Ishihara Complex Plasma Entanglement 2024

1 S e p e m b e r

ViewOnlineExportCitationRESEARCH ARTICLE| MARCH 27 2024Entanglement in a complex plasmaOsamu Ishihara Phys. Plasmas 31, 032118 (2024)https://doi.org/10.1063/5.0192854Articles You May Be Interested InQuantum entanglement for helium atom in the Debye plasmasPhys. Plasmas (March 2015)Dual monogamy inequality for entanglementJ. Math. Phys. (January 2007)Entanglement, Bell inequality and all thatJ. Math. Phys. (July 2012)

ABSTRACT

AFFILIATIONS

ARTICLE

Osamu Ishiharaa)

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a)Present address: Nerima-ku, Tokyo, Japan. Author to whom correspondence should be addressed: [email protected]

Entanglement in a complex plasma

Faculty of Engineering, Yokohama National University, Yokohama, 240-8501, Japan and Chubu University, Kasugai, Aichi-Pref., 487-8501, Japan

Cite as: Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 Submitted: 20 December 2023 . Accepted: 1 March 2024 . Published Online: 27 March 2024

Quantum mechanical approach had been taken to understand the collective nature of phenomena in a plasma.14-18 Waves in a plasma characterized by wave vector k and frequency x are considered as a collection of quasi-particles which possess momentum (cid:1)hk and energy (cid:1)hx, where (cid:1)h ¼ h=2p and h is a Planck constant. The plasma instabilities or plasma wave damping may be thought as a result of the emission or absorption of quasi-particles by particles. When the emis- sion exceeds the absorption, the number of quasi-particles grows, indi- cating the instability with the growth of the amplitude of the wave. When absorption exceeds emission, the wave is damped. On the other hand, a collision process of particles of one species s and the other spe- cies s0 in a plasma may be characterized by an emission of a quasiparti- cle by a species s followed by the absorption of the quasiparticle by a species s0. The quasi-particles exchanged by the particles s and s0 may be called as virtual waves which are not observable, but play an essen- tial role to understand the interaction of particles in the plasma with collective nature. The virtual wave carries the information of the back- ground plasma and characterizes the collective nature involving the interaction of plasma particles. Thus, there is a conceptual advantage in taking quantum mechanical approach by introducing quasi- particles carrying the collective nature of plasmas. From the quantum mechanical viewpoint, it becomes easier to write down the equation

Quantum mechanical approach is extended to the interaction of dust particles in a complex plasma. Massive and highly charged dust particles interact each other through the exchange of quasi-particles (virtual waves) in a quantum mechanical viewpoint. The interaction is described by the Hamiltonian, which describes the two-particle system as uncoupled harmonic oscillators. When the pair of dust particles are embedded in the injected plasma wave, the Hamiltonian is found to show the presence of coupled harmonic oscillator indicating the emergence of the entanglement in semiclassical nature. The entanglement of a pair of dust particles is encapsulated in the Hamiltonian, which is formulated by the method of sec- ond quantization. The frequency of the wave to trigger the emergence of the entanglement is found to be one-half of the dust plasma frequency. The interaction between a pair of dust particles is formulated as a scattering process and is described by the transition probability. Measure of the semi- classical entanglement is shown by the entropy, and the resulting entropy is found to increase with time.

Recent developments in the study of quantum entanglement,5,6 especially observations of the entanglement in a macroscopic world,7-9 have motivated the present study. The quantum entanglement is a property for two physical systems in which a measurement of one sys- tem determines the state of the other. In other words, the entangle- ment allows one or more particles to exist in a shared state even though they are far away each other. Two-particle entanglements have been studied for quantum states including two-electron system, two- photon system, and atom-photon system.10 The entanglement genera- tion in the scattering process of two-particle system was studied by defining the entanglement fidelity.11 The concept of the entanglement fidelity was applied to classical non-ideal plasmas12 as well as the ion- wake field in a complex plasma.13 I start wondering if we can entangle

A complex plasma is a plasma with dust particles and has intro- duced many new aspects in the area of traditional plasma physics.1-4 A complex plasma is ubiquitous in the universe, in the cosmic environ- ment as well as on earth. Dust particles embedded in a plasma are known to be charged and massive, allowing easy observation of par- ticles themselves or dynamics of collection of particles by shining the light on particles or even by naked eyes. The light could be photons in cosmic environment or lasers in laboratory environment. The direct observation helped to understand fundamental physics involving par- ticles in macroscopic ways.

VC 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0192854

the motion of macroscopic pair of dust particles in a complex plasma as a classical system.

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

I. INTRODUCTION

1 S e p e m b e r

31, 032118-1

(cid:5)

Þ2

(5)

(4)

(3)

(2)

(cid:4) p

Þ2 þ

Þ (cid:3) p

where

Hk ¼

Hosc ¼

A x; t ð

HI ¼ (cid:2)

2mj

Zje 2mj

j (cid:3) A xj; t ð

Þ2 Zje 2mj

d3x E x; t ð ð

Þ þ A xj; t ð

ARTICLE

H ¼ Hk þ HI þ Hosc;

is the interaction Hamiltonian and

is the kinetic energy of dust particles,

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is the oscillating energy accompanied by the fields. We describe the vector potential by using the time-dependent field coordinate qjkðtÞ associated with plasma waves of wavenumber k as

describing the change of the particle distribution function and the change of the entropy in a system of particles-quasiparticles due to the interactions among them.

A quantum/semiclassical approach reveals the effect of a pair interaction as a result of emission of a quasiparticle by a particle, fol- lowed by absorption of the quasiparticle by another particle. In other words, the approach reveals the effect as a result of exchange of virtual plasma wave between a pair of dust particles.27 In the present study, we take an approach for the pair interaction of dust particles as a con- nected oscillator in a complex plasma through the exchange of quasi- particles.

A theoretical approach from a quantum mechanical viewpoint was extended to the complex plasma and was successful in explaining the macroscopic phenomena observed as a wake formation in a com- plex plasma,19-23 where a pair of dust particles with negative charges are formed because of the deformed screening effect in the presence of supersonic ion flow. In a plasma, the long-range Coulomb potential inversely proportional to the distance from the charge is shielded in a typical distance known as a Debye length. The wake theory, however, shows that the Debye shielding around a dust particle in the presence of ion flow is deformed and extends like a bow wave known as a wake. The modified Debye potential forms oscillatory potential wells and extends far behind the dust particle beyond the Debye length. The presence of such a long-range interaction between dust particles is manifested experimentally as a formation of dust chain.24-26

j where we used the relation eik(cid:3)xj k (cid:3) p eik(cid:3)xj with p j ¼ (cid:2)i(cid:1)hrj. It was shown27 that the interaction Hamiltonian describes the interaction of two dust particles through the exchange of virtual longitudinal plasma waves (ion acoustic waves), resulting in the forma- tion of wake potential in the presence of ion flow. Figure 1 shows that collisions of two particles in space are modified as collisions exchanging virtual waves in a plasma. A pair of dust particles (designated as d) car- rying momenta p and p0 interact through the exchange of a virtual pho- non characterized by (k; xk), resulting in the new momenta p (cid:2) (cid:1)hk and p0 þ (cid:1)hk. The formation of wake potential by a dust particle placed in the ion acoustic wave with ion flow19-23 was interpreted as a result of a pair of dust particles exchanging virtual plasma waves. The theoretical work27 showed a successful example of Hamiltonian approach applied to a classical phenomenon.

In Sec. II, Hamiltonian description is reviewed for a complex plasma in which dust-wave interaction takes place. In Sec. III, the sec- ond quantization is introduced to describe wave-particle interaction in a complex plasma and the semiclassical entanglement between a pair of dust particle is introduced. In Sec. IV, the properties of the semiclas- sical entanglement are described through the comparison with a quan- tum entanglement. In Sec. V, the entropy is introduced to measure the semiclassical entanglement involved in the interaction between a pair of dust particles. Section VI concludes the paper with discussion.

where mj is the mass of the jth dust particle with charge Zje placed at xj at time t with momentum p j, A xj; t Þ is the vector potential of the ð field and the summation is taken for all the particles in the system, E x; t Þ is the electric field, e0 is the permittivity of free space, and ð (cid:3) (cid:3) (cid:2) j (cid:2) ZjeA xj; t j (cid:2) ZjeA xj; t p

Þ Þ ð Þ (cid:3) E x; t ¼ E x; t Þ. The Hamiltonian can be expressed as ð ð

As was pointed out in the recent paper,28 the present approach with quantum mechanical viewpoint describes the interaction of dust particles-plasma waves or of dust particles themselves. We emphasize here that our approach is for classical plasma system, not for the so- called misleading quantum dusty plasmas,29 with the application of quantum viewpoints.

We consider a collection of dust particles placed in a plasma, where electrostatic plasma waves are present. The Hamiltonian for charged dust particles placed in an electrostatic field carried by a plasma wave is given by (cid:2)

where k2 ¼ k (cid:3) k and V is the volume of the system. The electric field E is given in terms of the vector potential by r

with (cid:4) indicating complex conjugate. The interaction Hamiltonian is expressed as

II. HAMILTONIAN DESCRIPTION FOR DUST-WAVE INTERACTION14,27

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

The conditions for real E and A require

@A xj; t ð @t

j(cid:2)k; qjk ¼ (cid:2)q(cid:4)

j (cid:2) ZjeA xj; t p

(cid:2) j ¼ k (cid:3) p

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

e0Vk2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

e0Vk2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

e0Vk2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

e0Vk2

pj(cid:2)k tð Þkeik(cid:3)xj ;

@qjk tð Þ @t

j (cid:2) ZjeA xj; t

(cid:7) k (cid:3) p

_qjk tð Þkeik(cid:3)xj

qjk tð Þkeik(cid:3)xj ;

(cid:2) , E xj; t ð

¼ (cid:2)pj(cid:2)k tð Þ:

1 S e p e m b e r

j (cid:2) (cid:1)hk (cid:3) k

d3x E x; t ð ð

pjk ¼ (cid:2)p(cid:4)

_qjk tð Þ ¼

(cid:2) 2 ¼ p

31, 032118-2

qjkeik(cid:3)xj ;

k r X

2mj

(cid:3) 2 þ

A xj; t ð

HI ¼ (cid:2)

E xj; t ð

Zje mj

Þ ¼ (cid:2)

(cid:2) (cid:3) p

(cid:1)hk

where

j(cid:2)k;

H ¼

¼ (cid:2)

(10)

Þ ¼

j (cid:2)

Þ2;

(1)

(7)

(8)

(9)

(6)

(cid:8)

(cid:3)

ARTICLE

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Here, in the present study, we focus on the oscillatory part of the Hamiltonian. The oscillating part of the Hamiltonian can be expressed as

FIG. 1. Dust-dust interaction through the exchange of virtual plasma wave. The interaction of two particles in space (left) can be expressed as the interaction of two particles in the presence of a virtual wave in a plasma (right). A pair of dust particles designated as d with momenta p and p0 interact by exchanging a virtual plasma wave with wavenum- ber k and frequency xk. After the interaction, the dust particles carry momenta p (cid:2) (cid:1)hk and p0 þ (cid:1)hk. The interaction forms a classical wake potential in the presence of ion flow. The pair of dust particles shows the uncoupled harmonic oscillation.

is a dust plasma frequency and we used (cid:5) ¼ (cid:2)qkq(cid:2)kk2 for the first term and ¼ Vd(cid:2)k;k0 for the second term in the right- hand side of Eq. (11). Since the longitudinal plasma waves are heavily damped for the large wavenumber even in a complex plasma, the uni- tary transformation may introduce the wavenumber summation for k < kc; a critical wavenumber known as a Debye wavenumber. Our procedure here follows the one introduced by Bohm and Pines in 195314 and later applied to a complex plasma by Ishihara and Vladimirov in 1998.27 We now introduce destruction operator ajk tð Þ and creation † jk tð Þ through linear combination of pjk and qjk as operator a

(cid:1)h

(cid:1)h The harmonic oscillator Hamiltonian for a system of two dust particles (j ¼ 1,2) with oscillatory frequencies x1k and x2k is found as

(21) The transformation changes the variables from ðx; p; a; a†Þ to ðX; P; A; A†Þ. We note that X

Here, we introduced the screening effect of plasma waves with a plasma dielectric function e ¼ eðxjk; k) and used xj(cid:2)k ¼ xjk. We insert a factor to make ajk tð Þ dimensionless in Eqs. (12) and (13). A commutation relation is given by

where A; B (cid:5) ¼ AB (cid:2) BA and dkk0 is a Kronecker delta indicating zero ½ for k 6¼ k0 and unity for k¼ k0. We postulate that pjk and qjk are oper- ators satisfying

2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q Þ2=e0Vmj Zje ð k0 qkqk0 k (cid:3) k0exp i k þ k0 ½ ð d3x exp iðk þ k0Þ (cid:3) x

The oscillatory part of the Hamiltonian is now expressed as h † † jkajk þ ajka a jk

We now introduce a canonical transformation to see the nature of col- lective oscillations more clearly. We write

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s @xe=@x ð Þxjk 4(cid:1)hxjk ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s @xe=@x ð Þxjk 4(cid:1)hxjk

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

k (cid:3) ðPj (cid:2) (cid:1)hk=2Þ xjk (cid:2) k (cid:3) Pj=mj þ (cid:1)hk2=2mj

A generating function of the canonical transformation is introduced as

(cid:9) k ajkAjkeik(cid:3)Xj þ eik(cid:3)Xj A

† † ½ajk; aj0k0 (cid:5) ¼ ½a jk; a i

† † † jk (cid:2) ajkajk (cid:2) a jkajk þ ajka jka a

(cid:4) xjkqj(cid:2)k tð Þ þ ipjk tð Þ

(cid:4) xjkqjk tð Þ (cid:2) ipj(cid:2)k tð Þ

where xjd ¼ P Ð

ajkAjkeik(cid:3)Xj (cid:2) e(cid:2)ik(cid:3)Xj A

e0Vk2xjk @xe=@x ð

(cid:1)hxjk @xe=@x

@xe=@x

jk (cid:6) Pjk (cid:2) p

(cid:1)hS (cid:6) Onew (cid:2)

† jk tð Þ ¼ (cid:2)

h ajk; a

1 S e p e m b e r

Oold ¼ e(cid:2) i

(cid:5) ¼ pjk; pjk0 ½

e(cid:2)ik(cid:3)Xj ajk;

jdqjkqj(cid:2)k (cid:2)

j0k0 (cid:5) ¼ 0;

¼ djj0 dkk0 ;

ajk (cid:6) Ajk þ

† jk (cid:6) A a

ajkeik(cid:3)Xj :

31, 032118-3

pjkpj(cid:2)k ;

Hosc ¼ (cid:2)

(cid:1)hSOnewe

S; Onew

x2 jd x2 jk

ajk tð Þ ¼

Þxjk ! (cid:9)

(cid:5) ¼ 0;

i(cid:1)hdkk0 :

qjk; qjk0

qjk; pjk0

Zje mj

Hosc ¼

† jkajk

† jk þ

ajk ¼

where

S ¼ i

Þ (cid:3) x

(cid:10) ;

(16)

(24)

(17)

(18)

(19)

(20)

(22)

(23)

(13)

(12)

(15)

(14)

(11)

Þxjk

(cid:5) ¼

1 (cid:2)

† j0k0

i (cid:1)h

† jk

(cid:2)

(cid:5):

j;k

(cid:1)h

(cid:10)

(cid:9)

(cid:10)

”

(cid:4)

(cid:5)

;

(cid:5)

(cid:4)

(cid:2)

”

(cid:8)

(cid:7)

(26)

(25)

Þx2k

=x2

(cid:12) ;

=@x

Þx1k (cid:7)

HHO ¼

HHO (cid:6)

h ¼ A

† 1kA1k þ

(cid:8)

2 (cid:3)

† 2kA2k þ A

ARTICLE

† 1kA1k and A

2(cid:1)hx1k @xe=@x

2(cid:1)hx2k @xe=@x

(cid:11) (cid:7) (cid:1)hx1k A

Þ and p0 þ (cid:1)h k þ q=2

2 (cid:8)#

;

where x2 † jk; A

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(cid:7) † þ (cid:1)hx2k A 2kA2k þ

III. ENTANGLEMENT FOR A PAIR OF DUST PARTICLES

2 Þxk ¼ @x 1 (cid:2) x2

jd (cid:6) x2 i † j0k0 sion may be simplified as X

jk was used and commutation relations, ½Ajk; Aj0k0 (cid:5) † ¼ 0 and ½Ajk; A j0k0 (cid:5) ¼ djj0 dkk0 were used. The expres-

Equations (29) and (30) are known as normalized simple harmonic oscillator wave functions.32 In a quantum mechanical system, the wave functions are used to find probability of the results of any measurements.

We have seen in Sec. II that a system of a pair of particles (dust particles) can be described as uncoupled oscillators. The wave function is given by a product of two independent wave functions. The presence of such a wave function indicates that two particles are in the state not entangled each other. We now consider a situation in which a pair of particles are embedded in the background waves characterized by ðxk; kÞ and interact each other in the presence of external wave (k) characterized by a wavenumber q and frequency xq. As shown in Fig. 2, a pair of particles with momenta p and p0 will have momenta p (cid:2) (cid:1)h k (cid:2) q=2 ð Þ after the interaction. In the interaction described in Fig. 2, a dust particle with momentum p emits a quasiparticle (a virtual wave) with frequency x1k ¼ xk (cid:2) xq=2 and wavenumber k (cid:2) q=2, which is absorbed by an incoming quasiparticle [the external wave (k)] characterized by ðxq; qÞ, then we write

with the approximation @xe=@x xk(cid:6)xd (cid:6) 2. In a complex plasma characterized by charge neutrality as a whole system, a dust acoustic wave is characterized by the dust acoustic veloc- ity defined by the dust mass and plasma temperature,30 allowing the matching of the frequencies between the waves and dust oscillations. † The operators A 2kA2k may be interpreted to have eigen- values of positive integers in quantum mechanical description with the energy eigenvalue En ¼ (cid:1)hxk n þ 1=2 Þ, where n is a positive integer. two This expression indicates that the system is composed of † uncoupled oscillators. We note here that the operators Ajk and A jk are assumed to obey the commutation relations, resulting in the factors 1/2 in Eq. (26), but even without the commutation relations, the harmonic † oscillator expression with Ajk and A jk will remain in the classical sys- tems considered in the present study.

where l is a mass of an oscillating body, K is a spring constant, p is a momentum and x is a displacement. The corresponding one- dimensional wave function for a system of two oscillators of one with coordinates X1 and the other with X2 as a solution of Schr€odinger equation may be given by the product of two functions wn;m X1; X2 ð

While the virtual wave k (cid:2) q=2 is accompanied by the creation of a dust particle with momentum p (cid:2) (cid:1)h k (cid:2) q=2 Þ and destruction of a dust particle with momentum p, the virtual wave absorbs the external wave characterized by ðxq; qÞ and travels as a virtual wave character- ized by x2k ¼ xk þ xq=2 and wavenumber k þ q=2. A dust particle with momentum p0 absorbs the virtual wave. We write

The virtual wave k þ q=2 is accompanied by the creation of a dust particle with momentum p0 þ (cid:1)h k þ q=2 Þ and destruction of a dust particle with momentum p0. The harmonic oscillator Hamiltonian given by Eq. (26) is studied in detail by following the procedure devel- oped earlier.33-38 We define operators B in terms of A as

where the displacement X is normalized by (cid:1)h2=lK , Hn Xð Þ is the Hermite polynomials of the n-th degree with the normalization relation

To further discuss the problem of oscillators, we restrict our model to a simple one-dimensional harmonic oscillator given by a Hamiltonian,31

FIG. 2. A pair of dust particles interact each other in the presence of background waves with frequency xk and the injected wave with frequency xq. The pair shows the coupled harmonic oscillation.

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

(cid:7) (cid:8)1

p (cid:7) (cid:8)1

vm Xð Þvn Xð ÞdX ¼ dmn:

† 1k þ A2k ﬃﬃﬃ p

† 1kA1k ¼ A A

† 2kA2k ¼ A A

Hm X2ð (cid:2)

w0;0 X1; X2 ð

1ﬃﬃﬃ p e(cid:2)1 p

For m ¼ n ¼ 0,

1 S e p e m b e r

A1k þ A p

1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2mm!

Þ ¼ vn X1ð

31, 032118-4

1ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2nn!

2 ; Þe(cid:2)

† 1k ¼ B

Þvm X2ð

vm X2ð

B1k ¼

vn X1ð

Hn X1ð

Kx2;

p2 þ

† q k(cid:2)

† q kþ

(cid:3) 1=4

Ak(cid:2)

Akþ

H ¼

with

(33)

(34)

(35)

(30)

(31)

(32)

(27)

(28)

(29)

1 þX2

Þ ¼

ð 2 X2

† 2k

ﬃﬃﬃ

Þ:

;

(cid:3)

(cid:2)

(cid:3)

(cid:2)

n¼0

ﬃﬃﬃ

n0¼0

† 2k

ð 2 X2

1 þX2

(38)

(37)

(36)

(50)

(49)

2 þX02

1 þX02

Þe(cid:2) 1

X ¼

Hn X1ð

vn X1ð

B2k ¼

† 2k ¼

Ann0 ¼

HHO ¼

dX1dX2

ÞHn0 X2ð

G s; Xð

Þvn0 X2ð

ð nn0 Ann0

(cid:1)hxq

Ann0 vn X1ð

(cid:2) v0 X0

to another set

ÞdX1dX2;

A1k (cid:2) A p

ARTICLE

Þ ¼ e(cid:2)s2þ2sX ¼

(cid:7) (cid:8) X1 X2

† 1k (cid:2) A2k ﬃﬃﬃ p

1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2nþn0 n!n0!

which can be evaluated by

(cid:2) v0 X0

† † 2kB1k þ 1 1kB2k þ B

with a condition ated by

Physics of Plasmas

2Þ ¼ 1. To evaluate the integration

(cid:2) Þv0 X0 Þvn0 X2ð

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Þ2 ¼ 1. The coefficient Ann0 may be evalu-

and using the commutation relations for B operators, we obtain

1Þ ¼ H0ðX0 where we used H0ðX0 with respect to X1 and X2, we note the generating function

(cid:11) (cid:2) † † (cid:1)hxk B 2kB2k þ 1 1kB1k þ B (cid:12) :

To see the coupling situation, we introduce a rotational transfor- mation of coordinates known as a squeeze transformation, related to the representation of Lorentz group.34-36 We introduce transformation from one set of coordinates

† We note here that the operators Bjk and B jk are assumed to obey the commutation relations, resulting in the factors 1 appeared in Eq. (37). † † The presence of terms B 2kB1k involving the cross-products 1kB2k and B suggests the two coupled oscillators.

2 (cid:3) where we used 1 þ b2 (cid:2) and 2b= 1 (cid:2) b2 ¼ 2 sinh g cosh g ¼ e2g (cid:2) e(cid:2)2g ; X0 ; X0 tion is now expressed as wgðX0

h (cid:2) e(cid:2)2g X0

Noting that X2 and further coordinate transformation X1 ¼ ðu (cid:2) vÞ= þ vÞ= makes the integration evaluate as ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) b2

Equation (56) shows that the wave function is not separable in varia- bles X0 2 resulting in the entanglement property. Equation (57) agrees with the two-mode squeezed states introduced to study entan- glement of formation.39 For the limit of g ! 0 (or b ! 0), we obtain

with a constant b ((cid:2)1 (cid:7) b (cid:7) 1) expressed through a parameter g, known as a squeezing parameter, as

which can be expressed in terms of Ann0 given by Eq. (54) as X1

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

Þ=2 Þ=2. Our wave func- 2Þ or (cid:2)

b ¼ tanh g; 1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) b2 bﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) b2

The wave function is now (cid:2) w0;0 X0

ð 2Þ ¼ w0;0ðX0 (cid:3)

h (cid:2) e(cid:2)2g X0 (cid:2) = 1 (cid:2) b2

¼ cosh 2g þ sinh2g ¼ e(cid:2)2g þ e2g

n0¼0 n¼0 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q X1 1 (cid:2) b2

=ð1 (cid:2) b2Þ and X2 ¼ ðu

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q X1 1 (cid:2) b2

1 þ X2 X2 (cid:2) (cid:3)

1ﬃﬃﬃ p exp (cid:2) p

! X0

ÞdX1dX2: (cid:5)

and evaluate the integration

(cid:4) (cid:2) 2 ¼ 2 X2

sn n! Hn Xð Þ

We obtain the relation

1 (cid:2)b (cid:2)b

We note the relation

(cid:2) ¼ v0 X0

1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) b2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) b2

(cid:2) 2bX1X2 p

1 (cid:2) X0 (cid:3)

1 S e p e m b e r

(cid:2) wg X0

which gives us

(cid:2) v0 X0

X ¼ T0X0;

Ann0 ¼ bn

X0 ¼ TX;

¼ cosh g;

1 b b 1

Ann0 vn X1ð

¼ sinh g;

31, 032118-5

bnvn X1ð

1 and X0

e2bst ¼ p

ÞG t; X2 ð

þ e2g X0

2bst n!

1 þ X02

G s; X1 ð

2 þ X02

(cid:14) i ;

1 þ X0

1 (cid:2) X0

1 þ X0

1 þ X2

Þvn0 X2ð

1 þ X2

Þvn X2ð

TT0 ¼

X0 ¼

Þ ¼ p

T0 ¼

I s; t ð

dnn0 :

where

T ¼

Þe(cid:2)1

; X0

1 þX02

2 ¼

2 þX02

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

i ;

1 þX2

Þ ¼

and

ð 2 X2

ﬃﬃﬃ

Þn

n¼0

Þ:

(cid:13)

(cid:7)

(cid:8)

(cid:7)

(cid:8)

(cid:7)

(cid:8)

(cid:3)

(cid:2)

(cid:3)

;

(cid:3)

(cid:2)

(cid:3)

2 ;

(cid:15) (cid:15) (cid:15) (cid:15) (cid:15)

(cid:12)

Þ2;

g ¼

(60)

(59)

(58)

(69)

(68)

(67)

1 þ

H ¼

Þe(cid:2)1

(cid:15) (cid:15) (cid:15) (cid:15) (cid:15):

; X0

Þ2 þ

m (cid:2)

k0 1x2

k0 2x2

2X0 e(cid:2)1

1 2l

3x1x2

2 þ k0

g ! k0¼1

K ! k0¼1

2m1

2m2

1 and X0

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l=mj

k1 x1 (cid:2) x2

k2 x1 þ x2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mj=l

Þ for X0

ARTICLE

(cid:2) p2 1 þ p2

(cid:2) wg!0 X0

1ﬃﬃﬃ 2X0 p e(cid:2)1

reduced mass k0

(cid:11) 1 þ e(cid:2)1

xj (j ¼ 1, 2), in a form as

ﬃﬃﬃﬃﬃﬃﬃﬃ 3em 1 þ em q

We find that the state of

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m þ e(cid:2)2 1 (cid:2) e(cid:2)1 m

x1;2 ¼ xk6 xq

is a ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p m1m2 coordinate

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(cid:11) em 4 (cid:2) k02 1 þ e(cid:2)1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 (cid:2) 2 (cid:2) k02 ð m þ e(cid:2)2 m

which is separable in coordinates X0 entanglement appears if g 6¼ 0.

with k0 ¼ Dk=k and em ¼ m1=m2, which in the limiting case of k0 ¼ 1,

The parameter g depends only on the mass ratio and is given by Þ=8 for em (cid:8) 1. We find two Þ=8 for em ¼ 1 and g (cid:6) ln 4em g ¼ ln 3ð ð normal modes from Fig. 2 together with Eq. (37) as

where k1 and k2 are coupling constants. The model equation may be rewritten, after coordinate transformation of pj ! pj, xj !

Now we show that the wave function given by Eq. (56) is a solu- tion for a coupled harmonic oscillator.34-38 For a notational simplicity, (cid:3) we put x1; x2 . The physical model for a coupled har- ð monic oscillator of two masses of m1 and m2 located at x1 and x2 may be given by a model Hamiltonian

1 ¼ lk=m1, where l ¼ m1m2=ðm1þm2Þ Þ=2 and 3 ¼ lDk= 2 ¼ lk=m2, and k0 k0 Dk ¼ k2 (cid:2) k1. transformation p1 ! ðcos aÞ p1 þ ðsin aÞ p2; p2 ! (cid:2)ðsin a Þp1 þ ðcos aÞ p2, x1 and ! ðcos aÞ x1 þ ðsin aÞ x2; x2 ! (cid:2)ðsin aÞ x1 þ ðcos aÞ x2 with angle (cid:2)m1Þ(cid:5)g=2 gives a Hamiltonian without a ¼ tan(cid:2)1 Dk the cross product of x1x2. On the other hand, it is straightforward to show, by noting that p1 ¼ (cid:2)i(cid:1)h@=@x1, p2 ¼ (cid:2)i(cid:1)h@=@x2 and keeping in mind that spatial coordinates are normalized by (cid:1)h2=lK

The present quantum mechanical approach (semiclassical approach), especially the second quantization approach, reveals a new aspect in the classical coupling of a pair of dust particles. While a clas- sical interaction between two particles involves direct encounter of two particles, a semiclassical picture of the interaction between negatively charged dust particles in a plasma includes the exchange of a virtual wave between two dust particles. Figure 3 shows schematically the pro- posed process resulting in the semiclassical entanglement in a complex plasma. Figure 3(a) shows a system of uncoupled oscillators described by the wave function wn;mðX1; X2Þ with coordinates X1 and X2 of two

In Sec. III, we showed that a system of a pair of dust particles placed in a complex plasma becomes the state of the entangle- ment only if the system is exposed to the external wave. In this section, we elaborate the present process through the comparison with conventional quantum entanglements. Since the nature of the entanglement in this study is not quite the same as the con- ventional quantum entanglement, the present entanglement is called as a semiclassical entanglement.

where we set xk (cid:6) xd (dust plasma frequency) and ln em ¼ 0 for a pair of dust particles with equal mass. Thus, we find that a pair of dust particles are entangled only when the external wave with frequency given by Eq. (73) is injected into the complex plasma in consideration.

By taking the ratio of two normal mode frequencies, we obtain (cid:15) (cid:15) (cid:15) (cid:15) (cid:15)

which, with the approximation lnð16xÞ (cid:6) 6x (cid:2) x2=26x3=3 (cid:2) …, results in

By setting the conditions of vanishing cross products x1x2, the parame- ters are found as

With the help of Eq. (69), we obtain required frequency of the external wave to introduce the entanglement as

IV. SEMICLASSICAL PROPERTIES OF THE ENTANGLEMENT IN A COMPLEX PLASMA

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

K e(cid:2)4g x1 (cid:2) x2 ð s ﬃﬃﬃﬃ K l

which can be used to express Schr€odinger equation as

We find that the system has two normal mode frequencies

(cid:2) xk þ xq=2 (cid:2) xk (cid:2) xq=2

with k ¼ k1 þ k2 rotational

1 (cid:2) k02 Þk02em ð m (cid:2) 2(cid:2)k0

h e(cid:2)4g x1 (cid:2) x2 ð

ﬃﬃﬃﬃﬃ em 1 þ em

Þ2 þ e4g x1 þ x2 ð

(cid:2) p2 1 þ p2

k 4 (cid:2) 2k0 (cid:2) 1 ð

1 S e p e m b e r

Hwg ¼ Ewg

Þxd ! em¼1

e(cid:2)2g þ e2g

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m1m2

31, 032118-6

cosh 2g:

1 þ ln em

1 þ p2 p2

xq 4xk

=k m2 ð

p (cid:1)h

E ¼ (cid:1)h

x1;2 ¼

Further

ﬃﬃﬃﬃﬃﬃﬃ lK

e4g ¼

ﬃﬃﬃﬃ K l

1 þ e2

wg ¼

Þk0 (cid:2)

where

xq (cid:6)

(cid:3) 1=4

, that

xd;

H ¼

K ¼

62g:

(70)

(73)

(71)

(72)

(66)

(61)

(62)

(63)

(64)

(65)

g (cid:6)

i ;

i(cid:14)

Þem

(cid:3) (cid:3)

(cid:15) (cid:15) (cid:15) (cid:15) (cid:15)

Þ2

(cid:2)

(cid:13)

”

(cid:2)

(cid:3)

(cid:4)

;

Þ and vm X2ð

ARTICLE

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FIG. 3. Model of coupled oscillators for a pair of dust particles placed in a complex plasma. (a) Uncoupled oscillators. Two dust particles, X1 and X2 in the coordi- nates X, interact each other through the exchange of virtual wave with wave num- ber k. (b) Coupled oscillators in the pres- ence of external wave with wave number q. The coordinates X are transformed into X0 through the squeeze transformation: X0 ¼ TX; T ¼ T(g), and g ¼ tanh(cid:2)1b. The squeezing parameter g controls the entanglement.

dust particles embedded in a background wave (k) in a complex plasma. The wave function of the two-particle system is described by the product of two functions vn X1ð Þ indicating a separable wave function. The probability to find a particle at X1 and another at X2 is given by the square of the absolute value of product of the two independent wave functions, indicating no entanglement. Keeping in † kAkðk ¼ 1k; 2kÞ in Eq. (26) corresponds to the mind that (cid:1)hxkA energy spectrum of plasma oscillations, the Hamiltonian for the har- monic oscillators is expressed in a semiclassical way as (cid:2)

The direct observation of the present semiclassical entanglement may be difficult to achieve because of the squeezed state in a trans- formed coordinates. One of the important quantities to identify the emergence of the entanglement is the entropy and will be discussed in detail in Sec. V. Here, we briefly comment on the difference of the semiclassical entanglement from the quantum entanglement, especially related on the measurement. In the semiclassical entanglement, a pair of dust particles move in an interconnected manner like a Cooper pair, but not like composite quantum objects, e.g., electrons with spins or photons with polarizations. A pair of dust particles are not composite quantum objects which can be split apart to make quantum entangle- ment. Classical processes, including the present semiclassical entangle- ment, are inherently local allowing the measurement independently, while the quantum entanglement is inherently nonlocal. The quantum entanglement involves the EPR (Einstein-Podolsky-Rosen) pairs and a wave function collapse,40 while the recent theory on the uncertainty predicts the possible simultaneous measurements of position and momentum.41,42

characterizing the collective nature of background plasma. Thus, the present system is dealing with macroparticles (dust particles) in the sea of microsystem (composed of plasma particles). The present system is written, with the help of second quantization, in a non-separable wave function form only when the system is subject to the external wave. Then, the semiclassical entanglement appears only in coordinates in a squeeze transformation or in a squeezed state, where the modes A (A†; A) change to modes B(B†; B). As is seen in Eq. (74), the process described by mode A is closest to the classical state, while the mode B, as is shown in Eq. (75), describes the process next to closest to the clas- sical state. Thus, the entanglement is characterized by a semiclassical nature, not by quantum nature.

Figure 3(b) shows the coupled oscillators embedded in a background wave (k) in the presence of external wave (q) injected into a complex plasma. The coordinates are rotationally transformed from X to X0 (cid:2) and the wave function becomes wg X0 which is shown to be not

separable. Such a non-separable situation is manifested only through the second quantization by changing modes A(A1k, A2k) to B(B1k, B2k). Since the term (cid:1)hxk (cid:2) (cid:1)hxq=2 in the square bracket of Eq. (37) is sim- ply a pure quantum effect by the use of commutation relations, the harmonic oscillator Hamiltonian in the semiclassical expression is given by

The wave function associated with the semiclassical entanglement is given by Eq. (56) which represents a probability wave and the ampli- ; X0 tude of wg X0 is the probability amplitude of finding two dust

2 1 and X0 particles at X0 2. The two negatively charged dust particles are forming a pair like a Cooper pair in a superconductivity and were described in the context of wake potential formation.27 Formation of the pair is symbolized by the presence of a virtual wave connecting two dust particles. The wave function described by Eq. (56) is the manifes- tation of non-separable nature if g 6¼ 0.

† † 1kB2k þ B The B mode has interacting parts expressed by B 2kB1k, indi- cating the correlation of two dust particles. The nonzero squeezing parameter g, which is defined by the frequency ratio between the exter- nal wave xq and the virtual wave xk, controls the emergence of the semiclassical entanglement. Figure 3 is reminiscent of the Maxwell’s Demon, who operates a trapdoor to control fast-moving and slow- moving gas molecules, sitting at the origin of the coordinates who switches on the radiation to initiate the entanglement.

A complex plasma is formed by charged dust particles sur- rounded by ions, electrons, and neutrals. We focus on a pair of dust particles in the complex plasma. The interaction of two dust particles is accompanied by the interchange of a quasiparticle, a wave

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

(cid:11) (cid:2) † (cid:1)hxk B 1kB1k þ B

1kA1k þ (cid:1)hx2kA

† 1kB2k þ B B

1 S e p e m b e r

(cid:1)hxq

31, 032118-7

(cid:1)hx1kA

† 2kA2k

† 2kB1k

† 2kB2k

HHO ¼

; X0

(74)

(75)

(cid:2)

(cid:12)

(cid:3)

(cid:2)

(cid:3)

(cid:2)

(cid:3)

†

(cid:5)

(cid:4)

(cid:17)

(cid:16)

Þ!

(cid:5) Þ

(76)

(81)

Þlog 1 (cid:2) p ð

H Xð Þ ¼ (cid:2)

ARTICLE

H pð Þ (cid:6) (cid:2) p log p þ 1 (cid:2) p

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V. ENTROPY FOR ENTANGLEMENT

pubs.aip.org/aip/pop

HðpÞ is known as entropy function in the theory of information. When the ensemble of n letters each of which has a distribution X, the entropy is given by

First, briefly review the concept of entropy. Consider a system of strings with a total of n binary digits in which 0 occurs with probability 1 (cid:2) p and 1 occurs with probability p. Typical binary digits will con- tain nð1 (cid:2) pÞ zeros and np ones. The number of typical strings is of order 2nHðpÞ, where HðpÞ ¼ log n!= ðnpÞ! nð1 (cid:2) pÞ with base 2 in the logs. The Stirling approximation, log n! ’ n log n (cid:2) n for n (cid:8) 1, is used to obtain

Two-particle entanglements were studied for quantum states including two-electron system, two-photon system and atom-photon system.10 In Secs. II-IV, we have shown that two-particle semiclassical entanglement is possible in a complex plasma. Our question is how to measure the entanglement in our macroscopic system. We evaluate the entropy of the entanglement in our system as a measure of entanglement.

where in the schematic representation the virtual waves are omitted and the external wave is designated as k. The first term in the right-hand side indicates the process in which a dust particle with momentum (cid:1)hkð¼ pÞ, together with a dust particle with momentum (cid:1)hk0ð¼ p0Þ, is produced through the interaction between two dust par- ticles by absorbing external waves with energy (cid:1)hx0 and momentum (cid:1)hq 0. The second term shows the process in which a dust particle with momentum (cid:1)hk is removed accompanied by the emission of the wave with energy (cid:1)hx0 and momentum (cid:1)hq 0. The mathematical expression for the equation may be obtained by replacing each schematic diagram by the corresponding transition probability per unit time from an initial state i to a final state f together with the application of the Fermi golden rule.46-48 Thus, we obtain

The analogy can be applied to an interaction between particles in the system in consideration. The n binary digits may be considered as energy levels. The binary digits may correspond to the Fermion eigenval- ues of the number operators. Let N(k) be the number of particles with momentum pð¼ (cid:1)hkÞ in a second quantization formalism. If the particles are fermions N(k) ¼ 0 or 1. At each energy level a factor 1 (cid:2) NðkÞ appears if they are created, while NðkÞ appears if they are destroyed. If the particles are bosons N(k) ¼ 0,1,2,…. At each energy level, a factor N kð Þ þ 1 appears if they are created, while NðkÞ appears if they are destroyed. If the created particles are fermions, N(k) ¼ 1 then the factor 1 (cid:2) NðkÞ for the creation is zero suggesting that the transition to occu- pied states are forbidden. The entropy is given by45,46

=2; and Nk q where k1 ¼ k þ q (cid:2) q Þ is the 0ð

number of quasi-particles (k) with momentum (cid:1)hq 0. Here, we consider particles as fermions and quasi-particles as bosons. For the case particles are bosons, the procedure follows by replacing 1 (cid:2) N kð Þ by 1 þ N kð Þ etc., but the choice of fermions or bosons for particles will have no real consequences in the classical systems considered in the present study. Ef (cid:2) Ei are the energy difference between the final energy and the initial energy in the system given by

in agreement with Shannon entropy given by Eq. (77). We consider the interaction between dust particles as described in Fig. 2. We change the notations of variables and write xq; q Þ for the virtual wave and xq Þ for the external wave. An equation for the rate of change of ð NðkÞ may be written schematically as

for fermions, where kB is a Boltzmann’s constant and ln is a natural logarithm. It is obvious that there is a similarity between Eqs. (76) and (79). We note that in a classical limit NðkÞ (cid:9) 1, so the entropy given by Eqs. (78) and (79) is given by

where v qð Þ is the Fourier transform of the interaction potential depict- ing the potential energy between dust particles and (cid:3)

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

where Mfi is the matrix element for the transition given by

ÞNk q 0ð (cid:4) Þ 1 þ Nk q 0ð

(cid:1)h2 kj j2 2m1 (cid:1)h2 2m1

f1 k; k0; q (cid:3) (cid:2) f2 k; k0; q

known as Shannon entropy.43,44

(cid:2)N kð Þln N kð Þ þ 1 þ N kð Þ

N kð Þln N kð Þ þ 1 (cid:2) N kð Þ

=2; k2 ¼ k0 (cid:2) q (cid:2) q

(cid:2) ¼ f1 k; k0; q

(cid:2) (cid:2) f2 k; k0; q

(cid:15) (cid:15)2F k; k0; q

(cid:15) (cid:15) j2 ¼ v qð Þ

(cid:2) F k; k0; q

j2 (cid:1)h2 k0 j 2m2

N kð Þln N kð Þ;

¼ 1 (cid:2) N k1ð

(cid:5)ln 1 þ N kð Þ

(cid:5)ln 1 (cid:2) N kð Þ

¼ 1 (cid:2) N kð Þ

1 S e p e m b e r

for bosons and

p Xð Þlog p Xð Þ

(cid:5) 1 (cid:2) N k2ð ½

j2d Ef (cid:2) Ei

(85) (cid:5) ;

(cid:5)N kð ÞN k0ð

(cid:5) 1 (cid:2) N k0ð ½

31, 032118-8

k0 (cid:2) q (cid:2)

(cid:1)h2 2m2

k þ q (cid:2)

N kð Þ ¼

!(cid:15) (cid:15) (cid:15) (cid:15) (cid:15)

S ¼ (cid:2)kB

Ef (cid:2) Ei j

þ (cid:1)hxq

ÞN k2ð

(cid:5)N k1ð

S ¼ kB

2p (cid:1)h

@ @t

with

(82)

(83)

(84)

(86)

(87)

(77)

(78)

(79)

(80)

0 þ

(cid:17) ;

j ¼

Mfi

(cid:15) (cid:15) (cid:15) (cid:15) (cid:15)

; q

Þ;

(cid:15) (cid:15) (cid:15) (cid:15)

Þ;

(cid:2)

(cid:16)

(cid:2)

(cid:3)

(cid:2)

(cid:3)

;

(cid:5)

(cid:3)

(cid:2)

(cid:15) (cid:15)

(cid:16)

(cid:8)

(cid:7)

(cid:8)

(cid:7)

(cid:15) (cid:15) (cid:15) (cid:15)

(cid:16) (cid:3)

i (cid:4)

k (cid:3)

Þ ¼

k0 (cid:3)

(cid:17) :

(90)

(89)

(88)

(96)

0 ¼

(cid:17) (cid:5)

; x0

(cid:2) (cid:1)h

q þ

q (cid:2)

dx2;

dS dt

v qð Þ

2p (cid:1)h

¼ (cid:2)kB

(cid:1)h m2

(cid:1)h m1

. x; x0 ð

2m2

2m1

wg x; x2 ð

@N kð Þ @t

þ q (cid:15) (cid:15) (cid:15) (cid:15)

ARTICLE

(cid:2) q (cid:15) (cid:15) (cid:15) (cid:15)

Using Eq. (82), we get

(cid:15) (cid:15)2d Ef (cid:2) Ei

h Þ wg x0; x2 ð

ln N kð Þ (cid:2) ln 1 (cid:2) N kð Þ

ate Eq. (79) with respect to time

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¼ (cid:2) kB k k0 (cid:2) (cid:10) F k; k0; q

Þ and the other at x0 (cid:3)

; x0

q ln N kð Þ (cid:2) ln 1 (cid:2) N kð Þ

We proceed to find the time change of the entropy. We differenti-

(cid:2) Þwg x0 (95)

1ð¼x0Þ, but not x2 and x0 2,

From the condition Ef ¼ Ei, we obtain the frequency of the external wave as

where the integration is extended only over the coordinates x2ð¼ x0 2Þ. The entropy may be defined by taking the sum of the diagonal element of a matrix or trace of the matrix

¼ wg x1; x2 ð If we can observe coordinates x1ð¼xÞ and x0 it is appropriate to define the density matrix defined by ð

We note that the number of particles NðkÞ is the particle number den- sity in a classical limit. We now introduce a density matrix as a statisti- cal description. We consider a system of coupled oscillators in which (cid:3) ; x0 one set of oscillators is located at x1; x2 ð (cid:2)

described by wave functions wgðx1; x2Þ and wg x0 . If we observe

two sets of coordinates in the system in consideration, we may define a pure-state density matrix as45,49 (cid:3) (cid:2) . x1; x2; x0

The sum over k, k0; and q may be carried out by the use of the change of variables. We call Eq. (90) as the first equation. The second equation is by changing variables in the first equation as k ! k þ q (cid:2) q =2, k0 ! k0 (cid:2) q (cid:2) q =2; and q ! (cid:2)q. The third equation is by changing variables in the second equation as k ! k0 and k0 ! k and q ! (cid:2)q. The fourth equation is by changing variables in the first equation as k ! k0 and k0 ! k and q ! (cid:2)q: X

in agreement with the entanglement entropy evaluated for a squeezed 1 þ g2=2! þ g4=4!…; state.50,51 With the approximations cosh g ! g!0 e (cid:2) e2=2 þ e3=3 (cid:2) …, g þ g3=3! þ g5=5!…; and lnð1 þ eÞ ! sinh g ! e!0 g!0

S ¼ (cid:2)kBTr . ln q When we perform the integration in Eq. (96) over x2 from (cid:2)1 to 1, we obtain

4 (cid:15) (cid:15) (cid:10) v qð Þ (cid:15) (cid:15) þ v (cid:2)qð ð Þ (cid:15) (cid:15) (cid:15)2F k2; k1; q (cid:15) þ v qð Þ (cid:2) (cid:15) (cid:15) þ v (cid:2)qð (cid:15) (cid:15)

where b ¼ tanh g as defined in Eq. (42) and vn xð Þ is the n-th excited state oscillator wave function defined by Eq. (29). We find the trace in Eq. (97) as

(cid:2) where f1 k; k0; q Þ ln f1 (cid:2) ln f2 Since f1 (cid:2) f2 ð ð Þ ¼ 0 for f1 ¼ f2, we obtain (cid:10) ln f1 (cid:2) ln f2

In the classical limit, N kð Þ (cid:9) 1 or 16N kð Þ ! 1 and Nk q Þ 0ð Þ. The classical external wave frequency,

(cid:8) 1 or 1 þ Nk q Þ ! Nk q 0ð 0ð by letting p ¼ (cid:1)hk, p0 ¼ (cid:1)hk0, q ! 0 and (cid:1)h ! 0, is given by

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

2p (cid:1)h (cid:16) (cid:3) ln N kð Þ (cid:2) ln 1 (cid:2) N kð Þ

which may be expressed in terms of the parameter b as

(cid:15) (cid:15)2F k1; k2; (cid:2)q (cid:16) Þ ln N k2ð (cid:16) (cid:3)

(cid:15) (cid:15)2F k0; k; (cid:2)q Þ (cid:15) (cid:15) (cid:15) ¼ v (cid:2)qð (cid:15) X (cid:15) 2p (cid:15) (cid:1)h

are given by Eqs. (85) and (86). Þ > 0 for any f1 and f2ð6¼ f1Þ and f1 (cid:2) f2 Þ

(cid:15) (cid:15)2d Ef (cid:2) Ei (cid:4) (cid:2) ln f1 k; k0; q

(cid:3) (cid:2) tanh2g ln tanh2g (cid:3)

1 (cid:2) tanh2g

b2 1 (cid:2) b2 ln b2 þ ln 1 (cid:2) b2

4 k (cid:2) (cid:10) f1 k; k0; q

(cid:3) ¼ (cid:2)2 cosh 2g (cid:2) sinh2g

Noting that v qð Þ X

h 2 x (cid:2) x0 ð Þ(cid:2)1

(cid:3) (cid:2) tanh2ng ln tanh2ng

Þ (cid:2) ln 1 (cid:2) N k1ð ½ (cid:17)

q (cid:2) (cid:3) (cid:2) f2 k; k0; q

To carry out the integration, we note

S ¼ 2kB cosh2g ln cosh g

(cid:2) (cid:2) ln f2 k; k0; q

k k0 (cid:2) (cid:15) (cid:15)2F k; k0; q

(cid:2) and f2 k; k0; q

1 (cid:2) tanh2g

(cid:3) ln cosh(cid:2)2g

Þ (cid:2) sinh2g ln sinh g Þ

(cid:2) Þ ¼ 1 (cid:2) b2

i cosh 22g

(cid:15) (cid:15), we obtain

b2nvn xð Þvn x0ð Þ;

(cid:16) Þ ln N k1ð

Þ (cid:2) ln 1 (cid:2) N k2ð

(cid:10) exp (cid:2) 4 cosh 2g ð

Þ (cid:2) ln 1 (cid:2) N k0ð ½

dx dx0. x; x0

Þ ¼ p cosh 2g

1 S e p e m b e r

tanh2ng ¼

Þln . x0; x ð

þ x þ x0

31, 032118-9

d Ef (cid:2) Ei ð

We obtain

ln cosh g

Tr . ln q

S ¼ (cid:2)kB

ln N k0ð

. x; x0 ð

v qð Þ

we get

dS dt

(cid:11) 0:

kB (cid:4)

(cid:3) X

(103)

(100)

(101)

(102)

(104)

(105)

ð (cid:5) (cid:3)

Þ2 (cid:5)

¼ (cid:2)

(cid:17) (cid:5)

0 ¼

Þ(cid:2) 1

(97)

(98)

(99)

(94)

(91)

(92)

(93)

Þ ¼

(cid:3) q

(cid:5) ;

Þ2

n¼0

Þ:

(cid:7)

(cid:8)

(cid:17)

”

(cid:17)

(cid:3)

(cid:2)

(cid:3)

(cid:4)

(cid:5)

;

(cid:5)

Þ:

(106)

S ! g!0

kBg2 1 (cid:2) 2 ln g

ARTICLE

Physics of Plasmas

pubs.aip.org/aip/pop

VI. DISCUSSION AND CONCLUSIONS

The quantum entanglement was considered to be manifestation of quantum phenomena involving subatomic particles. However, the recent observations of entanglements have extended the concept from microscopic to macroscopic-scale objects.9,63,64 Our study involves macroscopic dust particles in a classical system and the present research on semiclassical entanglement may shed a light on the devel- opment of the concept of entanglement in some new directions.

The entropy of the entanglement is always positive or S > 0 for non- zero g. We note that the criterion S ¼ 0 indicates no entanglement in the system. The entropy for a pair of dust particles with equal mass is estimated to be 0.08 in the unit of kB (Boltzmann constant). On the other hand, the entropy for the two-electron system is reported to be 0.02-0.08 depending on the model,52 while the theory for the two- electron system predicts to be ln 2 (cid:6) 0:693. The estimating entropy remains to be challenging experimentally or computationally.53

to a plasma theory or even to a complex plasma theory is a synthesis of the theories and allows the situation with the dispersive properties of ambient medium. The quantum mechanical calculations are often found to be more straightforward and easier to understand the underly- ing physics. In our quantum mechanical view applied to a classical sys- tem of plasmas, the charged particles could be electrons, massive ions or even macroparticles like dust particles, while the plasma waves are treated as quasi-particles. It is interesting to note that quantum mechani- cal approach has been successfully applied to point-like macro-objects including large cosmic objects such as black holes involved in gravita- tion.57-61 It is now shown that there is a long-range entanglement across the event horizon, or between the inside and outside of a black hole.62

We have introduced the concept of semiclassical entanglement in a complex plasma. Our quantum mechanical approach to the dust- wave interaction reveals that the injection of the external wave into a two-particle system in a complex plasma plays a key role in the emer- gence of entanglement. The two-particle system is composed of two dust particles, massive and highly charged. First, we showed that dust particles in a complex plasma form a pair by exchanging virtual waves between two dust particles and can be viewed as uncoupled harmonic oscillators, which are not entangled. Then, we showed that a pair of dust particles can be described as a coupled harmonic oscillator only when the external wave is injected. The pair of dust particles exposed in the wave with frequency of half of the dust plasma frequency is shown to be entangled. The entropy is introduced as a measure of the entanglement for the pair of dust particles.

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The second example is the wake potential formation in a complex plasma.19-22 The quantum mechanical approach revealed that a pair of charged dust particles is formed only when both charges move together, or alternatively stationary in a moving frame like in the ion flow, through the exchange of virtual phonons (quanta of ion acoustic waves) and the resulting attraction between two negatively charged dust particles.27 Two dust particles behave like a Cooper pair. In a clas- sical picture, a charged dust particle forms an oscillating wake potential due to the modification of Debye shielding in the presence of ion flow accompanied by the ion acoustic wave.

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The semiclassical entanglement described in the present paper is found only through the quantum mechanical approach to the complex plasma.

Phys. Plasmas 31, 032118 (2024); doi: 10.1063/5.0192854 VC Author(s) 2024

Author Contributions Osamu Ishihara: Investigation (equal).

AUTHOR DECLARATIONS Conflict of Interest

The data that support the findings of this study are available from

ity in elastic collisions in nonideal plasmas,” Phys. Scr. 72, 234 (2005).

the corresponding author upon reasonable request.

The authors have no conflicts to disclose.

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