Aharonov-Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap

Abstract

Quantum tunnelling is a common fundamental quantum mechanical phenomenon that originates from the wave-like characteristics of quantum particles. Although the quantum tunnelling effect was first observed 85 years ago, some questions regarding the dynamics of quantum tunnelling remain unresolved. Here we realize a quantum tunnelling system using two-dimensional ionic structures in a linear Paul trap. We demonstrate that the charged particles in this quantum tunnelling system are coupled to the vector potential of a magnetic field throughout the entire process, even during quantum tunnelling, as indicated by the manifestation of the Aharonov-Bohm effect in this system. The tunnelling rate of the structures periodically depends on the strength of the magnetic field, whose period is the same as the magnetic flux quantum [straight phi]₀ through the rotor [(0.99±0.07) × [straight phi]₀ ].

ARTICLE

Received 13 Dec 2013 | Accepted 10 Apr 2014 | Published 13 May 2014

DOI: 10.1038/ncomms4868

AharonovBohm effect in the tunnelling of a quantum rotor in a linear Paul trap

Atsushi Noguchi1, Yutaka Shikano2,3, Kenji Toyoda1 & Shinji Urabe1

Quantum tunnelling is a common fundamental quantum mechanical phenomenon that originates from the wave-like characteristics of quantum particles. Although the quantum tunnelling effect was rst observed 85 years ago, some questions regarding the dynamics of quantum tunnelling remain unresolved. Here we realize a quantum tunnelling system using two-dimensional ionic structures in a linear Paul trap. We demonstrate that the charged particles in this quantum tunnelling system are coupled to the vector potential of a magnetic eld throughout the entire process, even during quantum tunnelling, as indicated by the manifestation of the AharonovBohm effect in this system. The tunnelling rate of the structures periodically depends on the strength of the magnetic eld, whose period is the same as the magnetic ux quantum f0 through the rotor [(0.990.07) f0].

1 Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan. 2 Research Center of Integrative Molecular Systems (CIMoS), Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan. 3 Institute for Quantum Studies, Chapman University, 1 University Dr, Orange, California 92866, USA. Correspondence and requests for materials should be addressed to A.N. (email: mailto:[email protected]

Web End [email protected] ).

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The quantum dynamical phenomenon in which a particle tunnels through a barrier that it classically could not penetrate1 was rst investigated in the 1920s to explain

alpha decay2,3. There are also many other natural phenomena ranging from cosmological scales4 to microscopic scales5, such as chemical reactions6, that arise from the quantum tunnelling effect; quantum tunnelling effects in articial materials have been investigated using diodes (tunnel diodes7) and superconductors (Josephson junctions811) and have been achieved in quantum simulators12,13. Determining the dynamics of quantum tunnelling is a difcult task because the momentum during quantum tunnelling might be purely imaginary14, and it is the tunnelling probability that is usually calculated instead15,16. However, the dynamics of tunnelling particles can be investigated if one can construct a highly coherent tunnelling system.

Trapped ions form a highly controllable, isolated quantum system in which both the internal and motional degrees of freedom have long coherence times17. Using one-dimensional (1D) ion chains trapped in a linear Paul trap, the oscillation can be cooled to near the ground states, and phonon manipulation can be achieved through laser irradiation18. An important feature of a trapped-ion system is the coulomb interaction between ions, which generates an effective interaction between ions for quantum information processing17. By virtue of these characteristics, a trapped-ion system is one of the more promising candidates for quantum information processing17. However, the coulomb interaction also provides several types of spatial arrangements of the trapped ions, that is, Wigner crystals, including 2D structures19. So far, the phase transitions20,21 and dynamics22 of these Wigner crystals have been investigated both theoretically and experimentally. However, these experiments have not been performed in the quantum regime22,23 because the motions of such 2D structures have never been cooled to near the ground states.

The AharonovBohm (AB) effect24, in which a charged particle is affected by a magnetic eld even as it travels through a region in which the magnetic eld is zero, has been experimentally conrmed using an electron in a superconducting ring25,26 and other materials2730. Even in a non-localized magnetic eld, the wave function of a charged particle obtains a phase that is proportional to the area of a closed loop31,32. This effect is also referred to as the AB effect. Charged quantum tunnelling particles should be affected by the vector potential of a magnetic eld even during quantum tunnelling. Thus, the AB effect should occur for tunnelling particles and is an example of the dynamics of tunnelling particles.

