�k
�k
�k
�k
Lasing BEC
Lasing BEC
Source BEC
Source BEC
(a)
(b)
Freely-falling
atoms
0�k
momentum-transfer
process
2�k
momentum-transfer
process
Figure 4.4: Illustration of the two potential pumping processes that could occur in the continuous
pumping experiment. The freely-falling atoms form the atom laser in the |2,0i state and can
undergo stimulated transitions into the lasing condensate in one of two processes. If the atoms are outcoupled from the lower part of the source condensate, they may immediately absorb a pumping photon from below and emit one in the same direction to decay into the lasing condensate with no net change in momentum (a). If the atoms are outcoupled from the centre of the source condensate,
they may fall under gravity for approximately 1 ms gaining approximately 2~kof momentum before
absorbing a pumping photon from below and emitting one downwards to decay into the source
condensate with a net momentum transfer of 2~k(b).
4.4
Simple single-mode model
We begin our theoretical investigation of the pumping mechanism behind the previously- described experiment by considering the simplest-possible model, a single spatial-mode mean-field approximation to the process illustrated in Figure 4.1(a). The equations of motion for this model are
d
dtcsource=−iΩ ∗c
excited, (4.4a)
d
dtcexcited =−iΩcsource−igαclasing−i∆cexcited− Γ 2cexcited, (4.4b) d dtclasing=−ig ∗α∗c excited, (4.4c) d dtα=−ig ∗c∗ lasingcexcited− γ 2α, (4.4d)
wherecsource, clasing andcexcited are the amplitudes of the source|1i, lasing |2iand excited
|3imodes respectively, Ω is the complex Rabi frequency due to the pumping laser coupling the source and excited modes with detuning ∆,α is the amplitude of the optical mode into which the excited atoms emit when decaying into the lasing mode, andg is the complex coupling constant for this transition. The optical mode α will decay as photons propagate
away from the system. This process is modelled phenomenologically with a loss rate γ from the optical mode α. The spontaneous decay of the excited state into modes other than the lasing mode occurs at a rate Γ. It has been assumed that the source mode is sufficiently dilute that the pumping laser is negligibly absorbed. This assumption is relaxed in Section 4.5. The evolution equations (4.4) are given in a rotating frame in which the energy difference between the source, excited and lasing modes have been appropriately removed.
In deriving (4.4) it has been assumed that atoms that undergo spontaneous decay from the excited mode will have no further impact upon the system. In particular, this means that the absorption of photons in the α mode by atoms in modes other than the lasing mode has been neglected. The absorption of these photons by the lasing mode is, however, retained. As discussed in Section 4.2 this is a valid approximation in the boson accumulation regime in which an excited atom is significantly more likely to decay into the lasing mode than into any other mode.
The long-term dynamics of (4.4) will determine the usefulness of the process as a pump- ing mechanism. The fast-timescale behaviour of this system can therefore be eliminated. By far the fastest process in the system is the decay of the photons in theα mode as they leave the system. The time taken for a photon to cross the width of a typical condensate (∼10µm) is ∼10−13s, giving γ ∼1013s-1. By comparison, the spontaneous decay rate of the excited mode is Γ∼108s-1 for the F0 = 1 manifold of 87Rb. As the α mode reaches a quasistationary value on the fastest timescale in the system (γ), it can be adiabatically eliminated and replaced with its quasistationary limit,
α≈ −2ig ∗ γ c
∗
lasingcexcited. (4.5)
We next assume that the pump laser is driving the atoms in the weak-field regime, Ωmax (∆,Γ). In this limit, the excited mode cexcited evolves on a more rapid timescale than either the source or lasing modes. The excited mode may therefore also be adiabatically eliminated and replaced with its long-term average
cexcited ≈ − iΩcsource 1 2Γ + 2 |g|2 γ Nlasing+i∆ , (4.6)
whereNlasing=|clasing|2 is the number of atoms in the lasing mode.
