{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 18 "Josephson Junction" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "We inves tigate a simple Hamiltonian that is used to describe the conduction pr operties of a Josephson junction. The discussion follows D.V. Averin's article in " }{TEXT 257 26 "Scalable Quantum Computers" }{TEXT -1 82 " , S.L Braunstein and H.-K. Lo (eds), Wiley-VCH (Berlin 2001), ISBN 3 -527-40321-3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 507 "A Josephson junction is a remarkable device: it is of a \+ mesoscopic scale, i.e., involves many particles, yet it displays quant um behaviour. This is a feature common to Bose condensed states (recen tly Bose-Einstein condensation has become possible for ensembles of at oms which are super-cooled in magneto-optical traps). This is remarkab le, because normally ensembles of quantum systems behave classically o n macroscopic scales. Here we can observe (almost) the entire ensemble being in a pure quantum state." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 176 "The junction Hamiltonian is in a number \+ representation with respect to Cooper pairs and has tridiagonal form ( nearest-neighbour coupling). It is controlled by three parameters:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 15 "q = C*V_e /(2*e)" }{TEXT -1 52 " is the charge induced on the junction capacita nce " }{TEXT 19 1 "C" }{TEXT -1 18 " by the potential " }{TEXT 19 3 "V _e" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT 19 3 "E_J" }{TEXT -1 68 " \+ is the energy of Cooper pair tunneling = Josephson coupling energy;" } }{PARA 0 "" 0 "" {TEXT 19 18 "E_C = (2e)^2/(2*C)" }{TEXT -1 48 " is th e charging energy of a single Cooper pair." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 47 "H = E_C*(n-q)^2 - 0.5*E_J*(|n>< n+1| + |n+1>> E_C" } {TEXT -1 60 " the junction can behave almost clasically. The capacitan ce " }{TEXT 19 1 "C" }{TEXT -1 145 " of the junction has to be small s o that the charging energy of a single Cooper pair is appreciable. Thi s can be achieved in a mesoscopic system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "The origin of the number part of \+ the Hamiltonian: it is an electrostatic energy associated with (a) the charging of the capacitance by the " }{TEXT 19 1 "n" }{TEXT -1 141 " \+ Cooper pairs, and (b) also by an external electric field which induces a potential difference between two adjacent superconducting islands, \+ " }{TEXT 19 3 "V_e" }{TEXT -1 26 ". The polarization charge " }{TEXT 19 13 "q=C*V_e/(2*e)" }{TEXT -1 36 " can be varied continuously throug h " }{TEXT 19 3 "V_e" }{TEXT -1 28 ". The charge is in units of " } {TEXT 19 4 "(2e)" }{TEXT -1 61 " so that it can be compared with the c harge of a Cooper pair." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The number of Cooper pairs " }{TEXT 19 1 "n" }{TEXT -1 253 " is quantized, and it is this characteristic difference which \+ results in interesting dynamics. The form of the coupling term derives from the fact that the tunneling of a Cooper pair from one supercondu cting island to the other results in the change of " }{TEXT 19 1 "n" } {TEXT -1 4 " by " }{TEXT 19 5 "+/- 1" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "One wants to find the g round state energy in units of " }{TEXT 19 3 "E_J" }{TEXT -1 18 " as a function of " }{TEXT 19 1 "q" }{TEXT -1 19 " for a fixed ratio " } {TEXT 19 7 "E_C/E_J" }{TEXT -1 78 ". Also of interest is the average n umber of Cooper pairs under this condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We pick the unit:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart; E_J:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E_C:=10*E_J;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "From Fig.1 in the article we read off that we should get \+ for the ground state approximately " }{TEXT 19 6 "E0 = 2" }{TEXT -1 6 " when " }{TEXT 19 7 "q=1/2, " }{TEXT -1 33 "and slightly less than ze ro when " }{TEXT 19 3 "q=0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We truncate the number represen tation at some finite value " }{TEXT 19 1 "N" }{TEXT -1 43 " and will \+ carry out a convergence analysis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:='q': N:=15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "HM:=Matrix(N+1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We ma ke room for the " }{TEXT 19 3 "n=0" }{TEXT -1 20 " Cooper pairs state. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for i from 1 to N+1 do: n:=i-1: HM[i,i]:=E_C*(n-q)^2: od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "What do we get when we sandwich the Hamiltonian between number \+ states " }{TEXT 19 3 "" } {TEXT -1 41 " ? Assume that the states are normalized." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 27 " + " }{TEXT -1 105 " is the off-diagonal coupling matrix element. \+ The number states are orthogonal and normalized, so we get:" }}{PARA 0 "" 0 "" {TEXT 19 49 "delta(i,n)*delta(n+1,j) + delta(i,n+1)*delta(n, j)" }{TEXT -1 8 " where " }{TEXT 19 5 "delta" }{TEXT -1 24 " is the K ronecker delta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "KD:=(mu::nonnegint,nu::nonnegint)->if mu=nu th en 1 else 0 fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "KD(2,2), KD(3,0),KD(0,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "KD(1.2, 2.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "KD(-2,0),KD(-2,-2) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We carry out a brute-force c alculation. The pair number parameter " }{TEXT 19 1 "n" }{TEXT -1 61 " is being summed over in the ket-bra part of the Hamiltonian." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for i from 1 to N+1 do: for j from 1 to N+1 do: HM[i,j]:=HM[i,j]-1/2*E_J*add(KD(i,n)*KD(n+1,j)+KD (i,n+1)*KD(n,j),n=0..N); od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(SubMatrix(HM,1..5,1..5));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 89 "The truncated Hamiltonian matrix has a tridiagonal structure. We compute the eigenvalues." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "EVs:=Eigenvalues(subs(q=0.5,HM)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "print(seq(Re(EVs[i]),i=1..N+1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "The eigenvalues come out sorted in value. The fround sta te is close to what can be read off from Fig. 1a in the reference." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 87 "Observe the convergence behaviour \+ of the two lowest eigenenergies with truncation size " }{TEXT 19 1 "N " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Two issues emerge:" }} {PARA 0 "" 0 "" {TEXT -1 55 "1) A question: can this matrix be diagona lized exactly?" }}{PARA 0 "" 0 "" {TEXT -1 141 "2) The calculation of \+ a tridiagonal matrix should be done more efficiently by an operator me thod which utilizes the sparseness of the matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "So let us construct a gr aph of the behaviour of the fround and first excited states with the c harge value " }{TEXT 19 1 "q" }{TEXT -1 68 ". We place the eigenvalue \+ calculation into a loop for a sequence of " }{TEXT 19 1 "q" }{TEXT -1 97 "-values, and sort the list of eigenvalues in order to extract the \+ first two as the lowest states." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "ET0:=[]: ET1:=[]: for i_q from 1 to 80 do: q0:=evalf(0+(i_q-1 /2)/40); EVs:=convert(map(Re,Eigenvalues(subs(q=q0,HM))),list): EVs:=s ort(EVs);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "E0:=evalf(EVs[1]): ET 0:=[op(ET0),[q0,E0]]: E1:=evalf(EVs[2]): ET1:=[op(ET1),[q0,E1]]: od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "PL0:=plot(ET0,color=re d): PL1:=plot(ET1,color=blue): plots[display](PL0,PL1,title=\"energy g ap vs charge q\",labels=[q,E]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "We observe periodic behaviour in the energy values as a function o f " }{TEXT 19 1 "q" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 227 "How do we get the average Cooper pair nu mber in the ground state? Averin refers to the junction free energy, w hich can be studied as a function of temperature. We do the simple qua ntum mechanics calculation at zero temperature." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "We use information from \+ the eigenstates given that we should know the composition of the groun d state in terms of the number states." }}{PARA 0 "" 0 "" {TEXT -1 481 "The average Cooper pair number can be calculated by using the eig envectors from the matrix diagonalization. Their interpretation is tha t they provide us with the expansion coefficients for the eigenstates \+ in terms of the number state basis. The magnitude squared of the coeff icient gives the occupation probability, and therefore we can add thes e probabilities weighted with the particle number to calculate the ave rage Cooper pair number in the energy eigenstate as a function of " } {TEXT 19 1 "q" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 364 "On a Maple technical note: the sorting i s more involved in this case, as we need to re-sort not just a list of eigenvalues, but the entire construct associated with the eigenvector s. For a first look at the ground-state behaviour it appears to be suf ficient to just track the first eigenvalue/eigenvector result; this is misleading, however, for certain values of " }{TEXT 19 1 "q" }{TEXT -1 38 " of does not get away without sorting!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "AN:=[]: for i_q from 1 to 90 do: q0:=evalf((i_q- 1/2)/80); EVs:=Eigenvectors(subs(q=q0,HM)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "j0:=1: E0:=Re(EVs[1][1]): for j from 2 to N do: if R e(EVs[1][j]) " 0 "" {MPLTEXT 1 0 26 "plot(AN,labels=[q,\"\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "The step-like behaviour is a manifestation of th e quantization of Cooper pairs in the mesoscopic junction. For smaller " }{TEXT 19 8 "E_C/E_J " }{TEXT -1 145 "values the step behaviour is \+ somewhat smoothed out. This reflects the physical situation that for l arge values of the tunneling energy of a pair " }{TEXT 19 3 "E_J" } {TEXT -1 115 " the junction electrodes are so strongly coupled that th e junction behaves classically and the average pair number " }{TEXT 19 3 "" }{TEXT -1 71 " just tracks the induced polarization charge, i.e., is proportional to " }{TEXT 19 1 "q" }{TEXT -1 72 ". Another wa y how this can be phrased is: large quantum fluctuations of " }{TEXT 19 1 "n" }{TEXT -1 134 " induced by the tunneling term in the Hamilton ian drown out the quantization effects observed in the step like behav iour in this case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 47 "Repe at the calculation for different values of " }{TEXT 19 8 "E_C/E_J " } {TEXT -1 55 "and observe the approach towards the continuous result " }{TEXT 19 5 "=q" }{TEXT -1 28 " as the ratio becomes small." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Now we re peat the findings for the eigenenergy vs " }{TEXT 19 1 "q" }{TEXT -1 37 ", and for the average pair number vs " }{TEXT 19 1 "q" }{TEXT -1 102 " for the first excited state and compare the finding with the gro und-state result. We also extend the " }{TEXT 19 1 "q" }{TEXT -1 7 "-r ange." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "AN0:=[]: AN1:=[]: for i_q from 1 to 240 do: q0:=evalf((i_q-1/2)/80); EVs:=Eigenvectors( subs(q=q0,HM)): j0:=1: E0:=Re(EVs[1][1]): for j from 2 to N do: if Re( EVs[1][j])j0 and Re(EVs[1][j]) " 0 "" {MPLTEXT 1 0 119 "AN0:=[op(AN0),[q0,add(Re(EVs[2][j,j0])^2*(j-1),j=1..N)]]; AN1:=[o p(AN1),[q0,add(Re(EVs[2][j,j1])^2*(j-1),j=1..N)]]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "PL3:=plot(AN0,color=red): PL4:=plo t(AN1,color=blue): plots[display](\{PL3,PL4\},labels=[q,\"\"],title =\"average Cooper pair number vs polarization charge q\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This result is cute. It shows that as one moves across the point " }{TEXT 19 10 "q=0.5*(2e)" }{TEXT -1 128 " fo r the induced polarization charge the ground and first excited states \+ trade status as far as Cooper pair number is concerned." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "Due to the mesoscop ic quantum behaviour of JJs they are considered a candidate for the im plementation of quantum computing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }