Complete particle-pair annihilation as a dynamical signature of the spectral singularity

0.16 MB·

by  G. R. Li

Complete particle-pair annihilation as a dynamical signature of the spectral
singularity
G. R. Li, X. Z. Zhang, and Z. Song
∗
School of Physics, Nankai University, Tianjin 300071, China
Motivated bythephysicalrelevanceofaspectralsingularityofinteractingmany-particlesystem,
weexplorethedynamicsoftwobosonsaswellasfermionsinone-dimensionalsystemwithimaginary
deltainteractionstrength. Basedontheexactsolution,itshowsthatthetwo-particlecollisionleads
toamplitude-reductionofthewavefunction. Forfermion pair,theamplitude-reductiondependson
the spin configuration of two particles. In both cases, the residual amplitude can vanish when the
relative group velocity of two single-particle Gaussian wave packets with equal width reaches the
magnitudeoftheinteractionstrength,exhibitingcompleteparticle-pairannihilationatthespectral
4
singularity.
1
0
PACSnumbers: 03.65.-w,03.65.Nk,11.30.Er
2

n
a
J

4 I. INTRODUCTION
1

Non-Hermitian operator has been introduced phenomenologically as an effective Hamiltonian to fit experimental
]
h data in various fields of physics [1–5]. In spite of the important role played non-Hermitian operator in different
p branchesofphysics,ithasnotbeenpaiddueattentionbythephysicscommunityuntilthediscoveryofnon-Hermitian
- Hamiltonians withparity-timesymmetry, whichhavea realspectrum[6]. It hasboostedthe researchonthe complex
t
n extension of quantum mechanics on a fundamental level [7–17]. Non-Hermitian Hamiltonian can possess peculiar
a
feature that has no Hermitian counterpart. A typical one is the spectral singularity (or exceptional point for finite
u
system), which is a mathematic concept. It has gained a lot of attention recently [18–26], motivated by the possible
q
[ physical relevance of this concept since the pioneer work of Mostafazadeh [27].
The majority of previous works focus on the non-Hermitian system arising from the complex potential, mean-

1
field nonlinearity [20, 28–37] as well as imaginary hopping integral [38]. In this paper, we investigate the physical
v
relevanceofthespectralsingularitiesfornon-Hermitianinteractingmany-particlesystem. Thenon-Hermiticityarises
8
from the imaginary interaction strength. For two-particle case, the exact solution shows that there exist a series of
0
1 spectral singularities, forming a spectrum of singularity associated with the central momentum of the two particles.
3 We consider dynamics of two bosons as well as fermions in one-dimensional system with imaginary delta interaction
. strength. Itshowsthatthe two-particlecollisionleads toamplitude-reductionofthe wavefunction. Forfermionpair,
1
0 the amplitude-reduction depends on the spin configuration of two particles. Remarkably, in both cases, the residual
4 amplitude can vanish only when the relative group velocity of two single-particle Gaussian wave packets with equal
1 width reaches the magnitude of the interaction strength. This phenomenon of complete particle-pair annihilation is
: the direct result of the spectral singularity. We also discuss the complete annihilations of a singlet fermion pair and
v
i a maximally two-mode entangled boson pair based on the second quantization formalism.
X
This paper is organizedas follows. In Section II, we present the model Hamiltonian and exact solution. In Section
r III, we construct the local boson pair initial state as initial state which is allowed to calculate the time evolution.
a
Based on this, we reveal the connection between the phenomenon of complete pair annihilation and the spectral
singularity. In Section IV, we extend our study a singlet fermion pair and a maximally two-mode entangled boson
pair based on the second quantization formalism. Finally, we give a summary in Section V.
II. HAMILTONIAN AND SOLUTIONS
Westartwithanone-dimensionaltwo-distinguishableparticlesystemwithimaginarydeltainteraction. Thesolution
can be used to construct the eigenstates of two-fermion and boson systems. The Hamiltonian has the form
∗Electronicaddress: [email protected]
2
1 ∂2 ∂2
H = + i2γδ(x x ) (1)
2p −2 ∂x2 ∂x2 − 1− 2
(cid:18) 1 2(cid:19)
where γ >0 and we use dimensionless units e=~=m=1 for simplicity.
Introducing new variables R and r, where
R = (x +x )/2, (2)
1 2
r = x x ,
1 2
−
we obtain the following Hamiltonian
H =H +H , (3)
2p R r
with
∂2
H = , (4)
R −4∂R2
∂2
H = i2γδ(r).
r −∂r2 −
Here R is the center-of-mass coordinate and r is the relative coordinate. The Hamiltonian is separatedinto a center-
of-mass part and a relative part, and can be solvable exactly.
