Sauter-Schwinger effect in a Bardeen-Cooper-Schrieffer superconductor
SSchwinger effect in a Bardeen-Cooper-Schrieffer superconductor
P. Solinas,
1, 2
A. Amoretti,
1, 2 and F. Giazotto Dipartimento di Fisica, Universit`a di Genova, via Dodecaneso 33, I-16146, Genova, Italy INFN - Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy NEST, Instituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
From the sixties a deep and surprising connection has followed the development of superconduc-tivity and quantum field theory. The Anderson-Higgs mechanism and the similarities between theDirac and Bogoliubov-de Gennes equations are the most intriguing examples. In this last analogy,the massive Dirac particle is identified with a quasiparticle excitation and the fermion mass energywith the superconducting gap energy. Here we follow further this parallelism and show that it pre-dicts an outstanding phenomenon: the superconducting Schwinger effect (SSE). As in the quantumelectrodynamics Schwinger effect, where an electron-positron couple is created from the vacuum byan intense electric field, we show that an electrostatic field can generate two coherent excitationsfrom the superconducting ground-state condensate. Differently from the dissipative thermal exci-tation, these form a new macroscopically coherent and dissipationless state. We discuss how thesuperconducting state is weakened by the creation of this kind of excitations. In addition to shed adifferent light and suggest a method for the experimental verification of the Schwinger effect, our re-sults pave the way to the understanding and exploitation of the interaction between superconductorsand electric fields.
Among the unexpected predictions of the Dirac equa-tion there is the existence of antiparticles and the factthat the vacuum consists in a dynamical equilibriumbetween particles and antiparticles, which are continu-ously created and annihilated. This is at the origin ofthe Schwinger effect. Originally discovered by Sauter,Heisenberg and Euler [1, 2], the Schwinger effect refersto the creation of an electron and a positron pair fromthe QED vacuum as a consequence of its instability un-der the presence of an external electric field (see Fig. 1 a)[3]. After the development of quantum electrodynamics(QED), in 1951 Schwinger gave a complete treatment ofthe effect [3]. He computed the critical electric field abovewhich the vacuum becomes unstable thereby resulting inthe creation of an electron-positron pair. Despite beingpredicted almost seventy years ago, the Schwinger effecthas never been observed so far because of the ultra-highelectric fields ( ∼ V/m) needed.In the sixties, Nambu and Jona-Lasinio noticed astriking similarity between the Dirac equation and theBogoliubov-de Gennes equations describing the elemen-tary excitations in a superconductor [4–8]. They are for-mally identical if one identifies the Dirac particle witha quasiparticle excitation in the superconductor and thefermion mass with the superconducting energy gap [7].In the spirit of the superconductor-QED analogy [4, 7],we would expect to observe a kind of vacuum instability,and a Schwinger effect in superconductors exposed to anelectric field. Notably, if we replace the electron massenergy (0 . ∼ µ eV-1 meV for a conventional superconductors),we await to drastically reduce the critical electric field ( ∼ V/m) needed to activate the SSE. In this perspective,this makes superconductors ideal candidates to realizeand measure the Schwinger effect.Despite the appeal and the strength of the superconductor-Dirac particle analogy, in realistic im-plementations and experiments we must keep in mindthe important differences between the two physical sys-tems. First, the superconducting phase is a macroscopicquantum state that lives in a complex environment. Inmetallic superconductors (i.e., the ones well-described bythe Bardeen, Cooper, and Schrieffer (BCS) theory [8])screening effects can limit the penetration of the elec-tric field in the interior of the system [9]. Secondly,it is known that a superconductor immersed in an ex-ternal electric field must be treated like an open quan-tum system, as the effects of the environment and dis-sipation processes become relevant [10, 11]. Regardlessof their importance, a detailed analysis of these pointswould complicate the discussion and drain the attentionaway from the real focus of the paper. For these reasons,we constraint our discussion to a very specific situation.We consider a film superconductor thin enough to becompletely penetrated by the electric field (or, alterna-tively, we refer to the effect on the edge of a superconduc-tor), and we analyse the unitary evolution arguing thatthe environment can affect the dynamics only on longertimescales. These assumptions allow us to focus on amore precise question: can a static electric field inducea Schwinger-like effect in a BCS superconductor by ex-citing the condensate ground state?
Below we will showthat the answer is indeed affirmative.Our starting point is the effective Hamiltonian describ-ing a standard BCS superconductor in the presence of anexternal electric field [8, 12]. We assume that the electricfield E f = { , , E f } is applied to a thin film supercon-ductor along the z direction (see Fig. 1b), and that thesuperconductor thickness L allows full electric field pene-tration in the sample. By decomposing the fermion fieldin plane waves and choosing the proper gauge, we obtainthe effective Hamiltonian in second quantization formal- a r X i v : . [ c ond - m a t . s up r- c on ] J u l a b BCS Superconductor
Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " (2)1 Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " L (2)1 Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lz (2)0 . . .
55 (3)1
Schwinger QEDQED vacuume + e ~Emc (1) Superconductorsingle excitationCondensateh + e k k " (2)1 Schwinger QEDQED vacuume + e ˜E (1)1 Schwinger QEDQED vacuume + e ˜E (1)1 QED SchwingereffectQED vacuum
Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " (2)1 Schwinger QEDQED vacuume + e ˜E (1)1 Schwinger QEDQED vacuume + e ~E (1) SCsingleexcitationCondensateh + e k k " (2)1 Schwinger QEDQED vacuume + e ~E (1) SCsingleexcitationCondensateh + e k k " (2)1 Condensate
Schwinger QEDQED vacuume + e ~E (1) Superconductorsingle excitationCondensateh + e k k " (2)1 BCS SuperconductorSchwinger effect
Schwinger QEDQED vacuume + e ~Emc (1) Superconductorsingle excitationCondensateh + e k k " (2)1 Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " (2)1 FIG. 1.
