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ORIGINAL RESEARCH article

Front. Phys., 17 September 2021
Sec. Nuclear Physics​
Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.730761

Equivalence of Dissipative and Dissipationless Dynamics of Interacting Quantum Systems With Its Application to the Unitary Fermi Gas

  • 1Department of Physics, Tohoku University, Sendai, Japan
  • 2Strangeness Nuclear Physics Laboratory, RIKEN Nishina Center, Wako, Japan
  • 3Frontier Research Institute for Interdisciplinary Science, Tohoku University, Sendai, Japan

We analytically study quantum dissipative dynamics described by the Caldirola-Kanai model with inter-particle interactions. We have found that the dissipative quantum dynamics of the Caldirola-Kanai model can be exactly mapped to a dissipationless quantum dynamics under a negative external harmonic potential, even when the particles are strongly interacting. In particular, we show that the mapping is valid for the unitary Fermi gas, which is relevant for cold atoms and nuclear matters.

1 Introduction

Dissipation plays essential roles in the non-equilibrium dynamics of quantum matters. There has been rapid growth of research interests in the dissipative quantum dynamics as it is relevant to macroscopic quantum tunneling [1], low-energy nuclear reactions [24], dynamics of cold atoms [57], and quantum information processings [8, 9]. Milestones in this research field is the Caldeira-Leggett model [1] and the Lindblad type equations [10, 11], where dissipation in the quantum system originates from the coupling of the system with the environment.

While these approaches to the quantum dissipative dynamics have been widely and successfully used, it is rather difficult to obtain an exact result in an analytical manner for most practical problems. To deal with the quantum dissipative dynamics in a more analytically feasible manner, we can alternatively resort to a simpler quantum equation of motion. One of the elementary models describing such a dissipative quantum motion is the Caldirola-Kanai model [12, 13] characterized by the Lagrangian

LCK({xi},{ẋi},t)=eγt[i=1Nmi2ẋi2U(x1,,xN,t)],(1)

and its semi-classical equation of motion.

ẍi(t)+γẋi(t)+1miUxi(x1(t),,xN(t),t)=0,(2)

where mi, xi and ẋi=(d/dt)xi are the mass, the position, and the velocity of the i-th particle respectively. U denotes an interaction or an external potential, and γ is the dissipation rate (see Equation 26 for the Schrödinger equation and the Hamiltonian for the Caldirola-Kanai model). Taking advantage of its simplicity, the Caldirola-Kanai model has been applied to various dissipative quantum phenomena, including damping of electromagnetic fields in a plasma medium [14], dissipative quantum tunneling in low-energy nuclear fusion reactions [15], dynamics of a damped charged oscillator in the presence of the Aharanov-Bohm effect [16], and so on. While the Caldirola-Kanai model has usually been applied to a single-particle motion regarding U as an external potential as in the original setting [12, 13], we can also consider the dynamics of interacting quantum particles under the dissipation regarding U as inter-particle interactions.

In this paper, we analytically study the quantum dissipative motion of interacting particles with the Caldirola-Kanai model. We show that this dissipative system can be exactly mapped to a dissipationless system with an inverted harmonic potential. Similar mappings have been discussed for single-particle systems in the literature [1719]. In this paper, we extend the mapping to many-particle systems, particularly to a strongly interacting system with a divergently large s-wave scattering length, i.e., unitary interaction. Such a system has attracted increasing interests in terms of the unitary Fermi gas in cold atoms. Furthermore, as neutron matters are well described by the unitary Fermi gas [2026], our exact mapping should be useful for understanding non-equilibrium dynamics of nuclear matters.

The paper is organized as follows: in Section 2, we introduce the Caldirola-Kanai model and its Hamiltonian, and then show the formal mapping procedure from the Caldirola-Kanai model to a dissipationless Hamiltonian with an inverted harmonic barrier. In Section 3, we study the exact mapping for interacting systems. In particular, we consider the Caldirola-Kanai model with the zero-range interaction, and show that it can be mapped to the dissipationless Hamiltonian with an inverted harmonic barrier when the particles are interacting via the unitary interaction. We conclude and discuss physical implications of our exact mapping in Section 4.

2 Mapping the Caldirola-Kanai Model to a Dissipationless Motion With an Inverted Harmonic Barrier

In this section, we show the exact mapping from the Caldirola-Kanai model to a dissipationless Hamiltonian with an inverted harmonic potential. To make it self-contained, we reformulate the arguments in Refs. [1719] and begin our discussion with a single-particle system in a one-dimensional space for classical mechanics in Section 2.1 and for quantum mechanics in Section 2.2. We see that the mapping between the dissipative and dissipationless dynamics can be understood as a transformation of the coordinate from one system to the other. We then extend this idea to a many-particle system in a d-dimensional space in Section 2.3. We find that the mapping can formally remain true even in the presence of inter-particle interactions.

2.1 Transformation of the Classical Coordinates

To see the mapping heuristically, let us first consider the classical dynamics of a damped harmonic oscillator in one spacial dimension. The equation of motion reads

ẍ(t)+γẋ(t)+ω2x(t)=0,(3)

with the frequency ω and the dissipation rate γ. Its analytical solution can be obtained easily. For example, for an under-damped case, it reads

x(t)=eγt2x(0)cos(Ωt)+ẋ(0)+(γ/2)x(0)Ωsin(Ωt)],(4)

with Ω=ω2(γ/2)2.

Notice that the solution is similar to that of a simple harmonic oscillator. This similarity can be understood in the following way. We first introduce a new variable y(t),

x(t)=eγt/2y(t).(5)

Substituting this relation into the equation of motion for x(t), Eq. 3, one finds that y(t) satisfies the equation of motion of a simple harmonic oscillator,

ÿ(t)+Ω2y(t)=0.(6)

The general solution is given by

y(t)=y(0)cos(Ωt)+ẏ(0)Ωsin(Ωt).(7)

From the relation Eq. 5, one finds

y(0)=x(0),ẏ(0)=ẋ(0)+γ2x(0).(8)

Substituting Eqs. 7 and 8 into Eq. 5 leads to the solution of the damped harmonic oscillator Eq. 4.

In the above discussion, we observe that the transformation Eq. 5 replaces the damping term γẋ(t) with the −(γ2/4)y(t) term. This replacement remains true even in the presence of an arbitrary external potential: suppose that x(t) describes a dissipative motion with an arbitrary external potential V(x, t), that is, x(t) satisfies the following equation of motion,

ẍ(t)+γẋ(t)+1mdVdx(x(t),t)=0.(9)

By the transformation Eq. 5, the equation of motion for y(t) is given as

ÿ(t)+1meγt/2dVdz(z=y(t)eγt/2,t)γ24y(t)=0.(10)

This equation of motion is derived from the following Lagrangian,

Ly(y,ẏ,t)=m2ẏ2eγtV(yeγt/2,t)mγ28y2.(11)

This means that the transformation Eq. 5 maps a dissipative system to a system with the rescaled potential eγtV(yeγt/2, t) together with the inverted harmonic potential − 2y2/8. The damped harmonic oscillator discussed above is a special example where the potential is given by V(x, t) = 2x2/2 and the time-dependence of the rescaled potential disappears because of eγtV(yeγt/2)=mω22y2.

With the transformation Eq. 5, one can find an effective Lagrangian for a dissipative motion described by Eq. 9:

Lx(x,ẋ,t)=Ly(y(x),ẏ(x,ẋ),t)=eγt[m2ẋ2V(x,t)]+mγ4ddtxeγt/22.(12)

Neglecting the last term which does not affect the equation of motion, one obtains the Lagrangian of the Caldirola-Kanai model under an arbitrary time-dependent external potential

LCK(x,ẋ,t)=eγt[m2ẋ2V(x,t)].(13)

From these Lagrangians, we can derive the Hamiltonians for the motions of x(t) and y(t). For a dissipative system described by the Caldirola-Kanai model, x(t), it should be noted that the canonical momentum, πx(t), is different from the kinetic momentum, mẋ(t),

πx(t)=LCKẋ=mẋ(t)eγt.(14)

One then obtains the Caldirola-Kanai Hamiltonian

HCK(x,πx,t)=πx22meγt+V(x,t)eγt.(15)

For a system described by y(t), on the other hand, the canonical momentum is equal to the kinetic momentum,

πy(t)=Lyẏ=mẏ(t),(16)

and the Hamiltonian is given by

Hy(y,πy,t)=πy22m+eγtV(yeγt/2,t)mγ28y2.(17)

We note that the essence of this mapping between the dissipative motion of x(t) and the dissipationless motion of y(t) is the scale transformation Eq. 5. The slowing-down effect of the damping γ is captured by the eγt2 scale factor when relating the dissipative motion of x(t) and dissipationless motion of y(t) [1719].

2.2 Mapping in Quantum Mechanics

Let us extend the mapping discussed in the previous subsection to quantum mechanics [1719]. From the Hamiltonians Eqs 15 and 17, one can derive the corresponding Schrödinger equations. For the Caldirola-Kanai model, Eq. 15, the Schrödinger equation reads

itϕ(x,t)=− 22meγt2x2+eγtV(x,t)ϕ(x,t),(18)

while for the other Hamiltonian without dissipation, one obtains

itψ(x,t)=22m2x2+eγtV(xeγt/2,t)mγ28x2ψ(x,t).(19)

The Schrödinger equation Eq. 18 for ϕ(x, t) serves as a phenomenological modeling of quantum dissipative systems [12, 13]. Indeed, one can show that the Heisenberg equation of motion with Eq. 18 has the same form as Eq. 9. The equivalence of Eq. 18 and Eq. 19 can be understood clearly by showing that the wavefunctions ϕ and ψ are related by a scale transformation as follows: we recall that the two models’ solutions in the classical cases can be mapped to each other by the following transformation

x(t)=y(t)eγt/2,πx(t)=πy(t)mγ2y(t)eγt/2.(20)

In quantum mechanics, these relations should be satisfied as operators, not merely as expectation values. In other words, ϕ(x, t) and ψ(x, t) should satisfy

dzϕ*(z,t)znϕ(z,t)=dzψ*(z,t)zeγt/2nψ(z,t),(21)

and

dzϕ*(z,t)iznϕ(z,t)=dzψ*(z,t)izmγ2zeγt/2nψ(z,t),(22)

for any natural number n. From these conditions, we can find a relation between ϕ(x, t) and ψ(x, t) up to a time-dependent phase, which should be determined to be consistent with the Schrödinger equations Eqs. 18, 19. This consideration leads to the following relation between ϕ(x, t) and ψ(x, t),

ϕ(x,t)=expimγ4eγtx2+γt4ψ(xeγt/2,t).(23)

Indeed, one can directly show that ϕ(x, t) defined by Eq. 23 satisfies the Schrödinger equation with the Caldirola-Kanai Hamiltonian Eq. 18 if ψ(x, t) is a solution of Eq. 19. Therefore, if the initial conditions are related by

ϕ(x,0)=expimγ4x2ψ(x,0),(24)

one can find ϕ(x, t) from ψ(x, t) through the equivalence Eq. 23, and vice verse.

The mapping dictates that the dissipative and dissipationless models obey essentially the same quantum dynamics through Eq. 23 and thus physically equivalent, as pointed out in the previous literature [19]. The effect of the dissipation in Eq. 18 is represented as a scale transformation between x(t) and y(t), and as the inverted harmonic barrier − 2x2/8. In particular, the time-dependent scale transformation of the coordinate captures the slowing-down effect of the damping γ. This can be understood more directly by taking V = 0: the dissipative quantum motion in free space of Eqs. 9 and 18 are equivalent to the dissipationless quantum motion under an inverted harmonic barrier − 2x2/8 in Eqs. 10 and 19. Another important example is the harmonic oscillator potential V(x, t) = 2x2/2, which was discussed in the previous subsection. In this case, Eq. 19 reads

itψ(x,t)=22m2x2+mΩ22x2ψ(x,t).(25)

On the other hand, the corresponding Caldirola-Kanai model describes a quantum damped harmonic oscillator. Therefore, one can map the quantum dynamics of a simple harmonic oscillator to that of a damped harmonic oscillator, as was found in Refs. [1719].

2.3 Extension to Many-Particle Systems

Let us extend the results in the previous subsections and in Refs. [1719] for a single particle to an N-particle system in d-spacial dimensions in the presence of inter-particle interactions. Our starting point is the dissipative equation of motion given by Eq. 2. Instead of Eq. 5, we now consider the transformation xi(t) = eγt/2yi(t) for i = 1, , N (note that all particles feel the same damping force, see Eq. 2). As we derive Eq. 13 in Section 2.1, the transformation leads to the Lagrangian given by Eq. 1. Therefore, the Schrödinger equation for the dissipative motion reads

itϕ(x1,,xN,t)=eγti=1N2xi22mi+eγtU(x1,,xN)ϕ(x1,,xN,t).(26)

On the other hand, the Schrödinger equation corresponding to Eq. 19 reads

itψ(x1,,xN,t)=i=1N2xi22mi+eγtU(x1eγt/2,,xNeγt/2,t)i=1Nmiγ28xi2ψ(x1,,xN,t).(27)

As in the derivation of Eq. 23, we can find the relation between ϕ and ψ,

ϕ(x1,,xN,t)=expieγti=1Nmiγ4xi2+dNγt4ψ(x1eγt/2,,xNeγt/2,t).(28)

This mapping for many-body quantum systems is a natural extension of what was studied for a single-particle motion in the previous literatures [1719]. The single-particle case in one dimension in Eq. 23 is indeed contained as N = 1, d = 1. When U(x1,,xN,t)=i=1NV(xi,t) and hence each particle moves independently, we can regard them as a collection of non-interacting particles, each obeying the Schrödinger equations Eqs. 18, 19. When the potential U cannot be represented as such a sum of one-body potentials, on the other hand, Eq. 26 describes interacting N quantum particles moving under the dissipation, while Eq. 27 describes dissipationless motions of interacting quantum particles in the presence of the inverted harmonic barrier mγ2xi2/8.

3 APPLICATION TO STRONGLY INTERACTING QUANTUM SYSTEMS

We show in this section that the exact mapping formulated in the previous section can be utilized for a physical problem of a strongly interacting quantum system, namely the unitary Fermi gas [2022, 2733]. The unitary Fermi gas is a system of spin-1/2 fermions interacting with an infinitely large s-wave scattering length. It has been recently realized with ultracold atoms [2732], and it has been extensively studied because it is important for understanding nuclear matters and neutron star physics [2026]. The exact mapping between the dissipative and dissipationless motions should therefore be useful for understanding not only the dynamics of the unitary Fermi gas in ultracold atoms, but also for nuclear phenomena.

In Section 3.1, we explain the interaction between the particles to model the unitary two-body interaction. In Section 3.2, we show that the exact mapping holds for a quantum system with the unitary interaction. In Section 3.3, we discuss the effects of dimensions and statistics of the particles. We argue that our exact mapping is non-trivial and useful for a three-dimensional system of fermions, while it is either trivial or breaks down for low-dimensional systems and for bosonic systems.

3.1 Zero-Range Interaction

Solutions of interacting quantum systems in general sensitively depends on details of the interaction potentials. However, for low-energy quantum systems, details of the interaction become irrelevant, so that any Hamiltonian can be universally described by a so-called pseudo-potential. This universality originates from the fact that the scattering between the particles at low energy are dominated by the s-wave scattering, which is universally characterized solely by the s-wave scattering length a for non-relativistic low-energy quantum systems interacting via short-range interactions. As long as this condition holds, the interaction potential between the i-th and j-th particles Vij can be legitimately replaced by the following pseudo potential introduced by Huang and Yang in three spacial dimension [34].

Vij(rij)=2πaij2μijδ(3)(rij)rijrij,(29)

where rij = |rirj|, and aij and μij = mimj/(mi + mj) are the s-wave scattering length and the reduced mass of the i-th and j-th particles, respectively. The Hamiltonian can then be wrriten as the sum of the independent-particle Hamiltonian H0 and the interaction term U, H = H0 + U with

U(x1,,xN,t)=i<jVij(rij).(30)

An alternative low-energy universal approach can be found by noting that the solution of the Huang and Yang pseudo-potential has the following asymptotic form when the particles get close together

limrij0Ψ(r1,r2,,rN)=1rij1aijA(Rij,r1,,ri1,ri+1,rN),(31)

where Rij = (miri + mjrj)/(mi + mj) is the center of mass between the i-th and j-th particles, and A is the regular part of the wavefunction Ψ when the limit rij → 0 is taken. The singular term 1/rij − 1/aij represents the universal s-wave scattering behavior of the two-body scattering. Therefore, we can simply replace the effect of the interaction term Vij with this boundary condition: we solve the free particles’ equation of motion of H = H0 supplemented with the boundary condition Eq. 31

The above two approaches, the Huang-Yang pseudo-potential Eq. 29 and the Bethe-Peierls boundary condition Eq. 31, are equivalent, and have both been widely used for low-energy dilute systems where s-wave interaction is predominantly large. In particular, they have been successfully used to study ultracold atoms close to the unitary limit |aij| → [21, 22, 24, 25, 35, 36]. It has also been used in low-energy nuclear physics as the s-wave scattering length between the nucleons are very large [20, 23, 26]. We note however that Eqs. 29 and 31 are only valid in three spacial dimension: they need to be modified in the other spacial dimension, as will be discussed in Section 3.3. We also note that it is required that three- and higher-body interactions are negligibly small to legitimately use the pseudo-potential and the boundary condition methods, as will also be discussed in more details in Section 3.3.

3.2 Exact Mapping for the Unitary Interacting System

Let us first consider N-particle system where the inter-particle interactions are modeled by the Bethe-Peierls boundary condition Eq. 31. As the boundary condition method deals with the non-interacting Hamiltonian U = 0, the Schrödinger equations for the Caldirola-Kanai model and the corresponding dissipationless model are the same as Eqs. 26, 27 with U = 0. Thus, we can naturally expect that the mapping Eq. 28 should also relate these two problems. This turns out to be true if the s-wave scattering length of the two systems satisfy a certain relation (see Eq. 34).

To show this, let us consider the Bethe-Peierls boundary condition for the Caldirola-Kanai model under the unitary interaction

ϕ(x1,x2,,xN,t)1xij1aij(CK)(t).(32)

Here aij(CK)(t) is the s-wave scattering length between the ij particles, which is now allowed to vary with time for later purposes. Note that the Caldirola-Kanai model with U = 0 describes the motion of non-interacting particles under the influence of a damping force (see Eq. 2 with U = 0). With the boundary condition Eq. 32, those damped particles interact through the s-wave scattering length aij(CK)(t) once two particles come to the same position. Assuming that the mapping Eq. 28 holds and noting that the exponential phase factor does not affect the singular short-distance behavior of the wavefunction, we obtain the following boundary condition of the corresponding dissipationless system

ψ(y1,y2,,yN,t)1yij1eγt2aij(CK)(t).(33)

Therefore, we find that the s-wave scattering length of the ψ system, aij(H)(t), must be given by

aij(H)(t)=eγt2aij(CK)(t).(34)

While we have used above the Bethe-Peierls boundary condition to derive the condition Eq. 34 for the equivalence of the two systems, we can alternatively derive it with the Huang-Yang pseudo-potential Eq. 29. Indeed, Eqs. 26, 27 with the Huang-Yang interaction read

itϕ(x1,,xN,t)=eγti=1N2xi22mi+eγti<j2πaij(t)2μijδ(3)(xij)xijxijϕ(x1,,xN,t).(35)
itψ(x1,,xN,t)=i=1N2xi22mi+e5γ2ti<j2πaij(t)2μijδ(3)(xij)xijxiji=1Nmiγ2xi28ψ(x1,,xN,t).(36)

We then need to relate aij(t) with the s-wave scattering lengths of the two systems aij(CK)(t), aij(H)(t). This can be done by substituting the Eqs. 32, 33 to the right-hand sides of Eq. 35 and Eq. 36, respectively. We then find

aij(CK)(t)=e2γtaij(t),aij(H)(t)=e52γtaij(t),(37)

from which we arrive at Eq. 34.

As the condition Eq. 34 must be satisfied for all pairs of particles ij at any time, our exact mapping is difficult to hold true in general. However, there are two special cases where Eq. 34 is easily satisfied. The first case is the non-interacting system aij(H)=aij(CK)=0. This corresponds to the system considered in the previous section and in Refs. [1719]. The other more interesting case is the unitary gas aij(H)=aij(CK)=±. We thus arrive at the following non-trivial conclusion for the two strongly interacting quantum dynamics: a dissipative quantum motion of the unitary gas described by Eq. 35 is equivalent to the dissipationless quantum dynamics of the unitary gas under an inverted harmonic potential in Eq. 36via the mapping Eq. 28.

The physical reason for the two seemingly opposite conditions, the non-interacting system and the unitary system, can be ascribed to the scale invariance. The mapping between the Caldirola-Kanai system and the dissipationless inverted harmonic system can be regarded as a sort of scale transformation in Eq. 5. With this transformation, the interaction term is transformed as (see Eqs. 26, 27)

U(x1,,xN)eγtU(x1eγt/2,,xNeγt/2,t).(38)

In general, they show rather different Hamiltonian dynamics. However, if the interaction term satisfies the scale invariance

eγtU(x1eγt/2,,xNeγt/2,t)=f(t)U(x1,,xN,t)(39)

with f(t) an arbitrary time-dependent function, the Hamiltonian keeps its form with this scale transformation, showing essentially the same dynamics. Indeed, the non-interacting system and the unitary system are known to be scale invariant because the length scale characterizing the interaction, the s-wave scattering length, disappears. In addition to the zero-range type interaction, we can also consider more general classes of scale-invariant interactions as will be discussed in the Supplementary Appendix A. We can indeed show that the exact mapping holds true for various classes of scale-invariant interactions as long as the interaction strength satisfies a similar relation as Eq. 34 (see Supplementary Appendix A). We also note that our mapping for the unitary system can be alternatively proved using the scaling solution of the unitary Fermi gas in the time-dependent harmonic trap [37, 38] (see Supplementary Appendix B).

3.3 Effects of Dimensions, Quantum Statistics of the Particles, and Higher-Body Interactions

We note that the Huang and Yang pseudo-potential Eq. 29 and the Bethe-Peierls boundary condition Eq. 31 are only valid in three spacial dimensions. For low-dimensional systems, the Bethe-Peierls boundary condition should be modified as [22, 36, 37, 39].

limrij0Ψ(r1,r2,,rN)=lnrijaij(2D)A(Rij,r1,,ri1,ri+1,rN)(40)

for a two-dimensional system, and

limrij0Ψ(r1,r2,,rN)=|xij|aij(1D)A(Rij,r1,,ri1,ri+1,rN)(41)

for a one-dimensional system, respectively. Here, aij(2D) and aij(1D) are two-dimensional and one-dimensional scattering lengths, respectively. By using these boundary conditions, we can follow almost the same argument as in Section 3.2. We then arrive at the same conclusion: the two-dimensional and one-dimensional systems of the Caldirola-Kanai model Eq. 26 and the corresponding inverted harmonic model Eq. 27 are equivalent via the mapping Eq. 28 if and only if the s-wave scattering lengths of the two systems satisfy Eq. 34 for any ij particles at any time. This can be also shown by using the Huang-Yang pseudo potential in two and one dimension [40, 41].

