where L is the Lindblad superoperator, which acts on

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p in the following way24:

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AtA A AAtA} (8.116) L P == -1- {2A AAt - a ap - pa a . apa 2tcav We can integrate (8.115) formally and write the reduced-density matrix for the field at the time of the arrival of the (i + l)th atom at the cavity as

p(ti+d

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etpLp(ti + tint).

(8.117)

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22 A superoperator is an operator that operates on ordinary operators. 231t does not remain diagonal if the atoms are injected in a quantum superposition of upper and lower levels. 24 For simplicity, we assume that the temperature is low enough for the average number of thermal photons to be negligible.

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THE MASER, THE LASER, AND THEIR CAVITY QED COUSINS

It is very difficult to control experimentally the time between successive atoms in an atomic beam. Here we follow Filipowicz et aL [209] and assume that the time interval tp has an exponential probability distribution (ljlp) exp( -tjlp), with mean lp. Assuming that the fluctuations of each tp are statistically independent. exp( tpL) and p(ti + tint) in (8.117) will not be correlated and the average yields the product of two separate averages: one over the present t p, the other over previous tp's on which p(ti + tint) alone depends (see the appendix of [209] for a formal proof of this). Calling p such an averaged p, we can write the evolution of the field in the micromaser as

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(8.118) The steady state of the mapping (8.118) is given by25 (8.119) Using that the ({nl, In))-matrix element of (8.116) is

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we find that (Problem 8.8)

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(8.120)

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II ;x sin

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(8.121)

where we have also approximated lp = lHI -ti - tint ~ tHI -ti = tat. Notice that N ex is the counterpart in a micromaser of the number of excited atoms of the gain medium in a laser. In a micromaser, however, the interaction time can be varied by selecting a different speed for the atomic beam, whereas in a laser it is always fixed by the short lifetime of the atomic energy levels. So, in the micromaser, the quantity that plays the role of dimensionless pumping parameter is (8.122)

Figure 8.6 shows the average photon number and its fluctuations as functions of () obtained numerically using (8.121). The narrow dips in the curve of the average photon number are the signature of trapping states. A trapping state is a unique

25This steady-state equation holds even under more general assumptions than we have made here, see [417].

THE SINGLE-ATOM MASER

1\ t! V

50 40 30 20

.....,

0 0 2 4 6 9ht 8

8 6 4 2 0 0 2 4 9ht 6 8

Figure B.6 The plot on the left shows the average photon number as a function of the dimensionless pumping parameter 6 (8.122) in a micromaser with N ex fixed at SO. On the ((n2) - (n}2) / (n). The right-hand side you see the corresponding Fano parameter f several dips on the average photon number curve, reflected also in the behavior of the Fano parameter, are the tail signatures of trapping states.

consequence of the coherent quantum dynamics of the micro maser that is not found in ordinary masers and lasers. It is formed when Nex and () are such that each atom undergoes an integer number of Rabi oscillations in the cavity and exits in the excited state, without changing the number of photons in the cavity. All subsequent atoms will also exit in the excited state (unless N ex and () are varied), stopping the growth of the photon population at a given maximum photon number. For this reason, the micromaser fields corresponding to trapping states are highly nonclassical. They are also very sensitive to thermal photons, as any fluctuation in the photon number increasing it beyond the maximum photon number will take the micrornaser out of a trapping state. Fluctuations in the interaction time, tint, can also take the micromaser out of a trapping state. This need for very low temperatures and for precise control of the interaction times has prevented the realization of trapping states until very recently. With recent technological advances, however, trapping states are even being exploited in schemes for the generation of Fock states. Trapping states can be characterized by two numbers: the maximum number of photons and the number q of complete (21r) Rabi flips each atom undergoes when the micrornaser reaches that trapping state. The trapping-state condition

(8.123)

then gives us the interaction time that would yield that particular trapping state. Now, to fix N ex and vary e, as in Figure 8.6, amounts to fixing the number ofexcited atoms in the gain medium and varying the interaction time. In a laser, however, it is the (effective) interaction time that is fixed by the short lifetime of the atomic energy levels involved in optical transitions, as we mentioned above. Only the number of excited atoms in the gain medium (pump rate) can be varied in a laser. So to make a comparison with the typical laser features discussed in Section 8.3, we will break