Exponential speedup of incoherent tunneling via dissipation
Dominik Maile, Sabine Andergassen, Wolfgang Belzig, Gianluca Rastelli
EExponential speedup of incoherent tunneling via dissipation
D. Maile,
1, 2
S. Andergassen, W. Belzig, and G. Rastelli
1, 3, 4 Fachbereich Physik, Universit¨at Konstanz, D-78457 Konstanz, Germany Institut f¨ur Theoretische Physik and Center for Quantum Science,Universit¨at T¨ubingen, Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Zukunftskolleg, Universit¨at Konstanz, D-78457, Konstanz, Germany INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, I-38123 Povo, Italy (Dated: February 5, 2021)We study the escape rate of a particle in a metastable potential in presence of a dissipativebath coupled to the momentum of the particle. Using the semiclassical bounce technique, we findthat this rate is exponentially enhanced. In particular, the influence of momentum dissipationdepends on the slope of the barrier that the particle is tunneling through. We investigate alsothe influence of dissipative baths coupled to the position, and to the momentum of the particle,respectively. In this case the rate exhibits a non-monotonic behavior as a function of the dissipativecoupling strengths. Remarkably, even in presence of position dissipation, momentum dissipationcan enhance exponentially the escape rate in a large range of the parameter space. The influence ofthe momentum dissipation is also witnessed by the substantial increase of the average energy lossduring inelastic (environment-assisted) tunneling.
Introduction—
In minimization methods, the researchof the absolute minimum becomes a challenging problemwhen the landscape is characterized by many relativeminima and energy barriers of comparable size. Clas-sical methods used by classical computers (e.g. MonteCarlo simulations) generally require exponentially longtimes. Quantum adiabatic annealing methods proposean alternative strategy which is based on the idea to el-evate the classical system to the quantum domain [1–5].However, such a strategy also poses time constraints. Analternative and optimal strategy could be to exploit thequantum tunneling effect as irreversible process [6] byavoiding possible quantum coherent oscillations betweendifferent minima. This can be in principle realized if thequantum system is not closed but is dissipatively cou-pled to an external bath. The influence of dissipationon quantum annealing has been recently studied [7–9].In open quantum systems, contrary to intuition, dissi-pation and dephasing can even enhance the rate of someprocesses, as in quantum computation via engineered dis-sipation [10] or in transport phenomena assisted by noisein quantum networks and biomolecules [11] and in photo-synthetic biomolecules [12]. Environment-assisted quan-tum transport has been recently studied in a controlledfashion in a spin network formed by trapped ions [13].Contrary to the classical case, an isolated quantumparticle at zero temperature in a metastable minimumcan escape into the unbound region of the potential, witha continuous energy spectrum, beyond the energy barrier[14–17]. In order to investigate the crossover from thequantum to the classical regime, Caldeira and Leggettfirst considered the metastable escape in presence of adissipative bath coupled to the position [18] finding thata such coupling leads, indeed, to a reduction of the es-cape rate, as intuitively expected. Subsequent works inthe literature analyzed in details this seminal problem [19–23]. On the other hand, the problem of the quantumescape from a metastable well in presence of a dissipa-tive bath coupled to the momentum of the particle has Figure 1. (a) The considered model of a particle in a (semi-double parabolic) potential and coupled to an external bathvia the momentum operator with coupling constant τ p . Theslope of the energy barrier on the right can be changed byvarying the minimum Σ, the different colored lines correspondto different values of Σ. (b) Exponential enhancement E of theescape rate Γ as a function of τ p . Different lines correspondto different regimes: Σ = 0 . V , Σ = 10 V , Σ = ∞ ; thedotted lines correspond to analytic expansions (see text). Thefrequency of the harmonic well is ω whereas ω c is the highfrequency cutoff of the bath spectral density. In all cases theescape rate increases exponentially. Parameters V / ~ ω =12 . ω c = 8000 ω . a r X i v : . [ qu a n t - ph ] F e b not been studied so far, although previous studies brieflymentioned a possible enhancement [24, 25].We here investigate the escape rate via quantum tun-neling of a particle trapped in a metastable well and cou-pled to an external bath via the momentum operator,see Fig. 1. This kind of dissipative interaction has beendiscussed in other potentials [25–31]. We here show thatthe coupling to an external bath via the momentum op-erator increases the escape rate. The enhancement isobtained even in presence of a second dissipative bathcoupled to the position operator of the particle. We notethat although we analyze an irreversible escape process,our results are valid also for asymmetric double well po-tentials [31, 32] as long as tunneling is completely in-coherent (irreversible) in the regime of strong dissipa-tive coupling with the two baths and in the limit of lowtemperature. We also calculate the average energy losstowards the environments during the tunneling processand find a crossover in the dominance of the respectiveenvironmental couplings as a function of the slope of thebarrier. Theoretical model—
In the semiclassical limit V (cid:29) ~ ω , where V is the energy barrier and ω the harmonicfrequency associated to the relative minimum, the escaperate Γ of the particle through the barrier is of the formΓ = K e − ~ S cl . (1)In the path integral formalism, the exponent S cl repre-sents the Euclidean action on the minimizing (classical)path x cl ( τ ) in the imaginary time τ and the prefactor K is related to the first order corrections due to fluctuationsaround this path [33–35]. In absence of any coupling toan external bath, we denote the bare escape associatedto the potential by Γ = K e − ~ S (0) cl . Generally, the pref-actor depends on the dissipative couplings. While K canbe enhanced in presence of momentum dissipation affect-ing the trapped particle, i.e. K (cid:29) K (see SupplementalMaterial [36]), the leading dependence is due to the ex-ponential term. For this reason the present analysis isfocused on the ratio between the two exponential terms E = e − ~ (cid:16) S cl − S (0) cl (cid:17) . (2)For E > E < V ( x ) = mω x / x < a and V ( x ) = mω ( x − x m ) / − Σ for a < x < x m , as shownin Fig. 1a. From the second minimum the potential re-mains flat, with V ( x ) = − Σ for x ≥ x m which controlsthe slope on the right side of the barrier. The matchingcondition at x = a yields mω ( a − x m ) / − Σ = V , re-lating x m to a . Integration out the baths to which theparticle is coupled via the momentum and position op-erators, the action in the exponential of Eq. (1) can be split into two parts S = S + S dis , with S [ x ( τ )] = Z β − β dτ h m x ( τ ) + V [ x ( τ )] i (3)and the dissipative part as [31] S dis [ x ( τ )] = 12 Z Z β − β dτ dτ F ( x ) ( τ − τ ) x ( τ ) x ( τ )+ 12 Z Z