- Email: [email protected]
Dynamical Casimir effect at finite temperature
10 April 2000
Physics Letters A 268 Ž2000. 174–177 www.elsevier.nlrlocaterphysleta
Dynamical Casimir effect at finite temperature Jing Hui a,) , Shi Qing-Yun b, Wu Jian-Sheng a a
Theoretical Physics DiÕision, Nankai Institute of Mathematics, Nankai UniÕersity, Tianjin 300071, PR China b Department of Physics, Zhengzhou UniÕersity, Zhengzhou, Henan 450052, PR China Received 23 December 1999; received in revised form 28 February 2000; accepted 28 February 2000 Communicated by P.R. Holland
Abstract The dynamics of the Maxwell field in a cavity with movable perfectly reflecting boundaries is investigated at nonzero temperature, applying the techniques of the thermo field dynamics ŽTFD.. For a resonantly vibrating cavity, the thermal effect on the creation of photons and the temperature correction to the mechanic response of the intervening vacuum are all derived, based on the previous work at zero temperature by Dodonov and Klimov wPhys. Rev. A 53 Ž1996. 2664x. The strong amplification of the Casimir effect due to finite temperature, which was first predicted by Plunien et al. wPhys. Rev. Lett. 84 Ž2000. 1882x in response theory, is also observed. The squeezing effect at finite temperature is probed too. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv; 03.65.Sq
In recent years, the dynamical Casimir effect, describing the force and radiation from moving mirrors has garnered much attention w1–3x. In particular, the generation of photons by the parametric resonance has great promise to be observed by the experiments w4x. Most theoretical studies have focused on the ideal zero temperature course, however, the existence of finite temperature should be considered for any actual experiments. Recently, Plunien et al. presented a realistic calculation of thermal effects on quantum radiation within the framework of quantum field theory of time-dependent system at finite temperature w5x, predicting a strong enhancement of the dynamical Casimir effect at finite temperature.
)
Corresponding author. E-mail address: [email protected] ŽJ. Hui..
In this Letter, we present a thermo field dynamic ŽTFD. formalism, to calculate the number of created photons as well as the Casimir force in a resonantly vibrating cavity at finite temperature. We begin by introducing the important concept of the thermo vacuum w6x into the model of mono-dimensional cavity with which mirrors undergo the parametric resonance w7,8x. The modified Casimir energy density at non-zero temperature is given, as extending of an earlier literature w9x. Then, through the Bogoliubov transformation, we produce two kinds of quasi-photon operators characterizing the thermal and dynamical effects, respectively. The calculations leading to the final results are based on the previous work at zero temperature by Dodonov and Klimov w4x, in which the Bogoliubov coefficients were already given explicitly. In this method, we can not only restore the important results about the enhance-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 6 5 - 1
J. Hui et al.r Physics Letters A 268 (2000) 174–177
175
ment effect pointed out by Plunien et al., but also derive the modified Casimir force at finite temperature. In addition, the squeezing effect is also investigated, leading to a squeezing coefficient independent of the temperature. As is usual, we consider a cavity with ideal mirrors, one of which is fixed at the origin while the other is allowed to move according to the special parametric resonance law:
where v n s nprq0 . The photons of the thermal vacuum obey the Bose-Einstein distribution, that is: Nn ' ²00˜ < bn† bn <00˜ : s sinh2u s wexpŽ nrg . y 1xy1. Žg s q0 Trp , and T is the cavity temperature.. It is just this expression which determines the heating coefficient u in the heating operator. From Eq. Ž4., we can easily obtain the temperature-dependent nth mode energy density in agreement with Hacyan et al. w10x:
q Ž t . s q0 Ž 1 q e sin Ž 2 v t . .
e Ž v n . ' ²00˜ < e Ž v n . <00˜ :
Ž 1.
here v s prq0 and 0 F t F t . The temperature effect is introduced by the thermo vacuum state w6x: <0: T ' H Ž u .<00˜ :, where the vacuum state <00˜ : belongs to the double Hilbert space determined by the tilde conjugate, and the heating operator: H Ž u . s expwyu Ž a n a˜ n y a†n a˜†n .x provides the thermal Bogoliubov transformation: H † Ž u . a n H Ž u . s ua n q Õa˜†n , H † Ž u . a†n H Ž u . s ua†n q Õa˜ n ,
Ž u s cosh u ,Õ s sinh u . .
where is the annihilation Žcreation. operator for the nth mode of the cavity field. The new operators bn s ua n q Õa˜†n and bn† s ua†n q Õa˜ n are called the quasi-photon operators, which satisfy w bn ,bn† x s 1. Then, the Casimir energy density of the cavity field, which was introduced in Ref. w9x, should be modified as:
e c s lim ²00˜ < Gx x <00˜ :a y ²0 M 0˜ M < Gx x <0 M 0˜ M :a .
