Quantum theory of matter-wave lasers

Transcription

1 Chapter 8 Quantum theory of matter-wave lasers If a cooling time is not negligible compare to a conensate particle escape time from a trap, such a system is necessarily ynamical an must be treate as a non-equilibrium open issipative system rather than a thermal equilibrium close system. The steay state population of a groun state in such a system has the pump (reservoir particle injection) rate epenence as shown in Fig At very low pump rates, injecte particles are share by many states so that the probability of fining them in the groun state, which is efine here as a quantum efficiency, is much lower than one. At a certain pump rate, the average population in the groun state reaches one, which is calle a quantum egeneracy point. Once the pump rate excees this critical point a bosonic final state stimulation accelerates a cooling process of the reservoir particles into the groun state an the groun state population increases nonlinearly. Eventually the quantum efficiency approaches to one at well above threshol (quantum egeneracy point). In this case all injecte particles are conense into the groun state before they escape from a trap. This behavior is similar to the laser phase transition base on the stimulate emission of photons also shown in Fig. 8.1, so that such a ynamical conensate is often referre to as a matter-wave laser [1, 2]. A matter-wave laser is istinct from a photon laser, since the temperature an chemical potential can be efine an play important roles in the formation of an orer, while such concepts o not exist in a photon laser. A matter-wave laser is also istinct from BEC, since the amplitue an phase fluctuate ynamically so that the finite spectral linewith (first-orer temporal coherence) an higher-orer temporal coherence functions can be efine [3]. How is the quantum theory of matter-wave lasers istinct from the classical treatment? So far we have treate the conensate orer parameter ψ 0 as a c-number since there is a macroscopic population in the groun state. The populations of the Bogoliubov quasiparticles at all energies are ientically equal to zero at T = 0. Is this treatment compatible with quantum mechanics even if we consier a finite lifetime for the conensate particles? Suppose the conensate particles ecay from a trap with a rate γ. Then the Heisenberg equation of motion for the fiel operator ˆψ 0 must satisfy t ˆψ 0 = iω ˆψ 0 γ 2 ˆψ 0, (8.1) 1

2 Polaritons per Moe at k polariton laser E LP = ev photon laser E CAV = ev polariton laser Quantum egeneracy threshol no inversion photon laser inversion Photons per Cavity Moe at k injecte exciton ensity (cm -2 ) Figure 8.1: Groun state population N 0 vs. pump rate P of a matter-wave (polariton) laser an photon laser [2]. t ˆψ 0 + = iω ˆψ 0 + γ 2 ˆψ 0 +. (8.2) Here µ = ω is the chemical potential of the conensate. From (8.1) an (8.2), we have the time epenence of the commutator bracket: [ ˆψ0 (t), ˆψ ] [ 0 + (t) = ˆψ0 (0), ˆψ ] 0 + (0) e γt. (8.3) [ Even though we assume a proper bosonic commutation relation at t = 0, i.e. ˆψ0 (0), ˆψ ] 0 + (0) = 1, the commutator bracket is not conserve as a time evolves (t > 0). What was wrong with our Heisenberg equations (8.1) an (8.2)? A mistake is that we have neglecte the coupling between the conensate an external reservoir fiels, which are responsible for a particle loss from a trap. The reservoirs inject noise whenever the system (the conensate in our case) issipates into the reservoirs. This relation is the quantum mechanical analogue of the fluctuation-issipation theorem [4, 5]. The classical counterpart of this problem is well establishe in satistical mechanics [6]. The quantum noise injecte by the external reservoirs is an ultimate origin for the spectral linewith an incomplete temporal coherence of the conensate forme in a matter wave laser. 8.1 Heisenberg-Langevin equation for an open issipate trap Derivation of the equation The Hamiltonian of a total system consisting of the conensate in a trap an the external reservoir fiel outsie of a trap shoul have the same form as that for a photon laser system with an open-issipate cavity [7], Ĥ = ω 0 ˆψ+ 0 ˆψ0 + ω p ˆφ+ p ˆφp + (g p ˆψ+ 0 ˆφp + g ˆψ ) p 0 ˆφ+ p. (8.4) p p 2

