Abstract
The complementarity principle^{1} demands that a particle reveals wavelike properties only when the different paths that it can take are indistinguishable^{2,3}. The complementarity has been demonstrated in optics with pairs of correlated photons^{4,5} and in twopath solidstate interferometers with phasecoherent electrons^{6}. In the latter experiment, a charge detector embedded near one path of a twopath electron interferometer provided whichpath information^{3}. Here, we report on electron dephasing in an Aharonov–Bohm ring interferometer^{7} via a charge detector adjacent to the ring. In contrast to the twopath interferometer, charge detection in the ring does not always provide path information. The interference was suppressed only when path information could be acquired, even if only in principle. This confirms that dephasing is not always induced by ‘disturbing’ the interfering particle through the interferometer–environment interactions: path information of the particle must be available too. Our experiment suggests that acquisition of whichpath information is more fundamental than the backaction in understanding quantum mechanical complementarity.
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The previous electronic ‘whichpath’ experiments^{6} were based on a solidstate twopath interferometer, fabricated in the plane of a highmobility twodimensional electron gas (2DEG). The interferometer consisted of an open Aharonov–Bohm ring, with a source and a drain of electrons weakly coupled to the open ring. In one path of the interferometer, a coherent quantum dot was embedded^{6,7,8}, being electrostatically coupled to a quantumpointcontact (QPC) charge detector (in the immediate proximity to the quantum dot). An electron trapped in the quantum dot modified the conductance of the nearby QPC and thus enabled charge detection by the QPC^{9,10,11}. Being an open geometry, with multiple grounded drains (bases) along the paths of the electron, assured that only two paths interfered while the backscattered electrons were drained out by the grounded bases. Thus, the detection of a charge inside the quantum dot (by the QPC) provided path information, which led to the suppression of the Aharonov–Bohm interference oscillations.
Here, we used a ‘closedloop’ Aharonov–Bohm interferometer^{7,12}, as shown in Fig. 1a, with a quantum dot and a QPC detector placed in a similar manner as in Buks and coworkers^{6}. However, in contrast to the previous schemes, the closed geometry allowed an electron to encircle the interferometer loop many times before it reached the drain, making the interferometer an analogue of the Fabry–Perot interferometer, where in the closed ring the forward propagating and the backward propagating paths are spatially separated. Let us look at a couple of examples: among various possible electron trajectories contributing to the interference, the most probable trajectories, which lead to source–drain conductance oscillations with periods of one (h/e; first harmonic) and a half (h/2e; second harmonic) flux quantum, are illustrated in Fig. 1b and c, respectively^{13,14}. In principle, there are an infinite number of other possible trajectories that give rise to the first and the secondharmonic interferences. However, as will be discussed below, these two sets of trajectories (shown in Fig. 1b,c) are the dominant ones. For the trajectories plotted in Fig. 1b, path information can be acquired by detecting the presence of an added electron inside the quantum dot. Alternatively, for the trajectories plotted in Fig. 1c, charge detection in the quantum dot does not, in general, provide path information to distinguish between the blue and red trajectories, as both pass the quantum dot once^{12}. Hence, we would expect strong dephasing of the first Aharonov–Bohm harmonic but not of the second harmonic. Adding an element of time in the detection process may distinguish between trajectories of an electron trapped in the quantum dot, and also suppress the second harmonics—as we describe below—making this experiment clearly different to its predecessors.
Our closedloop interferometer (Fig. 1a) was fabricated in a GaAs/AlGaAs heterojunction wafer containing a 2DEG with an elastic mean free path of l_{e}∼20 μm. Four side gates (M_{1}, M_{2}, M_{3}, M_{4}) together with an island gate (P) defined an Aharonov–Bohm ring of about 550 nm in radius (Fig. 1a), with approximately five conducting channels (N>5) in each path. A quantum dot and a QPC detector were placed to detect charges trapped inside the quantum dot.
