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Friction and fluctuations produced by the quantum ground state
Physics Letters 161 (1991) 197-201
PHYSICS LETTERS A
North-Holland
Friction and fluctuations produced by the quantum ground state V.B. Braginsky a n d F.Ya. Khalili Department of Physics, Moscow State University, Moscow 119899, USSR
Received 10 October 1991;accepted for publication 31 October 1991 Communicatedby V.M. Agranovich
Fundamental friction produced by zero-point quantum oscillations is predicted. This friction may be enhanced in resonant conditions. The limits of the couplingof e.m. and mechanical oscillators due to zero-pointoscillations are defined.
I. Introduction
For more than forty years, the Casimir effect is known: there is a dc force of attraction between two plates produced by zero-point oscillations of an e.m. field [ 1 ]. This effect is analyzed in detail both theoretically and experimentally (see refs. [ 2-4 ] ). Recently Levitov [ 5 ] predicted the existence of a mechanical friction produced by zero-point oscillations when one plate with finite conductivity moves parallel to another one. The force of friction in this case is parallel to the plates. Also recently Barton [ 6 ] has shown that the vacuum e.m. pressure on an ideally conductive semispace must have a fluctuational component which is perpendicular to the surface. He obtained estimates for the variance of the force of pressure as a function of the averaging time. From these results, the following several problems can be logically derived: (i) Does there exist a friction associated with the abovementioned fluctuating force? (ii) Is it possible to measure this friction under real conditions? (iii) Can one provide conditions under which this friction is enhanced? (iv) Are there any additional effects connected with this friction and quantum ground states? The present article represents an attempt to solve these problems.
Elsevier SciencePublishers B.V.
2. Friction in the free space
Using the results obtained by Barton [6] it is not difficult to show (the calculations are omitted here) that the spectral density of the fluctuations of the pressure of e.m. field zero-point oscillations P~ acting on an ideally conductive surface is equal to ~/2to7 2 P , o - 240x2c 6 ,
(1)
where h is the Planck constant, to is the frequency of observation, c is the speed of light. This fluctuational force is pumping the energy from electromagnetic degrees of freedom into mechanical degrees of freedom. It is evident that in the equilibrium state (when all e.m. and mechanical modes of the continuum possess energy ½hto) the opposite flux of energy also exists. In other words, the motion of the plate must be associated with mechanical friction. The physical "mechanism" of this friction is evident: the e.m. mode with frequency 09 interacts with mechanical mode 209 (because the ac part of the ponderomotive force has frequency 209). This is the parametric pumping of the energy from mechanical oscillations into e.m. oscillations. It is important to note that this friction (it is appropriate to call it the friction in e.m. vacuum) is qualitively different from that in ref. [ 5 ]: the latter one appears when one plate with finite conductivity moves parallel to another plate. It is logical to establish a quantitative relation between the spectral density of the fluctuational force of the ponderomotive pressure F 2 produced by zero19 7
Volume 16 1, number 3
PHYSICSLETTERSA
30 December 199 1
point e.m. oscillations and the coefficient of friction H(09) on the basis of the fluctuation-dissipation theorem:
conditions is presented in the next section.
F 2 = 2h09H ( 09 ) .
3. Friction and fluctuations under resonance conditions
(2)
Suppose we have an ideally conductive plate with dimensions much larger than c/09 (09 is the frequency of observation). In this case one can neglect diffraction effects on plate edges. Averaging of the fluctuations ( 1 ) over the whole surface yields h2095S 2 F,o = 60~4C4 ,
(3)
where S is the plate area. Thus, if the surface of the plate moves as a whole in the transverse direction, then
h094S H(09) = 1207~4c4 .
