Physicists have long studied Landau states and energy levels of electrons in a stable magnetic field. However, when the magnetic field dissipates, the question of how Landau states evolve remains unanswered. This study investigates the magnetic dissipative effect from a quantum mechanics perspective. We establish a quantum master equation that describes a dissipative magnetic field. Subsequently, we solve the master equation using an entangled state representation. Consequently, we found that the Landau state evolves into a binomial quantum state (mixed state) when the magnetic field dissipated. This is in sharp contrast to the case where the Landau state becomes a negative binomial state when an additional harmonic oscillator potential is applied. We hope that this finding will be helpful for the evolution of quantized plasma physics.

The Lorentz force influences the motion of the charged particles in a magnetic field. Landau was the first to consider the quantum effect of electrons in a stable, uniform magnetic field. Landau discovered a discrete energy state called the Landau state.1 An interesting question that has long been overlooked is how the Landau state dissipates when the magnetic field is damped, which differs from the situation in which an additional harmonic potential is applied. In this study, we addressed these two questions.

Maxwell’s equations indicate that a time-varying magnetic field is accompanied by a spatially-varying electric field. The E field associated with the time-dependent B field satisfies
×E=Bt.
(1)
How does a dissipative magnetic field affect the static Landau state? In this study, we address this question using quantum mechanical concepts. In Ref. 2, the electron’s position is described via the coordinate eigenstates λ,
λ=exp12λ2iλπ++iπ+K++λ*K+00,λ=λ1+iλ2,
(2)
where the ladder operators
π±=12MΩπx±iπy,π,π+=1.
(3)
Here, Ω = eB/M refers to the synchronous rotational frequency of electrons moving in a magnetic field, and πx and πy are the canonical momenta (instead of the mechanical momentum), given by
πx=px+eAx,πy=py+eAy.
(4)
The vector potential A is related to the magnetic field B (in the z direction) by
A=12By,12Bx,0,B=B.
(5)
Then we explain K+ in Eq. (2) as
K±=MΩ2x0iy0,
(6)
where x0 and y0 are another pair of dynamic variables, called the guiding center by Johnson and Lippmann in Ref. 3,
x0=xπyMΩ,y0=y+πxMΩ.
(7)
x0 and y0 cannot be accurately measured simultaneously because their commutation relationship is
x0,y0=iMΩ.
(8)
Other commutative relationships between those operators are
π,π+=1,K,K+=1,πx,πy=iMΩ,K±,π±=0,x0,π±=1,y0,π±=0,x,y=0.
(9)
Then the Hamilton H describing this system is expressed as (setting = c = 1)
H=12Mp+eA2=12Mπx2+πy2=π+π+12Ω.
(10)
Since the vacuum state 00 is annihilated by π and K, i.e.,
π00=0,K00=0,
(11)
from Eq. (2), we have
K++iπ|λ=λ|λ,Kiπ+|λ=λ*|λ,
(12)
and we see that |λ⟩ is an entangled state representing the electronic coordinate’s eigenstate,
x|λ=2MΩλ1|λ,y|λ=2MΩλ2|λ.
(13)
One can further prove that |λ⟩ is orthonormal and complete,2 so |λ⟩ makes up the electron’s coordinate representation in the presence of magnetic field. It should be noted that because of the presence of a magnetic field, the electrons are not in free motion.
The form of |λ⟩ is characteristic of the properties of entangled states, revealing the entanglement between the electromagnetic field and electrons. The eigenstates of π+π and K+K are as follows:
n,m=π+nK+mn!m!00=π+nn!0,mn,m=0,1,2,.
(14)
From Eq. (10), we see
Hn,nml=n+12Ωn,nml.
(15)
Let Lz = xpyypx. From Eqs. (4), (7), and (6), we obtain
Lz=x0+πyMΩπyeAyy0πxMΩπxeAx=π+πK+K,Lzn,nml=mln,nml,
(16)
where ml is the angular momentum quantum number; therefore, n,nml is a Landau state with definite energy and angular momentum. The following formula can be derived easily:
π+n,m=n+1n+1,m,πn,m=nn1,m,