3 The Heisenberg equations of motion for the conensate fiel operator ˆψ 0 an the reservoir fiel operator ˆφ p are t ˆψ 0 = 1 [ Ĥ] ˆψ0, = iω 0 ˆψ0 i i p g p ˆφp, (8.5) t ˆφ p = 1 [ Ĥ] ˆφp, = iω p ˆφp igp i ˆψ 0. (8.6) We can formally integrate (8.6) an substitute the result, ˆφ p (t) = ˆφ p (0)e iωpt igp t 0 t ˆψ0 (t )e iωp (t t ) into (8.5) to obtain the integro-ifferential equation: t ˆψ 0 (t) = iω 0 ˆψ0 (t) t 0 t p g p 2 ˆψ0 (t )e iωp(t t ) i p g p ˆφp (0)e iωpt. (8.7) The summation over the momentum p in (8.7) can be approximate by the integral over p, p V (2π ) 3 p, where V is a quantization volume. We can introuce the slowly 3 varying fiel operator by ˆψ 0 (t) = ˆΦ 0 (t)e iω0t. Substituting these relations, (8.7) can be rewritten as t ˆΦ 0 (t) = V t (2π ) 3 3 p g p 2 t ˆΦ0 (t )e i(ω p ω 0 )(t t ) V i (2π ) pg p ˆφp (0)e i(ω p ω 0 )t. (8.8) The first term of R.H.S of (8.8) is simplifie by introucing V Γ(τ) = (2π ) 3 3 p g p 2 e i(ω p ω 0 )τ = ω p D(ω p ) g(ω p ) 2 e i(ω p ω 0 )τ = 2πD(ω 0 ) g(ω 0 ) 2 δ(τ), (8.9) V (2π ) 3 where τ = t t an D(ω p ) = is the energy ensity of states for reservoir moes. Substituting (8.9) into the first term of R.H.S of (8.8) results in 0 τγ(τ)ˆφ 0 (t τ) = γ 2 ˆΦ 0 (t), (8.10) where γ = 2πD(ω 0 ) g(ω 0 ) 2 is the Fermi s golen rule ecay rate of the conensate particles [7]. The secon term of R.H.S. of (8.8) constitutes the noise operator V ˆF (t) = i (2π ) 3 3 pg p ˆφp (0)e i(ωp ω0)t. (8.11) 3

4 The two-time correlation function of this operator can be calculate as ˆF + (t) ˆF (t ) = gpg p ˆφ+ p (0) ˆφ p (0) e i(ωp ω 0)t i(ω p ω 0 )t p p = g p 2 ˆNR e i(ω p ω 0 )(t t ) p = γ ˆNR δ(t t ), (8.12) 1 where ˆNR = exp[(ε 0 µ R )/k B T R ] 1 is the reservoir particle population at the energy ε 0. Here µ R an T R are the chemical potential an temperature of the reservoir. Similarly we obtain ( ) ˆF (t) ˆF + (t ) = γ 1 + ˆNR δ(t t ). (8.13) Using (8.10) an (8.11), (8.8) is rewritten as t ˆΦ 0 (t) = γ 2 ˆΦ 0 (t) + ˆF (t), (8.14) where the noise operator ˆF (t) satisfies the two-time correlation functions (8.12) an (8.13). This is the Heisenberg-Langevin equation, in which the ynamics of the external reservoirs is ecouple from the system (conensate). An important an implicit assumption behin the above erivation is that the average population ˆNR of the reservoir can be evaluate inepenently from the system ynamics an it is etermine by the thermal equilibrium properly of the reservoirs. This is calle a reservoir approximation Commutator bracket conservation We can write the time evolution of the conensate fiel operator as t ˆψ 0 (t) = ˆψ 0 (t τ) + = ˆψ 0 (t τ) + t τ t t τ [ ] t t ˆψ 0 (t ) [( t iω 0 + γ ) ˆψ0 (t ) + 2 ˆf(t ] ), (8.15) where ˆf(t ) = ˆF (t )e iω 0t. Using the above expression, we can evaluate the expectation values of the following prouct operators at equal time: ˆψ+ 0 (t) ˆf(t) = = ˆf(t) t ˆψ+ 0 (t τ) + t t τ t ˆf + (t ) ˆf(t) t τ [( t iω 0 + γ ) ˆψ+ 2 0 (t ) + ˆf ] + (t ) ˆf(t) = 1 2 γ ˆNR, (8.16) ˆf + (t) ˆψ 0 (t) = 1 2 γ ˆNR, (8.17) 4

