Introduction to non-perturbative cavity quantum electrodynamics - - PowerPoint PPT Presentation

Introduction to non-perturbative cavity quantum electrodynamics Simone De Liberato

Quantum Theory and Technology

Fundamental interactions

Mass of up quark: 2.3 MeV Mass of down quark: 4.8 MeV Mass of a proton: 938 MeV 99% of proton mass is due to interaction In light-matter interaction the dimensionless coupling constant is Strong interaction Electromagnetic interaction

α ' 1 137

Low order perturbation theory works well (photon absorption and emission) The interaction strength is much smaller than the bare frequency (virtual quark-gluon plasma)

ω0 ΩR

Ultrastrong coupling

Ultrastrong light-matter coupling regime: non negligible

η = ΩR/ω0

Ultrastrong coupling between light and matter,

• A. F. Kockum et al., Nat. Phys. Rev. 1, 19 (2019)

Non-perturbative CQED phenomenology

• 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
• …

Temperature Pressure Chemical composition EM vacuum fluctuations

From weak to the ultrastrong

Purcell effect

Photonic density of states Free space: Cavity:

ω ρ(ω) ω ρ(ω)

Enhancement Suppression

|g |e ω0 Γsp = 2π ~ |i|Hint|f⇥|2ρ(~ω0) ω0 Γsp Γsp = !3

0d2 ge

3⇡✏0~c3

Strong coupling (time domain)

If the emitted photons is trapped long enough to be reabsorbed

ΩR ΩR ΩR Γ

Γ Γ

Fermi golden rule: first order perturbation. It cannot account for higher order processes, i.e. reabsorption. Valid if ΩR < Γ

ΩR > Γ ΩR ω0 |g |e

Strong coupling (frequency domain)

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

Γ ΩR > Γ |1, g |0, e |1, ⇥ |1, + 2ΩR ω0 ω0

|1, +i |2, +i |2, i |0, gi |1, gi |0, ei |2, gi |1, ei

|1, i Γ Γ ΩR ω0 |g |e

The Polariton 0 = +

p† x y

Half light and half matter excitation Matter excitation Cavity photon Lower polartion Upper polariton

q/qres

2ΩR

ω/ω0

Modes that are: easy to excite and observe interact strongly

Perturbation theory

Let us do perturbation using the full Hamiltonian First order perturbation: Second order perturbation: ∆E(2)

φ

= X

|ψ⇥=|φ⇥

|⇥φ|Hint|ψ⇤|2 Eφ Eψ H0 Hint

Higher-order effects are observable when is non negligible

∝ ω

Ultrastrong coupling regime

∆E(1)

φ

∝ ΩR ∝ Ω2

R

∝ Ω2

R

ω0 HQRM = ~ω0a†a + ~ω0|eihe| + ~ΩR(a + a†)(|eihg|+|gihe|) = ΩR × ΩR ω0 ΩR ω0

Coupling regimes

WR

Weak coupling Strong coupling Ultrastrong coupling

Fermi Golden rule Dressed states New physics

ω0 Γ

Is ultrastrong coupling possible?

En = −Ry n2 ˜ V = V (λ/2)3

Hydrogen atom Dimensionless volume We end up with

λ = 2πc ω0

Wavelength

ΩR ω0 = α3/2 nπ p ˜ V

• M. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys. 16, 767 (2007)

Coupling Overlap

• Reducing
• Increasing the number of dipoles
• Coupling to currents ( )

Three ways to ultrastrong coupling

α−1/2 ˜ V

Reducing the mode volume

Mode confinement: smaller cavity = larger coupling

ΩR ∝ 1 √ V

N dipoles of length d 1 dipole of length Collective coupling: more dipoles = larger coupling

ΩR ∝ √ N

Virtual photons & Decoupling

The coupled ground state

|Gi hG|a†a|Gi / Ω2

R

ω2 + O(Ω4

R

ω4 ) |0i

Energy Excited states

ΩR ω0

Renormalised excited states

|Gi

0.2 0.4 0.6 0.8 1 0.01 0.02 0.03 0.04 0.05 0.06 0.6 0.8 1 1.2 1.4 0.2 0.4

The coupled ground state has a population of virtual photons Stable against losses

• S. De Liberato

Nature Communications 8, 1465 (2017)

Open quantum systems

nout = Γha†ai

Emission of photons out of the ground state. Wrong!

ΩR Γ Γ

Except that: Number of photons inside the cavity Escape rate

hG|a†a|Gi / Ω2

R

ω2 + O(Ω4

R

ω4 )

• 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)

L(ρ) = Γ 2 (2aρa† − a†aρ − ρa†a)

This is also wrong These are white baths

nout = Γha†ai

No negative frequency modes All a baths are colored

Coupled vacuum Nonadiabatic quantum dynamics Standard vacuum Time t trelax

Quantum vacuum emission

Photon emission

Free system: Coupled oscillators: The coupling changes the ground state

|G |0

Standard vacuum Coupled vacuum

• 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

If the last term, always positive, becomes dominant

ΩR ω0 > 1

H = Hfield + p2 2m + V (r) − epA(r) m + e2A(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 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 The wall becomes a metallic mirror

• S. De Liberato, Phys. Rev. Lett. 112, 016401 (2014)

η = 0 η = 1 η = 2

In the non perturbative regime the Purcell effect fails Ultimate limit to switchingfrequency Observed and exploited in microcavity fabrication

Thank you for your attention