Discussion

7: Dynamical Casimir effect has been observed with a SQUID ar-ray attached to the terminal of a transmission line, which when a rapidly oscillating magnetic field is applied, it acts as a rapidly oscil-lating mirror changing the bound-ary condition of the transmission line.[109]

8: Since𝜌(−𝜔)has dimensions of energy [1/s] and𝐼_F(𝜔)/𝑒is dimen-sionless in natural units,~=1.

system is given by the Fourier transform of the wavefunc-tion, 𝐺(𝜔) =−𝑖∫ _𝑑𝑡

𝜓(𝑡)exp(𝑖𝜔𝑡).[102] Thus, rescaling the Green’s function implies renormalizing the wavefunction.

Typically, finite wavefunction renormalization factors appear calculating quasi-particle energies when the spectrum of excitations is not in the exact eigenstate of the system.[104] In this case, the various techniques used to diagonalize quan-tum mechanical operators and the Hamiltionian include Bogoliubov transformations and Feynman diagrams. [104, 105]

6. Discussion

Significance of renormalization

In the case considered here, this renormalization implies that the junction will spontaneously emit photons at different rates for the dc and the ac voltage. In fact, this observation is reminiscent of the rescaling of𝐴(Ω)/𝐵(Ω)= Ω³/2𝜋³𝑐² → Ω³(1+ 𝑎²/Ω²)/2𝜋³𝑐² = 𝐴_eff(Ω)/𝐵(Ω) in Unruh-Hawking radiation[98,99] predicted for a non-inertial observer under uniform acceleration 𝑎relative to an inertial observer.[106, 107] Such a renormalization factor is related to the nature of the quantum vacuum in QED and the fact that quantum operators of the inertial observer versus the accelerated observer can be related by a Bogoliubov transformation.[105, 108] In fact, Unruh-Hawking radiation is analogous to the dynamical Casimir effect where an oscillating mirror relative to a static one, spontaneously radiates photons from the quantum vacuum.⁷ For a system at its ground state,𝐴(𝜔)is zero. However, due to this driving of the mirror, the 𝐴(𝜔) coefficient is rendered finite.

Note that we could have easily chosen to renormalize𝐵(𝜔) instead of𝐴(𝜔), or to renormalize both. The motivation to choose to renormalize 𝐴(𝜔)instead is in analogy with the above argument of the casimir effect. Moreover, observe that 𝐵(𝜔) ∝ 𝐼_F(𝜔)𝐼_F(−𝜔)/𝑒² has the right dimensions the rate of stimulated emission.⁸ Defining the unknown proportionality constant for𝐵(𝜔)as a dimensionless coefficient𝛾, we write 𝐴(𝜔)= 𝑖𝜔²𝛾 _𝐼

F(−𝜔)𝜙(𝜔) −𝜙(−𝜔)𝐼_F(𝜔)_/ 2𝜋𝑒.

62 3. Introduction

9: Such a factor has been reported describing the effect of excited en-vironmental modes by an alter-nating voltage in single normal junctions[27] in an extended𝑃(𝐸) theory approach. Such a factor that may lead to amplification ef-fects has also been reported in ref. [28, 29] and ref. [53] More-over, the Caldeira-Leggett form of the environmental impedance neglects the back-action of the Josephson junctions on the en-vironment (with the bath and the junction becoming entangled) which has been reported to dra-matically change the predictions of the𝑃(𝐸)theory.[31,34,35] This back-action manifests through the non-linear inductive response of the junction where the Josephson coupling energy is renormalized and the insulator-superconductor phase transition conditions for the single Josephson junction are al-tered.[35] Novel features not yet observed experimentally with ar-rays for Josephson junction arar-rays include the renormalization of the radio frequency (RF) power ab-sorbed by the array junction as well as higher harmonic modifi-cations of the time-averaged cur-rent[32,33]. We thus view the ob-servation of the renormalization effect detailed in chapter2as a crucial first step extending𝑃(𝐸) theory to Josephson junction ar-rays.

Table 3.2.:Transforming the sum of infinite modes𝜔^𝑖in the Joseph-son junction Cooper-pair tunnel-ing rate.

P𝑖 → ∫ _𝑑

𝜔 𝛿(Ω−𝜔), 𝑎_𝑖 →𝑒𝑉(𝜔), 𝑎^†_𝑖 → 𝑒𝑉(−𝜔).

10: 𝑎_𝑖 and𝑎^†

𝑖 account for tunnel-ing of a Cooper-pair accompanied by stimulated photon emission or absorption processes.

Recall that in chapter2we found that the applied microwave 𝑉_RFwas rescaled by a factorΞ_A(𝜔)related to the difference of dc and ac response of the equivalent circuit of the array.

This also implies the renormalization of𝐴(𝜔)as exemplified by our discussion in this chapter. Although the arguments presented in this chapter for the renormalization of the𝐴(𝜔) coefficient by considering dc versus ac response may appear new, the results herein are familiar. In fact, the impedance in the𝑃(𝐸)function for the single small junction is always effective𝑍_eff =1/[𝑖𝜔𝐶+𝑍⁻¹(𝜔)].[16,100] The rescaling of 𝑉_RF(𝜔)is therefore by the factorΞ(𝜔)= 𝑍_eff(𝜔)/𝑍(𝜔)which in turn is related to the rate which the junction spontaneously emits a photon back to the environment.⁹

Determining A and B for small Josephson junctions

In eq. (3.3), we used Fermi’s golden rule to calculate the 𝐵(𝜔)coefficient. From𝑃(𝐸)theory, we know that stimulated emission processes in small junctions must be accompanied by emission or absorption of photons. The𝐴(𝜔) and𝐵(𝜔) coefficients we have derived do not capture this intuitive observation. Nonetheless, we know they are valid due to fluctuation-dissipation requirements: we have expressed the coefficients using the phase𝜙(𝜔)and the fluctuation current 𝐼_F(𝜔). In chapter5, we shall see that𝐵(𝜔) ∝𝐼_F(𝜔)𝐼_F(−𝜔)and 𝐴(𝜔) ∝ 𝐼_F(−𝜔)𝜙(𝜔)appear in the Caldeira-Leggett action as the Coulomb interaction and coupling terms respectively suggesting a deeper connection than is offered in the thesis.

Nonetheless, we can go farther to determine the rate of stimulated emission Γ₂→₁(𝜔) = 𝐵(Ω)𝜌(Ω) using Fermi’s golden rule by following closely the dipole example. In fact, the transition Hamiltonian in eq. (3.3) should be re-placed with the tunneling Hamiltonian of Cooper-pairs 𝐻_𝑖 = 𝑎^†_𝑖𝐸_Jexp(−𝑖2𝜙)/2 + 𝑎_𝑖𝐸_Jexp(𝑖2𝜙)/2 with the slight modification where we include the photon creation𝑎^†_𝑖 and annihilation𝑎_𝑖 operators for the infinite𝜔^𝑖 modes satisfying [𝑎_𝑖, 𝑎^†_𝑗] = 𝛿^𝑖𝑗, analogous to 𝐸_𝑖 in the dipole case.¹⁰ Thus, Γ₂→₁(𝜔) = 𝐵(𝜔)𝜌(𝜔)simply becomes the standard tunnel-ing rate calculation for Cooper-pairs ustunnel-ing Fermi’s golden rule, e.g. the calculation, for the tunneling rate, given e.g.

in eq. (135) in ref. [16] where E₂ − E₁ = 2𝑒𝑉 + 𝐸⁰_𝑅 −𝐸_𝑅 and 𝐸⁰

𝑅, 𝐸_𝑅 are energies related to the environment. The

6. Discussion 63

calculation yields,Γ₂→₁ =𝜋P

𝑖h𝑎^†_𝑖𝑎_𝑖i𝐸²

J𝑃(2𝑒𝑉−𝜔^𝑖)/2. Here, 𝑃(2𝑒𝑉 −𝜔^𝑖) is the 𝑃(𝐸_𝑖) function given in eq. (3.5) with 𝐸_𝑖 =2𝑒𝑉−𝜔^𝑖. Finally, changing the sum to an integral by the transformation in Table3.2, we arrive at,

Γ₂→₁(Ω)=𝜋𝐸²

J𝑃(2𝑒𝑉−Ω)𝜌(Ω)/2Ω, (3.18) where we have used 𝜌(Ω) = 𝑒²Ωh𝑉(Ω)𝑉(−Ω)i. Recalling thatΓ₂→₁ =𝐵(Ω)𝜌(Ω), we discover that𝐵(Ω)=𝜋𝐸²

J𝑃(2𝑒𝑉− Ω)/2Ω = Γ(2𝑒𝑉−Ω)/Ω, which is proportional to the Cooper-pair tunneling rate Γ in eq. (1.14), albeit with the voltage shifted byΩ/2𝑒. Moreover, using𝐴(Ω)/𝐵(Ω)=(2𝑒)²Ω[𝑍(Ω)+

𝑍(−Ω)]/2𝜋, we find that,

𝐴(𝑓)=2𝑟(𝑓)Γ(2𝑒𝑉 −2𝜋𝑓) (3.19) with the definition 𝑟(𝑓) ≡ 4𝑒²[𝑍(𝑓) +𝑍(−𝑓)]/4𝜋 and 𝑓 = Ω/2𝜋. Thus, the rate of spontaneous emission,𝐴(𝑓) is re-lated to the spectral density of photon emission 𝛾(𝑓) in the dynamical Coulomb blockade regime provided in ref.

