Collective Dynamics of a Generalized Dicke Model J. Keeling, J. A. Mayoh, M. J. Bhaseen, B. D. Simons Harvard, January 2012 Funding: Jonathan Keeling Collective dynamics Harvard, January 2012 1 / 25 New relevance Superconducting qubits Quantum dots Nitrogen-vacancies in diamond Cavity κ κ Ultra-cold atoms Pump Pump Rydberg atoms Coupling many atoms to light Old question: What happens to radiation when many atoms interact “collectively” with light. Superradiance — dynamical and steady state. Jonathan Keeling Collective dynamics Harvard, January 2012 2 / 25 Cavity κ κ Pump Pump Coupling many atoms to light Old question: What happens to radiation when many atoms interact “collectively” with light. Superradiance — dynamical and steady state. New relevance Superconducting qubits Quantum dots Nitrogen-vacancies in diamond Ultra-cold atoms Rydberg atoms Jonathan Keeling Collective dynamics Harvard, January 2012 2 / 25 If jr − r j λ, use P S ! S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2 N/2 2 ΓN /2 z = j j = = /dt If S S N 2 initially: 〉 z 〉 S z z 2 0 〈 S d dhS i ΓN ΓN 〈 I / −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972] Dicke effect: Enhanced emission X y − −ik·ri Hint = gk k Si e + H.c. k;i Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 N/2 2 ΓN /2 z = j j = = /dt If S S N 2 initially: 〉 z 〉 S z z 2 0 〈 S d dhS i ΓN ΓN 〈 I / −Γ = sech2 t Γ dt 4 2 I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972] Dicke effect: Enhanced emission X y − −ik·ri Hint = gk k Si e + H.c. k;i If jr − r j λ, use P S ! S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2 Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 N/2 2 ΓN /2 /dt 〉 z 〉 S z 0 〈 S d 〈 Γ I=- -N/2 0 tD tD Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972] Dicke effect: Enhanced emission X y − −ik·ri Hint = gk k Si e + H.c. k;i If jr − r j λ, use P S ! S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2 If Sz = jSj = N=2 initially: dhSz i ΓN2 ΓN I / −Γ = sech2 t dt 4 2 Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 N/2 2 ΓN /2 /dt 〉 z 〉 S z 0 〈 S d 〈 Γ I=- -N/2 0 tD tD Dicke effect: Enhanced emission X y − −ik·ri Hint = gk k Si e + H.c. k;i If jr − r j λ, use P S ! S dρ Γ i j i i = − S+S−ρ − S−ρS+ + ρS+S− Collective decay: dt 2 If Sz = jSj = N=2 initially: dhSz i ΓN2 ΓN I / −Γ = sech2 t dt 4 2 Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972] Jonathan Keeling Collective dynamics Harvard, January 2012 3 / 25 If Sz = jSj = N=2 initially: 1600 __ __ 1600 T=2ln(√N)/√N N=2000 1400 1400 1200 1200 2 2 1000 1000 (t)| (t)| 800 800 ψ ψ | | 600 600 400 400 __ 200 1/√N 200 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Time Time Collective radiation with a cavity: Dynamics X y − + Single cavity mode: oscillations Hint = Si + Si i [Bonifacio and Preparata PRA ’70; JK PRA ’09] Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 1600 __ __ 1600 T=2ln(√N)/√N N=2000 1400 1400 1200 1200 2 2 1000 1000 (t)| (t)| 800 800 ψ ψ | | 600 600 400 400 __ 200 1/√N 200 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Time Time Collective radiation with a cavity: Dynamics X y − + Single cavity mode: oscillations Hint = Si + Si z i If S = jSj = N=2 initially: [Bonifacio and Preparata PRA ’70; JK PRA ’09] Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 1600 __ __ 1600 T=2ln(√N)/√N N=2000 1400 1400 1200 1200 2 2 1000 1000 (t)| (t)| 800 800 ψ ψ | | 600 600 400 400 __ 200 1/√N 200 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Time Time Collective radiation with a cavity: Dynamics X y − + Single cavity mode: oscillations Hint = Si + Si z i If S = jSj = N=2 initially: [Bonifacio and Preparata PRA ’70; JK PRA ’09] Jonathan Keeling Collective dynamics Harvard, January 2012 4 / 25 ω ⇓ SR 0 0 g-√N 2 Spontaneous polarisation if: Ng > !!0 Dicke model: Equilibrium superradiance transition y z y − + H = ! + !0S + g S + S . y + Coherent state: jΨi ! eλψ +ηS jΩi Energy: ! N jηj2 − 1 η∗λ + λ∗η E = !jλj2 + 0 + gN 2 jηj2 + 1 1 + jηj2 Small g, min at λ, η = 0 [Hepp, Lieb, Ann. Phys. ’73] Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 ω ⇓ SR 0 0 g-√N 2 Spontaneous polarisation if: Ng > !!0 Dicke model: Equilibrium superradiance transition y z y − + H = ! + !0S + g S + S . y + Coherent state: jΨi ! eλψ +ηS jΩi Energy: ! N jηj2 − 1 η∗λ + λ∗η E = !jλj2 + 0 + gN 2 jηj2 + 1 1 + jηj2 Small g, min at λ, η = 0 [Hepp, Lieb, Ann. Phys. ’73] Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 ω ⇓ SR 0 0 g-√N 2 Spontaneous polarisation if: Ng > !!0 Dicke model: Equilibrium superradiance transition y z y − + H = ! + !0S + g S + S . y + Coherent state: jΨi ! eλψ +ηS jΩi Energy: ! N jηj2 − 1 η∗λ + λ∗η E = !jλj2 + 0 + gN 2 jηj2 + 1 1 + jηj2 Small g, min at λ, η = 0 [Hepp, Lieb, Ann. Phys. ’73] Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 ω ⇓ SR 0 0 g-√N Dicke model: Equilibrium superradiance transition y z y − + H = ! + !0S + g S + S . y + Coherent state: jΨi ! eλψ +ηS jΩi Energy: ! N jηj2 − 1 η∗λ + λ∗η E = !jλj2 + 0 + gN 2 jηj2 + 1 1 + jηj2 Small g, min at λ, η = 0 2 Spontaneous polarisation if: Ng > !!0 [Hepp, Lieb, Ann. Phys. ’73] Jonathan Keeling Collective dynamics Harvard, January 2012 5 / 25 No go theorem:. Minimal coupling (p − eA)2=2m X e X A2 − A · p , g( yS− + S+); , Nζ( + y)2 m i 2m i i For large N, ! ! ! + 2Nζ. (RWA) 2 Need Ng > !0(! + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g =!0 < 2ζ. No transition No go theorem and transition 2 Spontaneous polarisation if: Ng > !!0 [Rzazewski et al PRL ’75] Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 For large N, ! ! ! + 2Nζ. (RWA) 2 Need Ng > !0(! + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g =!0 < 2ζ. No transition No go theorem and transition 2 Spontaneous polarisation if: Ng > !!0 No go theorem:. Minimal coupling (p − eA)2=2m X e X A2 − A · p , g( yS− + S+); , Nζ( + y)2 m i 2m i i [Rzazewski et al PRL ’75] Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 2 Need Ng > !0(! + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g =!0 < 2ζ. No transition No go theorem and transition 2 Spontaneous polarisation if: Ng > !!0 No go theorem:. Minimal coupling (p − eA)2=2m X e X A2 − A · p , g( yS− + S+); , Nζ( + y)2 m i 2m i i For large N, ! ! ! + 2Nζ. (RWA) [Rzazewski et al PRL ’75] Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 2 But Thomas-Reiche-Kuhn sum rule states: g =!0 < 2ζ. No transition No go theorem and transition 2 Spontaneous polarisation if: Ng > !!0 No go theorem:. Minimal coupling (p − eA)2=2m X e X A2 − A · p , g( yS− + S+); , Nζ( + y)2 m i 2m i i For large N, ! ! ! + 2Nζ. (RWA) 2 Need Ng > !0(! + 2Nζ). [Rzazewski et al PRL ’75] Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 No go theorem and transition 2 Spontaneous polarisation if: Ng > !!0 No go theorem:. Minimal coupling (p − eA)2=2m X e X A2 − A · p , g( yS− + S+); , Nζ( + y)2 m i 2m i i For large N, ! ! ! + 2Nζ. (RWA) 2 Need Ng > !0(! + 2Nζ). 2 But Thomas-Reiche-Kuhn sum rule states: g =!0 < 2ζ. No transition [Rzazewski et al PRL ’75] Jonathan Keeling Collective dynamics Harvard, January 2012 6 / 25 Cavity κ κ Pump Pump Dicke phase transition: ways out 2 Problem: g =!0 < 2ζ for intrinsic parameters. Solutions: Non-solution Ferroelectric transition in D · r gauge. [JK JPCM ’07 ] See also [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11] Grand canonical ensemble: z y I If H ! H − µ(S + ), need only: 2 g N > (! − µ)(!0 − µ) I Incoherent pumping — polariton condensation. Dissociate g;!0, e.g. Raman scheme: !0 !. [Dimer et al. PRA ’07; Baumann et al. Nature ’10. Also, Black et al. PRL ’03 ] Jonathan Keeling Collective dynamics Harvard, January 2012 7 / 25 Cavity κ κ Pump Pump Dicke phase transition: ways out 2 Problem: g =!0 < 2ζ for intrinsic parameters.