Light-matter coupling: from the weak to the ultrastrong coupling and beyond. Simone De Liberato Quantum Light and Matter Group

Transcription

1 Light-matter coupling: from the weak to the ultrastrong coupling and beyond Simone De Liberato Quantum Light and Matter Group

2 General theory Open quantum systems Material implementations Advanced topics

3 General theory

4 Light-matter coupling: free space e Γ sp Emission rate with Fermi golden rule: g Γ sp = ω3 0d 2 ge 3π 0 c 3 We can derive such a formula from the minimal coupling Hamiltonian: H = H field + (p ea(r))2 2m + V (r) H = H field + p2 2m epa(r) + V (r) m + e2 A(r) 2 2m H 0 H int

5 Light-matter coupling: free space Fermi golden rule: Initial state: Γ sp = 2π i H int f 2 ρ( )= i = e 0 ω3 0d 2 ge 3π 0 c 3 Final state: f = g 1 Interaction Hamiltonian: H int = (a + a)( eg + ge )+ Ω2 R (a + a)(a + a) ω 0 d 2 ge With = Vacuum Rabi frequency 2 0 V Density of photonic states: ρ( )= V ω2 0 3π 2 c 3

6 Rotating wave approximation Antiresonant terms Connect states whose energy difference is Do not contribute in the Fermi golden rule 2 H int = (a + a)( eg + ge )+ Ω2 R (a + a)(a + a) Resonant terms Connect states whose energy difference is Do contribute in the Fermi golden rule We can thus consider the simpler Hamiltonian: Hint RWA = (a ge +a eg )+ 2Ω2 R a a 0 Emission Absorption Photon renormalisation

7 Jaynes-Cummings model H JC = ω c a a + ee + (a ge +a eg ) n, g = n, e = n n, g and n 1, eform a closed subspace: HJC n = ω c nωr nωr whose eigenvalues are n, and n, +, split at resonance of 2 n 1,e 2,g 0,e 1,g 0,g 2, + 2, 1, + 2 1,

8 e Purcell effect Γ sp = 2π i H int f 2 ρ( ) Γ sp g Photonic density of states Free space: ρ(ω) Γ sp ω 3 0 Cavity: ρ(ω) ω Enhancement Suppression ω

9 Purcell effect The enhancement is given by the ratio between the densities of states in free space and inside the cavity F P = 3 4π 2 ( λ n )3 ( Q V eff ) Quality factor of the cavity Volume of the cavity Mode confinement: smaller cavity = larger coupling 1 Veff Importance of sub-wavelength confinement!

10 Toward strong coupling Fermi golden rule: first order perturbation. Γ Γ It cannot account for higher order processes, i.e. reabsorption. Valid if < Γ If > Γ the emitted photons is trapped long enough to be reabsorbed Γ

11 Strong coupling Γ Γ The coupling splits the degenerate levels, creating the Jaynes-Cummings ladder. The losses give the resonances a finite width 1,e 2,g 2, + 2, Strong coupling: > Γ Condition to spectroscopically resolve the resonant splitting. 0,e 1,g 0,g Γ 1, + 2 1, In the strong coupling regime we cannot consider transitions between uncoupled modes, e.g., 0,e 1,g. We are obliged to consider the dressed states, 1,, 1, +, etc

12 The Dicke model N 1 two level systems in a cavity H Dicke = ω c a a + N e j e j + (a g j e j +a e j g j ) j=1 All the two level systems couple to the same photonic field We can introduce coherent excitation operators State with n systems in the excited state b = 1 N N j=1 g j e j n [b, b ] n =1 2n N Bosons in the limit N n

13 Bosons for real N 1 distinguishable two level systems Partition function: If they are indistinguishable instead: Z = N m=0 Z =(1+e β ) N = N e mβ = m N m=0 e mβ N m=0 N e mβ m 1 1 e β Partition function of a bosonic field With n photons we cannot distinguish a n level system from a bosonic one

14 From the Dicke model: The Dicke model H int = We can then substitute the coherent operators: N (a g j e j +a e j g j ) j=1 b = 1 N N j=1 g j e j Obtaining: H int = N (a b + b a) Superradiance: more dipoles = larger coupling N N dipoles of length d 1 dipole of length R. H. Dicke, Phys. Rev. 93, 99 (1954)

15 The Polariton The full light-matter Hamiltonian has the form: H Dicke = ω c a a + b b + (a b + b a) Introducing the polaritonic operators: p j = x j a + y j b, j [LP,UP] With x j 2 + y j 2 =1, in order to have We can diagonalise the Hamiltonian as: [p j,p i ]=δ i,j H = ω j p j p j j [LP,UP] A linear system of N interacting bosonic fields is (almost) always equivalent to a system of N non-interacting bosonic fields J. J. Hopfield, Phys. Rev. 112, 1555 (1958)