β − β dτ dτ F ( p ) ( τ − τ ) ˙ x ( τ ) ˙ x ( τ ) , (4)where the limit β → ∞ has to be performed at theend. Assuming Ohmic spectral densities for the twobaths, the two time-dependent functions read F ( x ) ( τ ) = P l F ( x ) l e iω l τ /β and F ( p ) ( τ ) = P l F ( p ) l e iω l τ /β , with theMatsubara frequency components F ( x ) l = γm | ω l | f c ( ω l )and F ( p ) l = m [ − τ p | ω l | f c ( ω l )) − ] respectively,where ω l = 2 πl/β (with l integer) [31] and f c ( ω l ) =(1+ | ω l | /ω c ) − is a Drude cutoff function with a frequencycutoff ω c [37]. The parameters γ and τ p are the couplingconstants associated to the position and the momentumdissipation, respectively.In the semiclassical path integral method for the quan-tum decay, the minimizing path x cl ( τ ) of the action S is the solution of the classical equation of motionwith the inverted potential − V ( x ). At zero tempera-ture ( β → ∞ ) a non-trivial solution x cl ( τ ) = 0 is calledbounce path: The particle starts at the minimum of thewell at x ( τ = − β/ | β →∞ = 0, reaches the turning point x ( τ = 0) = x esc , and then returns to x ( τ = β/ | β →∞ = 0.Examples of x cl ( τ ) for the bare potential γ = τ p = 0 arereported in Fig. 2, for the ones in presence of dissipationwe refer to the Supplemental Material [36]. The imagi-nary time spent in the region a < x ( τ ) < x esc is calledbounce time ξ B , which turns out to depend strongly onthe slope of the potential and ultimately vanishes in thelimit of sharp potential. This characteristic time scale is Figure 2. (a) Example of the inverted potential − V ( x )for the motion of the minimizing (classical) path x cl ( τ ). (b)Different paths x cl ( τ ) for different values of Σ in the nondissipative case γ = τ p = 0. The path shrinks with increasingΣ. For Σ = ∞ the particle is instantly reflected at the turningpoint x esc (see also Sup. Mat. [36]). determined by the integral equation [38]1 ω p /V + 1 = (5)1 π Z ∞ dω sin( ωξ B ) ω (cid:16) ω τ p ωf c ( ω ) + ω + γωf c ( ω ) (cid:17) . The analytic solution for the action S cl is given by [36] S cl = − ω V (cid:16)p /V + 1 (cid:17) π × Z ∞ dω − cos ( ωξ B ) ω (cid:16) ω τ p ωf c ( ω ) + ω + γωf c ( ω ) (cid:17) + 2 V (cid:16) p /V (cid:17) ξ B . (6)We note that the dependence on the bounce time ξ B implies a change of the action when the slope of the rightside of the potential barrier is varied. Results—
Our main results are summarized in Figs. 1and 3. In Fig. 1, we show the results for pure momentumdissipation of coupling strength τ p , for different slopes ofthe right side of the barrier. In presence of pure momen-tum dissipation, the action decreases as a function of thecoupling parameter τ p leading to the enhancement of theescape rate observed in Fig. 1b. In contrast, for pureposition dissipation in a metastable well one obtains anexponential suppression of the escape rate [18–20, 22, 23].A simple picture for the observed exponential speedupof the escape rate can be given in the limit of infinite slope(see Fig. 1a), which is obtained by taking the limit Σ →∞ . In this limit, the barrier is infinitely steep [20]. As aconsequence, the particle is instantly reflected at x esc = a leading to a vanishing bounce time ξ B → S cl / ~ ≡ a / (2 (cid:10) x (cid:11) ), with a the position ofthe barrier maximum and (cid:10) x (cid:11) the harmonic quantumfluctuations of the particle in the well. As the momentumdissipation increases the quantum fluctuations (cid:10) x (cid:11) [26,27, 30], the escape rate is enhanced as a consequence. Inthe weak coupling limit ω τ p (cid:28)
1, the expansion of (cid:10) x (cid:11) in the action S cl yields E (Σ= ∞ ) γ =0 ≈ (cid:18) ω c ω √ e (cid:19) π ~ V τ p ω τ p (cid:28) , ω c ξ B (cid:28) , (7)The escape rate E is limited by the cutoff frequency ω c as (cid:10) x (cid:11) diverges in the limit ω c → ∞ [30]. This simple formprovides a good approximation also for a sharp barrierwith a finite slope (and small ξ B ) as long as ω c ξ B (cid:28) S cl / ~ = a / (2 (cid:10) x (cid:11) ). However, the exponen-tial enhancement still persists in this more realistic situ-ation shown in Fig. 1b and in particular is not affectedby the presence of conventional (position) dissipation, as will be discussed in the following. The exponential en-hancement of the rate in presence of momentum dissi-pation strongly depends on the steepness of the barrier.As shown in Fig. 1b, the effect is maximal for a sharpbarrier (Σ = ∞ ), which represents an upper theoreticalbound, and then becomes less pronounced. A qualitativeunderstanding is provided by the following physical pic-ture: The presence of the momentum dissipation givesrise to an anomalous friction force in the equation forthe minimal path x cl . Such a force is proportional tothe acceleration through a memory friction function (i.e.nonlocal in time), see Supplemental Material [36]. Asthe acceleration is strongly controlled by the conserva-tive force ∼ dV ( x ) /dx , the anomalous friction force be-comes important at the turning point x esc , viz. when dx ( τ ) /d τ becomes large, see example of Fig. 2. Noticethat the turning point x esc also depends on the dissipa-tive couplings and does not coincide with the zero of thepotential V ( x ) = V (0) = 0 as in the case of no dis-sipation [36]. Finally, we remark that even in the limit dV ( x ) /dx | x = x esc = −∞ , the results for the action remainfinite as the friction is nonlocal in time.For finite value of Σ, an intermediate regime is identi-fied by a reduced slope of the potential for Σ (cid:29) V and anincreased bounce time ω c ξ B (cid:29) ω ξ B (cid:28) ω c . The escape rate still depends expo-nentially on the dissipative momentum interaction, butwith an explicit dependence on Σ E (Σ (cid:29) V ) γ =0 ≈ (cid:18) k Σ V (cid:19) π ~ V τ p ω τ p (cid:28) , ω c ξ B (cid:29) , (8)where k = e − C ) / ≈ .
81 with C the Euler constant.Eq. (8) corresponds to the (gray) dotted line in Fig. 3b.A further interesting regime corresponds to Σ (cid:28) V ,for a finite slope and bounce time [39]. In the limitΣ (cid:28) V the problem becomes equivalent to the solutionof a slightly asymmetric double well in the incoherentoverdamped limit, discussed in [32] for pure position dis-sipation. In a such limit the exponential enhancement isdescribed by E (Σ (cid:28) V ) γ =0 ≈ e π ~ V τ p ω τ p (cid:28) , ω ξ B (cid:29) . (9)We note that the different regimes of Σ describedby Eqs. (7)-(9) present characteristic base functions,whereas the exponent controlling the effect of momen-tum dissipation is the same.The analytical solutions for E presented so far are re-stricted to pure momentum dissipation ( γ = 0). The re-spective curves are displayed in Fig. 1 and agree well withthe numerical results of Eqs. (5) and (6) in the respec-tive regimes. Further analytical expressions in presenceof both dissipative couplings are reported in the Supple-mental Material [36] in various regimes of ω ξ B , ω c ξ B and Figure 3. (a) Particle in a metastable well in presence of both dissipative couplings. (b) Logarithmic plot of E as a functionof γ , at fixed ratio τ p ω /γ = 0 . E as a function of γ , for Σ = 2 V and different values of the ratio τ p ω /γ . (d) E as a function of Σ and γ , at fixed ratio τ p ω /γ = 0 .