To examine the characteristics of the quantum dot, the left path was pinched off by applying a large negative voltage to the side gates M_{1} and M_{2}. The Coulombblockaded conductance peaks of the quantum dot were monitored by varying the voltage on the centre island gate (supplied by the air bridge P, see Figs 1a and 2a). The quantum dot was tuned to the Coulombblockaded conductance peak marked by the vertical arrow in Fig. 2a. The transmission probability of the QPC detector, defined as T_{d}=G_{d}/(2e^{2}/h), was set to T_{d}=0.1775, where the dephasing rate was found to be the highest^{6}. The close proximity between the QPC detector and the quantum dot lead to a strong modification of its transmission, ΔT_{d}, as illustrated in Fig. 2a, each time the number of electrons inside the quantum dot changed by one. Equivalently, the mere possibility of measuring the added charge to the dot, leads, by the inadvertent backaction of the detector on the quantum dot, to dephasing of the dwelling electron in the quantum dot. Following the abovedescribed tuning of the dot, the current through the detector was shut off and the left arm of the interferometer was opened. The interferometer was then tuned to exhibit both the first and the second harmonic of the Aharonov–Bohm interference.
The effect of the charge (or path) detection on the interference pattern, at different detector bias voltages (0–400 μV), is shown in Fig. 2b. The Aharonov–Bohm oscillations clearly exhibit two harmonics (with the field periodicity of ∼4.6 and ∼2.3 mT), which are being suppressed with increasing the bias of the detector, as expected. As the number of ‘detecting electrons’ that pass the detector during the dwelling of the electrons in the quantum dot increases, dephasing is enhanced. The inset of Fig. 2b clearly shows that the first Aharonov–Bohm harmonic is much more sensitive to the bias of the detector than the second harmonic. As alluded to above, the difference arises from the fact that the charge detection cannot distinguish between the two types of trajectory illustrated in Fig. 1c leading to the second harmonic.
To be more specific, there are two major sets of trajectories for the second harmonic. In the first set of trajectories shown in Fig. 1c, the partial wave of an electron starting with the left path at the source makes one and a half clockwise turns around the interferometer (blue), whereas that starting with the right path makes just a half turn (red). In the second set of trajectories (not shown), the directions of the two partial waves are switched: the shorter path goes though the left and the anticlockwise path makes one and a half turns passing through the quantum dot twice. In the latter case, path information is obtained via charge detection because only the longer path passes through the quantum dot (even twice), whereas the other path never passes through the quantum dot. The transmission probability through the quantum dot, T_{P}, can be estimated as follows: as the conductance peak is about G_{QD}/(2e^{2}/h)∼0.25 and the number of transverse channels in each path is about N∼5, the transmission probability is T_{P}=G_{QD}/(2e^{2}/h)/N∼0.05. Therefore, other more complicated sets of trajectories can be ignored owing to the low transmission probability through the quantum dot. This also explains the absence of higherorder harmonics with n>2. Furthermore, the second set of trajectories, compared with the first set shown in Fig. 1c, can also be neglected owing to the large dwell time in the quantum dot (t_{d}∼2 ns), which is much larger than the characteristic mean time interval, t_{f,}, between successive injection of electrons through the left (shorter) path. From the bias voltage of the interferometer, V =10 μV, we find that t_{f}∼h/2e V ∼0.2 ns. For the first set of trajectories shown in Fig. 1c, both the partial waves pass the quantum dot only once, and there is no considerable time delay in the arrival of the two wave packets. However, for the second set of trajectories, the characteristic time of the shorter path is about t_{f}∼0.2 ns and that of the longer path is about 2t_{d}∼4 ns. This large time delay strongly suppresses the interference owing to the lack of overlap of the wave packets taking two different paths. Therefore, the second harmonic is dominated by the trajectories shown in Fig. 1c, which do not provide path information in the charge detection process.
