6, Quantum theory of Fluorescence

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1 6, Quantum theory of Fluorescence 1 Quantum theory of Damping: Density operator 2 Quantum theory of Damping: Langevin equation 3 System-Reservoir Interaction 4 Resonance Fluorescence 5 Decoherence Ref: Ch 8, 9, 10 in Quantum Optics, by M Scully an M Zubairy Ch 7 in Mesoscopic Quantum Optics, by Y Yamamoto an A Imamoglu Ch 8 in The Quantum Theory of Light, by R Louon Ch 14, 15 in Elements of Quantum Optics, by P Meystre an M Sargent III Ch 8 in Introuctory Quantum Optics, by C Gerry an P Knight IPT5340, Fall 06 p1/47

2 Damping via oscillator reservoir consier a single-moe fiel, with frequency ω, â an â operators, the reservoir may be taken as any large collection of systems with many egrees (eg phonons, other photon moes, etc), with closely space frequencies ν k, ˆb k an ˆb k operators, the Hamiltonian for the fiel-reservoir system is Ĥ = ωâ â + k ν kˆb ˆb + k g k (ˆb kâ + â ˆbk ), the Heisenberg equations of motion for the operators are tâ(t) = i [Ĥ, â] = iωâ(t) i k g kˆbk (t), ˆbk (t) = iν kˆb(t) igk â(t), t IPT5340, Fall 06 p2/47

3 Damping via oscillator reservoir the close equation for the fiel operator â(t) is, ˆbk (t) = ˆb k (0)e iνkt ig k t e iν k(t t )â(t ), tâ(t) = iωâ(t) k 0 g k 2 t e iν k(t t )â(t ) + ˆf a (t), 0 where ˆf a (t) = i k g kˆbk (0)e iν kt, efine slowly varying operator â (t) = âe iωt, then tâ (t) = k ˆF a (t) = i k g k 2 t e i(ν k ω)(t t )â (t ) + ˆF a (t), 0 g kˆbk (0)e i(ν k ω)t, we use the notation â(t) to replace â (t) in the following formulations, IPT5340, Fall 06 p3/47

4 Markovian white noise a single-moe fiel interacting with reservoir, tâ(t) = k ˆF a (t) = i k g k 2 t e i(ν k ω)(t t )â(t ) + ˆF a (t), 0 g kˆbk (0)e i(ν k ω)t, with the Weisskopf-Wigner theory, G(t) = Γ 2 δ(t), k g k 2 t e i(ν k ω)(t t )â(t ) 0 0 t G(t t )â(t ) Γ 2 â(t), where G(t t ) = k g k 2 e i(ν k ω)(t t ) = ν D(ν) g(ν) 2 e i(ν k ω)(t t ), D(ν) is the ensity of state for the reservoir, an the equation of motion for the fiel interacting with the reservoir is tâ(t) = Γ 2 â(t) + ˆF a (t), there are issipation an fluctuation terms, IPT5340, Fall 06 p4/47

5 Commutation relation for a Markovian process tâ(t) = Γ 2 â(t) + ˆF a (t), ˆF a (t) = i k g kˆbk (0)e i(ν k ω)t, if one ismiss the fluctuation term ˆF a (t), we have the solution â(t) = â(0)e Γ/2t, tâ(t) = Γ 2 â(t), for the non-interacting fiel, the commutation relation at t = 0 is [â(0), â (0)] = 1, but as time evolves to t 0, the commutation relation is not satisfie, [â(t), â (t)] = exp( Γt), the noise operator with appropriate correlation properties helps to maintain the commutation relation at all time, IPT5340, Fall 06 p5/47

6 Thermal reservoir the noise operator is efine as, ˆF a (t) = i k g kˆbk (0)e i(ν k ω)t, suppose the reservoir is in thermal equilibrium, R = R= 0 R = 0 R = 0 R= n k δ kk R = ( n k + 1)δ kk where n t h = n = n nρ nn = 1 exp( ν k /k B T) 1, IPT5340, Fall 06 p6/47

7 Thermal reservoir the noise operator is efine as, ˆF a (t) = i k g kˆbk (0)e i(ν k ω)t, the Langevin noise operators have zero means ˆF a (t) R = ˆF a(t) R = 0, but non-zero variances, ˆF a (t) ˆF a (t ) R = k = k k g k g k ˆb kˆb k exp[i(ν k ω)t i(ν k ω)t ], g k 2 n k exp[i(ν k ω)(t t )], = Γ n th δ(t t ), where n th = n (νk ω), IPT5340, Fall 06 p7/47