[110] by𝐴(𝑓) = 𝛾(𝑓)/𝑓. This is an interesting result since it implies that, whenever the impedance is renormalized (𝑍(Ω) → 𝑍_eff(Ω)), only the 𝐴(Ω) coefficient (and not the 𝐵(Ω) coefficient) is trivially renormalized, where the non-trivial renormalization entails the rescaling of𝜌(Ω)and/or the 𝑃(𝐸) function. The succeeding chapters toil towards formulating in detail the renormalization effect using path integral formalism.[102]

Electromagnetic Environment in Josephson

junctions 4.

A Josephson junction is a Superconductor(S)-Insulator(I)-Superconductor(S) tunnel junction (Fig. 4.1) that admits a supercurrent across it by the coherent tunneling of Cooper pairs[58,59] even in the absence of applied voltages. Due to its non-linear response to applied electromagnetic fields, Josephson junctions have found varied metrology[4] and de-tector applications.[5] Moreover, the junction size determines the current-voltage characteristics observed in experiments.

In particular, mesoscopic (small) junctions are known to be greatly susceptible to quantum fluctuations and changes in the electromagnetic environment compared to large junc-tions. This leads to complex theoretical considerations albeit richer physics such as circuit-Quantum Electro-Dynamics in the small junction whose features are often captured by the so-called𝑃(𝐸)theory.[16]

Despite the existence of excellent reviews on the subject and techniques[15,16,100], the authors found much of the techniques and prior concepts useful in following the ar-guments in this thesis scattered in various literature[10,11, 89, 101, 111, 112]. In particular, the techniques used in the subsequent chapters include path integral formalism[102]

and Green’s functions[11,89] to calculate phase-phase correla-tion funccorrela-tions and four-vector notacorrela-tion[113] where Maxwell’s equations appear for compactness. Thus, we include this section as a preamble for completeness and/or compactness.

Hopefully it offers a more nuanced understanding of the Caldeira-Leggett model and 𝑃(𝐸)theory in the context of Green’s functions and generally a path integral framework.

Quantum Phase Dynamics in Large Josephson Junctions (The Josephson Effect)

The physics of Josephson junctions (schematic shown in Fig.

4.1) is described by the well known Josephson equations[59],

66 4. Electromagnetic Environment in Josephson junctions

1: where the subscript 𝑥 distin-guishes it from quantum phases of other circuit elements defined later in the chapter.

𝐼_S =2𝑒𝐸_Jsin 2𝜙x(𝑡) (4.1a)

𝜕𝜙x(𝑡)

𝜕𝑡 = 𝑒𝑉_x (4.1b)

Here, 2𝜙x(𝑡)denotes the phase difference across the junction

1 and𝐸_J is the Josephson coupling energy[60]. The simplest derivation of eq. (4.1) follows from the real and imaginary parts of these two coupled Schrodinger equations

𝑖𝜕𝜓1

𝜕𝑡 =𝜇1𝜓1 +𝑚₀𝜓2, (4.2a) 𝑖𝜕𝜓2

𝜕𝑡 =𝜇2𝜓2+𝑚₀𝜓1. (4.2b) Here,𝜇1 and𝜇2 and the chemical potentials of the left (1) and right (2) junction respectively,𝜓1and𝜓2are the Cooper-pair wavefunctions of the left and right superconductors respectively and𝑚₀is a coupling energy term characterizing magnitude of overlap for the two wavefunctions across the insulator. When a potential difference (voltage)𝑉_xis applied across the junction, the two chemical potentials shift relative to each other in order to accomodate this change. This means that we can set𝜇1−𝜇2 =2𝑒𝑉_x, where 2𝑒 is the Cooper pair charge. Based on this, it is instructive to define an average chemical potential,𝜇≡ (𝜇1+𝜇2)/2, and solve for𝜇1 and𝜇2

in terms of𝜇. This yields, 𝜇1 = 𝜇+𝑒𝑉_xand 𝜇2 = 𝜇−𝑒𝑉_x. Plugging this back to eq. (4.2) yields,

𝑖𝜕𝜓1

𝜕𝑡 =(𝜇+𝑒𝑉_x)𝜓1+𝑚₀𝜓2, (4.3a) 𝑖𝜕𝜓2

𝜕𝑡 =(𝜇−𝑒𝑉_x)𝜓2+𝑚₀𝜓1. (4.3b) From this, it is clear𝜇is simply the common chemical po-tential relative to which the voltage drop is measured. This observation implies we can set it to zero without loss of generality,𝜇=0.

The Cooper pair wavefunctions are defined as𝜓1 =√

𝑛₁exp(𝑖2𝜑1) and𝜓2 = √

𝑛₂exp(𝑖2𝜑2)where 𝑛₁ and𝑛₂ are the number of Cooper-pairs in the left (1) and right (2) superconductor respectively, and 2𝜑1and 2𝜑2is their respective macroscopic quantum phases. Plugging these definitions into eq. (4.3), we

67

find,

𝜕𝑛₁

𝜕𝑡 =2𝑚₀√

𝑛₁𝑛₂sin(2[𝜑2−𝜑1]) (4.4a)

𝜕𝑛₂

𝜕𝑡 =2𝑚₀√

𝑛₂𝑛₁sin(2[𝜑1−𝜑2]) (4.4b) and,

2𝜕𝜑1

𝜕𝑡 =−𝑚₀ r_𝑛

2

𝑛₁ cos(2[𝜑2−𝜑1]) −𝑒𝑉_x, (4.5a) 2𝜕𝜑2

𝜕𝑡 =−𝑚₀ r_𝑛

2

𝑛₁ cos(2[𝜑2−𝜑1]) +𝑒𝑉_x. (4.5b) Taking the approximation that any tunneling currents that arise have the effect of varying 𝑛₁ and 𝑛₂ by only a small amount(𝛿𝑛)² '0 from an equilibrium value given by¹₂(𝑛₁+ 𝑛₂) ≡ 𝑛, i.e.𝑛₁ = 𝑛 +𝛿𝑛 and 𝑛₂ = 𝑛−𝛿𝑛, we see that the supercurrent across the barrier is given by𝐼_S≡ 𝑒𝜕𝛿𝑛/𝜕𝑡 and the voltage drop by 𝜕(𝜑2 −𝜑1)/𝜕𝑡 = 𝑒𝑉_x, which yield eq.

(4.1) when𝐸_J =𝑛𝑚₀ and𝜑2−𝜑1 =𝜙x.

There is another advantage of setting𝜇=0. In particular, eq.

(4.3) becomes a spinor equation, 𝑖𝜕𝜓

𝜕𝑡 = 𝐻_cp𝜓 (4.6a)

𝐻_cp = 𝑒𝑉_x𝜎3+𝑚₀𝜎1, (4.6b) 𝜓 ≡

𝜓1

𝜓2

, (4.6c)

where𝜎1and𝜎3 are the Pauli matrices, 𝜎1 =

0 1 1 0

, 𝜎3 =

1 0 0 −1

. (4.7a)

Since𝐸_J ∝𝑚₀,𝜎1plays the role of the coupling term that re-sults to Cooper-pair tunneling. Using the last Pauli matrix,

𝜎2 =

0 −𝑖 𝑖 0

, (4.7b)

we can define two operators 𝜎⁺ ≡ (𝜎1 +𝑖𝜎2)/2 and 𝜎⁻ ≡ (𝜎1−𝑖𝜎2)/2. These operators form tunneling matrix elements with the spinor and a transpose conjugate spinor defined

68 4. Electromagnetic Environment in Josephson junctions

Figure 4.1.: A schematic of a Josephson junction (S: ductor, I: insulator, S: supercon-ductor) depicting a Cooper pair from the left/right electrode tun-neling through the insulator to the right/left electrode.

as,

𝜓^†≡𝜓^∗T =(𝜓^∗₁,𝜓^∗₂). (4.8) For instance, tunneling from left to right requires replacing𝜓1

with𝜓2and annihilating𝜓1, which corresponds to𝜎⁺𝜓. The inverse process process corresponds to𝜎⁻𝜓. These matrices will be useful when calculating Cooper-pair tunneling rates for small junctions (See eq.4.34).

The Electromagnetic Environment and Fluctuation-Dissipation in Single Large Josephson Junctions

Equation4.1only considers the superconducting current and thus neglects the environment that lead to effects such as Coulomb blockade. The environment consists of all sources of the electromagnetic field (including the field itself) which couple to the Cooper-pair wavefunction via the phase dif-ference thus determining the𝐼−𝑉characteristics satisfying eq. (4.1). Specifically, the environment arises from processes such as the alternating currents and voltages, thermal fluc-tuations in the form of Johnson-Nyquist noise and coupled high impedance circuit environments[7,73].