5 ˆψ0 (t) ˆf (t) + = ˆf(t) ˆψ+ 0 (t) = 1 ( ) 2 γ 1 + ˆNR. (8.18) Here we use the facts that the past system operator ˆψ 0 + (t ) oes not epen on the future reservoir operator ˆf(t) an that there is no orer parameter in the reservoir, so that ˆf(t) ˆψ+ 0 (t ) = ˆψ+ 0 (t ) ˆf(t) = 0. (8.19) Using (8.16)-(8.18), we can now evaluate the time evolution of the commutator bracket: [ ˆψ0 (t), t ˆψ ] 0 + (t) = ˆψ0 (t) ˆψ 0 + (t) + ˆψ 0 (t) ˆψ+ 0 (t) ˆψ+ 0 (t) ˆψ 0 (t) ˆψ 0 + (t) ˆψ0 (t) [ = γ ˆψ0 (t), ˆψ ] 0 + (t) + ˆf(t) ˆψ+ 0 (t) + ˆψ 0 (t) ˆf + (t) ˆf + (t) ˆψ 0 (t) + ˆψ 0 + (t) ˆf(t) { [ = γ 1 ˆψ0 (t), ˆψ ]} 0 + (t). (8.20) Equation (8.20) guarantees that if the commutator bracket is conserve at a time t, it is conserve at all later time. The noise operator ˆf(t) (or ˆF (t)) is neee to conserve the proper commutator bracket in such an open issipative system Einstein relation between rift an iffusion coefficients The Heisenberg-Langevin equation for a system operator µ has a general form tâµ(t) = ˆD µ (t) + ˆF µ (t), (8.21) where ˆD µ (t) an ˆF µ (t) are rift an iffusion (noise) operators, respectively. The coefficient for the rift operator can be obtaine from the classical equation of motion for the same system, i.e. ˆDµ (t) = µ (t). (8.22) t We can efine the iffusion coefficient D µν by ˆFµ (t) ˆF ν (t ) = 2D µν δ(t t ). (8.23) Let us evaluate the following prouct of two system operators that satisfy the Heisenberg- Langevin equation. µ (t)âν(t) t = = µ (t)âν(t) + µ(t)  ν (t) ˆDµ (t)âν(t) + ˆFµ (t)âν(t) + µ (t) ˆD ν (t) + µ (t) ˆF ν (t). (8.24) The secon an fourth terms of the above equation are alreay evaluate in (8.16) an equal to D µν. Therefore, the iffusion coefficient D µν can be given by 2D µν = ˆDµ (t)âν(t) µ (t) ˆD ν (t), (8.25) 5

