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Measurement process in a variable-barrier system
22 July 1999
Physics Letters B 459 Ž1999. 193–200
Measurement process in a variable-barrier system L. Stodolsky Max-Planck-Institut fur Ring 6, 80805 Munchen, Germany ¨ Physik (Werner-Heisenberg-Institut), Fohringer ¨ ¨ Received 11 February 1999; received in revised form 20 May 1999 Editor: P.V. Landshoff
Abstract The description of a measuring process, such as that which occurs when a quantum point contact ŽQPC. detector is influenced by a nearby external electron taking up different positions, provides a interesting application of the method of quantum damping. We find a number of new effects, due to the complete treatment of phases afforded by the formalism, although our results are generally similiar to those of related treatments, particularly to those of Buks et al. These are effects depending on the phase shift in the detector, effects which depend on the direction of the measuring current, and in addition to damping or dissipative effects, an energy shift of the measured system. In particular, the phase shift effect leads to the conclusion that there can be effects of ‘‘observation’’ even when the barriers in question pass the same current. Secondly, the nature of the current through the barriers and its statistics is discussed, and a description of the correlations in the current due to ‘‘measurement’’ is obtained. In particular it is possible to see the origin of ‘‘telegraphic’’ signals, which in a certain limit give behavior resembling a ‘‘collapse of the wavefunction’’. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The measurement process using a quantum point contact ŽQPC. detector w1x can be described as the modification of a barrier whose transmission varies w2x according to whether an external electron is nearby or farther away. When the external electron is close by there is a certain higher barrier, and when it is farther away, there is a reduced barrier. Given an incident or probing flux on the barrier, the modification of the resulting current through the barrier, which can be observed by conventional means, thus ‘‘measures’’ where the external electron is located. Experiments of this type give a fundamental insight into the nature of measurement. In an elegant
recent experiment Buks et al. w1x – stimulated by the work of Gurvitz w3x – saw the expected loss of fringe contrast in an electron interference arrangement when one of the paths in the interferometer was ‘‘under observation’’ by a QPC w4x. This effect is due to the ‘‘damping’’ or ‘‘decoherence’’ arising from the creation of correlations between the position of the electron in the interferometer and the coordinates of the ‘‘environment’’ or ‘‘observer ’’ w5,7x. A number of theoretical treatments of such measurement processes have been given in recent years. Here we would like to use the ‘‘quantum damping’’ method of Refs. w5,7x, which was devised to deal with precisely such questions, and which can be used to give a complete and
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 5 9 - 0
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transparent treatment of the variable barrier problem. Furthermore it yields a number of new results. These include the effects of a phase which although it has no effect on the measuring current nevertheless contributes to the damping or decoherence, effects which depend on the direction of the detector current, and an energy shift of the observed system induced by the observing process. To elucidate the method we consider the simplest situation, the two-state system: An external electron can be in two states corresponding to two different locations. It thereby influences a barrier controlling the measuring current. Hence our variable barrier has two possibilities. A physical realization of the two-state system could be provided by two adjoining quantum dots, ‘‘left’’ŽL. and ‘‘right’’ŽR. with a resident electron tunneling between them. This two-state system is then ‘‘observed’’ by a QPC or similar detector whose barrier varies according to whether the resident electron is near or far from the detector, on dot L or dot R. According to whether this barrier is higher or lower there is then a different current in the detector circuit, thus furnishing a reading of the position of the external electron. To avoid confusion between the electron of the two-state system and the electrons of the measuring circuit, we will refer to the former as the ‘‘resident’’ or ‘‘external’’ electron, and to the latter as the detector electrons. Also, the tunneling for the external electron between the states L and R should not be confused with the tunneling of the detector or probing electrons through the variable barrier of the detector. The tunneling of the external electron will be given essentially by a tunneling energy V Žsee below., while that for the detector electrons will be characterized by an S-matrix S. Our discussion will consist of two parts, first a description of the state of the external electron, in terms of the density matrix of the two-state system. Secondly, beginning in Section 4, we give a description of the current in the detector circuit. We should perhaps stress that the two issues are distinct. An understanding of the evolution of the state of the ‘‘measured’’ system, -the external electron- does not of itself lead immediately to an understanding of the current in the detector circuit, and in fact while there have been numerous discussions of ‘‘decoherence’’
in recent years, the latter issue would appear at present to be the more subtle one. Our first focus is then on the 2 x 2 density matrix r for the two-state system; in particular we wish to study the effect of the presence of the detector on the time development of r . The density matrix is characterized by a three component ‘‘polarization vector’’ P, via r s 12 Ž I q P P s ., where the s are the pauli matrices. Pz gives the probability for finding the electron on the left or right dot via Pz s ProbŽ L. y ProbŽ R .. The other components, Ptr , contain information on the nature of the coherence. < P < s 1 means the system is in a pure state, while < P < s 0 means the system is completely randomized or ‘‘decohered’’. P will both rotate in time due to the real energies in the problem and shrink in length due to the damping or decoherence. The time development of P is given by a ‘‘Bloch-like’’ equation w5,7x P˙ s V = P y DPtr
Ž 1.
The three real energies V have the following significance in the present problem, where the twodot system is thought of as a double- well potential for the resident electron: Vz gives a possible energy difference for the two quasi-stationary states on each dot; Vz / 0 means the the double well is asymmetric. Vx and Vy are tunneling energies; Vx conserves the parity of the electron wavefunction on the two dots and Vy flips it w6x. The second term of Eq. Ž1. describes the damping or decoherence. D gives the rate at which correlations are being created between the ‘‘system ’’ Žthe external electron on the dots. and the ‘‘environment’’ Žthe detector.. The label ‘‘tr’’ means ‘‘transverse’’ to the z axis. The damping only affects Ptr because the observing process does not induce electron jumps from one dot to another; the observing process conserves Pz s ² sz :.
2. Influence of the detector The effect of repeated probings by an ‘‘environment’’ or ‘‘observing device’’ is described by a quantity L whose imaginary part gives the damping
L. Stodolskyr Physics Letters B 459 (1999) 193–200
and whose real part gives an energy shift to the system being measured. Thus we expect that the ‘‘observation process’’ here will cause an energy shift for the external electron as well as a damping or dissipation. D is thus the imaginary part of L, while the real part of L gives an observation-induced contribution to V. L itself is given by
L s i Ž flux . ² i <1 y SL SR† < i :
Ž 2.