Using eq. (4.1), one can define a conserved energy by treating the junction as a capacitance

𝐸 =𝑄²

x/2𝐶−𝐸_Jcos(2𝜙x) (4.9a)

𝑄_x=−𝐶𝑉_x (4.9b)

𝜕𝑄_x

𝜕𝑡 =𝐼_S (4.9c)

69

where𝐶 is the capacitance of the junction. Modifying the last equation in (4.9) to

𝜕𝑄_x

𝜕𝑡 =X

a

𝐼_a, (4.10)

one can then include all the environmental sources of energy in the form of currents. In fact, to arrive at eq. (4.10), the phase-difference needs to couple to the electromagnetic field (Maxwell equations) in a straight-forward manner

𝜕𝜙x

𝜕𝑥𝜇 = 𝑒 𝑑_eff𝑁^𝜈𝐹_𝜇𝜈 (4.11a) 𝑁^𝜇𝑁_𝜇 =−X

𝑖𝑗

𝑁_𝑖𝑁_𝑗𝛿^𝑖𝑗 =−1 (4.11b)

𝜕𝐹^𝜇𝜈

𝜕𝑥𝜇 =− 1 𝜀0𝜀r

X

a

𝐽^𝜈

a (4.11c)

Here, 𝐹_𝜇𝜈 = 𝜕𝐴_𝜇/𝜕𝑥_𝜈 − 𝜕𝐴_𝜈/𝜕𝑥_𝜇 = −𝐹_𝜈𝜇 is the electro-magnetic tensor, 𝑑_eff is the thickness of the barrier and 𝑁^𝜇 =(0, 𝑁^𝑖)points in the direction𝑁^𝑖 normal to the tunnel barrier. We have used Einstein notation where only the Greek indices are summed over and the Minkowski space-time signature is diag(𝜂𝜇𝜈)= (+, -, -, -). Taking the total derivative 𝜂^𝜇𝜈𝜕/𝜕𝑥^𝜈 = 𝜕/𝜕𝑥_𝜈 of eq. (4.11a) (with 𝜂_𝜇𝛼𝜂^𝜈𝛼 = 𝛿^𝜈_𝜇) and using𝜕𝑁^𝜈/𝜕𝑥^𝜇 =0 , we arrive at

𝜂^𝜇𝜈 𝜕²𝜙x

𝜕𝑥^𝜇𝜕𝑥𝜈 = 𝜕²𝜙x

𝜕𝑥^𝜇𝜕𝑥_𝜇 =−𝑒 𝑑_eff 𝜀0𝜀r

X

a

𝑁_𝜇𝐽_a^𝜇 =−𝑤𝐽 (4.12)

which is eq. (4.10) in disguise. Note that 𝐹_0𝑖 = 𝐸® and

1 2

P𝑖,𝑗𝜀^{𝑖𝑗 𝑘}𝐹^𝑖𝑗 = 𝐵®where𝐸®and𝐵®are the𝑥, 𝑦, 𝑧 components of the electric and magnetic fields respectively and𝜀^{𝑖𝑗 𝑘}is the Levi-Civita symbol. eq. (4.12) is the sourced Klein-Gordon equation with 𝐽 = P

a𝑁_𝜇𝐽^𝜇

a the source and 𝑤 = 𝑒 𝑑_eff/𝜀0𝜀r

the coupling constant. The vector 𝑁_𝜇and the anti-symmetry of the electromagnetic tensor 𝐹_𝜇𝜈 guarantees that, unlike Maxwell equations, the coupled Klein-Gordon equation lives in 2+1 dimensions instead of 3+1. For instance, when the tun-nel barrier is aligned to the y-z direction,𝑁^𝜇 =(0,1,0,0)and eq. (4.11a) and by extension eq. (4.12) become independent of 𝑥.

𝜕²𝜙x

𝜕𝑥^𝜇𝜕𝑥_𝜇 = 𝜕²𝜙x

𝜕𝑡2 − 𝜕²𝜙x

𝜕𝑦2 − 𝜕²𝜙x

𝜕𝑧2 =−𝑤𝐽 (4.13) Furthermore, taking the limit for small junctions which

70 4. Electromagnetic Environment in Josephson junctions

2: See eq. (3.4) of chapter3 3: Also the Lagrangian for eq.

(3.9) given in chapter3with𝜙→ 𝜙xand∇®²

⊥𝜙x =0

corresponds to taking the area of the barrierAto be small such that the phase neither varies with𝑦nor𝑧, we arrive at eq. (4.10)

𝜕²𝜙x

𝜕𝑡2 =−4𝐸_c𝐸_Jsin(2𝜙x) − 1 𝑅𝐶

𝜕𝜙x

𝜕𝑡 −2𝐸_c𝐼_F/𝑒 (4.14a) X3

a

𝐽¹

a =𝐽¹

S +𝐽¹

N+𝐽¹

F (4.14b) h𝐼_F(𝑡)𝐼_F(𝑡⁰)i = 4𝛽⁻¹

𝑅 𝛿(𝑡−𝑡⁰) (4.14c) with𝐽¹

S = 𝐽_cpsin 2𝜙xthe supercurrent,𝐽¹

N= 𝜎^𝑥𝑥𝐹₀₁the nor-mal current and𝜎^𝑥𝑥 the effective conductivity of the barrier along the𝑥direction. Here, we have used the cross-sectional area of the junction,Ato define𝐽^𝜇

aA= 𝐼^𝜇

a, the junction capac-itance𝐶 =𝜀0𝜀rA/𝑑_eff, the charging energy𝐸_c= 𝑒²/2𝐶and the junction conductance 1/𝑅= 𝜎^𝑥𝑥A/𝑑_eff. Finally,𝛽⁻¹ = 𝑘_B𝑇 is the inverse temperature and we have assumed the fluctua-tion current𝐽¹

FA= 𝐼_F is Gaussian-correlated over a bath (B stands for bath or Boltzmann distribution) with the thermal correlation function given by eq. (4.14c).

It is straight forward to generalize the conductance 1/𝑅 in eq. (4.14) using a spectral function 𝐾(𝜔) = [𝑍⁻¹(𝜔) + 𝑍⁻¹(−𝜔)]/(2𝜋) describing the macroscopic physics of the microscopic degrees of freedom of the system undergoing Brownian motion due to a heat bath comprising𝑘 harmonic oscillators,[101]

𝐻_B=

𝑘

X

𝑛=₁

_𝑄2

𝑛

2𝐶_𝑛 +

(𝜙^𝑛 −𝜙x)² 2𝑒2𝐿_𝑛

(4.15a) 𝐾(𝑡) =

𝑘

X

𝑛=₁

𝐿⁻_𝑛¹cos(𝜔^𝑛𝑡)=

∫ ^+∞

−∞

𝑑𝜔𝐾(𝜔)exp−𝑖𝜔𝑡 (4.15b) where𝐻_Bis the Hamiltonian of the heat bath consisting of 𝐿_𝑛𝐶_𝑛 circuits in parallel where𝜔^𝑛 =1/𝐿_𝑛𝐶_𝑛,𝑄_𝑛,𝜙^𝑛 are the charges stored by and the phases of the elements and𝐾(𝑡)is referred to as the Kernel representing the dissipative nature of the circuit.² The generalized Lagrangian for the system³

71

is given by

L= 𝐶 2𝑒²

_𝜕𝜙

x(𝑡)

𝜕𝑡 2

− 1 2𝑒²

∫ ^+∞

−∞

𝜙x(𝑡)𝜕𝐾(𝑠−𝑡)

𝜕𝑠 𝜙x(𝑠)𝑑𝑠

− 1 𝑒

∫ ^+∞

−∞

𝐼_F(𝑡−𝑠)𝜙x(𝑠)𝑑𝑠+𝐸_Jcos(2𝜙x), (4.16) where the fluctuation current is given by,

𝐼_F(𝑡)=

𝑘

X

𝑛=₁

𝜔^𝑛𝑄_𝑛sin(𝜔^𝑛𝑡) +𝑒⁻¹𝐿⁻_𝑛¹(𝜙^𝑛−𝜙x)cos(𝜔^𝑛𝑡) h𝐼_F(𝑡)𝐼_F(𝑡⁰)i =

𝑘

X

𝑛=₁

2𝐿⁻_𝑛¹h𝐻_B(𝜔^𝑛)icos𝜔^𝑛(𝑡−𝑡⁰).

The average is over the thermal bath degrees of freedom. For the Ohmic conductance above, we have𝑍⁻¹(𝜔)= 1/𝑅and h𝐻_B(𝜔)i =𝛽⁻¹ where the continuous, large𝑘 limit

𝑘→∞lim

𝑘

X

𝑛=₁

𝐿⁻_𝑛¹× →

∫ ^∞

−∞

𝑑𝜔𝐾(𝜔)× (4.17a) is taken in accordance with eq. (4.15b) thus recovering eq.

(4.14c). The fluctuation current density certainly satisfies the Green-Kubo relation[111,112],

𝜎^𝑥𝑥 = 𝛽 4

∫

𝑑⁴𝑥𝑁^𝜈𝑁^𝜇_𝐽

F𝜈(𝑡)𝐽_F𝜇(0)

= 𝛽

4A⁻¹𝑑_eff∫

𝑑𝑡h𝐼_F(𝑡)𝐼_F(𝑡⁰)i = 𝑑_eff

𝑅A, (4.17b) where we have used eq. (4.14c) in the last line. Note that to obtain the correct equation of motion, integration by parts of the second term in eq. (4.16) should be performed after applying the Euler-Lagrange equations, then the boundary term is dropped

1 𝑒2

∫ ^+∞

−∞

𝜕

𝜕𝑠

_{𝐾(𝑠−𝑡)}

𝜙x(𝑠) _𝑑𝑠

=0 (4.18)

72 4. Electromagnetic Environment in Josephson junctions

4: That the effect of the environ-mental impedance𝑍(𝜔) can be represented by a single quantum phase 𝜙z defined by the volt-age drop over𝑍(𝜔)is not at all obvious. At this stage, we treat it as an ansatz. It will not ap-pear in the equations until we impose the topological constraint P𝑎𝜙^𝑎=𝑒Φon the circuit.