5. For (b), (c) and (d) we used V / ~ ω = 12 . Σ /V and they also agree well with the numerical results(an example is given in Fig. 3b). The main result ana-lyzed for pure momentum dissipation also holds in pres-ence of both baths when position dissipation dominates,as shown in Fig. 3. In general, the presence of both dissi-pative couplings (momentum and position) leads howeverto a non-monotonic behavior as a function of γ/ω (or τ p ω ) clearly visible in E shown in Fig. 3b-d for a fixed ra-tio τ p ω /γ . Again, a simple physical picture is obtainedin the limit Σ = ∞ in which E is determined uniquelyby the harmonic quantum fluctuations (cid:10) x (cid:11) of the har-monic well. Such harmonic quantum fluctuations exhibita non-monotonic behavior as a function of the couplingstrengths [30]. This is reflected in the results for E shownin Fig. 3b for a fixed ratio τ p ω /γ . and different valuesof Σ. We note that, in the regime of sharp potentialΣ (cid:29) V , the effect of momentum dissipation dominatesover the range of γ/ω shown in the figure. This is due tothe strong dependence of the momentum dissipation onthe slope as explained previously. In Fig. 3c we show E for a given value of Σ = 2 V and for different values of theratio τ p ω /γ . We observe that the exponential speed-upstill persists in a wide range of γ/ω which depends on τ p ω /γ .Finally we also calculate the average energy loss dur-ing the tunneling process. Such average energy inducedby the dissipative couplings is defined as h ∆ E i = V (0) − V ( x esc ) [32, 40]. In absence of dissipative interaction, thereturning point x esc of the bounce path is simply given bythe condition V ( x esc ) = V (0) = 0, see Fig. 4a. However,by increasing the dissipative coupling, x esc shifts to largervalues [36], indicating that the particle escapes the bar-rier, on average, at an energy V ( x esc ) < V (0). In Fig. 4awe plot h ∆ E i as a function of Σ for different dissipativecases. Considering pure position dissipation we find thatthe loss saturates already for moderate Σ ∼ V becom-ing independent of the barrier. For small Σ the energyloss into a single momentum dissipative bath is smallerthan in the presence of pure position dissipation. How- ever, the latter energy loss increases by increasing Σ andeventually becomes larger as in the position dissipativecounterpart. This is again a consequence of the explainedincreasing influence of momentum dissipation as a func-tion of Σ. By further increasing Σ, h ∆ E i eventually sat-urates to a value determined by the cutoff frequency ω c (see [36] for the details). In presence of both dissipativecouplings and Σ (cid:28) V the loss coincides with the value ofpure position dissipation as can be seen from Fig. 4a. Forlarger values of Σ, the energy losses simply add up. Fromthe crossing point in Fig. 4a we can determine the val-ues in parameter space for which both dissipative bathsbecome equally important. Further, the average energyloss as a function of the dissipative coupling saturatesto the same asymptotic value in the overdamped limit γ/ω (cid:29) τ p ω (cid:29) h ∆ E i a < Σ sincethe expected maximum energy loss is V (0) − V ( x m ) = Σ[40]. Figure 4. (a) Average energy loss during the tunneling pro-cess h ∆ E i = V (0) − V ( x esc ) for a particle coupled to twobaths via the momentum and the position operators. (b)The quantity h ∆ E i as a function of Σ for different dissipativecases. The solid (purple) line is for both dissipative couplingswhereas the dashed (red) line and the dotted (blue) line arein presence of a single bath with coupling through the mo-mentum and the position, respectively. Conclusions—
Momentum dissipation leads to an ex-ponential enhancement of the escape rate of a particlein a metastable potential. In presence of position dissi-pation, we find a non-monotonic behavior as a functionof the dissipative coupling strengths. Depending on thebarrier, momentum or position dissipation can be domi-nant. For a sharper barrier, the region of momentum dis-sipation induced enhancement increases. The particle’saverage energy loss during the tunneling process shows astrong dependence on the interplay between momentumdissipation and the slope of the potential.To summarize, we propose a method that rapidly re-leases the system from a relative (metastable) minimumexploiting quantum tunneling as a pure, irreversible andinelastic process, assisted by the environment. Our theo-retical findings can be directly tested in superconductingquantum circuits [41–43] in which dissipative positionand momentum interaction translate to dissipative phaseor charge couplings. In particular, momentum/chargedissipation can be readily implemented simply using ca-pacitances and resistances [44]. Further, our results areimportant for quantum numerical minimization methodsin which the escape rate from a relative minimum playsa key role in setting the computational time scale [45, 46].
Acknowledgments—
We acknowledge financial supportfrom the MWK-RiSC program. This research was alsopartially supported by the German Excellence Initia-tive through the Zukunftskolleg and by the DeutscheForschungsgemeinschaft (DFG) through the SFB 767 andProject-ID 425217212 – SFB 1432. G.R. thanks P. Haukefor useful discussions and relevant comments. [1] G. E. Santoro and E. Tosatti,
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1, 2
S. Andergassen, W. Belzig, and G. Rastelli
1, 3, 4 Fachbereich Physik, Universit¨at Konstanz, D-78457 Konstanz, Germany Institut f¨ur Theoretische Physik and Center for Quantum Science,Universit¨at T¨ubingen, Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Zukunftskolleg, Universit¨at Konstanz, D-78457, Konstanz, Germany INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, I-38123 Povo, Italy (Dated: February 5, 2021)In this supplemental material we provide the details of the calculations for determining the escaperate in the semiclassical limit. We present analytical formulas for the rate in different regimes andwe also analyse in detail the change of the average energy loss in the inelastic environment-assistedtunneling.