The factor flux is the flux or probing rate of the detector electrons, where in the QPC application one can use the Landauer formula flux s eVdrp ", with Vd the voltage in the detector circuit w8x.. The label i refers to the the initial or incoming state of the electrons in the detector and the S’s are the S matrices for the two barriers Žlabeled by L and R. corresponding to the different locations of the external electron.
unitary, SS † s 1, as may be found from explicit constructions. 2.2. Time reÕersal An additional constraint arises from time reversal invariance. This states that S j, i s Si T , jT , where the subscript T means the time-reversed state. Here we have simply k T s yk and vice-versa. For the diagonal elements this gives Sk k s Sykyk while for the off-diagonal elements there is no further constraint beyond that already given by unitarity. Note that we neglect electron spin and there is no magnetic field; these further complications might be interesting in some applications. We can now parameterize the S-matrix with the time reversal constraint in terms of three angles as S s e if
2.1. S matrix In order to apply Eq. Ž2., we review shortly the barrier penetration problem in S-matrix language. At a given energy there are just two S-matrix elements for the incoming flux with wave vector k, namely Sk k and Syk k , representing transmission and reflection respectively. These are the coefficients appearing in the wavefunction of unit incoming amplitude, which far away from the barrier on the incoming side is e i k z q Syk k eyi k z
Ž 3.
while on the other, outgoing, side of the barrier at large distances we have Sk k e i k z. We may also have the incoming detector current from the other direction with wave vector yk, and with Syk yk and Skyk for transmission and reflection. Thus our S-matrix is Ss
ž
Sk k Syk k
Skyk Sykyk
/
where the two columns correspond to the two possible directions of the detector current. This matrix is
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ž
cos u ie ih sin u
ieyi h sin u cos u
/
Ž 4.
which it will be seen fulfills all the conditions just discussed. The first column contains the reflection and transmission coefficients for incoming waves k and the second column those for incoming waves with yk. The angle u , which gives the magnitude of transmission and reflection, is the same as that used by Buks et al., while the other two parameters are phases. The phase angle h creates a difference between incident waves with k and yk, that is for different directions of the detector current. This reflects a possible asymmetry in the shape of the barrier Žsee below. and leads, in the case of non-zero h , to the interesting possibility of effects which depend on the direction of the measuring current. We note that although f appears as an overall phase it is physically relevant. The phases are fixed by the ‘‘1’’ in Eq. Ž2., or correspondingly by the fact that we have 1 as the coefficient of the incoming wave in Eq. Ž3.. 2.3. Parity If the barriers in question are even in shape, another constraint arises due to parity symmetry.
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When this operation, namely z ™ yz, is applied to the wavefunction Eq. Ž3., we get a solution corresponding to a wave coming in from the other direction. Comparing coefficients we conclude that Syk yk s Sk k and Skyk s Syk k . The first condition was already obtained from time reversal, the second, however, says that for symmetric barriers we should set h s 0 in Eq. Ž4. giving equal S matrix elements for both directions of the detector current. Note that a non-zero h only affects the reflection coefficients; even with non-symmetric barriers T ensures that the transmission coefficients have the same phase.
3. Damping D is the parameter giving the damping or loss of coherence of the system, and is given by D s Im L. We label the parameters of the S matrices with an L or R for the two barriers created when the electron is on the right dot or the left dot and call f L y f R s Df , hL y hR s Dh and u L y u R s Du . We then have from the matrix Ž4.,
effect. This may help to explain why the role of f seems unfamiliar 1 According to this equation, two detector barriers which have the same transmissibility, that is the same u , but different phase shifts can nevertheless induce damping. Although they apparently give the same detector current they nevertheless establish a distinction or ‘‘make a measurement’’. This may seem less mysterious when we recall that even pure phase shifts correspond to physical effects like the delay or change in shape of wave packets. Recall that in scattering theory the entire scattering is given by just a phase, Sl s e 2 i d l . Although a change in current is the most obvious ‘‘measurement’’, it is not necessarily the only one. Thus we apparently differ with other treatments w1,4,9,10x, where the damping or decoherence is related to the detector current only. Our results are of course proportional to the flux or probing rate, but not necessarily to the transmitted current alone. In effect, the situation concerning f
1
D s Ž flux . Re 1 y e
i Df
cos Du
qsin u L sin u R Ž e i Dh y 1 .
Ž 5.
4
This applies for a given direction of the detector current; for the other direction, reverse the sign of Dh. For a symmetric barrier, this simplifies to D s Ž flux . 1 y cos Df cos Du 4
Ž 6.
Except for the cos Df factor this is essentially the same damping effect as in Ref. w1x. This factor is interesting, however, in that even for small angles, 2
D f 1r2 Ž flux . Ž Df . q Ž Du .
2
4
Ž 7.
the phase f is present and enters on an equal footing with u . Note in the opposite limit cos Du s 0 which applies for a ‘‘projection’’, where one barrier transmits only and the other reflects only, that f has no
The absence of an effect due to the phase in other treatments may be traced to taking an absolute value too soon: We wish to find the decrease in the overlap of the different environment wavefunctions associated with the two states of the two-state system due to a probing by the environment. After one probing Žwe illustrate with the notation of w1x or w5x., the overlap changes as 1™1y² x L < x R :. therefore the new value of < P < is <1y ² x L < x R :<. It will be seen that the phase of ² x L < x R : plays a role. However, if we simply use 1y <² x L < x R :<, which in effect is what was done in the other treatments, the effect of the phase is lost, and with it both the effect of f on the damping as well as the real energy shift induced by the environment. A check on these arguments, at least as far as the real energy shift is concerned, is afforded by the agreement of the real energy shift so obtained with the usual formula for the index of refraction of a particle in a medium w7x. Another way of understanding that a just a phase plays a role in damping is to consider the phase as acting on the object being measured Žhere the external electron., as in w11x. If the probe is random in time, then, as these authors find, the resulting random phase results in a damping involving the phase squared. This is an expression of the familiar scattering theory result that damping or dissipation ; Im f ; d 2 . This argument suggests that perhaps in problems with degenerate electrons at low temperature effects involving the reduction of this randomness might have to be considered. On the other hand, the transmission process by itself introduces a certain noise or randomness into the problem Žsee the review in w12x..
L. Stodolskyr Physics Letters B 459 (1999) 193–200
may be viewed as a case of the well-known tree falling in the forest with nobody there to hear it.
4. Energy shift
Vzind s Ž flux . Im e i Df cos Du qsin u L sin u R Ž e
shift will in general depend on the direction of the detector current. To find the difference effect we exchange k for yk in the initial state. For the difference in D for the two directions we have
D D s 2 Ž flux . sin Df sin u L sin u R sin Dh
While the damping is given by the imaginary part of Eq. Ž2., there is also a significance to the real part. In the description of the propagation of a particle in a medium it gives the index of refraction for the states of the particle in the medium w7x. In the present problem this state-dependent energy shift will give a measurement-induced contribution to Vz in Eq. Ž1. governing the internal evolution of the measured system w7x, Vzind s Re L so that
i Dh
y 1.
4
Ž 8.
which for an even barrier with Dh s 0 simplifies to Vzind s Ž flux . sin Df cos Du
Ž 9.