In this study, we have realised a quantum tunnelling rotor, which consists of three ions and has a 2D structure that is cooled to near the motional ground state in a linear Paul trap. This rotor has two stable orientations and can transition from one to another though quantum tunnelling. Because of the controllability and simplicity of the trapped-ion system, this quantum tunnelling rotor may offer a novel approach to understanding quantum tunnelling dynamics, and we were successful in observing the AB effect of tunnelling particles using this system. Several recent technical advancements regarding the manipulation of ions, such as gate operations and coherent motional controls17,18, offer further means of controlling the quantum tunnelling rotor.

ResultsThe quantum tunnelling rotor. A linear Paul trap, whose normal-mode frequencies are comparable along two axes (x, z) and large along the remaining axis (y), can be used to trap a single triangular structure of three 40Ca ions (Fig. 1a)18,19.

The symmetry of the potential denes two stable orientations for the triangular structure, which are referred to here as up and down, and the motional modes of these structures are more complex than those of a 1D coulomb chain20. Of the four lowest energy modes represented in Fig. 1b, the mode indicated by the solid curve represents a rotational mode in which the ions rotate around the centre of the triangular structure. Here we demonstrate the cooling of the rotational mode, which is one of the 2D modes, to near the ground state. Theoretically, the rotational mode has interesting characteristics: its normal-mode frequencies become zero at critical points, which corresponds to ox

2p 1.119 MHz in Fig. 1b. Under such conditions, the

rotational mode becomes massless, and singular phenomena that are similar to other degrees of freedom of 2D structures in a linear Paul trap are expected21.

Near the critical point of the rotational mode (Fig. 1b), the wave functions of each ion in the rotational direction are elongated, and quantum tunnelling can occur between two degenerate triangular structures33. Because the other motional modes have much larger spring constants by virtue of the strong coulomb interaction, it is reasonable to assume that these modes are unaffected by the quantum tunnelling. These triangular structures become a quantum rigid rotor, which is a quantum system that has a rotational degree of freedom but maintains its structure. In this system, the rotation of the quantum rigid rotor is driven by the quantum tunnelling. We refer to this system as a quantum tunnelling rotor (Fig. 1). Ions also have internal degrees of freedom, that is, spins, and the tunnelling can depend on the spin states of the ions of the quantum tunnelling rotor33. Using this system, we can investigate and manipulate the tunnelling system itself by controlling the trap parameters and the spin state. Figure 1c,d illustrates the effective potential of the quantum tunnelling rotor in the linear Paul trap for the actual experimental parameters used. Each bound state in the periodical multi-well potential corresponds to a single stable orientation of the quantum tunnelling rotor. In addition, the overlap of the up and down wave functions of the ions observed in Fig. 1d indicates that quantum tunnelling is possible. The orientations of the atomic triangles in this condition are not stable under laser-cooled temperatures. However, under intermediate radial connement conditions, corresponding a normal frequency of ox

2p 1.523 MHz in Fig. 1b, both degenerate triangular

structures are stable even for hot Doppler-cooled temperatures. Thus, projective measurements of the triangle orientations can be performed.

Observation of quantum tunnelling. We experimentally observed the quantum tunnelling of the rotor using a magnetic eld of a certain magnitude. The three ions were trapped in a linear Paul trap with normal-mode frequencies of {ox, oy,

oz} 2p {1.523, 1.961, 1.119} MHz (further details of the

experimental system can be found in refs 34,35), and the orientation of the triangular structure could be set using intermediate connement conditions. Figure 2a shows the time sequence of the experiment, which began with the measurement of the initial state of the triangular structure using a charge-coupled device camera. For quantum tunnelling to be observed, it was necessary to cool the quantum tunnelling rotor to the rotational ground state. The rotational mode had the lowest heating rate among the 2D motional modes (see Methods section) and could be cooled to near the motional ground state (a mean phonon number of nrt 0.0880.007) using the sideband cool

ing technique. After cooling, the trap radio frequency (RF) was adiabatically reduced to create the quantum tunnelling rotor. At that time, the normal-mode frequency of the rotational mode

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Radius, r0 (nm)1.0

1.0

Potential (Hz)

Potential (Hz)

300

Up

200

Jei /6

Jei /6

100

6.8 m

1.0 1.0 0

Radius, r0 (nm)

Down

Normal frequency (MHz)

1.5

1.0

0.5

200

100

1.1 1.3 1.5 1.7 x (MHz)

