arXiv:1108.3258v4 [cond-mat.mes-hall] 30 Aug 2011 h ult a enkont eapwru oli aiu physical various in tool powerful a mechanics be statistical to as known such been has duality The Introduction 1 junction ad-ihk-h,Cioak,Tko aa mmitsuya@cc Japan Tokyo, Chiyoda-ku, Kanda-Nishiki-cho, aa [email protected] Japan a enas nw nsvrlsuiso h oeho ucins junction Josephson the of studies several in known also been has [email protected] OIHG YONEDA MORISHIGE u ffc devices effect are tum junction Josephson general the systems, superconducting h hs n h ubro atce nada ytmo ahothe each of system dual a in particles of number Keywords relatio the the and derived We phase ex junction. the we SI/SC/SI example, reverse simple its and a junction As SC(superconductor)/SI(superinsulato conditions. the curr between duality between duality the relationship the the assumed as we system, each, a of stu such to In formalism, canonical systems. dual dual of theory a constructed have We unu otxaekona h ult eaint ahother each to relation duality the as b known operates are which tunneling vortex devices flux) quantum effect vortex(quantum quantum quantum the single as known are systems eie hc prtsb sn h igeCoe-artneigcre tunneling Cooper-pair single qu the a blockade using as Coulomb used by a be operates can which junction devices Josephson mesoscopic the thermore, ahohradte umrz eut nSec.6. in results c summarize previous then the and system of resis other network generalization each quantum a superconductor-superinsulator the as the Sec.5, derived In introduce we method. addition, ways two In the e junction. basic SI/SC/SI and the model single of theoretical the form SC(supercnductor)/SI(superinsulator)/SC(supercndu formulate canonical we single dual f Sec.4, the canonical In build dual circuit. we follow the LC Sec.3, con show as In we have composed example, we estab oscillator. is simplest harmonic system, paper been the quantum This as yet dual Ho section, formalism. not the next canonical other. study has dual each to system of are with theory dual we device pa Ther a So dual such Cooper a general. type. between of as similar definition role understood of the clear be equation can replace the devices we satisfy two if systems even these conditions, vortices, certain under is, H ULCNNCLFRAIMIN FORMALISM CANONICAL DUAL THE ∗ ‡ † oy ek nvriyDprmn fIfrainadCommu and Information of Department University Denki Tokyo oy ek nvriyDprmn fPyis22Kanda-Ni Physics,2-2 of Department University Denki Tokyo esiGke aa lcrnc olg,12- Hyakunin- 1-25-4 College, Electronics Japan Gakuen Denshi H UNU UCINSYSTEMS JUNCTION QUANTUM THE daiy unu ucinsses CS/Cjnto,SI/SC/S junction, SC/SI/SC systems, junction quantum :duality, [6] , [7] [8] ncnrs,temssoi ulJspsnjunction Josephson dual mesoscopic the contrast, In . hc prtsb sn h oprpi unln.Fur- tunneling. Cooper-pair the using by operates which ∗ MSAINIWA ,MASAAKI [1] n edtheory field and 1 † N ISY MOTOHASHI MITSUYA AND [2] [9] , [3] h oprpi n the and Cooper-pair The . k.dendai.ac.jp utemr,teduality the Furthermore, . shiki-cho,Chiyoda-ku,Tokyo, h,Sijk-u Japan Shinjuku-Ku, cho, iainEngineering,2-2 nication nw stequan- the as known tr ucinand junction ctor) r)/SC(superconductor) fcmeewith compete of ytequantum the dy n n voltage and ent ystems si between nship [10] nu effect antum utosfor quations raimof ormalism fr,these efore, sn the using y atr,we hapters, r. , tutda structed [11] quantum .I the In s. ihdin lished ac by tance ee,a wever, tdby ated systems [4] amined r and irs That . , [5] .In ‡ I 2 Dual canonical formalism of the harmonic os- cillator
In this section, as the simplest example, we show the dual canonical formalism of harmonic oscillator. First, we introduce the following commutation relations of between the coordinate x(t) and momenta p(t)
x(t), p(t′) ≡ x(t)p(t′) − p(t′)x(t) = i~δ(t − t′), (1) h i− The Hamiltonian H of the harmonic oscillator is given by