™0 ž
/
Ž 3.
Here, M represents the unperturbed Minkowski space, a represents the regularization procedure to dispose of the zero-point divergences. The operator Gx x is the xx-component of the Maxwell stress tensor, which in the present case equals to the energy density: eŽ x . s 12 wŽ E t AŽ x,t .. 2 q Ž Ex AŽ x,t .. 2 x. Let first consider the initial static case Ž t - 0.. The quantized field operator AŽ x,t - 0. is well known to be expanded as: A Ž x ,t - 0 . s Ý bn n
ž(
1
q0 v n
/
sin Ž v n x . eyi v n t q h.c.
Ž 4.
2 q02
ž
1q
2 exp Ž nrg . y 1
/
np s
2 q02
cth
n 2g Ž 5.
where eŽ v n . is the nth mode energy density operator. From Eqs. Ž3. and Ž5., we can also obtain the simple low-temperature expression of Casimir energy density:
Ž 2.
a n Ž a†n .
a
np s
p ec s y
24 q02
p q
q02
1
g 2q 4sh
2
Ž 6.
1
ž / 2g
where, in order to get a finite result, we have used the summation formula of Euler or Abel-Plana w9x which enables us to extract the infinite part of the total energy in a very simple way. Obviously, in the zero-temperature limits, we can restore the wellknown results for the attractiÕe interaction between the mirrors w9x. Now let us consider the more important thermodynamic case Ž t ) 0.. According to Law w7,8x, the cavity field A nŽ x,t ) 0. could be expanded with respect to the instantaneous basis, that is: A n Ž x ,t ) 0 . s Ý Q mŽ n. Ž t . m
(
q0 qŽ t.
sin
mp x qŽ t.
Ž 7.
where Q mŽ n. Ž t . is the instantaneous expanding coefficients. What is our most concerned is when the vibrating mirror comes to its initial position again Ž t G t .. In this condition, the field operator AŽ x,t G t . could also be expanded with respect to A nŽ x,t 0.. These two equivalent expressions for the field
J. Hui et al.r Physics Letters A 268 (2000) 174–177
176
operator yield the ‘‘new’’ physical operators through the dynamical Bogobiubov transformation w2x: c m s Ý Ž bn a n m q bn† bn)m . ,
In the similar way, the dynamical Casimir force at finite temperature could also be obtained as the following:
n
c†m s
Ý Ž bn† a n)m q bn bn m . .
Ž 8.
w ec x D s ec q
n 2
2
with the unitary condition: Ý mŽ< a n m < y < bn m < . s 1. The operators c m and c†m thus include both the two effects of the temperature and the motion of mirror. According to Ref. w4x, the exact solution of the Bogoliubov coefficients can be obtained by solving the Klein-Fok wave equation for the field operator AŽ x,t G t .. Since the method has been clearly explained by Dodonov and Klimov and already applied to the situation at ideal zero temperature w4x, we will use the results here without going into much detail. Now, if the initial state is the vacuum state with respect to the operator bn , the number of created photons for the principal mode Ž n s 1. can be easily written as the following: ^ N Ž t ,u T . ' Ý ²00˜ < c†m c m <00˜ : b y n 0 m
s
ž
/
Ý Ž Ž 1 q 2 Nn . < bn m < . n, m
s
½
cth Ž 1r2g . Ž ev t . r4
if ev t < 1
cth Ž 1r2g . Ž 4ev trp 2 q const .
if ev t 4 1
Ž 9. 1 Žconst s 2ln4 . p 2 y 2 here, in order to compare with the zero temperature case, we only consider the photons from the principle mode which, as pointed out by Dodonov and Klimov w4x, is the most concerned case in the present experimental conditions. From Eq. Ž9., we can easily see that the creation rate of photons depends on the cavity temperature and transforms from linear law with respect to the motion time in the short time limits Ž ev t < 1. to the constant value in the long time limits Ž ev t 4 1.. The similar phenomenon were also observed for the zero temperature case w4x. The thermal effect on the creation of photons is shown to provide a thermal factor: 1 q 2 N1 s cthŽ1r2g ., as we can see from Eq. Ž9..
q0
vÝ m
1 4 q0
Ý Ž 1 q 2 Nn . m < bn m < 2 n
cth Ž 1r2g . e sinh2 Ž ev t . .