Theeigenfunctionsofthecenter-of-massmotionH aresimplyplanewaves,whiletheHamiltonianH isequivalent
R r
to that of a single-particle in an imaginary delta-potential, which has been exactly solved in the Ref.[39]. Then the
eigen functions of the original Hamiltonian can be obtained and expressed as
ψ (K,k,x ,x )=eiK(x1+x2)/2 cos[k(x x )] (5)
+ 1 2 1 2
{ −
iγ
sin[k(x x )]sign(x x ) ,
1 2 1 2
− k − −
(cid:27)
in symmetrical form, and
ψ (K,k,x ,x )=eiK(x1+x2)/2sin[k(x x )], (6)
1 2 1 2
− −
in antisymmetrical form. The corresponding energy is
E(K,k)=K2/4+k2, (7)
with the central and relative momenta K,k ( , ). The symmetrical wavefunction ψ(K,k,x ,x ) is the
1 2
∈ −∞ ∞
spatial part wavefunction for two bosons or two fermions in singlet pair, while the antisymmetrical wavefunction
ϕ(K,k,x ,x ) only for two triplet fermions.
1 2
Before starting the investigation on dynamics of two-particle collision, we would like to point that there exist
spectral singularities in the present Hamiltonian. It arises from the same mechanism as that in the single-particle
systems [39, 40].
We can see that the eigen functions with even parity and momentum k= γ can be expressed in the form
−
ψ (K) ψ(K, γ,x ,x ) (8)
ss 1 2
≡ −
= eiK(x1+x2)/2e iγx1 x2 ,
− | − |
with energy
E (K)=K2/4+γ2. (9)
ss
We note that function ψ (K) satisfies
ss
∂ψ (K)
ss
lim iγψ (K) =0, (10)
ss
x1−x2→±∞(cid:20)∂(x1−x2) ± (cid:21)
which accordswith the definition of the spectral singularity in Ref. [20]. It shows that there exist a series of spectral
singularities associated with energy E (K) for K ( , ), which constructs a spectrum of spectral singularities.
ss
∈ −∞ ∞
We will demonstrate in the following section that such a singularity spectrum leads to a peculiar dynamical behavior
of two local boson pair or equivalently, singlet fermion pair.
3
III. DYNAMICAL SIGNATURE
A. Construction of initial state
The emergence of the spectral singularity induces a mathematical obstruction for the calculation of the time
evolution of a given initial state, since it spoils the completeness of the eigen functions and prevents the eigenmode
expansion. Nevertheless,thecompletenessoftheeigenfunctionsisnotnecessaryforthetimeevolutionofastatewith
asetofgivencoefficientsofexpansion. Itdoesnotcauseanydifficultyinderivingthetimeevolutionofaninitialstate
with arbitrary combination of the eigen functions. Namely, any linear combination of function set ψ(K,k,x ,x )
1 2
{ }
or ϕ(K,k,x ,x ) can be an initial state, and the time evolution of it can be obtained simply by adding the factor
1 2
{ }
e iE(K,k)t.
−
In order to investigate the dynamical consequence of the singularity spectrum, we consider the time evolution of
the initial state of the form
1 ∞ ∞
Ψ(x ,x ,0)= G(K)g(k)ψ(K,k,x ,x )dKdk, (11)
1 2 1 2
√Λ
Z−∞Z−∞
where Λ is the normalization factor, which will be given in the following and
1
G(K) = exp (K K )2 iKR , (12)
−2α2 − 0 − 0
(cid:20) (cid:21)
1
g(k) = exp (k k )2 ikr . (13)
−2β2 − 0 − 0
(cid:20) (cid:21)
Here α,β,r ,k >0 and K is arbitrary real number. We explicitly have
0 0 0
παβ
Ψ(x ,x ,0)= [(k +γ)expθ +(k γ)expθ ] (14)
1 2 0 + 0
2√Λk − −
0
where
α2 x +x 2 β2
θ = 1 2 R (x x +r )2 (15)
0 1 2 0
± − 2 2 − − 2 | − |
(cid:18) (cid:19)
x +x
1 2
+i K k x x k r +K R .
