Physics Letters A 373 (2009) 716–719

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Physics Letters A

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Qubits from tight knots, bent nano-bars and nano-tubes

Victor Atanasov a,b, Rossen Dandoloff a,∗a Department of Physics, 1 University of Cergy-Pontoise, 2 rue Adolph Chauvin, F-95302 Cergy-Pontoise, Franceb Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 September 2008Received in revised form 9 December 2008Accepted 17 December 2008Available online 30 December 2008Communicated by A.R. Bishop

PACS:03.65.-w03.65.Ge02.10.Kn

Keywords:Trefoil knotCurvatureQuantum effective potentialTwo-level systemQubits

We propose a novel mechanism for creating a qubit based on a tight trefoil knot, that is an electronnano-waveguide system so small as to be quantum coherent with respect to curvature-induced effects.To establish tight trefoil knots as legitimate candidates for qubits, we propose an effective curvature-induced potential that produces the two-level system and identify the tunnel coupling between the twolocal states. The proposed two-level system is geometrical in nature and is macroscopic of origin. It alsorepresents new and peculiar property of the trefoil knot. We also propose a different realization of aqubit based on twisted nano-bars and nano-tubes.

© 2009 Elsevier B.V. All rights reserved.

Mathematics views knots as closed, self-avoiding curves em-bedded in a three-dimensional space [1], e.g. any knot can be tiedon a thread of any length. This definition leads to the conclusionthat a knot tied on a thread can be arbitrarily shaped and all itsconformations are essentially equivalent. Besides, the lack of char-acteristic length prevents the introduction of energy scale. Physicsviews real knots differently [2] since tying them requires the use ofthread of a finite diameter and tying a particular type of knot re-quires a thread piece with proper length. In the physically relevantsituation the diameter will play the role of characteristic length.

Suppose we are to tie the trefoil knot on a thread with circu-lar cross-section and pull the knot tight. This process has its limit.There exists a particular conformation of the knot at which thethread’s ends cannot be pulled apart any more without changingits cross-sectional shape. The final conformation will be referredto as tight open knot. If the loose ends are connected continuouslywe will refer to the resulting structure as tight closed knot. An ar-gument based on the minimization of the elastic energy in thematerial on which the knot is tied points to the symmetrical con-formation as most energetically favorable. Thus the existence of a

* Corresponding author.E-mail addresses: victor.atanasov@u-cergy.fr (V. Atanasov),

rossen.dandoloff@u-cergy.fr (R. Dandoloff).1 CNRS-UMR 8089.

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2008.12.044

Fig. 1. The geometry of a trefoil knot and its curvature profile taken from [6]. Themap of curvature of the tight open trefoil knot shows a mirror symmetry (the mir-ror plane is vertical and is in the center of the map). On the figure the curvatureis normalized with respect to the diameter (2ρ0) of the tightly knotted electronnano-waveguide, that is (2ρ0)−1 = 1. Superimposed on the image is the quantumdescription suggested in the present study. This description comes in naturally asa result of the exhibited mirror symmetry and the curvature-induced quantum po-tential.

plane of symmetry both for the open and the close tight knot issuggested (see Fig. 1).

V. Atanasov, R. Dandoloff / Physics Letters A 373 (2009) 716–719 717

Fig. 2. The curvature-induced quantum potential for the knot’s centerline.

As indicated by de Gennes [3], such knots are spontaneouslytied and untied by thermal fluctuations on long polymeric mole-cules and change the macroscopic properties of materials [4]which is one of the very few practical applications of knot theoryin physics. In this Letter we attempt to extend the scope of prac-tical applications of knot theory and the idea that a collection ofnano-bars and nano-tubes glued together can produce prescribedquantum behavior for a particle trapped inside [5].

The thread’s cross-sections are shaped as disks of radius ρ0which centers are located on a centerline whose tangent �t is con-tinuous (no cusps). Their normals coincide with the tangent �t atevery point. The disks are not allowed to overlap which is the self-avoiding condition. The effect of the self-avoiding condition on thecurvature κ of the thread’s centerline is the following ρ0κ < 1.