Phase Correlation Functions, P(E) function and Coulomb Blockade of Cooper-pairs and Quasi-Particles in Single Small Josephson Junctions

The Hamiltonian

Consider a mesoscopic tunnel junction with capacitance 𝐶 driven by a voltage source 𝑉_x via an environmental impedance𝑍(𝜔). Each circuit element is characterized by a phase𝜙a related to the voltage drop𝑉_aof the element in the circuit by𝜙a(𝑡)=∫ ^𝑡

−∞𝑒𝑉_a(𝜏)𝑑𝜏, where the subscript a = J, x or z corresponds to the junction, voltage source and environ-ment impedance and𝜅𝑒 =2𝑒 , 𝑒 corresponds to Cooper pair, quasi-particle charge respectively.⁴ The voltages𝑉_zand𝑉_J decrease as one moves clockwise along the circuit, whereas the value increases for the voltage source𝑉_xin the same di-rection. The corresponding charge on the junction is defined as𝑄_J =𝐶𝑉_J where𝐶is the capacitance of the junction. The circuit can store a topological flux𝑒Φ = 𝜙J+𝜙z+𝜙x =P

𝑎𝜙a

related to a topological potential∫ ^𝑡

−∞𝐴(𝜏)𝑑𝜏 = Φ(𝑡), which we will find out, in chapter 5, that it leads to RF power renormalization when present.

The total Hamiltonian,Hof the circuit (Fig.4.2) is given by the expression,

H= X2

𝜅=₁

𝐻_𝜅+𝐻_J+𝐻_z (4.19)

Here, P2

𝜅=₁𝐻_𝜅 = 𝐻₁ +𝐻₂ where the Cooper-pair Hamil-tonian𝐻₂ = 𝜇𝜓⁺𝜓 = 𝜓⁺𝑖

𝜕/𝜕𝑡+𝑖𝐻_cp

𝜓 depends on the chemical potential𝜇and the 2-spinor𝜓and the quasi-particle Hamiltonian𝐻₁is given by,

𝐻₁ =𝐻_L+𝐻_R=X

𝑝,𝜎 𝜖^𝑝𝜎𝛾^†𝑝𝜎𝛾^𝑝𝜎 +X

𝑞,𝜎 𝜖^𝑞𝜎𝛾𝑞^†𝜎𝛾^𝑞𝜎 (4.20) where 𝛾^𝑝𝜎 or 𝛾^𝑞𝜎 and 𝛾𝑝^†𝜎 or 𝛾𝑞^†𝜎 are the annihilation and creation operators respectively of a quasi-particle state with energy𝜖^𝑝_𝜎 or𝜖^𝑞_𝜎, momentum𝑝 or𝑞 and spin 𝜎in the left

73

5: This Hamiltonian basically cor-responds to eq. (21) of ref. [16]

with the topological terms,Φ =0 and𝐴=0.

or right electrode, 𝐻_J =

X2

𝜅=₁

Θ_𝜅exp(−𝑖𝜅𝜙J) +ℎ.𝑐. (4.21) is the tunneling Hamiltonian where

Θ₁ = X

𝑝≠𝑞,𝜎

𝑀_𝑝𝑞𝛾^𝑞𝜎𝛾^†𝑝𝜎 (4.22a)

Θ₂ = 𝐸_J

2 (𝜎1 −𝑖𝜎2) ≡ 𝐸_J

2 𝜎⁻ (4.22b) 𝜎1,₂ are the𝑥, 𝑦Pauli matrices acting on the 2-spinor given in eq. (4.7),𝑀_𝑝𝑞 is a dimensionful spin-conserving complex-valued quasi-particle tunneling matrix, 𝑝 ≠ 𝑞 enforces the condition [𝐻_L, 𝐻_R] = 0 and 𝐸_J is the Josephson coupling energy[60],

𝐻_z =

(𝑄_J+𝐶𝑉_x+𝐶𝐴)² 2𝐶

+X

𝑛=₁

_𝑄2

𝑛

2𝐶_𝑛 +𝑒⁻²(𝜙^𝑛−𝜙J+𝜙x+𝑒Φ)² 2𝐿_𝑛

(4.23) is the Hamiltonian⁵ describing the environmental impedance 𝑍(𝜔)and junction capacitance 𝐶, where𝑍(𝜔)is character-ized by an infinite number of parallel 𝐿_𝑛𝐶_𝑛 circuits coupled serially to the tunnel junction. One can define𝑄 =𝑄_J−𝐶𝑉_x and𝜙= 𝜙J−𝜙xas the fluctuation variables of the junction charge𝑄_J = 𝐶𝑉_Jand junction phase 𝜙J(𝑡) = 𝑒∫^𝑡

−∞𝑑𝑡⁰𝑉_J(𝑡⁰) around the mean value determined by the voltage source𝑉_x, where𝑉_J(𝑡⁰)is the voltage drop across the junction. (See ref.

[16] on page 27. Note that𝜙Jand𝜙are related by a suitable unitary transformation Uof the Hamiltonian,

H⁰ =𝑖U^† 𝜕

𝜕𝑡 U+ UHU^†, (4.24) whereH⁰ =𝐻⁰

1+𝐻₂+𝐻_J+𝐻_z,𝐻_J =P2

𝜅=₁Θ_𝜅exp(−𝑖𝜅𝜙J) + ℎ.𝑐., 𝐻⁰

1 = P

𝑝≠𝑞,𝜎𝜖⁰𝑝𝜎𝛾𝑝^†𝜎𝛾^𝑝_𝜎 +P

𝑝≠𝑞,𝜎𝜖^𝑞_𝜎𝛾𝑞^†𝜎𝛾^𝑞_𝜎 and 𝜖⁰𝑝𝜎 = 𝜖^𝑝_𝜎+𝑒𝑉_x.𝑄 =𝑄_J−𝐶𝑉_x,𝑄_𝑛 are the conjugate variables to 𝜙 = 𝜙J−𝜙x, 𝜙n satisfying the charge-phase commutation relation,

[𝜙^𝑛, 𝑄_𝑚]= 𝑖𝛿^𝑚𝑛𝑒 , (4.25)

[𝜙, 𝑄] =𝑖𝑒 (4.26)

74 4. Electromagnetic Environment in Josephson junctions

6: As before in eq.4.3.

where 𝛿^𝑎𝑏 is the Kroneker delta. Operators, 𝑂(𝑡) in the Heisenberg picture are related to the ones in the Schrödinger picture, 𝑂(0) by 𝑂(𝑡) = 𝑈₀(𝑡)^†𝑂(0)𝑈₀(𝑡) with the unitary evolution operator𝑈₀(𝑡)given by𝑈₀(𝑡)=exp_−𝑖P2

𝜅=₁𝐻_𝜅𝑡 in the absence of tunneling.

In what follows, we assume the Cooper-pair ground state energy𝜇=0.⁶ The tunneling current𝐼(𝑉)at the junction is given by

𝐼(𝑉 , 𝑡⁰)=tr

T_𝑈†𝐼_J(0)𝑈

(4.27) Here,𝑈 =𝑈₀+𝑈_intwhere,

𝑈_int =exp

−𝑖

∫ ^+𝑡0

−𝑡⁰

𝐻_J(𝑡)𝑑𝑡

=exp

−𝑖

∫ 0

−𝑡⁰

𝐻_J(𝑡)𝑑𝑡

exp

−𝑖

∫ ^𝑡0

0

𝐻_J(𝑡)𝑑𝑡

=exp

+𝑖

∫ ^−𝑡0

0

𝐻_J(𝑡)𝑑𝑡

exp

−𝑖

∫ ^𝑡0

0

𝐻_J(𝑡)𝑑𝑡

=𝑈^†

int(−𝑡⁰)𝑈_int(+𝑡⁰) andTis the time ordering operator with the property given byTΘ_𝜅(𝑡)Θ^†_𝜅(0) = Θ_𝜅(𝑡)Θ^†_𝜅(0),TΘ_𝜅(0)Θ^†_𝜅(𝑡)= Θ^†_𝜅(𝑡)Θ_𝜅(0). Here,𝑡⁰is the elapsed time after switching on the interaction term𝑈_int(𝑡⁰) and takes the range 0 ≤ 𝑡⁰ ≤ +∞. [Note that 𝑈^†

int(𝑡⁰)takes care of𝑐.𝑐.term in eq. (4.29), updating the inte-gral range as discussed:∫^𝑡0

0 𝑑𝑡 → ∫0

−𝑡⁰𝑑𝑡+∫^𝑡0

0 𝑑𝑡 =∫^+𝑡0

−𝑡⁰ 𝑑𝑡.]

We shall be interested in the current𝐼(𝑉 , 𝑡⁰→ +∞)= 𝐼(𝑉) at equilibrium [eq. (4.45)].

The tunneling current operator is

𝐼_J(0)=−𝑖[𝑄_J(0), 𝐻_J(0)], (4.28) and the averageh...iis over, the quasi-particle equilibrium states, whose density matrix is given by 𝜌1 = 𝜌L𝜌R = Z₁⁻¹exp(−𝛽𝐻₁)withZ₁ =Z_L×Z_R =Q

𝑝[1+exp(−𝛽𝜖^𝑝𝜎)] × Q𝑞[1+exp(−𝛽𝜖^𝑞𝜎)], and the environment𝜌env =Z_env⁻¹ exp(−𝛽𝐻_z) where𝛽 =1/𝑘_B𝑇is the inverse temperature while the trace (tr) is over the Pauli matrices.

75

Figure 4.2.: A mesoscopic tun-nel junction, J with capacitance 𝐶driven by a voltage source𝑉_x via an environmental impedance 𝑍(𝜔)composed of infinite num-ber of parallel 𝐿_𝑛 𝐶_𝑛 circuits.

The circuit stores a flux 𝑒Φ = P𝑖𝜙^𝑖 = 𝜙J+𝜙x+𝜙z related to a topological potential 𝐴(𝑡) by

∫^𝑡

−∞ 𝑑𝑠𝐴(𝑠) = Φ(𝑡). (This figure corresponds to Fig. 2 of ref. [16]

without the topological flux.)

