4 March 2002

Physics Letters A 294 (2002) 271–277

www.elsevier.com/locate/pla

Testing wave–particle duality with the hot-ion quantumcomputation

Mang Fenga,b,∗, Xiwen Zhua, Kelin Gaoa, Ximing Fangc

a Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Academia Sinica,Wuhan 430071, PR China

b Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Street 38, D-01187 Dresden, Germanyc Department of Physics, Hunan Normal University, Changsha 410081, PR China

Received 25 October 2001; accepted 30 January 2002

Communicated by A.R. Bishop

Abstract

Wave–particle duality refers to quantum mechanical objects behaving as waves or particles under different physical con-ditions. We propose a scheme for testing the wave–particle duality by means of the hot-ion quantum computation. Withan efficient controlled-NOT gate, the ‘which-path’ information can be obtained with the method of bichromatic field. Theappearance and disappearance of interference fringes due to deletion and production of the entanglement between the observedion and the marker are analyzed, respectively. The scheme can be regarded as both a more efficient hot-ion quantum computationone and an application of quantum computation. 2002 Elsevier Science B.V. All rights reserved.

PACS: 42.50.Vk; 32.80.Pj

The wave–particle dual behavior of matter contains the basic mystery of quantum mechanics [1]. When aquantum object moves from one place to another along several different paths simultaneously, the wave characteris illustrated with the interference fringes. Any attempt to make the paths distinguishable, which is considered tobe the particle behavior of the object, will destroy the interference pattern. In one word, the wave–particle dualitymeans that the simultaneous observation of wave-like and particle-like behavior is prohibited [2].

The most famous classic gedanken experiments for testing the wave–particle duality with ‘which-path’ markerare Einstein’s recoiling slit [3] and Feynman’s light microscope [1], where the loss of fringes can be resortedto the Heisenberg’s uncertainty relation because labeling or measuring the paths unavoidably introduces a randommomentum transfer or unpredictable phase shift. However, the new gedanken experiment [4] proposed based on thelatest development in atomic physics and quantum optics denied the uncertainty relation to be always responsiblefor the loss of the interference pattern. Instead, it is the quantum correlation, i.e., entanglement, in these schemes

* Corresponding author. Present address: Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 65, I-10133,Torino, Italy.

E-mail address: feng@isiosf.isi.it (M. Feng).

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00125-1

272 M. Feng et al. / Physics Letters A 294 (2002) 271–277

that makes evolution paths of the quantum entity distinguishable, and thus destroys the fringes. The new explanationbrought about some dispute [5,6], and some experiments [7–11] have been made for testing the new explanation.With two cold ions well localized in the trap, NIST group [7] reported their first observation of interference fringesin the light scattered from these two ions. Although simple ‘which-path’ consideration can explain the loss of thefringes, the work was criticized due to the double roles of the two ions, which mixed the influence from the doubleslit and the ‘which-path’ detector [5]. While the experiments [8–11] with atoms and photons clearly demonstratedthat the quantum state entanglement between the ‘which-path’ detector and the observed quantum object destroyedthe fringes, where in [9] the amount of the ‘which-path’ information was measured. Recently, an experiment withnuclear magnetic resonance (NMR) [12] was carried out, in which the wave-like and particle-like behavior of thequantum object are observed simultaneously. According to the viewpoint of that paper, the wave–particle duality isvalid in and only in the case that merely the population measurement is carried out in the experiments. Thereforeas long as the coherence and population can be observed simultaneously, the wave and particle behavior can befound in the same observation.