Vij(rij)=− π2δ(2)(rij)μijln(qΛaij(2D))×1ln(qΛrij)rijrij(2D),(42)
Vij(xij)=2μijaij(1D)δ(xij)(1D),(43)

where q=12eC with C = 0.577… is the Euler gamma, and Λ is an arbitrary momentum scale.

It is then tempting to conclude that our exact mapping is also valid for the strongly interacting two-dimensional and one-dimensional systems. However, this is not the case: our mapping for the unitary interacting system should be non-trivial only for three-dimensional system. Indeed, the unitary system a = ± is known to be non-interacting and thus trivial in two and one dimensions, which is in stark contrast to the three dimensional system where it is genuine strongly interacting system [21, 22, 37, 42]. This well-known fact can be easily understood from Eqs. 42 and 43, where the interaction becomes non-interacting Vij = 0 when a = ±. This is in contrast to Eq. 29 in three dimension where Vij ≠ 0 when a = ±. Therefore, our mapping for low dimensional quantum systems is trivial and has limited range of applications.

We note that we have solely considered two-body interaction, neglecting three- and higher-body interactions. In most interacting quantum systems, the two-body interaction is much more relevant than the three- and higher-body interactions, even when it is present. Thus, our assumption seems plausible. We should remark however that the unitary interacting system may essentially require the three- and higher-body interactions. A prime example is a system of identical bosons interacting via the unitary interaction a = ±, where the Efimov effect occurs [35, 36, 43, 44]. When the Efimov effect occurs, the Hamiltonian with only the unitary two-body interaction in Eq. 29 becomes singular, and one needs to introduce the three-body boundary condition to make the Hamiltonian well-defined [35, 36, 45, 46]. As this three-body boundary condition is generally not scale invariant, our exact mapping breaks down. The Efimov effect generally occurs for three-dimensional unitary interacting bosonic systems, and thus our result is not applicable. On the other hand, the Efimov effect does not tend to occur for fermionic system due to the Pauli exclusion principle. In particular, it is shown that the spin-1/2 identical fermions with the unitary interaction Eq. 29, i.e., the unitary Fermi gas, is well-defined as there is no three-body nor higher-body Efimov effect [36, 4752]. Our result should therefore be useful for the unitary Fermi gas. Furthermore, as the mass-imbalanced Fermi system does not show the Efimov effect either when the mass imbalance is M/m ≲ 13 [36, 47, 48, 52, 53], we can state that we can readily apply our exact mapping to such mass-imbalanced two-component unitary Fermi gas systems [54, 55].

We also note that the Efimov effect does not occur for two-dimensional and one-dimensional systems [3537, 42, 56]. The Hamiltonian with the two-body interaction Eqs. 42, 43 are thus well-defined even without the higher-body interaction, and our argument on the two- and one-dimensional systems above are valid regardless of the quantum statistics of the particles. We repeat however that the two-dimensional and one-dimensional unitary systems correspond to non-interacting systems, so that the mapping would not be so useful.

4 Conclusion and Discussions

We have studied a quantum dissipative dynamics of the Caldirola-Kanai model (see Eqs. 2 and 26), and shown that it can be rigorously mapped to a dissipationless quantum dynamics under an inverted-harmonic potential (see Eq. 27). While this mapping has been known for a single-particle and non-interacting systems [1719], we have shown that it also holds true for strongly interacting systems. In particular, we have found that the dissipative dynamics of the unitary Fermi gas can be exactly mapped to a dissipationless dynamics of the unitary Fermi gas under an inverted harmonic potential.

The unitary Fermi gas has been recently realized in cold atom experiments [6, 7, 2732]. It has attracted a lot of research interests, because unveiling the properties of the unitary Fermi gas is important for understanding nuclear matters and neutron star physics [2026]. Our work therefore should be particularly useful for cold atoms and nuclear systems. The external potential can be well-controlled in cold atom experiments with laser and magnetic field, and it is possible to engineer the harmonic or inverted-harmonic potential [21, 22] and observe its quantum dynamics. It is thus a promising system where our mapping would be useful. We also note that we can control the s-wave scattering length in a time-dependent manner in cold atoms [57]. It is thus possible to realize a situation where our condition in Eq. 34 holds, so that our exact mapping is valid for systems with finite scattering length or low-dimensional systems.

A more challenging but interesting system to apply our result is nuclear physics. Nuclear systems composed of protons and neutrons can be approximately regarded as the unitary Fermi system as the s-wave scattering lengths between the nucleons are large. With recent studies on nuclear reactions, it has been pointed out that the dissipative quantum tunneling may play an important role in low-energy fusion reactions [5860]. It is also suggested that quantum dissipations of the nuclear matters are indispensable for understanding the dynamics of the neutron stars [61, 62]. Although we should remark that the Caldirola-Kanai model is a rather simple toy model and that the range corrections and three-body interaction neglected in our study often turns out to be relevant in the nuclear systems [46, 63, 64], our analytical mapping between the dissipative and dissipationless systems can provide us with novel qualitative perspective to understand those nuclear phenomena.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

Author Contributions

The contribution of each of the authors has been significant and the manuscript is the result of an even effort of both the authors. They share the first authorship.

