Abstract
We consider a particle trapped by a generic external potential and under the influence of a quantumthermal Ohmic bath. Starting from the Langevin equation, we derive the corresponding SchwingerKeldysh action. Then, within the pathintegral formalism, we obtain both the semiclassical FokkerPlanck equation and the quantum FokkerPlanck equation for this outofequilibrium system. In the case of an external harmonic potential and in the underdamped regime, we find that our FokkerPlanck equations contain an effective temperature \(T_{\text {eff}}\), which crucially depends on the interplay between quantum and thermal fluctuations in contrast to the classical FokkerPlanck equation. In the regime of high temperatures, one recovers the classical FokkerPlanck equation. As an application of our result, we also provide the stationary solution of the semiclassical FokkerPlanck equations for a superconducting Josephson circuit and for a Bose Josephson junction, which are experimentally accessible.
1 Introduction
Inspired by the works of Einstein [1] and Smoluchowski [2] about the Brownian motion of a mesoscopic particle in a fluid, in 1908 Langevin introduced its stochastic equation [3]. Some years later, Johnson [4] and Nyquist [5] observed that, in addition to thermal effects, also the quantum mechanical noise plays a relevant role in the electric current of conductors. In 1951, the quantum version of the fluctuationdissipation theorem of Callen and Welton [6] paved the way to the quantum Langevin equation [7,8,9]. In the first part of this paper, we explicitly show that, from the Langevin equation of a confined particle in a quantumthermal Ohmic bath [10], one derives the corresponding semiclassical SchwingerKeldysh action [11, 12]. The SchwingerKeldysh action is usually obtained by adopting a quite different approach, which involves a closed time contour with forward and backward branches in time and where the dynamical variables of the system are doubled to take into account the two branches [10,11,12]. Moreover, it is well known that from the SchwingerKeldysh action one obtains in the hightemperature regime the classical MartinSiggiaRose action [13]. Here, we follow a quite different path: for a particle in contact with a quantumthermal bath, its semiclassical SchwingerKeldysh action is obtained directly from the Langevin equation through a MartinSiggiaRose action, which is indeed the semiclassical SchwingerKeldysh action functional.
In the second part, we derive the main results of the paper: the FokkerPlanck equations of our specific system. These FokkerPlanck equations are partialdifferential equations describing the probability density P(q, v, t) of finding a particle with position q and velocity v at time t [14,15,16,17,18,19,20] in a quantumthermal Ohmic bath and trapped by a deterministic external potential V(q). From the shorttime propagator of the transition probability associated with the SchwingerKeldysh action, we obtain both the semiclassical and the quantum FokkerPlanck equation for a confined particle under the effect of a quantumthermal Ohmic bath. Remarkably, our FokkerPlanck equations are fully analytical and contain an effective temperature \(T_{\text {eff}}\). This effective temperature crucially depends on the interplay between quantum fluctuations, characterized by \(\hbar \Omega\) with \(\hbar\) the reduced Planck constant and \(\Omega\) the frequency of harmonic potential, and thermal fluctuations, characterized by the thermal energy \(k_{\text {B}}T\) with \(k_{\text {B}}\) the Boltzmann constant and T the temperature. Following our approach, we also obtain quantum FokkerPlanck equations for the superconducting phase in a Josephson circuit and for the population imbalance in an atomic Bose Josephson junction, which are described by generalized Langevin equations including quantum and thermal fluctuations.
2 Langevin equation for a particle in a quantumthermal Ohmic bath
Let us consider a particle of mass m and coordinate q(t) under the action of a deterministic potential V(q(t)) but also of a thermal bath which induces a dissipative force \(\gamma {\dot{q}}(t)\) with damping coefficient \(\gamma\) and a Gaussian stochastic force \(\xi (t)\). The stochastic coordinate q(t) of the particle satisfies the equation of motion
