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    Journals >Chinese Physics B >Volume 29 >Issue 10 >Page > Article
    • Chinese Physics B
    • Vol. 29, Issue 10, (2020)
    Damping of displaced chaotic light field in amplitude dissipation channel
    Ke Zhang1, Lan-Lan Li1, and Hong-Yi Fan2,†
    Author Affiliations
  • 1School of Electronic Engineering, Huainan Normal University, Huainan 232038, China
  • 2University of Science and Technology of China, Hefei 30031, China
    • show less
    DOI: 10.1088/1674-1056/ab99b2 Cite this Article
    Ke Zhang, Lan-Lan Li, Hong-Yi Fan. Damping of displaced chaotic light field in amplitude dissipation channel[J]. Chinese Physics B, 2020, 29(10): Copy Citation Text show less

    Abstract

    We explore how a displaced chaotic light (DCL) behaves in an amplitude dissipation channel, and what is its time evolution formula of photon number distribution. With the use of the method of integration within ordered product product of operator (IWOP) and the new binomial theorem involving two-variable Hermite polynomials we obtain the evolution law of DCL in the channel.

    Keywords

    ρc=(1−ef)efa†a,(1)

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    tr(ρca†a)=(1−ef)∂∂fTr[efa†a∑n=0∞|n⟩⟨n|]=(1−ef)∂∂f∑n=0∞efn=(1−ef)∂∂f11−ef=(e−f−1)−1,(2)

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    ρd=Ceλa†efa†aeλ∗a,(3)

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    ∫d2απ|α⟩⟨α|=1,(4)

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    |α⟩=exp⁡[−12|α|2+αa†]|0⟩,a|α⟩=α|α⟩.(5)

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    1=Trρd=Tr[∫d2απ|α⟩⟨α|ρd]=C∫d2απ⟨α|eλa†efa†aeλ∗a|α⟩=C∫d2απeλα∗+λ∗α⟨α|:exp⁡[(ef−1)a†a]:|α⟩=C∫d2απeλα∗+λ∗αexp⁡[−(1−ef)|α|2]=C1−efexp⁡[|λ|21−ef],(6)

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    C=(1−ef)exp⁡[−|λ|21−ef].(7)

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    ρd=(1−ef)exp⁡[−|λ|21−ef]eλa†efa†aeλ∗a.(8)

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    efa†a=:exp⁡[(ef−1)a†a]:,(9)

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    (1)

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    Tr(ρda†a)=Tr(ρdaa†)−1=Tr[ρda∫d2απ|α⟩⟨α|a†]−1=(1−ef)exp⁡(−|λ|21−ef)×∫d2απ|α|2⟨α|eλa†efa†aeλ∗a|α⟩−1=(1−ef)exp⁡(−|λ|21−ef)∫d2απ|α|2eλα∗+λ∗α×exp⁡[(ef−1)|α|2]−1=1e−f−1+|λ|2(1−ef)2,(10)

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    dρdt=k(2aρa†−a†aρ−ρa†a),(11)

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    ρ(t)=∑n=0∞T′nn!e−kta†aanρ0a†ne−kta†a=∑n=0∞Mnρ0Mn†,(12)

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    Mn=T′nn!e−kta†aan,(13)

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    eλa†aae−λa†a=ae−λ(14)

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    ∑n=0∞MnMn+=∑n=0∞T′nn!a†ne−2kta†aan=∑n=0∞T′nn!e2nkt:a†nan:e−2kta†a=:eTe2kta†a:e−2kta†a=:e(e2kt−1)a†a:e−2kta†a=1.(15)

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    Trρ(t)=Tr∑n=0∞Mnρ0Mn†=Trρ0.(16)

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    ρd(t)=∑n=0∞MnρdMn†=C∑n=0∞T′nn!e−kta†aaneλa†efa†aeλ∗aa†ne−kta†a,(17)

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    aneλa†=eλa†(a+λ)n=eλa†∑l=0(nl)λn−lal,(18)

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    aneλa†efa†aeλ∗aa†n=eλa†∑l=0∑k=0(nl)(nk)λn−lalefa†aa†kλ∗n−keλ∗a.(19)

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    efa†a=e−f⋮exp⁡[(1−e−f)aa†]⋮(20)