BCS superconductor Schwinger effect: a ,Analogy between the Schwinger effect in QED and in a BCSsuperconductor. In QED (left panel), the electromagneticvacuum is excited by an electric field E f leading to the cre-ation of an electron and a positron. In a BCS superconductor(right panel), the electric field excites a couple of quasipar-ticles with momentum and spin ( k , ↑ ) and ( − k , ↓ ) from thecondensate, which acts as the vacuum. The excited quasipar-ticles are still paired and correlated, and preserve the dissipa-tionless character of the ground state. Moreover, they starklydiffer from usual thermal single quasiparticle excitations thatare related to the destruction of a Cooper pair. b , Sketch ofthe physical system. A BCS superconducting film of thick-ness L is subject to an electric field E f . We consider L to besmall enough to assure a complete penetration of the electricfield. ism [12–15] H eff = (cid:88) k (cid:110) h k − ( a † k ↑ a k ↑ + a † k ↓ a k ↓ ) − ∆ a † k ↑ a †− k ↓ − h.c. (cid:111) , (1)where h k ± = m (cid:104) (cid:126) k ⊥ +( (cid:126) k z ∓ eE f t ) (cid:105) − µ , k ⊥ = k x + k y , µ is the chemical potential and m is the electron mass.The order parameter is ∆ = V (cid:80) k (cid:104) a k ↑ a − k ↓ (cid:105) and, underthis gauge choice, it acquires a superconducting phase∆ = | ∆ | e iχ with χ = e (cid:126) E f tz [8, 11, 12]. Notice thatthe phase factor depends on z so does the Hamiltonian(1), which is in the standard BCS form thereby allowinga great simplification. The price to pay is, however, todeal with a time-dependent problem.As in the homogeneous case, only ( k , ↑ ) and ( − k , ↓ )are coupled. Separating the negative k contributions inthe kinetic terms of Eq. (1), we have h − k − = h k + [12] and, as usual, reversing the momentum is equivalent tochange the sign of the particle charge. Defining ξ k ( t ) =( h k − + h k + ) / k = k ⊥ + k z ),we can cast Eq. (1) into a matrix form by means of theAnderson pseudospin representation [13–17]: H eff = 2 (cid:88) k (cid:18) ξ k − ∆ − ∆ ∗ − ξ k (cid:19) = 2 (cid:88) k B k · Σ k = 2 (cid:88) k H k , (2)where B k = {− Re(∆) , − Im(∆) , ξ k } is a pseudomagneticfield and Σ k = { τ x, k , τ y, k , τ z, k } is the Pauli operator vec-tor [12–14].In the QED Schwinger effect, electrons and positronscan be viewed as excitations of the vacuum induced bythe presence of an electric field. In a similar way, in thesuperconducting Schwinger effect we expect excitationsof the condensate, namely quasiparticles, to be generatedby the presence of the electric field. This implies thatthe proper basis to highlight the SSE and the creation ofquasiparticles is the one that diagonalizes (2) [18].Even though H k is time dependent, it can be diago-nalized as in the standard homogeneous case [8] by in-troducing the usual (now time-dependent) operators thatcreates and annihilates the excitations γ k ↑ = u k ( t ) a k ↑ − v k ( t ) a †− k ↓ and γ †− k ↓ = v ∗ k ( t ) a k ↑ + u ∗ k ( t ) a †− k ↓ [8, 12, 19].The eigenvalues are ± (cid:15) k ( t ) = ± (cid:112) ξ k ( t ) + | ∆ | and theground and the excited states, expressed in in the orig-inal { a k ↑ , a †− k ↓ } basis, are | ψ k, − ( t ) (cid:105) = { v k ( t ) , u k ( t ) } and | ψ k, + ( t ) (cid:105) = { u ∗ k ( t ) , − v ∗ k ( t ) } respectively, with u k = √ (cid:113) ξ k (cid:15) k e − iχ/ and v k = √ (cid:113) − ξ k (cid:15) k e iχ/ [12].If U k ( t ) is the time-dependent unitary transformationthat diagonalizes H k , namely U † k H k U k = H D,k , the dy-namics is determined by the Schr¨odinger equation i (cid:126) ∂ t | φ k ( z ) (cid:105) = ( H D,k − i (cid:126) U † k ∂ t U k ) | φ k ( z ) (cid:105) . (3)The contribution U † k ∂ t U k induces the transition betweeneigenstates of H D,k that are associated to the excitationof the ground state and the creation of quasiparticles.There is a crucial difference between these kind of excita-tions and the well known thermal ones. The off-diagonalterms in the operator U k ∂ t U † k are associated to γ † k ↑ γ †− k ↓ and γ k ↑ γ − k ↓ [12]. Therefore, U † k ∂ t U k creates or annihi-lates simultaneously two quasiparticles with ( k , ↑ ) and( − k , ↓ ). This is different from the creation of a singlequasiparticle through the operators γ † k ↑ and γ †− k ↓ exten-sively discussed in textbooks [8, 20]. While a single ex-citation destroys a Cooper pair, double excitations pre-serve the coherent interaction of the pair and, in thissense, they still possess superconducting properties, asdiscussed below [19, 21].The pairing potential in Eqs. (2) is ∆ = (cid:80) k ∆ k ,and is calculated self-consistently during the dynamics[12]. In the numerical simulations, we set µ = 1 eVand the initial pairing potential to ∆ = 100 µ eV [22–26]. Moreover we assumed the working temperature to a bc d � / � �� ✏ k ( t ) | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . ✏ k ( t ) | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . ✏ k ( t ) | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . ✏ k ( t ) ✏ k ✏ k | k , | | k , + | | k , | , | k , + | | k , + ( t m a x ) | .
99 0 .
999 1 ( ) / k , x , k , y , | k | . ( ) Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
55 (3)1 � / � �� ✏ k ( t ) ✏ k ✏ k | k , | | k , + | | k , | , | k , + | | k , + ( t m a x ) | .
99 0 .
999 1 ( ) / k , x , k , y , | k | . ( ) Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
55 (3)1 - - � / � �� ✏ k ( t ) ✏ k ✏ k | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
55 (3)1 � / � � ✏ k ( t ) | k , | | k , + | | k , | , | k , + | | k , + ( t m a x ) | .
99 0 .
999 1 ( ) Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
51 (3)1
Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
51 (3)1
Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
51 (3)1
Schwinger QEDQED vacuume + e ~Emc E f (1) Superconductorsingle excitationCondensateh + e k k " 2 Lzt min (2)0 . . .
51 (3)1
FIG. 2.
Superconducting Schwinger effect: a , Spectrum of the superconductor as a function of time t . The minimalenergy gap 2∆ is reached for t = t min . The arrows show the effect of the Landau-Zener excitation: the k -th mode initially inthe ground state (cyan) can undergo a transition to the excited state (blue) at the avoided level crossing. The inset shows azoom of the minimal gap region. b , Populations of the ground (cyan) and excited state (blue) as a function of time. Startingfrom the ground state the system is excited when the minimal energy is reached. c , Population of the excited state as a functionof time for k/k F = 0 . , .
999 and 1. In the panels a and b numerical simulations are performed setting k/k F = 0 .
999 and in a , b and c setting E f /E C = 0 . d , Final population (at t = t max ) of the excited state calculated as a function of k/k F forselected electric fields values, E f /E C = 0 . , . , . ,
1. Here the evolution time is the same for all the k and it is the maximumtime allowed in the model, i.e., associated to the minimum k . Simulations are performed setting µ = 1 eV, ∆ = 100 µ eV and L = 2 nm and z/L = 0 .