Funding

This work was supported by Tohoku University Graduate Program on Physics for the Universe (GP-PU), and JSPS KAKENHI Grant Numbers JP18J20565 and 21H00116.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We thank Kouchi Hagino, Yvan Castin, Hajime Togashi, and Yusuke Tanimura for fruitful discussions. In particular, some discussions in the Supplementary Appendix B are inspired by private communication with Yvan Castin.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2021.730761/full#supplementary-material

References

1. Caldeira AO, Leggett AJ. Influence of Dissipation on Quantum Tunneling in Macroscopic Systems. Phys Rev Lett (1981) 46:211–4. doi:10.1103/PhysRevLett.46.211

CrossRef Full Text | Google Scholar

2. Diaz-Torres A, Hinde DJ, Dasgupta M, Milburn GJ, Tostevin JA. Dissipative Quantum Dynamics in Low-Energy Collisions of Complex Nuclei. Phys Rev C (2008) 78:064604. doi:10.1103/PhysRevC.78.064604

CrossRef Full Text | Google Scholar

3. Tokieda M, Hagino K. Time-dependent Approaches to Open Quantum Systems. Front Phys (2020) 8:8. doi:10.3389/fphy.2020.00008

CrossRef Full Text | Google Scholar

4. Jeung DY, Hinde DJ, Williams E, Dasgupta M, Simpson EC, du Rietz R, et al. Energy Dissipation and Suppression of Capture Cross Sections in Heavy Ion Reactions. Phys Rev C (2021) 103:034603. doi:10.1103/PhysRevC.103.034603

CrossRef Full Text | Google Scholar

5. Diehl S, Micheli A, Kantian A, Kraus B, Büchler HP, Zoller P. Quantum States and Phases in Driven Open Quantum Systems with Cold Atoms. Nat Phys (2008) 4:878–83. doi:10.1038/nphys1073

CrossRef Full Text | Google Scholar

6. Bardon AB, Beattie S, Luciuk C, Cairncross W, Fine D, Cheng NS, et al. Transverse Demagnetization Dynamics of a Unitary Fermi Gas. Science (2014) 344:722–4. doi:10.1126/science.1247425

PubMed Abstract | CrossRef Full Text | Google Scholar

7. Cao C, Elliott E, Joseph J, Wu H, Petricka J, Schäfer T, et al. Universal Quantum Viscosity in a Unitary Fermi Gas. Science (2011) 331:58–61. doi:10.1126/science.1195219

PubMed Abstract | CrossRef Full Text | Google Scholar

8. Azouit R, Chittaro F, Sarlette A, Rouchon P. Towards Generic Adiabatic Elimination for Bipartite Open Quantum Systems. Quan Sci Technol (2017) 2:044011. doi:10.1088/2058-9565/aa7f3f

CrossRef Full Text | Google Scholar

9. Blais A, Grimsmo AL, Girvin SM, Wallraff A. Circuit Quantum Electrodynamics. Rev Mod Phys (2021) 93:025005. doi:10.1103/RevModPhys.93.025005

CrossRef Full Text | Google Scholar

10. Gorini V, Kossakowski A, Sudarshan ECG. Completely Positive Dynamical Semigroups of N-Level Systems. J Math Phys (1976) 17:821. doi:10.1063/1.522979

CrossRef Full Text | Google Scholar

11. Lindblad G. On the Generators of Quantum Dynamical Semigroups. Commun Math Phys (1976) 48:119–30. doi:10.1007/bf01608499

CrossRef Full Text | Google Scholar

12. Caldirola P. Forze Non Conservative Nella Meccanica Quantistica. Nuovo Cim (1941) 18:393–400. doi:10.1007/BF02960144

CrossRef Full Text | Google Scholar

13. Kanai E. On the Quantization of the Dissipative Systems. Prog Theor Phys (1948) 3:440–2. doi:10.1143/ptp/3.4.440

CrossRef Full Text | Google Scholar

14. Choi JR, Lakehal S, Maamache M, Menouar S. Quantum Analysis of a Modified Caldirola-Kanai Oscillator Model for Electromagnetic Fields in Time-Varying Plasma. PIER Lett (2014) 44:71–9. doi:10.2528/PIERL13061601

CrossRef Full Text | Google Scholar

15. Tokieda M, Hagino K. Quantum Tunneling with Friction. Phys Rev C (2017) 95:054604. doi:10.1103/PhysRevC.95.054604

CrossRef Full Text | Google Scholar

16. Nascimento JPG, Aguiar V, Cortez DS, Guedes I. Dissipative Dynamics and Uncertainty Measures of a Charged Oscillator in the Presence of the Aharonov-Bohm Effect. Revista Brasileira de Ensino de Fisica (2021) 43:e20200464. doi:10.1590/1806-9126-rbef-2020-0464

CrossRef Full Text | Google Scholar

17. Kerner EH. Note on the Forced and Damped Oscillator in Quantum Mechanics. Can J Phys (1958) 36:371–7. doi:10.1139/p58-038

CrossRef Full Text | Google Scholar

18. Baskoutas S, Jannussis A, Mignani R. Dissipative Tunnelling of the Inverted Caldirola-Kanai Oscillator. J Phys A Math Gen (1994) 27:2189–96. doi:10.1088/0305-4470/27/6/039

CrossRef Full Text | Google Scholar

19. Huang MC, Wu MC. The Caldirola-Kanai Model and its Equivalent Theories for a Damped Oscillator. Chin J Phys (1998) 36:566–87.

Google Scholar

20. George A, Baker jr. Neutron Matter Model. Phys Rev C (1999) 60:054311. doi:10.1103/PhysRevC.60.054311

CrossRef Full Text | Google Scholar

21. Zwerger W. The BCS-BEC Crossover and the Unitary Fermi Gas, Vol. 836. Springer Science & Business Media (2011).

22. Giorgini S, Pitaevskii LP, Stringari S. Theory of Ultracold Atomic Fermi Gases. Rev Mod Phys (2008) 80:1215–74. doi:10.1103/RevModPhys.80.1215

CrossRef Full Text | Google Scholar

23. Horikoshi M, Kuwata-Gonokami M. Cold Atom Quantum Simulator for Dilute Neutron Matter. Int J Mod Phys E (2019) 28:1930001. doi:10.1142/s0218301319300017

CrossRef Full Text | Google Scholar

24. van Wyk P, Tajima H, Inotani D, Ohnishi A, Ohashi Y. Superfluid Fermi Atomic Gas as a Quantum Simulator for the Study of the Neutron-star Equation of State in the Low-Density Region. Phys Rev A (2018) 97:013601. doi:10.1103/PhysRevA.97.013601