(4)
It is important that H(09) is proportional to 094 and correspondingly the force of friction is proportional to the fifth time derivative of the coordinate. Thus there is no contradiction with the relativity principle: at a constant speed, the force of this friction is equal to zero. This friction is very small. To demonstrate this, let us calculate the quality factor for the lowest transverse mechanical oscillation mode of the plate which is determined by this friction. The frequency is 09m=nV/a, where v is the speed of sound, a is the thickness of the plate. The quality factor of this mode Qm is m09m 60npc 4a 4 Qm = 2H(09m ) by3
(5)
With a = l 0 - 4 cm, p = 4 g / c m 3, 9 = 10 6 cm/s, the quality factor is a m ~ 5 X 1037. This estimate indicates that, with the existing state of the art of elimination of other sources of dissipation, the effect of friction in free space is practically nonmeasurable. In this respect the discussed effect is evidently similar to the effect of the heating of an oscillator, which is moving with acceleration in the e.m. vacuum [ 7,8 ]. It is not possible to directly detect both these effects. But the mechanical friction, produced by the interaction with the e.m. vacuum, may be substantially enhanced and thus possibly may be measured, if one provides resonance conditions. The analyses of these 198
Let us analyze the quantum ground state in an electromagnetic resonator (resonance circuit) formed by a localized inductance L and localized capacity C. Let us suppose that the coupling of this resonator to the heat-bath is small, thus the quality factor of the resonator Qe >> 1. In this case the density of the heat-bath modes within the band A09e= 09e/Qe << 09e= ( L C ) -1/2 may be considered as a constant. Therefore one can suppose that the heat-bath is a semi-infinite e.m. line (wave guide) with impedance rl= Q g ~( L / C ) 1 / 2 . Assuming that the temperature, T, of the resonator and of the line is zero, one can write down the equation for the operator of electrical charge in the capacity in the following form: d20(t)
L --dT
d0(t)
1
+ ,7 --d?- + ~ O( t )
=
6(09)e-i°Jt dw+h.c. ,
(6)
o where h.c. stands for Hermitian conjugate, 6(09) is the annihilation operator in the line which satisfies the following commutation relation:
[a(09), 6 + (09') ] =,~(09-09').
(7)
(6 + is the creation operator.) The solution of eq. (6) has the following form:
q(t)=z
1 i (h09,~ 1/2 6(09) e_i~,,d09+h.c. \ x )' z(w)
0
z(09) =09e2 _092_ i09w.
Qe "
(8)
The mechanical force which attracts the plates of the capacity is P(t)-
q2(t) 2Cd '
(9)
where d is the distance between the plates. The mean value o f F ( t ) is
Volume 161, number 3
PHYSICSLETTERSA co
hco~3 f
Fo== 2nQ=dj°
codco Iz(co)l 2 ,
(10)
and if Qe >> l, then hcoe Fo-~ 4 d "
(11)
It is important to note that this force is an addition to the Casimir force, which appears due to the coupling of the inductance L to the plates. The correlation function of the ac component of the force AJO(t) = F ( t ) - F0 is
B(t,t')=2(~e~d) 2 ×
i cos [ (co+ co') ( t ' - t) ]coco' dco dco' iz(co) 121z(co,) 12 ,
(12)
0
1 (hco~]2~ co'(co-co') dco' F2=~\Q~d]
d° [z(co)12lz(co')l 2"
(13)
If Qe >> 1, then _
physical meaning. Since the ponderomotive force is proportional to the square of the sum of amplitudes, the force acting on the plate is created by the beating of modes with close frequencies. To calculate the dynamical action of zero-point e.m( oscillations in the resonator on the mechanical degree of freedom (including the introduced friction) let us suppose that one of the plates in the capacity is movable and that its position is described by the coordinate x(t). Then the value q(t) described by formula (8) is only the zero-order approximation for small x(t)/d. For x(t)/d¢O it is necessary to modify eq. (6) by adding the factor 1-x(t)/d to the third term in the equation. Since we need only the additional force FI (t) linearly depending on x(t), in the first approximation the additional charge ql (t) has to satisfy the equation
Ld2(h(t) + q d0~(t) dt--------T -
and the spectral density of fluctuations of this force correspondingly:
F2
30 December 1991
h2co~Qe 4d 2
(14)
near the frequency 2coe. The characteristic time of correlation of the fluctuations of AF(t) is equal to QJcoe. It is evident that for a clystron-type resonator (with well-defined localized inductance and capacitance) and (for example) co~---l0 ~° s -~, Q,-~ l0 ~°, d = 10-4 cm, the spectral density of this force is many orders larger than for a plate in free space (see formula (3) ). In the example under discussion it is possible to qualitively describe the origin of the fluctuational force in the following way. In the system "resonator + lithe" each e.m. mode has strictly the energy ½hco, which has one part in the resonator and another part in the line. From the whole continuum of the modes the resonator "chooses" only those whose frequencies are mainly in the bandwidth Aco,~-co~/Qe. The averaged energy in the capacity turns out to be close to ]hco~. The phase of each mode is not defined, but the difference of the phase between the modes has a
T
1 (lo(t)Yc(t) + ~0~(t)= ~ ,
(15)
and the value F~ (t) of interest is equal to ~¢,(t) = ~
1
[Oo(t):h(t)+~,(t)Oo(t)] •
(16)
Solving eq. ( 15 ) and substituting 0~ (t) into formula (16) we may obtain the relation between F~ (t) and x(t), which it is appropriate to represent in spectral form:
P, (co) = z ( c o ) x ( c o ) , where P, (co) and ~'(co) are the spectra ofP~ (t) and ~(t), and oo
Z(CO)- 2nQed 2 Jo z*(co-co') + z*(co-+co') to' do)' X iz(co,) 12
(17)
is the generalized susceptibility, produced by the dynamical action of zero-point oscillations, and z* is the value conjugated with z. The imaginary part of the generalized susceptibility is equal to
hco~
Img(co)=
to
f
co'(co--co') dco'
2nQEdEJo iz(co)lEiz(co,)l 2.
(18)
From formulas ( 18 ) and (14) one can see that 199
Volume 161, n u m b e r 3
P H Y S I C S LETTERS A
2 F~,=2h Ilmz(09) I ,
in complete correspondence with the fluctuationdissipation theorem. From the structure of the integral ( 18 ) one can see that 2(09) has a sharp maximum near 209e. If Q~ >> 1 and 1o9- 209e I << 09~, then
hw2e 09- 209e + i09e/Qe Z(09)~- - 8d 2 (09_209~)2 + (09jQ~) 2 .
(19)
If the movable plate has mass m and is attached to the rigidity m09 2 and 09m=209e, and the intrinsic quality factor of this mechanical oscillator (Ore)intrinsic >> a e (in other words, if the mechanical heat-bath is more weakly coupled to the mechanical oscillator than the e.m. resonator coupled to the e.m. heat bath), (20)
8d 2
in the bandwidth Am ~ 09~/Q~ near the frequency 09m. This friction corresponds to the loaded quality factor (Qm)loaded of the mechanical oscillator determined by its coupling to the e.m. heat bath under resonant conditions, ( Qm ) Ioadecl "~
m09 2m 21Im Z(09~) I
16m09md 2 --
hQ~
(21)
(if (Qm)~oaded>> Q~). Suppose next that we put a dielectric plate with square area S and thickness d between the plates of the capacitor C and the eigenfrequency 09m= Kv/d of the lowest transverse mechanical mode of this plate is equal to 209,. Then the loaded quality factor of this mode will be approximately equal to
4xzYV
( Qm)l°aded '~ h09,Qe'
(22)
where Y=vZp is the Young modulus; V=Sd is the volume of the dielectric. For S = 10-3 cm 2, d = 10 -4 cm, 09,= 10 1° s -l, Qe=10 11 the value (Qm)loaa,d= 1.6 × 10 13. This is not very far from the recently obtained values of the mechanical quality factor. It is necessary to emphasize that formula (22) is valid at T~=0. If T~>0, then (am)loaded can be shown to be 2 (nT + ½) times smaller (nT is the mean number of thermal quanta). 200
In the case when the losses of the e.m. resonator are negligible and thus the coupling of the resonator with the heat bath is realized through the mechanical oscillator with finite Q~ only, the product (Qe)loadedQm is also described by formula (22). Another dynamical effect produced by zero-point oscillations in the described example appears if the 09m is close to 209e, but 09m is not equal exactly to 209~: Wm--209e=~', 171 <<09,,. In addition to the e.m. friction, an e.m. rigidity k~.m. can be observed:
h092
Y
k .... = - R e z ( 0 9 ) -~ 8d 2 y2+(09~/Qe)2.