I. GENERAL FORMULA OF THE DECAY RATE IN THE SEMICLASSICAL APPROXIMATION
In this section, we recall the theoretical method for calculating the amplitude (or matrix element) Z which canbe related to the escape rate of a particle placed in the metastable well, in the zero temperature limit. We generalizethis approach in presence of position and momentum dissipation. A. The semiclassical method
Because the potential is metastable, Z has an imaginary part Γ which corresponds to the escape rate. One cancalculate Z via the imaginary time path integral method, specifically using the instanton-bounce method [S1–S5].The starting point of this theoretical approach is the amplitude in the imaginary time Z = h x | e − β ~ H | x i = I D [ x ( τ )] e − ~ S [ x ( τ )] , (S1)where S [ x ( τ )] = S + S diss is the action of the open quantum system given in the main text. In particular S dis is thedissipative action and x ( τ ) is a generic periodic path with x = x ( β/
2) = x ( − β/ β → ∞ , one can set x = 0 such that Z is proportional to the density of probability (in the imaginary time) to find the particle at theorigin which corresponds to the minimum of the metastable well.One can calculate Z in the semiclassical approximation by finding the so-called classical path x cl ( τ ) that minimizesthe action and then one applies the expansion x ( τ ) = x cl ( τ ) + δx ( τ ) leading to S [ x ( τ )] = S cl [ x cl ( τ )] + S δ [ δx ( τ )]. Thispath, in the zero temperature limit β → ∞ , starts and ends in the minimum of the well, x ( ± β/ | β →∞ = 0. Beyondthe trivial solution x (0) cl ( τ ) = 0, there exists the so-called bounce path x (1) cl ( τ ) in which the particle moves from theminimum x = 0, gets reflected at the returning point x esc and comes back to its origin, see Fig. S1. The matrixelement of a single bounce path x (1) cl can be written in the semiclassical limit as z (1)0 = e − ~ S cl [ x (1) cl ( τ )] I D [ δx ( τ )] e − ~ S (1) δ [ δx ( τ )] , (S2)with the fluctuations around the classical path satisfying δx ( − β/
2) = δx ( β/
2) = 0 and S (1) δ being the expansion ofthe action over the single bounce path x (1) cl S (1) δ [ δx ( τ )] = Z β − β dτ (cid:18) m δ ˙ x ( τ ) + 12 d V ( x ( τ )) d x (cid:12)(cid:12)(cid:12) x (1) cl δx ( τ ) (cid:19) + S dis [ δx ( τ )] . (S3)One can express the generic fluctuations path as δx ( τ ) = P ∞ q =0 c q y q ( τ ) in which we use as basis the eigenfunctions ofthe following eigenvalue equation (cid:20) − m d dτ + V [ x (1) cl ] (cid:21) y q ( τ ) + Z β − β dτ F ( x ) ( τ − τ ) y q ( τ ) − Z β − β dτ F ( p ) ( τ − τ ) d dτ y q ( τ ) = λ ( B ) q y q ( τ ) , (S4) a r X i v : . [ qu a n t - ph ] F e b Figure S1. (a) Metastable well potential discussed in the present article. (b) Example of a single bounce path which minimizesthe action without dissipation. The bounce time ξ B denotes the imaginary time interval in which the path is in the region x > a . in which we set V [ x (1) cl ] = d V [ x ( τ )] dx (cid:12)(cid:12)(cid:12) x (1) cl and λ ( B ) q are the eigenvalues. Using the eigenvectors decomposition, we canwrite S (1) δ = ∞ X q =0 λ ( B ) q c q . (S5)A priori, the full matrix element is the sum of all possible n -bounce paths Z = ∞ X n =0 z ( n )0 . (S6)As discussed below, we will use the so-called dilute gas approximation for the bounces such that the quantities z ( n )0 for n ≥ z (1)0 . B. Discussion on the zero and on the negative eigenvalues
The spectrum { λ ( B ) q } contains a zero eigenvalue λ ( B )0 = 0 with the eigenfunction y ( τ ) = A τ ˙ x (1) cl ( τ ) because of thetranslational invariance of the bounce on the whole τ axis. A τ is a constant and ˙ x cl ( τ ) = dx cl ( τ ) /dτ . There existsalso a negative eigenvalue λ ( B )1 due to the fluctuations of the bounce time ξ which corresponds to the imaginary timein which the path is in the region a < x < x esc . From a mathematical point of view, one can consider Eq. (S4) asthe Schr¨odinger differential equation in which the τ axis is the space and y q ( τ ) the wavefunction. Then the negativeeigenvalue λ ( B )1 can be seen as the bound energy of the localized ground state [S3]. This leads to an imaginary partof the amplitude z (1)0 as we show below.We use the following change of variable by expressing the generic path (in the semiclassical limit) as x ( τ ) = x (1) cl ( τ ) + ∞ X q =0 c q y q ( τ ) −→ x ( τ, ξ ) = x (1) cl ( τ − τ , ξ ) + ∞ X q =2 ˜ c q y q ( τ − τ , ξ ) , (S7)in which we consider τ , the center of the bounce, and ξ , the bounce time, as the new variables instead of c and c .The Jacobian | dc /dτ | is obtained via the overlap of x ( τ, ξ ) with y ( τ ) = A τ ˙ x (1) cl ( τ ) yielding c ( τ ) = Z β − β dτ x ( τ, ξ ) y ( τ, ξ ) = Z β − β dτ [ x (1) cl ( τ − τ , ξ ) A τ ˙ x (1) cl ( τ, ξ ) + ∞ X q =2 ˜ c q y q ( τ − τ , ξ ) y ( τ )] . (S8)Because of the translational invariance we can expand x (1) cl ( τ − τ , ξ ) ≈ x (1) cl ( τ, ξ ) + ˙ x (1) cl ( τ, ξ ) τ (S9)and analogously y q ( τ − τ , ξ ) ≈ y q ( τ, ξ ) + ˙ y q ( τ, ξ ) τ . Inserting these expressions into (S8) we find c ( τ ) = Z β − β dτ " A τ (cid:18) x (1) cl ( τ, ξ ) ˙ x (1) cl ( τ, ξ ) + (cid:16) ˙ x (1) cl ( τ, ξ ) (cid:17) (cid:19) + ∞ X q =2 ˜ c q ˙ y q ( τ, ξ ) y ( τ, ξ ) τ (S10)and hence (cid:12)(cid:12)(cid:12)(cid:12) dc ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) = 1 A τ + A τ Z β − β dτ ∞ X q =2 ˜ c q ˙ y q ( τ, ξ ) ˙ x (1) cl ( τ, ξ ) , (S11)where we used that A τ = sZ β/ − β/ dτ (cid:16) ˙ x (1) cl ( τ ) (cid:17) ! − , (S12)which follows from the normalization condition R β/ − β/ y ( τ ) dτ = 1. C. Transformation of the fluctuations path integral for a single bounce