CrossRef Full Text | Google Scholar

25. Endo S. Virial Expansion Coefficients in the Unitary Fermi Gas. Scipost Phys Proc (2020) 3:049. doi:10.21468/SciPostPhysProc.3.049

CrossRef Full Text | Google Scholar

26. Shen G, Horowitz CJ, Teige S. Equation of State of Nuclear Matter in a Virial Expansion of Nucleons and Nuclei. Phys Rev C (2010) 82:045802. doi:10.1103/PhysRevC.82.045802

CrossRef Full Text | Google Scholar

27. Bourdel T, Khaykovich L, Cubizolles J, Zhang J, Chevy F, Teichmann M, et al. Experimental Study of the Bec-Bcs Crossover Region in Lithium 6. Phys Rev Lett (2004) 93:050401. doi:10.1103/PhysRevLett.93.050401

PubMed Abstract | CrossRef Full Text | Google Scholar

28. Nascimbène S, Navon N, Jiang KJ, Chevy F, Salomon C. Exploring the Thermodynamics of a Universal Fermi Gas. Nature (2010) 463:1057–60. doi:10.1038/nature08814

PubMed Abstract | CrossRef Full Text | Google Scholar

29. Horikoshi M, Nakajima S, Ueda M, Mukaiyama T. Measurement of Universal Thermodynamic Functions for a Unitary Fermi Gas. Science (2010) 327:442–5. doi:10.1126/science.1183012

PubMed Abstract | CrossRef Full Text | Google Scholar

30. Ku MJH, Sommer AT, Cheuk LW, Zwierlein MW. Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas. Science (2012) 335:563–7. doi:10.1126/science.1214987

PubMed Abstract | CrossRef Full Text | Google Scholar

31. Gaebler JP, Stewart JT, Drake TE, Jin DS, Perali A, Pieri P, et al. Observation of Pseudogap Behaviour in a Strongly Interacting Fermi Gas. Nat Phys (2010) 6:569–73. doi:10.1038/nphys1709

CrossRef Full Text | Google Scholar

32. Carcy C, Hoinka S, Lingham MG, Dyke P, Kuhn CCN, Hu H, et al. Contact and Sum Rules in a Near-Uniform Fermi Gas at Unitarity. Phys Rev Lett (2019) 122:203401. doi:10.1103/PhysRevLett.122.203401

PubMed Abstract | CrossRef Full Text | Google Scholar

33. Levinsen J, Massignan P, Endo S, Parish MM. Universality of the Unitary Fermi Gas: a Few-Body Perspective. J Phys B: Mol Opt Phys (2017) 50:072001. doi:10.1088/1361-6455/aa5a1e

CrossRef Full Text | Google Scholar

34. Huang K, Yang CN. Quantum-mechanical many-body Problem with Hard-Sphere Interaction. Phys Rev (1957) 105:767–75. doi:10.1103/physrev.105.767

CrossRef Full Text | Google Scholar

35. Nielsen E, Fedorov DV, Jensen AS, Garrido E. The Three-Body Problem with Short-Range Interactions. Phys Rep (2001) 347:373–459. doi:10.1016/s0370-1573(00)00107-1

CrossRef Full Text | Google Scholar

36. Naidon P, Endo S. Efimov Physics: a Review. Rep Prog Phys (2017) 80:056001. doi:10.1088/1361-6633/aa50e8

PubMed Abstract | CrossRef Full Text | Google Scholar

37. Werner F, Castin Y. General Relations for Quantum Gases in Two and Three Dimensions: Two-Component Fermions. Phys Rev A (2012) 86:013626. doi:10.1103/PhysRevA.86.013626

CrossRef Full Text | Google Scholar

38. Werner F, Castin Y. Unitary Gas in an Isotropic Harmonic Trap: Symmetry Properties and Applications. Phys Rev A (2006) 74:053604. doi:10.1103/PhysRevA.74.053604

CrossRef Full Text | Google Scholar

39. Olshanii M. Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons. Phys Rev Lett (1998) 81:938–41. doi:10.1103/PhysRevLett.81.938

CrossRef Full Text | Google Scholar

40. Olshanii M, Pricoupenko L. Rigorous Approach to the Problem of Ultraviolet Divergencies in Dilute Bose Gases. Phys Rev Lett (2001) 88:010402. doi:10.1103/PhysRevLett.88.010402

PubMed Abstract | CrossRef Full Text | Google Scholar

41. Rontani M, Eriksson G, Åberg S, Reimann SM. On the Renormalization of Contact Interactions for the Configuration-Interaction Method in Two-Dimensions. J Phys B Mol Opt Phys (2017) 50:065301. doi:10.1088/1361-6455/aa606a

CrossRef Full Text | Google Scholar

42. Werner F, Castin Y. General Relations for Quantum Gases in Two and Three Dimensions. Ii. Bosons and Mixtures. Phys Rev A (2012) 86:053633. doi:10.1103/PhysRevA.86.053633

CrossRef Full Text | Google Scholar

43. Efimov V. Energy Levels Arising from Resonant Two-Body Forces in a Three-Body System. Phys Lett B (1970) 33:563–4. doi:10.1016/0370-2693(70)90349-7

CrossRef Full Text | Google Scholar

44. Kraemer T, Mark M, Waldburger P, Danzl JG, Chin C, Engeser B, et al. Evidence for Efimov Quantum States in an Ultracold Gas of Caesium Atoms. Nature (2006) 440:315–8. doi:10.1038/nature04626

PubMed Abstract | CrossRef Full Text | Google Scholar

45. Thomas LH. The Interaction between a Neutron and a Proton and the Structure ofH3. Phys Rev (1935) 47:903–9. doi:10.1103/PhysRev.47.903

CrossRef Full Text | Google Scholar

46. Naidon P, Endo S, Ueda M. Microscopic Origin and Universality Classes of the Efimov Three-Body Parameter. Phys Rev Lett (2014) 112:105301. doi:10.1103/PhysRevLett.112.105301

