Modulated superradiance and synchronized chaos in clouds of atoms in a bad cavity

Transcription

1 Modulated superradiance and synchronized chaos in clouds of atoms in a bad cavity Emil Yuzbashyan Collaborators: Aniket Patra (Rutgers) and Boris Altshuler (Columbia) International orkshop Dissipative Quantum Chaos: from Semi-Groups to QED Experiments, Daejeon, South Korea, October 3-7, 017

2 Ultra-cold atoms coupled to a bad cavity: superradiant laser ) Raman superradiant laser (c) Experimental optical cavity e g d eg i MOT Location Bohnet et. al., EPJ eb of Conferences, v. 57, cm Optical Cavity Mirrors Ultra-stable, sub-milihertz linewidth optical laser (quality factor ~ ) Ultra-high precision atomic clock Extreme bad cavity limit (cavity photon decay rate largest rate) N ~10 6 of 87 Rb atoms Atom + Cavity Hamiltonian in the rotating frame of the cavity field: Ĥ = Ŝz + d Collective spin: Ŝ z = 1 â Ŝ + Ŝ+â NX ˆzj, Ŝ ± = 1 j=1 NX j=1 ˆ± j Cavity mode: â, â Detuning from the cavity mode:

3 Two (or more) clouds in the cavity: synchronization of atomic clocks Xu et. al., PRL 113, , 014 Christiaan Huygens an odd kind of sympathy in maritime clocks Letters to de Sluse, 1665 Oliveira & Melo, Scientific Reports 5:11548, 015

4 Two (or more) clouds in the cavity: synchronization of atomic clocks Xu et. al., PRL 113, , 014 Two collective spins: Ŝz A,B = 1 NX ˆA,B jz, Ŝ A,B ± = Ĥ = ŜA z j=1 Ŝ B z + NX j=1 ˆA,B j± Christiaan Huygens an odd kind of sympathy in maritime clocks Letters to de Sluse, 1665 â Ŝ A + ŜA +â +â Ŝ B + ŜB + â (Can always set! A +! B =0=)! A,B = ± /) Oliveira & Melo, Scientific Reports 5:11548, 015

5 Rich collective dynamics much beyond synchronization of atomic clocks synchronization (fixed point) δ Two types of limit cycles Chaos via quasiperiodicity Synchronized chaos Regions of coexistence limit cycle (periodically oscillating superradiance) Nonequilibrium phase diagram for two ensembles of atoms in a bad cavity (mean-field) pumping, d detuning from the cavity mode d

6 Master equation = i hŝa z Ŝz B, i J = S A + S B Lindblad super-operators: i h â Ĵ L[Ô] = 1 Cavity intensity decay rate Pump rate i + Ĵ+â, + applel[â] + X NX Ô Ô Ô Ô Ô Ô =A,B j=1 L[ˆ j+]

7 Master equation = i hŝa z Ŝz B, i J = S A + S B Lindblad super-operators: i h â Ĵ L[Ô] = 1 Cavity intensity decay rate Pump rate i + Ĵ+â, + applel[â] + X NX Ô Ô Ô Ô Ô Ô =A,B j=1 L[ˆ j+] Adiabatic elimination: (bad cavity limit) dhâi dt = apple hâi i hĵ i =) â i apple Ĵ, apple 1 Bonifacio et. al., Phys. Rev. A (1971)

8 Master equation = i hŝa z Ŝz B, i J = S A + S B Lindblad super-operators: i h â Ĵ L[Ô] = 1 Cavity intensity decay rate Pump rate i + Ĵ+â, + applel[â] + X NX Ô Ô Ô Ô Ô Ô =A,B j=1 L[ˆ j+] Adiabatic elimination: (bad cavity limit) dhâi dt = apple hâi i hĵ i =) â i apple Ĵ, apple 1 Bonifacio et. al., Phys. Rev. A (1971) = i Effective atomic master equation: hŝa z Ŝ B z, i + c L[Ĵ ] + X Collective decay rate, c = apple =A,B NX j=1 L[ˆ j+]

9 System size expansion = i hŝa z Ŝ B z, i + c L[Ĵ ] + X NX L[ˆ j+] Set N c =1 =A,B j=1 Carmichael: Statistical Methods in Quantum Optics 1 - Master Equations and Fokker-Planck Equations (1999)

