Table Of ContentAutangle: A case of Quantum Narcissism?
Rong Zhou,1,∗ Ryan O. Behunin,2,† Shih-Yuin Lin,3,‡ and Bei-lok Hu1,§
1Joint Quantum Institute and Maryland Center for Fundamental Physics,
University of Maryland, College Park, Maryland 20742, USA
2Center for Nonlinear Studies and Los Alamos National Laboratory,
Theoretical Division, Los Alamos, New Mexico 87545, USA
3Department of Physics, National Changhua University of Education, Changhua 50007, Taiwan
(Dated: January 5, 2012)
In this paper we ask a common psychological question and provide a physics answer: “Looking
2 into a mirror can one get entangled with one’s image?" This is not a frivolous question; rather, it
1 bears on the effect of boundaries on the behavior of quantum entanglement between a harmonic
0 oscillatorandaquantumfield,abasicproblemofinterestinproposedmirror-fieldsuperpositionand
2 relatedexperimentsinmacroscopicquantumphenomena,aswellasatomicfluctuationforcesneara
conductingsurface. Theobject’sinternaldegreeoffreedomismodeledbyaharmonicoscillatorand
n
a thepresenceofaperfectlyreflectingmirrorenforcestheDirichletboundaryconditionsonthequan-
J tum field, restricting the latter to a half space. By assuming a bilinear oscillator-field interaction,
we derive a coupled set of equations for the oscillator’s and the field’s Heisenberg operators. The
4
former can be cast in the form of a quantum Langevin equation, where the dissipation and noise
] kernels respectively correspond to the retarded and Hadamard functions of the free quantum field
h in half space. We use the linear entropy as measures of entanglement between the oscillator and
p the quantum field under mirror reflection, then solve the early-time oscillator-field entanglement
- dynamics and compare it with that between two inertial oscillators in free space. At late times
t
n when the combined system is in a stationary state, we obtain exact expressions for the oscillator’s
a covariance matrix and show that theoscillator-field entanglement decreases as theoscillator moves
u closer to the mirror. We explain this behavior qualitatively with the help of a mirror image and
q provideananswertothequestionraisedabove. Wealsocomparethissituationwiththecaseoftwo
[
real oscillators and explain the differences.
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∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
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I. INTRODUCTION
In this note we ask a common psychological question and provide a physics answer: “Looking into a
mirror can one get entangled with one’s image"? According to storybooks the answer for the evil queen
Q seems to be yes, a case of typical royal narcissism. We want to find out the answer from physical
considerations: here entanglement refers to quantum entanglement, and autangle means self-entanglement
1 , here referring to the entanglement of a real physical object Q with its mirror image. We hasten to
add that since the image is not a physical object one cannot define, let alone calculate, the entanglement
between a physical object and an unphysical construct. However, there exists entanglement between an
oscillator and a quantum field, and we can ask how this is altered if a mirror is present. A perfectly
reflecting mirror imposes Dirichlet boundary conditions on the field along the mirror surface, restricting its
existence to a half space. These are well-defined problems, belonging to the broader inquiry into the effects
of boundaries and topology on the quantum field and on quantum entanglement of objects, oscillators and
mirrors,coexistingwithand/ormediatedbythequantumfield. Infact,theybearonissuesofimportanceto
quantuminformationandmacroscopicquantumphenomena. Amongstresearchproblemsofcurrentinterest,
wemention: 1)Entanglementbetweentwotwo-level(2LA)atomsviaacommonquantumfieldalreadyshows
diverse quantum entanglement dynamics behavior [2], ranging from sudden death, touch of death, revival
to staying alwaysalive,andfeatures suchas dynamicalgeneration,protection, andtransferof entanglement
between subsystems. Model studies of oscillator-field entanglement with experimental ventures have been
carried out [3] as well as oscillator-field entanglement in conjunction with measurements in LIGO detectors
[4], the latter also serving as preparatory studies for mirror-field superposition [5] in macroscopic quantum
phenomena. 2)Entanglementnotonlychangesintime butitalsodependsonspatialseparationasshownin
model studies for 2LAs [6–8] and harmonic oscillators [9] interacting with a common quantum field. These
results are of both theoretical interest in understanding what quantum nonlocality means, and practical
value, such as for the design of quantum gates and applications to quantum teleportation. Extending these
findings to two and three dimensional systems would enable one to define quantum entanglement domains
[12,13]andgeometriceffectsofentanglement[14,15]. Theformerreferstotheeffectivedomainonasurface
orbodywhereentanglementisinducedbythepresenceofanoscillator,notunliketheinducedsurfacecharge
densityonaconductorordielectricplate. Thelatterreferstospecificgeometricpatternsinthearrangements
of oscillators where their entanglement strength can be maximized.