PubMed Abstract | CrossRef Full Text | Google Scholar

47. Efimov V. Energy Levels of Three Resonantly Interacting Particles. Nucl Phys A (1973) 210:157–88. doi:10.1016/0375-9474(73)90510-1

CrossRef Full Text | Google Scholar

48. Petrov DS. Three-body Problem in Fermi Gases with Short-Range Interparticle Interaction. Phys Rev A (2003) 67:010703. doi:10.1103/PhysRevA.67.010703

CrossRef Full Text | Google Scholar

49. Endo S, Naidon P, Ueda M. Universal Physics of 2+1 Particles with Non-zero Angular Momentum. Few-body Syst (2011) 51:207–17. doi:10.1007/s00601-011-0229-6

CrossRef Full Text | Google Scholar

50. Endo S, Naidon P, Ueda M. Crossover Trimers Connecting Continuous and Discrete Scaling Regimes. Phys Rev A (2012) 86:062703. doi:10.1103/physreva.86.062703

CrossRef Full Text | Google Scholar

51. Endo S, Castin Y. Absence of a Four-Body Efimov Effect in The2+2fermionic Problem. Phys Rev A (2015) 92:053624. doi:10.1103/PhysRevA.92.053624

CrossRef Full Text | Google Scholar

52. Bazak B, Petrov DS. Five-body Efimov Effect and Universal Pentamer in Fermionic Mixtures. Phys Rev Lett (2017) 118:083002. doi:10.1103/PhysRevLett.118.083002

PubMed Abstract | CrossRef Full Text | Google Scholar

53. Castin Y, Mora C, Pricoupenko L. Four-body Efimov Effect for Three Fermions and a Lighter Particle. Phys Rev Lett (2010) 105:223201. doi:10.1103/PhysRevLett.105.223201

PubMed Abstract | CrossRef Full Text | Google Scholar

54. Jag M, Zaccanti M, Cetina M, Lous RS, Schreck F, Grimm R, et al. Observation of a strong Atom-Dimer Attraction in a Mass-Imbalanced Fermi-Fermi Mixture. Phys Rev Lett (2014) 112:075302. doi:10.1103/PhysRevLett.112.075302

PubMed Abstract | CrossRef Full Text | Google Scholar

55. Endo S, García-García AM, Naidon P. Universal Clusters as Building Blocks of Stable Quantum Matter. Phys Rev A (2016) 93:053611. doi:10.1103/PhysRevA.93.053611

CrossRef Full Text | Google Scholar

56. Bruch LW, Tjon JA. Binding of Three Identical Bosons in Two Dimensions. Phys Rev A (1979) 19:425–32. doi:10.1103/PhysRevA.19.425

CrossRef Full Text | Google Scholar

57. Clark LW, Gaj A, Feng L, Chin C. Collective Emission of Matter-Wave Jets from Driven Bose-Einstein Condensates. Nature (2017) 551:356–9. doi:10.1038/nature24272

PubMed Abstract | CrossRef Full Text | Google Scholar

58. Dasgupta M, Hinde DJ, Diaz-Torres A, Bouriquet B, Low CI, Milburn GJ, et al. Beyond the Coherent Coupled Channels Description of Nuclear Fusion. Phys Rev Lett (2007) 99:192701. doi:10.1103/PhysRevLett.99.192701

PubMed Abstract | CrossRef Full Text | Google Scholar

59. Hagino K, Takigawa N. Subbarrier Fusion Reactions and many-particle Quantum Tunneling. Prog Theor Phys (2012) 128:1061–106. doi:10.1143/ptp.128.1061

CrossRef Full Text | Google Scholar

60. Rafferty DC, Dasgupta M, Hinde DJ, Simenel C, Simpson EC, Williams E, et al. Multinucleon Transfer inO16,18,F19+Pb208reactions at Energies Near the Fusion Barrier. Phys Rev C (2016) 94:024607. doi:10.1103/PhysRevC.94.024607

CrossRef Full Text | Google Scholar

61. Cutler C, Lindblom L. The Effect of Viscosity on Neutron star Oscillations. ApJ (1987) 314:234–41. doi:10.1086/165052

CrossRef Full Text | Google Scholar

62. Alford MG, Bovard L, Hanauske M, Rezzolla L, Schwenzer K. Viscous Dissipation and Heat Conduction in Binary Neutron-star Mergers. Phys Rev Lett (2018) 120:041101. doi:10.1103/PhysRevLett.120.041101

PubMed Abstract | CrossRef Full Text | Google Scholar

63. Epelbaum E, Nogga A, Glöckle W, Kamada H, Meißner U-G, Witała H. Three-nucleon Forces from Chiral Effective Field Theory. Phys Rev C (2002) 66:064001. doi:10.1103/PhysRevC.66.064001

CrossRef Full Text | Google Scholar

64. Hammer H-W, Nogga A, Schwenk A. Colloquium: Three-Body Forces: From Cold Atoms to Nuclei. Rev Mod Phys (2013) 85:197–217. doi:10.1103/RevModPhys.85.197

CrossRef Full Text | Google Scholar

Keywords: dissipative quantum system, Caldirola-Kanai model, cold atoms, unitary fermi gas, nuclear dynamics

Citation: Tokieda M and Endo S (2021) Equivalence of Dissipative and Dissipationless Dynamics of Interacting Quantum Systems With Its Application to the Unitary Fermi Gas. Front. Phys. 9:730761. doi: 10.3389/fphy.2021.730761

Received: 25 June 2021; Accepted: 31 August 2021;
Published: 17 September 2021.

Edited by:

Jinjin Li, Shanghai Jiao Tong University, China

Reviewed by:

Ignazio Licata, Institute for Scientific Methodology (ISEM), Italy
Joaquin Drut, University of North Carolina at Chapel Hill, United States

Copyright © 2021 Tokieda and Endo. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Masaaki Tokieda, masaaki.tokieda@inria.fr; Shimpei Endo, shimpei.endo@nucl.phys.tohoku.ac.jp

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