Introduction to non-perturbative cavity quantum electrodynamics Simone De Liberato Quantum Theory and Technology
Fundamental interactions Strong interaction Mass of up quark: 2.3 MeV 99% of proton mass is due to Mass of down quark: 4.8 MeV interaction Mass of a proton: 938 MeV (virtual quark-gluon plasma) Electromagnetic interaction 1 In light-matter interaction the dimensionless coupling constant is α ' 137 Low order perturbation theory works well (photon absorption and emission) The interaction strength is much smaller than the bare frequency Ω R ω 0
Ultrastrong coupling Ultrastrong light-matter coupling regime: non negligible η = Ω R / ω 0 Ultrastrong coupling between light and matter, A. F. Kockum et al. , Nat. Phys. Rev. 1 , 19 (2019)
Non-perturbative CQED phenomenology Temperature Pressure Chemical composition EM vacuum fluctuations • Quantum phase transitions • Quantum vacuum radiation • Topologically protected ground states • Increase in electrical conductivity • Modified electroluminescent properties • Change in chemical properties • Change in structural molecular properties • Modified lasing • Vacuum nonlinear processes • …
From weak to the ultrastrong
Purcell effect | e � Γ sp = 2 π ~ | � i | H int | f ⇥ | 2 ρ ( ~ ω 0 ) Γ sp ω 0 Photonic density of states | g � ! 3 0 d 2 ρ ( ω ) Free space: ge Γ sp = 3 ⇡✏ 0 ~ c 3 ω ω 0 Enhancement ρ ( ω ) Cavity: Suppression ω
Strong coupling (time domain) Fermi golden rule: first order perturbation. Ω R | e � It cannot account for higher order processes, Γ Γ ω 0 i.e. reabsorption. Valid if Ω R < Γ | g � If the emitted photons is trapped Ω R > Γ long enough to be reabsorbed Ω R Ω R Ω R Γ
Strong coupling (frequency domain) The coupling splits the degenerate levels, Ω R | e � creating the Jaynes-Cummings ladder. Γ Γ ω 0 | g � The losses give the resonances a finite width Strong coupling: Ω R > Γ | 2 , + i | 1 , e i Condition to spectroscopically resolve the resonant splitting. | 2 , g i | 2 , �i ω 0 In the strong coupling regime we cannot | 1 , + i consider transitions between uncoupled | 0 , e i Γ 2 Ω R modes, e.g. , . | 0 , e � | 1 , g � | 1 , g i | 1 , �i ω 0 We are obliged to consider the dressed | 0 , g i states, , , etc… | 1 , + � | 1 , �⇥
The Polariton p † 0 = + y x Half light and half matter excitation easy to excite and observe Modes that are: interact strongly Upper polariton ω / ω 0 Cavity Matter 2 Ω R photon excitation Lower polartion q/q res
Perturbation theory Let us do perturbation using the full Hamiltonian H QRM = ~ ω 0 a † a + ~ ω 0 | e ih e | + ~ Ω R ( a + a † )( | e ih g | + | g ih e | ) H int H 0 ∆ E (1) First order perturbation: ∝ Ω 2 ∝ Ω R R φ ∝ Ω 2 = Ω R × Ω R | ⇥ φ | H int | ψ ⇤ | 2 R Second order perturbation: ∆ E (2) X = φ ω 0 E φ � E ψ ω 0 | ψ ⇥� = | φ ⇥ ∝ ω 0 Ω R Higher-order effects are observable when is non negligible ω 0 Ultrastrong coupling regime
Coupling regimes Fermi Golden rule Dressed states New physics Strong Ultrastrong Weak coupling coupling coupling W R 0 Γ ω 0
Is ultrastrong coupling possible? E n = − Ry Hydrogen atom n 2 λ = 2 π c Wavelength ω 0 V Dimensionless volume ˜ V = ( λ / 2) 3 Coupling α 3 / 2 Ω R We end up with = p Overlap ˜ ω 0 n π V • Reducing ˜ V Three ways to ultrastrong • Increasing the number of dipoles coupling • Coupling to currents ( ) α − 1 / 2 M. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys. 16 , 767 (2007)
Reducing the mode volume Mode confinement: smaller cavity = larger coupling 1 Ω R ∝ √ V Collective coupling: more dipoles = larger coupling √ Ω R ∝ N 1 dipole of length N dipoles of length d
Virtual photons & Decoupling
The coupled ground state Renormalised Excited states excited states Energy ω 0 Ω R | 0 i | G i The coupled ground state has a population of virtual photons | G i 0.06 h G | a † a | G i / Ω 2 + O ( Ω 4 0.4 R R ) 0.05 ω 2 ω 4 0 0 0.2 0.04 Stable against losses 0.03 0 0.6 0.8 1 1.2 1.4 0.02 S. De Liberato 0.01 Nature Communications 8 , 1465 (2017) 0 0 0.2 0.4 0.6 0.8 1
Open quantum systems Number of photons inside the cavity Ω R Γ Γ n out = Γ h a † a i Escape rate h G | a † a | G i / Ω 2 + O ( Ω 4 Except that: R R ) ω 2 ω 4 0 0 Emission of photons out of the ground state. Wrong! L ( ρ ) = Γ This is also wrong 2 (2 a ρ a † − a † a ρ − ρ a † a ) n out = Γ h a † a i No negative frequency modes All a baths are colored These are white baths S. De Liberato, D. Gerace, I. Carusotto, and C. Ciuti, Phys. Rev. A 80 , 053810 (2009) S. De Liberato, Phys. Rev. A 89 , 017801 (2014)
Quantum vacuum emission The coupling changes the ground state | 0 � Free system: Standard vacuum Coupled oscillators: Coupled vacuum | G � Coupled vacuum Nonadiabatic quantum dynamics Standard vacuum t relax 0 Time t Photon emission S. De Liberato, C. Ciuti and I. Carusotto, Phys. Rev. Lett. 98 , 103602 (2007) G. Guenter et al., Nature 458 , 178 (2009)
Purcell effect breakdown + e 2 A ( r ) 2 H = H field + p 2 2 m + V ( r ) − e pA ( r ) 2 m m Intensity of the field at the location of the dipoles Ω R If the last term, always positive , becomes dominant > 1 ω 0 The low energy modes need to minimize the field location over the dipoles Light and matter decouple in the deep strong coupling regime S. De Liberato, Phys. Rev. Lett. 112 , 016401 (2014)
Purcell effect breakdown Example: a two-dimansional metallic cavity enclosing a wall of in-plane dipoles η = 1 η = 2 η = 0 The wall becomes a metallic mirror In the non perturbative regime the Purcell effect fails Ultimate limit to switchingfrequency Observed and exploited in microcavity fabrication S. De Liberato, Phys. Rev. Lett. 112 , 016401 (2014)
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