New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation

Optics Communications 281 (2008) 5571–5573

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New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation Hong-yi Fan, Li-yun Hu * Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China

a r t i c l e

i n f o

Article history: Received 23 April 2008 Received in revised form 27 July 2008 Accepted 2 August 2008

PACS: 03.65.Yz 03.65.Ud

a b s t r a c t We analyze the time evolution of mixed state q0 in a dissipative channel, characteristic of a decay constant j, by virtue of the elegant properties of entangled state representation hgj. We find that the matrix element of the mixed state qðtÞ at time t in hgj representation is proportional to that of the initial q0 in the 2 1e2jt decayed entangled state hgejt j representation, accompanying with a Gaussian damping factor e 2 jgj . Thus we have a new insight about the nature of the dissipative process. Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved.

1. Introduction In nature most systems are immersed in reservoir, so the phenomenon of damping and decoherence always happen. For a lossy channel (or cavity at zero temperature) the evolution of the density operator is described by [1]

dqðtÞ ¼ j½2aqðtÞay  ay aqðtÞ  qðtÞay a; dt

ð1Þ

where ½a; ay  ¼ 1; j is the rate of decay. Usually, as shown in the literature before, solving master equations is using either the Langevin equation or the Fokker–Planck equation after recasting the density operators into some definite representations, e.g. particle number representation (Q-function), coherent state representation (P-representation) or the Wigner representation. Here we alternatively treat this equation by virtue of the newly developed entangled state representation jgi defined in a two-mode Fock space, of which one mode is a fictitious one representing the effect of enviroment. It is Takahashi–Umezawa [2–4] who first introduced the fictitious Fock space to treat ensemble average as a pure state average, this pure state (thermo vacuum state) is also a two-mode squeezed state. Thus we guess that the damping of a system in a lossy channel may also be treated in an enlarged space. As we have experienced that many quantum optics phenomena are related to entangled states, for example, the two-mode squeezed state is simultaneously

* Corresponding author. Address: Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China. Tel./fax: +86 2162932080. E-mail addresses: [email protected], [email protected] (L.-y. Hu).

an entangled state, so it would be convenient and effective for us to study damping from the point of view of squeezing in the entangled state representation. Lukily, the merit of our approach, as one can see shortly later, lies in that the matrix element of the mixed state qðtÞ at time t in hgj representation is proportional to that of the initial state q0 in the decayed entangled state hgejt j representation, 2 1e2jt accompanying with a Gaussian damping factor e 2 jgj . The jt transformation hgj ! hge j is governed by a two-mode squeezing operator. Thus we have a new insight about the nature of the dissipative process. The entangled state representation has been recently used in discussing quantum optical version of classical circular harmonic correlation [5]. 2. Brief review of the jgi representation In Ref. [6] based on the theory of thermo field dynamics invented by Takahashi and Umezawa, we have introduced the twomode entangled state representation jgi,

  1 ~ ~y þ ay a ~y j0; 0i; jgi ¼ exp  jgj2 þ gay  g a 2

g ¼ g1 þ ig2 ;

ð2Þ

~y is a fictitious mode (tilde mode) accompanying the real where a ~ is annihilated by a ~, ½a ~; a ~y  ¼ 1. jgi photon creation operator ay , j0i obeys the eigenvector equations

~y Þjgi ¼ gjgi; ða  a

~  ay Þjgi ¼ g jgi; ða

ð3Þ

so

~y Þðay  a ~Þjgi ¼ jgj2 jgi: ða  a

0030-4018/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.002

ð4Þ

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H.-y. Fan, L.-y. Hu / Optics Communications 281 (2008) 5571–5573

When g ¼ 0,

The formal solution of Eq. (15) is

y y

~ ¼ jg ¼ 0i ¼ ea a~ j0; 0i

1 X

~ i; jn; n

ð5Þ

ð6Þ

and DðgÞ ¼ expðgay  g aÞ is the displacement operator. ~y Þjgi and can see that jgi is From Eq. (3) we calculate hg0 jða  a orthonormal,

hg0 jgi ¼ pdðg0  gÞdðg0  g Þ:

ð7Þ

~ is similar to the sin~y Þj0; 0i Noting that the state jg ¼ 0i ¼ expðay a y2 gle-mode coordinate eigenvector jq ¼ 0i ¼ p1=4 expð a2 Þj0i so far as the normalization of the state is concerned, its introduction should not be a problem. In fact, the state jgi can be constructed by a Fourier integration over the direct product of two independent single-mode coordinate eigenstates as the following:

jgi ¼ eig1 g2 =2

Z

1 1

g dqjqi  j q  g1 ieig2 q :

ð8Þ

One can check that Eq. (8) is equal to Eq. (2) by substituting

 pffiffiffi 1 1 jqi ¼ p exp  q2 þ 2qay  ay2 j0i; 2 2   pffiffiffi 1 1 y2 ~ ~a ~y  a ~ 2 þ 2q ~ j0i: ~i ¼ p1=4 exp  q jq 2 2 1=4



2

d g

p

jgihgj ¼

Z

ð9Þ

p

~y þ g a  ga ~ : expðjgj2 þ gay  g a

~y þ a ~Þ :¼ 1; ~ a  ay a  a ~y a þ ay a

and the commutative relation

~a ~y ay ; ðay  a ~Þða  a ~y Þ ¼ 2½ðay  a ~Þða  a ~y Þ: ½aa Using the operator identity

ekðAþrBÞ ¼ ekA exp½rð1  eks ÞB=s;

ð10Þ

~Þða  a ~y Þ þ aa ~a ~y ay þ 1gjq0 i jqðtÞi ¼ expfjt½ðay  a y y

~Þða  a ~y Þ=2jq0 i: ¼ ejtðaa~a~ a þ1Þ exp½ð1  e2jt Þðay  a

~  a a Þ is the twoIt is encouraging to remember that exp½kðaa mode squeezing operator [8–10], and has its natural expression in hgj representation [11],

Z

2

d g

p

jek gihgj;

ð11Þ

Note that jqðtÞi is introduced for the writing’s convenience and Eq. (11) should not be understood as an eigenvector equation. Operating the both sides of Eq. (1) on jg ¼ 0i yields

ð12Þ

From (2) it is easy to see that

ð13Þ

and

~Þn jg ¼ 0i: ~y a ðay aÞn jg ¼ 0i ¼ ða

ð14Þ

Using Eqs. (13), (14) and noting that the density operator qðtÞ is defined in the original space which is commutative with operators ~y ; a ~) in the tilde space, we can convert Eq. (12) into (a

d ~ÞjqðtÞi: ~  ay a  a ~y a jqðtÞi ¼ jð2aa dt

ð21Þ

which together with Eq. (7) leads to

~a ~y ay g hgj expfjt½aa Z 2 0 d g hgjejt g0 ihg0 j ¼ ejt p Z 2 ¼ ejt d g0 pdðg0 ejt  gÞdðg0 ejt  g Þhg0 j ¼ ejt hgejt j;

ð22Þ

thus projecting Eq. (20) onto hgj and using Eq. (4) we see

2

or 1

2

hgjqðtÞjg ¼ 0i ¼ e2Tjgj hgejt jq0 jg ¼ 0i;

T ¼ 1  e2jt :

~jg ¼ 0i; ay jg ¼ 0i ¼ a

ð20Þ

~y y

ð23Þ

where

At this point, it is convenient to introduce the state jqðtÞi [6] for density operator qðtÞ,

~y jg ¼ 0i; ajg ¼ 0i ¼ a

ð19Þ

(valid for ½A; B ¼ sB), and Eqs. (17) and (18) we can reform Eq. (16) as

1

3. The time evolution of jqðtÞi in the dissipative channel

dqðtÞ jg ¼ 0i ¼ j½2aqðtÞay  ay aqðtÞ  qðtÞay ajg ¼ 0i: dt

ð18Þ

¼ e2Tjgj hgejt jq0 i;

this merit provides us with a possibility to express density operators in terms of jgi.

jqðtÞi  qðtÞjg ¼ 0i:

ð17Þ

~Þða  a ~y Þ=2jq0 i hgjqðtÞi ¼ hgejt j exp½ð1  e2jt Þðay  a

2

d g

~ ¼ ðay  a ~  ay a  a ~y a ~Þða  a ~y Þ þ aa ~a ~y ay þ 1; 2aa

~a ~y ay Þ ¼ ek exp½kðaa

into Eq. (9) and then performing the integration. Thus the introduction of jgi stands on solid ground since the coordinate eigenstates are widely used. Using the normally ordered form of vacuum projector ~ ~ ¼: expðay a  a ~Þ :, (:: stands for the normal ordering), ~y a j0; 0ih0; 0j and the technique of integration within an ordered product (IWOP) of operators [7], we can show the completeness,

Z

ð16Þ

where q0 is the initial density operator, jq0 i ¼ q0 jg ¼ 0i. To deal with Eq. (16), we notice

n¼0

~ thus ~y Þn j0; 0i, ~ i ¼ n!1 ðay a where jn; n

jgi ¼ DðgÞjg ¼ 0i;

~Þgjq0 i; ~  ay a  a ~y a jqðtÞi ¼ expfjtð2aa

ð15Þ

ð24Þ

Eq. (23) reveals a new relationship between jq0 i and jqðtÞi, which manifestly shows that the matrix element of the mixed state qðtÞ in hgj and jg ¼ 0i representations is proportional to that of the initial state q0 in the decayed entangled state hgejt j and jg ¼ 0i representations, accompanying with a Gaussian damping factor 2 T e2jgj . The damping is embodied as a squeezing in hgj representation. Thus we see clearly how the dissipative channel plays its role in time evolution of mixed states. In the entangled state representation, the solution of this density operator’s master equation is equivalent to the evolution of thermo field dynamics if a fictitious freedom (thermo freedom) is introduced. R 2 Further, multiplying dpg jgi from the left to Eq. (23) and using the completeness relation Eq. (10) we have

jqðtÞi ¼

Z

2

d g

p

1

e2ð1e

2jt Þjgj2

jgihgejt jq0 i:

ð25Þ

Then using the operator identity

expðkay aÞ ¼: exp½ðek  1Þay a :;

ð26Þ

and the IWOP technique to perform the integration in Eq. (25) we have

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H.-y. Fan, L.-y. Hu / Optics Communications 281 (2008) 5571–5573

jqðtÞi ¼ ¼

Z

Z

2

d g

p

1

e2ð1e

2jt Þjgj2

jgihgejt :jq0 i

  j  ge2tðjgÞ 2 ~Þ : exp  jgj þ gðay  eðjgÞt a p jg 

 ~y þ aa ~ : jq ~y þ ay a ~  ay a  a ~y a  0i  exp g eðjgÞt a  a y ~y y y ~ þ a aÞg : exp½jT 1 aa ~a ~jq  0i ¼ T 3 exp½gT 1 a a  : expfðT 2  1Þða

 ðtÞi ¼ jq

2

d g

p

~Þ þ g ðejt a  a ~y ~y Þ þ ay a : exp½jgj2 þ gðay  ejt a

~ : jq0 i ~  ay a  a ~y a þ aa ~Þ exp½Taa ~jq0 i: ~y a ¼ exp½jtðay a þ a

where

ð27Þ

Since jq0 i ¼ q0 jg ¼ 0i, we can use Eqs. (13), (14) to rewrite Eq. (27) as y

1 X

y

ejta~ a~

n¼0

Tn n n ~ q0 jg ¼ 0i a a n!

Z

eðjgÞt ðj  gÞ; j  ge2tðjgÞ ð37Þ

expðfjzj2 þ nz þ gz Þ ¼ 

  1 ng exp  ; f f

ReðnÞ < 0:

ð38Þ

Eq. (36) can be further rewritten as

ð28Þ

~y  expfða ~ þ ay aÞ ln T 2 g exp½jT 1 aa ~y a ~ q  ðtÞi ¼ T 3 exp½gT 1 ay a  0 jg ¼ 0i jq 1 l j X j g lþj yj ay a ln T 2 l  yl ay a ln T 2 j a q0 a e a jg ¼ 0i; ð39Þ T a e ¼ T3 l!j! 1 l;j¼0 so