5. The same dynamical features are observed for any z/L . be much smaller than superconducting critical tempera-ture T C , eventually neglecting thermal excitations. Forlater convenience, we introduce two reference scales: atime scale t sc = (cid:126) /µ and a characteristic electric field E C = 5 × V / m.We suppose that a constant electric field is applied attime t in = 0, and the dynamical transients are negligible.The numerical simulations are performed in a finite timeinterval 0 < t < t max , where t max = mL/ ( (cid:126) k ) is thetime needed for a particle of mass m and momentum k to move from one side to the opposite of a sample ofthickness L (see Figure 1a). This sets the time scale forthe numerical simulations. In the latter, we set L = 2 nm[27], so that we can assume a complete penetration of theelectric field [28].The simulated spectrum of the Hamiltonian (2) as afunction of time is plotted in Fig. 2a. The minimumgap 2∆ is reached when the kinetic energy ξ k in Eq. (1) vanishes, namely for t min = √ µm (cid:112) − ( k/k F ) eE f . (4)The dynamics of the populations of the ground andexcited state | ψ k, − | and | ψ k, + | is shown in figure 2b fora fixed k/k F and E f /E C , and presents a clear signatureof the SSE. The k -th mode undergoes a sudden transitionto the excited state close to the minimal energy gap. Thiscorresponds to the superconducting Schwinger effect, andto the creation of two excited quasiparticles, as discussedabove.The dynamics changes considerably for different initialparticle momenta, as shown in Fig. 2c. Away from theFermi momentum, there is no quasi-particle excitationbut moving closer to k F the system is completely excited.In a small window very close to k F the system is onlypartially excited.A more complete picture can be inferred from Fig.2d where the final population of the excited state | ψ k, + ( t max ) | is shown as a function of the momentumfor different normalized electric fields E f /E C . For smallelectric field ( E f /E C = 0 . k F are excited. By increasing theelectric field strength, the excited population fraction in-creases up to a complete excitation for any k ≤ k F and E f /E C = 1. These results suggest that the electric fieldat which the excitations are produced is indeed close to E C and is remarkably similar to the one used in severalrecent experiments [22–26].To understand how the characteristic electric field E C appears naturally in the SSE, we have to reinterpret theexcitation production as a Landau-Zener transition [18,29–31]. The evolution of the system can be divided intothree different regions; an initial one at t = 0 where thedynamics is mostly dominated by the kinetic term, i.e. ξ k (0) (cid:29) ∆ and H k (0) ≈ ξ k (0) τ z ; an intermediate regionwith t ∼ t min , in which, on the contrary, ξ k (0) (cid:28) ∆and H k ( t min ) = ∆ τ x ; and eventually a late time region, t (cid:29) t min , where the kinetic energy dominates again and H k ( t ) ≈ ξ k ( t ) τ z [12]. This is the general scheme of theLandau-Zener model, where the transitions are producedwhen the system crosses the minimum energy gap withhigh energy velocity V k = √ eE (cid:113) − ( k/k F ) m µ [31]. Asshown in [12] the relevant parameter is Γ k = π ∆ V k (cid:126) = . × − E f /E C √ − ( k/k F ) . Since the Landau-Zener transitionsoccur when Γ k (cid:28) k . However, we have to take intoaccount that the Landau-Zener model cannot be appliedvery close or far away from the Fermi momentum.For initial momentum very close to k F , i.e., 1 − k/k F ≤ − , the pairing energy dominates at t = 0, i.e., H k (0) ≈ ∆ τ x , and the Landau-Zener model cannot be applied di-rectly. It turns out that the dynamics is frozen, resultingin a partial population of the excited state as in Fig. 2c[12]. In the opposite case, i.e., for initial momentum awayfrom k F , the electric field has not enough time to inducethe transition to the excited state or, in other terms,the minimal energy gap is reached later than t max . Thissuggests to use the condition t min ≤ t max to approxi-mate the electric field needed to produce the excitations: E f /E C = (2 µ ) / ( eE C L ) [12]. For L = 2 nm, we obtain E f = 2 E C , namely close to E C , as discussed before.The above analysis confirms the analogy between theQED and superconductivity, and holds also when we con-sider the vacuum instability and the Schwinger effect.Yet, while in QED the vacuum structure is unaffected bythe creation of particles, in a superconductor the creationof excitations can affect its electronic properties.We stress once again that the double excitation gener-ated by the electric field are deeply different from thesingle excitation due, for example, to thermal effects.The latter is related to breaking of a Cooper pair, and leads to an emptying of the pairing potential. Moreformally, if a single k excitation is produced, the state | ψ k ↑ (cid:105) = γ † k ↑ | BCS (cid:105) (where | BCS (cid:105) = Π k | ψ k, − (cid:105) is the BCSground state) does not give contribution to the pairingpotential, since ∆ k = (cid:104) ψ k ↑ | a k ↑ a − k ↓ | ψ k ↑ (cid:105) = 0.By contrast, the two excitations produced by theelectric field are still correlated, and contribute to thepairing potential but with opposite sign with respectto their contribution to the ground state. More pre-cisely, ∆ GS,k = (cid:104) BCS | a k ↑ a − k ↓ | BCS (cid:105) = u k v ∗ k , ∆ EX,k = (cid:104) ψ k, + | a k ↑ a − k ↓ | ψ k, + (cid:105) = − u k v ∗ k [12, 21]. This has two ma-jor implications. First, a fully excited state would bestill superconducting, preserving all the spectral prop-erties of the ground state, since ∆ EX = (cid:80) k ∆ EX,k = − ∆ GS . Secondly, the additional minus sign accumu-lated in ∆ EX,k can be interpreted as π -shift of the su-perconducting phase that, for the excited state, becomes e i ( χ + π ) . Even though this phenomenon has been dis-cussed in a abstract way in a few books [19, 21], this is,to our knowledge, the first time that these elusive corre-lated excitations could be related to macroscopic effects,and indirectly observed. In this direction, it is importantto understand how superconducting features are changedby their presence. If the k -th mode of the ground state isexcited, its negative contribution to ∆ can decrease thepairing potential. We expect a weakening of supercon-ductivity related to these interferences, even though it isworth to mention that in a more general picture the k -thmode could be in a coherent superposition of ground andexcited states.In the Anderson pseudo-spin formalism, the order pa-rameter for the k -th mode is ∆ k = ∆ k,x + i ∆ k,y = (cid:104) τ x (cid:105) + i (cid:104) τ y (cid:105) where the average (cid:104)(cid:105) is calculated with stateobtained by the dynamical evolution [13, 14, 16, 17]. Thenumerical calculation displayed in Fig. 3a shows thatwhile | ∆ k | is constant, ∆ k,x and ∆ k,y change in time sig-naling an accumulated phase. The pairing potential at t = t max is shown in Fig. 3b for different electric fieldvalues. As the electric field increases the pairing poten-tial is reduced because of the interference effects. Westress again that the effect of the environment should beincluded in the model in order to have a more quantita-tive picture of the impact of the electric field. Howeverour analysis gives strong indications that the presence ofa static electric field drastically weakens superconductiv-ity.Finally it is worth to mention that the SSE should beassociated with non-equilibrium phenomena which canbe eventually measured. To understand this point weanalyse the simplified master equation shown schemat-ically in Fig. 3 (see [12] for technical details). There,dissipative effects with an external environment are in-troduced [10, 11] assuming that they can occur in twodifferent ways: momentum scattering leading a reductionof kinetic energy, which is fairly well understood [32], andthe destruction of the excited states with no momentum � / � �� a ✏ k ( t ) ✏ k ✏ k | k , | | k , + | | k , | , | k , + | | k , + ( t m a x ) | .