In what follows the thread represents a thin nano-tube follow-ing the centerline and acts as an electron waveguide system withdiameter 2ρ0.

The curvature profile of the unique tightest conformation of thetrefoil open knot tied on a perfect thread is discussed within thenumeric experiment in [6]. The result is conveyed in Fig. 1.

We will consider possible quantum mechanical implications ofthis profile for κ , with regard to electron transport and formationof bound states in the bulk of an electron waveguide. It has beenshown by [8] that an electron in an electron waveguide whose axisfollows a space curve with curvature κ feels an effective potential[8] of the form

V eff(s) = − h̄2

2m

κ2(s)

4(1)

where m is the particle mass, h̄ is the Planck’s constant and s isthe arclength.

Other study [7] of elastic rods bent into open loose knots withinthe Kirchhoff equations for rods report similar symmetric profilefor the curvature with respect to the point of symmetry of theknot. Thus we believe that the simplified potential that we adoptand is depicted in Fig. 2 is justified in view of the numerical data[6] and the analytic solution [7].

In view of Goldstone and Jaffe work [8] on bound states in thintubes we note that, since the quantum dynamics along the cen-terline and in the cross-section are coupled the following resultsapply for the zero-mode in the circular cross-section of the nano-waveguide. This effect concerns only low-energy electrons.

The problem of quantum dynamics for an electron in the pres-ence of an idealized curvature-induced quantum potential of bind-ing quality depicted in Fig. 2 leads us to the striking observationthat due to the symmetry of the trefoil knot a two-well potentialalong the arclength of the knot’s central curve emerges.

A particle once prepared in a state occupying only one of thewells with certain energy E , that is |L〉 or |R〉 state, possesses tun-neling probability to filter through the barrier between the twowells along the arclength of the knot’s central curve. We can con-struct an approximate solution in this energy field due to curvatureusing the quasi-classical wave function ψ(s) describing the motionwith energy E in one of the wells and exponentially decreasingon both sides of the well’s boundaries. Using this approximatesolution and a model potential well of depth U0 = h̄2κ2/8m =

h̄2/(25mρ20 ) and width D = 5ρ0 (where we have taken model’s

values from observation of Fig. 1) we can quantify the energy splitdue to tunneling.

The bound state energies E = − h̄2

2m q2 corresponding to suchcurvature-induced effective potential are given by

En = −U0[1 − [

(kn D)/(2C)]2]

,

where U0 = h̄2k20/2m is the depth of the well in terms of k2

0 =κ2/4 for the knot, D is the width of the well, that is the lengthof the stretched section where the quantum particle relaxes dueto Heisenberg’s uncertainty principle, and C = κ D/4 is a geo-metrically determined quantity qualifying the potential well. Ac-cording to Fig. 1 we can take D = 5ρ0 and κ = 1/(2ρ0) whichfixes C = 5/8. Here the discrete values kn of the wave numberk2 = k2

0 − q2 are solutions to the algebraic equations

tan kD

2=

√C2 − (k D

2 )2

k D2

(= − k D

2√C2 − (k D

2 )2

)(2)

for the even (and for the odd) eigenstates. It is interesting to noticethat the existence of at least one even ground state is guaranteed. It is

k1 ≈ 1

5ρ0, E = −

(3

5

)2

U0 ≈ − 32h̄2

2552mρ20

. (3)

With the above value C = 5/8 we have only one even state (3). Inpractice, changing knot’s conformations we may adjust the valueof C and generate more levels.