Here, we assume the Markovian dynamics such that the damping term \(\gamma \dot{q}(t)\) does not include any memory effect and the equation of motion (1) involves only one time variable t. Given a generic observable O which depends explicitly on the Gaussian random variable \(\xi (t)\), the stochastic average has the following path integral representation
which crucially depends on the choice of the correlation function C(t), which, in general, is that \(C(tt')=\langle \xi (t) \xi (t')\rangle\).
Equation (1) is called semiclassical quantum Langevin equation [7,8,9] provided that the correlation function C(t) is given by
where T is the absolute temperature, \(k_{\text {B}}\) is the Boltzmann constant, and \(\hbar\) is the reduced Planck constant. The correlation function of Eq. (3) is the one of a stochastic quantumthermal Ohmic bath [7,8,9]. The presence of \(\gamma\) both in Eqs. (1) and (3) is a consequence of the fluctuationdissipation theorem (FDT) [10]. Moreover, in Eq. (3) the term \(e^{i\omega \, t}\) can be substituted by \(\cos ({\omega \, t})\) because the imaginary part is odd and its integral gives zero. Equation (1) with Eq. (3) is called semiclassical because the dynamical variable q(t) is not a quantum operator. The quantum nature of Eq. (1) is however encoded in the correlator C(t) through Eq. (3). Note that in the hightemperature regime \(k_{\text {B}}T\gg \hbar \omega\), where \(\coth \left( {\hbar \omega /(2 k_{\text {B}} T)}\right) \rightarrow {2k_{\text {B}}T/(\hbar \omega )}\), Eq. (3) gives \(C(t) = 2 \gamma \ k_{\text {B}} T \ \delta (t)\) with \(\delta (t)\) the Dirac delta function and Eq. (1) becomes the familiar classical Langevin equation [3].
3 MartinSiggiaRose action
To derive the MartinSiggiaRose action [13] (see also Refs. [21,22,23,24,25,26]) from the semiclassical quantum Langevin equation (1), we observe that the expectation value of the generic observable O can also be written as
where the Dirac delta function \(\delta (x)\) appears because one considers the path integral over all possible q(t) but with the constraint that q(t) satisfies Eq. (1). This constraint ensures the correct implicit dependence of q(t) with respect to \(\xi (t)\). Taking into account the path integral representation of \(\delta (x)\), we have
where \({\tilde{q}}(t)\) is an auxiliary response field. We now use Eq. (2) and the properties of Gaussian integrals obtaining
where
remembering that the stochastic noise \(\xi (t)\) is timetranslation invariant, namely \(C(tt')=\langle \xi (t) \xi (t') \rangle = \langle \xi (tt') \xi (0)\rangle\). The functional of Eq. (7), with \(C(t)=\langle \xi (t) \xi (0)\rangle\) given by Eq. (3), is our MartinSiggiaRose action [13].
We have seen that in the hightemperature regime \(C(t)=2\gamma k_{\text {B}} T \delta (t)\). In this regime, Eq. (7) is much simpler and the action is known as the classical MartinSiggiaRose action [13]. Quite remarkably, in this classical hightemperature regime, one can easily perform the path integral over the dual variable \({\tilde{q}}(t)\), obtaining
with the real effective action
that is called OnsagerMachlup [27] action according to Olender and Elber [28].
4 Semiclassical SchwingerKeldysh action
Let us restrict ourselves to a harmonic potential \(V(q)=m\Omega ^{2}q^{2}/2\). In the underdamped regime \(\gamma \ll m\Omega ^{2}/\omega _{\text {cut}}\) with \(\omega _{\text {cut}}\) being the ultraviolet cutoff frequency associated with the zeropoint fluctuations in Eq. (3), we apply the approximation [29, 30]
which is white noise including quantum fluctuations. The effective temperature is defined by
This approximation is justified for the following reason. The Langevin equation (1) in the longtime limit gives [31]
where \(\tilde{\xi }(\omega )=\int ^{\infty }_{\infty }dt\xi (t)e^{i\omega t}\) and
is the response function in the frequency domain. Equation (12) leads to the correlation function
with \(\tilde{C}(\omega )=\int ^{\infty }_{\infty }dte^{i\omega t}C(t)=\gamma \hbar \omega \coth {[\hbar \omega /(2k_{\text {B}}T)]}\). The autocorrelation function of the conjugate momentum also involves the combination of \(\tilde{C}(\omega )\left\ {\tilde{\chi }(\omega )}\right\ ^{2}\). In the underdamped limit \(\gamma \ll m\Omega ^{2}/\omega _{\text {cut}}\), the response function (13) is dominant only around \(\omega =\pm \Omega\) as