1 mω2 H = p2(t)+ x2(t), (2) 2m 2 where m and ω are mass and angular frequency of harmonic oscillator respec- tively. From the Heisenberg’s equation of motion, the equations of motion are given by
≡ dx(t) i p(t) x˙ = ~ H, x(t) = , (3a) dt h i− m ≡ dp(t) i − 2 f(t) = ~ H, p(t) = mω x(t), (3b) dt h i− wherex ˙(t) and f(t) are velocity and force of harmonic oscillator respectively. As a next step, we introduce the following commutation relations of between the dual coordinate x(t) and dual momenta p(t)
e x(t), p(t′) = i~eδ(t − t′). (4) h i− e e Then the dual Hamiltonian H of eq.(2) is given by
e 1 mω2 H = p2(t)+ x2(t), (5) 2m 2 e e e e e where m and ω are dual mass ande dual angular frequency of harmonic oscillator respectively. Thee Heisenberg’se equation of motion are given by
dx(t) i p(t) x˙ ≡ = H, x(t) = , (6a) dt ~ − m e h i e e dp(t) ei e 2 f(t) ≡− = − H, p(t) =e mω x(t). (6b) dt ~ − e h i e e e e e e where x˙ (t) and f(t) are dual velocity and dual force of harmonic oscillator respectively. e e
2 Between eqs.(3a), (3b) and eqs.(6a), (6b) we assume the following the duality conditions:
x˙(t)= f(t), (7a) f(t)= x˙e(t). (7b)
By imposing the duality conditions, we obtaine the next two type relationship, One of them are relationship between coordinate and momenta
x(t)= −p(t), (8a) x(t)= p(t). (8b) e Another one of them are relationshipe between mass and angular frequency 1 m = , (9a) mω2 1 e = mω2 (9b) m From the above, we derived the dualitye conditionse and relationship of relations of variables and constants. In particular, the duality conditions of (7a) and (7b) are important. These conditions can also be applied to the harmonic oscillators as well as general dynamical systems.
3 Dual canonical formalism of the quantum LC circuit
In this section, using the methods of the previous section, we build the dual canonical form of the quantum LC circuit, the commutation relations of between the electric charge Q(x) and magnetic flux Φ(x) are as follows:
Φ(x),Q(x′) = i~δ(x − x′), (10) h i−
The Hamiltonian HLC of the quantum LC circuit is given by 1 1 H = Q2(t)+ Φ2(t), (11) LC 2C 2L where C and L are capacitance and inductance of LC circuit respectively. The Heisenberg’s equations of motion are given by
≡ dQ(t) i −Φ(t) I(t) = ~ HLC,Q(t) = , (12a) dt h i− L ≡−dΦ(t) i Q(t) V (t) = ~ HLC, Φ(t) = . (12b) dt h i− C where I(t) and V (t) are current and voltage of LC circuit respectively.
3 As a next step, we introduce a dual LC circuit system and construct a com- mutation relation of between the dual electric charge Q(x) and dual magnetic flux Φ(x) as e ′ ~ − ′ e Φ(x), Q(x ) = i δ(x x ). (13) h i− e e The dual Hamiltonian HLC of eq.(11) is given by
e 1 2 1 2 HLC = Q (t)+ Φ (t), (14) 2C 2L e e e where C and L are a dual capacitancee and ae dual inductance of the dual LC circuit respectively. The Heisenberg’s equations of motion are given by e e
≡ dQ(t) i −Φ(t) I(t) = ~ HLC, Q(t) = , (15a) edt h i− eL e e e ≡ dΦ(t) i Q(t)e V (t) = ~ HLC, Φ(t) = . (15b) edt h i− eC e e e where I(t) and V (t) are dual current and dual voltage ofe dual LC circuit respec- tively. Between eqs.(12a), (12b) and eqs.(15a),(15b), we assume the following e e duality conditions:
I(t)= V (t), (16a) Ve(t)= I(t). (16b)
By imposing the duality conditions,e we derived the next two type relationship, One of them are relationship between charge and flux
Φ(t)= −Q(t), (17a) Φ(t)= Q(et) (17b)
Another one of them are relationshipe between capacitance and inductance
C = L, (18a) Le = −C (18b)
The duality conditions of (16a) ande (16b) are is very important, when we will treat with the Josephson junction systems in the subsequent sections.
4 4 Dual canonical formalism of between the sin- gle SC/SI/SC junction and the single SI/SC/SI junction
Based on the results of the previous sections, we examined the duality[10],[12],[13] between the SC/SI/SC junction and SI/SC/SI junction. First, we propose a small SC/SI/SC junction[9] and its equivalent circuit are shown in Fig.1.