16 The Polariton p 0 = x + y Half light and half matter excitation Modes that are: easy to excite and observe interact strongly Upper polariton ω/ω0 Cavity photon 2 Matter excitation Lower polartion q/q res

17 Ultrastrong coupling!! > Γ We can have either Large Small (strong coupling) (good cavity) Is the relevant small parameter in perturbation theory It measures the intrinsic strength of the transition Γ Weak coupling When 1 we can expect non-perturbative effects Ultrastrong coupling regime ( > 0.1 ) Fermi Golden rule Polaritons New physics Γ Strong coupling Ultrastrong coupling! "! 0" Loss rate Transition frequency

18 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

19 Ultrastrong coupling Let us consider the bosonised light-matter Hamiltonian without the rotating wave approximation:!! H = ω c a a + b b + (a + a)(b + b)+ Ω2 R (a + a) 2 H 0 H int First order perturbation: E (1) φ The second order contribution is due to antiresonant terms ( ab, a b, etc ) Second order perturbation: E (2) φ = ψ= φ φ H int ψ 2 E φ E ψ Ω 2 R Ω2 R In the ultrastrong coupling regime the antiresonant terms are not negligible!

20 Ultrastrong coupling Lower polariton RWA Upper polariton RWA Lower polariton Upper polariton!! ω

21 Root mean square error (mev) Energy (mev) (a) ! (mev) R (b) LP Angle (deg) Energy (mev) Ultrastrong coupling Data fitted with different Hamiltonians, with (c) UP Angle (deg) H int = (a + a)(b + b) H RWA int as only free parameter + Ω2 R (a + a) 2 = (a b + b a)+ 2Ω2 R H RWA int = (a b + b a) The best fit gives: =0.11 First observation of the ultrastrong coupling regime A. Anappara et al., Phys. Rev. B 79, (R) (2009) a a

22 j [LP,UP] Ultrastrong coupling We have thus to consider the full light-matter Hamiltonian: H = ω c a a + b b + (a + a)(b + b)+ Ω2 R (a + a) 2 Can we put it in diagonal form? H = ω j p j p j The previous transformation: p j = x j a + y j b is not enough, as we cannot generate the antiresonant terms multiplying and p j We need instead a transformation that mixes creation and annihilation operators: p j = x j a + y j b + z j a + w j b In order to have [p j,p i ]=δ x j 2 i,j, the coefficients have to respect the normalisation condition + y j 2 z j 2 w j 2 =1 The minuses imply that the coefficients are not bounded! p j

23 Ultrastrong coupling The ground state is the state annihilated by the annihilation operators We call 0 the ground state of the uncoupled light-matter system a 0 = b 0 =0 From the decomposition p j 0 =0 p j = x j a + y j b + z j a + w j b The coupling modifies the ground state We introduce the ground state of the coupled system G p j G =0 We have then G a a G = z LP 2 + z UP 2 =0 Ω2 R ω O( Ω3 R ω 3 0 ) The ground state contains a population of bound photons

24 What about the Purcell effect? = Purcell factor S. De Liberato and C. Ciuti, Phys. Rev. B 77, (2008) C. Ciuti and I. Carusotto, Phys. Rev. A 74, (2006) Strong coupling regime: quadratic dependency upon Ultrastrong coupling regime: saturation

25 and beyond What happens if > 1? The coupling becomes completely non-perturbative It has been called deep strong coupling regime Fermi Golden rule Dressed states New physics!! 0" #$%&! '()*+,-.! Γ /01(-.! '()*+,-.! 2+01%301(-.! '()*+,-.! 4$$*!301(-.! '()*+,-.!! "! Do we expect qualitatively new phenomena? J. Casanova, et al., Phys. Rev. Lett. 105, (2010)

26 and beyond If > 1 the last term, always positive, becomes dominant!! H = ω c a a + b b + Ω(a + a)(b + b) + Ω2 (a + a) 2 H = H field + p2 2m epa(r) + V (r) m + e2 A(r) 2 2m Intensity of the field at the location of the dipoles The low energy modes need to minimize the field location over the dipoles Polariton modes will be pure photon modes that avoid the dipoles pure matter mode Light and matter decouple in the deep strong coupling regime

27 and beyond!! The photonic field avoids the dipoles Light-matter interaction is due to local interactions Light and matter do not exchange energy

28 An example!! Example: a two-dimansional metallic cavity enclosing a wall of in-plane dipoles The wall becomes a metallic mirror

29 What about the Purcell effect? = Purcell factor C. Ciuti and I. Carusotto, Phys. Rev. A 74, (2006) S. De Liberato, arxiv: Breakdown of the Purcell effect!

