A Fundamental Lower Bound of Actuating Energy for Broadband Photon Switching
aa r X i v : . [ qu a n t - ph ] M a r A Fundamental Lower Bound ofActuating Energyfor Broadband Photon Switching
Masahiro Hotta
Department of Physics, Faculty of Science, Tohoku University,Sendai, 980-8578, [email protected]
Abstract
We derive a universal lower bound of actuating energy E s forbroadband photon switching by using an uncertainty relation betweentime and the negative energy density of quantum fields. We find thatbroadband photon switching between perfect reflection and perfecttransmission over a time t s should satisfy E s ≥ ~ πt s . Introduction
Considerable effort has been made to realize a network of long-distancequantum communication by photons. If the range of available wavelengths isvery broad, more information can be sent on a quantum communication net-work using encoding strategies than by monochromatic-beam schemes. Fornetwork applications, the technology of photon switching [1] (or similarly,switchable mirrors [2]) are expected to play an important role. From thetechnological point of view, amount of energy required to operate photonswitches should be as small as possible. The energy to actuate the switchdepends on the implementation method and, in principle, various kinds ofimplementation of photon switching may be considered. Then a nontriv-ial question arises. How can we reduce the actuating energy E s of photonswitching in a fundamental level ? In this paper, we consider switching de-vices, which take an actuating time t s , designed to switch between perfectreflection and perfect transmission for incident broadband photons and givea fundamental lower bound of E s . During the switching process, the switchbody emits undesired photons inevitably due to interactions between the de-vice and the electromagnetic field. The generation of photons by changingboundary conditions is a familiar mechanism in quantum field theory, whichcause, for example, the dynamical Casimir effect[3]. We evaluate the mini-mum work done by the switch, utilizing a gedanken experiment based on auncertainty relation between time and the negative energy of multi-particlestates. Of course, the actuating energy E s is bounded below by the work togenerate the undesired photons. Using the evaluation of the work, we showthat E s ≥ ~ πt s . It should be emphasized that the lower bound is satisfiedfor arbitrary broadband photon switching between perfect penetration andperfect reflection with actuating time t s . In this paper, we assume that thephoton beams propagate in vacuum and the bandwidth is so broad that ar-bitrary spatial configuration of electromagnetic field can be realized with agood precision. 1 Analysis of modes propagating along the xaxis
As well known, free electromagnetic field can be expanded as A µ = Z Z Z s ~ (2 π ) | k | X λ e µλ ( k ) (cid:2) a λ ( k ) e i ( kx −| k | t ) + a λ † ( k ) e − i ( kx −| k | t ) (cid:3) d k, where we set the light velocity equal to one, e µλ ( k ) is polarization vectorsand a λ ( k ) , a λ † ( k ) are annihilation and creation operators. In principle, allmodes can be coupled to and excited by photon switches during switchingoperations. The actuating energy E s of a photon switch is bounded belowby a sum of inevitable works for all modes to generate undesired excitations.Among the modes, let us concentrate on analysis of the modes which prop-agate along the x axis because we put the mirror boundary of the switch at x = 0 in the