/3 /3

0

Angle (rad)

Figure 1 | The quantum tunnelling rotor. (a) Charge-coupled device images of the two degenerate triangular crystal structures. (b) Calculated energies of the four lowest eigenmodes. The solid curve corresponds to the rotational mode, and the dashed curve corresponds to the zigzag mode. The dot-dashed and dotted curves represent the energies of the centre-of-mass modes of the radial and axial directions, respectively. The left- and right-hand insets display the sets of displacement vectors for the rotational and zigzag modes, respectively. The normal-mode frequency in the axial direction is oz 2p 1.119 MHz. (c) The quantum tunnelling rotor indicated the potential energy and pattern diagrams of the quantum tunnelling of

a three-ion triangular structure. (d) The solid blue curve represents the potential energy as a function of the rotational angle. The red (purple dashed) triangle corresponds to one of the bound states in the well, as denoted by up (down) in a, and the dashed red (purple) curve represents the corresponding wave function. (e) A charged quantum rotor in a magnetic eld. The sign of the AB phases is dependent on the direction of rotation, where y( 6p(F/f0)) is the AB phase for tunnelling.

changed to 2p 0.18 kHz from 2p 750 kHz (Fig. 1b). There was

a small amount of heating of the quantum tunnelling rotor caused by noise, micromotion and diabaticity when the trap potential was changed. Under tunnelling conditions, we assumed the mean phonon number to be nrtB4 (see Methods section). After a time t, the trap RF was increased, and we again measured the orientation of the triangular structures. Near the critical point, the quantum tunnelling rotor in the rotational ground state oscillated between the two degenerate orientations because of the quantum tunnelling (Fig. 2b).

Magnetic eld dependency of quantum tunnelling. The AB effect must occur in the quantum tunnelling rotor in a magnetic eld because the dynamics of the quantum tunnelling rotor are governed by the quantum interference of two-directional rotations; see Methods section for details. Thus, the tunnelling of the quantum tunnelling rotor can be controlled by adjusting the strength of the magnetic eld. The dependency of the transition probability on the number of magnetic ux quanta was investigated by xing the waiting time to 50 ms and adiabatically varying the magnetic eld, with the eld held constant during tunnelling (Fig. 3). The detail of the experimental setup regarding the magnetic elds are presented in Methods section. The oscillation period estimated from a sinusoidal t is (0.990.07)f0, for the single magnetic ux quantum f0 h/e, where h is the Planck

constant and e is the elementary charge. This tting result agrees with the theoretically predicted oscillation period obtained from equation (6), which is presented in the Methods section.

DiscussionFigure 2b presents the experimental results for the tunnelling dynamics of the rotor. These dynamics can be divided into quantum tunnelling and classical rotation. The ions are not completely in the ground state; ions in the rotational excited states rotate without tunnelling because the energies of these states are greater than the potential barrier, whose height is h 250 Hz (Fig. 1d) where h is the Planck constant. The

transition probabilities of the tunnelling consist of two components. One component represents the tunnelling between the up and down states and can be described by a sinusoidal curve, and the other represents the rotation of the excited states and can be described by an exponential decay. A t of the experimental data to a sum of these two curves yields a tunnelling rate of7.60.3 Hz and a decay time constant of 5.40.3 s 1, for example, the mean rates of the classical rotation. The amplitude of the oscillation, which was determined to be 0.19 based on the thermal distribution of the phonon number, is consistent with the population of the ground state. Furthermore, our tunnelling rates for different radial connements (Fig. 2c) exhibit good agreement with the numerical solution of the Schrdinger equation for the potential illustrated in Fig. 1d. The oscillation in the transition probability can therefore be attributed to quantum tunnelling.

Figure 3 shows the strength of the magnetic eld dependency of the tunnelling rate. This gure demonstrates that the ions couple to the vector potential even under quantum tunnelling conditions; that is, the system exhibits the AB effect. The amplitude of the oscillation is also consistent with the quantum tunnelling result (Fig. 2b). Note that the AB effect does not occur

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100 ms 100 ms

5 ms

Transition probability

M1

SB

Waiting time

M2

0.3

0.2

0.1

Time (ms)

0.5

Transition probability Tunnel rate (Hz)

0.25

0

0 1.0 1.0

0 Magnetic flux ( 0)

Figure 3 | The AharonovBohm effect during quantum tunnelling.