It should be noted that the sets of electron trajectories such as those described in Fig. 1c may be distinguished via charge detection in the quantum dot if the difference in their dwell time in the interferometer can be differentiated, hence dephasing also the second Aharonov–Bohm harmonic. The difference in the path lengths of the two possible trajectories shown in Fig. 1c, going from the source to the quantum dot, is about 1.7 μm, and for a Fermi velocity of ν_{F}∼1.62×10^{5} m s^{−1} we get a time difference of ∼10 ps. This must be compared with the time difference between consecutive electron arrivals in the QPC detector, h/2e V_{d}, where V_{d} is the applied bias voltage to the QPC detector^{6}. This simple argument leads to the conclusion that path information for the trajectories shown in Fig. 1c is provided when the bias on the detector will exceed ∼210 μV.
To observe this ‘timeresolving’ detection, the effect of the detector bias V_{d} was monitored up to 1.5 mV. Applying a high bias to the detector without modifying the transmission through the quantum dot is by no means a trivial task, as the electrostatic coupling between the detector and the quantum dot is strong^{15}. Hence, the genuine dephasing was excluded from the inadvertent electrostatic effect by setting the detector to a regime where it is not sensitive to the potential in the quantum dot (Fig. 3b). Compensating for the electrostatic gating effect led to the dependence of the two Aharonov–Bohm harmonics on detector bias as shown in Fig. 3c. With increasing detector bias, the firstharmonic dropped monotonically; however, the second harmonic remained unaltered in a lowbias regime, altering its declining slope at V_{d}∼500 μV. Still the suppression rate of the second Aharonov–Bohm harmonic remained lower than that of the first Aharonov–Bohm harmonic.
Because of the low transmission probability through the quantum dot, the major contribution of the first harmonic comes from the two direct paths through the interferometer. Thus, the dephasing rate^{16,17,18} of the first Aharonov–Bohm harmonic can be analysed in a similar manner to that in the previous experimental work with an openloop interferometer^{5}. The expected visibility (in terms of its fastFouriertransformed (FFT) amplitude A_{1}) has the form, ν=A_{1}/A_{1}(V_{d}=0)=1−Γ_{d}/Γ_{e}, where Γ_{e} is the natural broadening of the state in the quantum dot due to coupling to the leads, and Γ_{d} is the dephasing rate induced by the charge detection. Γ_{e}∼0.33 μeV in our experiment^{7} and Γ_{d} is given by an algebraic sum of two different contributions Γ_{d}=Γ_{T}+Γ_{φ}, where Γ_{T} and Γ_{φ} correspond to the current and the phasesensitive dephasing rates, respectively^{19,20,21,22,23,24}, expressed as:
The phase sensitivity, Δφ, is defined as the relative phase shift of the transmitted and reflected partial waves induced by an extra charge in the quantum dot. Recently, an unexpectedly large dephasing rate was observed and interpreted^{15} only in terms of Γ_{T}. It has been theoretically proposed that this can be understood by taking into account Γ_{φ} as well, which is much larger than Γ_{T} in a generic situation with a nonnegligible asymmetry in the charge response of the QPC potential^{21}. In practice, Δφ cannot be directly extracted from our measurement setup. The best fit to the data for the first Aharonov–Bohm harmonic, with Δφ (=0.031) as a fitting parameter, is given by the solid line in Fig. 3c. The fits reveal that the phasesensitive dephasing mechanism is more effective than that of the currentsensitive dephasing (namely, Γ_{φ}/Γ_{T}∼35), which is consistent with the previous observation^{15,21}.
The question of why the dephasing rate of the second Aharonov–Bohm harmonic at higher detector bias is so low may still be raised. We speculate that the lower rate may originate from the finite size of the electron wave packet in the detector channel, being larger than the interferometer. The timeresolving detection is effective only when the size of the electron wave packets is infinitesimally small. The dephasing can be alternatively understood in terms of the backaction^{3}, which is the randomization of the phase of an electron passing through the quantum dot due to the fluctuations of the quantum dot potential induced by the current noise in the QPC detector^{3}. For the trajectory in Fig. 1b (firstharmonic interference), the random phase is collected for the right path during the entire passage of a wave packet through the quantum dot, whereas no random phase is collected for the left path, which leads to the suppression of the firstharmonic interference. However, for the trajectory in Fig. 1c (secondharmonic interference), both wave packets taking the left and the right paths dwell some time in the quantum dot simultaneously and collect a common random phase, which does not suppress the interference. The random phase collected while only one of the packets occupies the quantum dot suppresses the interference. Thus, the random phase accumulated for the second harmonic should be smaller than that of the first harmonic, which gives a qualitative explanation for the lower dephasing rate of the second harmonic.