Perturbation Expansion

We can then expand eq. (4.27) as a perturbation series in the tunneling Hamiltonian𝐻_J(𝑡)by applying the perturbation expansion formula in AppendixC

𝐼 =

𝐼_J(0) −𝑖∫ ^+∞

−∞

[𝐼_J(0), 𝐻_J(𝑡)]𝑑𝑡 +𝑂(𝐻²

J)

(4.29) UsingTΘ_𝜅(𝑡)Θ^†_𝜅(0)= Θ_𝜅(𝑡)Θ^†_𝜅(0),TΘ_𝜅(0)Θ_𝜅^†(𝑡)= Θ^†_𝜅(𝑡)Θ_𝜅(0), hΘ_𝜅(𝑡)i=0 and

Θ_𝜅⁰(𝑡)Θ^†_𝜅(0)

=

Θ_𝜅^†0(𝑡)Θ_𝜅(0)

=𝛼𝜅(𝑡)𝛿𝜅⁰𝜅, we find that

𝐼 '

∫ ^+∞

−∞

[𝐻_J(𝑡),[𝑄_J(0), 𝐻_J(0)]]𝑑𝑡

(4.30)

=𝑖𝑒X²

𝜅=₁

∫ ^+∞

−∞

𝛼𝜅(𝑡) sin

𝜅Δ𝜙J(𝑡)

𝜙J

𝑑𝑡 (4.31) whereΔ𝜙J(𝑡) = 𝜙J(𝑡) −𝜙J(0). Thus, the particle degrees of freedom 𝛼𝜅(𝑡)and the environment are decoupled and the trace over the environment _...

𝜌env

has been re-written as h...i_𝜙

J in terms of the junction phase𝜙J degree of freedom.

For the quasi-particle current, the kernel𝛼1(𝑡)scales with the dimensionless tunneling conductance 𝑒⁻²𝑅⁻¹

T but its func-tional form depends on the gap, reflecting the corresponding structures in the quasi-particle𝐼−𝑉 characteristics. It can be

76 4. Electromagnetic Environment in Josephson junctions

computed as, 𝛼1(𝑡)=

Θ₁(𝑡)Θ₁^†(0)

= X

𝑝≠𝑞,𝜎

X

𝑝⁰≠𝑞⁰,𝜎

𝑀_𝑝𝑞𝑀^∗_𝑞0𝑝0× h𝑅, 𝑠| h𝐿, 𝑠|𝛾^𝑞𝜎(𝑡)𝛾𝑝^†𝜎(𝑡)𝛾^𝑝0𝜎(0)𝛾^†𝑞⁰𝜎(0)𝜌1|𝐿, 𝑠i |𝑅, 𝑠i

=2X

𝑝≠𝑞

X

𝑝⁰≠𝑞⁰

𝑀_𝑝𝑞𝑀_𝑞^∗0𝑝⁰h𝑅|𝛾^𝑞(𝑡)h𝐿|𝛾𝑝^†(𝑡)𝛾^𝑝0(0)𝜌L|𝐿i𝛾𝑞^†0(0)𝜌R|𝑅i

=2X

𝑝≠𝑞

𝑓(𝜖^𝑝)exp_−𝑖

𝜖^𝑝𝑡 X

𝑝⁰≠𝑞⁰

𝑀_𝑝𝑞𝑀^∗_𝑞0𝑝0𝛿^𝑝𝑝0h𝑅|𝛾^𝑞(𝑡)𝛾𝑞^†0(0)𝜌R|𝑅i

=2X

𝑝≠𝑞

𝑓(𝜖^𝑝)[1−𝑓(𝜖^𝑞)]exp_𝑖(

𝜖^𝑞 −𝜖^𝑝)𝑡 X

𝑝⁰≠𝑞⁰

𝑀_𝑝𝑞𝑀^∗_𝑞0𝑝⁰𝛿^𝑞𝑞0𝛿^𝑝𝑝0

=2X

𝑝≠𝑞

𝑓(𝜖^𝑝)[1− 𝑓(𝜖^𝑞)]𝑀_𝑝𝑞𝑀^∗_𝑞𝑝exp_𝑖(

𝜖^𝑞−𝜖^𝑝)𝑡

→ 1

𝜋𝑒²𝑅_T

∫ ^+∞

−∞

∫ ^+∞

−∞

𝑑𝜖^𝑝𝑑𝜖^𝑞N_L(Δ) N_L(0)

N_R(Δ) N_R(0)× 𝑓(𝜖^𝑝)(1− 𝑓(𝜖^𝑞))exp_𝑖(

𝜖^𝑞−𝜖^𝑝)𝑡 , (4.32) by taking the continuous limit 2𝜋𝑒²𝑅_TN_L(0)N_R(0)𝑀_𝑝𝑞𝑀^∗_𝑞𝑝 → 1. Here,𝑒²𝑅_Tis the dimensionless tunnel resistance, 𝑓(𝐸)= [1+exp(𝛽𝐸)]⁻¹is the Fermi-Dirac function andN_L(Δ),N_R(Δ) is the left, right BCS density of states which reduce to the elec-tron density of statesN_L(0),N_R(0)when the superconducting gapΔ = 0 vanishes,

𝑑𝐸_𝑝 𝑑𝜖^𝑝

𝑑𝐸_𝑞

𝑑𝜖^𝑞 = N_L(Δ) N_L(0)

N_R(Δ)

N_R(0), (4.33a) 𝐸_𝑝 =

q𝜖²𝑝−Δ², 𝐸_𝑞 =

q𝜖²𝑞 −Δ² (4.33b) where𝐸_𝑝 = 𝑝²/2𝑚, 𝐸_𝑞 = 𝑞²/2𝑚is the kinetic energy of the electrons above the Fermi sea.

Likewise, calculating𝛼2(𝑡), we find, 𝛼2 =

Θ₂^†(𝑡)Θ₂(0)

=

Θ₂^†(0)Θ₂(0)

= _𝐸

J

2 2

tr{(𝜎1+𝑖𝜎2)(𝜎1−𝑖𝜎2)}

= 𝐸²

J

4 tr{2𝜎0+𝑖[𝜎2,𝜎1]}= 𝐸²

J

2 tr{𝜎0+𝜎3} =𝐸²

J, (4.34) where𝜎0is the 2×2 identity matrix. We discover that, unlike 𝛼1(𝑡), the function𝛼2(𝑡) = 𝛼2(0) = 𝐸²

J is time independent and only depends on the strength of Cooper pair tunneling, 𝐸_J.

77

Path Integrals and Phase Correlations

To calculate the remaining average over𝜙Jin eq. (4.31), we work in Minkwoski time at zero temperature (thus by-passing a rigorous but otherwise tedious Wick rotation to Euclidean time) since the finite temperature propagator is trivially related to the zero temperature result [eq. (4.43b) for the trivial relation and AppendixBfor the formalism].

In this formalism: Given an observable𝑂(𝜙J), its average at zero temperature is given by the functional/path integral

𝛽lim→∞

h𝑂(𝜙J)i_𝜙

J = Z⁻¹

𝑘

Y

𝑛=₁

∫

𝐷𝜙n𝐷𝜙J𝑂(𝜙J)exp𝑖𝑆_z(𝜙n,𝜙J) (4.35) whereZ=Q^𝑘

𝑛=₁

∫ _𝐷

𝜙n𝐷𝜙Jexp𝑖𝑆_z(𝜙n,𝜙J)is the partition function normalizing eq. (4.35) and the Lagrangian in the action for the environment𝑆_z(𝜙n,𝜙J)is given by the (inverse) Legendre transform of the environment Hamiltonian in eq.

(4.23) 𝑆_z =

∫

L_z𝑑𝑡 =

∫

(𝑄_J+𝐶𝑉_x+𝐶𝐴)𝜕𝐻_z

𝜕𝑄_J −𝐻_z

𝑑𝑡, (4.36a) 𝐶𝜕𝜙x(𝑡)

𝜕𝑡 =𝑒𝑄_x, 𝐶𝜕𝜙J(𝑡)

𝜕𝑡 =𝑒𝑄_J, 𝜕Φ(𝑡)

𝜕𝑡 =𝐴(𝑡).

(4.36b) The effective action 𝑆⁰

z(𝜙J) resulting from performing first the functional integral product over 𝜙^𝑛 is given by

𝑆⁰

z(𝜙+𝑒Φ)= 𝐶 2𝑒2

∫ ^+∞

−∞

_{𝜕[𝜙(𝑡) −𝑒Φ(𝑡)]}

𝜕𝑡

2

𝑑𝑡− 1 2𝑒2

∫ ^+∞

−∞

𝜙(𝑡) −𝑒Φ(𝑡)2

P𝑛𝐿_𝑛 𝑑𝑡

− 1 4𝜋𝑒²

∫ ^+∞

−∞

∫ ^+∞

−∞

[𝜙(𝑡)−𝑒Φ(𝑡)]𝜕𝑍⁻¹(𝑠−𝑡)

𝜕𝑠 [𝜙(𝑡)−𝑒Φ(𝑡)]𝑑𝑠𝑑𝑡 + 1

𝑒

∫ ^+∞

−∞

𝐼_F(𝑡)[𝜙(𝑡) −𝑒Φ(𝑡)]𝑑𝑡 (4.37) with a fluctuation current𝐼_F(𝑡) = 0 and we have used𝜙 = 𝜙J−𝜙x. Here,𝑍⁻¹(𝑡)is the Fourier transform of a generalized

78 4. Electromagnetic Environment in Josephson junctions

admittance function𝑍⁻¹(𝜔)given by 𝑍⁻¹(𝜔)=

𝑘

X

𝑛=₁

𝜔^𝑛 𝑖𝜔𝐿_𝑛

_𝜔

𝑛

(𝜔+𝑖𝜀)²−𝜔^2𝑛

(4.38a)

=

𝑘

X

𝑛=₁

𝜔^𝑛 𝑖𝜔𝐿_𝑛

1

𝜔−𝜔^𝑛+𝑖𝜀 − 1 𝜔+𝜔^𝑛+𝑖𝜀

, (4.38b) 1

𝜔+𝜔^𝑛 ±𝑖𝜀 =∓𝑖𝜋𝛿(𝜔+𝜔^𝑛) +p.p. 1

𝜔+𝜔^𝑛

(4.38c) where eq. (4.38c) is the Sokhotski-Plemelj formular and p.p.

stands for Cauchy principal part. eq. (4.38) is related to the spectral function 𝐾(𝜔) = [𝑍⁻¹(𝜔) +𝑍⁻¹(−𝜔)]/(2𝜋)where 𝜀 is the infinitesimal satisfying 𝜔𝜀 = 𝜀 and the nilpotent condition𝜀² = 0. Note, the spectral function is the sum of negative and positive frequency impedance accounting for emission and absorption processes respectively by the circuit.