This effect only contributes to the ‘‘z direction’’ of V for the same reason D only effects Ptr , the observation process is presumed not to cause any jumps from one dot to the other. We note that the effect persists even if Du s 0, i.e. equal transmissibilities, as long as Df / 0. For small phases it is linear in Df , while the damping Eq. Ž7. is quadratic. This induced energy offers the intriguing opportunity of ‘‘tuning’’ the properties of the two-state system. The tunneling behavior of the external electron between the two dots depends strongly on how exact the degeneracy of the two wells is. Thus the ability to adjust it via the induced Vz , which is proportional to Vd and so easily adjustable, is quite interesting. For example, if it is difficult to fabricate identical dots, the induced Vz could be used to nevertheless make the two dots degenerate in energy. On the other hand a large Vz , by lifting an initial near-degeneracy tends to suppress transitions, offering another qualitative test of the theory.
5. Dependence on current direction For barriers that are not symmetric in shape, h can be non-zero, and the damping and the energy
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Ž 10 .
and for the induced energy shift the difference for the two directions
D Vz s 2 Ž flux . cos Df sin u L sin u R sin Dh
Ž 11 .
Both these effects are linear in small Dh.
6. Nature of the current The question of the current in the detector circuit poses some intriguing questions concerning the nature of ‘‘measurement’’. Should we expect a smooth current of some kind, reflecting some average transmission probability given by r ? Or should we view each transmission as a measurement which ‘‘collapses the wavefunction’’ to one dot or the other dot, leading to a series of ‘‘telegraphic’’ signals with different currents corresponding to one barrier or the other? If so, what determines the duration of these signals? The above considerations for the determination of r cannot alone answer such questions. From r we can only find the probability of a single transmission at a certain time, Žunconditional probability. but not if this event was, say, part of a long series of transmissions or a mixed series of transmissions and reflections Žconditional probability.. This sort of question arises because successive probings are in general not statistically independent; one transmission may imply an increased probability for the next one. To understand this, a helpful analogy may be a hypothetical sequence of successive Stern-Gerlach ‘‘measurements’’, performed on a single atom. There is a strong biasing effect from one observation to the next: if the spin is ‘‘up’’ passing through a first magnet, it will also be ‘‘up’’ after passing through the next one. One sees that the successive measurements are far from independent. In our present problem the Žspin 1r2. ‘‘atom’’ being measured is the two-state system of the resident electron, while the
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repeated probings by the detector circuit are the sequence of Stern-Gerlach ‘‘magnets’’. Indeed, the analogy is complete if we suppose some agency Žsay some additional magnetic field. that rotates the ‘‘spin’’ while it is undergoing the successive SternGerlach measurements. This new agency plays the role of the tunneling energy Vtr . An explicit treatment is perhaps most easily performed in an amplitude formulation 2 . Let 1 represent a transmission and 0 a reflection for the detector electrons. We write the probability amplitude that after N probings the external electron is on dot L, and that the first probing electron was transmitted, the second reflected, . . . the Ž N y 1.th transmitted and the Nth transmitted, as AŽ L,w11 . . . 01x.. Similarly there is an amplitude AŽ R,w . . . x. for the external electron to be on dot R after a given sequence of transmissions and reflections w . . . x. We consider the situation where many probing electrons are incident during the time 1rVtr it takes the external electron to tunnel between L and R : fluxrVtr 4 1
Ž 12 .
This assumption allows us, in discussing the current, to neglect ‘‘dot jumps’’ where an amplitude AŽ L,w . . . x. receives a contribution from an amplitude AŽ R,w . . . x.. Then for time periods d t such that Vtr d t < 1 we can simply write L L A Ž L, w 11 . . . 01 x . ; SkLk SkLk . . . Syk k Sk k
Ž 13 .
where we have suppressed phase factors associated with the time, and also the probability of the starting configuration. If we restore the latter, in the form of r L L for the probability of starting with L, and r R R for the probability of starting with R, we find by
squaring the amplitude the probability for a given sequence of transmissions and reflections: Prob w 11 . . . 01 x s r L LŽ pL pL . . . q L pL . q r R R Ž pR pR . . . q R pR .
where p s < Sk k < and q is the probability of no transmission, q s 1 y p s < Syk k < 2 . ŽNote these quantities are independent of the current direction k, due to T invariance..
7. ‘‘Telegraphic’’ behavior and correlations Eq. Ž14. shows how a tendency to ‘‘telegraphic’’ behavior with sequences of reflections or sequences of transmissions can arise. For example, let pL and pR be very different, say close to zero and close to one respectively. Then neither the first term nor the second term of Eq. Ž14. can be big for mixed sequences like w010 . . . 01x; but for ‘‘telegraphic’’ sequences w111 . . . 11x or w000 . . . 00x, one or the other term can be big. One type of sequence or the other will be selected. In this limit we would get a behavior resembling a ‘‘collapse of the wavefunction’’: a ‘‘1’’ leads to a further ‘‘1’’, and a ‘‘0’’ leads to a further ‘‘0’’ indefinitely, until the resident electron tunnels to the other dot. On the other hand, let pL and pR be nearly equal. Then this tendency favoring signals far from the mean will be weak; little correlation between the ‘‘measured’’ and the ‘‘measurer’’ is introduced. A quantitative expression of this point is given by Eq. Ž16. below. The two terms of Eq. Ž14. are essentially those leading to the binomial distribution in statistics. Hence if we now ask for the probability ProbŽ Q, N . for Q transmissions in N probings, the combinatorics are that of the binomial distribution, and we obtain Prob Ž Q, N . s r L L PLŽ Q . q r R R PR Ž Q .
2
Although we prefer the amplitude point of view, see w13x one could also arrive at Eq. Ž13. by collapsing the wavefunction of the dot electron after every probing: Let the wavefunction of the dot electron initially be a < L:q b < R :. After a probing it becomes L R < :. < :Ž R < : < :. a < L:Ž SkLk < k :q Syk k y k q b R S k k k q Syk k y k . If, say, a transmission takes place, we collapse to a < L: SkLk q b < R : SkRk for the new wavefunction. Doing this repeatedly and summing over all starting configurations then leads to Eq. Ž13..
Ž 14 .
2
Ž 15 .
where PLŽ Q . is the binomial expression for the probability of Q transmissions in N trials given the single trial probability pL . ŽFor N large and p small this is approximated by the poisson distribution Q PLŽ Q . f Ž n L . eyn L , with n L s pL N .. Q!