(23)
If ~=09~/Q~, then the value ke.m. is maximal and the corresponding shift of the resonant frequency of the mechanical oscillator is equal to ke.m. m09m/09 m =
ih09eQ~ X ( O ) ) --
30 December 1991
2m092m
hQ~
h09~Q~
- 64md209m - 16x2yv.
(24)
The sign of the frequency shift is defined by the sign of ~,. For the numerical example described above with a dielectric plate A09 m/(.D m ~---1.5 X 1 0 - 14.
4. Conclusion
As follows from the above discussion, the friction caused by the zero-point e.m. fluctuations typically does not impose limitations for the increase of the quality factor of oscillators up to very high values of Q. Meanwhile, under resonance conditions the effects considered in this paper are not too small and can probably be measurable in future. It is also appropriate to note that the obtained expressions for (Qm)loaded and similar ones include only the Planck constant h and (through the boundary conditions) the volume, frequency and Young modulus. The Young modulus in turn is determined only by h, the charge, the electron mass and dimensionless parameters characterizing the structure of the atomic cell. And this is what distinguishes the obtained limits from those based on semi-classical models.
Volume 161, number 3
PHYSICS LETTERS A
References [ 1 ] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51 ( 1948 ) 793. [2] L.D. Landau and E.M. Lifshits, Theoretical physics, Vol. 9 (Moscow, 1983). [ 3 ] Yu.S. Barash and V.L. Ginzburg, Usp. Fiz. Nauk I 16 ( 1975 ) 5.
30 December 1991
[4] V.M. Mostepanenko and N.N. Trunov, Usp. Fiz. Nauk 156 (1988) 385. [5] L.S. Levitov, Europhys. Lett. 8 (1989) 499. [6] G. Barton, J. Phys. A 24 ( 1991 ) 991. [7] W.G. Unruh, Phys. Rev. D 14 (1976) 870. [8] D.N. Klyshko, Phys. Lett. A 154 ( 1991 ) 440.
201
PHYSICS LETTERS A
North-Holland
Friction and fluctuations produced by the quantum ground state V.B. Braginsky a n d F.Ya. Khalili Department of Physics, Moscow State University, Moscow 119899, USSR
Received 10 October 1991;accepted for publication 31 October 1991 Communicatedby V.M. Agranovich
Fundamental friction produced by zero-point quantum oscillations is predicted. This friction may be enhanced in resonant conditions. The limits of the couplingof e.m. and mechanical oscillators due to zero-pointoscillations are defined.
I. Introduction
For more than forty years, the Casimir effect is known: there is a dc force of attraction between two plates produced by zero-point oscillations of an e.m. field [ 1 ]. This effect is analyzed in detail both theoretically and experimentally (see refs. [ 2-4 ] ). Recently Levitov [ 5 ] predicted the existence of a mechanical friction produced by zero-point oscillations when one plate with finite conductivity moves parallel to another one. The force of friction in this case is parallel to the plates. Also recently Barton [ 6 ] has shown that the vacuum e.m. pressure on an ideally conductive semispace must have a fluctuational component which is perpendicular to the surface. He obtained estimates for the variance of the force of pressure as a function of the averaging time. From these results, the following several problems can be logically derived: (i) Does there exist a friction associated with the abovementioned fluctuating force? (ii) Is it possible to measure this friction under real conditions? (iii) Can one provide conditions under which this friction is enhanced? (iv) Are there any additional effects connected with this friction and quantum ground states? The present article represents an attempt to solve these problems.
Elsevier SciencePublishers B.V.
2. Friction in the free space
Using the results obtained by Barton [6] it is not difficult to show (the calculations are omitted here) that the spectral density of the fluctuations of the pressure of e.m. field zero-point oscillations P~ acting on an ideally conductive surface is equal to ~/2to7 2 P , o - 240x2c 6 ,
(1)
where h is the Planck constant, to is the frequency of observation, c is the speed of light. This fluctuational force is pumping the energy from electromagnetic degrees of freedom into mechanical degrees of freedom. It is evident that in the equilibrium state (when all e.m. and mechanical modes of the continuum possess energy ½hto) the opposite flux of energy also exists. In other words, the motion of the plate must be associated with mechanical friction. The physical "mechanism" of this friction is evident: the e.m. mode with frequency 09 interacts with mechanical mode 209 (because the ac part of the ponderomotive force has frequency 209). This is the parametric pumping of the energy from mechanical oscillations into e.m. oscillations. It is important to note that this friction (it is appropriate to call it the friction in e.m. vacuum) is qualitively different from that in ref. [ 5 ]: the latter one appears when one plate with finite conductivity moves parallel to another plate. It is logical to establish a quantitative relation between the spectral density of the fluctuational force of the ponderomotive pressure F 2 produced by zero19 7
Volume 16 1, number 3
PHYSICSLETTERSA
30 December 199 1
point e.m. oscillations and the coefficient of friction H(09) on the basis of the fluctuation-dissipation theorem:
conditions is presented in the next section.
F 2 = 2h09H ( 09 ) .
3. Friction and fluctuations under resonance conditions
(2)
Suppose we have an ideally conductive plate with dimensions much larger than c/09 (09 is the frequency of observation). In this case one can neglect diffraction effects on plate edges. Averaging of the fluctuations ( 1 ) over the whole surface yields h2095S 2 F,o = 60~4C4 ,
(3)
where S is the plate area. Thus, if the surface of the plate moves as a whole in the transverse direction, then
h094S H(09) = 1207~4c4 .
(4)
It is important that H(09) is proportional to 094 and correspondingly the force of friction is proportional to the fifth time derivative of the coordinate. Thus there is no contradiction with the relativity principle: at a constant speed, the force of this friction is equal to zero. This friction is very small. To demonstrate this, let us calculate the quality factor for the lowest transverse mechanical oscillation mode of the plate which is determined by this friction. The frequency is 09m=nV/a, where v is the speed of sound, a is the thickness of the plate. The quality factor of this mode Qm is m09m 60npc 4a 4 Qm = 2H(09m ) by3
(5)
With a = l 0 - 4 cm, p = 4 g / c m 3, 9 = 10 6 cm/s, the quality factor is a m ~ 5 X 1037. This estimate indicates that, with the existing state of the art of elimination of other sources of dissipation, the effect of friction in free space is practically nonmeasurable. In this respect the discussed effect is evidently similar to the effect of the heating of an oscillator, which is moving with acceleration in the e.m. vacuum [ 7,8 ]. It is not possible to directly detect both these effects. But the mechanical friction, produced by the interaction with the e.m. vacuum, may be substantially enhanced and thus possibly may be measured, if one provides resonance conditions. The analyses of these 198
Let us analyze the quantum ground state in an electromagnetic resonator (resonance circuit) formed by a localized inductance L and localized capacity C. Let us suppose that the coupling of this resonator to the heat-bath is small, thus the quality factor of the resonator Qe >> 1. In this case the density of the heat-bath modes within the band A09e= 09e/Qe << 09e= ( L C ) -1/2 may be considered as a constant. Therefore one can suppose that the heat-bath is a semi-infinite e.m. line (wave guide) with impedance rl= Q g ~( L / C ) 1 / 2 . Assuming that the temperature, T, of the resonator and of the line is zero, one can write down the equation for the operator of electrical charge in the capacity in the following form: d20(t)
L --dT
d0(t)
1
+ ,7 --d?- + ~ O( t )
=
6(09)e-i°Jt dw+h.c. ,
(6)
o where h.c. stands for Hermitian conjugate, 6(09) is the annihilation operator in the line which satisfies the following commutation relation:
[a(09), 6 + (09') ] =,~(09-09').
(7)
(6 + is the creation operator.) The solution of eq. (6) has the following form:
q(t)=z
1 i (h09,~ 1/2 6(09) e_i~,,d09+h.c. \ x )' z(w)
0
z(09) =09e2 _092_ i09w.