8 Fluctuation-issipation theory for a single-moe fiel interacting with thermal reservoir, tâ(t) = Γ 2 â(t) + ˆF a (t), the Langevin noise operators have zero means but non-zero variances, ˆF a (t) R = ˆF a (t) R = 0, ˆF a(t) ˆF a(t ) R = ˆF a (t) ˆF a (t ) R = 0, ˆF a (t) ˆF a (t ) R = Γ n th δ(t t ), ˆF a (t) ˆF a (t ) R = Γ( n th + 1)δ(t t ), the amping of the system is etermine from the fluctuating forces of the reservoir, in other wors, the fluctuations inuce by the reservoir give rise to the issipation in the system, Γ = 1 n th t ˆF a (t) ˆF a (t ) R, this is one formulation of the fluctuation-issipation theorem, IPT5340, Fall 06 p8/47

Mollow s triplet: Resonance Fluorescence Spectrum Ω ω a 2> 1> n+1> n> n-1> 0> S(ω) [AU] 100 90 80 70 60 50 40 30 20 10 Ω Ω system interaction reservoir -1-05 0 05 1 ω-ω a elastic

9 Mollow s triplet: Resonance Fluorescence Spectrum Ω ω a 2> 1> n+1> n> n-1> 0> S(ω) [AU] Ω Ω system interaction reservoir ω-ω a elastic Rayleigh scattering an inelastic Raman scattering Theory: B R Mollow, Phys Rev 188, 1969 (1969) Exp: F Y Wu, R E Grove, an S Ezekiel, Phys Rev Lett 35, 1426 (1975) IPT5340, Fall 06 p9/47

10 Reservoir Theory Photon-Atom Interaction in PhCs Ω ω a 2> 1> n+1> n> n-1> 0> S(ω) [AU] Ω Ω system interaction reservoir ω-ω a Ω 2> ω a 1>? system interaction reservoir IPT5340, Fall 06 p10/47

11 Hamiltonian of our system: Jaynes-Cummings moel H = 2 ω aσ z + k ω k a k a k + Ω 2 (σ e iω Lt + σ + e iω Lt ) + k (g k σ + a k + g ka k σ ) An we want to solve the generalize Bloch equations: σ (t) = i Ω 2 σ z(t)e i t + σ + (t) = i Ω 2 σ z(t)e i t + t t σ z (t) = iω(σ (t)e i t σ + (t)e i t ) + n z (t) 2 t t G(t t )σ z (t)σ (t ) + n (t) t G c (t t )σ + (t )σ z (t) + n + (t) t [G(t t )σ + (t)σ (t ) + G c (t t )σ + (t )σ (t)] IPT5340, Fall 06 p11/47

12 Remarks: 1 coupling constant: g k g k (ˆ, r 0 ) = ω a 1 2 ǫ 0 ω k V ˆ E k ( r 0 ) 2 memory functions: G(τ) G c (τ) k k g k 2 e i kt Θ(τ) g k 2 e i kt Θ(τ) 3 Markovian approximation: G(t) = G c (t) = Γδ(t) IPT5340, Fall 06 p12/47

13 Quantum noise operators n (t) = i k g k e i kt σ z (t)a k ( ) n + (t) = i k n z (t) = 2i k g ke i kt a + k ( )σ z(t) [g ke i kt a + k ( )σ (t) g k e i kt σ + (t)a + k ( )] where the mean an the correlation functions of the reservoir before interaction, < a k ( ) > R =< a k ( ) > R= 0 < a k ( )a k ( ) > R = 0 < a k ( )a k ( ) > R = 0 < a k ( )a k ( ) > R = n k δ kk < a k ( )a k ( ) > R = ( n k + 1)δ kk IPT5340, Fall 06 p13/47