Ž 10 .
where e c is just the value in the static case, as discussed above. In the last step in Eq. Ž10., we have used the zero-temperature results about the summation of Bogoliubov coefficients Žsee Ref. w4x.. From Eqs. Ž9. and Ž10., we can see that the cavity temperature T Žor g . and the motion time t are the two variables needed to describe the thermodynamic response of the intervening vacuum. At last, it is also interesting to investigate the squeezing effect of the cavity field at finite temperature. From Ref. w11x, we know that in order to obtain information about squeezing of the cavity field, we need to compute the variances of the field quadratures. The quadratures are defined by: Xˆ s
2
2
s ec q
1
jqj †
'2
,
Yˆs i
j †yj
'2
,
Ž 11 .
where j can be chosen as any one of the three kinds of operator: Ž1. a n , Ž2. bn , and Ž3. c m , representing the three different cases considered as above, respectively. The average values of Xˆ and Yˆ are zero in any case, as we could observe easily, but the variances require more additional calculations. Here, we only list our results as below: Ž1. sxŽ1. x s sy y s 1r2
Ž t - 0,g s 0 .
Ž2. sxŽ2. x s sy y s Nn q 1r2
Ž t - 0,g / 0 .
Ž 12 . Ž 13 .
2
† ² ˜<1 < ˜: sxŽ3. x s 00 2 Ž c m q c m . 00 s Ž Nn q 1r2 .
= 1 q 2 Ý < bn m < 2 q 2 Ý Re Ž a n m bn)m . n
n
Ž 14 .
J. Hui et al.r Physics Letters A 268 (2000) 174–177 2
² ˜<1 † < ˜: syŽ3. y s y 00 2 Ž c m y c m . 00 s Ž Nn q 1r2 . = 1 q 2 Ý < bn m < 2 y 2 Ý Re Ž a n m bn)m . n
n
Ž 15 . Ž t G T,g / 0.. Obviously, in the zero temperature limits Žg s 0., we can restore the same results as that of the previous literature w2x. Eqs. Ž12. – Ž15. leads to the important conclusions: the squeezing effect still exists at finite temperature and it is independent of the cavity temperature. In other words, the squeezing happens just in the stage of Õacuum. For the parameter case, the squeezing coefficient could be written as: w2x K'
sxŽ3. x syŽ3. y
s 1 q 2pe trq0 .
177
the mechanic response of the thermo vacuum and find that the modified Casimir force at finite temperature is determined by two variables: the cavity temperature and the motion time; however, the squeezing effect is only related to the vacuum state and is independent of the cavity temperature. Of course, our investigations have only provided the beginning point for a new approach to the dynamical Casimir effect, there still exits a problem about the three-dimensional cavity and the more difficult problem about the dynamical Casimir effect in the general external electromagnetic field, including the chaotic field as well as the laser field, which may comprise the challenge for the further works in the future.
Ž 16 .
which means that the squeezing effect only depends on the motion time, but not on the cavity temperature. Summing up, we have studied the dynamical Casimir effect at finite temperature using the approach of thermal field theory, especially for the case of parameter resonance. As we can see from Eq. Ž9. that temperature effects even dominate the pure vacuum effect at very low temperature and, in consequence, have to be taken into account when analyzing the data in measurements of the number of created photons. In particular, at the room temperature Žabout 290 K. and for a cavity of the typical size 1 cm, the thermal factor: 1 q 2 N1 s cthŽ1r2g ., will be the order 10 3 which means that the number of creative photons in the present case will be three orders of magnitude larger than the pure vacuum case, which is just the strong enhancement effect pointed out most recently by Plunien et al. w5x. In addition, we also derived the generalized formula for
Acknowledgements H. J. is grateful to Dr. J. L. Chen for his helpful discussions.