99 0 .
999 1 ( ) / k , x , k , y , | k | . ( ) b Double excitation Single excitationGround state = 1 k F /k D = 0 . = 0 . E f = 1 (1) eg ge se es sg gs (2)1 = 1 k F /k D = 0 . = 0 . E f = 1 (1) eg ge se es sg gs (2)1 = 1 k F /k D = 0 . = 0 . E f = 1 (1) eg ge se es se,es sg gs sg,gs (2)1 = 1 k F /k D = 0 . = 0 . E f = 1 (1) eg ge se es se,es sg gs sg,gs (2)1 c � / � ✏ k ( t ) ✏ k ✏ k | k , | | k , + | | k , | , | k , + | | k , + ( t m a x ) | .
99 0 .
999 1 ( ) | | / k , x , k , y , | k | . ( ) ✏ k ( t ) ✏ k ✏ k | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . | | / k,x , k,y , | k | . . .
41 (5) eg ge se es se,es sg gs sg,gs (6)2 ✏ k ( t ) ✏ k ✏ k | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . | | / k,x , k,y , | k | . . .
41 (5) eg ge se es se,es sg gs sg,gs (6)2 ✏ k ( t ) ✏ k ✏ k | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . | | / k,x , k,y , | k | . . .
41 (5) eg ge se es se,es sg gs sg,gs (6)2 ✏ k ( t ) ✏ k ✏ k | k, | | k, + | | k, | , | k, + | | k, + ( t max ) | . . | | / k,x , k,y , | k | . . .
41 (5) eg ge se es se,es sg gs sg,gs (6)2 FIG. 3.
Weakening the superconductivity and dissipative master equation: a , Time evolution of ∆ k,x = Re(∆ k )(blue), ∆ k,y = Im(∆ k ) (cyan) and | ∆ k | = (cid:113) ∆ k,x + ∆ k,y (green) for k/k F = 0 .
999 and E f /E C = 1. The order parameteraccumulates a phase during the dynamics but | ∆ k | is constant. b , The global order parameter ∆ normalized to the initial one∆ as a function of the normalized position in the superconductor z/L , and for different electric fields E f /E C = 0 . , . , . , c , The transition scheme associated to the dissipative dynamics of a generic momentum mode. The ground state is excitedwith rate Γ eg due to the effect of the electric field since this is the only non-thermal transition is denoted with a red arrow.The double excited can also decay directly to the ground state with rate Γ ge . Additional relaxation channels are the one dueto the destruction of the Cooper pairs lead to the single excited state. The latter is connected by thermal transition to boththe ground and the double excited state. change, which is the relevant process for the generationof non-equilibrium features. Eventually, reasonably as-suming that the time scales associated with these twoprocesses are well separated allows us to focus only onthe second one.A laser analogy can help to understand how non-equilibrium, i.e., non-thermal, distributions can arise inthis contest. As in a laser, the electric field acts as an ex-ternal pumps that excites the ground state directly to thedouble-excited state. This is unstable and can decay di-rectly to the ground state or through the single excitationstate with the Cooper pair destruction. In this way, thestandard single excitation are populated. The balancebetween the energy pumping due to the electric field, thedissipation of kinetic energy through momentum scatter-ing and relaxation will eventually lead to a steady state.However, while the transitions by and from the singleexcitation state are thermal, the transition between theground and double excited state is not. Therefore, as ina laser, the steady state is not related to a thermal ornon-equilibrium distribution [12].The fact that the single-excitation state are, in general,non-thermal, opens the way to a direct measure of theSSE through tunnel spectroscopy [20]. When two super-conductors with different gap, i.e., ∆ and ∆ with ∆ < ∆ , are connected with a tunnel junction and subject to avoltage V , at finite temperature T the I − V curve shows aresonance current peak at e V = | ∆ − ∆ | [20]. This is anindication of the presence of quasi-particle in the super-conductors. If ∆ , ∆ (cid:28) k B T and E f = 0, the thermalexcitation should be negligible and no current peak at | ∆ − ∆ | should be observed. If we apply a static elec-tric to the superconductor with gap ∆ , according to the SSE model with dissipation discussed above, the increasein E f should generate double and single excitation andthis should results in a current resonance at | ∆ − ∆ | .More importantly, the behaviour of the resonance peakas a function of E f could provide evidence of the non-thermal distribution of the excited quasi-particle.The SSE and its manifestations should be observablewith currently available laboratory techniques. The fab-rication of a single-layer FeSe superconductor has beenreported [27, 33]. For such thin films the electric fieldis expected to penetrate completely in the superconduc-tor [28] making them a perfect test-bench to confirm thepresence of the SSE. With thicker structures, the presentmodel should describe the physics on the edges of thesuperconductor before the damping due to the screen-ing effects. In this case, a detailed study of the non-homogeneous electric field is needed. However, due tothe spatial extension of the Cooper pairs that are excited,the perturbation on the edges could affect the supercon-ductor up to the coherence length. This would lead tomeasurable effects also in thicker superconductors.Indeed, the described phenomenology is compatiblewith recent experiments performed on superconductingwires and nanobridges subject to strong electric fields.The observed weakening of superconductivity, the invari-ance of the electric field direction [22–26, 34], and the cre-ation of non-thermal switching supercurrent distributions[35] are compatible with and could be the manifestationof the creation of exotic paired excitations.Surprisingly, our simple model predicts, with no fittingparameters, that the Schwinger effect in superconductorsshould manifest itself in the presence of an electric field ofthe order E C ∼ V / m, which is in striking agreementwith those used in these experiments [22–26, 36, 37]. ACKNOWLEDGEMENTS
The authors acknowledge S. Paraoanu and N. Magnolifor fruitful discussions. PS and AA acknowledge financialsupport from INFN. AA has been partially supported bythe INFN Scientific Initiative SFT: Statistical Field The-ory, Low-Dimensional Systems, Integrable Models andApplications. FG acknowledges the Horizon 2020 inno-vation programme under Grant Agreement No. 800923-SUPERTED for partial financial support. [1] F. Sauter, Zeitschrift f¨ur Physik , 742 (1931).[2] W. Heisenberg and H. Euler, Zeitschrift f¨ur Physik ,714 (1936).[3] J. Schwinger, Phys. Rev. , 664 (1951).[4] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345(1961).[5] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 246(1961).[6] Y. Nambu, Phys. Rev. , 648 (1960).[7] I. J. R. Aitchison and A. J. G. Hey, Gauge theories inparticle physics: A practical introduction. Vol. 2: Non-Abelian gauge theories: QCD and the electroweak theory (CRC Press, Bristol, UK, 2012), ISBN 9781466513075.[8] P. De Gennes,