yz plane. Of course, E s is also bounded below by the work toproduce undesired photons of the x -axis modes. A beam propagating alongthe x axis with its crosssection B is described [4] by A µ ( x, t ) = 1 √ B X λ e µλ φ λ ( x, t ) . Here B is set large enough. The constant vector e µλ denotes polarizations ofphoton and λ takes two values of the polarization. The massless fields φ λ satisfy the equation of motion given by (cid:20) ∂ ∂t − ∂ ∂x (cid:21) φ λ ( t, x ) = 0 . (1)For later convenience, we suppress the superscript λ of φ λ . It should benoted that φ has a residual electromagnetic gauge symmetry defined by φ → φ + const. . The general solution of Eq. (1) is written as a sum of left-and right-moving components: φ ( x, t ) = φ + ( x + ) + φ − ( x − ), where φ + ( x + )denotes the left-moving field and φ − ( x − ) the right-moving field with light-cone coordinates x ± = t ± x . Note that the usual plane-wave modes give acomplete and orthogonal basis. In the left-moving field system, for example,2n arbitrary function F ( x + ) of x + can be uniquely expanded by use of themodes defined by u ω ( x + ) = r ~ πω e − iωx + , ( ω ≥
0) (2)as follows. F ( x + ) = Z ∞ dω (cid:2) F ω u ω ( x + ) + F ∗ ω u ∗ ω ( x + ) (cid:3) . The mode functions are orthogonal to each other in terms of the norm definedby ( f, g ) = i ~ Z ∞−∞ (cid:2) f ∗ ( x + ) ∂ + g ( x + ) − ∂ + f ∗ ( x + ) g ( x + ) (cid:3) dx + . You can check directly the orthonormality such that( u ω , u ω ′ ) = δ ( ω − ω ′ ) , (3)( u ∗ ω , u ω ′ ) = 0 , (4)( u ∗ ω , u ∗ ω ′ ) = δ ( ω − ω ′ ) . (5)Due to this fact, each field is expressed by the plane-wave expansion: φ d = Z ∞ dω r ~ πω h a dω e − iωx d + a d † ω e iωx d i , where d = + or − . The creation and annihilation operators a d † ω and a dω obey the standard commutation relations given by h a dω , a d ′ † ω ′ i = δ dd ′ δ ( ω − ω ′ )because of Eq.(3)-Eq.(5). By using the annihilation operators, the normalizedvacuum state | i is defined by a dω | i = 0. T µν of the quantum field is defined byadopting the normal operator ordering of a d † ω and a dω in the classical ex-pression of the tensor. It is remarkable that the quantum interference ef-fect between multi-particle states is able to suppress quantum fluctuationof the field and to yield negative energy density of the field [5]. For exam-ple, even though the classical energy flux (cid:2) ∂ + φ + ( x + ) (cid:3) of the left-movingfield is non-negative, the expectation value of the corresponding quantumflux operator T ++ ( x + ) =: ∂ + φ L ( x + ) ∂ + φ L ( x + ) : can be negative. In spite ofthe negative energy density, the expectation value of the total energy flux R ∞−∞ T ++ ( x + ) dx + for an arbitrary state remains non-negative because thetotal flux is given by R ∞ ~ ωa + † ω a + ω dω .By taking an arbitrary monotonically increasing C function f ( x )of x ∈ ( −∞ , ∞ )satisfying f ( ±∞ ) = ±∞ , a new set of mode functions v ω ( x ) = r ~ πω e − iωf ( x ) , ( ω ≥
0) (6)is obtained that can uniquely expand the field. Actually, we can show that anaribitary function g ( x ) can be uniquely expanded by the new mode functions.Let us define a function given by G ( x ) = g ( f − ( x )) , where f − is the inverse function of f . Then, by the Fourier transformation, G ( x ) is expressed as G ( x ) = Z ∞−∞ dk ˜ G ( k ) s ~ π | k | e − ikx , where ˜ G ( k ) is uniquely determined by g ( x ) as follows.