Qe "
(8)
The mechanical force which attracts the plates of the capacity is P(t)-
q2(t) 2Cd '
(9)
where d is the distance between the plates. The mean value o f F ( t ) is
Volume 161, number 3
PHYSICSLETTERSA co
hco~3 f
Fo=
codco Iz(co)l 2 ,
(10)
and if Qe >> l, then hcoe Fo-~ 4 d "
(11)
It is important to note that this force is an addition to the Casimir force, which appears due to the coupling of the inductance L to the plates. The correlation function of the ac component of the force AJO(t) = F ( t ) - F0 is
B(t,t')=2(~e~d) 2 ×
i cos [ (co+ co') ( t ' - t) ]coco' dco dco' iz(co) 121z(co,) 12 ,
(12)
0
1 (hco~]2~ co'(co-co') dco' F2=~\Q~d]
d° [z(co)12lz(co')l 2"
(13)
If Qe >> 1, then _
physical meaning. Since the ponderomotive force is proportional to the square of the sum of amplitudes, the force acting on the plate is created by the beating of modes with close frequencies. To calculate the dynamical action of zero-point e.m( oscillations in the resonator on the mechanical degree of freedom (including the introduced friction) let us suppose that one of the plates in the capacity is movable and that its position is described by the coordinate x(t). Then the value q(t) described by formula (8) is only the zero-order approximation for small x(t)/d. For x(t)/d¢O it is necessary to modify eq. (6) by adding the factor 1-x(t)/d to the third term in the equation. Since we need only the additional force FI (t) linearly depending on x(t), in the first approximation the additional charge ql (t) has to satisfy the equation
Ld2(h(t) + q d0~(t) dt--------T -
and the spectral density of fluctuations of this force correspondingly:
F2
30 December 1991
h2co~Qe 4d 2
(14)
near the frequency 2coe. The characteristic time of correlation of the fluctuations of AF(t) is equal to QJcoe. It is evident that for a clystron-type resonator (with well-defined localized inductance and capacitance) and (for example) co~---l0 ~° s -~, Q,-~ l0 ~°, d = 10-4 cm, the spectral density of this force is many orders larger than for a plate in free space (see formula (3) ). In the example under discussion it is possible to qualitively describe the origin of the fluctuational force in the following way. In the system "resonator + lithe" each e.m. mode has strictly the energy ½hco, which has one part in the resonator and another part in the line. From the whole continuum of the modes the resonator "chooses" only those whose frequencies are mainly in the bandwidth Aco,~-co~/Qe. The averaged energy in the capacity turns out to be close to ]hco~. The phase of each mode is not defined, but the difference of the phase between the modes has a
T
1 (lo(t)Yc(t) + ~0~(t)= ~ ,
(15)
and the value F~ (t) of interest is equal to ~¢,(t) = ~
1
[Oo(t):h(t)+~,(t)Oo(t)] •
(16)
Solving eq. ( 15 ) and substituting 0~ (t) into formula (16) we may obtain the relation between F~ (t) and x(t), which it is appropriate to represent in spectral form:
P, (co) = z ( c o ) x ( c o ) , where P, (co) and ~'(co) are the spectra ofP~ (t) and ~(t), and oo
Z(CO)- 2nQed 2 Jo z*(co-co') + z*(co-+co') to' do)' X iz(co,) 12
(17)
is the generalized susceptibility, produced by the dynamical action of zero-point oscillations, and z* is the value conjugated with z. The imaginary part of the generalized susceptibility is equal to
hco~
Img(co)=
to
f
co'(co--co') dco'
2nQEdEJo iz(co)lEiz(co,)l 2.
(18)
From formulas ( 18 ) and (14) one can see that 199
Volume 161, n u m b e r 3
P H Y S I C S LETTERS A
2 F~,=2h Ilmz(09) I ,
in complete correspondence with the fluctuationdissipation theorem. From the structure of the integral ( 18 ) one can see that 2(09) has a sharp maximum near 209e. If Q~ >> 1 and 1o9- 209e I << 09~, then
hw2e 09- 209e + i09e/Qe Z(09)~- - 8d 2 (09_209~)2 + (09jQ~) 2 .