Consequently, in Eq. (14), we can safely use approximation that \(\tilde{C}(\omega )\simeq \tilde{C}(\Omega )\), which justifies the white noise approximation in Eq. (10) in the underdamped limit \(\gamma \ll m\Omega ^{2}/\omega _{\text {cut}}\). In the classical limit \(\hbar \Omega \ll k_{\text {B}}T\), we recover \(T_{\text {eff}}=T\). In the lowtemperature regime \(\hbar \Omega \gg k_{\text {B}}T\), on the other hand, Eq. (10) provides \(T_{\text {eff}}=\hbar \Omega /(2k_{\text {B}})\).
By using Eq. (10), the action (7) with the kernel reads
Remarkably, Eq. (16) is very similar to the SchwingerKeldysh action [11, 12] of a quantum particle in contact with an Ohmic bath. The only difference is due to the fact that instead of \(\hbar {\tilde{q}}\, \partial V(q)/\partial q\) in the exact SchwingerKeldysh action there is \([V(q+\hbar {\tilde{q}})V(q\hbar {\tilde{q}})]/2\). See, for instance, page. 33 of Ref. [10]. Clearly, \([V(q+\hbar {\tilde{q}})V(q\hbar {\tilde{q}})]/2\simeq \hbar {\tilde{q}}\, \partial V(q)/\partial q\) under the assumption of a small \(\hbar {\tilde{q}}\). It is important to stress that the EchernSchonAmbegaokar action [32] used for superconducting Josephson junctions is nothing else than the exact SchwingerKeldysh action. At zero temperature, the SchwingerKeldysh action has been used to study the effect of quantum noise in the quantum phase transition of a Josephson junction [33]. In the classical limit \(T_{\text {eff}}\rightarrow T\), one can readily find that Eq. (16) coincides with the classical dissipative action [10].
We call Eq. (16) semiclassical SchwingerKeldysh action. Indeed, the classical SchwingerKeldysh action, which is nothing else than the classical MartinSiggiaRose action, is obtained with \(T_{\text{eff}} \to T\). Then, by functional integrating over \({\tilde{q}}(t)\) one recovers again the OnsagerMachlup action (9).
5 FokkerPlanck equations
In this section, we derive the corresponding FokkerPlanck equation for the MSR action in Eq. (7) or the SchwingerKeldysh action in Eq. (16).
5.1 Semiclassical FokkerPlanck equation
Let us rewrite the action of Eq. (7) as
where \(S_{\text {e}}[q(t),v(t),\tilde{q}(t),\lambda (t)]\) is a new effective action with a velocity field v(t) and an auxiliary field \(\lambda (t)\) as a Lagrange multiplier that guarantees \(v(t)=\dot{q}(t)\) by \(\delta S_{\text {e}}/\delta \lambda =0\) [34], and
The effective Lagrangian \(L_{\text {e}}[q(\bar{t}),v(\bar{t}), \tilde{q}(\bar{t}),\lambda (\bar{t})]\) can be used to introduce the propagator [34]
which gives the transition probability from the initial configuration (q, v, t) to the final configuration \((q',v',t')\). It follows that the probability \(\mathcal {P}(q,v,t)\) of finding the system in the configuration (q, v, t) satisfies the convolution equation
In Appendix 1, we show how to derive the semiclassical FokkerPlanck equation from this expression taking into account Eq. (20). The final result of this derivation is [34, 35]
under Eq. (10) where V(q) is the harmonic potential of frequency \(\Omega\). In the classical limit \(\hbar \rightarrow 0\), or equivalently \(k_{\text {B}}T\gg \hbar \Omega\), Eq. (22) reduces to the familiar classical one [10, 36, 37]
On the other hand, at \(T=0\), the effective temperature has a finite minimum \(T_{\text {eff}}=\hbar \Omega /\left( 2k_{\text {B}}\right)\) as shown in Fig. 1. Figure 1 illustrates that the difference from the original temperature is significant in the lowtemperature regime due to the quantum effects.
Quite remarkably, the semiclassical FokkerPlanck equation (22) has the following stationary analytical solution
where
With a harmonic potential \(V(q)=m\Omega ^{2}q^{2}/2\), the stationary solution gives the second moments of the coordinate and velocity as
and
These second moments are exactly consistent with a Langevin analysis in the underdamped limit [31, 38]. In the classical limit \(T_{\text {eff}}\rightarrow T\), Eqs. (26) and (27) recover the equipartition of energy. Due to the quantum correction in Eq. (11), both of Eqs. (26) and (27) are proportional to \(\hbar \Omega\) in the lowtemperature regime.
Note that Eqs. (26) and (27) are the underdamped results which are, strictly speaking, justified in the underdamped limit \(\gamma \ll m\Omega ^{2}/\omega _{\text {cut}}\). With finite damping constant \(\gamma >m\Omega ^{2}/\omega _{\text {cut}}\), the second moments of the velocity should have a logarithmic ultraviolet divergence [38]. We stress that, however, we are restricting ourselves to the underdamped limit and do not consider such cases throughout this paper. Only in this underdamped limit, do the second moments of the coordinate or the momentum have clear physical interpretation as they recover the equipartition of energy in the classical limit [31, 38].
