Photodetachment microscopy in a magnetic field Christophe Blondel Laboratoire Aimé-Cotton, Centre national de la recherche scientifique, université Paris- sud, F Orsay, France CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

Introducing photodetachment microscopy V. D. Kondratovich V. N. Ostrovsky J. Phys. B: At. Mol. Opt. Phys. 23 (1990) With  =120  eV F = 300 V/m z 0 = 0.51 m

A double-pass scheme, to measure the Doppler shift C D negative ion neutral atom h  A Electron affinity A :

Acquisition movies (detachment of S - ) = pm F=258.6 V/m from 2 to 2000 s = pm F=258.4 V/m from 2 to 1200 s

Photodetachment microscopy with a molecule OH - Q3(4) threshold F = 295 V/m  = pm accumulation time 1500 s 4 February 2002

What about the influence of a magnetic field ? laserinterferograms negative ion Maximum enclosed area : m 2 Flux at 1  T : +/ Wb What is, more generally, the influence of a magnetic field in a charged- particle interferometer ? Dirac flux quantum : What happens with the interference pattern ?

Do fringes and trajectories shift equally in matter-wave interferometers ? What is the action of a force on interference patterns ? What have experiments already demonstrated ? ‘Electric’ vs. ‘magnetic’ forces Is the ‘magnetic’ case strictly analogous to rotation ? What actually happens to photodetachment microscopy in a transverse magnetic field ?

Forces are not required to produce fringe shifts ! (1959) Phys. Rev. 115 (1959) 485

Semi-classical description of a charged-particle interferometer with magnetic flux for an emitter of fixed energy , the phase of every amplitude is equal to 2  /h times the reduced action In the presence of a vector potential, the action difference receives an additional term source (energy  ) 1 2 electron current If trajectories are unperturbed, the momentum just receives an additional term The classical action being

Forces are not required to produce fringe shifts ! (1949) Proc. Phys. Soc. B 62 (1949) 8

The global shift beside the experimental demonstration of the A-B effect (1960) 4 Möllenstedt & Düker, Z. Phys. 145 (1956) 377 Phys. Rev. Lett. 5 (1960) 3

Global shift described normal for Young’s slits experiments … for the case where a deflecting field is a force applied in a small strip after the slits. Phys. Rev. D 8 (1973) 1679

An experimental demonstration of a global shift (Shimizu et al. 1992) Jpn. J. Appl. Phys. 31 (1992) 436

The effect of an additional potential-gradient force is just a shift of the pattern In photodetachment microscopy, an additional electric field is an additional force, just a change in z0z0

What the semi-classical description says of the general magnetic case The probability current is carried around classical trajectories, i.e. along those quantum paths that correspond to an extremum of the action : source (energy  ) 1 2 electron current Not only does the reduced action W=S+  T get an additional term but the trajectories are modified too !

Orders of magnitude of the action integrals and action differences 1  T → WW  Js  20 h  = 310  eV F = 300 V/m z 0 = 0.5 m  = 0.1 mm Area : m 2 At the  T level, B-induced phase-changes are not infinitesimal ones !

What does theory say for the magnetic case ? For parallel electric and magnetic fields (weak magnetic field case) T. Kramer, C. Bracher and M. Kleber, Europhysics Letters 56 (2001) 471 The surface enclosed between trajectories is perpendicular to the field, the magnetic flux is thus only a second- order quantity !  = 60.8  eV, F=116 V/m, B=1 mT

Beam S Parallel fields : the experiment = nm

Parallel fields with a stronger magnetic field Always with parallel electric and magnetic fields !  = 100  eV, F=15 V/m, B=20 mT mm

Rotating interferometers Phys. Rev. A 54 (1996) 8 The classical deflection, in the Moiré deflectometer, is equal to the fringe deflection of the Mach-Zehnder interferometer (both for rotation and linear acceleration).

Can we use the analogy with rotation ? Lorentz force Coriolis force Analogy : - This is only a 1 st order analogy. Even without moving in the rotating frame, you undergo a centrifugal force ! - What would the electric field become in the non-rotating frame ? - Rotation is centrifugal, magnetic forces are centripetal.

What about the transverse field case ? Relativity … For orthogonal electric and magnetic fields, a reference frame exists which renders the EM field purely electric (or magnetic). As F 2 - c 2 B 2 is a relativistic invariant, the effective electric field would be rescaled to a smaller value. z0z0 The velocity of the new reference frame being c 2 B/F, a field of 1  T for 300 V/m appears very critical ! z0z0

How can we know ? Let us perform the experiment !

Experiment with a transverse field ( 1 )  = 83.5(2)  eV, F=291 V/m B=0 S-S-

A vectorial demonstration ( 1 ) source (energy  ) 1 2 electron current Additional acceleration :Additional velocity : Magnetic shift : The fringes shift by a quantity such that the phase change inside the original interference pattern is compensated by the magnetic phase : Would make an acceptable ?

A vectorial demonstration ( 2 ) source (energy  ) 1 2 electron current Trajectories have the same final phase difference as with B=0 if With the same force applied along both trajectories the difference of velocities is a constant Calculating the effect of the Lorentz force along an average trajectory

A vectorial demonstration ( 3 ) area At 1 st order, pairs of trajectories are shifted just to keep a constant phase difference

DDP= Experiment with a transverse field ( 2 )  = 80.9(2)  eV, F=291 V/m S-S (5) rad 17.28(5) rad

The second order magnetic effect Phase maximum : Closed orbit

Conclusions Photodetachment microscopy offers a spectacular example of the identity of fringe and trajectory shifts in interference patterns. PDM interferograms are extremely robust against perturbations. Electron affinity measurements are even more reliable than initially thought.

Reference Europhysics Letters 82 (2008) 20005

Coworkers Cyril Drag Walid Chaibi Christian Delsart