotating Terms H int = H JC + g a + + a g r, q Ultra-Strong Coupling Regime Balancing Drive and Dissipation Critical Behavior for Driven-Dissipative Quantum Systems? Non-Markovian Effects? Dissipation-Induced Interactions? External Drive? Coherent vs Incoherent? Rate of Entropy Production?Effective Temperature? (cfr. Polariton BEC in Microcavities) M.Schiro et al (In preparation)

20 Photon Circuit QED Lattices with Rabi Non Linearity M. Schiro, M. Bordyuh, B. Oztop and H. Tureci, Phys. Rev. Lett. 109, (2012)

21 The Rabi Lattice Model H RL = X ij J ij a i a j + X i H i Rabi H i Rabi = r a i a i + q + i i + g a i + a i + i + i No Chemical Potential! Light-Matter interaction fix the average number of excitations in the ground state Can we have a non-trivial ground-state due to an effective drive? What is the effect of counter-rotating terms on the Mott-to-Superfluid Transition?

22 Insights from the Rabi Model H Rabi = r a a + q + + g a + a + + Symmetries: Parity Z2 symmetry enough for an exact solution! Z 2 = e i N pol D. Braak, Phys Rev Lett (2011) a = a x = x Non Trivial Ground-state Properties < N > < a > < a 2 > =! r! q =0 P n - = cn g = 1.5 r g/ r Finite Number of Excitations, Squeezed, Symmetric n Rabi = X n, =± c n n

23 A Low Energy Doublet at Strong Coupling H Rabi = r a a + q + + g a + a + + Low-energy spectrum =! r! q =0 2 Energy = + 1 = 1 = + 1 = 1 Exponentially Small Splitting! = r e 2(g/ r) g/ r Ground-state has always the same (even) parity First excited state, with opposite parity, is almost degenerate for g r

24 What happens at finite (small) Hopping? H RL = X ij J ij a i a j + X i H i Rabi Qualitative Phase Diagram in the J vs g plane: Small Hopping: Disordered Phase, Gapped, Insulating (but Not a Mott Insulator!) h x i i =0 h a i i =0 Large Hopping: Ordered Phase, Gapped, not a Superfluid! Z2 Parity Symmetry Breaking h x i i6=0 h a i i6=0 Quantum Phases and QPT are different from the physics of massive quantum particles on a lattice!

25 Photon-Induced Qubit Interactions Photons always appear quadratically and can be traced out S g = g Z 0 d X i x i ( ) x i ( ) Effective ferro-electric interaction between TLS S int X Z d d 0 i x ( ) K ij ( 0 ) j x ( 0 ) ij 0 K R ( ) R = 1 R = 2 R = 3 R = 4 J/ r = Kernel depends from photon band-structure K ij ( )= h T x i ( ) x j ( 0 )i 0 Short-Ranged (space/ time) unless gapless modes Qubit Frequency as a Transverse Field

26 Mean Field Phase Diagram Mean Field Boundary Related to Static Susceptibility of Rabi model Z 1 = d h T x( ) x(0)i loc zj c 0 2 Ferro-electric Insulator h x i i6=0 h a i i6=0 1.5 δ = 0.5 δ = 0.0 δ = 0.5 δ = 1.0 δ = 2.0 Z2 Parity Symmetry Breaking g/ω 0 1 =! q! r Para-electric Insulator h x i i =0 h a i i = Negative detuning Universality Class?? favors the ordering phase J/ω 0

27 Universal Behavior at the QPT From Rabi single site: low-energy doublet with a tiny splitting = + 1 Opposite Parity: Isospin x ± = ± ± Energy = 1 = + 1 = 1 Rabi Hamiltonian in this subspace g/ r H i Rabi! 2 z i Project onto this low energy doublet: H eff = X h iji J x x i x j + J y y i y j + 2 X Anisotropic XY model! Quantum Ising Universality Class i z i Gapped phases, gapless at QCP -- Robust to weak dissipation!