6 where we assume the steay state conition, t µ (t)âν(t) = 0. Equation (8.25) is calle the Einstein relation between the rift an iffusion coefficients. The Einstein relation allows us to erive the Heisenberg-Langevin equation an the two-time correlation function of the noise operators if we alreay know the corresponing classical equation of motion. 8.2 Heisenberg-Langevin equations for ynamical conensates Derivation of the equations The classical equations of motion for a ynamical conensate are an open issipative Gross-Pitaevskii equation (7.30) for the conensate orer parameter an a rate equation (7.31) for the pump reservoir population. Please note that the pump reservoir is ifferent from the external reservoir treate in Sec The conceptual iagram for the particle flow is shown in Fig Before we quantize those equations, we separate the conensate System cooling pump reservoir external pump source conensate heating noise Reservoir loss external reservoir Figure 8.2: The ynamical conensate couple to the pump reservoir an the external reservoir. orer parameter into the prouct of the excitation amplitue an the spatial wavefunction: ψ 0 (r, t) = a(t)u 0 (r) = A(t)e iω 0t u 0 (r). (8.26) Here the groun state spatial wavefunction u 0 (r) an the oscillation frequency ω 0 (or the chemical potential ω 0 ) are etermine by the balance between the kinetic energy 2 2m 2, the confining potential V ext (r) an the particle-particle interaction. We neglect the conensate-conensate an conensate-reservoir interaction terms except for their contribution to u 0 (r) an ω, i.e. g c = g R = 0 for simplicity. Similarly we can split the pump 6

7 reservoir population into the prouct of the time epenent an space epenent parts: n R (r, t) = n(t)v R (r). (8.27) The normalization conitions for u 0 (r) an v R (r) are given by u 0 (r) 2 r = v R (r) 2 r = 1. (8.28) The classical equation of motions for A(t) an n(t) are now rewritten as t A(t) = 1 2 [γ c Γn(t)] A(t), (8.29) t n(t) = p(t) γ Rn(t) Γn(t) A(t) 2. (8.30) Here p(t) is the (time epenent) pump rate an Γ = R u 0 (r)v R (r)r is the effective gain coefficient. The Heisenberg-Langevin equation can be obtaine by replacing A(t), n(t) an N c = A(t) 2 by the operators Â(t), ˆn(t) an Â+ (t)â(t) + 1, respectively, an aing the noise operators: tâ(t) = 1 2 [γ c Γˆn(t)] Â(t) + ˆF A (t), (8.31) [Â+ ] t ˆn(t) = p(t) γ Rˆn(t) Γˆn(t) (t)â(t) ˆF n (t). (8.32) The two-time correlation functions of the noise operators can be obtaine using the Einstein relation: ˆF + A (t) ˆF A (t ) = δ(t t ) [γ Â+ ] c ˆNR + Γ ˆn Â, (8.33) ˆFA (t) ˆF + A (t ) ( = δ(t t ) [γ c ˆNR ˆFn (t) ˆF n (t ) = δ(t t )2 ) + 1 (Â+ + Γ ˆn  [ p + γ R ˆn + Γ ˆn (Â+  )] + 1 )] + 1, (8.34). (8.35) The Heisenberg-Langevin equation for the conensate population ˆN c Â+  can be erive for (8.31), t ˆN [ c (t) = γ c ˆNc (t) + Γˆn(t) ˆNc (t) + 1] + ˆF N (t). (8.36) Here the two-time correlation functions for the new noise operator are given by ˆFN (t) ˆF ( )] N (t ) = δ(t t )2 [γ c ˆNc + Γ ˆn ˆNc + 1, (8.37) ˆFN (t) ˆF n (t ) ( ) = δ(t t )2Γ ˆn ˆNc + 1. (8.38) Equations (8.32) an (8.36) constitute the quantum mechanical rate equations for the conensate population operator ˆN c (t) an the reservoir population operator ˆn(t), where the two corresponing noise operators are negatively correlate as shown in (8.38). We will see the important consequence of this fact in the next section. 7