Now, returning to the problem under study, one may wonder, is it really necessary to carry out such a
seriouscalculationinordertoanswera simple queryposedatthe beginning: Is there entanglementbetween
thequeenandherimage? Theanswerisyes,bothtothequeryandtothenecessityofabonafidecalculation.
As we cautioned ab initio, an image is not a physical object. Thus we try to avoid invoking an image in
this calculationso that no unphysical assumptions or intuition are brought in unnecessarily or unknowingly
in the derivations. However, at the end, by inspecting the results from the oscillator-field (under mirror
reflection)entanglementweseethatonecanusethenotionofanimagetodescribetheprocess,infact,come
up with a simple qualitative explanation of how it depends on distance.
Thepaperisorganizedasfollows: InSection. IIwesetuptheproblemandderiveaformalsetofequations
fortheHeisenbergoperatorsoftheoscillator’sinternaldegreesoffreedomandthequantumfield. Weimpose
theboundaryconditionsintroducedbythepresenceofthemirrorandexplainhowthecommutationrelations
arealtered. NextwecalculatetheretardedandHadamardGreenfunctionsofthealteredfieldconfigurations
and derive a quantum Langevin equation for Q(t) including the back-reaction of the altered field. We seek
solutions to this equation at late times and calculate the covariance matrix of the oscillator’s canonical
variablesatlatetimeswhenthe combinedsystemisinastationarystate. DetailsarecontainedinAppendix
A. In Sec. III we introduce the linear entropy and the von-Neumann entropy as measures of quantum
entanglement and calculate the early-time dynamics of entanglement between the oscillator and the field in
the oscillator-mirrorsetupdescribedabove. We obtainplots ofhowthe entanglementbetweenthe oscillator
and field in the half space with the mirror reflection evolves with time, depending on the distance between
the oscillator and the mirror. Finally in the discussions section we explain in what sense can one describe
this situation in terms of entanglement of the oscillator with its mirror image and why it decreases as the
oscillator moves closer to the mirror, a somewhat counterintuitive finding. We also compare this situation
1 We coin this term from two sources of inspiration: new words like 3-tangle and old ones like autism. A more learned word
‘ipso-tangle’wassuggested byProf. Brill.
3
with the case of two inertial oscillators in free space (calculations placed in Appendix B) and explain their
physical differences.
II. OSCILLATOR INTERACTING WITH FIELD UNDER MIRROR BOUNDARY CONDITION
A. Dynamics of Oscillator-Field System with Mirror
As described in the Introduction, the gist of the matter for this problem is to quantify the change in
some entanglement measure between the oscillator and the field with and without the presence of a mirror.
Technically our calculationinvolvestwo parts: 1)find the back-reactionofthe field configurationalteredby
the mirror on the oscillator. 2) find the correlationfunctions of the resultant oscillator’s canonicalvariables
includingthefield’sinfluences. Thusourtargetisthecovariancematrixfortheoscillatorwhichincorporates
allbackreactionsfromthequantumfield. WeworkwiththeHeisenbergequationsofmotionwhichefficiently
provides the oscillator’s field-influenced dynamics. We can then find the oscillator’s late time behavior and
quantify it’s entanglement with the modified field.
Consideranoscillatorlocatedatpositionx ataverticaldistanceofL/2fromthemirrorplaneatx =0.