14 Heisenberg-Langevin equations σ (t) = i Ω 2 σ z(t)e i t + σ + (t) = i Ω 2 σ z(t)e i t + t t σ z (t) = iω(σ (t)e i t σ + (t)e i t ) + n z (t) 2 t t G(t t )σ z (t)σ (t ) + n (t) t G c (t t )σ + (t )σ z (t) + n + (t) t [G(t t )σ + (t)σ (t ) + G c (t t )σ + (t )σ (t)] where the Langevin noise operators have zero means but non-zero variances, n (t) R = n + (t) R = n z (t) R = 0 n (t)n (t ) R = n + (t)n + (t ) R = 0 n (t)n + (t ) R = k g k 2 ( n k + 1)e i k(t t ) σ z (t)σ z (t ) n + (t)n (t ) R = k g k 2 n k e i k(t t ) σ z (t)σ z (t ) n z (t)n z (t ) R = 4 k g k 2 [( n k + 1)e i k(t t ) σ + (t)σ (t ) + n k e i k(t t ) σ (t)σ IPT5340, Fall 06 p14/47

15 Moeling DOS of PBCs 3 x DOS Frequency anisotropic moel: ω k = ω c + A k k i 0 2 ω ω D(ω) = c Θ(ω ω A 3 c ) S Y Zhu, et al, Phys Rev Lett 84, 2136 (2000) IPT5340, Fall 06 p15/47

16 Memory functions of PBCs anisotropic moel: ω k = ω c + A k k i 0 2 ω ω D(ω) = c Θ(ω ω A 3 c ) the memory functions uner this anisotropic moel also can be erive: G(ω) = β 3/2 i ωc + ω c ω a ω G c (ω) = β 3/2 i ωc + ω c ω a + ω where β 3/2 = ω2 a 2 6 ǫ 0 η, an we have use the space πa 3/2 average coupling strength η 3 8π erivation Ω E 2 in the IPT5340, Fall 06 p16/47

17 Memory functions of PBCs abs[g(ω)] arg[g(ω)] ω c ω/β Amplitue an phase spectrum of the memory function with ω a = ω c = 100β IPT5340, Fall 06 p17/47

18 Correlations of noise operators at zero temperature ñ (ω 1 )ñ + ( ω 2 ) R = πn(ω 1 )Θ(ω 1 + ω a ω c )δ(ω 1 ω 2 ) ñ z (ω 1 )ñ z ( ω 2 ) R = N(ω 1 )[4πδ(ω 1 ω 2 ) + σ z (ω 1 ω 2 ) R ] ñ z (ω 1 )ñ ( ω 2 ) R = 0 Θ(ω 1 + ω a ω c ) ñ (ω 1 )ñ z ( ω 2 ) R = N(ω 1 ) σ (ω 1 ω 2 ) R Θ(ω 1 + ω a ω c ) ñ z (ω 1 )ñ + ( ω 2 ) R = N(ω 1 ) σ + (ω 1 ω 2 ) R Θ(ω 1 + ω a ω c ) ñ + (ω 1 )ñ z ( ω 2 ) R = 0 with N(ω) 4β 3/2 ω a +ω ω c ω a +ω Quantum noises of the photonic bangap reservoir are not only color noises but also exhibit bangap behaviour IPT5340, Fall 06 p18/47

19 Liouville operator expansion σ ij (t) = e il(t t ) σ ij (t ) = n=0 [ i(t t )] n L n σ ij (t ) n! For zero-th orer Liouville operator expansion, we get σ (t) = i Ω 2 σ z(t)e i t σ + (t) = i Ω 2 σ z(t)e i t t t σ z (t) = iω(σ (t)e i t σ + (t)e i t ) t t G(t t )σ (t ) + n (t) t G c (t t )σ + (t ) + n + (t) t [G(t t ) + G c (t t )](1 + σ z (t )) + n z (t) vali for the case of atom with longer lifetime an uner weak pumping IPT5340, Fall 06 p19/47

20 Solve the optical Bloch equations Since it is now a linear problem, by using Fourier transform the moifie optical Bloch equations become: M(ω) X(ω) = X 0 (ω) where i(ω + ) + G(ω) 0 i Ω 2 M(ω) = 0 i(ω ) + G c (ω) i Ω 2 iω iω iω + G(ω) + G c (ω) σ (ω + ) X(ω) = σ + (ω ) σ z (ω) ñ (ω + ) X 0 (ω) = ñ + (ω ) 2π[ G(ω) + G c (ω)]δ(ω) + ñ z (ω) where ñ (ω), ñ + (ω), ñ z (ω), G(ω), an Gc (ω) are Fourier transforms of n (t), n + (t), n z (t), G(t), an G c (t), respectively IPT5340, Fall 06 p20/47