Superconductivity Of Metals And Alloys ,Advanced Books Classics (Westview Press, 1999), ISBN9780813345840.[9] P. Virtanen, A. Braggio, and F. Giazotto, Phys. Rev. B , 224506 (2019).[10] N. Kopnin,
Theory of Nonequilibrium Superconductivity ,International series of monographs on physics (ClarendonPress, 2001), ISBN 9780191709722.[11] N. B. Kopnin, Journal of Low Temperature Physics ,219 (2002).[12] P. Solinas, A. Amoretti, and F. Giazotto,
Supplementaryinformation (2020).[13] S. Tsuchiya, D. Yamamoto, R. Yoshii, and M. Nitta,Phys. Rev. B , 094503 (2018).[14] N. Tsuji and H. Aoki, Phys. Rev. B , 064508 (2015).[15] T. Lancaster and S. J. Blundell, Quantum field theory forthe gifted amateu (Oxford University Press, 2014).[16] P. W. Anderson, Phys. Rev. , 1900 (1958).[17] R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka,K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, andR. Shimano, Science , 1145 (2014), ISSN 0036-8075.[18] T. D. Cohen and D. A. McGady, Phys. Rev. D , 036008(2008).[19] J. Schrieffer, Theory Of Superconductivity , AdvancedBooks Classics (Avalon Publishing, 1999), ISBN9780738201207.[20] M. Tinkham,
Introduction to superconductivity (CourierDover Publications, 2012).[21] A. Leggett,
Quantum Liquids: Bose Condensation andCooper Pairing in Condensed-matter Systems , Oxfordgraduate texts in mathematics (OUP Oxford, 2006),ISBN 9780198526438. [22] G. De Simoni, F. Paolucci, C. Puglia, and F. Giazotto,arXiv preprint arXiv:1903.03435 (2019).[23] G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, andF. Giazotto, Nature Nanotechnology , 802 (2018).[24] F. Paolucci, F. Vischi, G. De Simoni, C. Guarcello,P. Solinas, and F. Giazotto, Nano Letters , 6263(2019).[25] F. Paolucci, G. De Simoni, E. Strambini, P. Solinas,and F. Giazotto, Nano Letters , 4195 (2018), pMID:29894197.[26] F. Paolucci, G. De Simoni, P. Solinas, E. Strambini,N. Ligato, P. Virtanen, A. Braggio, and F. Giazotto,Phys. Rev. Applied , 024061 (2019).[27] D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang,Z. Li, L. Wang, L. Zhao, S. He, et al., Nature Communi-cations , 931 (2012).[28] E. Piatti, D. Daghero, G. A. Ummarino, F. Laviano, J. R.Nair, R. Cristiano, A. Casaburi, C. Portesi, A. Sola, andR. S. Gonnelli, Phys. Rev. B , 140501 (2017).[29] L. Landau, Physikalische Zeitschrift der Sowjetunion ,46 (1932).[30] C. Zener, Proceedings of the Royal Society of London A , 696 (1932).[31] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports , 1 (2010), ISSN 0370-1573.[32] F. Giazotto, T. T. Heikkil¨a, A. Luukanen, A. M. Savin,and J. P. Pekola, Reviews of modern physics , 217(2006).[33] Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S.Zhang, W. Li, H. Ding, Y.-B. Ou, P. Deng, K. Chang,et al., Chinese Physics Letters , 037402 (2012).[34] M. Rocci, G. De Simoni, C. Puglia, D. D. Esposti,E. Strambini, V. Zannier, L. Sorba, and F. Giazotto,arXiv preprint arXiv:2006.07091 (2020).[35] C. Puglia, G. De Simoni, and F. Giazotto, Phys. Rev.Applied , 054026 (2020).[36] L. D. Alegria, C. G. Bøttcher, A. K. Saydjari, A. T.Pierce, S. H. Lee, S. P. Harvey, U. Vool, and A. Yacoby,arXiv preprint arXiv:2005.00584 (2020).[37] M. Ritter, A. Fuhrer, D. Haxell, S. Hart, P. Gumann,H. Riel, and F. Nichele, arXiv preprint arXiv:2005.00462(2020).[38] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys.Rev. , 162 (1957). Supplemental Information
GENERAL FRAMEWORK
The general effective Hamiltonian describing a standard superconductor is [S8] H eff = (cid:90) d r (cid:110) (cid:88) α (cid:104) Ψ † ( α r ) H e ( r )Ψ( α r ) + U ( r )Ψ † ( α r )Ψ( α r ) (cid:105) + ∆( r )Ψ † ( r ↑ )Ψ † ( r ↓ ) + ∆ ∗ ( r )Ψ( r ↓ )Ψ( r ↑ ) (cid:111) (S1)where α is the spin index, Ψ is the fermionic field satisfying the usual anti-commutation rules and, with V set as acoupling energy, [S8] ∆( r ) = − V (cid:104) Ψ( r ↓ )Ψ( r ↑ ) (cid:105) = V (cid:104) Ψ( r ↑ )Ψ( r ↓ ) (cid:105) U ( r ) = − V (cid:104) Ψ † ( r ↑ )Ψ( r ↑ ) (cid:105) = − V (cid:104) Ψ † ( r ↓ )Ψ( r ↓ ) (cid:105) (S2)are the self-consistent pair potential and the Hartree-Fock potential, respectively.The single particle Hamiltonian operator is rescaled over the Fermi energy (chemical potential) µ and it reads H e ( r ) = 12 m (cid:16) − i (cid:126) ∇ − ec A (cid:17) + U ( r ) − µ (S3)where A is the electromagnetic vector potential and U ( r ) is a scalar potential independent on the particle spin.We consider thin superconducting wires and with limited screening so that the electric field penetrates the super-conductor and it is constant inside it. Alternatively, this model can describe the effect of the electric field on the edgeof a metallic superconductor. The electric field E f is applied to a superconductor along the, say, z direction; i.e., theelectric field vector is E f = { , , E f } . Under these hypothesis, we have A = 0 and U ( r ) = eE f z . However, by agauge transformation, we can set U ( r ) = 0 and A = { , , − cE f t } .We expand the fermionic fields in Eq. (S1) asΨ( r α ) = (cid:88) k e i k · r a k α Ψ † ( r α ) = (cid:88) k e − i k · r a † k α . (S4)By performing the spatial integration we arrive at [S13–S15] H eff = (cid:88) k (cid:110) h k − ( t )( a † k ↑ a k ↑ + a † k ↓ a k ↓ ) − ∆ a † k ↑ a †− k ↓ − ∆ ∗ a k ↑ a − k ↓ (cid:111) (S5)where we have included the U ( r ) contribution in the redefinition of µ and put k ⊥ = k x + k y . The pairing potentialreads ∆ = V (cid:80) k (cid:104) a k ↑ a − k ↓ (cid:105) [S13–S15].It is convenient to simplify the notation but, at the same time, keep track of the presence of the vector potential A . For this reason, we introduce the kinetic energy h k − ( t ) = h k − e (cid:126) c A ( t ) = m (cid:104) (cid:126) k ⊥ + ( (cid:126) k z + eE f t ) (cid:105) − µ .In the sum in Eq. (S5) both positive and negative k contributions are present. We can separate the negative termslike h − k − a †− k ↑ a − k ↑ . We have h − k − = h − k − e (cid:126) c A ( t ) = 12 m (cid:104) (cid:126) k ⊥ + ( (cid:126) k z − eE f t ) (cid:105) − µ = h k + (S6)Thus, formally reversing the momentum is equivalent to change the charge to the particle.The superconductor pair potential can be written as ∆ = | ∆ | e iχ where χ is the