References w1x V.V. Dononov, A.B. Klimov, D.E. Nikonov, J. Math. Phys. 34 Ž1993. 2742. w2x V.V. Dodonov, A.B. Klimov, V.I. Man’ko, Phys. Lett. A 149 Ž1990. 225. w3x R. Golestanian, M. Kardar, Phys. Rev. Lett. 78 Ž1997. 3421. w4x V.V. Dodonov, A.B. Klimov, Phys. Rev. A 53 Ž1996. 2664. w5x G. Plunien, R. Schuetzhold, G. Soff, Phys. Rev. Lett. 84 Ž2000. 1882. w6x H. Umezawa, Y. Yamanaka, Adv. Phys. 37 Ž1988. 531. w7x C.K. Law, Phys. Rev. Lett. 73 Ž1994. 1931. w8x C.K. Law, Phys. Rev. A 49 Ž1994. 433. w9x V.M. Mostepanenko, N.N. Trunov, Sov. Phys. Usp. 31 Ž1988. 965. w10x S. Hacyan, R. Jauregui, F. Soto, C. Villarreal, J. Phys. A 23 Ž1990. 2401. w11x V.V. Dodonov, M.A. Andreata, J. Phys. A 32 Ž1999. 6711.
Physics Letters A 268 Ž2000. 174–177 www.elsevier.nlrlocaterphysleta
Dynamical Casimir effect at finite temperature Jing Hui a,) , Shi Qing-Yun b, Wu Jian-Sheng a a
Theoretical Physics DiÕision, Nankai Institute of Mathematics, Nankai UniÕersity, Tianjin 300071, PR China b Department of Physics, Zhengzhou UniÕersity, Zhengzhou, Henan 450052, PR China Received 23 December 1999; received in revised form 28 February 2000; accepted 28 February 2000 Communicated by P.R. Holland
Abstract The dynamics of the Maxwell field in a cavity with movable perfectly reflecting boundaries is investigated at nonzero temperature, applying the techniques of the thermo field dynamics ŽTFD.. For a resonantly vibrating cavity, the thermal effect on the creation of photons and the temperature correction to the mechanic response of the intervening vacuum are all derived, based on the previous work at zero temperature by Dodonov and Klimov wPhys. Rev. A 53 Ž1996. 2664x. The strong amplification of the Casimir effect due to finite temperature, which was first predicted by Plunien et al. wPhys. Rev. Lett. 84 Ž2000. 1882x in response theory, is also observed. The squeezing effect at finite temperature is probed too. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv; 03.65.Sq
In recent years, the dynamical Casimir effect, describing the force and radiation from moving mirrors has garnered much attention w1–3x. In particular, the generation of photons by the parametric resonance has great promise to be observed by the experiments w4x. Most theoretical studies have focused on the ideal zero temperature course, however, the existence of finite temperature should be considered for any actual experiments. Recently, Plunien et al. presented a realistic calculation of thermal effects on quantum radiation within the framework of quantum field theory of time-dependent system at finite temperature w5x, predicting a strong enhancement of the dynamical Casimir effect at finite temperature.
)
Corresponding author. E-mail address: [email protected] ŽJ. Hui..
In this Letter, we present a thermo field dynamic ŽTFD. formalism, to calculate the number of created photons as well as the Casimir force in a resonantly vibrating cavity at finite temperature. We begin by introducing the important concept of the thermo vacuum w6x into the model of mono-dimensional cavity with which mirrors undergo the parametric resonance w7,8x. The modified Casimir energy density at non-zero temperature is given, as extending of an earlier literature w9x. Then, through the Bogoliubov transformation, we produce two kinds of quasi-photon operators characterizing the thermal and dynamical effects, respectively. The calculations leading to the final results are based on the previous work at zero temperature by Dodonov and Klimov w4x, in which the Bogoliubov coefficients were already given explicitly. In this method, we can not only restore the important results about the enhance-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 6 5 - 1
J. Hui et al.r Physics Letters A 268 (2000) 174–177
175
ment effect pointed out by Plunien et al., but also derive the modified Casimir force at finite temperature. In addition, the squeezing effect is also investigated, leading to a squeezing coefficient independent of the temperature. As is usual, we consider a cavity with ideal mirrors, one of which is fixed at the origin while the other is allowed to move according to the special parametric resonance law:
where v n s nprq0 . The photons of the thermal vacuum obey the Bose-Einstein distribution, that is: Nn ' ²00˜ < bn† bn <00˜ : s sinh2u s wexpŽ nrg . y 1xy1. Žg s q0 Trp , and T is the cavity temperature.. It is just this expression which determines the heating coefficient u in the heating operator. From Eq. Ž4., we can easily obtain the temperature-dependent nth mode energy density in agreement with Hacyan et al. w10x:
q Ž t . s q0 Ž 1 q e sin Ž 2 v t . .
e Ž v n . ' ²00˜ < e Ž v n . <00˜ :
Ž 1.