A priori, we must deal with the integration of the path integral for the fluctuations. We find for the transformation I δx ( ± β ) =0 D [ δx ( τ )] −→ N ∞ Y q =0 Z ∞−∞ dc q √ π ~ , (S13)where N is a constant. However, the above treatment of extracting the zero and the negative eigenvalue leads to ∞ Y q =0 Z ∞−∞ dc q √ π ~ −→ " ∞ Y q =2 Z ∞−∞ d ˜ c q √ π ~ β/ − β/ dτ √ π ~ (cid:12)(cid:12)(cid:12)(cid:12) dc ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) Z β dξ √ π ~ (cid:12)(cid:12)(cid:12)(cid:12) dc ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) . (S14)Now we can write for the full partition function element of a single bounce as z (1)0 = N " ∞ Y q =2 Z ∞−∞ d ˜ c q √ π ~ β/ − β/ dτ √ π ~ (cid:12)(cid:12)(cid:12)(cid:12) dc ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) Z β dξ √ π ~ (cid:12)(cid:12)(cid:12)(cid:12) dc ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) e − ~ S cl [ x (1) cl ( τ,ξ )] e − ~ ∞ P q =2 λ ( B ) q ˜ c q , (S15)The Jacobian | dc ( τ ) /dτ | is provided in Eq. (S11) in which the second part vanishes because it is linear in ˜ c q , Z ∞−∞ d ˜ c q ˜ c q e − ~ λ q ˜ c q = 0 q ≥ | dc ( τ ) /dτ | → /A τ and use Eq. (S12). For the integration over ξ we use the steepestdecent method, namely we find the fixed bounce time ξ B such that dS ( ξ ) /dξ | ξ B = 0 leading to z (1)0 = e − ~ S cl [ x (1) cl ( τ,ξ B )] N βA τ √ π ~ Q ∞ q =2 q λ ( B ) q Z ∞−∞ dϕ √ π ~ e − ~ λ ( B )1 ϕ , (S17)in which the last integral corresponds to the contribution of the breathing mode around ξ B with λ ( B )1 <
0. We denote | λ ( B )1 | = − λ ( B )1 >
0. As a consequence the integral of Eq. (S17) diverges. This is not surprising as we want tocalculate the eigenvalues for a system that is metastable at the position x = 0 [S6]. The integral has to be analyticallycontinued to avoid this divergence, as discussed in chapter 17 of the book of Kleinert [S3], in the paper of Langer [S7]and the one of Callan and Coleman [S6]. The basic idea is to deform a stable action into a metastable one by keepingtrack of the eigenvalue λ ( B )1 leading to 1 √ π ~ Z dϕ e ~ | λ ( B )1 | ϕ = 12 i q | λ ( B )1 | , (S18)and we obtain the imaginary part previously discussed. Finally, the single bounce amplitude reads z (1) L = e − ~ S cl [ x (1) cl ( τ,ξ B )] N βA τ √ π ~ Q ∞ q =2 q λ ( B ) q i q | λ ( B )1 | = iβ e − ~ S cl [ x (1) cl ( τ,ξ B )] A τ √ π ~ N Q ∞ q =0 q λ (0) q Q ∞ q =0 q λ (0) q Q ∞ q =2 q λ ( B ) q q | λ ( B )1 | (S19)where we have introduced the product of the eigenvalues for the harmonic potential associated to the metastable wellwith frequency ω . Then one can notice that N Q ∞ q =0 q λ (0) q = Z (0)0 = 1 p π h x i e − βEGS (S20)corresponds to the value of the amplitude for the case of a harmonic potential in presence of dissipation, with E GS the ground state energy and h x i the harmonic fluctuations. We also set K = 1 √ π ~ A τ vuut Q ∞ q =0 λ (0) q Q ∞ q =2 λ ( B ) q q | λ ( B )1 | . (S21)containing the ratio of determinants, the Jacobian prefactor A τ and the negative eigenvalue. The final formula reads z (1)0 = iβ Z (0)0 Ke − ~ S cl [ x (1) cl ( τ,ξ B )] . (S22)Recalling Eq. (S6), we must sum over many bounce paths. In the zero temperature limit we use the dilute gas ofbounces in which the different bounces do not interact with each other and S cl [ x ( n ) cl ( τ )] ’ n S cl [ x (1) cl ( τ )] and, in asimilar way, the integral over the fluctuations paths δx ( τ ) Z = Z (0)0 ∞ X n =0 n ! (cid:18) iβ (cid:19) n K n e − n ~ S cl [ x (1) cl ( τ,ξ B )] . (S23)Assuming that the amplitude decays exponentially as Z = 1 p π h x i e − β (cid:16) EGS ~ − i Γ (cid:17) , (S24)we finally find by comparison between Eq. (S23) and Eq. (S24)Γ = K e − ~ S cl . (S25) II. THE BOUNCE PATH AND THE CLASSICAL ACTION IN PRESENCE OF POSITION ANDMOMENTUM DISSIPATIONA. General integral formulas
The parametrization of the metastable potential displayed in Fig. S1a reads V ( x ) = mω x x < a mω ( x − x m ) − Σ a < x < x m − Σ x m < x . (S26)We set V = mω a / mω ( a − x m ) / − Σ = V . We obtain x m = a h p /V i for the point where the potential becomes flat.We use the Matsubara frequency decomposition for the classical bounce path that minimizes the action S = S + S dis defined in the main text. Hereafter we use the notation x (1) cl ( τ, ξ ) ≡ x cl ( τ, ξ ) x cl ( τ, ξ ) = 1 β ∞ X ‘ = −∞ x l ( ξ ) e iω l τ , (S27)with the Matsubara frequency ω l = 2 πl/β ( l integer) and ξ is the bounce time as defined in the previous section.From the condition dS/dx l = 0, we obtain the solution x l ( ξ ) = 2 ω a (cid:16) p /V (cid:17) sin( ω l ξ ) ω l (cid:18) ω l + ω + F ( x ) l m + ω l F ( p ) l m (cid:19) . (S28)Inserted in Eq. (S27) the path solves the equation of motion − m d dτ x cl ( τ, ξ ) + dV ( x ( τ )) dx | x cl + Z β − β dτ F ( x ) ( τ − τ ) x cl ( τ , ξ ) − Z β − β dτ F ( p ) ( τ − τ ) d dτ x cl ( τ , ξ ) = 0 , (S29)where we see that the kernel for momentum dissipation couples to the acceleration, as mentioned in the main text.Then, we insert the path into the action S and use the condition dS cl /dξ = 0 to find the relation determining thesaddle point of the bounce time, which we denoted ξ B , in presence of both dissipative couplings. In the limit β → ∞ ,the quantity ξ B solves the following integral equation1 π Z ∞ dωω ω sin( ωξ B ) (cid:16) ω τ p ωf c ( ω ) + ω + γωf c ( ω ) (cid:17) = 1 q Σ V + 1 . (S30)The integral equation determining the action reads S cl = − ω V (cid:16)q Σ V + 1 (cid:17) π Z ∞ dω (1 − cos ( ωξ B )) ω (cid:16) ω τ p ωf c ( ω ) + ω + γωf c ( ω ) (cid:17) + 2 V r V ! ξ B . (S31)The Eqs. (S30) and (S31) for ξ B and S cl correspond to the equations reported in the main text, which we computednumerically in the general case.Examples of the bounce path are shown in Fig. S2 for two fixed values of Σ. The dissipative interaction determinesthe path trajectory, in particular the bounce time ξ B and the returning point x esc . In Fig. S2, the black solid linecorresponds to the non dissipative case discussed in the main text, the blue dashed line shows the influence of positiondissipation, the red dotted line of the momentum dissipation and the green dotted-dashed line to both dissipativeinteractions. While the bounce becomes wider in presence of pure position dissipation, it shrinks for pure momentumdissipation. However, in both cases, the returning point x esc shifts to larger values. When both dissipative couplingsare present, for Σ = V in Fig. S2, the bounce is similar to the one of position dissipation only. By increasing Σ thisbehavior changes: in Fig. S2b the bounce almost coincides with the one of momentum dissipation. Hence, there is acrossover in the influence of the two different dissipative interactions which depends on the steepness of the potential. B. Analytical formulas and expansions