(17)
so π+ and π can be seen as creation and annihilation operators, respectively.
When the magnetic amplitude dissipates, the Landau state (pure state) evolves into a mixed state, which is denoted as the density operator ρ. In this section, with the accompanying classical magnetic field amplitude varying, we shall introduce the quantum master equation of ρ and then solve it. For this purpose, we introduced eigenstates for π,
πζ=ζζ,
(18)
where ζ belongs to the coherent states
ζ=expζπ+ζ*π0,π0=0
(19)
and satisfies
d2ζπζζ=1.
(20)
Thus, any density operator ρ can be expanded by utilizing the completeness of the coherent states, that is, Pζ,t, where
ρt=d2ζπPζ,tζζ,
(21)
where the classical function Pζ,t is a P representation of ρt. First, the amplitude decay of the coherent state is discussed,
ζζζeκtζeκt,
(22)
where κ is the decay rate. Using the normal product form of 00=:exp(π+π):, we rewrite
ζeκtζeκt=:expζ2e2κt+ζeκtπ++ζ*eκtππ+π.
(23)
It then follows
ddtζeκtζeκt=ddt:expζ2e2κt+ζeκtπ++ζ*eκtππ+π:=2κζ2e2κtζeκtζeκtκπ+ζeκtζeκtζeκtκζeκtζeκtζ*eκtπ=2κπζeκtζeκtπ+κπ+πζeκtζeκtκζeκtζeκtπ+π.
(24)
Let ζeκtζeκt=ρt. Then Eq. (24) is equivalent to
dρdt=κ2πρπ+π+πρρπ+π.
(25)
Because we can expand any density operator in the coherent state representation, Eq. (25) is the master equation for the density operator, which represents the dissipated magnetic field.
In order to solve Eq. (25), we introduce another entangled state,
η=exp12η2+ηπ+η*π̃++π+π̃+00̃.
(26)
Eq. (25) differs from Eq. (2), where π̃+ is a fictitious mode accompanying the real mode π+, π̃,π̃+=1, π̃0̃=0, and η obeys the eigenvector equations
ππ̃+|η=η|η,π+π̃|η=η*|η,ηπ+π̃=η*η,ηππ̃+=ηη.
(27)
Using the normal ordering form of the vacuum projector
00̃00̃=:eπ+ππ̃+π̃:
(28)
and the technique of Integration Within an Ordered Product (IWOP)4–8 of the operator, we can show the orthogonality as
ηη=πδηηδη*η*,
(29)
as well as the completeness relation
d2ηπηη=d2ηπ:×exp|η|2+ηπ+η*π̃++η*πηπ̃+π+π̃++ππ̃π+ππ̃+π̃:=1.
(30)
Here we employed integration formula,9 
d2απexphα+sα+gα=1hexpsgh,Reh<0.
(31)
Let
η=0=eπ+π̃+00̃I.
(32)
We have
πI=π̃+I,π+I=π̃I,(π+π)nI=(π̃+π̃)nI.
(33)
Acting on both sides of Eq. (25) on the state I and letting ρ=ρI, we get
ddtρ=κ2πρπ+π+πρρπ+πI.
(34)
Using Eq. (33) and noting that the density operators defined in real space are commutative with those in tilde space, Eq. (34) becomes
ddtρ=κ2ππ̃π+ππ̃+π̃ρ,
(35)
so its formal solution is
ρ=expκt2ππ̃π+ππ̃+π̃ρ0,
(36)
where ρ0ρ0I, with ρ0 being the initial density operator.
It is noteworthy that the operators in Eq. (35) obey the following commutative relations:
ππ̃,π+π=ππ̃,π̃+π̃=π̃π,π+π+π̃+π̃2,ππ̃=π̃π.
(37)
Using the operator identity10 
eλA+σB=eλAexpσ1eλτB/τ
(38)
(valid for [A, B] = τB), we have
expκt2ππ̃π+ππ̃+π̃=expκtπ+π+π̃+π̃expTππ̃,
(39)
where
T=1e2κt.
(40)
Then substituting Eq. (39) into Eq. (36) and using Eq. (33) yield
ρ=expκtπ+π+π̃+π̃j=0Tjj!πjπ̃jρ0=eκtπ+πj=0Tjj!πjρ0π+jeκtπ̃+π̃I=j=0Tjj!eκtπ+ππjρ0π+jeκtπ+πI,
(41)
which leads to
ρt=j=0Tjj!eκtπ+ππjρ0π+jeκtπ+π=j=0Mjρ0Mj,
(42)
where
Mj=Tjj!eκtπ+ππj,
(43)
satisfying j=0MjMj=1, which is trace conservative Trρt=1 (see the  Appendix). Therefore, ρt in Eq. (42) qualifies as a density operator.
We now consider how Landau states dissipate. In Eq. (42), let the initial state ρ0=n,mn,m, and n,m=π+nn!0,m. Substituting ρ0 into Eq. (42) and using
πjn,m=n!nj!nj,m,
(44)
we obtain
ρt=j=0Tjj!eκtπ+ππjn,mn,mπ+jeκtπ+π=j=0nTjj!eκtπ+πn!nj!nj,mnj,meκtπ+π=j=0nTjj!e2κtnjn!nj!nj,mnj,m=j=0n1e2κtjCnje2κtnjnj,mnj,m=j=0nCnnje2κtj1e2κtnjj,mj,m.
(45)
Noticing Cnnj=Cnj, we have
ρt=j=0nCnje2κtj1e2κtnjj,mj,m.
(46)
Therefore, we conclude that ρt is a binomial state (a mixed state) where e−2κt is the binomial parameter and the initial pure Landau state evolves into a binomial state. The binomial state interpolates between the coherent state and the Landau state.11 