28 The Role of Counter-Rotating Terms: From Jaynes-Cumming to Rabi M. Schiro et al, (in preparation)

29 From Jaynes-Cumming to Rabi H Rabi = r a a + q + + g (a + hc)+g (a + + hc) Structure of the Ground State vs g 0 X /g = c P n = c n 2 n n n, =± Probabilty of having n polaritons P n g /g = g /g = Π = 1 Π = P n g /g = g /g = 0.75 g/ω Π = 1 Π = n n g /g Level Crossings between different parity sectors at finite g 0 /g

30 The Fate of Mott Lobes What happens to the Jaynes-Cumming Mott Lobes as we tune g /g? g = 0.5 ω g = 1.1 ω 0 J c /ω Π = +1 < a > = 0 < σ x >= Π = 1 Π = J c /ω Π = 1 g = 3.5 ω Π = +1 g = 2.5 ω 0 1e-06 Π = J c /ω 0 1e-08 1e-10 Π = +1 Π = 1 1e-06 Π = 1 J c /ω 0 1e g /g g /g Lobes at fixed Parity survive at any finite g 0 /g 6= 1 but shift up

31 The Role of Counter-Rotating Terms Effective Action Z =Tre H = Z i D i D i e S eff [ i, i] S eff [ i, i ]= Z 0 d X h iji i J 1 ij j + X i [ i, i ] M. Fisher et al, PRB(1989) for Bose-Hubbard [, ] = logh e R 0 d ( ( ) a( )+a ( ) ( )) i Local Physics (Rabi Non-Linearity) Standard Field Theory of U(1) Superfluid-to-Mott Transition... S eff = Z 0 d K 1 + K K r 2 + u Relevant Operators (explicit breaking down to Z2) Z S CRT = 0 d g 0 ( + ) K. Damle & S. Sachdev, PRL(96)

32 Summing Up Rabi Hubbard Lattice Model Use Counter-Rotating Terms to polarize the vacuum Non trivial Ground State, no driving, no chemical potential Relevant perturbation, drives the system toward a Z2 QCP Perspectives: Effect of Driving and Dissipation Real Circuit QED architecture (flux qubits?) Effect of Disorder

33 Acknowledgements In collaboration with: Hakan Tureci, Mykola Bordyuh, Baris Oztop (Princeton, EE) David Huse (Princeton, Physics) Discussions with: Andrew Houck, D.Underwood, D. Sadri (Princeton) Takis Kontos(ENS), Nicolas Roch (Berkeley) K. Le Hur(Polytechnique)

34 Thanks!

35 Basics of Superconducting Circuits (I) H r = Transmission Line Resonator Z d 0 q 2 (x) 2 (x) dx + 2c 2l Superconducting-based to give dissipation-less currents Circuit Quantization M. Devoret, Les Houches (1995) X q(x) = A n (x) a n + a n n X (x) = i B n (x) a n a n n H r = X n n = n a n a n n d p lc

36 Qubits come in different flavors (charge, flux, phase qubits) Basics of Superconducting Circuits (II) Qubit: Mesoscopic SC system with atom-like spectrum The Josephson Junction is the crucial (non-linear) ingredient! Ex:The Cooper Pair Box H CPB =4E C (n n g ) 2 E J cos

37 Ultra-Strong Coupling of Light and Matter How large can be the coupling between atom and photon in circuit QED? M. Devoret, S. Girvin and R. Schoelkopf, Ann Phys (2007) Ultra-Strong Coupling Regime with Superconducting Circuits T. Niemczyk et al, Nat Phys (2010) Inductively Coupled Flux Qubit g/ r ' 0.12 Deviations from JC Physics Many Other Platforms: Quantum Dots, 2DEG,WG Superlattices,.. Todorov et al, PRL (2010), G. Scalari et al Science (2012), A.Crespi et al PRL (2012),.. Is the standard CQED picture working? S. Nigg et al, Phys. Rev. Lett (2012) P. Nataf & C. Ciuti, Nat. Comm (2010), O. Viehnmann et al, PRL (2011)