Thus, eq. (4.16) differs slightly from eq. (4.37) where the real-valued spectral function𝐾(𝑡)in the classical Lagrangian gets replaced with the complex valued admittance𝑍⁻¹(𝑡)/(2𝜋) in the quantum case.

Introducing the Dirac delta function 𝛿(𝑥) for functional integrals with the property

∫

𝐷𝑥 𝑓(𝑥)𝛿(𝑥−𝑦)= 𝑓(𝑦) (4.39) for any functional𝑓(𝑥), we may proceed to insert∫ _𝐷

𝜙z𝛿(𝜙J+ 𝜙x+𝜙z−𝑒Φ)= 1 into eq. (4.35) thus introducing the con-straintP

𝑎𝜙a =𝜙J+𝜙x+𝜙z =𝑒Φguaranteed by the circuit in Fig. (4.2). Consequently, the average in eq. (4.31) is now taken over both𝜙Jand 𝜙z:

𝛽→+∞lim hsin

𝜅Δ𝜙J(𝑡)_i

𝜙J𝜙z

= Z⁻¹

∫ 𝐷𝜙J

∫

𝐷𝜙z𝛿

X

𝑎 𝜙a−𝑒Φ

×sin

𝜅Δ𝜙J(𝑡)

exp𝑖𝑆⁰

z(𝜙−𝑒Φ) (4.40)

79

We find,

−hsin

𝜅Δ𝜙J(𝑡)_i

𝜙J𝜙z =hsin[𝜅Δ𝜙x(𝑡)+𝜅𝑒∫ ^𝑡

0

𝐴(𝜏)𝑑𝜏+𝜅Δ𝜙z(𝑡)]i_𝜙

z

= hsin

𝜅Δ𝜙z(𝑡)_i

𝜙zcos

𝜅Δ𝜙x(𝑡) +𝜅𝑒

∫ ^𝑡

0

𝐴(𝜏)𝑑𝜏

+ hcos

𝜅Δ𝜙z(𝑡)_i

𝜙zsin

𝜅Δ𝜙x(𝑡) +𝜅𝑒

∫ ^𝑡

0

𝐴(𝜏)𝑑𝜏

(4.41) with Δ𝜙a(𝑡) = 𝑒∫ ^𝑡

0 𝑉_a(𝑡⁰)𝑑𝑡⁰ and ΔΦ(𝑡) = ∫^𝑡

0 𝐴(𝑡⁰)𝑑𝑡⁰. We have assumed Fubini’s theorem for interchange of integra-tion order applies and thus performed first the integral over 𝜙z. Using the fact that 𝑆⁰

z is quadratic, the result-ing functional integral over 𝜙z in eq. (4.41) is Gaussian resulting in hsin

𝜅Δ𝜙z(𝑡)_i

𝜙z =0 term vanishing. Likewise, hcos

𝜅Δ𝜙z(𝑡)_i

𝜙z satisfies Wick’s theorem[16]

hcos

𝜅Δ𝜙z(𝑡)_i

𝜙z

=exp 𝜅²_[

𝜙z(𝑡) −𝜙z(0)]𝜙z(0)

𝜙z

, (4.42a)

∫

𝐷𝜙zexp𝑖𝑆⁰

z(𝜙z, 𝐼_F) =exp𝑖𝑆⁰⁰

z(𝐼_F), (4.42b) 𝑆⁰⁰

z(𝐼_F)= 2𝜋 2𝑒²

∫ ^+∞

−∞

𝐼_F(−𝜔)𝐺_eff(𝜔)𝐼_F(𝜔)𝑑𝜔 (4.42c) 𝐺_eff(𝜔)=−𝑒²𝑖𝜔⁻¹𝑍_eff(𝜔), (4.42d)

𝑍_eff(𝜔)= 1

𝑍⁻1(𝜔) +𝑦(𝜔) (4.42e) where𝑦(𝜔)= 𝑖𝜔𝐶−𝑖𝜔⁻¹P

𝑛𝐿⁻_𝑛¹.

We introduce the zero temperature propagator𝐷_+∞(𝑡)given by

𝐷_+∞(𝑡)= 1 2𝜋

∫ ^+∞

−∞

𝑑𝜔

𝜔 exp−𝑖𝜔𝑡_𝑍

eff(𝜔) +𝑛. 𝑓 . , (4.43a) where 𝑛. 𝑓 . stands for negative frequency. The finite tem-perature propagator is related to𝐷_+∞(𝑡)by a sum over the

80 4. Electromagnetic Environment in Josephson junctions

photon number states 𝐷_+∞(𝑡) →𝐷_𝛽(𝑡)=

+∞

X

𝑛=₀

𝐷_+∞(𝑡−𝑖𝑛𝛽)

= 1 2𝜋

∫ ^+∞

−∞

𝑑𝜔 𝜔

exp−𝑖𝜔𝑡 1−exp(−𝛽𝜔)

_𝑍

eff(𝜔) +𝑛. 𝑓 . (4.43b) Thus, computing the phase–phase correlation function, we find

𝜙z(𝑠)𝜙z(𝑡)

𝜙z = Z⁻¹𝛿²exp𝑖𝑆⁰⁰

z(𝐼_F) 𝑒⁻²𝛿𝐼_F(𝑠)𝛿𝐼_F(𝑡)

_𝐼

F=₀,Z=1

= 𝑒²𝐷_+∞(𝑠−𝑡) →𝑒²𝐷_𝛽(𝑠−𝑡), (4.44) which satisfies the well-know fluctuation-dissipation theo-rem.[9]

Cooper Pair and Quasi-Particle Tunneling Current

Finally, plugging in results (4.41) and (4.43b) in eq. (4.31), and usingΔ𝜙x(𝑡)=∫^𝑡

0 𝑉(𝑡⁰)𝑑𝑡⁰=𝑉𝑡where𝑉_x =𝑉 is a constant external voltage andΔΦ(𝑡)=0, the total𝐼−𝑉characteristics is given by

𝐼₀(𝑉) =𝐼₁(𝑉) +𝐼₂(𝑉)

=𝑒⁻¹𝑅⁻¹

T

∫ ^+∞

−∞

𝑑𝜖^𝑝𝑑𝜖^𝑞N(𝜖^𝑞)N(𝜖^𝑝)

N²(0) 𝑓(𝜖^𝑝)(1− 𝑓(𝜖^𝑞))

×_𝑃

1(𝜖^𝑞−𝜖^𝑝+𝑒𝑉) −𝑃₁(𝜖^𝑞 −𝜖^𝑝−𝑒𝑉) +𝑒𝜋𝐸²

J {𝑃₂(2𝑒𝑉) −𝑃₂(−2𝑒𝑉)} (4.45) where we have introduced the so called𝑃(𝐸)function[16]

𝑃_𝜅(𝐸)= 1 2𝜋

∫ ^+∞

−∞

𝑑𝑡 exp𝜅²J(𝑡)exp𝑖𝐸𝑡, (4.46a) 𝑒⁻²J(𝑡)= 𝐷_𝛽(𝑡) −𝐷_𝛽(0) (4.46b) with𝐸 some arbitrary energy. It gives the probability that the junction will absorb energy 𝐸 from the environment.

Note that eq. (4.45) reduces to the normal junction 𝐼 −𝑉

81

characteristics 𝐼(𝑉)|_Δ=0 =𝑒⁻¹𝑅⁻¹

T

∫ ^+∞

−∞

𝑑𝐸_𝑝𝑑𝐸_𝑞𝑓(𝐸_𝑝)(1− 𝑓(𝐸_𝑞))

×_𝑃

1(𝐸_𝑞−𝐸_𝑝+𝑒𝑉) −𝑃₁(𝐸_𝑞−𝐸_𝑝−𝑒𝑉)

= 𝑒⁻¹𝑅⁻¹

T

∫ ^+∞

−∞

𝑑𝐸 𝐸

1−exp(−𝛽𝐸)

× {𝑃₁(−𝐸+𝑒𝑉) −𝑃₁(−𝐸−𝑒𝑉)} (4.47) when the superconducting gap vanishesΔ =0, since𝐸_J(Δ = 0)=0,N(Δ = 0)/N(0) =1 and𝐸_𝑝 =𝜖^𝑝, 𝐸_𝑞 = 𝜖^𝑞.

Here, we have used path integral formalism and demon-strated how to reproduce the𝐼–𝑉characteristics of the single small Josephson junction. Finally, recall we introduced a flux stored by the circuit, Φ only to set it equal to zero. In the next chapter, we discuss how the microwave amplitude and Green’s function are renormalized in a manner analogous to vacuum polarization in QED. We show that this implies that Φ ≠ 0 is non-vanishing.