5.2 Quantum FokkerPlanck equation
As mentioned in Section 4, the dissipative SchwingerKeldysh action is different from the MSR action in terms of the potential term in the higher order of \(\tilde{q}\). Let us call the corresponding FokkerPlanck equation derived from the SchwingerKeldysh action the quantum FokkerPlanck equation since it includes higher order of the quantum component \(\tilde{q}\), which should coincide with the semiclassical one to the first order of \(\tilde{q}\).
To obtain the quantum FokkerPlanck equation, we can simply substitute the potential term in Eq. (18) as
where \(V^{(k)}(q)\) denotes the kderivative of V(q) in q. Since it is timelocal, one can proceed to the FokkerPlanck equation in a similar manner as in the last section. Eventually, one obtains the quantum FokkerPlanck equation
where \(\tilde{F}\), that gives the difference from semiclassical equation (22), is defined as
For a harmonic potential \(V(q)=m\Omega ^{2}q^{2}/2\), as in the last equality in Eq. (30), the higherorder derivatives in Eq. (30) vanish. With a generic potential, the presence of infinite derivatives makes it quite difficult to solve it in full generality. While a truncation of the series with \(n=1\) or \(n=1,2\) could be used to find reliable corrections to the semiclassical result, it may lead to negative probability distribution [39].
6 FokkerPlanck equation for Josephson junctions
In a resistively and capacitively shunted Josephson (RCSJ) junction, the superconducting phase \(\phi (t)\) obeys the generalized Langevin equation [31, 40]
where \(E_{C}=(2e)^{2}/(2C)\) is the charging energy with a capacitance C, \(\alpha =R_{Q}/R\) is the ratio between the critical resistance \(R_{Q}=h/(2e)^{2}\) and the resistance R, \(E_{J}\) is the Josephson energy, \(I_{\text {ext}}\) is the external current, and \(\Phi _{0}=h/(2e)\) is the magnetic flux quantum. The current noise \(\xi (t)\) originates from the shunted resistor and is assumed to satisfy the FDT in Eq. (3). It has been theoretically predicted that the system is superconducting below the critical resistance \(\alpha >1\) while it is insulating above the resistance \(\alpha <1\) [41, 42]. The ultraviolet cutoff frequency can be chosen as \(\omega _{\text {cut}}=\Delta /\hbar\) with \(\Delta\) being the superconducting gap. The approximation (10) is valid within the underdamped limit \(\alpha \ll 2\pi E_{J}/\Delta\) [31].
We can identify this Langevin equation as Eq. (1) by replacing \(q(t)\rightarrow \phi (t)\), \(v\rightarrow \dot{\phi }=2\pi V/\Phi _{0}\), \(m\rightarrow \hbar ^{2}/(2E_{C})\), \(\gamma \rightarrow \hbar \alpha /(2\pi )\), and \(V(q)\rightarrow U_{\text {wash}}[\phi ]\equiv E_{J}\cos {\phi }\Phi _{0}I_{\text {ext}}/(2\pi )\cdot \phi\). Consequently, following the procedures in the last sections, we obtain the quantum FokkerPlanck equation for the RCSJ junction as
with \(V(t)=\Phi _{0}\dot{\phi }/(2\pi )\) the voltage. The function \(\tilde{F}[\phi ,V]\) is given by
where, in the last row, we used \(U_{\text {wash}}[\phi ]\simeq E_{J} \Phi _{0}I_{\text {ext}}/(2\pi )\cdot \phi +E_{J}\phi ^{2}/2\). We can find the semiclassical stationary solution as
which is illustrated in Fig. 2(a) for \(E_{J}/E_{C}=1\), \(k_{\text {B}}T/E_{C}=0.1\), and \(\Phi _{0}I_{\text {ext}}/(2\pi E_{J})=0.3\). Also in Fig. 2(a), we use \(U_{\text {wash}}[\phi ]\simeq E_{J}\Phi _{0}I_{\text {ext}}/(2\pi )\cdot \phi +E_{J}\phi ^{2}/2\). The effective temperature is given by Eq. (11) with \(\Omega =(2E_{C}E_{J})^{1/2}/\hbar\). Equation (34) indicates that, in the stationary configuration, the voltage V is more localized as one increases \(E_{C}/(k_{\text {B}}T_{\text {eff}})\), and the superconducting phase \(\phi\) is localized with a large \(E_{J}/E_{C}\). Experimentally, one can observe \(E_{J}/E_{C}\simeq 3.8\times 10^{5}\) in a RCSJ circuit [43], which reflects the highly localized phase in the superconducting circuit.
In a onedimensional Bose Josephson junction (BJJ) in a headtotail configuration, we have two onedimensional Bose gasses in contact through a tunnel coupling \(J(x)=J_{0}L\delta (x)\) at a point \(x=0\) where \(J_{0}\) is the strength of the Josephson coupling and L is the system size. The zeromode of the population imbalance \(\zeta _{0}\) also obeys the Langevin equation [44, 45]