10 System size expansion = i hŝa z Ŝ B z, i + c L[Ĵ ] + X NX L[ˆ j+] Set N c =1 =A,B j=1 Carmichael: Statistical Methods in Quantum Optics 1 - Master Equations and Fokker-Planck Equations (1999) Define the characteristic function and the associated quasiprobability distribution N ( A, A, A, B, B, B ) tr h e ı AŜA + e ı A Ŝ A z e ı A Ŝ A e ı BŜB + e ı B Ŝ B z e ı B Ŝ Bi Z Z Z Z d v A dm A d v B dm B P (va,v A,m A,vB,v B,m B ) 1 Expand in p (system size expansion) N e ı A v A e ı A v A e ı Am A e ı B v B e ı B v B e ı Bm B Glauber Sudarshan P representation, PRL 10, 77; Phys. Rev. 131 (1963)

11 0 th order semiclassical equations of motion Ensemble A: ṡ A x = s A y sa x + 1 sa z J x ṡ A y = s A x sa y + 1 sa z J y ṡ A z = (1 s A 1 1 z ) sa x J x sa y J y Ensemble B: ṡ B x = s B y sb x + 1 sb z J x ṡ B y = s B x sb y + 1 sb z J y ṡ B z = (1 s B 1 1 z ) sb x J x sb y J y s = N hŝi, J = sa + s B, N c =1

12 0 th order semiclassical equations of motion Ensemble A: ṡ A x = s A y sa x + 1 sa z J x ṡ A y = s A x sa y + 1 sa z J y ṡ A z = (1 s A 1 1 z ) sa x J x sa y J y Ensemble B: ṡ B x = s B y sb x + 1 sb z J x ṡ B y = s B x sb y + 1 sb z J y ṡ B z = (1 s B 1 1 z ) sb x J x sb y J y s = N hŝi, J = sa + s B, N c =1 ṡ = s H s (s H) H = ± ẑ

13 0 th order semiclassical equations of motion Ensemble A: ṡ A x = s A y sa x + 1 sa z J x ṡ A y = s A x sa y + 1 sa z J y ṡ A z = (1 s A 1 1 z ) sa x J x sa y J y Ensemble B: ṡ B x = s B y sb x + 1 sb z J x ṡ B y = s B x sb y + 1 sb z J y ṡ B z = (1 s B 1 1 z ) sb x J x sb y J y s = N hŝi, J = sa + s B, N c =1 Landau-Lifshitz damping 1 s (J ẑ) Single ensemble, = 0 LL equation ṡ = s H s (s H) H = ± ẑ Two coupled LL eqs + pumping

14 Next order in system size expansion (leading quantum correction) Fokker-Planck equation P A, A,µ A, B, B,µ B,t = N 3 P v A,v A,m A,v B,v B,m B,t S A,B ±,z (t) = 1 N SA,B ±,z (t) 1 N hŝa,b ±,z (t)i

15 Symmetries Ensemble A: ṡ A x = s A y sa x + 1 sa z J x ṡ A y = s A x sa y + 1 sa z J y ṡ A z = (1 s A 1 1 z ) sa x J x sa y J y Ensemble B: ṡ B x = s B y sb x + 1 sb z J x ṡ B y = s B x sb y + 1 sb z J y ṡ B z = (1 s B 1 1 z ) sb x J x sb y J y Eqs of motion invariant with respect to: Rotations around z-axis (choice of axes in xy-plane is arbitrary) symmetry (similar to particle-hole) s A x! s B x s A y! s A z! s B z s B y

16 Phase diagram: symmetric attractors: fixed points Trivial steady state: s A,B x,y =0, s A,B z =1 Ensemble A: ṡ A x = s A y sa x + 1 sa z J x ṡ A y = s A x sa y + 1 sa z J y ṡ A z = (1 s A 1 1 z ) sa x J x sa y J y Ensemble B: ṡ B x = s B y sb x + 1 sb z J x ṡ B y = s B x sb y + 1 sb z J y ṡ B z = (1 s B 1 1 z ) sb x J x sb y J y Eqs of motion invariant with respect to: Rotations around z-axis (choice of axes in xy-plane is arbitrary) symmetry (similar to particle-hole) s A x! s B x s A y! s A z! s B z s B y

17 Phase diagram: symmetric attractors: fixed points Trivial steady state: s A,B x,y =0, s A,B z =1 Maximally pumped. All atoms excited. No emission. Exists for all values of δ and Trivial steady state d Nonequilibrium phase diagram for two ensembles of atoms in a bad cavity (mean-field) pumping, d detuning from the cavity mode

18 Phase diagram: symmetric attractors: fixed points Trivial steady state: s A,B x,y =0, s A,B z =1 Maximally pumped. All atoms excited. No emission. Exists for all values of δ and Looses stability via supercritical pitchfork (PF) and super-critical Hopf (H+) bifurcations. z PF Trivial steady state H+ d Nonequilibrium phase diagram for two ensembles of atoms in a bad cavity (mean-field) pumping, d detuning from the cavity mode