Eq. Ž15., for distinct pL and pR , leads to a twopeaked distribution, and generally describes the statistics of the current for short times. For example
L. Stodolskyr Physics Letters B 459 (1999) 193–200
the quantity ProbŽ Q1 ,Q2 . for Q1 transmissions in N1 probings, followed by Q 2 transmissions in N2 probings gives a measure of the correlations induced by the ‘‘measurement’’ process. In particular ProbŽ Q1 ,Q2 . y ProbŽ Q1 .ProbŽ Q2 . is zero if the currents in the two intervals are independent. Now ProbŽ Q1 ,Q2 . may be calculated as simply the weighted average of two parallel processes, each one calculated as a statistically independent sequence: ProbŽ Q1 ,Q 2 . s r L L PLŽ Q1 . PLŽ Q 2 . q r R R PR Ž Q1 . PR Ž Q2 ., while ProbŽ Q . is given by Eq. =P Ž15.. Thus Prob Ž Q1 ,Q2 . y Prob Ž Q1 . Prob Ž Q2 . s r L LŽ 1 y r L L . Ž PLŽ Q1 . y PR Ž Q1 . . = Ž PLŽ Q 2 . y PR Ž Q2 . .
Ž 16 .
We expect this expression to vanish when only one dot is occupied, for then successive probings are independent; it is in fact zero for r L L one or zero. Similarly it vanishes when the current provides no information on the state, when pL and pR are equal or PL s PR . The correlation defined in Eq. Ž16. thus gives a characterization of the strength of ‘‘collapse’’ or ‘‘telegraphic’’ effects. These considerations hold, as said, for times short relative to the tunneling time 1rVtr . For longer times an amplitude AŽ L,w . . . x. receives contributions from an amplitude AŽ R,w . . . x., thus we anticipate the time scale for the duration of a ‘‘telegraphic’’ signals on the order of 1rVtr .
8. Strong damping In the previous section the time scale was the tunneling time 1rVtr . However, there is a another regime w5x of behavior, which although it may not be relevant in problems where the damping is relatively weak, as when the two barriers differ little, but which is of interest in itself. This is the case of strong damping, when DrVtr 4 1. In the limiting case of strong damping the amplitude for any configuration AŽ L,w . . . x. is reached by only one ‘‘path’’ w . . . x and the situation resembles a classical diffusion problem. The solutions of Eq. Ž1. show the ‘‘Turing-Watched-Pot-Zeno’’ behavior w14x where
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‘‘measurement’’ inhibits the time evolution of the external electron and the characteristic time scale for transitions between L and R becomes the longer DrVtr2 . In this case the condition Eq. Ž12. can be weakened to flux Ž DrVtr2 . 4 1, or since D s flux in the limit, to Ž fluxrVtr . 2 4 1. Similarly the time scale for the duration of telegraphic signals becomes the longer time DrVtr2 .
9. Conclusions and applications If it were possible to fabricate the two dot system or its analogs and to carry out experiments with them, the various effects could be tested through their effect on the diagonal elements of r , or Pz . Pz Ž t . gives the probability that a dot is occupied, and through Eq. Ž1. it is influenced by the real energies and D. One way of determining it experimentally would be from the current described by Eq. Ž15., that is from the relative strengths of the two peaks of the Q distribution. It should be stressed that Eq. Ž15. represents an average over many repeated runs from the same initial condition, say the injection of the external electron onto one of the dots, and not an average over time in one run. It should also be noted that even if the evolution of r cannot be followed in detail, the time scale of the relaxation of correlations yields information on Vtr and D. For experiments of the type of Ref. w1x, the real energy Eq. Ž8. induced by the measurement will show up as a shift in the interference fringes. In Eq. Ž8. the effect is expressed as an energy, that is as a phase per unit time, which in an experiment such as Ref. w1x corresponds to a phase shift given by the dwell time of the measured electron times the energy Eq. Ž8.. Hence Eq. Ž8. predicts, for non-zero f , a Vd-dependent fringe shift. For the asymmetric barrier, Eq. Ž11. predicts a component to this fringe shift which reverses with Vd . Similarly, for the asymmetric barrier, Eq. Ž10. predicts that the loss of fringe contrast or visibility of Ref. w1x can depend on the current direction. Finally, there is the role of f . According to Eq. Ž6., if we can arrange for two detector barriers to have the same transmission but different f there still should be a loss of fringe contrast or damping.
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The fact that the fluctuations of the current in the detector circuit are related to the measurement or the damping suggests a kind of fluctuation-dissipation relation. This leads to the idea of a measurement-related noise, related to the dissipative parameter D. This will be studied elsewhere w15x.
Acknowledgements I am grateful to the Weizmann quantum dot group for introducing me to this subject and especially to E. Buks, S. Gurvitz and M. Heiblum for several discussions, and to R.A. Harris for help in clarifying the ideas. Finally, I am grateful to a number of these colleagues, and in particular A.J. Leggett, for criticisms concerning the subject of Section 6, finally leading to the viewpoint developed there. I would also like to acknowledge the hospitality of the Institute of Advanced Studies at the Hebrew University of Jerusalem, where this work was begun.
References w1x E. Buks, R. Schuster, M. Heiblum, D. Mahalum, V. Umanksy, Nature 391 Ž1998. 871.
w2x M. Field, C.G. Smith, M. Pepper, D.A. Ritchie, J.E.F. Frost G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 Ž1993. 1311. w3x S. Gurvitz, quant-phr9607029. w4x I.L. Aleiner, N.S. Wingreen, Y. Meier, Phys. Rev. Lett. 79 Ž1997. 3740. w5x R.A. Harris, L. Stodolsky, J. Chem. Phys. 74 Ž1981. 2145; Phys. Lett. B 116 Ž1982. 464. w6x J.A. Cina, R.A. Harris, Science 267 Ž1995. 832. w7x For a general introduction and review of these concepts see L. Stodolsky, Quantum Damping and Its Paradoxes in Quantum Coherence, J.S. Anandan ŽEd.., World Scientific, Singapore, 1990; for some recent applications see L. Stodolsky, Acta Physica Polonica B 27 Ž1996. 1915; G. Raffelt, G. Sigl, L. Stodolsky, Phys. Rev. Lett. 70 Ž1993. 2363. w8x See Topological Quantum Numbers and Nonrelativistic Physics by David J. Thouless, World Scientific, Singapore, 1998, p. 79. w9x S.A. Gurvitz, Phys. Rev. B 56 Ž1997. 15215. w10x Y. Levinson, Europhys. Lett. 39 Ž1997. 299. w11x A. Stern, Y. Aharanov, Y. Imry, Phys. Rev. A 41 Ž1990. 3436. w12x M. Reznikov, R. de Picciotto, M. Heiblum. D.C. Glattli. A. Kumar, L. Saminadayar, Quantum Shot Noise, Superlattices and Microstructure 23 Ž1998. 901. w13x B. Kayser, L. Stodolsky, Phys. Lett. B 359 Ž1995. 343. w14x See the second of Ref. w5x for how this effect arises from Eq. Ž1.. Some early references are M.T. Parkinson Nucl. Phys. B 69 Ž1974. 399; A.M. Wolsky, Found. Phys 6 Ž1976. 367; C. Chiu et al., Phys. Rev. D 16 Ž1977. 520; V. Gorini et al., Rep. Math. Physics 13 Ž1978. 149; A. Peres, Am. J. Phys. 48 Ž1980. 931. According to Wolsky, the idea may have been brought up by Alan Turing. w15x L. Stodolsky, quant-phr9903072, to be published in Physics Reports.