in the linear regime \(\left\ {\zeta _{0}(t)}\right\ \ll 1\) with \(\Omega\) the Josephson frequency, \(\gamma\) the damping constant associated with the Josephson coupling \(J_{0}\), M the effective mass related to the interparticle interaction strength g, and \(\bar{\rho }\) the average atomic density. The stochastic noise \(\xi (t)\) satisfies the FDT in Eq. (3). The ultraviolet cutoff can be chosen as \(\omega _{\text {cut}}=2\pi c\bar{\rho }\simeq 10^{4}\Omega\) where \(c=(g\bar{\rho }/m)^{1/2}\) is the speed of sound and m is the atomic mass [45, 46]. Then, the approximation (10) is valid if \(\bar{N}J_{0}/Mc^{2}\ll 10^{8}\) with \(\bar{N}\equiv \bar{\rho }L\) being the average number of atoms, which is the Josephson regime in which the tunneling energy \(\bar{N}J_{0}\) is much smaller than the kinetic energy \(Mc^{2}\). For this onedimensional BJJ, we find the quantum FokkerPlanck equation as
with \(\Omega _{\text {R}}\equiv J_{0}/\hbar\) the Rabi frequency and \(\phi _{0}(t)=\dot{\zeta }_{0}(t)/\Omega _{\text {R}}\) the zeromode of the relative phase. The function \(\tilde{F}(\zeta _{0},\phi _{0})\) is given by
The linearized equation (35) involves a harmonic potential and gives no quantum correction that stems from the higherorder derivatives of the external potential. The stationary solution is given by
We show the stationary solution in Fig. 2(b) for \(\Omega /\Omega _{\text {R}}=0.5\) and \(k_{\text {B}}T/(\hbar \Omega )=0.1\). Figure 2(b) shows that the relatively localized \(\phi _{0}\) around the origin and the delocalized \(\zeta _{0}\) are realized as a stationary configuration. As one decreases the ratio between the interaction energy and the tunneling energy \(\Omega /\Omega _{\text {R}}=\sqrt{2g\bar{\rho }/J_{0}}\) less than one, the population imbalance \(\zeta _{0}\) is delocalized and the relative phase \(\phi _{0}\) is highly localized in the stationary configuration given by Eq. (38).
7 Conclusions
In the first part of this paper, we have derived the SchwingerKeldysh action of a particle under the effect of a deterministic external potential and a stochastic Ohmic bath, which contains both thermal and quantum fluctuations. Contrary to previous papers [10,11,12], our derivation has been performed starting from the Langevin equation of the system. In the second part of the paper, we have then adopted the SchwingerKeldysh action to include the velocity of the particle by using a HubbardStratonovich transformation and to derive the fully analytical semiclassical and quantum FokkerPlanck equations for the timedependent probability of the particle in the quantumthermal Ohmic bath. The semiclassical FokkerPlanck equation involves the effective temperature associated with the frequency of harmonic potential. The obtained results can be applied to various contexts. In Section 6, we wrote down the quantum FokkerPlanck equations for the Josephson mode in an atomic Josephson junction and for the superconducting phase in a superconducting Josephson circuit. We showed the stationary solution of the semiclassical FokkerPlanck equation for each of the Josephson systems. These Josephson systems have been experimentally realized and attracted marked attention. We expect that our work would also contribute to the understanding of such a noisy Josephson junction. For instance, the escape rate of the superconducting phase from a local potential minimum is related to the temperature that appeared in the FokkerPlanck equation [10]. Experimental measurements imply that the escape temperature deviates from the absolute temperature at a lowtemperature regime. The deviation of the escape temperature is explained by macroscopic quantum tunneling [47,48,49]. In addition to the macroscopic quantum tunneling, within the underdamped limit, our obtained effective temperature originating from the quantum fluctuations would give a considerable contribution to this escape temperature in a superconducting Josephson circuit, which would be useful to verify our result. To obtain a quantum FokkerPlanck equation without higherorder derivatives, it could be useful to apply the approach of effective action [50]. The effective action includes the quantum fluctuations, and it would enable us to derive a quantum FokkerPlanck equation.
Availability of data and materials
The data generated during the current study are available from the contributing author upon reasonable request.
References