19 s A x = s B x = J? s A y = s B y = J? s A z = s B z = Phase diagram: symmetric attractors: fixed points Non-trivial steady state [NTSS] (synchronized, superradiant): + J? = p p 1 ( 1) + rotation around z-axis z Non-trivial steady state (constant superradiance) PF Trivial steady state H+ d Nonequilibrium phase diagram for two ensembles of atoms in a bad cavity (mean-field) pumping, d detuning from the cavity mode

20 Stability analysis of NTSS: Coexistence Loses stability via sub-critical Hopf bifurcation on the boundary between green and blue regions PF TSS coexistence NTSS H+ H- Unstable limit cycle exists in the blue region. Acts as the separatrix between the NTSS and the already existing limit cycle from the super-critical Hopf bifurcation of the TSS. Hence coexistence

21 Phase diagram: symmetric attractors: limit cycle symmetric limit cycle s A x = s B x, s A y = s B y, s A z = s B z s A y s A z s A x s A z Constant superradiance (NTSS) PF Trivial steady state (TSS) H+ symmetric limit cycle (modulated superradiance) pumping, d detuning from the cavity d s B z + rotation around z-axis (1-parameter family of limit cycles)

22 Phase diagram: symmetric attractors: limit cycle symmetric limit cycle s A x = s B x, s A y = s B y, s A z = s B z Superradiance emission spectrum: 3 s A z s A y s A x E(!) 3 Constant superradiance (NTSS) PF Trivial steady state (TSS) H+ symmetric limit cycle (modulated superradiance) Equidistant peaks, odd harmonics of the limit cycle frequency ~100 Hz, periodically oscillating superradiance (cf. single peak at w 0 ~1 GHz for NTSS) d 5! ω Can generate unusual, orders of magnitude different frequencies in this superradiant laser!

23 Phase diagram: symmetric attractors: limit cycle symmetric limit cycle analytic solution in several limits: s A x = a cos( t ), s A y = a sin t, s A z = a sin( t ) = p, a = p (1 ), tan = Constant superradiance (NTSS) PF Trivial steady state (TSS) H+ symmetric limit cycle (modulated superradiance) d pumping, d detuning from the cavity

24 Phase diagram: symmetric attractors: limit cycle symmetric limit cycle analytic solution near triple point d = = 1 s x = s y = acn(bt, k), s z =1 s y Jacobi elliptic function cn b = 1 (1 k ), a = k ( 1) 1 k Period: T = 4K(k)p 1 k p 1 Constant superradiance (NTSS) PF Trivial steady state (TSS) H+ symmetric limit cycle (modulated superradiance) pumping, d detuning from the cavity d 4 5(1 k ) (k 4 k + 1)E(k) ( k )(1 k )K(k) (k 1)E(k)+(1 k )K(k) = 1 1

25 Phase diagram: symmetry breaking: asymmetric limit cycles s A z s A? s B z Constant superradiance Trivial steady state symmetry breaking line symmetric limit cycle d =.38, =.055 Asymmetric limit cycles (yellow region) Projections of the asymmetric limit cycle s B? Note: for symmetric limit cycle s A z = s B z, s A? = s B?

26 Floquet stability analysis of the symmetric limit cycle Change to symmetric and antisymmetric variables s x = sa x + s B x, s y = sa y s B y, s z = sa z + s B z m x = s A x s B x, m y = s A y + s B y, m z = s A z s B z For symmetric limit cycle: m(t) =0, s(t) =s A (t) periodic

27 Floquet stability analysis of the symmetric limit cycle Change to symmetric and antisymmetric variables s x = sa x + s B x, s y = sa y s B y, s z = sa z + s B z m x = s A x s B x, m y = s A y + s B y, m z = s A z s B z For symmetric limit cycle: m(t) =0, s(t) =s A (t) periodic Linearize eqs of motion in m around the symmetric limit cycle ṁ x m y ṁ y = m x m x + s x (t)m x m y + s z (t)m y ṁ z = m z s x (t)m x s y (t)m y Linear diff eqs with periodic coefficients use Floquet theorem m(t) = 3X c k e µ kt p k (t) k=1 periodic Floquet exponents

28 Phase diagram: symmetry breaking: asymmetric limit cycles m(t) = µ 1 =0 µ,3 < 0 3X c k e µ kt p k (t) k=1 due to symmetry w.r.t. rotations around z-axis symmetric limit cycle stable µ or µ 3 > 0 unstable Constant superradiance Trivial steady state symmetry breaking line symmetric limit cycle d Asymmetric limit cycles (yellow region)

29 Signatures of asymmetric limit cycle in the emission spectrum ω Even harmonics of the limit cycle frequency appear Shift of the carrier frequency Stokes anti-stokes asymmetry! ω Asymmetric LC for d =.4, = ω Symmetric LC for d =.44, =.056!!