Q 3
ThetotalactionofthesystemdescribingthisoscillatorwithinternaldegreeoffreedomQinteractinglinearly
with a massless scalar field confined to the half space defined by x >0 is given by:
3
1 1
S[Q,Q˙;Φ,∂ Φ]= M dt (Q˙2 Ω2Q2)+ dt d3x ∂ Φ∂µΦ+λ dt Q(t)Φ(x ,t). (2.1)
µ 2 Q − 2 µ Q Q
Z Z Zx3>0 Z
Due to the linearityofoursystemthe equationsofmotionforHeisenbergoperators(carryinghats)havethe
same form as the equations for the corresponding classical variables:
M Q¨ˆ(t)+M Ω2Qˆ(t)=λ d3x Φˆ(x ,t), (2.2)
Q Q Q Q
Zx3>0
(cid:3)Φˆ(x,t)=λ δ3(x x )Qˆ(t), and Φˆ(x ,x =0,t)=0. (2.3)
Q − Q k 3
Equation (2.3) specifies the Dirichlet boundary (DBC) conditions imposed on the field where the mirror
surfaceis located,namely, inthe x =0 plane and x =(x ,x ). Toobtainthe back-reactionofthe fieldon
3 k 1 2
the oscillator we first solve (2.3) and then plug the solution into (2.2). The solution for Φˆ(x,t) is given by
t
Φˆ(x,t)=Φˆ (x,t)+λ dt′GΦ (t,x;t′,x )Qˆ(t′) (2.4)
0 Q ret Q
Zti
whereΦˆ (x,t)isthehomogeneoussolutiontoequation(2.3)whichdescribesthedynamicsofthesource-free
0
field (without Q) in the presence of the mirror and GΦ (x,t;y,t′) is the retarded Green’s function for the
ret
field. Pluggingthe solutionforthe fieldoperatorinto the equationofmotionfor the oscillatorweobtainthe
followingequationgoverningthe dynamicsofthe oscillatorwiththe effectsofthe fieldalreadyincorporated.
This is what we mean by the ‘field-influenced’ dynamics of the oscillator
t
M Q¨ˆ(t)+M Ω2Qˆ(t) λ2 dt′GΦ (t,x ;t′,x )Qˆ(t′)=λ Φˆ (x ,t). (2.5)
Q Q − Q ret Q Q Q 0 Q
Zti
As will be explained in further detail later the term containing the retarded Green’s function describes
how the oscillator transfers energy to the field and the right hand side acts similarly to a Langevin forcing
term describing how quantum field fluctuations drive the oscillator.
B. Field Modified by Mirror and Propagators in Half-space
Inordertosolvetheequationofmotionfortheoscillator(2.5)weneedtoknowthefieldoperatorsdefined
in the half space. Following this procedure we first write down the free field operator Φˆ which satisfies the
0
Klein-Gordon equation and vanishes in the x =0 plane,
3
1
Φˆ0(x,t)= d3k 4π3ωe−iωt+ikk·xksink3x3ˆbk+H.c. (2.6)
Zk3>0 r
4
where ω = k.
| |
Notice that the normalization factor is different for field in half space.
Writing (Πˆ (x′,t) = Φˆ˙ (x′,t)) and demanding that the equal-time commutation relations are satisfied,
0 0
that is,
[Φˆ (x,t),Πˆ (x′,t)]=iδ(3)(x x′) iδ(2)(x x′)δ(x +x′), (2.7)
0 0 − − k− k 3 3
we can infer that the mode expansion coefficients can be identified with the creation and annihilation oper-
ators
[ˆb ,ˆb†]=δ , [ˆb ,ˆb ]=0, [ˆb†,ˆb†]=0. (2.8)
k k k,k′ k k k k
Note that the commutation relation above (2.7) is not in the form normally encountered. For the case of
a field satisfying the Dirichlet boundary conditions on the x = 0 plane it is necessary to add the "image
3
term" to the right hand side such that for points restricted to the x < 0 half space where the image term
3
has no support the relation (2.7) reduces to the standard form.
With the expression for the free field operator in hand we now calculate the retarded and Hadamard
propagators. Theretardedpropagatorquantifiesthefieldsourcedbytheoscillatoranddescribeshowenergy
is dissipated from the oscillator into the field
GΦ (x,t;y,t′) iθ(t t′)[Φˆ (x,t),Φˆ (y,t′)]
ret ≡ − 0 0
1 1
=θ(t−t′)2π3 d3kω sin(ω(t−t′))eikk·(xk−yk)sin(k3x3)sin(k3y3) (2.9)
Zk3>0
∞
=θ(t t′) dωsin(ω(t t′)) I(ω;x,y).