˜ G ( k ) = r | k | π ~ Z ∞−∞ dxG ( x ) e ikx = r | k | π ~ Z ∞−∞ dxg ( f − ( x )) e ikx . Hence we obtain 4 ( x ) = Z ∞ dk ˜ G ( k ) r ~ πk e − ikx + Z −∞ dk ˜ G ( k ) s ~ π ( − k ) e − ikx = Z ∞ dω " ˜ G ( ω ) r ~ πω e − iωx + ˜ G ( − ω ) r ~ πω e iωx . Substitution of f ( x ) into G yields the following result which we really want. g ( x ) = G ( f ( x ))= Z ∞ dω " ˜ G ( ω ) r ~ πω e − iωf ( x ) + ˜ G ( − ω ) r ~ πω e iωf ( x ) = Z ∞ dω h ˜ G ( ω ) v ω ( x ) + ˜ G ( − ω ) v ∗ ω ( x ) i . The orthonormality in terms of the normal product can be also derivedstraightforwardly. For example, we can calculate ( v ω , v ω ′ ) and obtain a rightresult as follows. ( v ω , v ω ′ ) = i ~ Z ∞−∞ [ v ∗ ω ∂v ω ′ − ∂v ∗ ω v ω ′ ] dx = 12 π Z ∞−∞ e i ( ω − ω ′ ) f ( x ) dfdx dx = 12 π Z ∞−∞ e i ( ω − ω ′ ) x ′ dx ′ = δ ( ω − ω ′ ) . (7)In moving from the second line to the third line, we have used the coordinatetransformation x ′ = f ( x ). Similarly, it is proven that( v ∗ ω , v ω ′ ) = 0 , (8)( v ∗ ω , v ∗ ω ′ ) = δ ( ω − ω ′ ) . (9)For the left-moving field φ + , the new expansion is given by φ + (cid:0) x + (cid:1) = Z ∞ dω (cid:2) b + ω v ω ( x + ) + b + † ω v ∗ ω ( x + ) (cid:3) . (10)5ecause of Eq.(7)-Eq.(9), b + † ω , b + ω are new creation and annihilation operatorswhich satisfy h b + ω , b + † ω ′ i = δ ( ω − ω ′ ) and depend linearly on the operators a + † ω and a + ω . A normalized quantum state | Φ i defined by b + ω | Φ i = 0 is a squeezedstate. For an arbitrary | Φ i , the expectation value of the energy flux of φ + isevaluated as h Φ | T ++ (cid:0) x + (cid:1) | Φ i = − ~ π ... f ( x + )˙ f ( x + ) − ¨ f ( x + )˙ f ( x + ) ! , (11)where the dot means a derivative in terms of x + . The formula in Eq.(11) canbe given in several ways[5]. One simple way is the point splitting method[6] which is based on the relation: h Φ | T ++ (cid:0) x + (cid:1) | Φ i = lim δ → (cid:20) h Φ | ∂ + φ L ( x + + δ ) ∂ + φ L ( x + ) | Φ i−h | ∂ + φ L ( x + + δ ) ∂ + φ L ( x + ) | i (cid:21) . (12)By substituting the field expansion in Eq.(10) into the first term of the right-hand-side of Eq.(12), we obtain the result in Eq.(11).An important example of negative energy flux is generated by a mono-tonically increasing C function f ε ( x ) given by f ε ( x ) = Θ ( x i − x ) x + Θ ( x f − x ) Θ ( x − x i ) (cid:20) x i − √ ε + 1 √ ε − ε ( x − x i ) (cid:21) + Θ ( x − x f ) " ε ( √ ε − ε ( x f − x i )) ( x − x f ) + x i − √ ε + 1 √ ε − ε ( x f − x i ) , (13)where x i ≤ x f , Θ ( x ) is the step function and ε = (cid:16) π | E n | ~ (cid:17) is a nonnega-tive constant. For the squeezed state | Φ shock i corresponding to f ε ( x ), theexpectation value of the left-moving energy flux is computed as h Φ shock | T ++ ( x + ) | Φ shock i = − | E n | δ ( x + − x i ) + | E n | − π ~ | E n | l δ (cid:0) x + − x f (cid:1) , (14)6here l = x f − x i ( > − | E n | . Because R ∞−∞ h Φ shock | T ++ ( x ) | Φ shock i dx is positive, we obtain | E n | l ~ π − | E n | l ≥ . (15)Because the numerator is definitely positive, the denominator must be non-negative, which leads to the inequality | E n | ≤ ~ πl = ~ π ( x f − x i ) . (16)The length l is the arrival interval between two shock waves with negativeenergy and positive energy. Thus, Eq. (16) shows an uncertainty relationbetween time and the negative energy of the quantum field.In the above argument, one might worry about whether nonanalytic be-havior of the step function which appears in f ε ( x ) leads to a unphysicalresult or not. However there is no need to worry. In real situations, we canalways consider a smooth function f Λ ,ε ( x ) which depends on a physical cut-off parameter Λ and approaches f ε ( x ) when Λ becomes large. Because thebandwidth is assumed so broad that the cutoff Λ is big enough, the expressionof energy flux in Eq.