Physics Letters B 459 Ž1999. 193–200
Measurement process in a variable-barrier system L. Stodolsky Max-Planck-Institut fur Ring 6, 80805 Munchen, Germany ¨ Physik (Werner-Heisenberg-Institut), Fohringer ¨ ¨ Received 11 February 1999; received in revised form 20 May 1999 Editor: P.V. Landshoff
Abstract The description of a measuring process, such as that which occurs when a quantum point contact ŽQPC. detector is influenced by a nearby external electron taking up different positions, provides a interesting application of the method of quantum damping. We find a number of new effects, due to the complete treatment of phases afforded by the formalism, although our results are generally similiar to those of related treatments, particularly to those of Buks et al. These are effects depending on the phase shift in the detector, effects which depend on the direction of the measuring current, and in addition to damping or dissipative effects, an energy shift of the measured system. In particular, the phase shift effect leads to the conclusion that there can be effects of ‘‘observation’’ even when the barriers in question pass the same current. Secondly, the nature of the current through the barriers and its statistics is discussed, and a description of the correlations in the current due to ‘‘measurement’’ is obtained. In particular it is possible to see the origin of ‘‘telegraphic’’ signals, which in a certain limit give behavior resembling a ‘‘collapse of the wavefunction’’. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The measurement process using a quantum point contact ŽQPC. detector w1x can be described as the modification of a barrier whose transmission varies w2x according to whether an external electron is nearby or farther away. When the external electron is close by there is a certain higher barrier, and when it is farther away, there is a reduced barrier. Given an incident or probing flux on the barrier, the modification of the resulting current through the barrier, which can be observed by conventional means, thus ‘‘measures’’ where the external electron is located. Experiments of this type give a fundamental insight into the nature of measurement. In an elegant
recent experiment Buks et al. w1x – stimulated by the work of Gurvitz w3x – saw the expected loss of fringe contrast in an electron interference arrangement when one of the paths in the interferometer was ‘‘under observation’’ by a QPC w4x. This effect is due to the ‘‘damping’’ or ‘‘decoherence’’ arising from the creation of correlations between the position of the electron in the interferometer and the coordinates of the ‘‘environment’’ or ‘‘observer ’’ w5,7x. A number of theoretical treatments of such measurement processes have been given in recent years. Here we would like to use the ‘‘quantum damping’’ method of Refs. w5,7x, which was devised to deal with precisely such questions, and which can be used to give a complete and
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 5 9 - 0
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transparent treatment of the variable barrier problem. Furthermore it yields a number of new results. These include the effects of a phase which although it has no effect on the measuring current nevertheless contributes to the damping or decoherence, effects which depend on the direction of the detector current, and an energy shift of the observed system induced by the observing process. To elucidate the method we consider the simplest situation, the two-state system: An external electron can be in two states corresponding to two different locations. It thereby influences a barrier controlling the measuring current. Hence our variable barrier has two possibilities. A physical realization of the two-state system could be provided by two adjoining quantum dots, ‘‘left’’ŽL. and ‘‘right’’ŽR. with a resident electron tunneling between them. This two-state system is then ‘‘observed’’ by a QPC or similar detector whose barrier varies according to whether the resident electron is near or far from the detector, on dot L or dot R. According to whether this barrier is higher or lower there is then a different current in the detector circuit, thus furnishing a reading of the position of the external electron. To avoid confusion between the electron of the two-state system and the electrons of the measuring circuit, we will refer to the former as the ‘‘resident’’ or ‘‘external’’ electron, and to the latter as the detector electrons. Also, the tunneling for the external electron between the states L and R should not be confused with the tunneling of the detector or probing electrons through the variable barrier of the detector. The tunneling of the external electron will be given essentially by a tunneling energy V Žsee below., while that for the detector electrons will be characterized by an S-matrix S. Our discussion will consist of two parts, first a description of the state of the external electron, in terms of the density matrix of the two-state system. Secondly, beginning in Section 4, we give a description of the current in the detector circuit. We should perhaps stress that the two issues are distinct. An understanding of the evolution of the state of the ‘‘measured’’ system, -the external electron- does not of itself lead immediately to an understanding of the current in the detector circuit, and in fact while there have been numerous discussions of ‘‘decoherence’’
in recent years, the latter issue would appear at present to be the more subtle one. Our first focus is then on the 2 x 2 density matrix r for the two-state system; in particular we wish to study the effect of the presence of the detector on the time development of r . The density matrix is characterized by a three component ‘‘polarization vector’’ P, via r s 12 Ž I q P P s ., where the s are the pauli matrices. Pz gives the probability for finding the electron on the left or right dot via Pz s ProbŽ L. y ProbŽ R .. The other components, Ptr , contain information on the nature of the coherence. < P < s 1 means the system is in a pure state, while < P < s 0 means the system is completely randomized or ‘‘decohered’’. P will both rotate in time due to the real energies in the problem and shrink in length due to the damping or decoherence. The time development of P is given by a ‘‘Bloch-like’’ equation w5,7x P˙ s V = P y DPtr
Ž 1.
The three real energies V have the following significance in the present problem, where the twodot system is thought of as a double- well potential for the resident electron: Vz gives a possible energy difference for the two quasi-stationary states on each dot; Vz / 0 means the the double well is asymmetric. Vx and Vy are tunneling energies; Vx conserves the parity of the electron wavefunction on the two dots and Vy flips it w6x. The second term of Eq. Ž1. describes the damping or decoherence. D gives the rate at which correlations are being created between the ‘‘system ’’ Žthe external electron on the dots. and the ‘‘environment’’ Žthe detector.. The label ‘‘tr’’ means ‘‘transverse’’ to the z axis. The damping only affects Ptr because the observing process does not induce electron jumps from one dot to another; the observing process conserves Pz s ² sz :.
2. Influence of the detector The effect of repeated probings by an ‘‘environment’’ or ‘‘observing device’’ is described by a quantity L whose imaginary part gives the damping
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and whose real part gives an energy shift to the system being measured. Thus we expect that the ‘‘observation process’’ here will cause an energy shift for the external electron as well as a damping or dissipation. D is thus the imaginary part of L, while the real part of L gives an observation-induced contribution to V. L itself is given by
L s i Ž flux . ² i <1 y SL SR† < i :
Ž 2.