− − ·
Z0
The Hadamard function, which will be encountered in the next section, describes the quantum fluctuations
of the field and is given formally by the anti-commutator of the field operator:
GΦ(x,t;y,t′) Φˆ (x,t),Φˆ (y,t′)
H ≡h{ 0 0 }i
1 1
= 2π3 d3kω cos(ω(t−t′))eikk·(xk−yk)sin(k3x3)sin(k3y3) (2.10)
Zk3>0
∞
= dωcos(ω(t t′)) I(ω;x,y),
− ·
Z0
where
ω π/2 2π
I(ω;x,y)= 2π3 dθ dφ eikk·(xk−yk)sin(ωcosθx3)sin(ωcosθy3). (2.11)
Z0 Z0
III. EARLY TIME DYNAMICS OF OSCILLATOR-FIELD ENTANGLEMENT
In [9] the dynamics of the entanglement between two inertial oscillators with identical couplings to a
commonmasslessscalarfieldwasinvestigatedanditexhibitedseveralinterestingfeatures. Thereareroughly
threetemporalregimesifthetwooscillatorsareproperlyseparatedandweaklycoupledtothefield. Thefirst
regime is at early time, up to t O(1/λ2) when accumulated effects of field-mediated mutual influences,
≈ Q
which manifest themselves in the field’s retarded propogators,between the oscillators are still weak and the
reduced dynamics of the oscillators are dominated by influences of vacuum fluctuations of the field. As the
effects of field-mediated mutual influences between the oscillators gradually gain strength, the system will
enter an intermediate regime in which the oscillators’ reduced dynamics become quite complicated. In the
late-time limit, the oscillator approaches a time-stationary state.
In particular, in [9] it was shown that at early time, after causal contact has been established between
two detectors, an oscillatory pattern of entanglement emerges which varies with the distance between the
oscillators. This is not due to mutual influences since in the weak coupling limit the mutual influence effect
is always weak compared to effect of vacuum fluctuation.
5
Forourmodel, weexpectasimilardivisionintothree temporalregimes. Hereinthis sectionwe study the
earlytime behaviorofourmodel,assumingthatinitiallyboththe oscillatorandthe fieldareintheirground
states and are uncorrelated. We show that the oscillator-field entanglement also develops an oscillatory
pattern which varies with the distance between the oscillator and the mirror. At a given instant of time,
the oscillator-field entanglement exhibits spatial oscillations characterized by frequency Ω (we adopt the
conventionc=1). Additionally, we find that the growthrate at which the oscillator-fieldbecome entangled
oscillates as a function of the oscillator-mirrorspacing.
A. Measure of Oscillator-Field Entanglement
In our setup, the oscillatorand the field together form a closed quantum system which undergoes unitary
evolution. Since the initial state of the total system is Gaussian and the action is quadratic, the reduced
quantum state of the oscillator at any time will remain Gaussian. For such systems the amount of bipartite
entanglement between the oscillator and the field can be quantified by the linear entropy of the oscillator’s
reduced density matrix, which takes on values different from 0 when the oscillator and field are entangled.
Here the linear entropy is defined as
S =1 , (3.1)
L
−P
where ,Trρˆ2 isthepurityoftheoscillator’sreducedquantumstate. HereforourgeneralmixedGaussian
P a
state, the purity is given by
1 1
= = (3.2)
P 2√detV 2 <Qˆ,Qˆ ><Pˆ,Pˆ > <Qˆ,Pˆ >2
−
q
where <Oˆ ,Oˆ >,Tr(ρˆ Oˆ ,Oˆ )/2, Oˆ =(Qˆ,Pˆ).
i j a i j
·{ }
Mathematically we know that is always less than or equal to 1. Smaller purity means the reduced
P
density matrix of the oscillator is less pure, and correspondingly the oscillator is more entangled with the
field.
Another measure of bipartite entanglement between the oscillator and the field is von Neumann entropy,
which for a single-mode mixed Gaussian state is
1 1+ 2
S = −P ln( P) ln( P ) (3.3)
ν
2 1 − 1+
P −P P
andisamonotonicallyincreasingfunctionofthelinearentropy. BothS andS yieldthesamecharacteriza-
L ν
tionof mixedness andare equivalentas entanglementmeasuresin the case ofa single-mode mixedGaussian
state.