(14) can be realized in a good precision. Next we mention a local fluctuation property of flux-vanishing states.The general form of the function f ( x ) which satisfies h Φ | T ++ | Φ i = 0 is givenby f ( x ) = c + dxa + bx , (17)where a, b, c, d are real constants. In a space–time region where the function f ( x ) takes the form of Eq. (17), there is no difference between the state | Φ i and the vacuum | i about quantum field fluctuation, which will be coupled7o and excited by the switch body in the later analysis. Due to the residualgauge symmetry, the fundamental many-point functions of the input statesare not h Φ | φ + ( x +1 ) · · · φ + ( x + N ) | Φ i , but are instead h Φ | ˙ φ + ( x +1 ) · · · ˙ φ + ( x + N ) | Φ i .First of all, it can be easily checked that both the one-point functions of ˙ φ + vanish: h Φ | ˙ φ + ( x +1 ) | Φ i = h | ˙ φ + ( x +1 ) | i = 0. The two-point function of | Φ i iscalculated as h Φ | ˙ φ + ( x +1 ) ˙ φ + ( x +2 ) | Φ i = − ~ π ˙ f ( x +1 ) ˙ f ( x +2 ) (cid:0) f ( x +1 ) − f ( x +2 ) (cid:1) . (18)Substituting eq(17) into eq(18), it is possible to show by explicit calculationsthat h Φ | ˙ φ + ( x +1 ) ˙ φ + ( x +2 ) | Φ i = h | ˙ φ + ( x +1 ) ˙ φ + ( x +2 ) | i . (19)Note that the many-point functions of in-field, which describe the initialconditions of the system, can be decomposed via the Wick’s theorem into asum of the products of the two-point functions of the state. Therefore, whenthe input state is | Φ i , all the local properties in a space–time region whereEq. (17) holds are equal to those of the vacuum input. As an instructiveexample, let us consider a situation in Figure 1. For the state | Φ i , localquantum fluctuation of a spacetime region R in which the energy flux vanishes( h Φ | T µν | Φ i = 0) are the same as that of the vacuum state | i . Hence anylocal event P which takes place in R evolves as if the initial state were thevacuum state | i (Figure 2), as far as the sequential events originated fromevent P happen in the region R . This property comes from the conformalsymmetry of the massless field and is very crucial to derive the lower boundof actuating energy of photon switching in the later discussion. We consider now the switching process. A perfect mirror boundary ofthe switch is located at x = 0 in the yz plane. The mirror body lies onthe left-hand side of the boundary. Until t = 0, the mirror reflects incidentleft-moving beams with arbitrary shapes arriving at x = 0 . From t = 0,8he mirror and its body gradually become transparent. During the switch-ing interval, the switch generates undesired photons, which energy gives alower bound of E s . Incident beams arriving at x = 0 after t = t s penetratecompletely and continue to propagate freely in the spatial region x ≤ φ in evolves freely and inthe infinite future the out-asymptotic field φ out evolves freely. In the pastbefore the mirror begins to be transparent, plane-wave mode functions of φ in are given by U inω ( t, x ) = Θ( x ) (cid:2) u ω ( x + ) − u ω ( x − ) (cid:3) , where u ω is defined by Eq.(2). Note that the mirror boundary condition issatisfied as follows. U inω ( t,