8 8.2.2 Linearization an noise spectrum When the pump rate is well above the critical (quantum egeneracy) point (see Fig. 8.1), the conensate forms an orer parameter with well-stabilize amplitue an phase except for small fluctuations, so that we can expan the excitation amplitue as ] Â(t) = [A c + Â(t) e i ˆφ(t), (8.39) when A c is a c-number average amplitue while Â(t) an ˆφ(t) are the amplitue an phase noise operators. Using (8.39) we can also linearize the conensate population operator as ˆN c (t) = A 2 c + 2A c Â(t) = N c + ˆN(t). (8.40) The pump reservoir population operator can be similarly linearize ˆn(t) = n R + ˆn(t). (8.41) If we substitute (8.40) an (8.41) into (8.32) an (8.36), take the ensemble averages an assume t ˆNc = t ˆn = 0, we have the following steay state solutions γ c = Γn R, (8.42) p = Γn R A 2 c = γ c A 2 c, (8.43) where we assume γ R ΓA 2 c, i.e. the pump reservoir population is mostly eplete by the stimulate scattering into the conensate. The physical meaning of these relations is clear. Equation (8.42) inicates that the loss rate of the conensate particles must be balance by the gain rate (stimulate scattering rate) from the pump reservoir particles. Equation (8.43) shows that all injecte particles per secon, p, is extracte as the leakage conensate particles per secon, γ c A 2 c, which suggests the quantum efficiency is equal to one at well above conensation threshol. The equations of motion for the fluctuation operators Â, ˆφ an ˆn can be obtaine by using (8.39), (8.41), (8.42) an (8.43) in the Heisenberg-Langevin equations, t Â(t) = 1 ˆn(t) + 1 [ ] ˆFA (t)e i ˆφ(t) + e i ˆφ(t) + ˆF A 2A c τ st 2 (t), (8.44) t ˆφ(t) = t ˆn(t) = i [ ] ˆFA (t)e i ˆφ(t) e i ˆφ(t) + ˆF A 2A (t), (8.45) c ( ) ˆn(t) 2γ c A c τ st τ Â(t) + ˆF n (t). (8.46) sp Here τ sp = 1 γ R is the spontaneous ecay time of the pump reservoir population an τ st = 1 is the stimulate ecay time. The Fourier analysis of (8.44)-(8.46) provies ΓA 2 c the amplitue an phase noise spectra as well as the pump reservoir population spectrum. At well above threshol, the amplitue spectrum is reuce to the Lorentzian S A (Ω) = γ c Ω 2 + γc 2. (8.47) 8

9 Â2 Using the Parseval theorem = Ω 0 2π S A(Ω), the variance of the amplitue noise operator  (or the population noise operator ˆN) is equal to Â2 = 1 4, (8.48) ˆN 2 = 4A 2 c Â2 = N c. (8.49) The amplitue noise is equal to that of a coherent state an the conensate particle statistics obey the Poisson istribution. This result supports posteriori the assumption an argument on the phase-locking between the conensate Â2 an excitations that are presente in the previous chapter. The finite amplitue noise = 1/4 an particle number noise ˆN 2 = N c in the conensate are trace back to the zero-point fluctuation (vacuum fiel) injecte from outsie of the trap, which appears as the iffusion coefficient γ c in eq. (8.34), an the pump fluctuation, which appears as the iffusion coefficient p in (8.35). Their contributions are equal at well above threshol. We assume that no particles are injecte from the external reservoir outsie the trap, ˆNR = 0, an the pump rate obeys the Poisson point process with full shot noise. All the other noise sources are cancele out ue to the negative correlation between the conensate noise source an the pump reservoir noise source, represente by (8.38) At well above threshol, the phase noise spectrum is given by γ c S φ (Ω) = Ω 2 A 2. (8.50) c The phase noise spectrum is inversely proportional to square frequency, which is a characteristic of the ranom walk phase iffusion an calle the Wiener-Levy process [7]. The couple equations (8.44) an (8.46) for Â(t) an ˆn(t) provie the stabilization force for the amplitue which counteracts the noise riving forces ˆF A an ˆF n. As a result of this competition, the amplitue noise becomes ientical to that of the coherent state. The equation of motion (8.45) for ˆφ(t) oes not have such a restoring force an consequently the phase iffuses freely: ˆφ 2 (t) = 2D φ t, (8.51) where the iffusion constant is given by 2D φ = γ c 2A 2 c. (8.52) The spectral lineshape of the conensate is calculate by the correlation function, I(ω) = τe iωτ  + (t)â(0) 0 τe iωτ e ˆφ(τ) 2. (8.53) Because of the above phase iffusion process with an exponential ecay, the conensate energy (or frequency) is broaene to a Lorentzian shape with a full with at half-maximum (FWHM) of 2D φ [7]. 9