0 0 1 2 0 0 0 0
2 ∓ | − |−
(cid:18) (cid:19)
Furthermore, from the identity
α2(x +x +A)2+4β2(x x +B)2 =α2 (x +A)2+(x +A)2 A2 (16)
1 2 1 2 1 2
− −
h i
+4β2 (x +B)2+(x B)2 B2 + α2 4β2 x x
1 2 1 2
− − −
h i (cid:0) (cid:1)
we can see that the cross term x x vanishes if we take α=2β. The initial state can be written as a separable form
1 2
πβ2
Ψ(x ,x ,0) = (k +γ)[ϕ (x )ϕ (x )u(x x )+ϕ (x )ϕ (x )u(x x )] (17)
1 2 0 + 1 2 2 1 + 2 1 1 2
√Λk { − − − −
0
+(k γ)[ϕ (x )ϕ (x )u(x x )+ϕ (x )ϕ (x )u(x x )] ,
0 + 2 1 2 1 + 1 2 1 2
− − − − − }
where u(x) is Heaviside step function and
r 2 i
ϕ (x)=exp β2 x 0 + (K 2k )x . (18)
0 0
± − ∓ 2 2 ±
(cid:20) (cid:16) (cid:17) (cid:21)
In this case, Λ can be obtained as
4π3β2(k γ)2
0
Λ= − .
k2
0
4
0.3 t=0 t=1 0.4 t=0 t=1 0.3 t=0 t=1
0.3
0.2 0.2
0.2
0.1 0.1
2φ|(r, t)| 0.03 t=2 t=3 φ2|(r, t)| 00..140 t=2 t=3 φ2|(r, t)| 0.30 t=2 t=3
0.3
0.2 0.2
0.2
0.1 0.1
0.1
0 0 0
−12 −6 0 6 −12 −6 0 6 −12 −6 0 6 −12 −6 0 6 −12 −6 0 6 −12 −6 0 6
r r r
(a) (b) (c)
2
FIG. 1: (Color online) The profiles of |ϕ(r,t)| of the two-boson Gaussian wavepackets are plotted for different values of k0
and α=2β: (a) γ =5.0, k0 =5.0,α=2β =2.0;(b) γ =5.0, k0 =5.0,α=2β =3.0;(c) γ =2.0,k0 =5.0,α=2β =2.0. One
canseethattheperfectpairannihilation inthecaseof(a)andimperfectpairannihilation inthecasesof(b)and(c)whenthe
widthof theinitial wavepacketsbecomes narrower, andtherelativegroup velocity deviatesfrom γ,respectively. Itshowsthat
perfect pair annihilation can bea signature of thesingularity spectrum.
Withoutlossofgeneralitywehavesettheinitialcenter-of-masscoordinateR =0anddroppedanoverallphasek r .
0 0 0
We note that functions ϕ (x) and ϕ (x) representGaussianfunctions with centers at r /2 and r /2, respectively.
+ 0 0
Obviously, the probability contributi−ons of ϕ (x )ϕ (x )u(x x ) and ϕ (x )ϕ (x )u(x −x ) are negligible
+ 2 1 2 1 + 1 2 1 2
under the condition βr 1. We then yield − − − −
0
≫
1
Ψ(x ,x ,0) β [ϕ (x )ϕ (x )+ϕ (x )ϕ (x )], (19)
1 2 + 1 2 + 2 1
≈ r2π − −
whichrepresentstwo-bosonwavepacketstatewiththesamewidth,groupvelocityK /2 k ,andlocation r /2. Here
0 0 0
± ∓
the renormalization factor has been readily calculated by Gaussian integral. So far we have construct an expected
initial state without using the biothogonal basis set. The dynamics of two separated boson wavepackets can be
described by the time evolution as that in the conventional quantum mechanics.