0) = 0 . In the past region, φ in is expanded as φ in ( t, x ) = Z ∞ dω (cid:2) A inω U ω + A in † ω U ∗ ω (cid:3) , where A inω , A in † ω are annihilation and creation operators of the in-particles.The in -vacuum state is defined by A inω | , in i = 0 . The initial state is | , in i for the switching process. In order to solve the scattering problem by themirror for two shock waves with negative and positive energy density, let usintroduce another set of mode functions, which satisfies the mirror boundarycondition. For an arbitrary monotonically increasing C function f ( x )of x ∈ ( −∞ , ∞ )satisfying f ( ±∞ ) = ±∞ , the mode functions are given by V inω ( t, x ) = Θ( x ) (cid:2) v ω ( x + ) − v ω ( x − ) (cid:3) , (20)where v ω is given by Eq.(6) and V inω ( t,
0) = 0. It is possible to expand φ in as φ in ( t, x ) = Z ∞ dω (cid:2) B inω V ω + B in † ω V ∗ ω (cid:3) , A squeezed in-state | Φ , in i is defined by B inω | Φ , in i = 0 . If we take the samefunction f as Eq.(13), | Φ , in i gives the expectation value of the in-comingenergy flux in Eq.(14). If x i <
0, the negative flux in Eq.(14) is firstlyreflected to the right direction by the mirror and propagating freely. As seen9n Eq.(20), the mirror does not make any tail of the wavepacket and thenegative energy flux keeps its shock-wave shape. The switching process ofthe mirror after t = 0 does not affect the evolution of the wavepacket withnegative energy because the wave runs away at light velocity and causalityof the system prevent the switching event from disturbing the wavepacket’sevolution. Hence even in the remote future the negative flux propagates tothe right with its localized shape and is spatially separated from positive-energy right-moving excitations. Here a comment is added that if spatialsupport of the mirror is finite and given by [ x m , U ′ inω ( t, x ) = Θ( x m − x ) (cid:2) u ω ( x − − x m ) − u ω ( x + + x m ) (cid:3) , which satisfy the boundary conditions U ′ inω ( t, x m ) = 0. Then φ in is expandedas φ in ( t, x ) = Z ∞ dω (cid:2) A inω U ω + A in † ω U ∗ ω (cid:3) + Z ∞ dω (cid:2) A ′ inω U ′ ω + A ′ in † ω U ′∗ ω (cid:3) . The in-vacuum state is redefined by A inω | , in i = A ′ inω | , in i = 0, and thesqueezed in-state | Φ , in i is redefined by B inω | Φ , in i = A ′ inω | Φ , in i = 0.In the remote future region after the mirror has been completely removed,two sets of free plane-wave mode functions expands the out-asymptotic field φ out as follows. φ out ( t, x ) = φ out + (cid:0) x + (cid:1) + φ out − (cid:0) x − (cid:1) = Z ∞ dω (cid:2) A + outω u ω ( x + ) + A + out † ω u ∗ ω ( x + ) (cid:3) + Z ∞ dω (cid:2) A − outω u ω ( x − ) + A − out † ω u ∗ ω ( x − ) (cid:3) , where A ± out † ω and A ± outω are creation and annihilation operators of the out-particles. The out-vacuum state is defined by A ± outω | , out i = 0 . It shouldbe stressed that the energy-momentum tensor operators is normal-orderedso as to its expectational value for | , out i vanishes. Hence, the state | i inEq.(12) is regarded as | , out i . 10et us consider a gedanken experiment in order to evaluate the inevitablework for the switch to produce undesired photons. Instead of the vacuumstate | in, i , we take as an input quantum state the squeezed state | Φ , in i which satisfies Eq. (14) with x i ≤ x f ≥ t s . The negative flux inEq. (14) reaches x = 0 when t = x i and is perfectly reflected. From t = x i + 0 to t = 0 , no input or output beams reach the switch body. From t = 0, the mirror boundary and its body gradually become transparent. Theswitch generates undesired photons, which energy E Φ gives a lower bound ofactuating