§4.4 Simple single-mode model 115
the remaining two modes are obtained by substituting (4.5) and (4.6) into (4.4), d dtNsource=−(Γ + Γ 0)N excited=− | Ω|2(Γ + Γ0) 1 4(Γ + Γ0)2+ ∆2 Nsource, (4.7a) d dtNlasing= Γ0 Γ + Γ0 −dtdNsource , (4.7b) where Γ0= 4|g| 2
γ Nlasing is the rate constant with which atoms in the excited mode decay into the lasing mode.
The efficiency of the transfer of atoms in the pumping process is Γ0/(Γ + Γ0), which
behaves as expected: as the occupation of the lasing modeNlasingis increased, the efficiency increases due to bosonic stimulation. What is perhaps not expected is the behaviour of the transfer rate constant. In the limit of large detuning, ∆Γ + Γ0, the rate constant for atom transfer is
−d dtNsource Nsource ≈ | Ω|2 ∆2 Γ + Γ 0 , (4.8)
which increases as the lasing mode population increases (due to increasing Γ0). In the limit of small detuning (∆Γ + Γ0) however, very different behaviour is obtained
−d dtNsource Nsource ≈4 | Ω|2 Γ + Γ0, (4.9)
which decreases as the lasing mode population increases. This behaviour is due to the depletion of the excited state as Γ0 increases (and therefore the occupation of the lasing mode increases). In the limit of small detuning, spontaneous and stimulated decay are the fastest processes reducing the occupation of the excited state, hence increasing either of those rates will reduce the overall population of the excited state. In this limit, as the excited state population decreases proportionally to the squared sum of these rates
Nexcited ≈4 | Ω|2
(Γ + Γ0)2Nsource, (4.10)
the overall rate of atom transfer is suppressed [refer to (4.7a)]. In the opposite limit of large detuning, the spontaneous and stimulated decay processes are a perturbation on the Rabi-flopping process which populates the excited state. In this limit, the excited state
population is independent of the spontaneous and stimulated emission rates
Nexcited ≈ | Ω|2
∆2 Nsource, (4.11)
and the overall atom transfer rate into the lasing mode can be Bose-enhanced.
R
The simple model developed in this section can be applied to the continuous pumping experiment described in Section 4.3 to give a limit for the efficiency of the ‘2~k’ momentum- transfer process. In this process, falling atoms outcoupled from the upper condensate reach a momentum of 2~k vertically downwards before absorbing a pump photon of momentum
~k from below and emitting a photon of similar momentum directed downwards to decay into the lasing mode. If it is assumed that the intensity of the pump mode is approximately constant throughout the system, then the rate of transfer of atoms into the lasing mode cannot be faster than the rate of spontaneous emission while the source atoms are not momentum-resonant with the lasing mode1. An upper bound for the efficiency of the ‘2~k’ momentum-transfer process should therefore be given by the ratio of the time for which the source atoms are momentum-resonant with the condensate (∼100µs, see Section 4.3) to the fall time for the atoms to that point (∼1 ms). As the upper bound for this process (∼10%) is lower than the observed efficiency (∼35%), either the transfer of atoms into the lasing mode must significantly reduce the intensity of the pump mode (and therefore reduce the spontaneous losses experienced as the atoms fall), or it must be the ‘0~k’ process that operates in the continuous pumping experiment. The argument used to obtain an upper bound for the efficiency of the ‘2~k’ momentum-transfer process does not apply to the ‘0~k’ momentum-transfer process in which atoms outcoupled from the upper condensate
can be immediately pumped into the lower condensate.
The simple model considered in this section does not take into account the behaviour of the emitted photons in the α mode as they traverse the source and/or lasing modes as they leave the system. The investigation of this and other multimode effects is the subject of the next section and will be shown to lead to modifications of the behaviour described by the simple single-mode model.
1When the source atoms are not momentum-resonant with the lasing mode, it is equivalent to there
being no atoms in the lasing mode. In this case, Γ0= 0 and from (4.9) it can be seen that the spontaneous
§4.5 Multimode model 117