here v s prq0 and 0 F t F t . The temperature effect is introduced by the thermo vacuum state w6x: <0: T ' H Ž u .<00˜ :, where the vacuum state <00˜ : belongs to the double Hilbert space determined by the tilde conjugate, and the heating operator: H Ž u . s expwyu Ž a n a˜ n y a†n a˜†n .x provides the thermal Bogoliubov transformation: H † Ž u . a n H Ž u . s ua n q Õa˜†n , H † Ž u . a†n H Ž u . s ua†n q Õa˜ n ,
Ž u s cosh u ,Õ s sinh u . .
where is the annihilation Žcreation. operator for the nth mode of the cavity field. The new operators bn s ua n q Õa˜†n and bn† s ua†n q Õa˜ n are called the quasi-photon operators, which satisfy w bn ,bn† x s 1. Then, the Casimir energy density of the cavity field, which was introduced in Ref. w9x, should be modified as:
e c s lim ²00˜ < Gx x <00˜ :a y ²0 M 0˜ M < Gx x <0 M 0˜ M :a .
™0 ž
/
Ž 3.
Here, M represents the unperturbed Minkowski space, a represents the regularization procedure to dispose of the zero-point divergences. The operator Gx x is the xx-component of the Maxwell stress tensor, which in the present case equals to the energy density: eŽ x . s 12 wŽ E t AŽ x,t .. 2 q Ž Ex AŽ x,t .. 2 x. Let first consider the initial static case Ž t - 0.. The quantized field operator AŽ x,t - 0. is well known to be expanded as: A Ž x ,t - 0 . s Ý bn n
ž(
1
q0 v n
/
sin Ž v n x . eyi v n t q h.c.
Ž 4.
2 q02
ž
1q
2 exp Ž nrg . y 1
/
np s
2 q02
cth
n 2g Ž 5.
where eŽ v n . is the nth mode energy density operator. From Eqs. Ž3. and Ž5., we can also obtain the simple low-temperature expression of Casimir energy density:
Ž 2.
a n Ž a†n .
a
np s
p ec s y
24 q02
p q
q02
1
g 2q 4sh
2
Ž 6.
1
ž / 2g
where, in order to get a finite result, we have used the summation formula of Euler or Abel-Plana w9x which enables us to extract the infinite part of the total energy in a very simple way. Obviously, in the zero-temperature limits, we can restore the wellknown results for the attractiÕe interaction between the mirrors w9x. Now let us consider the more important thermodynamic case Ž t ) 0.. According to Law w7,8x, the cavity field A nŽ x,t ) 0. could be expanded with respect to the instantaneous basis, that is: A n Ž x ,t ) 0 . s Ý Q mŽ n. Ž t . m
(
q0 qŽ t.
sin
mp x qŽ t.
Ž 7.
where Q mŽ n. Ž t . is the instantaneous expanding coefficients. What is our most concerned is when the vibrating mirror comes to its initial position again Ž t G t .. In this condition, the field operator AŽ x,t G t . could also be expanded with respect to A nŽ x,t 0.. These two equivalent expressions for the field
J. Hui et al.r Physics Letters A 268 (2000) 174–177
176
operator yield the ‘‘new’’ physical operators through the dynamical Bogobiubov transformation w2x: c m s Ý Ž bn a n m q bn† bn)m . ,
In the similar way, the dynamical Casimir force at finite temperature could also be obtained as the following:
n
c†m s
Ý Ž bn† a n)m q bn bn m . .
Ž 8.
w ec x D s ec q
n 2
2
with the unitary condition: Ý mŽ< a n m < y < bn m < . s 1. The operators c m and c†m thus include both the two effects of the temperature and the motion of mirror. According to Ref. w4x, the exact solution of the Bogoliubov coefficients can be obtained by solving the Klein-Fok wave equation for the field operator AŽ x,t G t .. Since the method has been clearly explained by Dodonov and Klimov and already applied to the situation at ideal zero temperature w4x, we will use the results here without going into much detail. Now, if the initial state is the vacuum state with respect to the operator bn , the number of created photons for the principal mode Ž n s 1. can be easily written as the following: ^ N Ž t ,u T . ' Ý ²00˜ < c†m c m <00˜ : b y n 0 m
s
ž
/
Ý Ž Ž 1 q 2 Nn . < bn m < . n, m
s
½
cth Ž 1r2g . Ž ev t . r4
if ev t < 1
cth Ž 1r2g . Ž 4ev trp 2 q const .