B. Mode Decomposition and EOM for Mode Functions
We perform the following mode decompositions for Heisenberg operators of the oscillator Qˆ(t) and the
fieldΦˆ(x)with respecttothe creationandannihilationoperatorsofthe oscillatoraˆ,aˆ† andthoseofthe field
ˆb,ˆb† :
~ d3k ~
Qˆ(t)= q (t)aˆ+q∗(t)aˆ† + q (t,k)ˆb +q (t,k)ˆb† , (3.4)
r2Ωr a a Zk3>0 √2π3r2ω h + k − ki
~ (cid:2) (cid:3) d3k ~
Φˆ(x)= f (x)aˆ+f∗(x)aˆ† + f (x;k)ˆb +f (x;k)ˆb† . (3.5)
r2Ωr a a Zk3>0 √2π3r2ω h + k − ki
(cid:2) (cid:3)
By plugging (3.4) and (3.5) into (2.2) and (2.3), respectively, we find the following equations of motion
for the mode functions
6
2γ
∂2+2γ ∂ +Ω2 q (t)= Qθ(t L)q (t L), (3.6)
t Q t r a − L − a −
(cid:0) (cid:1) 2γ λ
∂2+2γ ∂ +Ω2 q (t,k) = Qθ(t L)q (t L,k)+ Q f (t,x ;k), (3.7)
t Q t r + − L − + − M 0+ Q
Q
(cid:0) (cid:1)
in which f0+(t,xQ;k)=e−iωt+ikk·xksink3x3 and γQ ≡λ2Q/(8πMQ).
Since we have assumed the oscillator’s position to be x (t) = x¯ (0,0,L/2), we have f (x¯ ,t;k) =
Q Q ≡ 0+ Q
e−iωtsink3L.
2
According to our initial condition, the solutions have to satisfy initial conditions that f (0,x;k) =
+
eikk·xksink3x3, ∂tf+(0,x;k) = iωeikk·xksink3x3, qa(0) = 1, ∂tqa(0) = iΩr, and fa(0,x) = ∂tfa(0,x) =
− −
q (0;k)=∂ q (0;k)=0.
+ t +
The solution to (3.6) can be written as the following expansion which is truncated at nL>t,
q (t)= q(n)(t), (3.8)
q q
n=0
X
where
q(0)(t)=q(h)(t), (3.9)
a a
2γ
q(n)(t)= dτ G (t,τ )( Q)θ(t L) (3.10)
a 1 r 1 − L − ×
Z
2γ
... dτ G (τ τ )( Q)θ(τ nL)q(h)(τ nL).
n r n−1− n − L n− a n−
Z
(h)
Here q (t) is the homogeneous solution satisfying the initial conditions
a
q(h)(t)= 1(1+ Ωr+iγQ)e−γQt−iΩt+ 1(1 Ωr +iγQ)e−γQt+iΩt. (3.11)
a 2 Ω 2 − Ω
The solution to (3.7) can be written in a similar fashion as above, also truncated at nL>t,
q (t,k)= q(n)(t,k), (3.12)
+ +
n=0
X
where
q(0)(t,k)= dτ G (t,τ )λ f (τ ,x¯ ;k) (3.13)
+ 1 r 1 Q 0+ 1 Q
Z
λ k L
= Q sin 3 [(M1 M2)e−iωt+(M2eiΩt M1e−iΩt)e−γQt],
M Ω 2 − −
Q (cid:18) (cid:19)
2γ
q(n−1)(t,k)= dτ G (t,τ )( Q)θ(τ L) (3.14)
+ 1 r 1 − L 1− ×
Z
2γ
dτ G (τ ,τ )( Q)θ(τ 2L)
2 r 1 2 2
− L − ×
Z
... dτ G (τ τ )θ(τ (n 1)L)λ
n r n−1 n n Q
− − −
M f (τ (n 1)L,x¯ ;k),
R Q 0+ n− − Q
where M = 1/[2( ω iγ +Ω)], M = 1/[2( ω iγ Ω)], G (t,τ) is the retarded Green’s function
1 − − Q 2 − − Q− r
which satisfies (∂2+2γ ∂ +Ω2)G (t,τ)=δ(t τ).
t Q t r r −
7
C. Zeroth-Order Correlation Functions
As explained before, in order to understand the dynamics of entanglement, it is sufficient to compute
the covariance matrix of the oscillator’s reduced quantum state. For a separable initial state, the covari-
ance matrix can be decomposed into two parts corresponding to the two sets of operators in the mode
decomposition:
<Qˆ(t),Qˆ(t)>=<Qˆ(t),Qˆ(t)> +<Qˆ(t),Qˆ(t)> (3.15)
a v
1 d3k 1
= q (t)2+ q (t,k)2,
2Ω | a | 2π32ω| + |
r Zk3>0
<Pˆ(t),Pˆ(t)>=<Pˆ(t),Pˆ(t)> +<Pˆ(t),Pˆ(t)> (3.16)
a v
1 d3k 1
=M2 ∂ q (t)2+ ∂ q (t,k)2 ,
Q 2Ω | t a | 2π32ω| t + |
(cid:20) r Zk3>0 (cid:21)
<Qˆ(t),Pˆ(t)>=<Qˆ(t),Pˆ(t)> +<Qˆ(t),Pˆ(t)> (3.17)
a v
M 1 d3kM 1
= Q [q∗(t)∂ q (t)+q∗(t)∂ q (t)]+ Q [q∗(t,k)∂ q (t,k)+q (t,k)∂ q∗(t,k)].