38 Polariton BEC in Semiconducting Micro-cavities Exp: J. Bloch, Yamamoto,D. Snoke... Theory: J.Keeling, C.Carusotto, C.Ciuti.. Non-Equilibrium Superfluidity Effect of Pump and Dissipation on the BEC? Vortices, Defects, Hydrodynamics

39 Disordered Arrays of Coupled Cavities H = X i ( r + i ) a i a i + X ij a t ij i a j + hc t ij =2Z 0 C ij ( r + i )( r + j ) D. Underwood, W. Shanks, J. Koch and A. Houck, PRA 86, (2012)

40 Quantum Impurity Physics with Light and Matter

41 Coupling a Microwave Resonator to Quantum Dot(s)+Leads M. Delbeq et al PRL (2011) (ENS-Paris, T. Kontos Lab) T. Frey et al PRL (2011) (ETH-Zurich, A. Wallraff Lab) Similar setup in Jason Petta experiment (Princeton)

42 Kondo Effect in the phase of the microwave signal? M. Delbeq et al PRL (2011) (ENS-Paris, T. Kontos Lab) Relation between phase of transmitted photon and current flowing into the dot Kondo Physics(?) - Dot and Resonator Capacitively Coupled (charge...rather than spin) But: is Light really quantum here? Many (thermal) photons in the cavity.. Coupling is global rather than local: AC Bias?

43 Phase Spectroscopy of a Resonator coupled to Dot+Leads H loc = U 2 (n 1)2 + 0 a a + gx(n 1)

44 Phase Spectroscopy of a Resonator coupled to Dot+Leads Input-Output Relations r( ) = = 1.0 d /dv tan ( ) = U=0 (Holstein Model) ev hv out L ( )i hvl in( )i =1 2iJ L( ) xx ( ) 0 = = = = ev R xx ( ) 2J L ( ) 1 2J L ( ) I xx ( ) Photon Propagator (knows about electrons) d ev Inelastic Features as the bias hits the photon frequency ev 0

45 Non-Equilibrium Effective Theory for Photon in the Resonator Can we integrate out the fermions? Z Z = Dx c Dx q exp (is eff [x c,x q ]) Yes, perturbatively at large bias ev 0 S eff = S loc [x c,x q ]+ 1 Z Z dt dt 0 2 X x (t) (t, t 0 ) x (t 0 ) =cl,q Fermions induce anharmonicity, friction and noise for the photon

46 Effective Potential and Photon Anharmonicity Z S loc [x c,x q ]= dt 2 x q ẍ c x c + F (x c ) 10 V eff (x) 5 0 λ = 0.25 λ = 5.25 V eff (x) = 2 0 x Z x 0 dx 0 F (x 0 ) V eff (x) ε 0 = 0.0 ε 0 = 1.0 ε 0 = 5.0 V eff (x) = 2 x x 3 + gx x Anisotropy and Anharmonicity controlled by gate/bias voltages

47 Non-Equilibrium Fermionic Environment At Finite Bias Noise and Friction do not satifsy Fluctuation- Dissipation Theorem... F ( ) = K ( ) R ( ) A ( ) 6= coth 2.. but low-frequency modes are effectively thermal! T eff (V ) F(Ω) ev = 0.1 ev = 0.5 ev = 2.0 ev = e ω 0 = 1.0 ω 0 = 2.0 ω 0 = 5.0 ω 0 = 7.5 T eff Ω δn ph ev W el = I*V T eff 100 T eff ~ V 4 1 T eff ~ V ev ev

48 Summing Up Biased Quantum Dot coupled to Resonator Phase Spectroscopy sensitive to Photon Green s function Non-Equilibrium Effective Theory for the Photon: Anharmonicity/friction/noise and Effective Temp Perspectives: Coulomb Blockade and Kondo Regime? Quantum-Classical Crossover with increasing Input Power?

49 Thanks!