All analytical formulas are calculated in the limit ω c (cid:29) γ, τ p ω , ω meaning that the cutoff ω c is the largest frequencyin the problem. We will see that, while for pure conventional dissipation the cutoff is irrelevant in the limit ω c (cid:29) γ, ω ,it plays an important role in the case of momentum dissipation.First, we consider Eq. (S30) and expand the integral using the roots of the denominator. This yields the exactresult 1 π Z ∞ dω sin( ωξ B ) ω (cid:16) ω τ p ωf c ( ω ) + ω + γωf c ( ω ) (cid:17) = 12 1 ω + 1 πzω c X i =1 T i f [ ξ B ω c Λ i ] , (S32) Figure S2. Example of bounce paths with the two dissipative couplings for (a) Σ = V and (b) Σ = 20 V . The returning point x esc shifts to higher values in both dissipative cases. where 1 /z = (1 + τ p ω c ) and the coefficients T i = z − Λ i ( z + 1) + Λ i Λ i (Λ i − Λ j )(Λ i − Λ k )(Λ i − Λ l ) with ( j, k, l ) = i (S33)for i, j, k, l = 1 , , ,
4, whereas the auxiliary function is defined as f ( zx ) = Ci ( zx ) sin( zx ) + 12 cos( zx )( π − Si ( zx )) = Z ∞ du sin( z u )( x + u ) . (S34)The roots of the denominator in the limit ω c (cid:29) γ, τ p ω , ω are the same as in [S8]Λ , = 1(1 + σ ) ω ω c (cid:18) P + ± q P − − (cid:19) Λ , = 1 − ω ω c P + (1 + σ ) ± iσ (cid:18) ω ω c P + (1 + σ ) (cid:19) , (S35)with σ = γτ p , P + = γ + τ p ω ω and P − = γ − τ p ω ω . (S36)In the limit ξ B ω (cid:28) P − >
1, we find ξ B ω c Λ , = ξ B ω σ ) (cid:18) P + ± q P − − (cid:19) (cid:28) P − < ξ B ω c Λ , = ξ B ω σ ) (cid:18) P + ± i q | P − − | (cid:19) , (S38)and find in the limit ξ B ω (cid:28) ξ B ω c Λ , ) (cid:28) ξ B ω c Λ , ) (cid:28)
1. For the other two roots, we obtain ξ B ω c Λ , = ξ B ω c (cid:18) ± iσ − ω ω c P + (1 + σ ) ± ω ω c iσ P + (1 + σ ) (cid:19) ξ B ω (cid:28) ≈ ξ B ω c (1 ± iσ ) = ( Re( ξ B ω c (1 ± iσ )) (cid:28) , Im( ξ B ω c (1 ± iσ )) (cid:28) ξ B ω c (cid:28) ξ B ω c (1 ± iσ )) (cid:29) , Im( ξ B ω c (1 ± iσ )) (cid:29) ξ B ω c (cid:29) ξ B strongly depends on the shape of the potential, the different regimes in Eq. (S39) depend on Σ and thecutoff ω c .We conclude this section giving an analytic formula for Eq. (S31). The integral can be expanded similarly via thesame roots as in Eq. (S35), yielding the action S cl = 2 V (cid:16)q Σ V + 1 (cid:17) ω π γω ( C + log( ξ B ω )) − ω zω c X i =1 T i Λ i (cid:18) ln (cid:18) Λ i ω c ω (cid:19) + g [ ω c ξ B Λ i ] (cid:19)! − ξ B Σ , (S40)where the auxiliary function g is defined as g ( zx ) = − cos( zx ) Ci ( zx ) − sin( zx ) Si ( zx ) + 12 sin( zx ) π = Z ∞ du cos ( z u )( x + u ) . (S41)We see that the arguments of g [ x ] in Eq. (S40) are the same as the ones of f [ x ] in Eq. (S32) and therefore the limitsdefined in Eqs. (S37) and (S39) can be used. In the following, we will use the approximated solution for the roots toexpand the auxiliary functions g [ x ] and f [ x ] for the analytical formulas for the action.
1. The limit ξ B ω (cid:28) and ξ B ω c (cid:28) In the case of Σ (cid:29) V , the bounce time becomes almost zero and the condition ξ B ω (cid:28) ξ B →
0. Moreover, at fixed ω c , increasing Σ eventually leads to ξ B ω c (cid:28)
1. Then we can expand g [ x ] and f [ x ] forsmall arguments and we find ξ B ω = ~ mω q Σ V + 1 1 h x i , (S42)where we introduced the quantum fluctuations in presence of both dissipative couplings [S8] h x i = h x i π (1 + σ ) τ p ω (cid:18) ln (cid:18) ω c ω (cid:19) + σ arctan( σ ) + ln (cid:0) σ (cid:1)(cid:19) + (1 + τ p ω P − ) q(cid:12)(cid:12) − P − (cid:12)(cid:12) Θ q,p , (S43)where Θ q,p = arctan( q − P − /P + ) for | P − | < q P − − /P + ) for | P − | > . (S44)Here h x i are the harmonic quantum fluctuations in the well without dissipation. This result we also find by simplyexpanding the numerator sin( ξω ) ≈ ξω of the integrand in Eq. (S30), since it is peaked around ω ≈ S cl = ~ a h x i . (S45)This result coincides with the action of the potential with an infinitely sharp barrier, as expected in the limit Σ → ∞ .Expanding Eq. (S45) in the pure momentum dissipative case to first order in τ p ω we find Eq. (7) of the main text.The cutoff frequency ω c in h x i is related to the environment coupled to the momentum, while the effect of the cutofffrequency for the position bath drops out of the calculation in the limit ω c (cid:29) γ, ω .
2. The limit ξ B ω (cid:28) (cid:28) ξ B ω c Since we are in the regime ω c (cid:29) ω , there exists a regime ξ B ω (cid:28) (cid:28) ξ B ω c , which we may reach by reducing Σwith respect to the case discussed above (note that Σ (cid:29) V still holds). In this limit, we can expand the functions f [ ξ B ω c Λ , ] and g [ ξ B ω c Λ , ] for small arguments, but f [ ξ B ω c Λ , ] and g [ ξ B ω c Λ , ] for large arguments (see Eqs. (S37),(S39).We find for the bounce time equation1 π ξ B ω (1 + σ ) (1 + τ p ω P − )2 q P − − (cid:18) Λ Λ (cid:19) − τ p ω (cid:18) ln( ω ξ B ) −
12 ln(1 + σ ) + ( C − (cid:19) = 1 q Σ V + 1 , (S46)which is, in presence of momentum dissipation, a nonlinear equation to be solved numerically. In the limit herediscussed, the action reads S cl = V ω r V + 1 ! ξ B ω − V ω (cid:16)q Σ V + 1 + 1 (cid:17) π (1 + σ ) ξ B ω τ p ω . (S47)Hence, by inserting Eq. (S46) into Eq. (S47), the action depends logarithmically on the bounce time ξ B and thereforeon Σ. We also see, that this dependence vanishes in the limit τ p = 0, meaning that momentum dissipation induces astronger dependence on the shape of the barrier as discussed in the main text. The combined Eqs. (S46) and (S47)are displayed via the dashed lines in Fig. 2b of the main text. Further, for the case γ = 0 expanding (S46) and (S47)to first order in τ p ω leads to Eq. (8) in the main text.