C SCSI SC J θ C θ 1 J 2 =
Vg
Vg Figure 1: Schematic of superconductor(SC)/superinsulator(SI)/superconductor(SC) junction and its equivalent circuit
Now we introduce an order field (original particles field) Ψ(x)
Ψ(x)= N(x) exp (iθ(x)) . (19) p For example, in superconducting systems, N = N1 − N2 is represent the relative number operator of Cooper pair, and θ = θ1 − θ2 is represent the relative phase of Cooper pair. The commutation relations between N(x) and θ(x′) are given by N(x), θ(x′) = i~δ(x − x′). (20) h i− The junction is characterized by its capacitance CJ and the Josephson energy EJ = ~IC/2e, where IC is critical current. Now, this junctions system when considered as the Josephson junctions, This system can be described by the following Hamiltonian of Cooper pair:
2 HJ =4N EC + EJ (1 − cos θ) . (21)
The first term describe the Coulomb energy of Josephson junction, N is the 2 number operator of Cooper pair, and EC = e /(2CJ) is charging energy per single-charge . The second term in eq.(21) describe the Josephson coupling energy. From eq.(21) two Josephson’s equations are given by
~ ∂θ(t) ~ i 4N V = = ~ HJ, θ = EC, (22a) 2e ∂t 2e h i− e ∂N(t) i − I =2e =2e~ HJ, N = IC sin θ, (22b) ∂t h i−
5 L SISC SI J L θ1 J θ2 = e e
I
I superconductive current source wire Figure 2: Schematic of superinsulator(SI)/superconductor(SC)/superinsulator(SI) junction and its equivalent circuit where V and I are voltage and current of Cooper pair, respectively. Then, we describe the theoretical model and basic equations for the single superinsulator(SI)/superconductor(SC)/superinsulator(SI) junction. A small SI/SC/SI junction and its equivalent circuit are shown in Fig.2. On the analogy of an order field(19), we introduce a disorder field[10],[12],[13],[14](dual particles field) Ψ(x)= N(x)exp iθ(x) (23) q e e e For example, in superconducting systems, N = N1 −N2 is represent the relative number operator of vortex, and θ = θ − θ is represent the relative phase of 1 e2 e e vortex. We suppose that the commutation relations between N(x) and θ(x′) e e e are given by e e N(x), θ(x′) = iδ(x − x′), (24) h i− e e and the dual Hamiltonian[15],[16] of eq.(1) is given by
2 2EC − HJ = N Ev + 2 1 cos θ . (25) π e e e The first term of eq.(25) describes the magnetic energy of dual Josephson junc- 2 tion, Ev = Φ0/(2Lc) is a magnetic energy per single-vortex, and Lc = Φ0/(2πIc) is a critical inductance. The second term in eq.(25) describes the dual Josephson coupling energy, where θ = θ1 − θ2 is dual phase difference across the junction. From eq.(25) two dual Josephson’s equations are given by e e e ~ ∂θ ~ i V = = ~ HJ, θ = Ic2πN, (26a) Φ0 ∂te Φ0 h i− e e e e − ∂N i 2EC I = Φ0 = Φ0 ~ HJ, N = sin θ, (26b) ∂te h i− πe e e e e where V and I are voltage of vortex and current of vortex respectively. From the conditions of (16a) and (16b), we derived the next two type relationship. e e
6 One of them is relationship between the phase of Cooper pair θ(x) and vortex number N(x) as follows: e 1 N(x)= − sin θ(x). (27a) 2π e Another one of them is relationship between the phase of vortex field θ(x) and Cooper pair number N(x) as follows: e 1 N(x)= sin θ(x). (27b) 2π e From the duality conditions of (16a) and (16b), we derived the quantum resistance[17],[18] by the next two ways method. One of those ways is to use the ratio between the fluctuation of the number of Cooper pair and the fluctuation of the number of vortex, ~ ≡ V − 8N EC R = 2 I (2e) sin θ EJ I − h ∆N = = 2 Ie (2e) ∆Ne h sin ∆θ = (28a) (2e)2 sin ∆θ Another way of these is to use the ratio betweene the fluctuation of the phase of Cooper pair and the fluctuation of the phase of vortex, we derived the quantum conductance ( dual quantum resistance R) as follows
e2 2 ≡ V (2e) π N EJ R = ~ eI 2 sineθ EC e 2 Ve (2e) ∆θ e = = ~ Ve ∆θe (2e)2 ∆N = − (28b) ~ ∆N
In the eqs.(28a) or (28b), a case of conditione of ∆N > ∆N (ie:∆θ > ∆θ) or E > E , it becomes insulator state. In particular, in this extreme case C J e e R → ∞, it becomes superinsulator state . If the reverse case, in the condition of ∆N < ∆N (ie:∆θ < ∆θ) or EJ > EC , it becomes conducting state. In particular, in this extreme case R → 0, it becomes superconductor state . As e e a special case of these conditions, in the case of ∆N ≃ ∆N or ∆θ ≃ ∆θ, it becomes self dual state. In this case, the resistance R becomes e e h R → R = (29) Q (2e)2 where RQ is a quantum resistance.