1 X T n jtay a n y a q0 ayn ejta a : e n! n¼0

ð29Þ

q ðtÞ ¼ T 3

y

where ejta a an q0 ayn ejta a is similar in form to the generalization of Keller-Mollow’s quantized photoncounting formula [12–17] for the case of non-equilibrium open photon field. Finally we mention that our new method also works for more than the single-mode damped oscillator, the following master equation:

 ðtÞ dq  ðtÞaay  þ j½2aq  ðtÞay  ðtÞa  aay q  ðtÞ  q ¼ g½2ay q dt  ðtÞ  q  ðtÞay a;  ay aq

ð30Þ

1 X l;j¼0

jl g j l!j!

References

We sincerely thank the referee for his useful suggestion. Work supported by Natural Science Foundation of China under Grant 10775097 and 10475056.

ð31Þ

whose formal solution is

~a ~y Þ þ jtð2aa ~  ay a  a ~y a ~y  aay  a ~Þgjq  ðtÞi ¼ expfgtð2ay a  0 i: jq ð32Þ Then noticing the relation in Eqs. (17), (18), we have

ð33Þ

Substituting Eq. (33) into (32) and using Eqs. (18) and (19) we see

ð34Þ

 ðtÞ in hgj and jg ¼ 0i is given by Thus the matrix element of q



jþg  0 i: ð1  e2ðjgÞt Þjgj2 hgeðjgÞt jq 2ðj  gÞ

ð40Þ

2jt

d ~y  aay  a ~Þjq ~a ~y  þ jð2aa ~  ay a  a ~y a  ðtÞi;  ðtÞi ¼ g½2ay a jq dt

~a ~y ay þ 1Þðj  gÞt  ðtÞi ¼ exp½ðaa jq    jþg  ~Þða  a ~ y Þ jq  0 i:  exp 1  e2ðjgÞt ðay  a 2ðj  gÞ

y

In particular, when g ¼ 0, then T 1 ¼ 1ej , T 2 ¼ ejt , T 3 ¼ 1, Eq. (40) reduces to Eq. (29). In sum, we have adopted the entangled state approach for treating the time evolution of density operator in the dissipative channel in Eq. (1), the result Eq. (23) help us to grasp the inward nature of dissipative things in an intuitive manner. Thus we may say that the entangled state representation is beneficial to quantum optics theory [18]. Acknowledgements

~y  aay  a ~Þ ~a ~y Þ þ jtð2aa ~  ay a  a ~y a gtð2ay a y y ~ ~ ~ ~ ¼tðj þ gÞða  a Þða  a Þ þ tðj  gÞðaa  ay ay þ 1Þ:

y

yj a a ln T 2 l  a q0 ayl ea a ln T 2 aj : T lþj 1 a e

representing the laser theory in the lowest-order approximation can also be solved by virtue of the entangled state representation, where g and j are the gain and the cavity loss, respectively. Eq. (30) reduces to Eq. (1) when g ¼ 0. Similar to the way of deriving Eqs. (23) and (29), we operate the both sides of (30) on jg ¼ 0i, which yields

  ðtÞi ¼ exp  hgjq

T2 ¼

2

d z

p

which implies

y

1  e2ðjgÞt ; j  ge2tðjgÞ jg ; T3 ¼ j  ge2tðjgÞ

T1 ¼

and we have used the following integral formula,

1 X T n jta~y a~ n a q0 ayn jg ¼ 0i ¼e e n! n¼0 1 X Tn n y y ¼ ejta a a q0 ayn ejta~ a~ jg ¼ 0i n! n¼0 1 X Tn n y y ¼ ejta a a q0 ayn ejta a jg ¼ 0i; n! n¼0 jtay a

qðtÞ ¼

2

d g

ð36Þ

~y a ~ : jq0 i ~Þ þ ð1  e2jt Þaa ¼: exp½ðejt  1Þðay a þ a

jqðtÞi ¼ ejta a

Z

ð35Þ

from which we see that in the bra hgeðjgÞt j the gain factor is egt , sharply in contrast to the damping factor ejt . Further, using the completeness relation in Eq. (10), we obtain

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