superconduting phase. It is relatedto the gauge-invariant scalar φ and vector A potentials by the equations [S11] A = A − (cid:126) c e ∇ χφ = V + (cid:126) e ∂χ∂t . (S7)These are related to the physical electric E and magnetic field h by the relations [S11] E = − c ∂ A ∂t − ∇V h = ∇ × A . (S8)By setting φ = 0 and A = 0, i.e., no magnetic field, we obtain χ = 2 e (cid:126) E f t z (S9)and A = { , , − cE f t } as above. Therefore, the superconducting phase, the pairing potential (S2) and the Hamiltonian(S5) depends on the spatial coordinate z .This gauge choice allows us to deal with a homogeneous problem where the spatial dependence has vanished inEq. (S5). This is a great simplification because allows to use the standard approach and techniques to describethe superconducting state and dynamics. The price to pay for this simplification is to deal with a time-dependentHamiltonian so that we are forced to solve the time-dependent dynamics. Because the problem is homogeneous, onlythe ( k , ↑ ) and ( − k , ↓ ) are coupled. This make the problem easily solvable but numerically and analytically.We can collect the terms in Eq. (S5) separating the k and the − k contributions. By using the state Φ = { a k ↑ , a †− k ↓ } ,the relation h − k − = h k + and the anti-commutation rules for fermionic operators a † k α and a k α , we can rewrite Eq.(S5) in matrix form as [S13–S15] H eff = 2 (cid:88) k (cid:18) ξ k − ∆ − ∆ ∗ − ξ k (cid:19) = 2 (cid:88) k B k · Σ k = 2 (cid:88) k H k (S10)where ξ k = h k − + h k + (cid:126) k m + e E f t m − µ, (S11) B k = {− Re(∆) , − Im(∆) , ξ k } is a pseudo-magnetic field and Σ k = { τ x,k , τ y,k , τ z,k } . This is nothing but the theAnderson pseudospin approach [S13, S14, S16, S17]. QUASI-PARTICLE CREATION: THE SUPERCONDUCTOR SCHWINGER EFFECT
To highlight the Schwinger effect and the creation of quasi-particles, it is convenient to use the representation thatdiagonalizes (S10). This is the approach used in an alternative derivation of the original Schwinger effect in quantumelectrodynamics in Ref. [S18].The operator H k has the same form of the standard homogeneous case and can be analytically diagonalized [S8].The eigenvalues are ± (cid:15) k = ± (cid:112) ξ k + | ∆ | and the ground and the excited states are, in the original { a k ↑ , a †− k ↓ } basis, | ψ k, − ( t ) (cid:105) = { v k ( t ) , u k ( t ) } and | ψ k, + ( t ) (cid:105) = { u ∗ k ( t ) , − v ∗ k ( t ) } , respectively, with u k ( t ) = 1 √ (cid:115) ξ k ( t ) (cid:15) k ( t ) e − iχ ( t ) / v k ( t ) = 1 √ (cid:115) − ξ k ( t ) (cid:15) k ( t ) e iχ ( t ) / . (S12)Be U k ( t ) the diagonalizing operator such that U † k H k U k = H D,k . Since U k is time dependent, the dynamics isdetermined by the Schroedinger equation i (cid:126) ∂ t | ψ k ( z ) (cid:105) = ( H D,k − i (cid:126) U † k ∂ t U k ) | ψ k ( z ) (cid:105) . (S13)The contribution U † k ∂ t U k derives from the fact that the Hamiltonian is time-dependent and induces the transitionbetween eigenstates of H D,k . Notice that Eq. (S13) depends on z . Thus, it gives us the dynamics of the k -th modein position z . Double excitations
The unitary operators U k ( t ) and U † k ( t ) can be written as U k = (cid:18) u ∗ k v k − v ∗ k u k (cid:19) U † k = (cid:18) u k − v k v ∗ k u ∗ k (cid:19) (S14)with u k and v k as in Eq. (S12). This leads to the transformation [S8, S20] γ k ↑ = u k a k ↑ − v k a †− k ↓ γ †− k ↓ = v ∗ k a k ↑ + u ∗ k a †− k ↓ γ † k ↑ = u ∗ k a † k ↑ − v ∗ k a − k ↓ γ − k ↓ = v k a † k ↑ + u k a − k ↓ . (S15)These are the creation-annihilation operators for a quasiparticle that is a superposition of electron and hole [S8, S15,S20].In this representation the diagonal element of H D,k are associated to the γ † k ↑ γ k ↑ and γ †− k ↓ γ − k ↓ . On the contrary, the U † k ∂ t U k off-diagonal terms are associated to γ † k ↑ γ †− k ↓ and γ k ↑ γ − k ↓ and, therefore, create or annihilate simultaneously two quasiparticles with ( k , ↑ ) and ( − k , ↓ ). These are what in the original BCS paper are called ”real” excited pairs[S19, S38] and Leggett associates with the natural excitation in the ( k , ↑ ) and ( − k , ↓ ) space since they are still partof the condensate [S21]. They must be distinguished by the conventional ”Bogoliubov quasiparticles” discussed inliterature [S8, S20] that are related to the destruction of a Cooper pair.Using the Anderson pseudo-spin formalism is easy to understand the nature of the excited state | ψ k, + (cid:105) . The pairingpotential is defined as ∆ = (cid:80) k ∆ k with ∆ k = V (cid:104) a k ↑ a − k ↓ (cid:105) . For the ground state | ψ k, − (cid:105) , we obtain ∆ k = V u k v ∗ k as expected [S8, S20]. For the excited state, we have ∆ k = − V u k ( t ) v ∗ k ( t ) [S21]. This can be seen as an additionalphase factor e iπ or, alternatively, a π shift in the superconducting phase associated to ∆ k due to the ground-excitedtransition.We conclude that the excited k states preserve the superconductive feature. While the single excitation states(Bogoliubov quasiparticles) are associated to a vanishing coherence factor, i.e., (cid:104) a k ↑ a − k ↓ (cid:105) = 0, the double excitationstates | ψ k, + (cid:105) are associated to the same coherent factor with a minus sign. This means that a fully excited state, i.e.,with all the k modes excited, would have the same pairing potential and the same gap. In this sense, the excited stateis still superconducting or, in Leggett’s words, is still part of the condensate [S21].The fact that the excited pair state are still superconducting have important implications. The single excitationBogoliubov quasiparticles are obtained by the destruction of a Cooper pair and vanishing coherence factor. This leadsto the emptying of the condensate since the excited modes do not contribute to the pairing potential ∆ = (cid:80) k ∆ k .On the contrary, the state generated by the SSE, i.e., the superposition of ground and excited state, is stillsuperconductive. But since the dynamics of the k modes is different, they accumulate a different phase factors.The coherence factor ∆ k is in general a complex number with a time dependent phase. The different phases cangenerate ”interference effects” in the sum ∆ = (cid:80) k ∆ k effectively leading to a suppression of the pairing potential andthe superconductivity as shown in Fig. 3b of the main text. We stress once more that the mechanism at the basis ofthe destruction of superconductivity in presence of an electric field is completely distinguished and new with respectto the usual thermal one.An important remark must be done. As discussed above, a fully excited state would be similar in every aspect tothe ground state. However, it turns out that the current associated to the ground and the excited state is the same.Therefore, the current is not a good observable to distinguish the SSE. Numerical simulations