2Ω 2 a t a a t a 2π3 2ω 2 + t + + t +
r Zk3>0
Here < Oˆ ,Oˆ > < (Oˆ Oˆ +Oˆ Oˆ ) > /2. In the weak coupling limit, the effect of reflected influences
i j i j j i
≡
correspond to terms in 3.8 and 3.12 which are of higher than zeroth order, and therefore at early time (up
to t 1/γ ) accumulated effect of reflected influences is always small. Thus for the purpose of studying
Q
≈
the early time behavior of entanglement, we can ignore the contribution of reflected influences and restrict
ourselves to lowest order correlations.
We have
<Qˆ(t),Qˆ(t)> = 1 q(h)(t)2
a 2Ωr| a |
= 1 1(1+ Ωr+iγQ)e−γQt−iΩt+ 1(1 Ωr+iγQ)e−γQt+iΩt 2,
2Ωr|2 Ω 2 − Ω |
<Pˆ(t),Pˆ(t)> = MQ2 ∂ q(h)(t)2
a 2Ωr| t a |
=MQ2 1(1+ Ωr+iγQ)(γ +iΩ)e−γQt−iΩt+ 1(1 Ωr+iγQ)(γ iΩ)e−γQt+iΩt 2,
2Ωr|2 Ω Q 2 − Ω Q− |
<Qˆ(t),Pˆ(t)> = MQ 1(q(h)∗(t) ∂ q(h)(t)+q(h)(t) ∂ q(h)∗(t))
a 2Ωr 2 a · t a a · t a
=MQRe [1(1+ Ωr+iγQ)(γ +iΩ)e−γQt−iΩt+ 1(1 Ωr+iγQ)(γ iΩ)e−γQt+iΩt]
2Ωr { 2 Ω Q 2 − Ω Q−
[1(1+ Ωr−iγQ)e−γQt+iΩt+ 1(1 Ωr−iγQ)e−γQt−iΩt] .
· 2 Ω 2 − Ω }
Notice that the zeroth order correlators <...>(0)’s do not depend on the distance between the oscillator
a
and the mirror, but the part induced by vacuum fluctuations does:
<Qˆ(t),Qˆ(t)> = d3k 1 q(0)(t,k)2 (3.18)
v k3>0 2π32ω| + |
=(MλQQΩ)2 k3>0 2dπ3k321ω(sink32L)2|(M1R−M2)e−iωt+(M2eiΩt−M1e−iΩt)e−γQt|2
=(MλQQΩ)24Rπ12 dωω2 1− sinωLωL |(M1−M2)e−iωt+(M2eiΩt−M1e−iΩt)e−γQt|2.
R (cid:0) (cid:1)
Similarly we have
8
<Pˆ(t),Pˆ(t)> =M2 d3k 1 ∂ q(0)(t,k)2 (3.19)
v Q k3>0 2π32ω| t + |
=(λQMQ)2 1 Rdωω 1 sinωL
Ω 4π2 2 − ωL ×
|−iω(M1−M2)e−iωt+((iΩ−γQ)RM2eiΩ(cid:0)t−(−iΩ−(cid:1)γQ)M1e−iΩt)e−γQt|2,
<Qˆ(t),Pˆ(t)> =M d3k 1 1(q(0)∗(t,k)∂ q(0)(t,k)+q(0)(t,k)∂ q(0)∗(t,k)) (3.20)
v Q k3>0 2π32ω2 + t + + t +
=R M (λQ)2 1 dωω 1 sinωL
Q Ω 4π2 2 − ωL ×
Re{[−iω(M1−M2)e−iωt+((iΩ−γRQ)M2e(cid:0)iΩt−(−iΩ(cid:1)−γQ)M1e−iΩt)e−γQt]
[(M1 M2)eiωt+(M2eiΩt M1e−iΩt)e−γQt] .