Renormalization

Rescaling of microwave amplitude in small

Josephson junctions 5.

1. Rescaling of Oscillating Voltages Applied on Single Junctions . . . . 86 2. Rescaling of Applied Os-cillating Voltages in Linear Arrays of Josephson Junc-tions . . . .100 3. Soliton Field Theory

Ori-gin of the Lehmann Weight in an Infinite Array . . . .103 4. Discussion. . . .105 In chapter 4, we introduced the 𝑃(𝐸)function in terms of

Green’s function in eq. (4.42e) for the equivalent circuit of the Josephson junction and the environment given in Fig.4.2.

It is worth noting that the early introduction of𝑃(𝐸)theory and its applications for single junctions is provided in ref. [16, 114] while the case of the single Josephson junction is given in ref. [114]. In particular,𝑃(𝐸) theory accurately predicts the current-voltage characteristics (given by eq. (4.45)) for a single junction biased by a dc voltage (𝑉) and strongly coupled to its electromagnetic environment in the form of an impedance function 𝑍(𝜔).

Typically, the theory accounts for the effect of an ac bias voltage by considering a shift 𝑉 → 𝑉 +𝑉_accosΩ𝑡 of the 𝐼–𝑉 characteristics, where 𝑉_ac is the amplitude and Ω is the angular frequency of the alternating voltage. Using a unitary transformation method equivalent to eqs. (25 - 31) of ref. [16], entailing the splitting off of the dc component 𝜙(𝑡)of the quantum phase from the ac component 𝜙RF(𝑡), the aforementioned shift was shown not to hold for a normal junction driven via its electromagnetic environment.[27]

Instead, the two phases 𝜙(𝑡)and 𝜙RF(𝑡)should satisfy the following constraint,

𝐶𝜕²𝜙RF

𝜕𝑡² −

∫

𝑑𝑡⁰𝑍⁻¹(𝑡−𝑡⁰) 𝜕

𝜕𝑡

𝜙RF(𝑡⁰) −𝜙(𝑡⁰)

=0, (5.1) equivalent to eq. (4.9) of ref. [27]. Taking its Fourier transform, it is clear that eq. (3.15) satisfies this constraint. The constraint requires that the shift𝑉 → 𝑉 +𝑉_accosΩ𝑡 be modified to 𝑉 →𝑉 +

1+𝑖Ω𝐶𝑍⁻¹(Ω)⁻1

^{𝑉accos(Ω𝑡+𝜂), where𝜂(Ω)} is the argument of

1+𝑖Ω𝐶𝑍⁻¹(Ω)⁻1

. Herein, we shall refer to the rescaling 𝑉_ac to |Ξ|𝑉_ac, where Ξ is a ratio of impedance functions, asrenormalization.

It is important to note that since the arguments presented in chapter3, consistent with eq. (5.1) through eq. (3.15), merely require the responses of the dc voltage vary from the ac voltage to be valid, the renormalization result of ref. [27] for

86 5. Rescaling of microwave amplitude in small Josephson junctions

1: A Lehmann weight is a factor which renormalizes the Green’s function of a quantum mechani-cal system. Lehmann weights are also referred to as wavefunction renormalization factors since the Green’s function of a quantum mechanical system is often the Fourier transform of the wave-function.

single normal junctions should also apply for single small Josephson junctions.

Thus, in this chapter, we introduce a method for calculating the renormalization factor using Lehmann weights.[103]¹ This method has the advantage of being amenable to var-ied situations ranging from single normal junctions, single Josephson junctions as well as arrays. Particularly, we con-sider renormalization effects of applied oscillating voltages due to wavefunction renormalization/Lehmann weights[103]

that rescale the environmental impedance of the single junction as well as the array. The array is treated as an infinitely long[51] effective circuit. As is the case for the single junction, we find a Lehmann weight of the general form,Ξ(𝜔) = exp(−Λ⁻¹)exp−𝛽𝑀(𝜔)exp(𝑖𝛽𝜀^𝑚), where Λ is the soliton length of the array[41,50–52], which also acts as a linear response function for oscillating electromagnetic fields, and can be interpreted as the probability amplitude of exciting a ‘particle’ of mass𝑀from the junction ground state by the RF field.[115] The quantum statistics of this

‘particle’ are determined by the argument 𝛽𝜀^𝑚, where 𝜀^𝑚 is identified as the Matsubara frequency.[116] In the case of the infinite array, this ‘particle’ corresponds to a bosonic charge soliton injected into the array. Possible application of these results is in accurately determining the absorbed RF power in dynamical Coulomb blockade experiments espe-cially where long arrays are used as on-site electromagnetic power detectors.[33,54]

1. Rescaling of Oscillating Voltages Applied on Single Junctions

Introduction

Within the Caldeira-Leggett model[10] introduced in chapter 4, the environment of a dissipative voltage-biased single junction shown in Fig. 4.2 is modeled by the action 𝑆_z =

∫ _𝑑𝑡

L_z, where the Lagrangian is given by,

L_z = 𝐶 2𝑒²

_𝜕𝜙0

𝜕𝑡 2

+

𝑘

X

𝑛=1

(𝐶_𝑛 2𝑒²

_𝜕𝜙

𝑛

𝜕𝑡 2

− 𝐿⁻_𝑛¹

2𝑒² 𝜙^𝑛 −𝜙⁰2

) , (5.2a)

1. Rescaling of Oscillating Voltages Applied on Single Junctions 87

2: 𝜙⁰is the phase defined for con-venience, to shorten the expres-sion of the Lagrangian to avoid always writing 𝜙J−𝜙x−𝑒Φ = 𝜙−𝑒Φin the calculations in chap-ter4.

3: A Lehmann weight renormal-izes a Green’s function.

4: AppendixD.

5: This co-efficient can be com-puted using the driven system’s equation of motion.

where𝑒is the electric charge,𝐶 =𝜀0𝜀rA/𝑑_effis the junction capacitance, 𝜙⁰ ≡ 𝜙J −𝜙x−𝑒Φwhere² 𝜙J, 𝜙x, 𝜙z and 𝑒Φ are the phases associated with the the voltage drop at the junction𝑉_J, voltage source (external voltage)𝑉_xand the flux stored by the circuit respectively,𝜙^𝑛 is the bath degrees of freedom represented by𝑘 coupled (via𝜙)𝐿_𝑛𝐶_𝑛 oscillators constituting the environment.

The effective action, 𝑆⁰

z =−𝑖ln

∫ ^𝑘 Y

𝑛=₁

𝐷𝜙^𝑛exp𝑖𝑆_z(𝜙^𝑛,𝜙⁰)

=

∫ 𝑑𝑡

( 𝐶 2𝑒2

_𝜕𝜙⁰

𝜕𝑡 2

− 1

4𝜋𝑒2

∫

𝑑𝑠𝜙⁰(𝑠)

_𝜕𝑍−₁(𝑡 −𝑠)

𝜕𝑡

𝜙⁰(𝑡) )

= 2𝜋 2𝑒2

∫

𝑑𝜔 𝜙⁰(𝜔)𝑖𝜔𝑍⁻¹

eff(𝜔)𝜙⁰(−𝜔) (5.2b) requires the impedance Green’s function in the𝑃(𝐸)function to be modified by a wavefunction renormalization (Lehmann) weight[103]),

𝑃_𝜅(𝐸) = 1 2𝜋

∫

𝑑𝑡exp 𝜅²J(𝑡) +𝑖𝐸𝑡_,

(5.3a) J(𝑡) = 2𝑒²

2𝜋

∫ _𝑑𝜔

𝜔 Re{𝑍_eff(𝜔)} exp(−𝑖𝜔𝑡) −1

1−exp(−𝛽𝜔), (5.3b) 𝑍_eff(𝜔)=[𝑍⁻¹(𝜔) +𝑖𝜔𝐶]⁻¹ = Ξ(𝜔)𝑍(𝜔), (5.3c) where Re{𝑍_eff(𝜔)} =[𝑍_eff(𝜔) +𝑍_eff(−𝜔)]/2 is the real part of 𝑍_eff(𝜔), 𝛽 is the inverse temperature, 𝜅𝑒 = 1𝑒 ,2𝑒 is the quasi-particle, Cooper-pair charge,𝐸is the energy exchanged between the junction,Ξ(𝜔)is a Lehmann weight³ and𝐿_𝑛𝐶_𝑛 circuits acting as the environment,𝜔is the Fourier transform frequency that also plays the role of the thermal photon frequency at finite temperature.

It is known – at least since the work of Callen and Welton[9] – that the (causal) response function⁴ Ξ(𝜔) ≡∫^+∞

−∞ 𝑑𝑡𝜃(𝑡)𝜒(𝑡) exp(𝑖𝜔𝑡)𝑑𝑡 for a system driven by oscillating electromag-netic fields appears as the coefficient⁵ of the black body spec-trum. Consequently, this requires that the response𝑉⁰

RF(𝑡) as seen by the junction J in Fig. 4.2be a weighted function of𝜒(𝑡)and the applied oscillating voltage𝑉_RF(𝑡):𝑉⁰

RF(𝑡)=

∫ ^𝑡

−∞𝑑𝑠𝜒(𝑡−𝑠)𝑉_RF(𝑠). Therefore, to accurately describe the𝐼– 𝑉 characteristics of J driven by an applied oscillating voltage

88 5. Rescaling of microwave amplitude in small Josephson junctions

𝑉_RF(𝑡), it is not enough to simply rescale the impedance𝑍(𝜔) in the𝑃(𝐸)function: the amplitude and phase of the applied oscillating voltage𝑉_RFhas to be renormalized accordingly.