It is interesting to point out the situation in which the electronwaveguide on which the knot is tied has a finite length with aninfinite potential barrier at both ends. This leads to the conditionof a node at the ends and modifies (2) which for a knot tied in themiddle of the rope’s extent is

kD =1∑

ε=0

tan−1{±

[q

kcoth q

(l + (−1)ε

D

2

)]±1}(4)

for the even (+) and for the odd (−) eigenstates. Here l mea-sures the distance from the knot to the either end. The boundaryconditions do not affect the existence of the guarantied negativeenergy even eigenstate for which we can again use (3). The hardwalls introduce quantization in the positive energy spectrum upto a finite number of positive energy eigenstates. The second eveneigenstate after (3) is at k2 = 7/(25ρ0) for l = 5D which corre-sponds to energy E2 = 3U0/25. The first odd eigenstate appearsat kodd

1 = 4/(25ρ0) or Eodd1 = 7U0/25. As l is increased they shift

slightly and for l → ∞ we are left with (3).An estimate of the number of states Ns as a function of the

geometry and the topology of the knot can be obtained evaluatingthe integral [9]

Ns =∫ √−2mV eff(s)ds

2π h̄= 1

4π

∫κ(s)ds. (5)

Now let us consider a tight closed knot. A property of knots onclosed curves [10] is the existence of a lower bound for∮

κ(s)ds � 4π.

Thus Ns � 1 and a bound state in the case of a tight knot tiedon a closed nano-waveguide does exist! Due to the symmetryof the tightest conformation it is again split into two levels �Eapart. Actually, the tight closed knot can serve as a simulation of aone-dimensional solid due to the periodicity of the closed config-uration.

The knot becomes a two-level quantum mechanical system orin other words a qubit, due to tunneling. The energy level E

718 V. Atanasov, R. Dandoloff / Physics Letters A 373 (2009) 716–719

Fig. 3. The ratio between the transmitted and the incident probability currents jt/ ji

as a function of the wave number q = √2mE/h̄. Here we have set D = 5/2, ρ0 =

1/2, κ = 1 and d = 10−2.

Table 1

i 1 2 3 4 5 6

qi = √2mEi/h̄ 0.38 1.15 1.81 2.46 3.06 3.74

splits into two E+ and E− . The zeroth-approximation wave func-tions corresponding to the two levels are symmetrical and anti-symmetrical combination of the quasi-classical wave function ψ(s)and ψ(−s), that is ψ±(s) = [ψ(s) ± ψ(−s)]/√2. The split is ex-

pressed as �E = E+ − E− = 2h̄2

m ψ(0)dψds (0), where the zero is cho-

sen so as the axis of symmetry of the potential passes through(see Fig. 2) it and the quasi-classical wave function is given in thestandard way [9]. The width of the energy split is

�E = Ωh̄

πe− 1

h̄

∫ a−a |p1|ds = h̄2|k1|

mDe−|k1|d, (6)

where the order of magnitude is h̄2

52mρ20

e−d/(5ρ0) and Ω is the fre-

quency of the classical periodic motion in the well. It is given interms of the period for the classical motion

Ω = 2π

T= π

m

( a∫b

ds

pn

)−1

= π h̄|k1|mD

∼ π h̄

52mρ20

,

where a and b are the turning points corresponding to the periodictrajectory in the well, D = |a − b| is the width of the potentialwell, the associated momentum p1 is p1 = √

2m(U0 − E) = h̄|k1| ∼h̄/(5ρ0). Here d is the width of the barrier between the two wells.

All quantities in (6) are geometry determined by the knot’s con-formation!

Now if we propagate a plane wave with positive energy fromone end of the knot to the other we may acquire insight into theproperties of the curvature-induced potential by measuring the ra-tio between the transmitted jt and incident ji currents. For themodel potential (see Fig. 2) theory gives a lengthy expression [11].Conveyed graphically in Fig. 3 is the behavior of jt/ ji . The fig-ure renders visible the energies for which the knot is completelytransparent. The first few energies for which jt/ ji = 1 with thedata from Fig. 3 are summarized in Table 1.