· − − }
In[9], forthe caseoftwoinerialoscillatorsdistanceLapart,whicharelocatedinfreespacewithidentical
couplings to the field, the zerothorder correlatorsdue to vacuum fluctuation are givenas (M 1 and d in
Q
≡
[9] is replaced by L):
<Qˆ (t),Qˆ (t)> =(λQ)2 1 dωω sinωL
A B v Ω 4π2 ωL ·
(M1 M2)e−iωt+(M2eiΩt M1Re−iΩt)e−γQt 2,
| − − |
<Qˆ (t),Pˆ (t)> =(λQ)2 1 dωω sinωL
A B v Ω 4π2 ωL ·
Re{[−iω(M1−M2)e−iωt+((iΩ−γQ)M2eiΩt−R(−iΩ−γQ)M1e−iΩt)e−γQt]·
[(M1 M2)eiωt+(M2eiΩt M1e−iΩt)e−γQt] ,
− − }
<Pˆ (t),Pˆ (t)> =(λQ)2 1 dωω sinωL
A B v Ω 4π2 ωL ·
|−iω(M1−M2)e−iωt+((iΩ−γQ)M2eiΩt−(R−iΩ−γQ)M1e−iΩt)e−γQt|2.
Physically, the v-part of the zeroth order correlators < ... > effectively measure the response of the
v
oscillator to vacuum fluctuations of the field. The similarity between the integrands of < Qˆ(t),Qˆ(t) > in
v
Eq.(3.18) and < Qˆ (t),Qˆ (t) > in Eq.(3.21) is not surprising. In Appendix B we show that, if we have
A B v
two inertial oscillators C and D in free space at a distance L apart, with the same coupling constants but
in opposite signs, then (Qˆ (t)+Qˆ (t))/2 obeys the same equation of motion as the one for Qˆ(t) in our
C D
model. Thus we see that the self correlator < Qˆ(t),Qˆ(t) > here has the same value as (< Qˆ ,Qˆ > + <
v C C
Qˆ ,Qˆ >)/4+<Qˆ ,Qˆ >/2attandcontainsthepartofcorrelationsofvacuumfluctuationinfreespace
D D C D
which is odd with respect to the z =0 plane, whereas <Qˆ (t),Qˆ (t)> ( λ )λ here has exactly the
3 C D v Q Q
∝ −
same value of < Qˆ (t),Qˆ (t) > in [9] because the two oscillators in [9] are identically coupled to the
A B v
−
field and so <Qˆ ,Qˆ > λ λ .
A B v Q Q
∝
D. Early-time Dynamics of Oscillator-field Entanglement
With the previous results we can now investigate the evolution of oscillator-field entanglement as the
distancebetweentheoscillatorandthemirrorchanges. Whereasitmaybepossibletoobtainanapproximate
analytical expression in the weak coupling limit, we can simply study the dependence of the linear entropy
on L and t numerically, as shown in Figure 1.
Our numerical results reveal the following behaviors:
For a given L, the entanglement between the oscillator and the field increases monotonically at early
•
times,showingthattheoscillatorisgettingmoreandmoreentangledwiththefieldaftertheinteraction
is turned on. (See Figure 1 (upper row) and (lower right).)
9
SLvs.L SLvs.t
t=5 L=1
SL SL
0.40
0.4
0.38
0.3
0.36
0.2
0.34
0.32 0.1
L t
2 4 6 8 10 2 4 6 8 10
Figure 1. Entanglement dynamics at early time where only zeroth order correlations contribute. Here γ = 0.02
Q
and Ω=5. The upperleft plot shows the linear entropy as a function of L and t, with theupperright plot being its
contour plot. The lowerleft plot shows the dependenceof linear entropy on L at a given instant of time whereas the
lower right plot exhibits how thelinear entropy evolves with time for oscillator located at a certain distance.
Within the light cone, at every given time in this stage, the oscillator-field entanglement exhibits an
•
oscillatorybehaviorwithspatialfrequency beingΩ. This showsthatthe growthrateofS withwhich
L
theoscillatorgetsentangledwiththefieldalsooscillatesasafunctionofL,asshowninFigure1(upper
row) and (lower left).
Such oscillatory behavior in L is solely due to the oscillatory behavior of the zeroth order correlatorscorre-
sponding to <Rˆ ,Rˆ > , R=Q,P, given in the last paragraph of the previous subsection. After cosθ in
C D v
k ωcosθ has been integrated over the interval [0,1] in the integration corresponding to <Qˆ ,Qˆ > in
3 C D v
≡
(3.18)or (3.21), the field modes onresonancewith the oscillatorhaveconstructive(destructive)interference
at the local maxima (minima) of S against L at fixed t at early times.