21 Solve the optical Bloch equations The solutions are where σ (ω + ) = (2g h + Ω2 ) ñ (ω) + Ω 2 ñ + (ω) + iωg ñ z (ω) i2πωg[ G(ω) + G c (ω)]δ(ω Ω 2 (f + g) + 2f g h σ + (ω ) = Ω2 ñ (ω) + (2f h + Ω 2 ) ñ + (ω) iωf ñ z (ω) + i2πωf[ G(ω) + G c (ω)]δ(ω Ω 2 (f + g) + 2f g h σ z (ω) = 2iΩg ñ (ω) 2iΩf ñ + (ω) + 2f g ñ z (ω) 4πf g[ G(ω) + G c (ω)]δ(ω) Ω 2 (f + g) + 2f g h f(ω) = iω i + G(ω) g(ω) = iω + i + G c (ω) h(ω) = iω + G(ω) + G c (ω) IPT5340, Fall 06 p21/47

22 Fluorescence spectrum Because the two-time correlation function of the atomic ipole is proportional to the first orer correlation function g (1) (τ), we can obtain the fluorescence spectrum by taking the Fourier transform of the first orer correlation function: S(ω) = τ g (1) (τ)e iωτ σ + (ω) σ ( ω) R It shoul be note that here we cannot irectly apply the quantum regression theorem since it is invali for non-markovian process IPT5340, Fall 06 p22/47

23 Mollow s triplet σ + (ω) + As a check, we first use our formulation to calculate the free space case At free space, one can assume the memory functions are elta functions since k g k 2 e i kt = Γδ(t) with Γ being the ecay rate of the excite atom The noise correlation functions at zero temperature are also elta-function correlate (ie, white noises) Therefore, the fluorescence spectrum at steay state is given by: σ ( ω) R = π2 Ω 2 ( Γ ) δ(ω + ) Ω Γ2 4 πγω 4 ( Ω2 2 + Γ2 + (ω + ) 2 ) 2( Ω Γ2 4 )[Γ2 ( Ω Γ2 4 2(ω + )2 ) 2 + (ω + ) 2 (Ω Γ2 (ω + ) IPT5340, Fall 06 p23/47

24 Mollow s triplet In the limit of strong on-resonance pumping (Ω Γ, = 0), Eq(1) can be reuce to: σ + (ω) σ ( ω) R = 2π{2π Γ2 3 4Ω 2 δ(ω) + 16 Γ (ω + Ω) Γ2 1 4 Γ ω Γ Γ (ω Ω) Γ2 } Then, the resonance fluorescence spectrum exhibits the Mollow triplets for white noise: three Lorentzian profiles with peaks in the ratio 1 : 3 : 1, an withs of 3 2 Γ, Γ, an 3 2 Γ Ω ω a 2> 1> n+1> n> n-1> 0> S(ω) [AU] Ω Ω system interaction reservoir ω-ω a IPT5340, Fall 06 p24/47

25 Resonance fluorescence spectra near the ban-ege Ω = 025 β ω c = 100 β S(ω) [AU] (ω-ω a )/β (ω a -ω c )/β 05 DOS Wc Wa w IPT5340, Fall 06 p25/47

26 Quarature spectra Define quarature fiel operator as: Ê θ (t) = e iθ Ê (+) (t) + e iθ Ê ( ) (t) θ = 0 ( π ) are the in-phase (out-of-phase) quarature fiels 2 Then the corresponing spectra with normally orer variance is: S θ (ω) < Ẽθ(ω),Ẽθ( ω) > 1 4 [< σ (ω) σ ( ω) > e 2iθ + < σ + (ω) σ ( ω) > + < σ + ( ω) σ (ω) > + < σ + ( ω) σ + (ω) > e 2iθ ] IPT5340, Fall 06 p26/47

27 Quarature spectra in free space Ω = 50Γ Ω = 20Γ Ω = 10Γ Ω = 05Γ Theory: D F Walls an P Zoller, Phys Rev Lett 47, 709 (1981) Theory: L Manel, Phys Rev Lett 49, 136 (1982) IPT5340, Fall 06 p27/47

28 Observation of squeezing fluorescence spectra with 147 Yb atoms Exp: Z H Lu, S Bali, an J E Thomas, Phys Rev Lett 81, 3635 (1998) IPT5340, Fall 06 p28/47