A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys 17, 549 (1905). https://doi.org/10.1002/andp.19053220806

A.M. Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 21, 756 (1906). https://doi.org/10.1002/andp.19063261405

P. Langevin, Sur la theórie du mouvement brownien. C. R. Acad. Sci. Paris. 146, 530 (1908). https://doi.org/10.1119/1.18725

J.B. Johnson, Thermal agitation of electricity in conductors. Phys. Rev. 32, 97 (1928). https://doi.org/10.1103/physrev.32.97

H. Nyquist, Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110 (1928). https://doi.org/10.1103/physrev.32.110

H.B. Callen, T.A. Welton, Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951). https://doi.org/10.1103/physrev.83.34

R.H. Koch, D.J. Van Harkingen, J. Clarke, Quantumnoise theory for the resistively shunted Josephson junction. Phys. Rev. Lett. 45, 2132 (1980). https://doi.org/10.1103/PhysRevLett.45.2132

G.W. Ford, M. Kac, On the quantum Langevin equation. J. Stat. Phys. 46, 803 (1987). https://doi.org/10.1007/BF01011142

H. Metiu, G. Schön, Description of quantum noise by a Langevin equation. Phys. Rev. Lett. 53, 13 (1985). https://doi.org/10.1103/PhysRevLett.53.13

A. Kamenev, Field Theory in NonEquilibrium Systems (Cambridge University Press, Cambridge, 2011). https://doi.org/10.1017/CBO9781139003667

J. Schwinger, Brownian motion of a quantum oscillator. J. Math. Phys. 2, 407 (1961). https://doi.org/10.1063/1.1703727

L.V. Keldysh, Diagram technique for nonequilibrium processes. Sov. Phys. JETP 20, 1018 (1965)

P.C. Martin, E. Siggia, H. Rose, Statistical dynamics of classical systems. Phys. Rev. A 8, 423 (1973). https://doi.org/10.1103/PhysRevA.8.423

A.D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys. 43, 810 (1914). https://doi.org/10.1002/andp.19143480507

M. Planck, Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsberichte Preuss. Akad. Wiss. Berlin 24, 324 (1917)

L.D. Chang, D. Waxman, Quantum FokkerPlanck equation. J. Phys. C Solid State Phys. 18, 5873 (1985). https://doi.org/10.1088/00223719/18/31/019