The numerical simulations are performed using the self-consistent relation for the pairing potential (S2). Thesolution scheme is presented for a given spatial point z and the procedure can be iterated for different z to obtain thespatial behaviour of the main superconductor quantities.The k involved in the superconductivity are the ones with k F − k D ≤ k ≤ k F + k D where k F and k D are theFermi and the Debye momentum, respectively [S8]. As a reference, we have taken µ = (cid:126) k F / (2 m ) = 1 eV ( m is theelectron mass) and a Debye temperature of 300 K [S8] corresponding to a normalized momentum k D /k F = 0 .
16 thatare standard values of BCS superconductors. The momentum space is divided in ∆ k intervals and the dynamics iscomputed for any k n = k F − k D + n ∆ k (with n integer) in the useful interval.The state of the system is initialized in the ground state. Thus, all the k -th modes are initially in the ground state | ψ k, − (0) (cid:105) of the initial Hamiltonian (S10). For all the relevant k , the numerical code calculates the solution | ψ k (∆ t ) (cid:105) of a discretized version of the Schroedinger equation (S13) for small time increment ∆ t . Then, with all the | ψ k (∆ t ) (cid:105) ,it calculates the new pairing potential ∆(∆ t ) = (cid:80) k ∆ k with ∆ k = (cid:104) ψ k (∆ t ) | a k ↑ a − k ↓ | ψ k (∆ t ) (cid:105) . The updated pairingpotential is inserted in the Schroedinger equation for the calculation of the following time evolution.All the numerical results presented are calculated in a self-consistent way but it turns out that the differences withthe non-self-consistent case, i.e., with ∆ constant in time, are small. LANDAU-ZENER TRANSITION AND RELEVANT PARAMETERS
The overall physical features of the dynamics can be understood with an analogy simple Landau-Zener problem[S29–S31]. We discuss the case in which | ∆ | is constant in time and not calculated with the self-consistent relation.This approximation not only allows us understand the main physical features of the dynamics but it turns out to bean excellent approximation of the full self-consistent dynamics.Let us consider first the case in which k < k F . At t = 0 the kinetic energy dominates and H k (0) ≈ ξ k τ z,k . Theminimum energy is reached at t min when ξ k ( t min ) = 0 and the Hamiltonian H k ( t min ) = Re(∆) τ x,k + Im(∆) τ y,k .Finally, for t > t min the kinetic energy terms h k − ( t ) dominates over ∆ and H k ( t ) ≈ h k − ( t ) τ z,k .Thus, we have a Landau-Zener problem [S29–S31] with i ) the typical Hamiltonian changes τ z,k → τ x,k → τ z,k , ii )and avoided crossing at t min with energy gap 2∆ and iii ) the system in the ground state H k (0) at the beginning ofthe evolution.To have an estimate of the transition probability between instantaneous eigenstates of H k (also called adiabaticstates), we use the approach discussed in Ref. [S31]. We set t = t min + δt with δt (cid:28) t min and expand and linearize ξ k ( t min ) for small δt to obtain the normalized (to µ ) energy velocity close to the minimum gap V k = √ eE (cid:118)(cid:117)(cid:117)(cid:116)(cid:34) − (cid:18) kk F (cid:19) (cid:35) µm . (S16)This would correspond to ∂(cid:15) k /∂t of the original Landau-Zener model for linear drive [S31].The probability to to have a transition from the ground to the excited state (so called Landau-Zener probability)with minimum energy gap 2∆ is [S31] P LZ = e − π ∆2 Vk (cid:126) . (S17)In our case, using the above expression and the scaling parameters, the Landau-Zener coefficient readsΓ k = 2 π ∆ V k (cid:126) = 1 . × − E f E C (cid:114) − (cid:16) kk F (cid:17) (S18)Therefore, for (almost) any k , we have Γ k (cid:28) P LZ ≈ k modes should undergo a ground-to-excitedstate transition (see also discussion below).The situation is different for initial momentum very close to the Fermi momentum, i.e., 1 − k/k F ≈ − where theLandau-Zener model s inadequate to describe the dynamics.In this case, the initial Hamiltonian is H k (0) ≈ − ∆ τ x and the system is in its ground state | ψ k, − (0) (cid:105) = 1 / √ { , } (since u k ≈ v k ≈ / √ H k ( t min ) ≈ − ∆ τ x,k . Thus, during the first part of evolution0 ≤ t ≤ t min , H k ( t ) ∝ τ x,k . A key ingredient of the Landau-Zener model, i.e., the first Hamiltonian change τ z,k → τ x,k ,is now missing. Since the system remains in an eigenstate for the Hamiltonian, the first part of the evolution accountsonly for a dynamical phase factor and no Landau-Zener transition between eigenstates occurs.For t > t min , the kinetic energy increases and start dominating so that the Hamiltonian is H k ( t ) ≈ ξ k τ z . Duringthis evolution, no Landau-Zener transition occurs and the system remains in the ground state of τ x,k . Decomposingthis in the egigenstates of τ z we obtain that the ground and excited states are equally populated as shown in Fig. 2cof the main text.The last relevant feature of the dynamics is that for k (cid:28) k F , there is no transition to the excited state (Fig. 2c inthe main text). This feature can be understood by comparing the time t min and t max . The first one sets the timein which the minimum energy gap is reached and the transition occurs and it is determined by the electric field; thesecond one sets the time for the dynamics and it is determined by the spatial length L of the system.We recall the expression t min = √ µm (cid:112) − ( k/k F ) eE f (S19)and t max = mL/ ( (cid:126) k ). The condition t min = t max gives the limit for the transition to occur. From this, we obtainan equation for the minimum electric field needed to excite the mode k (neglecting the k dependence) E f E C = 2 µeE C L . (S20)Below this value the dynamics is not long enough to reach the minimal gap or, rephrasing, the electric field has notenough time to give to the superconductor mode enough energy to undergo the ground-to-excite transition.For L = 2 nm, we obtain E f = 2 E C , namely close to E C = 5 × V/m that is the electric field at which thesuperconductivity is suppressed in several recent experiments [S22–S26, S34, S35].