10 8.2.3 Excess particle number noise an higher-orer coherence functions The Heisenberg-Langevin equations (8.32) an (8.36)[ assume that ] the stimulate an spontaneous cooling rate into the conensate, Γˆn(t) ˆNc (t) + 1 is proportional to the pump reservoir population ˆn(t). In orer to satisfy the energy conservation, the ifference in the pump reservoir particle energy an the conensate particle energy must be absorbe in a thir physical system. In the case of an exciton-polariton conensate, this energy conservation is satisfie by phonon emission [8]. In fact, this is not a sole cooling channel an there is an extra cooling mechanism, four wave mixing, in which two pump reservoir particles scatter into one conensate particle an one hot particle with the momentum an energy conservation satisfie. This secon cooling mechanism is a ominating channel over the first one an results in the heating of the pump reservoir particles when a quantum egeneracy point is exceee an the stimulate four wave mixing is switche on. As a result of the heating of the pump reservoir particles, the clumping of the pump reservoir population above the conensate threshol is not perfect but remains mil as shown in Fig. 8.3[8]. Figure 8.3: Pump reservoir population n R (left) an conensate population N c (right) vs. normalize pump rate p/γ c [8]. This imperfect gain clumping means that the negative correlation between ˆF N (t) an ˆF n (t) in (8.38) never cancel the noise associate with the stimulate an spontaneous cooling processes completely. Because of the imperfect gain clumping, the particle number noise ˆN 2 c higher than the Poisson limit, ( ) ˆNc ˆNc + 1, but consierably ˆNc, as shown in Fig. 8.4 even at well above the conensate is below that of the thermal state, threshol [8]. The excess particle number noise is experimentally characterize by the measurement 10

11 Figure 8.4: The particle number noise ˆN c 2 vs. the average particle number ˆNc of an exciton-polariton conensate [8]. of higher-orer coherence functions g (n) (0) efine by [9] g (n) (0) = ˆψ + 1 (0) ˆψ + 2 (0) ˆψ + n (0) ˆψ n (0) ˆψ 1 (0) ˆψ + 1 (0) ˆψ 1 (0) ˆψ + 2 (0) ˆψ 2 (0) ˆψ + n (0) ˆψ n (0), (8.54) where ˆψ i (0) is the fiel operator in front of a particle etector i at time t = 0. The avantage of measuring g (n) (0) rather than ˆN c 2 is that g (n) (0) is inepenent of particle loss between the trap an the particle etector, while the irect measurement of ˆN c 2 suffers from the evolution of the noise power accoring to ˆN 2 = L 2 ˆN 2 c + L(1 L) ˆN c, (8.55) where L is the particle loss incluing the quantum efficiency of the particle etector [10]. Figure 8.5 shows the measure g (2) (0) an g (3) (0) vs. the pump rate p/p th for an excitonpolariton conensate [11]. The experimental results agree well with the theoretical preictions at well above the conensate threshol. The iscrepancy between the experimental an theoretical results near the threshol is an experimental artifact: the response time of the particle etector is larger than the correlation time of the particle number noise so that the real g (2) (0) an g (3) (0) values of the conensate are washe out by the integration effect of measurement. If the conensate is in a coherent state, the particle number statistics obey a Poisson istribution an the coherence functions are coherent in all orers: g (n) (0) = 1 (coherent state). (8.56) On the other han, if the conensate is in a thermal state, the particle number statistics features a super-poisson istribution an the coherence functions are g (n) (0) = n! (thermal state). (8.57) 11