30 Light-matter coupling Fermi Golden rule Dressed states New physics 0" #$%&! '()*+,-.!!! Γ /01(-.! '()*+,-.! 2+01%301(-.! '()*+,-.! 4$$*!301(-.! '()*+,-.!! "! J. Casanova, et al., Phys. Rev. Lett. 105, (2010)

31 Light-matter decoupling 0"! "! 0.1" Γ

32 Open quantum systems

33 Open quantum systems a In order to calculate the system emission we have to couple it to the external world α j Extra-cavity photonic modes H SE = j ω j α j α j + κ(a α j + α j a) (plus eventually another bath coupled to the matter mode b ) We can then deploy all the arsenal of the theory of open quantum systems In the ultra and deep strong coupling regime we need to pay attention!

34 Open quantum systems a Γ The system-environment coupling gives a finite lifetime to cavity photons H SE = j ω j α j α j + κ(a α j + α j a) Emission rate per photon Number of emitted photons: n out = Γa a Number of photons in the cavity Except that: G a a G = z LP 2 + z UP 2 =0 Ω2 R ω O( Ω3 R ω 3 0 ) Emission of photons out of the ground state. Wrong!

35 Open quantum systems Master equation: ρ t = i[h, ρ]+l(ρ) Wrong! Lindblad operator: L(ρ) = κ 2 (2aρa a aρ ρa a) Integral Lindbad operator: L(ρ) = κ 2 (Uρa + aρu a Uρ ρu a) U = 0 dtg(t)e iht ae iht In order to obtain the simplified version: e iht ae iht ae iω ct Implying: U = 0 dtg(t)e iht ae iht = g(ω c )a False in the ultra and deep strong coupling S. De Liberato et al, Phys. Rev. A 80, (2009) F. Beaudoin, J. M. Gambetta, and A. Blais, Phys. Rev. A 84, (2011)

36 Open quantum systems We considered the photons as fundamental excitations, and coupled them to the external world. But we can also do the opposite. 1) We start from the system-environment Hamiltonian H SE = j ω j α j α j + κ(a α j + α j a) 2) We express the photon operators as a function of the polaritons ones a = x LP p LP + x UP p UP z LP p LP z UPp UP 3) We obtain a polaritonic system-environment Hamiltonian H SE = j ω j α j α j + κ(+x LP α j p LP + x UPα j p UP z LP α j p LP z UP α j p UP + x LP α j p LP Antiresonant term +x UP α j p UP z LP α j p LP z UPα j p UP ) Resonant term

37 Open quantum systems Problem: H SE G S 0 E = 0 Due to the antiresonant terms in the system-environment coupling the product of the system and environment ground state is not the total ground state We have to apply the rotating wave approximation Naïve way H SE = j ω j α j α j + κ(+x LP α j p LP + x UPα j p UP z LP α j p LP z UP α j p UP + x LP α j p LP +x UP α j p UP z LP α j p LP z UPα j p UP )

38 Open quantum systems H RWA SE = j ω j α j α j + κx LP (p LP α j + α j p LP)+κx UP (p UP α j + α j p UP) 10 1 κ 10 2 κ = κx UP UP The loss rate of the upper polariton is much larger than the loss rate of a photon κ = κx LP LP How can a the coupling with matter increase the mirror losses?

39 Open quantum systems H RWA SE = j ω j α j α j + κx LP (p LP α j + α j p LP)+κx UP (p UP α j + α j p UP) κ κ = κx UP UP An exact microscopic calculation, using Maxwell boundary conditions gives different results M. Bamba and T. Ogawa, Phys. Rev. A 88, (2013) 10 3 κ = κx LP LP Is the usual open quantum system approach flawed?

40 Open quantum systems We are using an Hopfield Bogoliubov transformation that mixes creation and annihilation operators a = x LP p LP + x UP p UP z LP p LP z UPp UP That needs to be normalised [a, a ] x LP 2 + x UP 2 z LP 2 z UP 2 =1 Doing the rotating wave approximation we are instead considering x LP 2 + x UP 2 z LP 2 z UP 2 > 1 The operators are non-normalised H SE We are removing the non-resonant terms of, while we should remove the ones of H. As a consequence H SE would have only resonant terms.