The factor flux is the flux or probing rate of the detector electrons, where in the QPC application one can use the Landauer formula flux s eVdrp ", with Vd the voltage in the detector circuit w8x.. The label i refers to the the initial or incoming state of the electrons in the detector and the S’s are the S matrices for the two barriers Žlabeled by L and R. corresponding to the different locations of the external electron.
unitary, SS † s 1, as may be found from explicit constructions. 2.2. Time reÕersal An additional constraint arises from time reversal invariance. This states that S j, i s Si T , jT , where the subscript T means the time-reversed state. Here we have simply k T s yk and vice-versa. For the diagonal elements this gives Sk k s Sykyk while for the off-diagonal elements there is no further constraint beyond that already given by unitarity. Note that we neglect electron spin and there is no magnetic field; these further complications might be interesting in some applications. We can now parameterize the S-matrix with the time reversal constraint in terms of three angles as S s e if
2.1. S matrix In order to apply Eq. Ž2., we review shortly the barrier penetration problem in S-matrix language. At a given energy there are just two S-matrix elements for the incoming flux with wave vector k, namely Sk k and Syk k , representing transmission and reflection respectively. These are the coefficients appearing in the wavefunction of unit incoming amplitude, which far away from the barrier on the incoming side is e i k z q Syk k eyi k z
Ž 3.
while on the other, outgoing, side of the barrier at large distances we have Sk k e i k z. We may also have the incoming detector current from the other direction with wave vector yk, and with Syk yk and Skyk for transmission and reflection. Thus our S-matrix is Ss
ž
Sk k Syk k
Skyk Sykyk
/
where the two columns correspond to the two possible directions of the detector current. This matrix is
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ž
cos u ie ih sin u
ieyi h sin u cos u
/
Ž 4.
which it will be seen fulfills all the conditions just discussed. The first column contains the reflection and transmission coefficients for incoming waves k and the second column those for incoming waves with yk. The angle u , which gives the magnitude of transmission and reflection, is the same as that used by Buks et al., while the other two parameters are phases. The phase angle h creates a difference between incident waves with k and yk, that is for different directions of the detector current. This reflects a possible asymmetry in the shape of the barrier Žsee below. and leads, in the case of non-zero h , to the interesting possibility of effects which depend on the direction of the measuring current. We note that although f appears as an overall phase it is physically relevant. The phases are fixed by the ‘‘1’’ in Eq. Ž2., or correspondingly by the fact that we have 1 as the coefficient of the incoming wave in Eq. Ž3.. 2.3. Parity If the barriers in question are even in shape, another constraint arises due to parity symmetry.
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When this operation, namely z ™ yz, is applied to the wavefunction Eq. Ž3., we get a solution corresponding to a wave coming in from the other direction. Comparing coefficients we conclude that Syk yk s Sk k and Skyk s Syk k . The first condition was already obtained from time reversal, the second, however, says that for symmetric barriers we should set h s 0 in Eq. Ž4. giving equal S matrix elements for both directions of the detector current. Note that a non-zero h only affects the reflection coefficients; even with non-symmetric barriers T ensures that the transmission coefficients have the same phase.
3. Damping D is the parameter giving the damping or loss of coherence of the system, and is given by D s Im L. We label the parameters of the S matrices with an L or R for the two barriers created when the electron is on the right dot or the left dot and call f L y f R s Df , hL y hR s Dh and u L y u R s Du . We then have from the matrix Ž4.,
effect. This may help to explain why the role of f seems unfamiliar 1 According to this equation, two detector barriers which have the same transmissibility, that is the same u , but different phase shifts can nevertheless induce damping. Although they apparently give the same detector current they nevertheless establish a distinction or ‘‘make a measurement’’. This may seem less mysterious when we recall that even pure phase shifts correspond to physical effects like the delay or change in shape of wave packets. Recall that in scattering theory the entire scattering is given by just a phase, Sl s e 2 i d l . Although a change in current is the most obvious ‘‘measurement’’, it is not necessarily the only one. Thus we apparently differ with other treatments w1,4,9,10x, where the damping or decoherence is related to the detector current only. Our results are of course proportional to the flux or probing rate, but not necessarily to the transmitted current alone. In effect, the situation concerning f
1
D s Ž flux . Re 1 y e
i Df
cos Du
qsin u L sin u R Ž e i Dh y 1 .
Ž 5.
4
This applies for a given direction of the detector current; for the other direction, reverse the sign of Dh. For a symmetric barrier, this simplifies to D s Ž flux . 1 y cos Df cos Du 4
Ž 6.
Except for the cos Df factor this is essentially the same damping effect as in Ref. w1x. This factor is interesting, however, in that even for small angles, 2
D f 1r2 Ž flux . Ž Df . q Ž Du .
2
4
Ž 7.
the phase f is present and enters on an equal footing with u . Note in the opposite limit cos Du s 0 which applies for a ‘‘projection’’, where one barrier transmits only and the other reflects only, that f has no
The absence of an effect due to the phase in other treatments may be traced to taking an absolute value too soon: We wish to find the decrease in the overlap of the different environment wavefunctions associated with the two states of the two-state system due to a probing by the environment. After one probing Žwe illustrate with the notation of w1x or w5x., the overlap changes as 1™1y² x L < x R :. therefore the new value of < P < is <1y ² x L < x R :<. It will be seen that the phase of ² x L < x R : plays a role. However, if we simply use 1y <² x L < x R :<, which in effect is what was done in the other treatments, the effect of the phase is lost, and with it both the effect of f on the damping as well as the real energy shift induced by the environment. A check on these arguments, at least as far as the real energy shift is concerned, is afforded by the agreement of the real energy shift so obtained with the usual formula for the index of refraction of a particle in a medium w7x. Another way of understanding that a just a phase plays a role in damping is to consider the phase as acting on the object being measured Žhere the external electron., as in w11x. If the probe is random in time, then, as these authors find, the resulting random phase results in a damping involving the phase squared. This is an expression of the familiar scattering theory result that damping or dissipation ; Im f ; d 2 . This argument suggests that perhaps in problems with degenerate electrons at low temperature effects involving the reduction of this randomness might have to be considered. On the other hand, the transmission process by itself introduces a certain noise or randomness into the problem Žsee the review in w12x..
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may be viewed as a case of the well-known tree falling in the forest with nobody there to hear it.
4. Energy shift
Vzind s Ž flux . Im e i Df cos Du qsin u L sin u R Ž e
shift will in general depend on the direction of the detector current. To find the difference effect we exchange k for yk in the initial state. For the difference in D for the two directions we have
D D s 2 Ž flux . sin Df sin u L sin u R sin Dh
While the damping is given by the imaginary part of Eq. Ž2., there is also a significance to the real part. In the description of the propagation of a particle in a medium it gives the index of refraction for the states of the particle in the medium w7x. In the present problem this state-dependent energy shift will give a measurement-induced contribution to Vz in Eq. Ž1. governing the internal evolution of the measured system w7x, Vzind s Re L so that
i Dh
y 1.
4
Ž 8.
which for an even barrier with Dh s 0 simplifies to Vzind s Ž flux . sin Df cos Du
Ž 9.