7 5 The superconductor-superinsulator network sys- tems of compete with each other
Until the previous chapter, we had been dealt with models of single junctions in order to simplify the problem. In this chapter, we had extended the theory from single junctions to periodic quantum dot network systems[19],[20]. Such a system, we considered two type competing systems. Two of those systems are superconducting dots systems (as shown in FIG.3(a))in superinsulator host. The remaining two systems are superinsulating dots systems(as shown in FIG.3(b)) in superconducting host.
(a) (b)
Figure 3: Schematic of superconductor-superinsulator network systems (a) Superconducting dots in superinsulator host. (b) Superinsulating dots in superconducting host.
The Hamiltonian (space 2d+imaginary time) of Josephson network systems (lattice systems) by superinsulator dots in superconducting host(as FIG.3(b)) is written as Q(i) Q(i) H =4E N(i) − M −1 N(i) − C 2e ij 2e Xi,j 2π + E 1 − cos θ(i) − θ(j)+ Aij (30a) J Φ Xi,j 0
8 −1 where M is a dimensionless potential matrix, and Aij is a vector potential of lattice version j Aij ≡ dl · A = − (Φ(i) − Φ(j)) . (30b) Zi
2E q(a) V (i)= C 2πN (a) − 2π M −1 , (31a) πe C 2e aj Xa 2e 2π I(i)= −E sin θ(i) − θ(a)+ Aia . (31b) J ~ Φ Xa 0 On the other hand, The Hamiltonian of dual Josephson network systems by superconducting dots in superinsulating host (as FIG.3(a)) is written as
2 Φ(i) −1 Φ(i) H =2π EJ N(i)+ M N(i)+ Φ ij Φ Xi,j 0 0 e e f e 2 2π + E 1 − cos θ(i) − θ(j)+ Aij (32a) π2 C 2e
2e Φ(a) −1 V (i)= EJ 2πN(a)+2π M , (33a) ~ Φ ai Xa 0 e e f 4EJ 2π I(i)= sin θ(i) − θ(a)+ Aia . (33b) 2πe 2e Xa e e e e As with the previous chapter, from the duality conditions of (16a) and (16b), we derived the next two type relationship,
1 2π q(a) N(a)= (M) sin θ(b) − θ(a)+ Aba + , (34a) 2π ab 2e 2e Xb e e e 1 2π Φ(a) N(a)= − (M) sin θ(b) − θ(a)+ Aba − . (34b) 2π ab Φ Φ Xb 0 0 e The relation of (34a) and (34b) can be considered as a generalization of the relation of (27a) and (27b).
9 6 Summary and Conclusion
In this paper, our results are described as follows. In the first result, we ex- amined the harmonic oscillator system as the simplest dual canonical systems. Here, we imposed the dual condition the between force and velocity in a dual system in each other, and we derived two type relationship. One of them is relationship between coordinate and momenta, another one of them is relation- ship between mass and angular frequency. In the second result, we studied the quantum LC circuit system as the simplest dual electromagnetic systems. Where, we imposed the dual condition the between electric current and voltage in a dual system in each other, and we derived two type relationship. One of them is relationship between charge and flux, another one of them is relation- ship between capacitance and inductance. In the third results, we studied the duality between the SC/SI/SC junction and SI/SC/SI junction. We imposed a dual condition similar to the second results, and we derived two type relation- ships. One of them is relationship between the phase of Cooper pair θ(x) and vortex number N(x). Another one of them is relationship between the phase of vortex field θ(x) and Cooper pair number N(x). In the fourth result, as a e generalization of the third results, we had extended the theory from single junc- e tions to periodic quantum dot network systems. We imposed a dual condition similar to the third results, and we derived the generalized relation of the third result. As mentioned above, our duality conditions will be become a universal and powerful tool for studying the generalized dual systems.
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