energy E s, Φ of the switch in the case. When the positive flux inEq. (14) reaches x = 0 at t = x f ( ≥ t s ), the mirror has been already removedat the point and the flux propagates freely to the left. After t = t s , theright-moving flux of the field is described as h Φ , in | T −− ( x − ) | Φ , in i = − | E n | δ ( x − − x i )+Θ (cid:0) x − (cid:1) Θ (cid:0) t s − x − (cid:1) T s ( x − ) , (21)where T s ( x − ) denotes the undesired flux induced by the switching. For thetotal energy of the undesired flux E Φ = R t s T s ( x − ) dx − , the following in-equality holds: E Φ ≥ | E n | − π ~ | E n | | x i | , (22)as will be proven later. Because no input or output energy flux reaches themirror body from t = x i + 0 to t = 0 and Eq. (17) is satisfied around themirror body during the interval | x i | , the conformal asymmetry argumentmentioned above can apply. We are able to regard production and evolutionof the undesired photons as event P and its sequential events of R in Figure1. Consequently, the actuating energy E s, Φ of the switch is independent of | Φ , in i and exactly equal to the actuating energy E s of the switch withthe input vacuum state | , in i . Therefore E s , which we are interested in, islower-bounded by the undesired photon energy E Φ of the state | Φ , in i . Itshould be noted that the inequality of Eq. (22) is satisfied by all possibleinput fluxes in Eq. (14). Therefore, the inequality E s ≥ max { | E n | , x f , x i } | E n | − π ~ | E n | | x i | (23)must hold. Here x f , x i and | E n | are constants which appear in Eq. (14). Forfixed x f and | E n | , maximizing | E n | − π ~ | E n || x i | is achieved by maximizing | x i | .11rom the uncertainty relation in Eq. (16), | x i | is bounded above as | x i | = | x f − x i | − | x f | ≤ ~ π | E n | − | x f | . (24)Thus the maximum of | x i | is ~ π | E n | − | x f | . Substituting the maximum valueinto | E n | − π ~ | E n || x i | , we obtain E s ≥ max { x f , | E n | } ~ π | x f | . (25)Because ~ π | x f | does not depend on | E n | , the inequality becomes E s ≥ max x f ~ π | x f | . (26)Note that the minimum of | x f | is t s . Hence we obtain a result, ignoringphoton polarization: E s ≥ ~ πt s . (27)In the real situations, photon beams have two massless fields correspondingto two polarizations. Hence, the lower bound of E s should be doubled: E s ≥ ~ πt s . (28)This result is our main result. The bound holds for the inverse operation toswitch perfect transmission to perfect reflection. Because the photon switchcan be coupled to other electromagnetic modes propagating to directionsdifferent from the x axis, the switch may generate undesired photons inthe other modes. We have also neglected the contribution of undesired left-moving excitations which might be generated during the switching. However,even if these additional contributions exist, the bound of Eq. (28) remainsvalid because the contributions just increase E s . Therefore we can concludethat the bound in Eq.(28) is universal.12 Inevitable work to produce undesired pho-tons