29 Fluorescence quarature spectra near the ban-ege Ω = 025 β ω c = 100 β 03 Ω = 025 β ω c =100 β 03 Quarature Spectrum S1 (ω) [AU] (ω-ω a )/β (ω-ω a )/β (ω a -ω c )/β DOS Wc Wa w IPT5340, Fall 06 p29/47

30 Resonance fluorescence quarature spectra near the ban-ege 1 Suppression an enhancement of the relative fluorescence peak amplitues varie at ifferent wavelength offsets 2 Squeezing occurs in the out-of-phase quarature for free space when Ω 2 < 4Γ 2 3 Squeezing occurs in the in-phase quarature for PhCs when Ω 2 > 4Γ 2 4 Resonance fluorescence squeezing spectra come from the interference between two siebans of Mollow s triplet R-K Lee an Y Lai, J Opt B, 6, S715 (2004) IPT5340, Fall 06 p30/47

31 Density operator metho Heisenberg-Langevin equation is a irectly corresponence to classical escription of stochastic system, in many cases, Heisenberg-Langevin equation is nonlinear, in general it is extremely ifficult to eal with the Heisenberg-Langevin equation, another metho is to evelop a Schröinger or interaction picture analysis, in this way we want to use a linear eterministic ifferential equation for the reuce system ensity operator, naturally, as the quantum system is open, there is statistical as well as quantum uncertainty an a true wave function escription is no longer possible, Ref: Ch 7 in Mesoscopic Quantum Optics, by Y Yamamoto an A Imamoglu Quantum Noise, by C W Gariner IPT5340, Fall 06 p31/47

32 Master equation we consier a system S interacting with a reservoir R via the interaction Hamiltonian ˆV, the combine ensity operator is enote by ˆρ(t), assume that at an initial time t = 0, the two systems are uncorrelate, ˆρ(t = 0) = ˆρ S (0) ˆρ R (0), in the interaction picture, the ynamics of ˆρ(t) is t ˆρ(t) = 1 i [ĤI(t), ˆρ(t)], since the number of egrees of freeom of the reservoir is very large, it is impossible to keep track of its quantum evolution, we can only focus on the system with a reuce ensity operator, by tracing over the reservoir egrees of freeom, t ˆρ S(t) = 1 i Tr R([ĤI(t), ˆρ(t)]), IPT5340, Fall 06 p32/47

33 Master equation we can only focus on the system with a reuce ensity operator, t ˆρ S(t) = 1 i Tr R([Ĥ I (t), ˆρ(t)]), where t ˆρ(t) = 1 i [ĤI(t), ˆρ(t)], without any approximation, the master equation for the reuce ensity operator is, t ˆρ S(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ(t )]]) i Tr R([ĤI(t), ˆρ(0)]), since Tr R ([ĤI(t), ˆρ(0)]) vanish for all the interaction Hamiltonians of interest in quantum optics, we have the master equation, t ˆρ S(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ(t )]]), 0 IPT5340, Fall 06 p33/47

34 Born-Markov approximation the master equation for the reuce ensity operator, without any approximation, t ˆρ S(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ(t )]]), 0 four key approximations use in the following, 1 Rotating-wave approximation, 2 Born approximation, ˆρ(t ) = ˆρ S (t ) ˆρ R (t ), 3 the initial raiation fiel ensity operator commutes with the free Hamiltonian an the reservoir is not affecte by the interaction with the system, ˆρ R (t) = Tr S [ˆρ(t)] = ˆρ R (0), 4 Markov approximation, ˆρ S (t ) ˆρ S (t), IPT5340, Fall 06 p34/47

35 Master equation the master equation for the reuce ensity operator, without any approximation, t ˆρ S(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ(t )]]), 0 with Born-Markov approximation, the master equation becomes t ˆρ S(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ S (t) ˆρ R (0)]]), 0 atom amping by fiel reservoirs, fiel amping by fiel reservoirs, fiel amping by atomic reservoirs, IPT5340, Fall 06 p35/47