if ev t 4 1
Ž 9. 1 Žconst s 2ln4 . p 2 y 2 here, in order to compare with the zero temperature case, we only consider the photons from the principle mode which, as pointed out by Dodonov and Klimov w4x, is the most concerned case in the present experimental conditions. From Eq. Ž9., we can easily see that the creation rate of photons depends on the cavity temperature and transforms from linear law with respect to the motion time in the short time limits Ž ev t < 1. to the constant value in the long time limits Ž ev t 4 1.. The similar phenomenon were also observed for the zero temperature case w4x. The thermal effect on the creation of photons is shown to provide a thermal factor: 1 q 2 N1 s cthŽ1r2g ., as we can see from Eq. Ž9..
q0
vÝ m
1 4 q0
Ý Ž 1 q 2 Nn . m < bn m < 2 n
cth Ž 1r2g . e sinh2 Ž ev t . .
Ž 10 .
where e c is just the value in the static case, as discussed above. In the last step in Eq. Ž10., we have used the zero-temperature results about the summation of Bogoliubov coefficients Žsee Ref. w4x.. From Eqs. Ž9. and Ž10., we can see that the cavity temperature T Žor g . and the motion time t are the two variables needed to describe the thermodynamic response of the intervening vacuum. At last, it is also interesting to investigate the squeezing effect of the cavity field at finite temperature. From Ref. w11x, we know that in order to obtain information about squeezing of the cavity field, we need to compute the variances of the field quadratures. The quadratures are defined by: Xˆ s
2
2
s ec q
1
jqj †
'2
,
Yˆs i
j †yj
'2
,
Ž 11 .
where j can be chosen as any one of the three kinds of operator: Ž1. a n , Ž2. bn , and Ž3. c m , representing the three different cases considered as above, respectively. The average values of Xˆ and Yˆ are zero in any case, as we could observe easily, but the variances require more additional calculations. Here, we only list our results as below: Ž1. sxŽ1. x s sy y s 1r2
Ž t - 0,g s 0 .
Ž2. sxŽ2. x s sy y s Nn q 1r2
Ž t - 0,g / 0 .
Ž 12 . Ž 13 .
2
† ² ˜<1 < ˜: sxŽ3. x s 00 2 Ž c m q c m . 00 s Ž Nn q 1r2 .
= 1 q 2 Ý < bn m < 2 q 2 Ý Re Ž a n m bn)m . n
n
Ž 14 .
J. Hui et al.r Physics Letters A 268 (2000) 174–177 2
² ˜<1 † < ˜: syŽ3. y s y 00 2 Ž c m y c m . 00 s Ž Nn q 1r2 . = 1 q 2 Ý < bn m < 2 y 2 Ý Re Ž a n m bn)m . n
n
Ž 15 . Ž t G T,g / 0.. Obviously, in the zero temperature limits Žg s 0., we can restore the same results as that of the previous literature w2x. Eqs. Ž12. – Ž15. leads to the important conclusions: the squeezing effect still exists at finite temperature and it is independent of the cavity temperature. In other words, the squeezing happens just in the stage of Õacuum. For the parameter case, the squeezing coefficient could be written as: w2x K'
sxŽ3. x syŽ3. y
s 1 q 2pe trq0 .
177
the mechanic response of the thermo vacuum and find that the modified Casimir force at finite temperature is determined by two variables: the cavity temperature and the motion time; however, the squeezing effect is only related to the vacuum state and is independent of the cavity temperature. Of course, our investigations have only provided the beginning point for a new approach to the dynamical Casimir effect, there still exits a problem about the three-dimensional cavity and the more difficult problem about the dynamical Casimir effect in the general external electromagnetic field, including the chaotic field as well as the laser field, which may comprise the challenge for the further works in the future.
Ž 16 .
which means that the squeezing effect only depends on the motion time, but not on the cavity temperature. Summing up, we have studied the dynamical Casimir effect at finite temperature using the approach of thermal field theory, especially for the case of parameter resonance. As we can see from Eq. Ž9. that temperature effects even dominate the pure vacuum effect at very low temperature and, in consequence, have to be taken into account when analyzing the data in measurements of the number of created photons. In particular, at the room temperature Žabout 290 K. and for a cavity of the typical size 1 cm, the thermal factor: 1 q 2 N1 s cthŽ1r2g ., will be the order 10 3 which means that the number of creative photons in the present case will be three orders of magnitude larger than the pure vacuum case, which is just the strong enhancement effect pointed out most recently by Plunien et al. w5x. In addition, we also derived the generalized formula for
Acknowledgements H. J. is grateful to Dr. J. L. Chen for his helpful discussions.
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