So far, it has been widely accepted that the quantum mechanical complementarity of a particle can be understood in terms of the momentum transfer (or backaction), which is inevitably caused by detecting the path of the particle, as explicitly stated by Feynman et al. ^{2} a few decades ago. Recently, however, it has been demonstrated that the particlelike behaviour also takes place by the whichpath information even for the sufficiently weak momentum transfer^{5}, which refutes the backaction picture of the dephasing. In clear contrast to previous studies, our work confirms that the wavelike behaviour is preserved unless the whichpath information can be acquired out of the detection process, even if it can be done only in principle, regardless of the strength of the finite ‘disturbance’ caused by the charge detection. This is verified by investigating the secondharmonic dephasing in a closedloop Aharonov–Bohmring interferometer, which has no analogue in the systems studied previously, including the optical^{4}, atomic^{5} and solidstate^{6} interferometers. Thus, this study demonstrates that acquisition of whichpath information is of prime importance for quantum mechanical complementarity.
Methods
The 2DEG in a GaAs/AlGaAs heterojunction wafer resided about 80 nm below the surface. The electron density was n_{s}=1.8×10^{11} cm^{−2} with a corresponding mobility of μ=3.3×10^{6} cm^{2} V^{−1} s^{−1} at 4.2 K, resulting in an elastic mean free path of l_{e}∼20 μm. Figure 1a shows a scanning electron micrograph of the device used in this study. Negative voltages were applied to the gates so as to pinch off the 2DEG underneath them. Four side gates (M_{1}, M_{2}, M_{3}, M_{4}) together with an island gate (P) defined an Aharonov–Bohm ring with a geometrical radius of about 550 nm, which was also confirmed by the Aharonov–Bohm interference oscillations. Two QPCs (Q_{1}, Q_{2}) at the source and the drain were used to configure the twoterminal measurement. A quantum dot and a QPC detector were placed to detect charges trapped inside the quantum dot. Two gates (F_{1}, F_{2}) defined the quantum dot and separated it from the QPC detector. The island gate was electrically controlled through the bridged electrode P. The measurements were made by applying a 10 μV r.m.s. excitation voltage to the source and monitoring the output current at the drain (the electron temperature was 140 mK).
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Acknowledgements
H.J.L. was supported by the Electron Spin Science Center in POSTECH and the Pure Basic Research Program (Grant No. R012006000112480) administered by the Korea Science and Engineering Foundation (KOSEF), by the Korea Research Foundation (Grant No. KRF2005070C00055) and by the POSTECH Core Research Program. Y.C. was supported by the Korea Foundation for International Cooperation of Science and Technology (KICOS; Grant No. 200604969), Nanoscopia Center of Excellence (NCoE; Grant No. M6050400024906A040024910) at Hanyang University through a grant provided by the Korean Ministry of Science & Technology, and the Priority Research Centers Program (Grant No. KRF2006005J02801) funded by KRF. K.K. was also supported by KRF (Grant No. KRF2006331C00016). M.H. wishes to acknowledge the partial support of the MINERVA foundation, the German Israeli foundation (GIF), the German Israeli project cooperation (DIP), the Israeli Science foundation (ISF) and the Korea Ministry of Science and Technology program.
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Chang, DI., Khym, G., Kang, K. et al. Quantum mechanical complementarity probed in a closedloop Aharonov–Bohm interferometer. Nature Phys 4, 205–209 (2008). https://doi.org/10.1038/nphys854
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DOI: https://doi.org/10.1038/nphys854
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