12 Figure 8.5: The theoretical an experimental secon- an thir-orer coherence functions g (2) (0) an g (3) (0) of an exciton-polariton conensate [11]. The theoretical an experimente results, g (2) (0) 1.4 an g (3) (0) 2.6, shown in Fig. 8.5 inicate the conensate is not a simple thermal state but not close to a coherent state, either. The results are compatible with the results shown in Fig Excess phase noise an spectral linewith So far we have neglecte the self-phase moulation term, g c ψ 0 (r, t) 2 ψ 0 (r, t), an the cross-phase moulation term, g R n R (r, t)ψ 0 (r, t), in the open issipative Gross-Pitaevskii equation. If we keep these two terms in the quantization of the Gross-Pitaevskii equation, the R.H.S. of the Heisenberg-Langevin equation (8.45) is supplemente by these two terms: t ˆφ(t) = i 2A c [ ˆFA (t)e i ˆφ(t) e i ˆφ(t) ˆF + A (t) ] + 3g c ˆN(t) + g R ˆn R(t). (8.58) The phase is not only moulate by the intrinsic Langevin noise source ˆF A (t) but also by the conensate population noise ˆN(t) an the pump reservoir population noise nˆ R (t) through the conensate-conensate an conensate-pump reservoir repulsive interactions. As shown in Fig. 8.4, both ˆN 2 an ˆN R 2 increase with the pump rate, while the intrinsic phase iffusion noise ecrease with the pump rate (see (8.52)). Such an expecte behavior is emonstrate in Fig. 8.6, where the spectral linewith ecreases accoring to the Schawlow-Townes linewith at just above the threshol but increases ue to the population fluctuations ˆN(t) an ˆn R (t) at well above the threshol. The unique behavior of the spectral linewith of a ynamical conensate was experimentally confirme as shown in Fig. 8.7[12]. In this particular experiment, the injecte pump reservoir particles (exciton-polaritons) are spin unpolarize (a) an polarize (b). When the pump reservoir has a mixture of up-spin an own-spin species, the cooling 12

13 Figure 8.6: The spectral linewith vs. conensate [8]. conensate population for an exciton-polariton process is more efficient so that the gain clumping becomes more perfect an the particle number noise is well suppresse. Consequently, the linewith broaening at well above the conensate threshol is rather mil (Fig. 8.7(a)). On the other han, when the pump reservoir has single spin species, the cooling process is less efficient so that the gain clumping becomes less perfect an the particle number noise is not well suppresse. As a result of this, the line broaening at well above the conensate threshol is severe (Fig. 8.7(b)). Figure 8.7: (a) The measure spectra near zero momentum ( k x < 0.55 µm 1 ) for linearly polarize pumping (θ p = 90, θ = 0 ) as a function of polariton ensity. (b) Same spectra for left-circularly polarize pump an right-circularly polarize etection [12]. 13