The transition probability as a function of the magnetic ux producedby the tunable coils expressed in magnetic quantum units(f0 4.14 10 15 Wb). We performed 800 experiments per data point.

The error bars are statistical errors calculated by assuming binomial distributions of the results. The solid curve represents a cosine t to the results.

100 200 300Time (ms)

0

20

10

0 1.0 2.0 3.0 x z (kHz)

Figure 2 | Quantum tunnelling in the rotor. (a) Experimental time sequences. M1: measurement of the initial state, SB: sideband coolingof the rotational mode and M2: measurement of the nal state. (b) The correlation between the initial and nal states from 400 experiments per data point as a function of the waiting time t. The t function is the sum of a sinusoidal curve and an exponential decay. The error bars represent the s.d. of binomial distributions. (c) The tunnelling rate as a function of the radial connement. The solid curve represents a numerical calculation performed using the experimental parameter oz 2p 1.119 MHz. The

vertical error bars represent the s.e. of tting. The horizontal error bars correspond to the s.d. of the amplitude of the trap RF.

F f0

; 2

in classical rotations of the excited states because of the incoherency. The quantum tunnelling rotor may be affected by the Lorentz force during quantum tunnelling. The Lorentz force can be calculated from a velocity and a magnetic eld. The expectation value of the particle position can be evaluated using Ehrenfests theorem; however, the velocity of a tunnelling particle is expected to be purely imaginary because E U is negative

(where E is the kinetic energy and U is the potential energy). One approach to circumventing this difculty is to take the absolute value |E U| to be the tunnelling kinetic energy. The validity

of this approach is supported by quantum clock conditions15, the Ehrenfest theorem36 and an analogous experimental demonstration using a single photon14. Using this approach, we estimate the effective maximal rotation speed v of the rotor to be (2|U0 E|/M)1/2 (refs 14,15,36). Here, U0 h 270 Hz is the

maximal potential energy (Fig. 1d), E h 90 Hz is the energy of

the rotational ground state and M 3 40 1.67 10 27 kg is the

mass of the quantum tunnelling rotor. This speed yields a maximal strength of the Lorentz force of evBB10 26 N for a magnetic eld of BB5 G. An alternative approach is to calculate the effective mean velocity vm from the tunnelling rate. In this case, vm Jr0p/3, where J 7.4 Hz is the tunnelling rate and

r0 3.42 mm is the radius of rotation. This velocity yields a mean

strength of the Lorentz force of evmBB10 27 N for the same magnetic eld. According to the results of both approaches, the

Lorentz force is quite small and can be ignored in our experiment because the Lorentz-force-induced change in the radius of the closed loop would be on the order of femtometres.

Methods

Quantum tunnelling rotor in a magnetic eld. The AB effect occurs in the quantum tunnelling rotor, and the wave functions of the ions gain a phase that is proportional to the homogeneous magnetic eld and the area of the closed loop drawn by the path of each ion of the quantum tunnelling rotor31,37. Because of the strong radial connement of each ion, this closed loop corresponds to the potential valley shown in Fig. 1c.

The wave function of the quantum tunnelling rotor can be expressed as38

cupt Dt cupt iJr JlDtcdownt; 1 where cup, down(t) are the wave functions of the two quantum tunnelling rotor orientations, Dt is a short time interval and Jr (Jl) is the complex amplitude of the tunnelling for clockwise (anti-clockwise) rotation. Because the AB effect introduces a phase difference between the clockwise and anti-clockwise rotations (Fig. 1e) and the quantum tunnelling rotor has sixfold rotational symmetry, the complex amplitudes are given by

Jr J exp 3 i

2p 6

Jl J exp 3 i

2p 6

F f0

; 3

where J is the strength of the tunnelling, F SB> is the magnetic ux that

penetrates the quantum tunnelling rotor, S is the area of the closed loop, B> is the magnetic eld component perpendicular to the plane of the quantum tunnelling rotor, f0 h/e is a quantum of magnetic ux, h is the Planck constant and e is the

elementary charge. Using these amplitudes, the wave function of the up orientation is given by

cupt Dt cupt iJeipF=f e ipF=f Dtcdownt 4

cupt i2J cospF=f0Dtcdownt; 5 where the transition probability P from cup to cdown (and vice versa) is

calculated as

P / j cospF=f0 j 2 1 cos2pF=f0 =2 6 using Fermis golden rule. The tunnelling dynamics depends on the magnetic ux with a period given by f0.

