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Interferometric test of the electron mass shift in a cavity

D. K. ROSS

Department of Physics and Astronomy, Iowa State University - Ames, IA 50011, USA (ricevuto il 26 Settembre 1996; approvato il 3 Aprile 1997)

Summary. — We show that an electron interferometer experiment can investigate

the QED mass shift of an electron in a plane parallel cavity. The mass shift for a localized Gaussian wave packet in a Penning trap is expected to be small and the same as the classical electrostatic potential energy. A plane wave interacting with the walls of an interferometer, however, gives a much larger and non-classical result. The experiment appears both feasible and interesting.

PACS 12.20.Fv – Experimental tests.

1. – Introduction

A number of measurable effects due to the electromagnetic vacuum are known. There are measurable vacuum effects that are realized when the vacuum interacts with charged particles as in, for example, the Lamb shift [1] and quantum noise in electronic devices [2], and when physical boundary conditions are altered as in the Casimir effect [3], the Casimir-Polder force [4], and spontaneous emission in cavities [5]. Hinds [6] and Haroche [7] have reviewed some aspects of the cavity QED experiments. A further vacuum effect that is, in principle, measurable is the Davies-Unruh effect [8, 9] which is due to an altered physical vacuum as perceived from an accelerated frame of reference. More closely related to the present proposed experiment is the paper by Brown et al. [10] in which QED effects modify the cyclotron motion of an electron in a microwave cavity. In their experiment, wavelengths corresponding to the cyclotron motion frequency are modified by QED, in a way similar to the shift in frequency of Rydberg atoms in cavities alluded to above [5].

In the present paper we propose a new experiment in which the mass of an electron is shifted to a new value when it is placed between two plane parallel conducting plates, and this mass shift is observed in an electron interferometer. This shift is due to a shift in the mass renormalization because of boundary conditions at the surface of the plates. We discuss the expected mass shift briefly in the next section and then look at a proposed interferometer experiment in sect. 3.

2. – Mass shift of an electron between conducting plates

The rest mass shift between parallel conducting plates was first calculated by Barton [11] using so-called nonrelativistic QED. The relativistic result was obtained by Babiker and Barton [12]. The calculation yielded ultraviolet-divergent terms, making a momentum cut-off necessary. Kreuzer and Svozil [13] used the Euler-Maclaurin formula for the case of a plane wave between the two conductors and found

Dm 42 a

2 a[ ln ( 4 am) 11] , (1)

where a is the separation of the two planes. Kreuzer [14] considered localized Gaussian wave packets with no overlap with the plates and found

Dm 4 a

4 a[c(t) 1c(12t)22c(1) ] , (2)

where t 4d/a and d is the distance of the wave packet from one of the conducting planes. Limiting cases are

Dm 42a aln 2 , for d 4a/2 (3) and Dm 42 a 4 d , for d b a . (4)

The result (2) for the localized Gaussian wave packets is the same as the classical result for a point charge between two parallel conducting plates. We can easily see this for the d 4a/2 case. If the electron is midway between two parallel conducting plates a distance a apart, the infinite sequence of image charges is the same on both sides of the plates. If the plates are assumed to be large compared to their separation, the image charges are a plus charge a distance a/2 from the nearest plate, a negative charge a distance 3 a/2 from the nearest plate, a plus charge 5 a/2 away, a negative charge 7 a/2 away, etc. The resulting potential energy of the electron interacting with these image charges is then

PE 42ah a , (5)

where a is the fine-structure constant and

h 4121/211/321/41. . . . 4ln 2 . (6)

(Note that the potential energy of a single electron a distance x from a single conductor is 2a/4x. ) We can write this potential energy as an effective mass shift which agrees with (3).

Another sure indication that the mass shift for a localized wave packet between conducting plates is classical is that (2) does not contain ˇ. If we look at the plane-wave case in (1), however, we see that it is not classical. ˇ explicitly appears in the ln term in

(1) since 4 am contains the Compton wavelength of the electron. This corresponds with our intuition. A plane-wave electron interacts directly with the plates and is intrinsically quantum mechanical, while the localized Gaussian electron is at least approximately classical in its behaviour. It is interesting that (1) follows from (2) in a limited sense, if (2) is averaged over t, the position of the localized electron. The average blows up when the electron is near either plate and a minimum distance must be inserted. If this minimum distance is the electron Compton wavelength divided by 4, (1) is obtained. This gives a clearer idea of the relationship between the Gaussian and the plane-wave results.

