Original paper can be found at: http://eproceedings.worldscinet.com/ Copyright World Scientific Publishing Co. DOI: 10.1142/9789812704474_0009Interfering electrons in a mesoscopic ring are irradiated with both classical and nonclassical microwaves. The average intensity of the charges is calculated as a function of time and it is found that it depends on the nature of the irradiating electromagnetic field. For various quantum states of the microwaves, the electron autocorrelation function is calculated and it shows that the quantum noise of the external field affects the interference of the charges. Two-mode entangled microwaves are also considered and the results for electron average intensity and autocorrelation are compared with those of the corresponding separable state. In both cases, the results depend on whether the ratio of the two frequencies is rational or irrational.Peer reviewe

Chong, C.C.

Tsomokos, D.I.

Vourdas, A.

English

arXiv

University of Hertfordshire Research Archive

arXiv:cond-mat/0209555v1  [cond-mat.mes-hall]  24 Sep 2002CORRELATION PROPERTIES OF INTERFERINGELECTRONS IN A MESOSCOPIC RING UNDERNONCLASSICAL MICROWAVE RADIATIOND. I. TSOMOKOS, C. C. CHONG AND A. VOURDASDepartment of Computing, School of Informatics,University of Bradford,Bradford BD7 1DP, United KingdomInterfering electrons in a mesoscopic ring are irradiated with both classical andnonclassical microwaves. The average intensity of the charges is calculated as afunction of time and it is found that it depends on the nature of the irradiatingelectromagnetic field. For various quantum states of the microwaves, the elec-tron autocorrelation function is calculated and it shows that the quantum noiseof the external field affects the interference of the charges. Two-mode entangledmicrowaves are also considered and the results for electron average intensity andautocorrelation are compared with those of the corresponding separable state. Inboth cases, the results depend on whether the ratio of the two frequencies is ratio-nal or irrational.1. IntroductionThe Aharonov-Bohm effect [1] manifests itself as a nontrivial quantumphase, whenever electric charges travel in a field-free region enclosing amagnetostatic flux. This ‘geometrical phase’ has been generalized [2] andthe original results have found applications in various contexts, for examplein conductance oscillations in mesoscopic rings [3] and ‘which-path’ exper-iments that use novel solid-state devices [4].A recent development of these ideas has been to replace the magne-tostatic flux by an electromagnetic field [5]. The objective here is verydifferent, since this ‘ac Aharonov-Bohm experiment’ constitutes a nonlin-ear device where the interaction between the interfering electrons and thephotons leads to interesting nonlinear phenomena [6]. For an overview ofrelated studies on the interaction of mesoscopic devices with microwaveswe refer the reader to [7].It is interesting to investigate the same phenomena with quantized elec-tromagnetic fields. This ‘quantum ac Aharonov-Bohm experiment’ withnonclassical microwaves, has been studied [5] and one can quantify how the12quantum noise destroys slightly the electron interference [8]. The aim is toinvestigate how various quantum phenomena and the quantum statistics ofthe nonclassical microwaves link to corresponding quantum phenomena onthe electrons.In what follows we study the interference of the electrons by calculatingtheir intensity, while they are being irradiated with classical or nonclassicalmicrowaves. The correlation properties of electron interference are thenstudied by calculating the autocorrelation function of the electron intensity(Sec. 2). We also consider two-mode microwaves with frequencies ω1 andω2 (Sec. 3). Two-mode microwaves can be factorizable, separable or en-tangled [9] and since the problem of entanglement is generally complex, weapproached it using an example. In particular, we assumed that the twomodes of the microwave field form a Bell state and calculated its effect onelectron interference. We found that the result is very different from that ofthe corresponding separable case. We conclude in Sec. 4 with a discussionof our results.2. One-mode microwaves2.1. Classical microwavesThe following system is considered: a beam of electric charges splits intotwo