This effect only contributes to the ‘‘z direction’’ of V for the same reason D only effects Ptr , the observation process is presumed not to cause any jumps from one dot to the other. We note that the effect persists even if Du s 0, i.e. equal transmissibilities, as long as Df / 0. For small phases it is linear in Df , while the damping Eq. Ž7. is quadratic. This induced energy offers the intriguing opportunity of ‘‘tuning’’ the properties of the two-state system. The tunneling behavior of the external electron between the two dots depends strongly on how exact the degeneracy of the two wells is. Thus the ability to adjust it via the induced Vz , which is proportional to Vd and so easily adjustable, is quite interesting. For example, if it is difficult to fabricate identical dots, the induced Vz could be used to nevertheless make the two dots degenerate in energy. On the other hand a large Vz , by lifting an initial near-degeneracy tends to suppress transitions, offering another qualitative test of the theory.
5. Dependence on current direction For barriers that are not symmetric in shape, h can be non-zero, and the damping and the energy
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Ž 10 .
and for the induced energy shift the difference for the two directions
D Vz s 2 Ž flux . cos Df sin u L sin u R sin Dh
Ž 11 .
Both these effects are linear in small Dh.
6. Nature of the current The question of the current in the detector circuit poses some intriguing questions concerning the nature of ‘‘measurement’’. Should we expect a smooth current of some kind, reflecting some average transmission probability given by r ? Or should we view each transmission as a measurement which ‘‘collapses the wavefunction’’ to one dot or the other dot, leading to a series of ‘‘telegraphic’’ signals with different currents corresponding to one barrier or the other? If so, what determines the duration of these signals? The above considerations for the determination of r cannot alone answer such questions. From r we can only find the probability of a single transmission at a certain time, Žunconditional probability. but not if this event was, say, part of a long series of transmissions or a mixed series of transmissions and reflections Žconditional probability.. This sort of question arises because successive probings are in general not statistically independent; one transmission may imply an increased probability for the next one. To understand this, a helpful analogy may be a hypothetical sequence of successive Stern-Gerlach ‘‘measurements’’, performed on a single atom. There is a strong biasing effect from one observation to the next: if the spin is ‘‘up’’ passing through a first magnet, it will also be ‘‘up’’ after passing through the next one. One sees that the successive measurements are far from independent. In our present problem the Žspin 1r2. ‘‘atom’’ being measured is the two-state system of the resident electron, while the
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repeated probings by the detector circuit are the sequence of Stern-Gerlach ‘‘magnets’’. Indeed, the analogy is complete if we suppose some agency Žsay some additional magnetic field. that rotates the ‘‘spin’’ while it is undergoing the successive SternGerlach measurements. This new agency plays the role of the tunneling energy Vtr . An explicit treatment is perhaps most easily performed in an amplitude formulation 2 . Let 1 represent a transmission and 0 a reflection for the detector electrons. We write the probability amplitude that after N probings the external electron is on dot L, and that the first probing electron was transmitted, the second reflected, . . . the Ž N y 1.th transmitted and the Nth transmitted, as AŽ L,w11 . . . 01x.. Similarly there is an amplitude AŽ R,w . . . x. for the external electron to be on dot R after a given sequence of transmissions and reflections w . . . x. We consider the situation where many probing electrons are incident during the time 1rVtr it takes the external electron to tunnel between L and R : fluxrVtr 4 1
Ž 12 .
This assumption allows us, in discussing the current, to neglect ‘‘dot jumps’’ where an amplitude AŽ L,w . . . x. receives a contribution from an amplitude AŽ R,w . . . x.. Then for time periods d t such that Vtr d t < 1 we can simply write L L A Ž L, w 11 . . . 01 x . ; SkLk SkLk . . . Syk k Sk k
Ž 13 .
where we have suppressed phase factors associated with the time, and also the probability of the starting configuration. If we restore the latter, in the form of r L L for the probability of starting with L, and r R R for the probability of starting with R, we find by
squaring the amplitude the probability for a given sequence of transmissions and reflections: Prob w 11 . . . 01 x s r L LŽ pL pL . . . q L pL . q r R R Ž pR pR . . . q R pR .
where p s < Sk k < and q is the probability of no transmission, q s 1 y p s < Syk k < 2 . ŽNote these quantities are independent of the current direction k, due to T invariance..
7. ‘‘Telegraphic’’ behavior and correlations Eq. Ž14. shows how a tendency to ‘‘telegraphic’’ behavior with sequences of reflections or sequences of transmissions can arise. For example, let pL and pR be very different, say close to zero and close to one respectively. Then neither the first term nor the second term of Eq. Ž14. can be big for mixed sequences like w010 . . . 01x; but for ‘‘telegraphic’’ sequences w111 . . . 11x or w000 . . . 00x, one or the other term can be big. One type of sequence or the other will be selected. In this limit we would get a behavior resembling a ‘‘collapse of the wavefunction’’: a ‘‘1’’ leads to a further ‘‘1’’, and a ‘‘0’’ leads to a further ‘‘0’’ indefinitely, until the resident electron tunnels to the other dot. On the other hand, let pL and pR be nearly equal. Then this tendency favoring signals far from the mean will be weak; little correlation between the ‘‘measured’’ and the ‘‘measurer’’ is introduced. A quantitative expression of this point is given by Eq. Ž16. below. The two terms of Eq. Ž14. are essentially those leading to the binomial distribution in statistics. Hence if we now ask for the probability ProbŽ Q, N . for Q transmissions in N probings, the combinatorics are that of the binomial distribution, and we obtain Prob Ž Q, N . s r L L PLŽ Q . q r R R PR Ž Q .
2
Although we prefer the amplitude point of view, see w13x one could also arrive at Eq. Ž13. by collapsing the wavefunction of the dot electron after every probing: Let the wavefunction of the dot electron initially be a < L:q b < R :. After a probing it becomes L R < :. < :Ž R < : < :. a < L:Ž SkLk < k :q Syk k y k q b R S k k k q Syk k y k . If, say, a transmission takes place, we collapse to a < L: SkLk q b < R : SkRk for the new wavefunction. Doing this repeatedly and summing over all starting configurations then leads to Eq. Ž13..
Ž 14 .
2
Ž 15 .
where PLŽ Q . is the binomial expression for the probability of Q transmissions in N trials given the single trial probability pL . ŽFor N large and p small this is approximated by the poisson distribution Q PLŽ Q . f Ž n L . eyn L , with n L s pL N .. Q!
Eq. Ž15., for distinct pL and pR , leads to a twopeaked distribution, and generally describes the statistics of the current for short times. For example
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the quantity ProbŽ Q1 ,Q2 . for Q1 transmissions in N1 probings, followed by Q 2 transmissions in N2 probings gives a measure of the correlations induced by the ‘‘measurement’’ process. In particular ProbŽ Q1 ,Q2 . y ProbŽ Q1 .ProbŽ Q2 . is zero if the currents in the two intervals are independent. Now ProbŽ Q1 ,Q2 . may be calculated as simply the weighted average of two parallel processes, each one calculated as a statistically independent sequence: ProbŽ Q1 ,Q 2 . s r L L PLŽ Q1 . PLŽ Q 2 . q r R R PR Ž Q1 . PR Ž Q2 ., while ProbŽ Q . is given by Eq. =P Ž15.. Thus Prob Ž Q1 ,Q2 . y Prob Ž Q1 . Prob Ž Q2 . s r L LŽ 1 y r L L . Ž PLŽ Q1 . y PR Ž Q1 . . = Ž PLŽ Q 2 . y PR Ž Q2 . .