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    alefa†aa†k=e−fal⋮exp⁡[(1−e−f)aa†]⋮a†k=e−f∫d2zπzle(1−e−f)|z|2|z⟩⟨z|z∗k=e−f∫d2zπzlz∗k:e−e−f|z|2+za†+z∗a−a†a:=(ef)(l+k)/2∫d2zπzlz∗k:e−|z|2+ef/2za†+ef/2z∗a−a†a:=(ef)(l+k)/2(−i)l+k:e(ef−1)a†a×∑m=0l!k!(−1)mm!(l−m)!(k−m)!(ief/2a†)k−m(ief/2a)l−m:=(−i)l+k(ef)(l+k)/2:e(ef−1)a†a×Hk,l(ia†ef/2,iaef/2):(21)

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    Hm,n(x,y)=∂n+m∂tm∂τnexp⁡(tx+τy−tτ)|t=τ=0=∂m∂tmetx∂n∂τnexp⁡[τ(y−t)]|t=τ=0=∂m∂tm[etx(y−t)n]|t=0=∑l=0(ml)∂l∂tl(y−t)n∂m−l∂tm−letx|t=0=∑l=0min(m,n)m!n!(−1)ll!(m−l)!(n−l)!xm−lyn−l(22)

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    ∑m,n=0tmτnm!n!Hm,n(x,y)=exp⁡(tx+τy−tτ).(23)

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    ∑l=0∑k=0(nl)(nk)λn−lalefa†aa†kλ∗n−k=∑l=0∑k=0(nl)(nk)λn−l(−i)l+k(ef)(l+k)/2:e(ef−1)a†aHk,l(ia†ef/2,iaef/2):λ∗n−k=|λ|2n:e(ef−1)a†a∑l=0n∑k=0n(nl)(nk)(−ief/2λ)l(−ief/2λ∗)kHk,l(ia†ef/2,iaef/2):.(24)

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    ∑r=0l∑q=0k(lr)(kq)Hr,q(x,y)frgq=fltkHl,k(x+1f,y+1g).(25)

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    a†qar=⋮Hr,q(a,a†)⋮.(26)

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    eta†et′a=et′aeta†e−tt′=⋮et′a+ta†−tt′⋮=∑r=0∑q=0(t)r(t′)qq!r!⋮Hr,q(a,a†)⋮,(27)

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    eta†et′a=∑r=0∑q=0t′qtrq!r!a†raq,(28)

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    ∑r=0l∑q=0k(lr)(kq)⋮Hr,q(a,a†)⋮frgq=∑r=0l∑q=0k(lr)(kq):a†qar:frgq=:(ga†+1)k(fa+1)l:,(29)

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    ∑l,k=0∞sltk:(ga†+1)k(fa+1)l:l!k!=et(ga†+1)es(fa+1)=esf(a+1/f)etg(a†+1/g)e−sftg=⋮esf(a+1/f)+tg(a†+1/g)e−sftg⋮=∑l,k∞(sf)l(gt)kl!k!⋮Hl,k(a+1f,a†+1g)⋮,(30)

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    :(ga†+1)k(fa+1)l:=fltk⋮Hl,k(a+1f,a†+1g)⋮.(31)

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    ∑r=0l∑q=0k(lr)(kq)⋮Hr,q(a,a†)⋮frgq=fltk⋮Hl,k(a+1f,a†+1g)⋮.(32)

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    ∑r=0l∑q=0k(lr)(kq)Hr,q(x,y)frgq=fltkHl,k(x+1f,y+1g).(33)

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    ∑l=0∑k=0(nl)(nk)λn−lalefa†aa†kλ∗n−k=∑l=0∑k=0(nl)(nk)λn−l(−i)l+k(ef)(l+k)/2:e(ef−1)a†aHk,l(ia†ef/2,iaef/2):λ∗n−k=|λ|2n:e(ef−1)a†a∑l=0n∑k=0n(nl)(nk)(−ief/2λ)l(−ief/2λ∗)kHk,l(ia†ef/2,iaef/2):=(−ef)n:e(ef−1)a†aHn,n(ia†ef/2+iλ∗e−f/2,iaef/2+ie−f/2λ):.(34)

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    Ln(xy)=(−1)nn!Hn,n(x,y),(35)

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    aneλa†efa†aeλ∗aa†n=eλa†∑l=0∑k=0(nl)(nk)λn−lalefa†aa†kλ∗n−keλ∗a=eλa†(−ef)n:e(ef−1)a†aHn,n(ia†ef/2+iλ∗e−f/2,iaef/2+iλe−f/2):eλ∗a=eλa†(−ef)n:e(ef−1)a†an!(−1)nLn[−(a†ef/2+λ∗e−f/2)(aef/2+λe−f/2)]:eλ∗a.(36)

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    W=∑n=0∞T′nn!aneλa†efa†aeλ∗aa†n,(37)