B. Annihilating collision
It is presumable that before the bosons start to overlap they move as free particles with the center moving in
their the group velocities K /2 k and the width spreading as function of time (4β4t2+1)/β2. We concern the
0 0
±
dynamic behavior after the collision. To this end, we calculatethe time evolutionof the giveninitial state, which can
p
be expressed as
Ψ(x1,x2,t)= 1 ∞ ∞ G(K)g(k)ψ(K,k,x1,x2)e−i(K2/4+k2)tdKdk. (20)
√Λ
Z−∞Z−∞
By the similar procedure as above, we find that the evolved wave function can always be written in the separated
form
Ψ(x ,x ,t)=Φ(R,t)φ(r,t), (21)
1 2
where
4β2 2β2(R K /2)2 i 16β4R2+4RK tK2
Φ(R,t)= 4 exp − 0 + 0− 0 , (22)
sπ(1+4β2t2) "− 1+4β2t2 (cid:0) 4+16β2t2 (cid:1)#
and
φ(r,t)= 1 ∞ exp 1 (k k )2 ikr cos(kr) iγ sin(k r) exp ik2t dk. (23)
√Ω −2β2 − 0 − 0 − k | | −
Z−∞ (cid:20) (cid:21)(cid:20) (cid:21) (cid:0) (cid:1)
5
where the normalization factor
π3/2β(k γ)2
0
Ω= − . (24)
k2
0
Straightforwardalgebra shows that
φ(r,t)=(k +γ)Θ +(k γ)Θ (25)
0 + 0
− −
where
√β k0 γ −1 β2[r (r0 2k0t)]2
Θ = | − | exp | |± − +i∆ , (26)
± 2π1/2(1+2iβ2t) (− 2(4β4t2+1) ±)
∆ = βp4(|r|±r0)2t−2k02t∓2k0(|r|±r0). (27)
± 2(4β4t2+1)
In the case of β4t2 1, k t r the probability distribution is
0 0
≫ ≫
φ(r,t)2 π(k0+γ)2 exp (|r|+2k0t)2 + π k02−γ2 e−k02/β2 exp r2 , (28)
| | ≈ 4Ωk02t (− 4β2t2 ) (cid:0) 2Ωk(cid:1)02t (cid:18)−4β2t2(cid:19)
which leads the total probability under the case k /β 1
0
≫
∞ φ(r,t)2dr (k0+γ)2. (29)
| | ≈ (k γ)2
Z−∞ 0−
We can see that, after the collision the residual probability becomes a constant and vanishes when k = γ. It
0
−
shows that when the relative group velocity of two single-particle Gaussian wave packets with equal width reaches
the magnitude of the interaction strength, the dynamics exhibits complete particle-pair annihilation.
Inordertodemonstratesuchdynamicbehaviorandverifyourapproximateresult,thenumericalmethodisemployed
tosimulatethetimeevolutionprocessforseveraltypicalsituations. Theprofilesof ϕ(r,t)2 areplottedinFig. 1. We
| |
wouldliketo pointthat the complete annihilationdepends onthe relativegroupvelocity,whichis the consequenceof
singularityspectrum. This enhancesthe probabilityofthe pair annihilationfor acloudofbosons,whichmayprovide
an detection method of the spectral singularity in experiment.
IV. SECOND QUANTIZATION REPRESENTATION
In this section, we will investigate the two-particle collision process from another point of view and give a more
extended example. By employing the second quantization representation, the initial state in Eq. (19) can be ex-
pressed as the form a†1a†2|0i, where a†i (i=1,2) is the creation operator for a boson in single-particle state with the
wavefunction
ϕ+(x)=hx|a†1|0i, ϕ−(x)=hx|a†2|0i, (30)
and 0 denotes the vacuum state of the particle operator. Similarly, if we consider a fermion pair, the initial state in
| i
Eq. (19) can be written as
1
√2 c†1,↑c†2,↓−c†1,↑c†2,↓ |0i, (31)
(cid:16) (cid:17)
where c†i,σ (i=1,2;σ=↑,↓) is the creation operator for a fermion in single-particle state with the wavefunction
ϕ+(x)χσ =hx|c†1,σ|0i, ϕ−(x)χσ =hx|c†2,σ|0i. (32)
6
t=0 t=1 0.6 t=0 t=1 t=0 t=1
0.2 0.2
0.4
0.1 0.1
0.2
φ2|(r, t)| 0 t=2 t=3 φ2|(r, t)| 0.60 t=2 t=3 φ2|(r, t)| 0 t=2 t=3
0.2 0.2
0.4
0.1 0.1
0.2
0 0 0
−25 0 −25 0 −25 0 −25 0 −25 0 −25 0
r r r
(a) (b) (c)
2
FIG.2: (Coloronline)Theprofilesof|ϕ(r,t)| ofamaximallytwo-modeentangledbosonpairareplottedfordifferentvaluesof
k0andα=2β: (a)γ =10.0,k0 =10.0,α=2β =1.0;(b)γ =10.0,k0 =10.0,α=2β =3.0;(c)γ =4.0,k0 =10.0,α=2β =1.0.
Onecanseethattheperfectpairannihilationinthecaseof(a)andimperfectpairannihilationinthecasesof(b)and(c)when
thewidth of the initial wavepackets becomes narrower, and therelative group velocity deviates from γ, respectively.
Here
1 0
χ = ,χ = , (33)
↑ 0 ↓ 1
(cid:18) (cid:19) (cid:18) (cid:19)
arethe spinpartofwavefunction. Wesee thatthe initialstateinEq. (31)issingletpairwithmaximalentanglement.