36 Atom amping by fiel reservoirs consier a two-level atom ampe by a fiel reservoir in free space, the interaction Hamiltonian is Ĥ I = k (g kˆσ â k e i(ω ω k)t + H C), assume the reservoir ensity operator is a multimoe thermal fiel, ˆρ R = k n exp( ω kn k B T ) 1 exp( ω kn k B T ) n kk n, the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ a (t) is, t ˆρ a(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ a (t) ˆρ R (0)]]), 0 = 1 2 Γ{n th[ˆσ ˆσ +ˆρ a ˆσ +ˆρ aˆσ ] + (n th + 1)[ˆσ +ˆσ ˆρ a ˆσ ˆρ aˆσ + ])} + H IPT5340, Fall 06 p36/47

37 Fiel amping by fiel reservoirs consier a single-moe fiel in a cavity with a finite leakage rate, assume the reservoir ensity operator is a multimoe thermal fiel, ˆρ R = k n exp( ω kn k B T ) 1 exp( ω kn k B T ) n kk n, the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, t ˆρ f(t) = ( 1 t i )2 = t 0 t 0 t k t Tr R ([ĤI(t),[ĤI(t ), ˆρ f (t) ˆρ R (0)]]), g 2 k {n th[ââ ˆρ f (t ) â ˆρ f (t )â]e i(ω ω k)(t t ) +(n th + 1)[â âˆρ f (t ) âˆρ f (t )â ])e i(ω ω k)(t t ) } + H C, IPT5340, Fall 06 p37/47

38 Fiel amping by fiel reservoirs the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, t t ˆρ f(t) = t t 0 k g 2 k {n th[ââ ˆρ f (t ) â ˆρ f (t )â]e i(ω ω k)(t t ) +(n th + 1)[â âˆρ f (t ) âˆρ f (t )â ])e i(ω ω k)(t t ) } + H C, again, by replacing sum k g 2 k with the integral ω k D(ω k )g(ω k ) 2, an t t 0 t k g 2 k e±i(ω ω k)(t t ) = t t ω k D(ω k )g(ω k ) 2 e ±i(ω ω k)(t t ), t 0 ω k D(ω k )g(ω k ) 2 πδ(ω ω k ), πd(ω)g(ω) ( ω Q e ), where ω/q e is the cavity photon ecay rate ue to leakage (output coupling) via a partially reflecting mirror, IPT5340, Fall 06 p38/47

39 Fiel amping by fiel reservoirs the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, t ˆρ f(t) = 1 2 ( ω Q e ){n th [ââ ˆρ f (t ) â ˆρ f (t )â] + (n th + 1)[â âˆρ f (t ) âˆρ f (t )â +H C, compare to the case of atom amping by fiel reservoirs, t ˆρ a(t) = 1 2 Γ{n th[ˆσ ˆσ +ˆρ a ˆσ +ˆρ aˆσ ] + (n th + 1)[ˆσ +ˆσ ˆρ a ˆσ ˆρ aˆσ + ])} + H IPT5340, Fall 06 p39/47

40 Fiel amping by atomic reservoirs consier the amping of an optical cavity moe by a two-level atomic beam reservoir, this is the reverse problem of a laser, the statistics of the atomic reservoir is etermine by the Boltzmann istribution, ˆρ R=atom (t = 0) = ρ aa 0 0 ρ bb = ρ aa a a + ρ bb b b, where ρ aa = exp( ω 0 /k B t), an ρ aa = exp( ω 0/k B t) 1 + exp( ω 0 /k B t), assume there is no quantum coherence between the upper an lower states, ρ ab = ρ ba = 0, IPT5340, Fall 06 p40/47

41 Fiel amping by atomic reservoirs the interaction Hamiltonian for a single atom is Ĥ I = g(ˆσ â + ˆσ + â) = g 0 â â 0, at t = 0, the atom-fiel ensity operator is, ˆρ(t) = ˆρ f (t) ˆρ R = ρ aaˆρ f (t) 0 0 ρ bbˆρ f (t), the terms for the commutator are [Ĥ I,[Ĥ I, ˆρ(t)]] = g 2 +H C, ââ ρ aaˆρ f (t) âρ bbˆρ f (t)â 0 0 â âρ bbˆρ f (t) â ρ eeˆρ f (t)â IPT5340, Fall 06 p41/47

42 Fiel amping by atomic reservoirs the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, t ˆρ f(t) = ( 1 t i )2 t Tr R ([ĤI(t),[ĤI(t ), ˆρ f (t) ˆρ R (0)]]), 0 assume that r atoms are injecte into the cavity per secon, an they spen an average time of τ secons insie the cavity, ie τ 0 t rt = 1 2 rτ2, then, t ˆρ f(t) = 1 2 R e[ââ ˆρ f â ˆρ f â] 1 2 R g[â âˆρ f âˆρ f (t)â ] + HC, where R e = rρ aa g 2 τ 2, an R g = rρ bb g 2 τ 2, IPT5340, Fall 06 p42/47