3. The limit ξ B ω (cid:29) and ξ B ω c (cid:29) Finally we analyze the regime ξ B ω (cid:29) ξ B ω c (cid:29) (cid:28) V . The action can then beapproximated as S cl ≈ (cid:15) ω + 8 π V ω γω ln ( ξ B ω ) − Σ ξ B , (S48)where ξ B ω ≈ π V Σ γω and (cid:15) ω = 8 π V ω γω C − ln (cid:0) σ (cid:1) ! − (cid:16) γω P − − (cid:17) q P − − (cid:18) Λ Λ (cid:19) , (S49)with C the Euler constant. In particular, in the limit γ = 0 and τ p ω (cid:28) S cl ≈ V ω − π V τ p (S50)yielding Eq. (9) for E in the main text. Note that for τ p = 0 and γ = 0 we recover the action for the incoherent decayin a (slightly) asymmetric parabolic double well in presence of dissipation. [S9] S10 . In particular, within the steepestdecent approximation, the quantity E can be approximated as E Σ (cid:28) V τ p =0 ≈ e − (cid:15) γ ) ~ ω (cid:18) π γω V Σ (cid:19) − π V ~ ω γω , (S51)with (cid:15) ( τ p = 0) ~ ω = 8 π V ~ ω γω ( C − − V ~ ω π (cid:16) γω P − − (cid:17) q P − − (cid:18) Λ Λ (cid:19) + 1 , (S52)and the decay is exponentially suppressed in presence of pure position dissipation. The Eq. (S51) shows that E depends on Σ only via the prefactor of the exponential function and is independent of the cutoff ω c . III. THE PREFACTOR K
In this section we present an overview of the calculation of the prefactor K , defined in Eq. (S21), for the potentialshown in Fig. S1a. For a more detailed introduction we refer to Ref. [S9]. Example of results for the prefactor K arereported in Fig. S3d in which K is scaled with its value in absence of dissipation K . Similarly to the exponentialfunction, K is enhanced in presence of momentum dissipation. A. The ratio between the determinants
We start by calculating the ratio of determinants defined in Eq. (S21), namely the ratio between by the two productsof the two sets of eigenvalues R = Q ∞ q =0 q λ (0) q Q ∞ q =2 q λ ( B ) q . (S53)The eigenvalues of the bounce path λ ( B ) q are defined via Eq. (S4), while the eigenvalues λ (0) q are associated to thefollowing equation. (cid:18) − m d dτ + mω (cid:19) y (0) q ( τ ) + Z β − β dτ F ( x ) ( τ − τ ) y (0) q ( τ ) − Z β − β dτ F ( p ) ( τ − τ ) d dτ y (0) q ( τ ) = λ (0) q y (0) q ( τ ) . (S54)Taking the second derivative of the potential in Eq. (S4) along a bounce trajectory yields V [ x (1) cl ] = d V [ x ( τ )] dx (cid:12)(cid:12)(cid:12) x (1) cl .Assuming that the time τ is the space and y ( τ ) the wavefunction, in absence of dissipation, the equation correspondsto the Schr¨odinger equation with a potential at constant value mω containing two delta-potential wells (at the timeswhen the periodic path crosses the discontinuity at x = a ). Each well has one bound state and the finite size of thebounce (determined by ξ B ) leads to a hybridization of the two wells yielding the two bound states λ ( B )0 and λ ( B )1 (denoting the zero mode due to translational invariance and the negative eigenvalue of the breathing mode). The restof the eigenvalues forms a continuum above mω .We rewrite the ratio of the determinants containing only the continuum eigenvalues defined in Eq. (S21) via R = Q ∞ q =0 q λ (0) q Q ∞ q =2 q λ ( B ) q = e R ∞ mω dλ ln( λ )( ρ ( λ ) − ρ ( λ )) , (S55)where we defined the spectral densities ρ ( λ ) = P ∞ q =2 δ (cid:16) λ ( B ) q − λ (cid:17) and ρ ( λ ) = P ∞ q =0 δ (cid:16) λ (0) q − λ (cid:17) . We rewrite thespectral densities using the (retarded) Greens functions of the respective problem ρ ( λ ) = 1 π Im (cid:16) G (0) λ (0) (cid:17) , and ρ ( λ ) = 1 π Im (cid:16) G ( B ) λ (0 , (cid:17) , (S56)in which G (0) λ is given by G (0) λ ( τ ) = 1 πm Z ∞ dω cos( ωτ ) ω τ p ωf c ( ω ) + ω + γω − λm − i(cid:15) , (S57)with (cid:15) →
0. For simplicity we do not consider a high frequency cutoff for the environment coupled to the positionas it is irrelevant. By contrast, the cutoff for the momentum bath has to be finite, otherwise R diverges in the limit ω ( p ) c → ∞ . Further, we determine G ( B ) λ in terms of G (0) λ via the Lippmann-Schwinger equation G ( B ) λ ( τ, τ ) = G (0) λ ( τ − τ ) − Z ∞−∞ dτ G (0) λ ( τ − τ ) V ( x (1) cl ( τ )) G ( B ) λ ( τ , τ ) . (S58)Following the calculation outlined in [S9 and S11] we findln( R ) = 12 π (cid:2) ln ( λ ) (cid:0) φ + λ + φ − λ (cid:1)(cid:3) ∞ mω − Z ∞ mω dλ λ (cid:0) φ + λ + φ − λ (cid:1)! , (S59)with φ ± λ = arg (cid:16) U − − (cid:16) G (0) λ (0) ± G (0) λ ( ξ B ) (cid:17)(cid:17) and U = mω ( a + x m ) / | ˙ x cl ( ξ B ) | . The phases satisfy φ ± mω = − π andlim λ →∞ φ ± λ = 0 (calculated in Sec. III D) leading to the resultln( R B ) = ln (cid:0) mω (cid:1) − π Z ∞ mω dλ λ (cid:0) φ + λ + φ − λ (cid:1) . (S60)We display the numerical results for R rescaled with the value R in absent of dissipation for different values of Σ inFig. S3a and as function of the dissipative coupling strength.0 B. Determination of the negative eigenvalue λ B We use the above Greens function to obtain the solution for the negative eigenvalue via the poles of the T-matrixdefined through the Lippmann Schwinger Eq. (S58), see Ref.[S9]. The negative eigenvalue is determined by theequation | ˙ x cl ( ξ B | − mω a r V ! (cid:18) G (0) −| λ ( B )1 | ( ξ B ) + G (0) −| λ ( B )1 | (0) (cid:19) = 0 . (S61)In the non dissipative case, we can solve the integral (S57) leading to the following equation for the negative eigenvaluewhich only depends on the bounce path via the bounce time ξ (0) B ω (cid:16) − e − ω ξ (0) B (cid:17) − e − ξ (0) B r ω + | λ ( B, | m q ω + | λ ( B, | m = 0 . (S62)In Fig. S3c we show numerical results for the value λ ( B )1 in presence of both dissipative couplings, for differentvalues of Σ and scaled with its value λ ( B, without dissipation. By increasing the dissipation the absolute value of | λ ( B, | / | λ ( B )1 | is enhanced, leading to a larger contribution to the prefactor. Figure S3. Change in the quantities contained in the prefactor K due to dissipation for τ p ω = 0 . γ/ω and ω c /ω = 8000. (a)Ratio of the determinants scaled with its value without dissipation, (b) prefactor of the Jacobian transformation scaled with itsvalue without dissipation, (c) nagative eigenvalue scaled with its value without dissipation. (d) The prefactor K of the escaperate scaled with its value K without dissipation. C. The Jacobian prefactor A τ Finally we discuss the result for the Jacobian prefactor [S9 and S11]1 A τ = sZ β/ − β/ dτ (cid:16) ˙ x (1) cl ( τ ) (cid:17) . (S63)In the non dissipative case