In sharp contrast to the case where the Landau state decays in a magnetic field and degenerates into a binomial state (mixed state), the Landau state becomes a negative binomial state when an additional harmonic oscillator potential is applied.

When an electron in a uniform magnetic field is subjected to a harmonic oscillator potential, the Hamiltonian is
H=12Mπx2+πy2+12Mω2x2+y2.
(47)
Using Eqs. (3), (6), and (7), we can rewrite H as
H=Gπ+π+12+FK+KiFK+π+Kπ+12F,
(48)
where G = Ω + ω2/Ω and F = ω2/Ω. Employing the squeezing operator expressed in the entangled state representation,
S=d2λπγλγλ,γ=ΩΩ1/2,Ω=Ω2+4ω2,
(49)
since Sλ=1γλγ, we can diagonalize H,
SHS1=12Ω+Ωπ+π+ΩΩK+K+Ω.
(50)
Hence, the Landau state becomes S1n,0; thus we can derive
S1n,0=n=0Cn+llsechfn+1itanhfln+l,l.
(51)
Here, f=14lnΩ2Ω2+4ω2; this is a negative binomial state.

In summary, quantum mechanical techniques were used for the first time to demonstrate the impact of a dissipative magnetic field on the Landau state. By establishing and solving a quantum master equation to dissipate the magnetic field, we found that the pure Landau state evolves into a binomial quantum state (mixed state) when the magnetic field dissipates. This is in sharp contrast to the case where the Landau state becomes a negative binomial state when an additional harmonic oscillator potential is applied. We hope that this finding will help further advance the relevant work in quantized plasma physics.

This work was sponsored by the Anhui Provincial Natural Science Foundation (Grant No. 2008085MA22) and the Natural Science Foundation of the Anhui Higher Education Institutions of China (Grant No. KJ2020A0673).

The authors have no conflicts to disclose.

Cheng Da: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Supervision (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Hong-Yi Fan: Conceptualization (supporting); Investigation (supporting); Methodology (supporting); Supervision (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

Here, we first prove j=0MjMj=1 for Mj in Eq. (43).

Using formula12 
:eef1π+π:=efπ+π,
(A1)
we can prove
j=0MjMj=j=0Tjj!π+je2κtπ+ππj=j=0Tjj!e2jκt:π+jπj:e2κtπ+π=:eTe2κtπ+π:e2κtπ+π=1.
(A2)
Then we have
Trρt=Trn=0Mnρ0Mn=Trρ0=1.
(A3)
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