arXiv:1805.00167v2 [nucl-th] 9 Jul 2018

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Quantum dissipation of a heavy quark from a nonlinear stochastic Schrodinger

equation

Yukinao Akamatsu,∗ Masayuki Asakawa,† and Shiori Kajimoto‡

Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

Alexander Rothkopf§

Institute for Theoretical Physics, Heidelberg University, 69120 Heidelberg, Germany and

Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway

(Dated: July 10, 2018)

We study the open system dynamics of a heavy quark in the quark-gluon plasma using a Lindbladmaster equation. Applying the quantum state diffusion approach by Gisin and Percival, we deriveand numerically solve a nonlinear stochastic Schrodinger equation for wave functions, which isequivalent to the Lindblad master equation for the density matrix. From our numerical analysis inone spatial dimension, it is shown that the density matrix relaxes to the Boltzmann distribution invarious setups (with and without external potentials), independently of the initial conditions. Wealso confirm that quantum dissipation plays an essential role not only in the long-time behavior ofthe heavy quark but also at early times if the heavy quark initial state is localized and quantumdecoherence is ineffective.

I. INTRODUCTION

The study of phases of strongly interacting matter un-der extreme conditions [1, 2] is attracting broad inter-est in recent years, extending far beyond nuclear matter.Fruitful interdisciplinary collaborations, among others,with the field of ultracold atomic gases (for a review seee.g. [3]) have broadened the scope of how to explore theorigins of the universe.One example of extreme conditions is very high tem-

peratures, which are particularly interesting, since theybear direct relevance to the birth and early history ofour universe. The properties of the hottest matter evercreated on the Earth are being investigated at currentcollider facilities, such as the Relativistic Heavy-Ion Col-lider (RHIC) and the Large Hadron Collider (LHC) andupcoming facilities, such as NICA and FAIR. There, twoheavy nuclei are collided at ultra-relativistic energies inorder to create a novel deconfined state of matter knownas the quark-gluon plasma (QGP) [4].The transition from hadronic matter to the QGP is

characterized by the liberation of colored degrees offreedom (quarks and gluons) otherwise confined insidehadrons. This qualitative picture is confirmed by lat-tice QCD calculations of, for example, the QCD entropy,which show a significant rise around the pseudocriticaltemperature [5–8].In analogy with the Debye screening phenomenon in

electromagnetic plasmas, the liberated colored particles

∗ [email protected]† [email protected]‡ [email protected]§ [email protected]

may rearrange themselves around a test color charge suchthat the test charge is screened with a finite screeninglength [9]. Compared to the confining string-like forcebetween color sources in the vacuum, the in-medium forcein a high temperature QGP is hence drastically modifiedand becomes short ranged [10–12].The in-medium modification of the QCD force has

been expected to have dramatic consequences on the be-havior of bound states of heavy quarks and antiquarks,so called heavy quarkonium [13, 14]. In turn the dynam-ics of quarkonium states observed in heavy-ion collisionspromises a direct window on the phase structure of thebulk matter, in which they are immersed.One classic prediction in this context is the enhanced

probability of dissociation of charm quark pairs in nuclearcollisions, in case that a QGP is created [15]. The qual-itative behavior of J/ψ and Υ yields observed at RHICand Υ yields at LHC support this idea [16–25].However, the enhanced production rate of charm quark

pairs at LHC complicates an interpretation of J/ψ yieldsat LHC [26, 27]. The reason is that now another pro-cess to create J/ψ needs to be taken into account, i.e.the non-negligible probability that initially uncorrelatedcharm quarks end up forming bound states at the phaseboundary [28], i.e. in the late stages of the collision atfreezeout. The successful prediction of J/ψ yields at theLHC by means of the statistical model of hadronization[29] is seen as support for the existence of such a produc-tion mechanism.At present it is still not clear at which collision energy

the two effects, i.e. the suppressed and enhanced yields ofquarkonium become comparable. In order to more clearlyinterpret the collected experimental results we thus needto better understand the dynamics of quarkonia in theQGP. I.e. the development of a unified quantum mechan-

2

ical description for the real-time equilibration of quarko-nia is called for.

Recently, the dynamics of heavy quark pairs in theQGP has been actively studied by various methods [30].Among them are kinetic descriptions [31–38] or dynam-ical models involving a complex potential [39–43] forquarkonium. One more recent promising approach is theopen quantum system formulation for the heavy quarkpairs [44–57], which developed concurrently to computa-tions of the in-medium complex potential [58–64].

In the open quantum system formulation [65], we dis-tinguish the subsystem of interest (quarkonium) from theenvironment (QGP), which is made possible by a hierar-chy of time scales in each sector. The dynamics of theenvironment is fast enough, so that we can trace it outand replace its coupling to the subsystem by the averageresponse to a slowly changing external source.

In this way a master equation for the reduced densitymatrix of a heavy quark pair can be obtained, which canbe expressed as arising from the contribution of threekinds of forces: the screened potential, thermal fluctua-tions, and dissipation [48]. The first two forces combineinto a fluctuating potential force (stochastic potential[46, 47, 50]), while the last one is an irreversible force.So far there exist only a few numerical analyses of thequantum dissipation of quarkonium in the QGP [55–57].However, quantum dissipation is an essential ingredientto understand the long-time behavior of a heavy quarkpair in the QGP. It is particularly important to knowhow and when quantum dissipation influences the timeevolution of the heavy quark pair because the lifetime ofthe QGP in heavy-ion collisions is not long enough forthe heavy quark pair to get fully equilibrated.

In this paper we consider as a first step the physicsof a single heavy quark immersed in a hot medium. Wenumerically solve the corresponding master equation andstudy its equilibrium solution as well as the effects of dis-sipation. The central conceptual result of this study is astochastic unravelling prescription for the master equa-tion, in which the wave function is evolved in terms of astochastic Schrodinger equation and the mixed states ofthe density matrix emerge from an ensemble average.

Our results are obtained following the “Quantum StateDiffusion” approach developed by Gisin and Percival[66, 67] and by subsequently solving the correspondingnonlinear stochastic Schrodinger equation. Note thatby applying the Quantum State Diffusion approach toa heavy quark master equation derived in the Lindbladform [48, 68] (see Section II for the definition) we obtainfor the first time in this context a non-linear Schrodingerequation with a clear connection to the underlying mi-croscopic theory.

For simplicity, we consider a heavy quark in the QGPin one spatial dimension. It is then shown that the mas-ter equation possesses a steady state solution consistentwith the Boltzmann distribution ρeq ∝ e−βH . Further-more, we observe that the approach to equilibrium de-pends on initial conditions and cannot be captured by

a single decay rate. Finally, we analyze the effect ofquantum dissipation by comparing with simulations inwhich the dissipative terms are dropped. Their effect, asexpected, is essential at later times but interestingly al-ready plays an important role at rather early times if theinitial wave function is well localized and decoherence bythermal fluctuations is ineffective.

This paper is organized as follows. In Sec. II, we in-troduce the Quantum State Diffusion approach applica-ble to a general Lindblad master equation. We then ap-ply this approach to the Lindblad equation for a singleheavy quark in the quark-gluon plasma at high temper-ature and derive the corresponding nonlinear stochasticSchrodinger equation. In Sec. III, we solve this nonlinearstochastic Schrodinger equation numerically and analyzethe relaxation process of the heavy quark. In addition westudy the effect of quantum dissipation on the time evo-lution of a heavy quark, before we summarize our workin Sec. IV.

II. QUANTUM STATE DIFFUSION FOR OPEN

QUANTUM SYSTEMS

A. Lindblad equation and quantum state diffusion

There is a particularly useful class of master equa-tions for open quantum systems, which is Markovianand fulfills basic physical requirements: the reduced den-sity matrix ρ is hermitian (ρ = ρ†), correctly normalized(Trρ = 1), and positive (〈α|ρ|α〉 ≥ 0 for any state |α〉)during its time evolution. Such master equations may bewritten in general as

d

dtρ(t) = −i [H, ρ] +

∑

n

(

2LnρL†n − L†

nLnρ− ρL†nLn

)

,

(1)

in the so called Lindblad form [68]. Here the evolution ofthe density matrix operator is described in terms of thefull dimension of the system Hilbert space. This makesa direct numerical simulation computationally highly de-manding, especially in realistic 3+1 dimensions. Thereare, however, several ways to solve the Lindblad equa-tion by what is known as stochastic unravelling. I.e. bycarrying out a stochastic evolution of the wave functionsof the system instead, whose ensemble average then cor-rectly reproduces the density matrix. One such stochas-tic unravelling corresponds to the quantum state diffu-sion (QSD) approach [66].

Given the Lindblad equation (1), the correspondingQSD equation is a stochastic nonlinear Schrodinger equa-

3

tion:

|dψ〉 = |ψ(t+ dt)〉 − |ψ(t)〉

= −iH |ψ(t)〉dt+∑

n

(

2〈L†n〉ψLn − L†

nLn

− 〈L†n〉ψ〈Ln〉ψ

)

|ψ(t)〉dt

+∑

n

(Ln − 〈Ln〉ψ) |ψ(t)〉dξn, (2)

with complex white noises dξn whose mean and varianceare given by

M (dξn) = M(ℜ(dξn)ℑ(dξm)) = 0, (3a)

M (ℜ(dξn)ℜ(dξm)) = M(ℑ(dξn)ℑ(dξm)) = δnmdt. (3b)

Here 〈O〉ψ ≡ 〈ψ|O|ψ〉 denotes the quantum expectationvalue of an operator with respect to a state ψ and M(O)denotes the statistical average of O.This stochastic evolution equation is solved in the

Ito discretization scheme. By using ψ(t) everywhere inEq. (2) we obtain the wave function at the next discretetime step ψ(t+ dt). In the limit dt → 0, the QSD equa-tion (2) preserves the norm of ψ in each stochastic up-date. The initial wave function is distributed accordingto the initial mixed (or pure) state of the density matrix.The density matrix is subsequently constructed from anensemble average of wave functions ψ(t),

ρ(t) = M(|ψ(t)〉〈ψ(t)|) , (4)

and obeys the Lindblad equation (1) in the dt→ 0 limit.The QSD equation can also be formulated for unnor-

malized wave functions φ

|dφ〉 = |φ(t+ dt)〉 − |φ(t)〉

= −iH |φ(t)〉dt+∑

n

(

2〈L†n〉φLn − L†

nLn)

|φ(t)〉dt

+∑

n

Ln|φ(t)〉dξn, (5)

with the same complex noise. Here 〈O〉φ ≡ 〈φ|O|φ〉/〈φ|φ〉denotes the quantum expectation value. The density ma-trix constructed by

ρ(t) = M

(

|φ(t)〉〈φ(t)|

〈φ(t)|φ(t)〉

)

(6)

is again a solution of the master equation (1). In ournumerical simulation, we implement eq.(5).

B. Quantum state diffusion for a heavy quark in

the quark-gluon plasma

Let us now consider the theory of open quantum sys-tems for a single heavy quark in the QGP and stochas-tically unravel its Lindblad equation via the QSD ap-proach. The Lindblad equation for heavy quarks hasbeen derived in [48] by treating the scattering between

heavy quarks and medium particles perturbatively, i.e.by assuming that the QCD coupling constant g is small.It is further assumed that the heavy quark mass M ismuch larger than the temperature T/M ≪ 1 so thatthere exists a time scale hierarchy between heavy quarksand medium particles.The Lindblad master equation for a single heavy quark

is given by the following operators [48]

H = −∇2

2M+ Vext(x), (7a)

Lk =

√

D(k)

2Veik·x/2

(

1 +ik ·∇

4MT

)

eik·x/2, (7b)

D(k) = g2Tπm2

D

k(k2 +m2D)

2, mD = gT

√

Nc3

+Nf6,

(7c)

with Nc and Nf being the numbers of colors and quarkflavors, respectively. The Lindblad operator Lk de-scribes the scattering process between a heavy quarkand medium particles with momentum transfer k, takingplace with rate D(k). The term ∝ eik·x in Lk describesthermal fluctuations, while the term ∝ eik·x/2 ik·∇

4MT eik·x/2

describes dissipation and originates in the recoil of theheavy quark during the collision. For simplicity, we ig-nore the effects of internal color degrees of freedom 1.We can calculate the parts of the QSD equation as

follows. The nonlinear term is given by

2∑

k

〈L†k〉φLkφ(x) =

1∫

d3y|φ(y)|2(8)

×

∫

d3y

[

nφ(y)f(x− y) +i

4Tjφ(y) · g(x− y)

]

φ(x),

where nφ and jφ denote the probability density and cur-rent:

nφ(x) ≡ φ∗(x)φ(x), (9a)

jφ(x) ≡1

2iM[φ∗(x)∇φ(x) − (∇φ∗(x)) φ(x)] , (9b)

and f and gi are operators defined as

f(x− y) ≡

(

1 +∇2

8MT

)

D(x− y) +∇D(x− y)

4MT·∇x,

(10a)

gi(x− y) ≡

(

1 +∇2

8MT

)

∇iD(x− y)

+∇∇iD(x− y)

4MT·∇x. (10b)

1 Also, we assume that it is admissible to set the second ordercoefficient in the derivative expansion of the Feynman-Vernoninfluence functional A(k) = D(k)/8T 2. This reduces the numberof Lindblad operators that need to be considered.

4

The function D(x) in the equations above is the inverse

Fourier transform of D(k). The linear deterministic termreads

∑

k

L†kLkφ(x) =

1

2

(

D(0) +∇2D(0)

4MT+

∇4D(0)

64M2T 2

)

φ(x)

+∇i∇jD(0)

32M2T 2∇i∇jφ(x), (11)

and the linear stochastic term amounts to∑

k

Lkφ(x)dξk

=

[

dζ(x) +∇2dζ(x)

8MT+∇dζ(x) ·

∇

4MT

]

φ(x), (12)

where the definition and the correlation of the complexnoise field dζ(x) is given by

dζ(x) ≡

√

V

2

∫

d3k

(2π)3

√

D(k)eik·xdξk, (13a)

M (dζ(x)dζ∗(y)) = D(x− y)dt, (13b)

M (dζ(x)dζ(y)) = M(dζ∗(x)dζ∗(y)) = 0. (13c)

Using Eqs. (8), (11), and (12), we may now perform theQSD simulation for a single heavy quark.

III. RESULTS OF NUMERICAL SIMULATION

In this section we explicitly check the application ofthe QSD approach and investigate the properties of theLindblad equation in three simple settings. We considera single heavy quark in the QGP in one spatial dimensioneither with or without external potentials.In particular, we study the equilibration of the heavy

quark in each setting and discuss the importance of quan-tum dissipation. As external potentials we deploy eitherthe harmonic potential or the regularized Coulomb po-tential:

Vext(x) =1

2Mω2x2, −

α√

x2 + r2c. (14)

where rc = 1/M . The noise correlation function D(x) isset to have correlation length ∼ 1/mD and is approxi-mated with a Gaussian dependence on distance

D(x) = γ exp[

−x2/l2corr]

. (15)

The parameters of our numerical setup are summarizedin Table I.The Hamiltonian time evolution is solved by the

fourth-order Runge-Kutta method and that for the otherparts is implemented via an explicit forward step accord-ing to the QSD equation (5) with dt = ∆t. We checkthat the discretization effects for both ∆t and ∆x arenegligible by comparing with results with smaller ∆t or

∆x. In the simulation periodic boundary conditions areemployed so that the noise correlation is replaced with

M (dζ(x)dζ∗(y)) = D(rxy)∆t, (16a)

rxy = min{|x− y|, Nx∆x− |x− y|}, (16b)

and the function D(x − y) in the QSD equation (10) isalso replaced by D(rxy). We confirm by changing Nxthat the volume Nx∆x/lcorr ≃ 13 is large enough so thatwe can neglect finite volume effects.

A. Equilibration of a heavy quark

The Lindblad equation itself does not guarantee theBoltzmann distribution ρ ∝ exp(−H/T ) to be the staticsolution (see Appendix A). In the derivation of the Lind-blad equation, the fluctuation-dissipation theorem for theenvironment, i.e. the QGP sector, constrains the termsimplementing the fluctuation and dissipation of the heavyquarks. To be specific, the coefficient i/4MT in Lk is de-termined from the fluctuation-dissipation theorem for athermal QGP medium. Therefore, we expect that theequilibrium density matrix is close to the Boltzmann dis-tribution ρ ∝ exp(−H/T ). Here we analyze how well theequilibration of the density matrix is achieved and studyits equilibrium properties.

1. In the absence of an external potential

The initial heavy quark wave function is taken to beuniform (plane wave with zero momentum). In Fig. 1,we show the profiles of a normalized wavefunction at dif-ferent times in one sample event. A wavefunction typetypically encountered here is a localized solitonic state,arising from the nonlinearity of the evolution equation.Figure 2 on the other hand contains the time evolutionof the momentum distribution of the heavy quark:

N(p, t) ≡ M

(

|φ(p, t)|2

〈φ(t)|φ(t)〉

)

, (17a)

φ(p, t) =

∫

dxe−ipxφ(x, t), (17b)

TABLE I. Numerical setup and parameters of the poten-tials. Nx = 128(127) is used for the harmonic (regularizedCoulomb) potential.

∆x ∆t Nx

1/M 0.1M(∆x)2 128, 127

T γ lcorr ω α rc

0.1M T/π 1/T 0.04M 0.3 1/M

5

0

0.02

0.04

0.06

0.08

0.1

-60 -40 -20 0 20 40 60

|Ψ|2 /M

M*x

M*t=1240620248124

62

FIG. 1. Profiles of a normalized wavefunction at differenttimes in one sample event.

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 0.5 1 1.5 2

M*N

(p,t)

p/M

M*t=124062024812462

Boltzmann

FIG. 2. Time evolution of the momentum distributionof a heavy quark. The bars denote statistical errors. Thedashed line corresponds to the Boltzmann distribution withT = 0.1M .

where p takes on values available on a periodic lattice ofsize Nx ×∆x.The corresponding classical dynamics of the heavy

quark is a Brownian motion with a drag force

dp

dt= −

γ

MT l2corrp. (18)

Its typical relaxation time is τrelax =MTl2

corr

γ = 100π/M .

We can see that the momentum distribution approachesthe Boltzmann distribution with temperature T = 0.1Mover a time scale ∼ τrelax. Note also that at late times(t = 620/M, 1240/M) slight deviations from the Boltz-mann distribution are observed above p >∼ 1.5M . Thereason lies in the poor convergence of the gradient ex-pansion, applied in evaluating the Lindblad operators,at high momenta (p >∼ M), when one takes into accountthe effects of dissipation. On the other hand, we shouldnot rely on the nonrelativistic description for a heavy

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

N0(

t)

Harmonic: ω/M=0.04, N0(0)=1N1(0)=1N2(0)=1

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

N1(

t)

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

N2(

t)

M*t

FIG. 3. Time evolution of the occupation number of theeigenstates in the harmonic potential with ω = 0.04M . Thebars denote statistical errors.

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

N0(

t)

regularized Coulomb: α=0.3, N0(0)=1N1(0)=1N2(0)=1

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

N1(

t)

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

N2(

t)

M*t

FIG. 4. Time evolution of the occupation number of theeigenstates in the regularized Coulomb potential with α = 0.3and rc = 1/M . The bars denote statistical errors.

quark with such a high momentum, so this limitation isnot so restrictive in practice.

2. In the presence of external potentials

Let us turn to the case with external potentials presentnext. The potential here is added by hand out of pure

6

theoretical interest and should not be confused with thepotential for a quarkonium state. One should ratherthink of a single heavy quark in a fictitious trap.The initial heavy quark wave function is taken to be

the ground state, the first, or the second excited statesof the corresponding Hamiltonian. The time evolution ofthe occupation number of these levels,

Ni(t) ≡ M

(

|〈ψi|φ(t)〉|2

〈φ(t)|φ(t)〉

)

, H |ψi〉 = Ei|ψi〉, (19)

is shown in Fig. 3 for the harmonic potential and in Fig. 4for the regularized Coulomb potential. Independent ofthe initial conditions, the occupation numbers convergeto their equilibrium values. In contrast, the relaxationtime depends on the initial condition.One might expect a naive relaxation process

Ni(t) = (N inii −N eq

i ) exp(−Γit) +N eqi , (20)

to describe the dynamics, as motivated and applied inthe rate equation approach to heavy quarks. However,from Fig. 3 and Fig. 4 it is obvious that a single decayrate cannot capture the relaxation of the eigenstate oc-cupation. Note that actually there are cases where theoccupation number shows a non-monotonic approach toequilibrium.In order to investigate the properties of the equilib-

rium density matrix, we show in Fig. 5 the equilib-rium distribution of the lowest ten levels as a func-tion of the eigenenergy for the harmonic and the reg-ularized Coulomb potentials. In the figures, we plotthe results for several different potential parameters:ω/M = 0.01, 0.04, 0.09 for the harmonic potential andα = 0.2, 0.3, 0.4 and rc = 1/M for the regularizedCoulomb potential. The initial condition is chosen to bethe ground state of each Hamiltonian. The equilibriumdistribution is calculated at late enough time for eachsetup: Mt = 1550, 3100, 4650 for ω/M = 0.01, 0.04, 0.09and Mt = 4650, 7750, 9300 for α = 0.2, 0.3, 0.4, respec-tively. In all cases, the distribution is close to the Boltz-mann distribution with T = 0.1M . We also calculate thereal and imaginary parts of the off-diagonal elements ofthe density matrix in equilibrium and check that they areconsistent with zero within the statistical uncertainty.

B. The effect of dissipation

As a last consideration let us now turn off the quan-tum dissipation. Since quantum dissipation is describedby the term ∝ eik·x/2 ik·∇

4MT eik·x/2, we can switch it off

by taking the M → ∞ limit in the QSD equation ev-erywhere, except in the Hamiltonian part. In Fig. 6, weshow the corresponding time evolution of the momentumdistribution as given by the QSD equation without dis-sipation. Clearly, the distribution does not approach theequilibrium Boltzmann distribution. Instead the energygained by the heavy quark from thermal fluctuations is

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Nieq

Harmonic: N0(0)=1, ω/M=0.09ω/M=0.04ω/M=0.01

0.05*exp(-E/T)

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Nieq

Tfit/M=0.0950.0980.104

0.01

0.1

1

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

Nieq

Ei/M

regularized Coulomb: N0(0)=1, α=0.2α=0.3α=0.4

0.02*exp(-E/T)

0.01

0.1

1

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

Nieq

Ei/M

Tfit/M=0.10070.10040.0966

FIG. 5. Equilibrium occupation of the lowest ten eigenstatesfor the harmonic potential (upper panel) and the regularizedCoulomb potential (lower panel). The parameters are var-ied as ω/M = 0.01, 0.04, 0.09 for the harmonic potential andα = 0.2, 0.3, 0.4 and rc = 1/M for the regularized Coulombpotential. The bars denote statistical errors. The data arefitted by C · exp(−Ei/Tfit) and the dashed lines indicate theslope of a Boltzmann distribution with T = 0.1M .

not dissipated back to the system and the heavy quarkoverheats.In Fig. 7, we show the time evolution of the eigenstate

occupation as given by the QSD equation without dissi-pation. The lowest three levels become equally occupiedregardless of the energy gaps 2. It is expected that notonly these levels but all the levels are eventually occupiedequally if the quantum dissipation is neglected. We alsoobserve that the effect of quantum dissipation at earlytime strongly depends on the external potential. Thisdependence can be understood by analyzing the Lind-blad equation as follows. The initial decay rate of aneigenstate ψi of the Hamiltonian is given by

Γi = −2∑

n

(

〈Ln〉ψi〈L†

n〉ψi− 〈L†

nLn〉ψi

)

. (21)

Using the Lindblad operators in Eq. (7), we obtain

Γi = D(0)−

∫

d3xd3yD(x− y)nψi(x)nψi

(y) (22)

+∇2D(0)

4MT+

∇4D(0)

64M2T 2+

∇i∇jD(0)

16M2T 2〈∇i∇j〉ψi

,

2 For the free case, the second and the fourth excited states areused because of the degeneracy of positive and negative momen-tum states.

7

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 0.5 1 1.5 2

M*N

(p,t)

p/M

M*t=124062024812462

Boltzmann

FIG. 6. Time evolution of the momentum distribution ofa heavy quark without dissipation. The bars denote statis-tical errors and the dashed line corresponds to a Boltzmanndistribution with T = 0.1M .

0

0.05

0.1

0.15

0.2

0.25

0.3

0 200 400 600 800 1000 1200

Ni(t

)

N0(0)=1, Free: w/o diss., N0(t)N2(t)N4(t)

w/ diss., N0(t)

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

Ni(t

)

Harmonic: ω/M=0.04, w/o diss., N0(t)N1(t)N2(t)

w/ diss., N0(t)

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

Ni(t

)

M*t

regularized Coulomb: α=0.3, w/o diss., N0(t)N1(t)N2(t)

w/ diss., N0(t)

FIG. 7. Time evolution of the occupation number of theeigenstates without dissipation. For comparison, the timeevolution with dissipation is also plotted. The bars denotestatistical errors.

in which the first (second) line arises from thermal fluctu-ations (dissipation). With the Gaussian approximation(15) for D(x), we get for our one-dimensional model

Γiγ

= 1−

∫

dxdy exp

[

−(x− y)2

l2corr

]

nψi(x)nψi

(y)

−1

2MTl2corr+

3

16M2T 2l4corr−

〈∇2〉ψi

8M2T 2l2corr. (23)

With the values from Table. I, the effect of dissipa-tion (the second line in Eq. (23)) is ≃ −0.048 −

0.125〈∇2〉ψi/M2. The ground state wave function yields

〈∇2〉ψ0/M2 ≃ 0,−0.02,−0.08 for the free case, the har-

monic potential, and the regularized Coulomb potential,respectively. Therefore, the effect of dissipation in thiscase ranges from −0.048 to −0.038 and only slightly de-pends on the potential.On the other hand, the decay rate due to thermal fluc-

tuation is quite sensitive to the size of the wave functionas can be seen from the first line in Eq. (23). In fact, thewave function size is M∆x ≃ 3.5, 1.9 < 10 = Mlcorr forthe ground states of the harmonic and the regularizedCoulomb potentials respectively. If the size is smallerthan the correlation length lcorr, the decay rate in theabsence of dissipation is already comparatively small sothat the relative importance of dissipation increases.

IV. CONCLUSION

In this paper, we investigate how quantum dissipationinfluences the time evolution of the density matrix of aheavy quark in the quark-gluon plasma (QGP). The mas-ter equation for heavy quark systems in the QGP hasbeen obtained in the Lindblad form [48, 68] and thus pos-sesses particularly useful properties: The density matrixρ stays hermitian, remains correctly normalized and pos-itive during the time evolution. We solve this Lindbladequation for a single heavy quark by stochastic unravel-ing. Applying the approach of quantum state diffusion(QSD) [66] to our Lindblad equation, we derive a nonlin-ear stochastic Schrodinger equation for the heavy quarkwave function.Subsequently we solve the QSD equation for a heavy

quark in simple settings, i.e. in one spatial dimension andwith or without external potentials (harmonic and regu-larized Coulomb potentials). We found that in both casesthe density matrix relaxes to ρeq ∝ e−H/T within statis-tical errors. This property is expected from a constraintin the Lindblad equation introduced by the fluctuation-dissipation theorem for the QGP sector but was not ex-plicitly guaranteed by the Lindblad equation itself (seeAppendix A). We also found that the relaxation processstrongly depends on the initial condition so that it is notcaptured by a simple rate equation.As a further topic we study the effect of quantum dissi-

pation by switching off the dissipative terms in the QSDequation. Without the dissipative terms, the heavy quarkis overheated because it only receives energy from thethermal medium which is not dissipated back. It is shownthat the importance of dissipation, as compared to thethermal fluctuations, strongly depends on the wave func-tion size. The relative importance of dissipation increaseswhen the wave function is small.In the future, we plan to extend our analysis to the

description of heavy quarkonium in the QGP. In thatcase, not only thermal fluctuations but also dissipationtakes place nontrivially because the collisions of a heavyquark and those of a heavy antiquark interfere with each

8

other. It would be interesting and phenomenologicallyrelevant to study the effects of quantum dissipation inthat case. The computations for quarkonium in threespatial dimensions and in evolving fluid background forheavy-ion collisions will be one of the ultimate goals ofour project.

ACKNOWLEDGMENTS

The work of Y. A. is partially supported by JSPSKAKENHI Grant Number JP18K13538. M. A. is sup-ported in part by JSPS KAKENHI Grant NumberJP18K03646. Y.A. thanks the DFG Collaborative Re-search Centre SFB 1225 (ISOQUANT) for hospitalityduring his stay at Heidelberg University and A.R. wassupported by SFB 1225 in full. Y.A. also thanks T. Hi-rano for recommending [67].

Appendix A: Approximate steady state solution of

the Lindblad equation

First we give a few algebraic relations of the Lindbladoperators in Eq. (7):

Lkp = (p− k)Lk, L†kp = (p+ k)L†

k, (A1a)

LkL†k =

D(k)

2V

(

1−kp

4MT+

k2

8MT

)2

, (A1b)

L†kLk =

D(k)

2V

(

1−kp

4MT−

k2

8MT

)2

. (A1c)

The steady state solution ρeq(p) of the Lindblad equationfor a heavy quark without external potential Vext = 0

satisfies

0 =∑

k

(

2Lkρeq(p)L†k − L†

kLkρeq(p)− ρeq(p)L†kLk

)

= 2∑

k

(

ρeq(p− k)LkL†k − ρeq(p)L

†kLk

)

= 2∑

k

(

ρeq(p− k)LkL†k − ρeq(p)L

†−kL−k

)

. (A2)

In a collision with momentum transfer k, the on-shellenergy of the heavy quark changes by an amount,

∆Ek =p2

2M−

(p− k)2

2M=

2kp− k2

2M. (A3)

In the momentum representation, LkL†k and L†

−kL−k aregiven by

LkL†k =

D(k)

2V

(

1−∆Ek4T

)2

, (A4a)

L†−kL−k =

D(k)

2V

(

1 +∆Ek4T

)2

. (A4b)

Then detailed balance in reactions between p ↔ p − kdictates

ρeq(p)

ρeq(p− k)=

LkL†k

L†−kL−k

≃ 1−∆EkT

+1

2

(

∆EkT

)2

+ · · ·

≃ e−∆Ek/T . (A5)

Therefore up to the order O[(∆Ek/T )3] ≪ 1, the steady

state solution is the Boltzmann distribution

ρeq(p) ∝ e−p2

2MT . (A6)

The above approximation is reliable for ∆Ek/T ≪ 1. Forp ∼ Mv, ∆Ek/T ∼ v/T lcorr + 1/MT l2corr ∼ v + 0.1 sothat our approximation is valid for v ≪ 1. Therefore, itis surprising that the numerical result in FIG. 2 is wellapproximated by the Boltzmann distribution for as largeas p ∼ 1.5M .

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