Since the plane-wave case includes true QED effects beyond what could be derived classically, it is the more interesting of the two cases to test experimentally. It is especially nice that the ln term in (1) can be large. Penning trap experiments by their nature involve localized electrons where (2) is appropriate. Since (2) is the same as the classical expression, any measurement of the mass shift in these experiments is not terribly interesting. An interferometer experiment lends itself to testing the more quantum mechanical plane wave case and we explore that below. An interferometer experiment can test the QED mass shift beyond what we would expect classically.

3. – Electron interferometer experiment

We envision using an electron interferometer to observe the QED mass shift for the plane-wave case in (1). The electron beam is split into two parts one of which passes between two conducting plates and the other of which does not. Let the separation of the conducting plates be a as above and the length of the electron path between the plates be D with a b D. The phase shift of the electron beam which passes between the plates relative to the one which does not can then be written

Df 4 D

ˇ (Pin2 Pout) , (7)

where Pin is the momentum of the electron between the plates and Pout is the momentum of the electron outside the plates. (7) follows from simple de Broglie wavelength considerations or from the detailed solution of the Schrödinger equation with an attractive «potential» V 42Dmc2. (7) can be written as

Df 4 DE

ˇc2(vin2 vout) , (8)

where the v’s are the respective velocities and E is the common energy of the two beams. Energy conservation gives

min

k

k

(9) can be solved for vinand the result put into (8) to give finally Df 4 DE ˇ_c [

k

We can write this approximately as Df 42D ˇ c Dm c vout , (11)

where we assumed that the velocity is non-relativistic, consistent with using the Schrödinger equation, and we also assumed that Dm/m b vout2 /c2 which is easily satisfied. Dm is given by (1) for the plane-wave QED case. We will see below that for practical electron interferometers and plates, the plane-wave approximation is much better than a localized Gaussian approximation.

Let us now look at electron interferometers and at some numbers to get a feel for the feasibility of the experiment. Chambers [15] used an electron interferometer with a small magnetic whisker between the beams to observe the Aharonov-Bohm effect. Subsequent Aharonov-Bohm experiments by Möllenstedt and Bayh [16-18] put a magnetic solenoid between the beams and achieved a beam separation of 60 mm. More recent electron interferometers [19-23] can achieve beam separations of 200 mm or more.

Let us look at some numbers and see if an interferometer experiment is feasible. Putting (1) into (11) gives

Df 4a D 2 a c vout ( ln 4 am 11) , (12)

where m is the inverse electron Compton wavelength. Let us assume a beam separation of 120 mm with each coherent beam about 25 mm wide using a 32 kV accelerating potential. Such an interferometer has been used previously [19]. One way to do the experiment would be to start the interferometer and then bring up a very tiny set of parallel plates in such a way that one beam passes between the plates and the other beam is relatively undisturbed. The phase shift would be observed as the plates were brought up. Let us take the conducting plates to be 250 mm long and 125 mm wide separated by 25 mm. Since the plate separation is the same as the coherent beam width and the plate dimensions are substantially larger, the plane-wave case (1) is a much better approximation than the localized Gaussian case (2). The plates could be fabricated on a non-conducting base using photolithography [24]. Putting these numbers and vout/c 40.34 (marginally non-relativistic) from the accelerating potential into (12) gives a phase shift of 2.2 rad or 0.35 fringes. Electron interferometers can be operated at much lower accelerating potentials [21]. If we use 1,000 volts then vout/c 4 0.063 and the phase shift is 11.9 rad or 1.9 fringes. The experiment looks feasible though very difficult. Note that the ln term in (1) is making a substantial contribution. (1) gives a result a factor of 14.7 larger than (3) would give for a Gaussian packet centered between the plates. The latter is also the classical result. Since this proposed experiment can measure true non-classical QED effects on the mass shift of the electron, it seems well worth doing.

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