In this Letter, we propose a scheme of ‘which-path’ experiment with hot-ion quantum computation (SM)model [13]. Our main idea is to use the operation of quantum computation [14] for demonstrating the appearanceand disappearance of the interference fringes, that is, the creation or deletion of the entanglement between themarker and the observed ion with quantum computation operations will present or remove the ‘which-path’information. There are some obvious advantages in using the ion trap quantum computation to study this problem:first, the manipulation is made on individual ions by quantum level [15], rather than in the NMR the operationperformed on the bulk molecules; second, the entanglement between ions is deterministic [16,17], which can avoidthe problem due to the selection of data from random process in experiments with photons and atoms; third, fourtrapped ions have been experimentally entangled [17] based on the SM model, and subsequent experiments [18,19]for testing the Bell inequality and constructing the dephasing-free qubits are also based on the SM model. So ourscheme may be readily achieved experimentally. In what follows, let us first briefly review quantum computationwith trapped ions. Then our scheme for testing the wave–particle duality will be proposed based on a practicalcontrolled-NOT (CN) gate. Finally, the requirement for experimentally realizing our scheme will be discussed.

The implementation of quantum computation is based on two basic operations [20]. One is for the single qubitand the other is for two qubits. The suitable combination of such two operations will in principle produce anyunitary operations desired by quantum computation. Single qubit operation can be easily performed on the trappedions with laser pulse of [21]

V k(φ) =(

cos(

kπ2

) −ieiφ sin(

kπ2

)−ie−iφ sin

(kπ2

)cos

(kπ2

) ),

wherek is the parameter proportional to the wave-vector of the laser andφ the phase of the laser. While therealization of two-qubit operation, such as the CN gate or phase gate, is relatively difficult, which is the maintopics of various quantum computation schemes [13,21,22]. In the famous (CZ) model of Ref. [21], the motionalstates of ions are introduced as the data bus, and the success of the model strongly depends on both the groundmotional state and some special requirements for the internal levels of the ion. However, as it is difficult to do sowith the existing ion trap technique, how to release the requirement in CZ model has been attracted much attention.Among the various modified models, the most promising scheme is the SM model [13]. For the simplest case ofthe model, two identical two-level ions in the string are both illuminated with two lasers of different frequenciesω1,2 = ω0 ± δ, whereω0 is the resonant transition frequency of the ions, andδ the detuning, not far from the trapfrequencyν. With the choice of laser detunings the only energy conserving transition is from|ggn〉 to |een〉 orfrom |gen〉 to |egn〉, where the first (second) letter denotes the internal statee or g of the ith (j th) ion andn isthe quantum number for the motional state of the ion. As we considerν − δ ηΩ with η being the Lamb–Dickeparameter andΩ the Rabi frequency, there is only negligible population being transferred to the intermediate stateswith motional quantum numbern ± 1. It has been proven that this two-photon process has nothing to do with the

M. Feng et al. / Physics Letters A 294 (2002) 271–277 273

Fig. 1. Configuration of the two ionsa andb, and the scheme of the CN gate, where|g〉, |e〉 and |r〉 are ground, metastable, and auxiliaryinternal levels, respectively, and|n〉 is an arbitrary motional Fock state. The dashed lines represent the virtually excited states. The thin arrowdenotes the single qubit operation on the ionb, and the thick arrows are for the bichromatic field radiated on the two ions. The dashed linearrow between|g〉a and |r〉a denotes the weak detection light with ESA. For the radiation of bichromatic field, the case of blue detuning forthe ionb and red detuning for the iona is also available (see details in Ref. [13]).

motional state|n〉. So quantum computation with such configuration is valid even for hot ions, and we will ignorethe motional state in the following description.

The basic procedure of our scheme is based on the SM model. Two identical ionsa andb are supposed to betrapped in a linear trap. Different from the SM model, we consider three levels in each of them, as shown in Fig. 1,where|g〉, |e〉 and|r〉 are respectively ground, metastable, and auxiliary levels. We use the ionb as the marker, andion a to be the observed ion. Suppose that the ions are initially in|g〉a |g〉b . By applying a laser pulseV k0

a (0) andthenV

1/2a (φ) sequentially with the subscript denoting the radiated ion, we have the following process:

|g〉a |g〉b →[cos

(k0π

2

)|g〉a − i sin

(k0π

2

)|e〉a

]|g〉b

(1)→ 1√2

[cos

(k0π

2

)− eiφ sin

(k0π

2

)]|g〉a − i

[e−iφ cos

(k0π

2

)+ sin

(k0π

2

)]|e〉a

|g〉b.