possible paths C0 and C1. The charges enter a region that is irradiatedwith microwaves (using a suitable waveguide). The microwaves propagatein the waveguide with the time-dependent magnetic field perpendicular tothe plane of the two paths and the electric field parallel to it. Let ψ0, ψ1be the electron wavefunctions with total winding equal to 1, in the absenceof magnetic field. The effect of the electromagnetic field is the phase factorexp[ieφ(t)] and the intensity isI(t) = |ψ0+ψ1 exp[ieφ(t)]|2 = |ψ0|2+|ψ1|2+2|ψ0||ψ1|ℜ{exp[i(σ+eφ(t))]}(1)where σ = arg(ψ1) − arg(ψ0). Units in which kB = ~ = c = 1 are usedthroughout. For simplicity we consider the case of equal splitting, in which|ψ0|2 = |ψ1|2 = 1/2 and let σ = 0. In this case we getI(t) = 1 + cos[eφ(t)]. (2)We calculate the autocorrelation function of the electron intensity:Γ(τ) = limT→∞12T∫ T−TR(t, τ)dt; R(t, τ) ≡ I(t)I(t+ τ). (3)3An expansion of Γ(τ) into a Fourier series gives the spectral density SK :SK =Ω2pi∫ 2pi/Ω0Γ(τ) exp(−iKΩτ)dτΓ(τ) =∞∑K=−∞SK exp(iKΩτ). (4)Firstly, we consider the case where the classical time-dependent flux isgiven byφ(t) = φ1 sin(ωt) (5)and using Eqs. (2) and (3) we find the autocorrelation function:Γcl(τ) = [1 + J0(eφ1)]2 + 2∞∑K=1[J2K(eφ1)]2 cos(2Kωτ), (6)where JK are Bessel functions. Comparison of Eqs. (4) and (6) shows thatΩ = 2ω andS0 = [1 + J0(eφ1)]2; SK = [J2K(eφ1)]2. (7)2.2. Nonclassical microwavesA monochromatic electromagnetic field of frequency ω is considered, attemperatures kBT << ~ω. We quantize the electromagnetic field by con-sidering the vector potential Ai and the electric field Ei as dual quantumvariables. The loop C = C0 −C1 is small in comparison to the wavelengthof the microwaves, hence the Ai and the Ei can be integrated around it andyield the magnetic flux φ and the electromotive force VEMF, respectively,as dual quantum variables. The annihilation operator can be introducedas a = 2−12 ξ−1(φ+ iω−1VEMF), and similarly the creation operator, whereξ is a constant proportional to the area enclosed by C. The flux operatoris consequently written as φ(t) = exp(itH)φ(0) exp(−itH), where H is theHamiltonian that contains the ωa†a term and an interaction term. Thisinteraction term can be neglected for small currents.Under these conditions the magnetic flux, which defines the phase factor,becomes the operator φˆ(t) = (ξ/√2)[exp(iωt)a† + exp(−iωt)a] .Hence thisphase factor exp(ieφ) now isexp[ieφˆ(t)]= D [iq exp(iωt)] , q =ξe√2(8)where D(λ) is the displacement operator D(λ) = exp(λa† − λ∗a). Theinterference between the two electron beams is described by the intensity4operatorIˆ(t) = 1 + cos[eφˆ(t)]= 1+12D [iq exp(iωt)] +12D [−iq exp(iωt)] . (9)Let ρ be the density matrix describing the external nonclassical microwaves.The expectation value of the electron intensity is〈I(t)〉 ≡ Tr[ρIˆ(t)]= 1 +12W˜ (λ) +12W˜ (−λ); λ = iq exp(iωt), (10)where Tr [ρD(λ)] ≡ W˜ (λ) is the Weyl (or characteristic) function which hasbeen studied by various authors including ourselves (e.g. [10] and referencestherein).0 0.5 1 1.5 2 2.5 300.20.40.60.811.21.41.61.82< I(t) >ω tCl Coh Th Sq Figure 1. 〈I(t)〉 as a function of ωt for ω = 10−4, 〈N〉 = 200, r = 6.4. We use unitswhere ~ = kB = c = 1. Continuous line represents the case of irradiation with classicalmicrowaves; line of circles, coherent states; line of crosses, squeezed states; and dottedline, thermal states.We have calculated 〈I(t)〉 for various quantum states of the microwaves(using results for W˜ (λ) in Ref. [11]). In order to find the Γ(τ) from Eq.(3), one needs to calculate the quantityR(t, τ) ≡ Tr[ρIˆ†(t)Iˆ(t+ τ)](11)Numerical results are presented for different quantum states that wecalculated. In particular, we plot four cases: classical microwaves and50 0.5 1 1.5 2 2.5 30.50.60.70.80.91ℜ [γ(τ)]0 0.5 1 1.5 2 2.5 3−6−4−20246 x 10−3ℑ [γ(τ)]ω τClassicalCoherentSqueezedThermal(a) (b) Figure 2. γ(τ) as a function of ωτ for ω = 10−4, 〈N〉 = 200, r = 6.4. Part (a) shows thereal part of γ(τ); part (b) shows the imaginary part. We use units where ~ = kB = c = 1.nonclassical microwaves in coherent, squeezed, and thermal states. For ameaningful comparison, we consider the case where the average number ofphotons 〈N〉 in coherent, squeezed, and thermal states is the same:〈N〉 = |A|2 =[sinh( r2)]2+[cosh(r2)− sinh(r2)]2B2=1exp(βω)− 1 . (12)For the classical case we