14 8.2.5 Input-output formalism When we probe the quantum statistical properties of a ynamical conensate, our etector interacts with the output fiel leake out of the trap rather than internal fiel insie the trap. This fact requires us to take one more step of calculation. The result of such theoretical analysis is somewhat surprising: the quantum statistics of the output fiel is ifferent from those of the internal fiel, which is truly a quantum mechanical effect. The output fiel operator ˆr(t) is relate to the internal fiel operator Â(t) an the incient vacuum fiel ˆf(t) by [5, 10] ˆr(t) = γ c Â(t) ˆf(t), (8.59) where ˆr +ˆr an ˆf + ˆf are normalize to the average particle flux per secon while Â+ Â is normalize to the imension-less particle number. We can express the output fiel operator as ˆr(t) = Âe(t)e i ˆφ e(t). At well above threshol, the spectrum of the amplitue noise Âe(t) is reuce to the white noise S Ae (Ω) = 1 2. (8.60) A shown in Fig.8.8(a), the amplitue noise spectrum at frequencies lower than γ c stems from the pump noise, while that at frequencies higher than γ c is trace back to the incient vacuum fiel fluctuation. We can calculate the spectrum of the particle flux operator ˆN e = ˆr +ˆr by using the linearization, ˆNe γ c A 2 c + 2 γ c A c Âe = N e + ˆN e S Ne (Ω) = 4γ c A 2 cs Ae (Ω) = 2N e. (8.61) This is the full shot noise without any cut-off frequency. This means that if we integrate the output particles for an arbitrary time interval τ, the statistics is always Poissonian. (a) (b) pump noise vacuum fluctuation vacuum fluctuation phase iffusion noise Figure 8.8: (a) The amplitue noise spectrum an (b) phase noise spectrum of the output particle fiel from the matter-wave laser. At well above threshol, the spectrum of the phase noise ˆφ e (t) is given by N e S φe (Ω) = 1 ( ) 1 + 2γ2 c 2 Ω 2, (8.62) 14

15 where the first term of R.H.S. of (8.62) is ue to the incient vacuum fiel fluctuation an the secon term is contribute by the ranom walk phase iffusion noise stuie in the previous section. In the sprit of the linearization, i.e. the small fluctuations aroun the large average value, the phase noise can be calculate using (8.62). At frequencies well above γ c, the prouct of S Ae (Ω) an N e S φe (Ω) is equal to 1/4, which is the spectral minimum uncertainty prouct an the unique property of a broaban coherent state [10]. At frequencies well below γ c, the ranom-walk phase iffusion noise enhances the phase noise so that the minimum uncertainty prouct is not satisfie. 15

16 Bibliography [1] A. Imamoglu, R. J. Ram, S. Pau, an Y. Yamamoto. Nonequilibrium conensates an lasers without inversion: Exciton-polariton lasers. Phys. Rev. A, 53: , [2] H. Deng, G. Weihs, G. Snoke, J. Bloch, an Y. Yamamoto. Polariton lasing vs. photon lasing in a semiconuctor microcavity. Proc. Natl. Aca. Sci., 100: , [3] H. Deng, G. Weihs, C. Santori, J. Bloch, an Y. Yamamoto. Conensation of semiconuctor microcavity exciton polaritons. Science, 298: , [4] R. L. Stratonovich. Nonlinear nonequilibrium thermoynamics I. Linear an nonlinear fluctuation-issipation theorems. Springer-Verlag, Berlin, [5] C. W. Gariner an P. Zoller. Quantum Noise. Springer-Verlag, Heielberg, [6] F. Rief. Funamentals of Statistical an Thermal Physics. McGraw-Hill Science, New York, [7] M. Sargent, M. O. Scully, an W. E. Lamb. Laser Physics. Aison-Wesley, New York, [8] F. Tassone an Y. Yamamoto. Lasing an squeezing of composite bosons in a semiconuctor microcavity. Phys. Rev. A, 62:063809, [9] Roy J. Glauber. The quantum theory of optical coherence. Phys. Rev., 130(6): , [10] Y. Yamamoto an A. Imamoglu. Mesoscopic Quantum Optics. Wiley-Interscience, [11] T. Horikiri, P. Schwenimann, A. Quattropani, S. Hofling, a. Forchel, an Y. Yamamoto. Higher orer coherence of exciton-polariton conensates. Phys. Rev. B, con-mat/ v2, (to be publishe). [12] G. Roumpos, C-W. Lai, T. C. H. Liew, Y. G. Rubo, A. V. Kavokin, an Y. Yamamoto. Signature of the microcavity exciton polariton relaxation mechanism in the polarization of emitte light. Phys. Rev. B, 79:195310,