Measurement of the population on|g〉a(|e〉a) will result in 1− cosφ sin(k0π)(1 + cosφ sin(k0π)). That means,with the fixed values ofk0, repeating the process above by changing the phaseφ will produce the Ramsey fringes.It is the demonstration of wave behavior of the ion due to the indistinguishability of evolution paths of the iona

from |g〉a to the final state|g〉a or |e〉a . The ionb only acted as a spectator in the whole process. For distinguishingthe evolution paths of iona, we should entangle the ionsa andb. So a CNab gate has to be introduced with thetruth table of|g〉a |g〉b(|e〉b) → |g〉a |e〉b(|g〉b), and|e〉a|g〉b(|e〉b) → |e〉a |g〉b(|e〉b). We note that the realization ofCN operation with the hot-ion quantum computation has been discussed in the SM model. While for achieving ourscheme, we have to modify it, which will be discussed later. We still suppose that the two ions are initially in thestates|g〉a |g〉b. Then we apply a laser pulseV k0

a (0), then CNab operation, and then another laser pulseV1/2a (φ).

We will have the following steps:

|g〉a |g〉b →[cos

(k0π

2

)|g〉a − i sin

(k0π

2

)|e〉a

]|g〉b → cos

(k0π

2

)|g〉a |e〉b − i sin

(k0π

2

)|e〉a|g〉b

274 M. Feng et al. / Physics Letters A 294 (2002) 271–277

→ 1√2|g〉a

[cos

(k0π

2

)|e〉b − eiφ sin

(k0π

2

)|g〉b

]

(2)− i√2|e〉a

[e−iφ cos

(k0π

2

)|e〉b + sin

(k0π

2

)|g〉b

].

The measurement of the population of|g〉a(|e〉a) will not find Ramsey fringes because the possibility of|g〉a(|e〉a)becomes the constant, which means the particle-like behavior of the iona. The reason for the loss of fringes isthat evolution paths of the iona can be distinguished with the labeling of the ionb. As the last laser pulse is onlyapplied on the iona, the entanglement of the ionsa andb guarantees that we can know ‘which-path’ informationabout the evolution of the iona by means of the ionb.

The CN operation plays essential role in our scheme. For achieving our scheme, we will use the proposal of thebichromatic field, whereas we modify the CN operation scheme to make the CN implementation more simplifiedthan that in the SM model. We first apply a laser pulse ofV

1/2b (3π/2), which makes|g〉b be(1/

√2)(|g〉b + |e〉b).

Then we illuminate the two ions with the laser beams of two different frequencies,ωa,b = ωgr ± δ, whereωgr isthe transition frequency between|g〉 and|r〉, andδ the detuning not far from the trap frequencyγ . With the resultsin the SM model, the evolution of the two ions will be

(3)|gg〉ab → cos

(Ωt

2

)|gg〉ab + i sin

(Ωt

2

)|rr〉ab, |rr〉ab → cos

(Ωt

2

)|rr〉ab + i sin

(Ωt

2

)|gg〉ab,

whereΩ = −(Ωη)2/2(ν − δ) with Ω andη the Rabi frequency and Lamb–Dicke parameter, respectively. If wecan set the evolution time to bet = 2π/Ω , which makes|gg〉ab be−|gg〉ab (i.e., the phase gate), then after applyinganother laser pulseV 1/2

b (π/2), we finally obtain|ge〉ab. This can be easily demonstrated by following equations:

(4)|g〉a |g〉b → 1√2|g〉a

(|g〉b + |e〉b) ⇒ 1√

2

(−|g〉a |g〉b + |g〉a |e〉b) → |g〉a |e〉b,

(5)|e〉a|g〉b → 1√2|e〉a

(|g〉b + |e〉b) ⇒ 1√

2

(|e〉a|g〉b + |e〉a|e〉b) → |e〉a |g〉b,

where the short arrows and long arrows denote single qubit operations and bichromatic field operations, respec-tively. Compared with the CN gate in the SM model which includes nine operations, our CN gate only has threesteps. If for simplicity, we neglect the time for single qubit operation, the time for the CN gate in the SM modeland in the present work areπ/|Ω| and 2π/|Ω|, respectively. However, in actual experiments, there exists the delaytime between any two neighboring operations. It means that, the more the operations, the longer the time of theCN gate. More importantly, there are some uncontrollable factors in each operations which might lead to errors,and the accumulation of the errors would probably destroy the correct results of the quantum computation. Thebest way to reducing these errors is to decrease the number of the operation. Therefore, in this sense, our CN gatescheme is more practical than in the SM model.

Besides the strict requirement for the laser-cooling and confinement of the trapped ions, the another problemof the CZ model is that the realization of the CN operation needs an auxiliary (metastable) level besides the logicstates, which increases the sensitivity of the CN operation to external magnetic fields fluctuation, and decreases thespeed of quantum computation [22]. In our scheme, we also need three levels. However, the difference is obvious.With specific values of the parameters [23], we can obtainΩ ≈ 5× 10−4ν, and the time for the phase gate will beapproximately 2.3 × 10−4 s if ν = 2π × 8.8× 106 Hz [17]. So as long as the lifetime of|r〉 is about 10−3 s, ourscheme can be achieved since the states of the ions have returned to|g〉a |g〉b before the spontaneous emission takesplace. Under this circumstance, although the auxiliary state|r〉 still belongs to a metastable state, as its lifetime(with 10−3 s) is much shorter than that of|e〉a (generally with the lifetime of 1–10 s),|r〉a can play a role in thefinal measurement of the population on|e〉a and|g〉a with the method of electronic shelving amplification (ESA)[24,25].

M. Feng et al. / Physics Letters A 294 (2002) 271–277 275

The quantum interference is essential to quantum computation. It is quantum interference of different computa-tional paths that enhances the final results we desired. Although the interference effect we discussed above is notequivalent to that in the process of quantum computation, above discussion might leave misleading impression thatquantum entanglement by quantum computation operation removed the quantum interference. This is not true. Letus use the density operator for a discussion. For the process testing the particle-like behavior, if we set

ρ|g〉a |g〉b =(

1 00 0

)⊗

(1 00 0

),

the final state will have the form

(6)ρ = 1

2

sin2 k0π2 −1

2 sink0π −ieiφ sin2 k0π2 − i

2eiφ sink0π

−12 sink0π cos2 k0π

2i2eiφ sink0π ieiφ cos2 k0π

2

ie−iφ sin2 k0π2 − i

2e−iφ sink0π sin2 k0π2

12 sink0π

i2e−iφ sink0π −ie−iφ cos2 k0π

212 sink0π cos2 k0π

2

.

So if we only measure the population of the states, we will only obtain the diagonal terms, which is irrelated to thephaseφ, and the fringes thereby disappears. However, from above equation, we know that the off-diagonal termsinclude the information aboutφ, which means that the wave character (i.e., the coherence) still remains after thepath-labeling. So if we measure the coherence of the entangled state, fringes will reappear. With NMR technique,one can easily measure the population and the coherence simultaneously, so it is not surprising that the wave-likeand particle-like behavior could be observed simultaneously in Ref. [12]. However, it is hard to do so with currention trap technique. Therefore the wave-like and particle-like behavior are presented in different procedures in ourscheme. Nevertheless, as it has been proven [26] that the correlated state can suppress the fluctuation, such as thephase uncertainty, to be smaller than the Heisenberg limit, and our scheme has excluded the motional states of ions(i.e., no random momentum transfer in our scheme), the only reason for the disappearance of fringes in our schemeis the entanglement between the observed iona and the markerb. We can strengthen this claim by introducingsome additional operations, for example CNba · V k0

b (φ + π), to be the quantum eraser [10,11,27]. Performing thequantum eraser after the manipulation for testing the particle-like behavior of the ion, we will obtain