took φ21 = 2|A|2 = 2〈N〉. In all results of Figs. 1to 3, ω = 10−4 (which in our units is eV ), 〈N〉 = 200, r = 6.4.The results show that the quantum noise in the irradiating microwavesaffects the electron interference. All microwaves that we have consideredhave the same average number of photons, but differ in the quantumnoise. These four types of microwaves lead to different electron interferenceresults and different autocorrelation functions. Irradiation of the electronsby nonclassical microwaves leads to nonzero value of the imaginary partof the electron autocorrelation function. This is not so (i.e. the imaginarypart of Γ(τ) vanishes) when the ring is irradiated with classical microwaves.6−2 −1 0 1 200.20.40.60.811.2KS KCl Coh Sq Th Figure 3. SK coefficients for the electrons with 〈N〉 = 200, r = 6.4. We use units where~ = kB = c = 1. The bars correspond to irradiation of the ring by (from left to right):classical, coherent, squeezed, and thermal states.3. Two-mode nonclassical microwavesWe consider two-mode nonclassical microwaves. We are particularly in-terested to study how entangled two-mode microwaves affect the electroninterference. For this reason we consider a Bell state |s〉 = 2−1/2(|01〉+|10〉)where |01〉 , |10〉 are two mode number eigenstates. For comparison we alsoconsider the separable (disentangled) stateρsep =12(|01〉〈01|+ |10〉〈10|). (13)Clearly, the density matrix of the entangled state ρent = |s〉〈s| can bewritten asρent = ρsep +12(|01〉〈10|+ |10〉〈01|). (14)In this case the phase factor exp[ieφ(t)] becomes the product of two dis-placement operators and, consequently, the intensity becomesIˆ(t) = 1 +12D1(λ1)D2(λ2) +12D1(−λ1)D2(−λ2); λj = iq exp(iωjt)(15)for two modes (j = 1, 2). Therefore, we find that〈I(t)〉sep = 1 +(1− q2) exp (−q2) , (16)〈I(t)〉ent = 〈I(t)〉sep − q2 exp(−q2) cos[(ω1 − ω2)t]. (17)7It is seen that for this example, the 〈I(t)〉sep is constant in time, whilethe 〈I(t)〉ent is an oscillatory function of time. Clearly, different correlationsamong the two irradiating modes of the microwaves may lead to differentaverage electron intensities.4. DiscussionThe subject of mesoscopic devices interacting with microwaves has receivedattention in the last few years (e.g., Ref. [7]). Our contribution has been toconsider that these microwaves are prepared in various nonclassical states[5,8,12]. Here we have quantified the effect of the quantum noise on elec-tron interference. More specifically we have calculated both the electronaverage intensity and the spectral density for several types of nonclassi-cal microwaves and a comparison of the results with the case of classicalmicrowaves (Figs. 1-3), demonstrates clearly that the presence of both clas-sical and quantum noise in the nonclassical microwaves affects the electronintensity. What is more, when the ring is irradiated with two-mode mi-crowaves, then entanglement among these two modes (i.e., the formationof a Bell state) leads to a time-dependent expectation value of the electronintensity.References1. M. Peshkin and A. Tonomura, The Aharonov-Bohm effect, Lecture notes inPhysics Vol. 340, Berlin: Springer (1989).2. A. Shapere and F. Wilczek (ed), Geometric Phases in Physics, Singapore:World Scientific (1989).3. S. Washburn and R.A. Webb, Adv. Phys. 35, 375 (1986)A.G. Aronov and Y.V. Sharvin, Rev. Mod. Phys. 59, 755 (1987).4. G. Hackenbroich, Phys. Rep. 343, 464 (2001).5. A. Vourdas, Phys. Rev. B54, 13175 (1996)A. Vourdas and B.C. Sanders, Europhys. Lett. 43, 659 (1998).6. M.P. Silverman, Nuovo Cimento B97, 200 (1987)M. Buttiker, Phys. Rev. B46, 12485 (1992).7. M. Buttiker, J. Low Temp. Phys. 118, 519 (2000)R. Deblock et. al., Phys. Rev. B65, 075301 (2002).8. P. Cedraschi, V.V. Ponomarenko, M. Buttiker, Phys. Rev. Lett. 84, 346 (2000)A. Vourdas, Phys. Rev. A64, 053814 (2001).9. R.F. Werner, Phys. Rev. A40, 4277 (1989); A. Peres, Phys. Rev. Lett. 77,1413 (1996); R. Horodecki and M. Horodecki, Phys. Rev. A54, 1838 (1996);V. Vedral et. al., Phys. Rev. Lett. 78, 2275 (1997).10. S. Chountasis and A. Vourdas, Phys. Rev. A58, 848 (1998).11. A. Vourdas, Phys. Rev. B49, 12040 (1994).12. C.C. Chong, D.I. Tsomokos, A. Vourdas, Phys. Rev. A66, 33813 (2002).

Correlation properties of interfering electrons in a mesoscopic ring under nonclassical microwave radiation

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