The entire knot can be placed in an external field that couldbe time dependent Eext(t). If we place a charged particle in one ofthe wells, say |L〉 or |R〉 state, the knot becomes electrically asym-metrical and an effective electric dipole moment μ = e(d + D) can

Fig. 4. A collection of circular and straight nanobars which recreates the double-wellcurvature-induced potential due to confinement as depicted in Fig. 2.

be introduced. External fields can couple with it thus reminiscethe way ammonia molecule is driven by external electric field atresonance frequency. For the knot this resonance frequency is cal-culated to be

ω = �E

h̄= h̄|kn|

mDe−|kn|d ∼ h̄

52mρ20

e−d/(5ρ0). (7)

The knot qubit can be prepared in the |R〉 state by setting a lon-gitudinal electric field with respect to Fig. 1 towards the right(assuming the particle in the interior of the knot is negativelycharged, say an electron). By suddenly turning off this electricfield the knot’s state is prepared in a coherent superposition of(|L〉 ± |R〉)/√2. Because of the tunnel splitting, the system thenstarts to oscillate coherently, with a frequency given by (7). For ρ0in nm the characteristic frequency is of the order of 250 GHz.

After an amount of time t , the knot can be in either |L〉 or|R〉 states. Therefore, by detecting the particle’s distribution, as afunction of t , we can determine the coherent oscillation frequencyexperimentally. Driven transitions between the two states can beachieved by adding a sinusoidal component at the resonance fre-quency.

Quantum computing requires two-qubit operations. For chargedknots, the inter-qubit interaction comes naturally in terms of theelectric dipole interaction due to μ between the knots. The appli-cation of longitudinal electric field can be used to tune selectedknots into resonance, then microwaves can be used to performconditional rotations and other operations.

If instead of an electron nano-waveguide we consider a quan-tum wire one could easily manipulate the two-level system byapplying an electric field along the wire. This would lead to ashift in the potential energy making the two wells asymmetrical.The potential difference between the two ends of the well will be�V = eE D . Here E is the field’s strength, e = ∓|e| is the chargeof the particle assumed to be an electron or a hole respectively.The critical value at which the level disappears is to be found

by the standard inequality [9] cos κ D2 � −

√1 + 8m

h̄2κ2 eE D which

for weak field leads to an estimate for the non-destructive fieldstrength

0 � E � h̄2κ2

4meD

(1 + cos D

κ

2

)� h̄2

20meρ20

.

A quantum wire, however, may pose a series of problems withdecoherence of the system due to scattering on possible defects,electron–electron interactions, etc.

Now let us turn our attention to the description of the devicedepicted in Fig. 4. It is made up of a collection of straight and bent(into quarter circles with radius R) nano-rods. The quantum effec-tive potential within the glued rods is exactly the same in form asthe one for the knot (Fig. 2) with D = π R/2. Here d = ‖BC‖ andD can be adjusted by picking rods with appropriate (with respectto our resolving power in measuring E , since the energy split is�E ≈ D−2 exp [−d/D]) lengths and radius of curvature. Due to thesimilarity in the quantum potential this device can be manipulatedin exactly the same manner as the knot qubit.

It is interesting to note also that if we choose d = 0, that is weremove the central rod and actually merge the points B and C ,we can produce another device which can serve as a qubit. By ap-plying symmetrically electric field, provided the rod is conducting

V. Atanasov, R. Dandoloff / Physics Letters A 373 (2009) 716–719 719

Fig. 5. The quantum potential within the nano-bars’ interior in the presence of suit-ably applied electric field E .

with respect to the point of merger (also a point of symmetry),say (−), and points A and D , say (+), in the manner depictedin Fig. 5 we can create controllable energy split of the quasi-stationary states in the well created by the electric field and theinsulator at the end points on the figure. Such a device is vulnera-ble to noise from the driving electrical generator.

In conclusion we have expanded the application of knot theoryin physics by recognizing that a tight trefoil knot tied on an elec-tron waveguide with nanoscale diameter creates in its interior aquantum potential due to curvature of binding quality. An energylevel in this potential is split into two due to the symmetry of theknot. Similar potential can also be created by a collection of bent

nano-bars and nano-tubes glued together. Electric fields can be ap-plied to manipulate and tune such a system due to its electricalasymmetry if a charged particle is occupying the left or the rightwell of the double-well potential producing the two-level system.This electrical asymmetry also couples identical knots and servesas inter-qubit interaction. All of this suggests experimental setupswhich could be realized with present technology.

References

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