OUT-OF-EQUILIBRIUM QUASI-PARTICLES
It is known that even a constant electric field in a superconductor generates out-of-equilibrium phenomena [S10].The fundamental reason for this is that charged particles in the superconductor are accelerated and increase theirenergy. As a direct consequence, the effect of the environment must be included in the treatment otherwise the energyincrease would inevitably lead to the destruction of superconductivity.The SSE can naturally lead to non-equilibrium or out-of-equilibrium phenomena but also in this situation we mustinclude some dissipative channel. The electric field acts as a pump increasing of the superconductor energy while thedissipation tends to reduce it. The steady state would be reach when the injected and dissipated energy rates areequal.The energy dissipation can occur in two ways. The first one is the scattering where a particle with momentum k isscattered into a particle of momentum k (cid:48) and it partially dissipates its kinetic energy. The second is the destruction (orcreation) of excited states. These are, in general, distinguished dissipation mechanisms. For example, the destructionof a double excitation state is associated to the decrease in energy of 2∆ but there is no scattering of momentum sincethe transition does not change k of the particles, i.e., it remains within in the ( k ↑ , − k ↓ ) space.Without pretending to be a quantitative description, a simple model can help us to understand how non-equilibrium,i.e., non-thermal, distributions can arise in this contest. We make the working hypothesis that the two momentumscattering and creation/annihilation time scales are well separated. This allows us to treat separately the energyprocess due to the creation/destruction of excited states and the k scattering the dissipated the kinetic energy andfocus on the first one. In fact, the dissipation through a scattering process is fairly well understood [S32] while thenon-equilibrium features (the most interesting for us) are consequences of the first process.We use a simple master equation to describe the transition between the state in the ( k ↑ , − k ↓ ) as shownschematically in Fig. 3c of the main text. From the ground state | ψ k, − (cid:105) we can excite a double excited state | ψ k, + (cid:105) = γ † k ↑ γ †− k ↓ | ψ k, − (cid:105) and, in presence of an environment, the latter can relax back to the ground state.In addition to these, we must include the possibility to break a Cooper pair and generate a single excitation state | ψ s (cid:105) = γ † k ↑ | ψ k, − (cid:105) = γ k ↑ | ψ k, + (cid:105) and | ψ s (cid:105) = γ †− k ↓ | ψ k, − (cid:105) = γ − k ↓ | ψ k, + (cid:105) . Formally these are obtained with a singledestruction ( γ ) or creation ( γ † ) operator applied to the excited or ground state, respectively. Notice that, without theelectric field and the double exited state, these are the only ones that are thermally excited and result in a Fermi-Diracdistribution for the quasi-particles (the thermal double excitation is exponentially suppressed).We denote with Γ eg and Γ ge , the excitation and relaxation rates between the excited and ground state, respectively.Since the environment does not distinguish between the creation or destruction of a quasiparticle with ( k , ↑ ) or ( − k ↓ ),the two single excited state | ψ s (cid:105) and | ψ s (cid:105) are indistinguishable. It is convenient to describe these transitions in termsof single transition to a single excitation state | ψ s (cid:105) . The environment induces transitions from and to the excitedstates with rates Γ es and Γ se , respectively, and from and to the ground state with rates Γ gs and Γ sg , respectively (seeFig.3c of the main text).For a given mode k , the corresponding master equations for the ground P g , single excitation P s and excited statepopulations P e read ∂ t P g = − (Γ ge + 2Γ gs ) P g + Γ eg P e + Γ sg P s ∂ t P s = − (Γ se + Γ sg ) P s + 2Γ es P e + 2Γ gs P g ∂ t P e = − (Γ eg + 2Γ es ) P e + Γ ge P g + Γ se P s . (S21)By using the normalization condition P e + P s + P g = 1, we can reduce these to two differential equations for P g and P s ∂ t P g = − (Γ eg + Γ ge + 2Γ gs ) P g + (Γ sg − Γ eg ) P s ∂ t P s = − (2Γ es + Γ se + Γ sg ) P s + 2 (Γ gs − Γ es ) P g . (S22)The stationary solution is obtained for ∂ t P g = 0 and ∂ t P s = 0. The solution of these algebraic equations expressedin terms of the ratio between the ground and the single excitation state is P g P s (cid:12)(cid:12)(cid:12) steady = Γ eg Γ se + Γ eg Γ sg + 2Γ es Γ sg ge Γ es + 2Γ eg Γ gs + 4Γ es Γ gs . (S23)We further assume that i ) for the thermal transitions , the relaxation and excitation rates are related by theBoltzmann rules, i.e., Γ se = e − (cid:15)kkBT Γ es and Γ gs = e − (cid:15)kkBT Γ sg ( k B is the Boltzmann constant and (cid:15) k is single excitationenergy gap ) and ii ) Γ es = Γ sg since the energy gap and the transition amplitudes are the same. We arrive to P g P s (cid:12)(cid:12)(cid:12) steady = 12 Γ eg Γ sg + 2Γ sg + e − (cid:15)kkBT Γ eg Γ sg Γ ge Γ sg + e − (cid:15)kkBT (Γ eg Γ sg + 2Γ sg ) . (S24)The rates Γ eg and Γ ge are not Boltzmann-related and, in particular, Γ eg depends only on the electric field effect (if2∆ (cid:28) k B T ). Therefore, the above expression cannot be related to a simple decay or thermalization and would resultin a non-thermal distribution of quasi-particles.To have a more direct comparison we can consider the same model without electric field by taking the limit Γ ge → eg → P g P s (cid:12)(cid:12)(cid:12) no E f = 12 e − (cid:15)kkBT (S25)that leads, indeed, to a thermal distribution of quasi-particle.Even more evident is the limit with a few thermal excitations (cid:15) k (cid:29) k B T , in which we have P g P s (cid:12)(cid:12)(cid:12) steady → Γ eg + 2Γ sg ge P g (cid:12)(cid:12)(cid:12) no E f → P s (cid:12)(cid:12)(cid:12) no E f →→