Detection. Images were acquired from the 1=

2

p ; 1=

p ; 0

direction by

directing the incident Doppler cooling laser along the 1=2; 1=2; 1=

2

2

p

direction. The setup and trapped structures are depicted in Fig. 4. A 1D coulomb chain was observed for strong connement conditions (a normal-mode frequency of ox42p 1.75 MHz in Fig. 1b), whereas near intermediate connement

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(Fig. 4c,d), the orientation of the triangular structure did not change during Doppler cooling. The orientations of the triangles in these two cases could be distinguished within a detection time of 50 ms.

Even under intermediate connement conditions, the triangular orientation was not stable against background-gas collisions (300 K). However, because of the ultra-high vacuum, the collision rate of such collisions is B1 every few minutes and was thus sufciently slow that the orientations could be detected.

Stabilization of the trap RF. Because the tunnelling frequency depends on the normal-mode frequencies, the trap RF and radial potential must be stabilised. The amplitude of the trap RF was monitored using a pick-up coil and an RF power metre, and the trap RF was stabilised using a feedback technique. The uctuations of the normal-mode frequencies were less than a few hundred hertz.

Sideband cooling. The sideband cooling of 2D structures in a linear Paul trap is difcult to achieve because the ions are subject to large micromotions when they

are not located at the node of the trap RF. However, we were able to cool the rotational mode to near the ground state using sideband cooling with a red sideband transition laser (2S1/2 2D5/2; 729 nm) and a quenching laser (2D5/2 2P3/2;

854 nm). The lasers were simultaneously incident on the trap for 5 ms. The asymmetry between the red and blue sideband spectra after sideband cooling (Fig. 5) indicated cooling to near the ground state.

Adiabatic cooling. The ground state of the rotational mode must be prepared to observe quantum tunnelling, and because the normal-mode frequency of the rotational mode is quite small under tunnelling conditions, an extremely low temperature is necessary to achieve this goal. To obtain a ground-state population larger than 0.1, a temperature of o90 nK is required; this temperature is much lower than the recoil limit in conventional sideband cooling. Adiabatic cooling, which has been demonstrated with trapped ions39, was therefore performed to achieve the ground state. Reducing the energy of a single motional quantum o under adiabatic conditions do

dt =o2 1

reduces the temperature, even though the phonon entropy is conserved. The mean phonon number

n at temperature T is

calculated using boson statistics as follows:

Ion trap

1

exp o=kBT 1

;

x

where o is the secular frequency of the mode, kB is the Boltzmann constant and is the reduced Planck constant.

In the experiment, the mean phonon number of the rotational mode just after sideband cooling was 0.08, which corresponds to 10 mK. We reduced the secular frequency adiabatically from 750 to 0.18 kHz to reduce the temperature to 40 nK, although there was some heating, as indicated by an increase in the entropy of the motional mode. This temperature was sufciently cold for quantum tunnelling to be observed.

Heating. Heating of the trapped ions causes a decrease in the population of the ground state of the quantum tunnelling rotor. For the intermediate connement state, we observed a heating rate after sideband cooling that was much less than 1 quantum s 1, and no heating of the rotational mode was observed within 1 s under these conditions. Determining the mean phonon number after a time interval of 1 s was difcult because the other modes were subject to stronger heating. This lack of sensitivity can be attributed to the symmetry of the rotational mode. It is difcult for electric eld noise to heat the rotational mode because the electrodes are separated from the rotor by distances that are much larger than the rotors own length scale. The other modes, for example, the zigzag mode, could not be cooled by the sideband cooling because of its strong heating.

We measured the phonon number of the rotational mode after ramping down the potential to create the triangular structure of the quantum tunnelling rotor and then ramping it back up to its original magnitude. After the potential was ramped up, the mean phonon number was 8. This value was nearly independent of the waiting time between ramping down and up for times of up to 300 ms. Hence, we speculate that the heating of the rotational mode predominantly occurred during the variation of the potential. Although the trap frequencies were very low near the critical points, the heating rate for the rotational mode remained small. This low heating was also likely attributable to the symmetry of the rotational mode. However, this value depended on the position of the centre of the triangular structure. The origins of the heating of the rotational mode may have been trap RF, other hot modes and various sources of noise. These heating sources were carefully eliminated by changing the position of the rotor. Thus, it is possible that ne control of the DC voltages of the linear Paul trap in synchronisation with the variation of the amplitude of the trap RF may permit the preparation of a quantum tunnelling rotor that is colder than that of the current experiment.