41 Open quantum systems H RWA SE = j ω j α j α j + κx LP (p LP α j + α j p LP)+κx UP (p UP α j + α j p UP) x 2 LP + x 2 UP x 2 LP + x 2 UP κ κ = κx UP x 2 LP + x 2 UP UP Once the coupling is renormalised, we recover the same result obtained from Maxwell boundary conditions S. De Liberato, arxiv: κ = κx LP x 2 LP + x 2 UP LP

42 Material implementations

43 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

44 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

45 Flux qubit Flux qubit: a superconducting ring in which persistent currents can flow in both directions I -I 0 1

46 Circuit CQED Capacitors Flux qubit T. Niemczyk et al., Nat. Phys. 6, 772 (2010) One single dipole coupled to the electromagnetic field Jaines-Cummings modes, not Dicke model =0.12

47 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

48 Switch on

49 Organic molecules Largest observed splitting, =0.16 T. Schwartz et al., Phys. Rev. Lett. 106, (2011)

50 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

51 Doped quantum well Conduction band Subband dispersions Fermi level AlGaAs GaAs AlGaAs Valence Nanometric quantum confinement z In-plane wavevector

52 Excitations with Tunable Energy GaAs!" 12!150meV Mid-infrared nm 2DEG

53 Excitations with Tunable Energy GaAs!" 12!15meV Far-infrared nm

54 Microcavity Planar structure that confines photons Duble metal configuration Metal-air configuration!!

55 Intersubband polaritons ω/ω12 q/q res Upper Polariton Microcavity photon 2 Intersubband transition D. Dini et al., Phys. Rev. Lett. 90, (2003) Lower polartion

56 Enhanced coupling Superradiant enhancement N 2DEG Sub-wavelength confinement V eff λ 3 < 10 6 C. Feuillet-Palma et al., Opt. Exp. 20, (2012).

57 First observation A. Anappara et al., Phys. Rev. B 79, (R) (2009)! R! 12 = 0.11 Y. Todorov et al., Phys. Rev. Lett. 105, (2010) " R # 12 = 0.24

58 Comparison with other systems Atoms in superconducting cavities Superconducting circuits (2010) Organic molecules (2011) < =0.12 =0.16 Higher density superradiance Intrinsically larger dipoles Better confinement Intersubband polaritons (2009) #!! "!! =0.24 Landau polaritons (2012)! =0.58

59 A naïf idea $! % &! % &! Rydberg atom r = n2 2 e 2 m Cyclotron orbit r = mv eb

60 A more realistic description ανn QW 1 B D. Hagenmüller, S. De Liberato, and C. Ciuti, PRB 81, (2010)

61 Sub-wavelength confinement

62 Experimental observation Cyclotron transition Photonic mode G. Scalari et al., Science 335, 1323 (2012) =0.58 ν(1.2t ) = 15

63 Advanced topics

64 Dynamical Casimir effect A mirror accelerated in vacuum emits photons (due to friction with vacuum fluctuations)!!

65 Dynamical Casimir effect A mirror, accelerated in the vacuum, emits photons L(t) We need the oscillation frequency comparable with the photon one Impossible for a mechanic spring! Or, we can keep the length fixed, and change the dielectric constant n(t) L opt =n(t)l No moving parts!

66 Ultrastrong coupling The ground state is the state annihilated by the annihilation operators We call 0 the ground state of the uncoupled light-matter system a 0 = b 0 =0 From the decomposition p j 0 =0 p j = x j a + y j b + z j a + w j b The coupling modifies the ground state We introduce the ground state of the coupled system G p j G =0 We have then G a a G = z LP 2 + z UP 2 =0 Ω2 R ω O( Ω3 R ω 3 0 ) The ground state contains a population of bound photons

67 Quantum vacuum emission The coupling changes the ground state Free system: 0 Standard vacuum Coupled oscillators: G Coupled vacuum Coupled vacuum Nonadiabatic quantum dynamics Standard vacuum '! # (%)*+! Time t Photon emission G a a G = z z 2 2 Ω2 ω 2 + O(Ω3 ω 3 ) S. De Liberato, C. Ciuti and I. Carusotto, Phys. Rev. Lett. 98, (2007)

68 Nonadiabatic modulation Reflected THz field amplitude (a.u Energy (mev) t D = -200 fs -150 fs -100 fs -75 fs -50 fs -25 fs 0 fs 25 fs 50 fs 75 fs 100 fs 150 fs 200 fs G. Guenter et al., Nature 458, 178 (2009)

69 Nonadiabatic modulation Reflected THz field amplitude (a.u Energy (mev) t D = -200 fs -150 fs -100 fs -75 fs -50 fs -25 fs 0 fs 25 fs 50 fs 75 fs 100 fs 150 fs 200 fs G. Guenter et al., Nature 458, 178 (2009)

70 First observation C. M. Wilson et al., Nature 479, 376 (2011)

71 Beyond the Dicke model b q a q b q a q These systems are modeled as Dicke models.

72 The Hilbert space is a large place d Dicke =2 N d TBM = 2N N 4N Nπ A flat band model is not a Dicke model! One mode approximation is justified only in the linear regime!

73 Quantum Phase Transitions Toward flat photonic dispersions Many modes resonant with the transition Toward steep photonic dispersions Few modes resonant with the transition S. De Liberato and C. Ciuti, Phys. Rev. Lett. 110, (2013)

74 Thank you for your attention