A. Arnold, J.L. Lopez, P.A. Markowich, J. Soler, An analysis of quantum FokkerPlanck models: a Wigner function approach. Rev. Mat. Iberoam. 20, 771 (2004)

S. Jang, Generalized quantum FokkerPlanck equation for photoinduced nonequilibrium processes with positive definiteness condition. J. Chem. Phys. 144, 214102 (2016). https://doi.org/10.1063/1.4952477

H. Collins, R. Holman, T. Vardanyan, The quantum FokkerPlanck equation of stochastic inflation. JHEP 11, 065 (2017). https://doi.org/10.1007/JHEP11(2017)065

P.J. Colmenares, FokkerPlanck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics. Phys. Rev. E 97, 052126 (2018). https://doi.org/10.1103/PhysRevE.97.052126

C. De Dominicis, Techniques de renormalisation de la theorie des champs et dynamique des phenomenes critiques. J. Phys. Colloq. 37, C1247 (1976). https://doi.org/10.1051/jphyscol:1976138

H.K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties. Z. Phys. B 23, 377 (1976). https://doi.org/10.1007/BF01316547

C. De Dominicis, L. Peliti, Fieldtheory renormalization and critical dynamics above \(T_{c}\): Helium, antiferromagnets, and liquidgas systems. Phys. Rev. B 18, 353 (1978). https://doi.org/10.1103/PhysRevB.18.353

C. Aron, G. Biroli, L. F. Cugliandolo, Symmetries of generating functionals of Langevin processes with colored multiplicative noise. J. Stat. Mech. P11018 (2010). https://doi.org/10.1088/17425468/2010/11/p11018

U.C. Täuber, Critical dynamics: a field theory approach to equilibrium and nonequilibrium scaling behavior (Cambridge University Press, Cambridge, 2014). https://doi.org/10.1017/CBO9781139046213

R. Araújo, S. Wald, M. Henkel, Axiomatic construction of quantum Langevin equations. J. Stat. Mech. 053101 (2019). https://doi.org/10.1088/17425468/ab11dc

L. Onsager, S. Machlup, Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953). https://doi.org/10.1103/PhysRev.91.1505

R. Olender, R. Elber, Yet another look at the steepest descent path. J. Mol. Struct. (Theochem) 398399, 63 (1997). https://doi.org/10.1016/S01661280(97)000389

B. Huard, Interactions between electrons, mesoscopic Josephson effect and asymmetric current fluctuations. PhD thesis, Université Pierre et Marie Curie  Paris VI (2006)

B. Huard, H. Pothier, N.O. Birge, D. Esteve, X. Waintal, J. Ankerhold, Josephson junctions as detectors for nonGaussian noise. Ann. Phys. Wiley 16(10–11), 736–750 (2007). https://doi.org/10.1002/andp.2007519101107

K. Furutani, L. Salasnich, Quantum and thermal fluctuations in the dynamics of a resistively and capacitively shunted Josephson junction. Phys. Rev. B 104, 014519 (2021). https://doi.org/10.1103/PhysRevB.104.014519

U. Eckern, G. Schön, V. Ambegaokar, Quantum dynamics of a superconducting tunnel junction. Phys. Rev. B 30, 6419 (1984). https://doi.org/10.1103/PhysRevB.30.6419

E.G. Dalla Torre, E. Demler, T. Giamarchi, E. Altman, Quantum critical states and phase transitions in the presence of nonequilibrium noise. Nat. Phys. 6, 806 (2010). https://doi.org/10.1038/nphys1754

A.K. Das, S. Panda, J.R.L. Santos, A path integral approach to the Langevin equation. Int. J. Mod. Phys. A 30, 1550028 (2015). https://doi.org/10.1142/S0217751X15500281

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, Singapore, 2004). https://doi.org/10.1142/7305

N. Hashitsume, A statistical theory of linear dissipative systems. Progr. Theoret. Phys. 8, 461 (1952). https://doi.org/10.1143/ptp/8.4.461

R. Kubo, The fluctuationdissipation theorem. Rep. Prog. Phys. 29, 255 (1966). https://doi.org/10.1088/00344885/29/1/306

H. Grabert, U. Weiss, P. Talkner, Quantum theory of the damped harmonic oscillator. Z. Phys. B Condens. Matter 55, 87–94 (1984). https://doi.org/10.1007/BF01307505