43 Fiel amping by atomic reservoirs the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, where t ˆρ f(t) = 1 2 R e[ââ ˆρ f â ˆρ f â] 1 2 R g[â âˆρ f âˆρ f (t)â ] + HC, R e = rρ aa g 2 τ 2, an R g = rρ gg g 2 τ 2, R e is the rate coefficient for photon emission by atoms per secon, R g is the rate coefficient for photon absorption by atoms per secon, the cavity photon ecay rate are efine by ν Q 0 an the thermal equilibrium photon number n th ν Q 0 R g R e, an R e (1+n th ) = R g n th n th = R e R g R e = 1 exp( ω/k b T) 1, the later one conition gives the thermal equilibrium photon number n th, R e (1 + n th ) is the sum of spontaneous an stimulate emission rate per secon, R g n th is (stimulate) absorption rate per secon, IPT5340, Fall 06 p43/47

44 Fiel amping by atomic reservoirs the equation of motion for the reuce ensity operator Tr R [ˆρ(t)] ˆρ f (t) is, t ˆρ f(t) = 1 2 R e[ââ ˆρ f â ˆρ f â] 1 2 R g[â âˆρ f âˆρ f (t)â ] + HC, = 1 ν {n th [ââ ˆρ f â ˆρ f â] + (n th + 1)[â âˆρ f âˆρ f (t)â ]} + HC, 2 Q 0 where we use that R e = ν Q 0 n th an R g = ν Q 0 (n th + 1), the iagonal elements of the reuce ensity matrix are t ρ n,n(t) = ν Q 0 {[n th (n + 1) (n th + 1)n]ρ n,n n th ρ n 1,n 1 (n th + 1)(n + 1)ρ n+1,n+1 }, = [ R e (n + 1) R g n]ρ nn + R e nρ n 1,n 1 + R g (n + 1)ρ n+1,n+1, IPT5340, Fall 06 p44/47

45 Detaile balance equilibrium is obtain when the net flow between all pairs of level vanishes, R g nρ n,n = R e nρ n 1,n 1, or ρ n,n = n th n th + 1 ρ n 1,n 1, is conition is referre to as etaile balance, the solution for etaile balance is, ρ n,n = [1 exp( ω/k B T)]exp( n ω/k B T), with n th = 1 exp( ω/k b T) 1, etaile balance in this case gives the thermal (Bose-Einstein) istribution with an average photon number, n = n ρ n,n n = 1 exp( ω/k B T) 1 = n th IPT5340, Fall 06 p45/47

46 Detaile balance etaile balance in this case gives the thermal (Bose-Einstein) istribution with an average photon number, n = n ρ n,n n = 1 exp( ω/k B T) 1 = n th this is the result we use for the thermal raiation fiel, although the file ˆρ f may initially be in a pure state, the process of tracing over the (unobserve) atomic states leas to a fiel in a mixe state, ˆρ f = n ρ n,n n n, the effect of the atomic beam is to bring the fiel to the same temperature as that of atoms, IPT5340, Fall 06 p46/47

47 Reservoir, ecoherence, an measurement the reservoir theory lies in the process of tracing over the reservoir coorinates, which inuces issipation an ecoherence of the system, at the same time, this is an irreversible ynamics for the system, this process correspons to the lack of measurement as to whether the atom is in the upper level or in the lower level after interaction with the fiel, if the initial an final states of the atom are know, ie if the information concerning the atomic beam is not iscare, the fiel remains in a pure state, the primary ifference between the reservoir an quantum measurement theories is whether information store in the environment (reservoir) that interacts with system is iscare or rea out IPT5340, Fall 06 p47/47

arxiv: v3 [quant-ph] 25 Sep 2012

arxiv: v3 [quant-ph] 25 Sep 2012 Three-Level Laser Dynamics with the Atoms Pumpe by Electron Bombarment arxiv:115.1438v3 [quant-ph] 25 Sep 212 Fesseha Kassahun Department of Physics, Ais Ababa University, P. O. Box 33761, Ais Ababa, Ethiopia