L
In [9] the early time dynamics of entanglement between the two oscillators exhibit similar oscillatory
dependence on the distance between them, which is also due to distance-dependent correlations of vacuum
fluctutaionsexperiencedbythe oscillators. Moreprecisely,from(3.21)onecanseethatthe fieldmodes with
ω =nπ/L,n=1,2,3, ,hasnocontributionatalltotheintegrationof<Qˆ ,Qˆ > ,andthosesatisfying
A B v
···
tanωL = ωL give the maximum and minimum values of the factor sinωL/(ωL) in the integrand. In the
weak coupling limit the integration in (3.21) is mainly contributed by the poles at ω Ω, namely, those
≈ ±
mode on resonance with the oscillators. So < Qˆ ,Qˆ > 0 for L nπ/Ω, while < Qˆ ,Qˆ > has local
C D v C D v
≈ ≈
maximum or minimum values when L is about the solution of tanΩL=ΩL in the weak coupling limit.
10
IV. LATE-TIME STATIONARY LIMIT OF OSCILLATOR-FIELD ENTANGLEMENT
A. Quantum Langevin Equation and Covariance Matrix at Late-times
Onecancomparethedynamicsoftheoscillatorundertheinfluenceofthefieldtothewell-studiedquantum
Brownian motion (QBM) model, namely, the retarded and Hadamard Green functions correspond to the
dissipationandnoisekernelsrespectivelyandthefunctionI(ω;x,y)representsthespectraldensity[see,e.g.,
[22, 23]) and the references therein.] In the same vein Eq. (2.5) can equivalently be written as a quantum
Langevin equation
t
M Q¨ˆ(t)+M Ω2Qˆ(t) dτµ(t,τ) Qˆ(τ)=ξˆ(t), (4.1)
Q Q
− ·
Zti
in which
Λ
µ(t,s)=λ2 θ(t s) dωI(ω;x ,x )sinω(t s), (4.2)
Q − Q Q −
Z0
and ξˆ(t) = λ Φˆ (x ,t). In the integrals over the frequency above and below we have assumed a high
Q 0 Q
frequency finite cutoff Λ for the quantum field Λ which regularizes the quantum field’s retarded Green’s
function from which we obtain the effective equations of motion of the oscillator [9]. In QBM language µ is
the dissipation kernel, ν(t,s) is the noise kernel which quantifies the two-time correlation of the Langevin
forcing term ξˆ(t),
Λ
ν(t,s) ξˆ(t),ξˆ(s) =λ2 dωI(ω;x ,x )cosω(t s). (4.3)
≡h{ }i Q Q Q −
Z0
[FromnowonwesimplydenoteI(ω),I(ω;x ,x ).] Herethe averageistakenwithrespecttothe initial
Q Q
state density matrix of the field.
By introducing the damping kernel γ(t,s) defined by
∂ ∂
µ(τ,s)=M γ(τ,s)=M γ(τ,s), (4.4)
Q Q
∂τ ∂s
we can bring (2.5) to the final form
t
M Q¨ˆ(t)+M Ω2Qˆ(t)+M dτγ(t,τ) Qˆ˙(τ)+2M γ(t,t ) Qˆ(t )=ξˆ(t), (4.5)
Q Q r Q · Q i · i
Zti
where Ω2 =Ω2 2M γ(t,t) is the renormalized frequency.
r − Q
The general solution to (4.1) is given by
t
Qˆ(t)=Qˆ (t)+ ds G˜(t,s)ξˆ(s), (4.6)
0
Zti
where G˜(t,s) is the retarded Green’s function for (4.1) satisfying
t
M G¨˜(t,s)+M Ω2G˜(t,s) dτµ(t,τ) G˜(τ,s)=δ(t s). (4.7)
Q Q
− · −
Zti
For here and below all the Fourier components are defined for positive frequency only, which correspond
to Fourier transformation of function defined on t>0. Accordingly, the Fourier space representation of the
above Green’s function can be written as:
G˜(ω) (M ( ω2+Ω2) µ(ω))−1
Q
≡ − −
,( ω2 iωγ(ω)+Ω2)−1M−1. (4.8)
− − r Q