This point of view and its implications has been discussed in detail in chapter3.

In subsequent sections, we first consider tracing our steps from standard quantum electrodynamics and ease our way into circuit-QED and hence𝑃(𝐸)theory. We then proceed to introduce the finite temperature propagator for the junc-tion and consider how the Lehmann weight arises for the impedance in𝑃(𝐸)theory, and its implications for single junc-tions and long arrays driven by𝑉_RF(𝑡). We find that, a finite time varying fluxΦ(𝑡)stored by the circuit consistently im-plements the aforementioned wavefunction renormalization by guaranteeing the circuit responds linearly to𝑉_RF(𝑡).

Connection of 𝑃(𝐸) theory to quantum electrodynamics (QED).

Here, we shall make the connection of the Caldeira-Leggett model to quantum electrodynamics (QED). We shall find out that circuit-QED is merely the 1 dimensional space time version of QED. This also allows us to link the propagator introduced in eq. (4.43), chapter3with the photon propagator in QED.

The Fourier transform of the summed terms in Caldeira-Leggett action given in eq. (5.2a) is𝑆₀+𝑆_intwhere,

𝑆₀ = 2𝜋 2𝑒²

X

𝑛

∫

𝑑𝜔𝐶_𝑛𝜙^𝑛(𝜔)

𝜔²−𝜔²𝑛

𝜙^𝑛(−𝜔) + 2𝜋

2𝑒2

X

𝑛

1 𝐿_𝑛

∫

𝑑𝜔𝜙^𝑛(𝜔)𝜙⁰(−𝜔) +𝑂(𝜙⁰²), (5.4) 𝑂(𝜙⁰²) is a term with 𝜙⁰² that we initially neglect, 𝜙^𝑛 are the Caldeira-Leggett phases of 𝐿_𝑛𝐶_𝑛 circuits in Fig. 4.2, 𝜔^𝑛 =1/𝐿_𝑛𝐶_𝑛 and the interaction term𝑆_int is given by,

𝑆_int = 2𝜋 2𝑒2

∫

𝑑𝜔𝜔²𝐶𝜙⁰(𝜔)𝜙⁰(−𝜔) + 2𝜋

𝑒

∫

𝑑𝜔𝐼_F(−𝜔)𝜙⁰(𝜔), (5.5)

1. Rescaling of Oscillating Voltages Applied on Single Junctions 89

6: Some of these conditions have been introduced in chapter3in eq. (3.8)

where𝐼_Fis the fluctuation current which we shall later set, 𝐼_F =0.

Integrating out the fluctuating degrees of freedom,𝜙^𝑛 as we did in eq. (4.35) of chapter3,

Y

𝑛

∫

𝐷𝜙^𝑛exp(𝑖𝑆₀)=exp(𝑖𝑆⁰

0), (5.6a) we arrive at,

𝑆⁰

0 = (2𝜋)² 4𝜋

∫

𝑑𝜔𝜙⁰(−𝜔)𝐺⁻¹(𝜔)𝜙⁰(𝜔), (5.6b) 𝐺⁻¹(𝜔)= 1

𝑒2

X

𝑛

1 𝐿_𝑛

𝜔²𝑛

𝜔²−𝜔^2𝑛 ≡𝑖𝜔𝑒⁻²𝑍⁻¹(𝜔), (5.6c) where we have transformed the Green’s function𝐺(𝜔), from the𝜙^𝑛 degrees of freedom, to the environmental impedance 𝑍(𝜔).

Proceeding to combine the two actions yields, 𝑆⁰

0+𝑆_int = 2𝜋 2𝑒²

∫

𝑑𝜔𝜙⁰(𝜔)_𝑒2𝐺−₁(𝜔) +𝜔²𝐶

𝜙⁰(−𝜔) + 2𝜋

𝑒

∫

𝑑𝜔𝐼_F(−𝜔)𝜙⁰(𝜔). (5.7) Note that eq. (5.7) is the action for eq. (3.13) with the Josephson coupling energy𝐸_J =0, already encountered in chapter3.

Moreover, we saw in chapter3that eq. (3.13) is consistent with Maxwell’s equations. In fact, by defining the electromagnetic vector potential𝐴_𝜇=(𝑉 ,𝐴)® , the electric field𝐸®=𝜕𝐴/® 𝜕𝑡−

∇𝑉® , the magnetic field𝐵®=∇ × ®® 𝐴and the fluctuation current density𝐽^𝜇

F, it can be seen that the interaction term given by 𝑆_intabove is actually Maxwell’s action in disguise,

𝑆_int ∝

∫

𝑑𝑡𝑑A𝑑𝑙 h_𝜀0𝜀r

2 ( ®𝐸·𝐸®− ®𝐵·𝐵) +® 𝑒 𝐽^𝜇

F𝐴_𝜇i

= 1 2

∫ _𝑑4𝑘

(2𝜋)4𝜀0𝜀r𝐴^𝜇(𝑘)𝐺⁻_𝜇𝜈¹𝐴_𝜈(−𝑘)+𝑒

∫ _𝑑4𝑘 (2𝜋)4𝐽^𝜇

F(−𝑘)𝐴_𝜇(𝑘), (5.8) with the conditions⁶ 𝐵® = 0, 𝜕𝜙⁰/𝜕𝑡 = −𝑒 𝑙𝑛®·𝐸®, 𝜙⁰ = 𝑒∫ _𝑑𝑙

𝑛®·𝐴®,𝐶 = 𝜀A/𝑙and∫ _𝑑

A𝑛®·®𝐽_F= 𝐼_Fwhere𝑛®=(1,0,0) is the normal vector to the junction barrier,𝑙 ≡ 𝑑_effthe effec-tive barrier thickness andAthe junction area. The last term

90 5. Rescaling of microwave amplitude in small Josephson junctions

7: Actually, the Green’s function takes the form,[89]

𝐺_𝜇𝜈 =lim

𝜀→0

−𝜂𝜇𝜈+𝑘_𝜇𝑘_𝜈/𝜀² 𝑘2−𝜀² .

(5.9)

8: The difference is the dimen-sionality of the theory: QED is in 1 + 2 space-time dimensions while circuit-QED is solely in the time dimension.

9: In self-energy interactions, the photon polarizes the QED vacuum by creating electron-positron pairs which subse-quently annihilate. Such pairs can be created an infinite number of times, thus the contribution to the amplitude of all the processes takes the form:𝐺_eff=𝐺+𝐺𝑈 𝐺+ 𝐺𝑈 𝐺𝑈 𝐺· · · = 𝐺/(1 −𝑈 𝐺) = 1/(𝐺⁻¹−𝑈), where𝐺is the pho-ton propagator and𝑈the vacuum polarization energy (interaction term). e.g. see [89]

is the Fourier transform of the action where𝑘 ≡ 𝑘^𝜇 =(𝜔,𝑘)® is the photon energy-momentum satisfying𝑘² ≡ 𝑘^𝜇𝑘_𝜇 =𝜀² and𝐺_𝜇𝜈(𝑘) ∝ 1/𝑘² is the photon Green’s function in 1 + 3 space-time dimensions.⁷

Integrating out𝜙⁰(𝜔)degrees of freedom in𝑆₀+𝑆_int given by eq. (5.7) as in eq. (5.6a) leads to a circuit-QED term,

𝜋 𝑒²

∫

𝑑𝜔𝐼_F(𝜔)𝐺_eff(𝜔)𝐼_F(−𝜔) (5.10) where 𝐺⁻¹

eff = 𝐺⁻_𝜔¹ + 𝑒⁻²𝜔²𝐶 is reminiscent of the famous Coulomb interaction term in QED,

𝛼

∫ _𝑑4𝑘 (2𝜋)3𝐽^𝜇

F(𝑘)𝐺_𝜇𝜈(𝑘)𝐽^𝜈

F(−𝑘), (5.11) where 𝛼 = 𝑒²/4𝜋𝜀0𝜀r is the fine structure constant. The QED term is obtained in a similar fashion by integrating out 𝐴_𝜇 instead of 𝜙⁰. Nonetheless, both expressions are essen-tially describing the same process.⁸ Thus, we have showed that 1/𝜔²𝐶 and hence𝐺(𝜔)are the photon propagator in 1 dimensions.

However, a question still remains: is there any significance of this trivial Fourier space transformation given by𝐺⁻¹(𝜔) → 𝐺⁻¹

eff(𝜔)? We notice that we can define a factor Ξ(𝜔) = 𝐺(𝜔)/𝐺_effand claim that this factor renormalizes the prop-agator𝐺(𝜔)(of the𝜙^𝑛 degrees of freedom) to𝐺_eff(𝜔)due to the presence of the Maxwell term,𝑆_int. Since this renor-malization takes a photon propagator into a different photon propagator, the Feynman rules to calculate it resemble photon self-energy interactions.⁹ Bearing this in mind, we formulate the following Feynman rules for the propagator:

1. The photon propagator 𝐺(𝜔) = 𝑒²𝑍(𝜔)/𝑖𝜔 is repre-sented by: ;

2. The vacuum polarization energy term,𝑈(𝜔)= 𝑒⁻²𝜔²𝐶 is represented by: ;

3. Therefore, the leading order interaction term, 𝐺(𝜔)𝑈(𝜔)𝐺(𝜔)is drawn as, , where time increases from left to right.

Note that diagram 1 reads as follows: A photon of energy 𝜔is created, propagates with a probability amplitude𝐺(𝜔) and annihilates at a later time. Thus the amplitude must

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