30 Phase diagram: funny region Constant superradiance Trivial steady state symmetric limit cycle d pumping, d detuning from the cavity mode

31 As we move along the white arrow in the d-plane another fundamental frequency emerges quasiperiodic motion with frequencies. Attractor: circle D torus Phase diagram: funny region s A? s B? =.115, =.055 pumping, d detuning from the cavity mode δ yellow asymmetric LCs green symmetric LC dark blue quasiperiodic orange chaos red synchronized chaos d

32 Quasiperiodic motion with frequencies. Attractor D torus E(!) s A x s A y! t 10 4 =.115, =.055 s A z Max Lyapunov exponent = 0 pumping, d detuning from the cavity mode t 10 4

33 Phase diagram: quasiperiodic (RTN) route to chaos As we move a bit further chaos ensues Ruelle Takens Newhouse (RTN) theorem: quasiperiodic motion with 3 or more frequencies not robust against small perturbations giving rise to chaos. RTN route to chaos is typical for strongly coupled systems. In our case, chaotic and -frequency quasiperiodic behaviors alternate pumping, d detuning from the cavity mode δ yellow asymmetric LCs green symmetric LC dark blue quasiperiodic orange chaos red synchronized chaos d

34 Phase diagram: chaos E(!) s A z! s B z Continuous emission spectrum =.1, =.055 s A z s A x s A y t 10 4 Max Lyapunov exponent > 0 t 10 4 pumping, d detuning from the cavity mode

35 eous magnetic field to induce a differential Zeeman The atoms in both ensembles are pumped incoherently excited state, as could be realized by driving a transition ird state that rapidly decays to e [3, 4]. s system is described by the Hamiltonian in the rotating of the cavity field: = δ As we move further chaos synchronizes Dynamics of each ensemble are individually chaotic, but they copy each ( Jˆ z A Jˆ z B ) + Ω (â Jˆ A + J ˆ Aâ + + â Jˆ B + ˆ+ J Bâ), (1) other most of the time Ω is the atom-cavity coupling, and â and â are anion and creation operators for cavity photons. Here 1 Nj=1 symmetry approximately restored in ˆσ z (A,B) j and Jˆ A,B = N j=1 ˆσ (A,B) j are the collective spin chaotic operators, dynamics written in terms of the Pauli operators two-level system ˆσ z (A,B) j and ˆσ (A,B) j = (ˆσ+ (A,B) j ). Phase diagram: synchronized chaos (color online) Twoensembles of driven two-level atoms coua single-mode cavity field. The atoms in ensemble A are debove the cavity resonance (dashed line). Ensemble B contains detuned below the cavity resonance by an equivalent amount. pumping, d detuning from the cavity mode δ yellow asymmetric LCs green symmetric LC dark blue quasiperiodic orange chaos red synchronized chaos d

36 Synchronized chaos s A z t 10 4 t 10 4 s B z s A x s A y s A x s A y s A? t 10 4 t 10 4 =.079, =.055 s B? pumping, d detuning from the cavity mode Max Lyapunov exponent > 0

37 Synchronized chaos via on-off intermittency Fixed =.05 in all plots s A z s B z =.080 s A? s B? =.080 =.0801 =.0801 =.0800 t 10 4 t 10 4 =.0800

38 Dynamics of each ensemble are individually chaotic, but they copy each other most of the time Phase diagram: synchronized chaos E(!) ! 0.05 Typically, directional coupling between the two parts drive and response. Here synchronization due to mutual coupling between two ensembles δ yellow asymmetric LCs green symmetric LC dark blue quasiperiodic orange chaos red synchronized chaos d

39 Potential application of synchronized chaos: steganography Cuomo and Oppenheim, 1993 Figure from A. Uchida: Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization 01. Cryptography = hide the meaning of the message Steganography = hide the existence of the message

40 Collaborators: Aniket Patra Rutgers University synchronization (fixed point) δ limit cycle (periodically oscillating superradiance) d Boris Altshuler Columbia University Thanks to Sergey Denisov & Yuri Rubo for discussions Nonequilibrium phase diagram for two ensembles of atoms in a bad cavity (mean-field) pumping, d detuning from the cavity mode