We now consider the inequality in Eq. (22). It is enough to discuss theout- asymptotic region in the remote future, where the negative flux andadditional flux produced by the mirror switch make free evolution. Thus noneed to take account of any boundary conditions in the whole space ( −∞ , ∞ ).This inequality comes from a fact that if we are given a negative-energy fluxin some spatial region, there must exist positive-energy flux in other regionsin order to guarantee the positivity of total energy in the whole space. Letus take an arbitrary out-state | Ψ , out i of φ − ( x − ) which satisfies h Ψ , out | T −− ( x − ) | Ψ , out i = − | E n | δ ( x − ) (29)for −∞ < x − < L . Here, the positive constant | E n | is fixed independent of | Ψ , out i . The states | Ψ , out i are not restricted to in-squeezed states. Eq.(29) can be expressed using x − ∈ ( −∞ , ∞ ) as h Ψ , out | T −− ( x − ) | Ψ , out i = − | E n | δ ( x − ) + Θ (cid:0) x − − L (cid:1) T s ( x − − L ) , (30)where T s ( x − − L ) depends on the details of the state | Ψ , out i and is con-strained such that E = R ∞−∞ h Ψ , out | T −− ( x ) | Ψ , out i dx is positive. We use theLagrange multiplier method to find a quantum state | Ψ , out i which minimizesthe total energy E with a constraint in Eq. (29). This requires minimizingthe quantity I , defined by I = Z ∞−∞ h Ψ , out | T −− ( x ) | Ψ , out i dx + Z L −∞ η ( x ) [ h Ψ , out | T −− ( x ) | Ψ , out i + | E n | δ ( x )] dx + λ [ h Ψ , out | Ψ , out i − , where η ( x ) and λ are multipliers. Because we do not need any conditionson the energy flux at x = L , we set η ( L ) = 0. From the variation of I interms of λ , we get the normalization condition of the state | Ψ , out i . Fromthe variation of I in terms of η ( x ), Eq. (29) is reproduced. The variation of I in terms of | Ψ , out i leads to the eigenvalue equation F | Ψ , out i = − λ | Ψ , out i , (31)13here F is a Hermitian operator defined by F = Z ∞−∞ [1 + η ( x )Θ ( L − x )] T −− ( x ) dx. (32)The operator F can be diagonalized by a method given by Flanagan [7]. Wedefine a C function f ( x ) such that f ( x ) = x (33)for x ∈ ( L, ∞ ) and f ( x ) = L − Z Lx du η ( u ) (34)for x ∈ ( −∞ , L ). Assuming a monotonic increase of f ( x ) and f ( −∞ ) = −∞ ,which will be justified later, we can expand the right-moving field φ − suchthat φ − (cid:0) x − (cid:1) = Z ∞ dω r ~ πω h b Rω e − iωf ( x − ) + b R † ω e iωf ( x − ) i , where b R † ω , b Rω are creation and annihilation operators. The operator F canbe rewritten [7] as F = Z ∞ ~ ωb R † ω b Rω dω − ~ π Z ∞−∞ (cid:16) ∂ x p η ( x )Θ ( L − x ) (cid:17) dx. (35)For a fixed η ( x ), the normalized eigenstate | vac, η i with the minimum eigen-value of F satisfies b Rω | vac, η i = 0 . This simplifies the problem because thestate | vac, η i is a squeezed state in the meaning mentioned above. Its gen-erating function is denoted by f η ( x ). We can find a function f η ( x ) whichgenerates plain-wave mode functions for both x > L ( >
0) and x < f η ( x ) = Θ( x − L ) x + Θ( L − x )Θ( x ) (cid:20) ρ ( L − x ) + ρ − ρ + L (cid:21) + Θ( − x ) (cid:20) x ( ρL + 1) + 1 ρ L + ρ − ρ + L (cid:21) , ρ is defined by ρ = | E n | ~ π −| E n | L . It is easy to check that thisfunction f η ( x ) is a monotonically increasing C function with f η ( ±∞ ) = ±∞ .The corresponding squeezed state | vac, η i takes a minimum of E . The energyflux of | vac, η i is calculated as h vac, η | T −− ( x − ) | vac, η i = − | E n | δ ( x − ) + | E n | − π ~ | E n | L δ (cid:0) x − − L (cid:1) . (36)By integrating the second term in the right-hand-side of Eq. (36), it is shownthat the minimum of E Φ is given by | E n | − π ~ | E n | L . By substituting L = | x i | , Eq.(22) is exactly derived. Acknowledgments
I would like to thank M. Ozawa, M. Morikawa and A. Shimizu for usefuldiscussions. This research was partially supported by the SCOPE project ofthe MIC.
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