1830 B. R. MOLLOW 5 This problem has been treated for the case n =2 by B. R. Mollow, Phys. Rev. A 4, 1666 (1971). 2A perturbation theory analysis of multiphoton processes may be found in W, Heitler, The Quantum Theory of Radiation, 3rd ed. (Clarendon, Oxford, England, 1957), Chap. 4. 3Multiphoton absorption from weak quantum-mechanical fields with arbitrary statistics has been discussed by B. R. Mollow, Phys. Rev. 175, 1555 (1968); G. S. Agarwal, Phys. Rev. A 1, 1445 (1970). A. 'Gold and I'. Hernandez, Phys. Rev. 139, A2002 (1965). T. Oka and T. Shimizu, Phys. Rev. A 2, 587 (1970), have observed strong (microwave ) 2-photon excitations in inversion nonsymmetric cases. In their analysis of the problem, the effect of states other than the resonantly coupled pair is ignored. This approximation requires a special justification, since in general the contribution of other states to the effective coupling constant is compar- able to that of the original pair, unless the other states are very far away in energy or are weakly coupled by the dipole-moment operator to both states of the resonantly coupled pair. 6R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948). PHYSICA L REVIEW A VOLUME 5, NUMBER 4 APRIL 1972 Analog of the dc Josephson Effect in Superfluid Helium J. P. Hulin, C. Laroche, A. Libchaber, and B. Perrin Groupe de Physique des &olides, Ecole Normale &uperieure, Paris 5, France (Received 2 August 1971) An experiment is described in which we study the flow of helium through a 20-p, orifice which connects two helium baths. Closed capacitors are used to measure the helium level. Flow-gravitational-head difference characteristics are traced for various temperatures from 1.15 to 2. 26'K. Subcritical and supercritical flow are clearly observed, and the effect of temperature on the relation between the chemical potential and the level difference is discus- sed. Abrupt appearance of supercritical flow is observed. With a quartz transducer placed in front of the coupling hole, a decrease and subsequent increase of the flow velocity with quartz voltage is measured. I. INTRODUCTION We report here various experiments on the flow of superfluid helium between two baths through a small hole(20 p diam and 15 g thickness). This setup approximates a Josephson junction. In such, a weak link, pressure gradients, vortex motion, and phase variations are localized in the very small volume of the hole and its vicinity. Richards, and later Khorana, have used this coupling system for a clear experimental study of the ac Josephson ef- fect; their experiment showed that the vortex mo- tion through the hole has an average frequency re- lated to the chemical-potential difference hv= rfIgbz, where M is the helium-level difference between the two baths; this motion is synchronized by a quartz transducer. Their results tend to indicate that there is no independent movement of vortices across the hole, which thus acts as a reasonable Josephson weak link. Other flow experiments through thin channels have been reported, using either powders, long and narrow slits, ' or long capillaries'; in all these systems vortex motion may appear randomly in different parts of the channel, and the phase varia- tion of the order parameter cannot be simply ana- lyzed. In our experiments with thin holes, we have stud- ied the superfluid flow under zero chemical poten- tial, the appearance of a critical velocity, and the variation of the dc current with a gravitational chemical-potential difference. These experiments have been done at various temperatures (1. 15 & T &2. 2'K) in order to study the effect of the normal fluid. The chemical-potential difference is created by a displacement of a plunger moving at different constant velocities. We observe a sharp transition from subcritical to critical flow for all tempera- tures. Above the critical value we have two re- gimes: For high level differences (2 mm or more) the flow rate reaches a nearly constant value of about 40 cm/sec for a 20- p hole; below this level difference one measures a smaller velocity, and the critical value obtained for a very small differ- ence in levels is v, = 25+ 5 cm/sec. Finally we have studied the effect of a quartz transducer on the dc Josephson current; the criti- cal current is strongly reduced for low voltages ap- plied to the quartz; one can reach a cancellation of the flow for a critical voltage. For higher quartz excitation its own pumping action is predominant.