In contract, state √12 c†1, c†2 +c†1, c†2 |0i should not lose any amplitude after collision.
↑ ↓ ↑ ↓
On the other hand,(cid:16)we canextend ou(cid:17)r conclusionto other types of initial state. For instance, we canconstruct the
initial state with
1
G(K) = exp (K K )2 iKR , (34)
−2α2 − 0 − 0
(cid:20) (cid:21)
1
g(k) = (k k )exp (k k )2 ikr ,
− 0 −2β2 − 0 − 0
(cid:20) (cid:21)
which are also local states in K and k spaces, respectively.
In coordinate space, the above wavefunction has the from
2iπβ4
(1) (1)
Ψ(x ,x ,0) = (k +γ) ϕ (x )ϕ (x ) ϕ (x )ϕ (x ) u(x x )
1 2 √Ξk0 0 + 1 − 2 − + 1 − 2 2− 1
+(x ⇄n x )] h(cid:16) (cid:17)
1 2
(k γ) ϕ(1)(x )ϕ (x ) ϕ(1)(x )ϕ (x ) u(x x )
− 0− − 1 + 2 − + 2 − 1 2− 1
+ (x ⇄xh(cid:16))] , (cid:17) (35)
1 2
}
which can be reduced to
2iπβ4(k γ)
Ψ(x ,x ,0) 0− ϕ(1)(x )ϕ (x ) ϕ (x )ϕ(1)(x ) +(x ⇄x ) , (36)
1 2 ≈ √Ξk0 + 1 − 2 − + 1 − 2 1 2
h(cid:16) (cid:17) i
under the approximation βr 1. Here Ξ is the normalized constant, and ϕ(1)(x)= x r0 ϕ (x).
By the same procedure, at0t≫ime t the evolved wavefunction is ± ∓ 2 ±
(cid:0) (cid:1)
φ(r,t)= (k +γ)Θ +(k γ)Θ , (37)
0 + 0
− − −
with
π iβ3[r (r 2k t)] β2[r (r 2k t)]2
0 0 0 0
Θ = | |± − exp | |± − +i∆ , (38)
± r2Ω′ k0(1+2iβ2t)3/2 (− 2(4β4t2+1) ±)
β4(r r )2t 2k2t 2k (r r )
∆ = | |± 0 − 0 ∓ 0 | |± 0 . (39)
± 2(4β4t2+1)
7
where the normalization factor
π3/2β3(k γ)2
0
Ω′ = 4k2 − .
0
In the case of β4t2 1, k t r the probability distribution is
0 0
≫ ≫
π(k +γ)2 (r +2k t)2
φ(r,t)2 0 (r +2k t)2exp | | 0 (40)
| | ≈ 16Ω′k02t3 | | 0 (− 4β2t2 )
π k02−γ2 e−k02/β2 r2 4k2t2 exp r2 ,
− 8Ωk2t3 − 0 −4β2t2
(cid:0) ′ (cid:1)0 (cid:18) (cid:19)
(cid:0) (cid:1)
which leads the total probability
∞ φ(r,t)2dr (k0+γ)2. (41)
| | ≈ (k γ)2
Z−∞ 0−
The profiles of ϕ(r,t)2 are plotted in Fig. 2. We can see that the same behavior occurs in the present situation.
| |
In order to clarify the physical picture, we still employ the second quantization representation by introducing
another type of boson creation operator b†i (i=1,2) with
ϕ(−0)(x) = hx|b†1|0i,
(0)
ϕ− (x) = hx|b†2|0i.
Then the initial state in Eq. (36) can be expressed as
1
√2 a†1b†2+a†2b†1 |0i, (42)
(cid:16) (cid:17)
which is maximally two-mode entangled state.
V. SUMMARY AND DISCUSSION
Insummaryweidentifiedaconnectionbetweenspectralsingularitiesanddynamicalbehaviorforinteractingmany-
particle system. We explored the collision process of two bosons as well as fermions in one-dimensional system with
imaginarydeltainteractionstrengthbasedontheexactsolution. Wehaveshowedthatthereisasingularityspectrum
which leads to complete particle-pair annihilation when the relative group velocity is resonant to the magnitude of
interaction strength. The result for this simple model implies that the complete particle-pair annihilation can only
occur for two distinguishable bosons, maximally two-mode entangled boson pair and singlet fermions, which may
predict the existence of its counterpart in the theory of particle physics.
Acknowledgments
We acknowledge the support of the National Basic Research Program (973 Program) of China under Grant
No.2012CB921900and CNSF (Grant No. 11374163).
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