Ž 16 .
We expect this expression to vanish when only one dot is occupied, for then successive probings are independent; it is in fact zero for r L L one or zero. Similarly it vanishes when the current provides no information on the state, when pL and pR are equal or PL s PR . The correlation defined in Eq. Ž16. thus gives a characterization of the strength of ‘‘collapse’’ or ‘‘telegraphic’’ effects. These considerations hold, as said, for times short relative to the tunneling time 1rVtr . For longer times an amplitude AŽ L,w . . . x. receives contributions from an amplitude AŽ R,w . . . x., thus we anticipate the time scale for the duration of a ‘‘telegraphic’’ signals on the order of 1rVtr .
8. Strong damping In the previous section the time scale was the tunneling time 1rVtr . However, there is a another regime w5x of behavior, which although it may not be relevant in problems where the damping is relatively weak, as when the two barriers differ little, but which is of interest in itself. This is the case of strong damping, when DrVtr 4 1. In the limiting case of strong damping the amplitude for any configuration AŽ L,w . . . x. is reached by only one ‘‘path’’ w . . . x and the situation resembles a classical diffusion problem. The solutions of Eq. Ž1. show the ‘‘Turing-Watched-Pot-Zeno’’ behavior w14x where
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‘‘measurement’’ inhibits the time evolution of the external electron and the characteristic time scale for transitions between L and R becomes the longer DrVtr2 . In this case the condition Eq. Ž12. can be weakened to flux Ž DrVtr2 . 4 1, or since D s flux in the limit, to Ž fluxrVtr . 2 4 1. Similarly the time scale for the duration of telegraphic signals becomes the longer time DrVtr2 .
9. Conclusions and applications If it were possible to fabricate the two dot system or its analogs and to carry out experiments with them, the various effects could be tested through their effect on the diagonal elements of r , or Pz . Pz Ž t . gives the probability that a dot is occupied, and through Eq. Ž1. it is influenced by the real energies and D. One way of determining it experimentally would be from the current described by Eq. Ž15., that is from the relative strengths of the two peaks of the Q distribution. It should be stressed that Eq. Ž15. represents an average over many repeated runs from the same initial condition, say the injection of the external electron onto one of the dots, and not an average over time in one run. It should also be noted that even if the evolution of r cannot be followed in detail, the time scale of the relaxation of correlations yields information on Vtr and D. For experiments of the type of Ref. w1x, the real energy Eq. Ž8. induced by the measurement will show up as a shift in the interference fringes. In Eq. Ž8. the effect is expressed as an energy, that is as a phase per unit time, which in an experiment such as Ref. w1x corresponds to a phase shift given by the dwell time of the measured electron times the energy Eq. Ž8.. Hence Eq. Ž8. predicts, for non-zero f , a Vd-dependent fringe shift. For the asymmetric barrier, Eq. Ž11. predicts a component to this fringe shift which reverses with Vd . Similarly, for the asymmetric barrier, Eq. Ž10. predicts that the loss of fringe contrast or visibility of Ref. w1x can depend on the current direction. Finally, there is the role of f . According to Eq. Ž6., if we can arrange for two detector barriers to have the same transmission but different f there still should be a loss of fringe contrast or damping.
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The fact that the fluctuations of the current in the detector circuit are related to the measurement or the damping suggests a kind of fluctuation-dissipation relation. This leads to the idea of a measurement-related noise, related to the dissipative parameter D. This will be studied elsewhere w15x.
Acknowledgements I am grateful to the Weizmann quantum dot group for introducing me to this subject and especially to E. Buks, S. Gurvitz and M. Heiblum for several discussions, and to R.A. Harris for help in clarifying the ideas. Finally, I am grateful to a number of these colleagues, and in particular A.J. Leggett, for criticisms concerning the subject of Section 6, finally leading to the viewpoint developed there. I would also like to acknowledge the hospitality of the Institute of Advanced Studies at the Hebrew University of Jerusalem, where this work was begun.
References w1x E. Buks, R. Schuster, M. Heiblum, D. Mahalum, V. Umanksy, Nature 391 Ž1998. 871.
w2x M. Field, C.G. Smith, M. Pepper, D.A. Ritchie, J.E.F. Frost G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 Ž1993. 1311. w3x S. Gurvitz, quant-phr9607029. w4x I.L. Aleiner, N.S. Wingreen, Y. Meier, Phys. Rev. Lett. 79 Ž1997. 3740. w5x R.A. Harris, L. Stodolsky, J. Chem. Phys. 74 Ž1981. 2145; Phys. Lett. B 116 Ž1982. 464. w6x J.A. Cina, R.A. Harris, Science 267 Ž1995. 832. w7x For a general introduction and review of these concepts see L. Stodolsky, Quantum Damping and Its Paradoxes in Quantum Coherence, J.S. Anandan ŽEd.., World Scientific, Singapore, 1990; for some recent applications see L. Stodolsky, Acta Physica Polonica B 27 Ž1996. 1915; G. Raffelt, G. Sigl, L. Stodolsky, Phys. Rev. Lett. 70 Ž1993. 2363. w8x See Topological Quantum Numbers and Nonrelativistic Physics by David J. Thouless, World Scientific, Singapore, 1998, p. 79. w9x S.A. Gurvitz, Phys. Rev. B 56 Ž1997. 15215. w10x Y. Levinson, Europhys. Lett. 39 Ž1997. 299. w11x A. Stern, Y. Aharanov, Y. Imry, Phys. Rev. A 41 Ž1990. 3436. w12x M. Reznikov, R. de Picciotto, M. Heiblum. D.C. Glattli. A. Kumar, L. Saminadayar, Quantum Shot Noise, Superlattices and Microstructure 23 Ž1998. 901. w13x B. Kayser, L. Stodolsky, Phys. Lett. B 359 Ž1995. 343. w14x See the second of Ref. w5x for how this effect arises from Eq. Ž1.. Some early references are M.T. Parkinson Nucl. Phys. B 69 Ž1974. 399; A.M. Wolsky, Found. Phys 6 Ž1976. 367; C. Chiu et al., Phys. Rev. D 16 Ž1977. 520; V. Gorini et al., Rep. Math. Physics 13 Ž1978. 149; A. Peres, Am. J. Phys. 48 Ž1980. 931. According to Wolsky, the idea may have been brought up by Alan Turing. w15x L. Stodolsky, quant-phr9903072, to be published in Physics Reports.