Chapter 9Master Equation Approach toMatter-Wave LasersIn the previous chapter we have presented the quantum theory of matter-wave lasers in theHeisenberg picture. We started with the classical equations of motion, Gross-Pitaevskiiequation for the (c-number) condensate order parameter and the rate equation for the (realnumber) pump reservoir population. In order to conserve the proper commutator bracketfor the corresponding operators (q-numbers) against dissipation processes, we introducedthe noise operators. The resulting equations are the Heisenberg-Langevin equations. Inspite of the transparency in their physical interpretation in terms of the correspondingclassical equations, it is generally impossible to solve the Heisenberg-Langevin equationsin a nonlinear regime. We introduced the linearization approximation to circumvent thisdifficulty and calculated the noise spectra. In this chapter we will present an alternativeapproach based on the master equations in the Sch¨odinger picture. Even though we losethe classical-quantum correspondence in this second approach, we can solve the masterequations exactly. Therefore, it is useful to study this alternative approach to gain morequantitative results.9.1 Four fundamental assumptions in the quantum theoryof an open-dissipative systemWhen we study a small quantum system which dissipatively couples to a large externalworld and in return receives a fluctuating force from the external world, we can usuallyintroduce the following assumptions [1].9.1.1 Born approximationIn spite of the mutual coupling between a small quantum system A and large reservoir(external world) B, the reservoir has a very fast internal relaxation process so that aquantum correlation between the two systems is quickly lost. Therefore, we can write thetotal density op erator in a product state:ˆρAB(t) = ˆρA(t) ⊗ ˆρB(t), (9.1)1where ˆρAand ˆρBare the density operators for the system and reservoir, respectively.9.1.2 Markov approximationA reservoir consists of many (or infinite) degrees of freedom, so that it has a very shortmemory (correlation) time. During such a short memory time τ of the reservoir, thechange in the system state can be neglected. Therefore, the system density operator in(9.1) satisfiesˆρA(t + τ) ' ˆρA(t). (9.2)9.1.3 Reservoir approximationThe reservoir is so large that the reservoir state is not affected by its coupling to thesystem. Therefore, the reservoir density operator is independent of time and often atthermal equilibrium condition:ˆρB(t) ' ˆρB(0). (9.3)9.1.4 Rotating wave approximationThe coupling KABbetween the system and the reservoir is orders of magnitude smallerthan the transition frequencies ωn− ωmof both systems:KAB¿ ωn− ωm, (9.4)where ωnand ωmare the adjacent discrete energy levels of the system or reservoir. There-fore, only energy conserving (or nearly energy conserving) transitions are taken into ac-count in the analysis.9.2 Master equation9.2.1 Basic modelingWe start with the Liouville-von Neumann equation for the total density operator [2]:ddtˆρAB=1i~hˆHint, ˆρABi, (9.5)whereˆHintis the total Hamiltonian of the coupled system and reservoir in the interactionpicture. The iterative solution for the reduced density operator for the system is writtenasˆρA(t+τ) ≡ T rB[ˆρAB(t + τ)] = T rB½ˆρA(t) ⊗ ˆρB(t) +1i~Zt+τtdt1hˆHint(t1) , ˆρA(t) ⊗ ˆρB(t)i+µ1i~¶2Zt+τtdt1Zt+τtdt2hˆHint(t1) ,hˆHint(t2) , ˆρA(t) ⊗ ˆρB(t)ii), (9.6)where we truncate the interaction at the second order.2Let us study first the particular interaction HamiltonianˆHintof the formˆHint= ~g³ˆaˆb++ ˆa+ˆb´, (9.7)where ˆa (ˆa+) andˆb³ˆb+´are the annihilation (creation) operator of the condensate particle(quantum system) and the pump reservoir particle (reservoir), respectively. The pumpreservoir density operator is given by the statistical mixture of the one particle state andthe vacuum state:ˆρres=µρ1100 ρ00¶|1i|0i= ρ11|1ih1| + ρ00|0ih0|. (9.8)The pump reservoir state does not have passes a quantum coherence (off-diagonal term)and is independent of time due to the reservoir approximation. The initial state for acombined system-reservoir density operator at a time t = 0 is given in the matrix form ofthe reservoir co ordinate {|0i, |1i} [3]:ˆρAB(0) =µρ11ˆρA(0) 00 ρ00ˆρA(0)¶. (9.9)If we recall the reservoir creation and annihilation operators are identical to the projectors:ˆb+= |1ih0| andˆb = |0ih1|, the interaction Hamiltonian (9.7) is rewritten in the samematrix form asˆHint= ~gµ0 ˆaˆa+0¶. (9.10)Because of the Born-Markov approximation, the commutatorhˆHint, ˆρA(t) ⊗ ρB(t)idoes not change during a reservoir relaxation time (or rather interaction time in thiscase) τ. Also because of the reservoir approximation, ˆρB(t) is time-independent. Basedon these considerations, we can rewrite (9.6)ˆρA(t + τ) = T rBnˆρA(t) ⊗ ˆρB(t) +τi~hˆHint, ˆρA(t) ⊗ ˆρB(t)i+τ22(i~)2hˆHint,hˆHint, ˆρA(t) ⊗ ˆρB(t)ii¾. (9.11)The first-order interation term (second term of R.H.S) of (9.11) is evaluated ashˆHint, ˆρA(t) ⊗ ˆρB(t)i= ~gµ0 ˆaˆa+0¶µρ11ˆρA00 ρ00ˆρA¶− ~gµρ11ˆρA00 ρ00ˆρA¶µ0 ˆaˆa+0¶= ~gµ0 ˆaρ00ˆρA− ρ11ˆρAˆaˆa+ρ11ˆρA− ρ00ˆρAˆa+0¶, (9.12)so that this term vanishes after taking the trace over the reservoir coordinateT rBnhˆHint, ˆρA⊗ ˆρBio= 0. (9.13)3Similarly we can evaluate the second-order interation term (third term of R.H.S) of (9.11)ashˆHint,hˆHint, ˆρA(t) ⊗ ˆρB(t)ii= (~g)2µˆaˆa+ρ11ˆρA− 2ˆaρ00ˆρAˆa++ ρ11ˆρAˆaˆa+00 ˆa+ˆaρ00ˆρA− 2ˆa+ρ11ˆρAˆa + ρ00ˆρAˆa+ˆa¶,(9.14)so that this term has a non-zero contribution to (9.11):T rBnhˆHint,hˆHint, ˆρA(t) ⊗ ˆρB(t)iio= (~g)2©ρ11£ˆaˆa+ˆρA+ ˆρAˆaˆa+− 2ˆa+ˆρAˆa¤+ ρ00£ˆa+ˆaˆρA+ ˆρAˆa+ˆa − 2ˆaˆρAˆa+¤ª(9.15)Using (9.15) in (9.11), we obtain the system operator evolution to the second orderˆρA(t + τ) = ˆρA(t) −12(gτ)2©ρ11£ˆaˆa+ˆρA+ ˆρAˆaˆa+− 2ˆa+ˆρAˆa¤+ρ00£ˆa+ˆaˆρA+ ˆρAˆa+ˆa − 2ˆaˆρAˆa+¤ª. (9.16)We can model the system-pump reservoir interaction in a matter wave laser by thefollowing picture. We inject a particle into the pump reservoir state at a rate r per secondwith a finite probability ρ11. With the probability of ρ00= 1 − ρ11, we do not injecta particle. The injected particles interact with the condensate for a time