R.F. Pawula, Approximation of the linear Boltzmann equation by the FokkerPlanck equation. Phys. Rev. 162, 186 (1967). https://doi.org/10.1103/PhysRev.162.186

F.T. Brandt, J. Frenkel, J.C. Taylor, Noise in resistively shunted Josephson junctions. Phys. Rev. B 82, 014515 (2010). https://doi.org/10.1103/PhysRevB.82.014515

A. Schmid, Diffusion and localization in a dissipative quantum system. Phys. Rev. Lett. 51, 1506 (1983). https://doi.org/10.1103/PhysRevLett.51.1506

S.A. Bulgadaev, Phase diagram of a dissipative quantum system. JETP Lett. 39, 315 (1984)

M.H. Devoret, J.M. Martinis, J. Clarke, Measurements of macroscopic quantum tunneling out of the zerovoltage state of a currentbiased josephson junction. Phys. Rev. Lett. 55, 1908 (1985). https://doi.org/10.1103/PhysRevLett.55.1908

J. Polo, V. Ahufinger, F.W.J. Hekking, A. Minguzzi, Damping of Josephson oscillations in strongly correlated onedimensional atomic gases. Phys. Rev. Lett. 121, 090404 (2018). https://doi.org/10.1103/PhysRevLett.121.090404

F. Binanti, K. Furutani, L. Salasnich, Dissipation and fluctuations in elongated bosonic Josephson junctions. Phys. Rev. A 103, 063309 (2021). https://doi.org/10.1103/PhysRevA.103.063309

M. Pigneur, T. Berrada, M. Bonneau, T. Schumm, E. Demler, J. Schmiedmayer, relaxation to a phaselocked equilibrium state in a onedimensional bosonic Josephson junction. Phys. Rev. Lett. 120, 173601 (2018). https://doi.org/10.1103/PhysRevLett.120.173601

J.M. Martinis, M.H. Devoret, J. Clarke, Energylevel quantization in the zerovoltage state of a currentbiased Josephson junction. Phys. Rev. Lett. 55, 1543 (1985). https://doi.org/10.1103/PhysRevLett.55.1543

J. Clarke, A.N. Cleland, M.H. Devoret, D. Esteve, J.M. Martinis, Quantum mechanics of a macroscopic variable: the phase difference of a Josephson junction. Science 239, 992 (1988). https://doi.org/10.1126/science.239.4843.992

J.A. Blackburn, M. Cirillo, N.G. Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures. Phys. Rep. 611, 1–33 (2016). https://doi.org/10.1016/j.physrep.2015.10.010

J. Stapmanns, T. Kühn, D. Dahmen, T. Luu, C. Honerkamp, M. Helias, Selfconsistent formulations for stochastic nonlinear neuronal dynamics. Phys. Rev. E 101, 042124 (2020). https://doi.org/10.1103/PhysRevE.101.042124
Acknowledgements
The authors acknowledge Pietro Faccioli, Amos Maritan, Fabio Sattin, and Sandro Azaele for useful suggestions.
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KF is supported by a PhD fellowship of the Fondazione Cassa di Risparmio di Padova e Rovigo.
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KF and LS equally contributed to all aspects of the manuscript. Both authors read and approved the final manuscript.
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Appendix A: Derivation of FokkerPlanck equation
Appendix A: Derivation of FokkerPlanck equation
In order to derive the semiclassical FokkerPlanck equation for \(\mathcal {P}(q,v,t)\) from Eqs. (20) and (21), let us consider an infinitesimal time interval \(\varepsilon =t't\). The probability \(\mathcal {P}(q',v',t+\varepsilon )\) satisfies, to \(\mathcal {O}({\varepsilon })\) [34, 35],
Taking into account Eq. (10), Eq. (39) gives
In the above calculation, we used \(\int D[\lambda (t)]e^{i\lambda (t)\left[ q'(t)q(t)\varepsilon v(t)\right] /\hbar }=\delta \left[ q'(t)q(t)\varepsilon v(t)\right]\). Hence, performing the integrals in Eq. (40), one obtains
For \(\varepsilon \rightarrow +0\) one finally finds Eq. (22), which is our semiclassical FokkerPlanck equation.