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    ρd(t)=(1−ef)exp⁡[−|λ|21−ef]e−κta†aWe−κta†a.(38)

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    (1−z)−1exp⁡[zxz−1]=∑l=0Ln(x)zn(39)

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    W=∑n=0∞T′nn!aneλa†efa†aeλ∗aa†n=eλa†∑n=0∞T′nn!(−ef)n:e(ef−1)a†an!(−1)nLn[−(a†ef/2+λ∗e−f/2)(aef/2+λe−f/2)]:eλ∗a=eλa†∑n=0∞(T′ef)n:e(ef−1)a†aLn[−(a†ef/2+λe−f/2)(aef/2+λ∗e−f/2)]:eλ∗a=11−T′efeλa†:e(ef−1)a†aexp⁡[−T′ef(a†ef/2+λ∗e−f/2)(aef/2+λe−f/2)T′ef−1]:eλ∗a,(40)

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    e−kta†a=a†e−kta†ae−kt(41)

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    ρd(t)=(1−ef)exp⁡[−|λ|21−ef]e−κta†aWe−κta†a=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]×exp⁡[(λa†1−T′ef)e−kt]×e−κta†a:exp⁡[(ef1−T′ef−1)a†a]:e−κta†a×[exp⁡(λ∗a1−T′ef)e−kt]=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]×exp⁡[(λa†1−T′ef)e−kt]×exp⁡[a†a(lnef1−T′ef−2kt)]×exp⁡[(λ∗a1−T′ef)e−kt]=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]×exp⁡[(λa†1−T′ef)e−kt]×exp⁡[a†alnef(1−T′)1−T′ef]×exp⁡[(λ∗a1−T′ef)e−kt].(42)

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    trρd(t)=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]∫d2zπ⟨z|exp⁡[(λa†1−T′ef)e−kt]×exp⁡[a†alnef(1−T′)1−T′ef]exp⁡[(λ∗a1−T′ef)e−kt]|z⟩=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]∫d2zπexp⁡[−(1−ef(1−T′)1−T′ef)|z|2+λz∗e−kt1−T′ef+λ∗ze−kt1−T′ef]=exp⁡{[e−2kt(1−ef)(1−T′ef)+T′−1(1−ef)(1−T′ef)]|λ|2}=1.(43)

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    tr[ρ(t)aa†]=tr[ρ(t)a∫d2zπ|z⟩⟨z|a†]=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]∫d2zπ|z|2⟨z|exp⁡[(λa+1−T′ef)e−kt]×exp⁡[a†alnef(1−T′)1−T′ef]exp⁡[(λ∗a1−T′ef)e−kt]|z⟩=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2]∫d2zπ|z|2exp⁡[−1−ef1−T′ef|z|2+λz∗1−T′efe−kt+λ∗z1−T′efe−kt]=1−ef1−T′efexp⁡[T′−1(ef−1)(T′ef−1)|λ|2](1−T′ef)2e2kt∂2∂λ∂λ∗×∫d2zπexp⁡[−1−ef1−T′ef|z|2+λz∗1−T′efe−kt+λ∗z1−T′efe−kt]=exp⁡[T′−1(ef−1)(T′ef−1)|λ|2](1−T′ef)2e2kt∂2∂λ∂λ∗exp⁡[e−2kt(ef−1)(T′ef−1)|λ|2],(44)

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    tr[ρ(t)aa†]=(T′ef−1)exp⁡[e−2kt(ef−1)(T′ef−1)|λ|2]×∂2∂λ′∂λ′∗exp⁡[1ef−1|λ′|2]=(T′ef−1)exp⁡[e−2kt(ef−1)(T′ef−1)|λ|2]∂∂λ′×{λ′ef−1exp⁡[1ef−1|λ′|2]}=(T′ef−1)exp⁡[e−2kt(ef−1)(T′ef−1)|λ|2]×{1ef−1+|λ′|2(ef−1)2}exp⁡[1ef−1|λ′|2]=(T′ef−1){1ef−1+|λ′|2(ef−1)2}=(T′ef−1){1ef−1+|λ|2e−2kt(ef−1)2(T′ef−1)}=T′ef−1ef−1+|λ|2e−2kt(ef−1)2.(45)

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    tr[ρ(t)a†a]=tr[ρ(t)aa†]−1=ef(T′−1)ef−1+|λ|2e−2kt(ef−1)2=[1e−f−1+|λ|2(ef−1)2]e−2kt.(46)

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    Ke Zhang, Lan-Lan Li, Hong-Yi Fan. Damping of displaced chaotic light field in amplitude dissipation channel[J]. Chinese Physics B, 2020, 29(10):
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