(19)
If the movable plate has mass m and is attached to the rigidity m09 2 and 09m=209e, and the intrinsic quality factor of this mechanical oscillator (Ore)intrinsic >> a e (in other words, if the mechanical heat-bath is more weakly coupled to the mechanical oscillator than the e.m. resonator coupled to the e.m. heat bath), (20)
8d 2
in the bandwidth Am ~ 09~/Q~ near the frequency 09m. This friction corresponds to the loaded quality factor (Qm)loaded of the mechanical oscillator determined by its coupling to the e.m. heat bath under resonant conditions, ( Qm ) Ioadecl "~
m09 2m 21Im Z(09~) I
16m09md 2 --
hQ~
(21)
(if (Qm)~oaded>> Q~). Suppose next that we put a dielectric plate with square area S and thickness d between the plates of the capacitor C and the eigenfrequency 09m= Kv/d of the lowest transverse mechanical mode of this plate is equal to 209,. Then the loaded quality factor of this mode will be approximately equal to
4xzYV
( Qm)l°aded '~ h09,Qe'
(22)
where Y=vZp is the Young modulus; V=Sd is the volume of the dielectric. For S = 10-3 cm 2, d = 10 -4 cm, 09,= 10 1° s -l, Qe=10 11 the value (Qm)loaa,d= 1.6 × 10 13. This is not very far from the recently obtained values of the mechanical quality factor. It is necessary to emphasize that formula (22) is valid at T~=0. If T~>0, then (am)loaded can be shown to be 2 (nT + ½) times smaller (nT is the mean number of thermal quanta). 200
In the case when the losses of the e.m. resonator are negligible and thus the coupling of the resonator with the heat bath is realized through the mechanical oscillator with finite Q~ only, the product (Qe)loadedQm is also described by formula (22). Another dynamical effect produced by zero-point oscillations in the described example appears if the 09m is close to 209e, but 09m is not equal exactly to 209~: Wm--209e=~', 171 <<09,,. In addition to the e.m. friction, an e.m. rigidity k~.m. can be observed:
h092
Y
k .... = - R e z ( 0 9 ) -~ 8d 2 y2+(09~/Qe)2.
(23)
If ~=09~/Q~, then the value ke.m. is maximal and the corresponding shift of the resonant frequency of the mechanical oscillator is equal to ke.m. m09m/09 m =
ih09eQ~ X ( O ) ) --
30 December 1991
2m092m
hQ~
h09~Q~
- 64md209m - 16x2yv.
(24)
The sign of the frequency shift is defined by the sign of ~,. For the numerical example described above with a dielectric plate A09 m/(.D m ~---1.5 X 1 0 - 14.
4. Conclusion
As follows from the above discussion, the friction caused by the zero-point e.m. fluctuations typically does not impose limitations for the increase of the quality factor of oscillators up to very high values of Q. Meanwhile, under resonance conditions the effects considered in this paper are not too small and can probably be measurable in future. It is also appropriate to note that the obtained expressions for (Qm)loaded and similar ones include only the Planck constant h and (through the boundary conditions) the volume, frequency and Young modulus. The Young modulus in turn is determined only by h, the charge, the electron mass and dimensionless parameters characterizing the structure of the atomic cell. And this is what distinguishes the obtained limits from those based on semi-classical models.
Volume 161, number 3
PHYSICS LETTERS A
References [ 1 ] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51 ( 1948 ) 793. [2] L.D. Landau and E.M. Lifshits, Theoretical physics, Vol. 9 (Moscow, 1983). [ 3 ] Yu.S. Barash and V.L. Ginzburg, Usp. Fiz. Nauk I 16 ( 1975 ) 5.
30 December 1991
[4] V.M. Mostepanenko and N.N. Trunov, Usp. Fiz. Nauk 156 (1988) 385. [5] L.S. Levitov, Europhys. Lett. 8 (1989) 499. [6] G. Barton, J. Phys. A 24 ( 1991 ) 991. [7] W.G. Unruh, Phys. Rev. D 14 (1976) 870. [8] D.N. Klyshko, Phys. Lett. A 154 ( 1991 ) 440.
201