1√2|g〉a

1

2sin(k0π)

(1− eiφ

)|g〉b − i

[sin2

(k0π

2

)+ e−iφ cos2

(k0π

2

)]|e〉b

(7)+ 1√2|e〉a

i

2sin(k0π)

(eiφ − 1

)|g〉b +[e−iφ sin2

(k0π

2

)+ cos2

(k0π

2

)]|e〉b

,

which means that the fringes reappear because the labeling of evolution paths of the iona has been wiped out bythe quantum eraser. Therefore, the success of our scheme can confirm without any doubt the new explanation forthe appearance and disappearance of interference fringes.

Some points related to the experimental implementation of our scheme need to be mentioned. First, as referredto above, a suitable metastable|r〉 should be chosen to accomplish both the phase gate and the ESA. Secondly,making a precise adjustment ofφ andk with existing techniques is essential to our scheme. It is also the challengefor realizing the ion trap quantum computer with the CZ model. Thirdly, the two ions need to be individuallyaddressed in the lights because we have the single-ion operation with the laser pulseV k(φ) besides the bichromaticfield. It is fortunately under the reach of the existing technique [15,19]. Fourthly, as we measure the iona withESA scheme, the weak detection light is only applied on the iona. As long as the emitted fluorescence from|r〉ato |g〉a has the different polarization from those from|e〉a to |g〉a and from|e〉b to |g〉b , we can precisely measurethe population in|g〉a and |e〉a . So from this result, the bichromatic field applied on both ions should also havedifferent polarization from that of the lasers applied on individual ions.

276 M. Feng et al. / Physics Letters A 294 (2002) 271–277

Before ending our discussion, we have to emphasize that the wave–particle duality indicated in the text book ofquantum mechanics is based on the interference of different trajectories of a single particle. While the discussion ofthe wave–particle duality made here involves two particles with internal degrees of freedom. Although the situationlooks more complicated than single particle case, the judgment of the former for the wave and particle charactersis actually consistent with that of the latter. Particularly the definition of ‘interference’ in original wave–particleduality is still valid in our work, which means the Ramsey fringes are produced by the interference of evolutionpaths of the observed ion itself, instead of by the interaction between two ions. However, as two ions are actuallyinvolved in our scheme, and non-demolition measurement of population is available in ion trap experiments, is itpossible to have a simultaneous measurement of wave- and particle-like behavior of trapped ions, like the NMRmethod in Ref. [12], by elaborately designed experiment? This would be the further investigation of the presentwork.

In conclusion, we have proposed a scheme for ‘which-way’ experiment based on the hot-ion quantumcomputation with the SM model and the CZ model. The CN gate in our scheme is more practical than that in theSM model and the requirement for motional states of ions in the CZ model is removed. Although our scheme stilluses three electronic levels in each ions, the third level|r〉 can play double roles, which is much different from thatin the CZ model. So this part of our scheme provides a more practical hot-ion quantum computation scheme, whichcan be realized experimentally with the existing technique. On the other hand, the test of the wave–particle dualitycan be considered as another application of quantum computation in the study of fundamental physics, besides thework of Ref. [18]. As the entanglement between the trapped ions is deterministic, and the read-out in such a systemis of high efficiency, the experimental test of the wave–particle duality with our scheme will be more convincingthan former work in this respect. Moreover, although quantum computation hardware is still in its infancy, we mayexpect small-scaled quantum computer with 20 trapped ion pairs to appear in the near future. If we suppose that thedata retrieved from 1000 times of experiments are enough to present the specific results of wave–particle duality,such a quantum computer can theoretically finish the job with only fifty times of computation.

Acknowledgement

The work is partly supported by the National Natural Science Foundation of China.

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