we find the analytic expression Z β/ − β/ dτ (cid:16) ˙ x (1) cl ( τ ) (cid:17) ≈ a ω r V + 1 ! (cid:16) − e − ω ξ (0) B (1 + ω ξ (0) B ) (cid:17) . (S64)The numerical value A τ scaled with A (0) τ is shown in Fig. S3b. D. Calculation of the phases φ ± λ In this section we give a detailed derivation of the quantities φ ± λ appearing Eq. (S59).We start by calculating the Greens function G (0) λ ( τ ). In presence of Ohmic momentum dissipation without highfrequency cutoff this quantity diverges similarly to the position quantum fluctuations. We recall the dissipative kernelin Matsubara space for momentum dissipation with cutoff ω c F ( p ) ( ω ) = − τ p | ω | mf c ( ω )1 + τ p | ω | f c ( ω ) , (S65)with f c ( ω ) = (1 + | ω | /ω c ) − . We rewrite Eq. (S57) and find G (0) λ ( τ ) = 1 πmω c Z ∞ dx (1 + (1 + τ p ω c ) x ) cos( xω c τ ) − p Ω c + χ ( − p ) x + αx + x − i(cid:15) , ( (cid:15) → , (S66)with p = − λ/mω , Ω c = ω /ω c , α = γ/ω c + τ p γ + 1, and χ ( − p ) = − p Ω c (1 + τ p ω c ) + γω c . (S67)The denominator of Eq. (S66) is a cubic polynomial with an imaginary part. We expand the polynomial into its roots˜ x , , and obtain, by introducing ˜ x = ν , ˜ x = − ν and ˜ x = − ν , − p Ω c + χ ( − p ) x + αx + x = ( x − ν )( x + ν )( x + ν ) , (S68)where ν , , >
0. Because the full form of the quantities ν , , is not important at this stage, we do not present themhere explicitly, but refer to the next section. With this definition we can perform a principle value integration inEq. (S66). We obtain the result G (0) λ ( τ ) = 1 πmω c X i =1 e U i g [ τ ω c ν i ] − e U π sin( τ ω c ν ) ! + imω c (1 + (1 + τ p ω c ) ν ) cos( ν ω c τ ) | χ ( − p ) + 2 αν + 3 ν | , (S69)where the prefactors e U i originate from a partial fraction expansion (defined in the next section) and the auxiliaryfunction g ( x ) is defined in the previous section. For the factor U − we have to calculate˙ x (2) cl (cid:18) ξ B (cid:19) = 2 ω a π Z ∞ dx (1 + (1 + τ p ω c ) x ) [1 − cos( ξ B ω )]Ω c + χ (1) x + αx + x , (S70)which has no imaginary part. We calculate the integral by rewriting the polynomial in the denominator asΩ c + χ (1) x + αx + x = ( x + k )( x + k )( x + k ) , (S71)2with Re( k i ) >
0, and find U − = − πmω c X i =1 e T i (ln ( k i ) + g [ k i ω c ξ ]) , (S72)where the prefactors e T i and the quantities k i are also defined in the next section. Inserting τ = ξ B in Eq. (S69) andcalculating G (0) p ( τ = 0) we obtain the result n ( ± ) λ = − πmω c X i =1 e T i (ln ( k i ) + g [ k i ω c ξ B ]) + 1 πmω c X i =1 e U i (ln( ν i ) ± g [ ξ B ω c ν i ]) ∓ e U π sin( ξ B ω c ν ) ! − imω c (1 + (1 + τ p ω c ) ν ) (1 ∓ cos( ν ω c ξ B )) | χ ( − p ) + 2 αν + 3 ν | . (S73)The phases are then calculated via φ ( ± ) λ = arg( n ( ± ) λ ) and we use φ ( ± ) mω = − π and lim λ →∞ φ ( ± ) λ = 0 in Eq. (S59). E. Further auxiliary variables
In the previous section we introduced the quantities e T i , e U i , ν i , and k i . The first two originate from the partial fractionexpansions of the integrands in Eqs. (S66) and (S70). The latter ones are related to the roots of the denominators of e G (0) λ ( ω ) and U − . The prefactors read e T = 1 − (1 + τ p ω c ) k ( k − k )( k − k ) e T = − τ p ω c ) k ( k − k )( k − k ) e T = − τ p ω c ) k ( k − k )( k − k ) (S74)and e U = 1 + (1 + τ p ω c ) ν ( ν + ν )( ν + ν ) e U = 1 − (1 + τ p ω c ) ν ( ν + ν )( ν − ν ) e U = 1 − (1 + τ p ω c ) ν ( ν + ν )( ν − ν ) . (S75)Using the basic formula for the roots of cubic polynomials, we find for the ones of Eq. (S71) − k = − α η (1)3 (cid:16) Σ(1) + p η (1) + Σ (cid:17) − (cid:16) Σ(1) + p η (1) + Σ (1) (cid:17) · − k = − α − (1 + i √ η (1)3 · (cid:16) Σ(1) + p η (1) + Σ (1) (cid:17) + (1 − i √ (cid:16) Σ(1) + p η (1) + Σ (1) (cid:17) · (S76) − k = − α − (1 − i √ η (1)3 · (cid:16) Σ(1) + p η (1) + Σ (1) (cid:17) + (1 + i √ (cid:16) Σ(1) + p η (1) + Σ (1) (cid:17) · , η (1) = 3 χ (1) − α and Σ(1) = 9 αχ (1) − α − c (and used p = − λ/mω , Ω c = ω /ω c , α = γ/ω c + τ p γ + 1). Further, the roots for the polynomial (S68) read ν = − α η ( p )3 (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) − (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) · − ν = − α − (1 + i √ η ( p )3 · (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) + (1 − i √ (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) · (S77) − ν = − α − (1 − i √ η ( p )3 · (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) + (1 + i √ (cid:16) Σ( p ) + p η ( p ) + Σ ( p ) (cid:17) · , where η ( p ) = 3 χ ( − p ) − α and Σ( p ) = 9 αχ ( − p ) − α + 27 p Ω c . IV. THE TUNNELING AVERAGE ENERGY LOSS IN PRESENCE OF DISSIPATION
As discussed in the main text the returning point can be used to calculate the average energy loss h ∆ E i of theparticle during the tunneling in the presence of the dissipative interaction with the environment according the equation h ∆ E i = V (0) − V ( x esc ) [S9].In Fig. S4a we show the results for the average energy loss as a function of the dissipative coupling strength definedas c S = γ/ω for the position dissipation and as c S = ω τ p for the momentum dissipation.In Fig. S4a the blue solid line corresponds to pure position dissipation ( c S = γ/ω , τ p = 0) while the red dashedline to the results for pure momentum dissipation ( c S = τ p ω , γ = 0). In the overdamped limit c S (cid:29) V position dissipation dissipates more energy than momentum dissipation whereas for Σ = 10 V the situation isreversed: momentum dissipation has a larger influence.In Fig. S4b we fix the dissipative coupling strength and show h ∆ E i as a function of Σ /V for pure momentumdissipation. The loss saturates, similarly to the escape rate, to a value determined by the high frequency cutoff ω c . Figure S4. (a) The average energy loss h ∆ E i as a function of the dissipative coupling strength, c s = γ/ω for position dissipationand c s = ω τ p , for two values of Σ, for pure position dissipation (solid blue line) and for pure momentum dissipation (dashedred line) In the overdamped limit h ∆ E i saturate to the same value. (b) Saturation of the energy loss in presence with puremomentum dissipation as a function of Σ, for different ω c at τ p ω = 0 . The Uses of Instantons, Instantons in gauge theories , Subnucl. Ser. , 805 (1979). [S2] H. Grabert, U. Weiss, and P. H¨anggi, Quantum Tunneling in Dissipative Systems at Finite Temperatures,
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