Lens

y

z

CCD

397 nm laser

5 m

Figure 4 | Experimental system. (a) The relative geometry of the ion trap and the imaging system. Images were acquired of three 40Ca ions under various conditions: (b) under strong connement conditions(ox 2p 2.1 MHz), (c) under slightly stronger than intermediate

connement (ox 2p 1.65 MHz), (d) under intermediate connement

conditions (ox 2p 1.523 MHz) and (e) near the critical point

(ox 2p 1.119 MHz).

a b

2.0

Excitation (No. of ions)

1.0

Excitation (No. of ions)

2.0

1.0

0

0 0.78 0.75 0.72 0.72 0.75 0.78 0.78 0.75 0.72 0.72 0.75 0.78

Frequency (MHz) Frequency (MHz)

Figure 5 | Sideband spectra of the rotational mode. Rotational blue and red sideband spectra (a) before and (b) after sideband cooling. The number of excitations of the three ions is presented as a function of the frequency. We performed 200 experiments per data point. The error bars represent the s.d. of binomial distributions.

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Magnetic eld. To ensure that the magnetic eld was sufciently stable for sideband cooling, we used two orthogonal sets of coils to generate the magnetic elds. The rst set created a constant magnetic eld (3.4 G) for the sideband cooling and an offset of the magnetic ux, whose directional vector was

1=2; 1=2; 1=

2

p

.

Here, the normal vector of the triangular structure was taken to be {0, 1, 0} (see Fig. 4a). The second set of coils created a tunable magnetic eld whose direction was 1=2; 1=2; 1=

2

p

to change the number of magnetic ux quanta penetrating the quantum tunnelling rotor. From the area of the closed loop (37 mm2) and the direction of the magnetic eld, the number of magnetic ux quanta that penetrated the quantum rotor was F/f0 1.50.2 when

there was no current owing through the tunable coils. This value is subject to certain errors because of the unknown residual magnetic elds and the angular error between the coils and the triangular structure of the ions.

Fitting. The t to the transition-probability data presented in Fig. 2b is given by

f p0; n; t0; u p0

1 e t=T cos 2pnt

2

!

1 e ut 2 ;

where p0 is the population of the rotational ground state of the quantum tunnelling rotor, n is the tunnelling frequency, T2 is the coherence time of the tunnelling and u is the average rotational velocity of the rotational excited states. Fitting parameters of p0 0.100.02, n 7.60.3 Hz, T2 300200 ms and u 5.40.3 s 1 were

used. The coherence time may have been limited by the heating of the rotational mode and by uctuations in the tunnelling rate. According to Fig. 2c in the main text, the tunnelling rate n was

n 21:5 Hz 0:008 ox oz=2p

near the experimental condition ox ozB2p 2 kHz. If the uctuation in the

radial connement is assumed to have been B300 Hz because of imperfections in the feedback control of the trap RF, the uctuations in the tunnelling frequency were 2.4 Hz. This indicates that if we consider only tunnelling rate uctuations, the coherence time was 130, which is consistent with the tting parameter.

The t function used to obtain the results presented in Fig. 3 in the main text was

ga; x; y0; h

1 p0

a2 cos2pxn y0 h; 7

where n is the number of magnetic ux quanta penetrating the quantum tunnelling rotor. The tting parameters were a 0.060.02, x 0.990.07,

y0

0.28p0.22p and h 0.150.01.

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Acknowledgements

We thank Shmuel Nussinov, Yasunobu Nakamura and Takuya Higuchi for their useful discussions. This work was supported by the MEXT KAKENHI Quantum Cybernetics Project, a Grant-in-Aid for Young Scientists (Grant No. 25800181), and the Japan Society for the Promotion of Science (JSPS) through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program), which was initiated by the Council for Science and Technology Policy (CSTP). A.N. was supported in part by the Japan Society for the Promotion of Science.

Author contributions

A.N. performed the experiment with assistance from K.T. and S.U., Y.S. provided theoretical guidance regarding the AB effect experiment and S.U. headed the project. All of the authors contributed to writing the manuscript.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Noguchi, A. et al. AharonovBohm effect in the tunnelling of a quantum rotor in a linear Paul trap. Nat. Commun. 5:3868 doi: 10.1038/ncomms4868 (2014).

6 NATURE COMMUNICATIONS | 5:3868 | DOI: 10.1038/ncomms4868 | http://www.nature.com/naturecommunications

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