Casimir-Polder force for a polarizable molecule near a dielectric substrate out of thermal equilibrium
aa r X i v : . [ qu a n t - ph ] M a y Casimir-Polder force for a polarizable molecule near a dielectricsubstrate out of thermal equilibrium
Wenting Zhou and Hongwei Yu , Center for Nonlinear Science and Department of Physics,Ningbo University, Ningbo, Zhejiang 315211, China Institute of Physics and Key Laboratory of Low DimensionalQuantum Structures and Quantum Control of Ministry of Education,Hunan Normal University, Changsha, Hunan 410081, China
Abstract
We demonstrate that the Casimir-Polder force for a molecule near the surface of a real dielectricsubstrate out of thermal equilibrium displays distinctive behaviors as compared to that at thermalequilibrium. In particular, when the molecule-substrate separation is much less than the moleculartransition wave-length, the CP force in the high temperature limit can be dramatically manipulatedby varying the relative magnitude of the temperatures of the substrate and the environment so thatthe attractive-to-repulsive transition can occur beyond certain a threshold temperature of eitherthe substrate or the environment depending on which one is higher for molecules both in the groundand excited states. More remarkably, when the separation is comparable to the wave-length, suchtransitions which are impossible at thermal equilibrium may happen for longitudinally polarizablemolecules with a small permittivity, while for isotropically polarizable ones the transitions can evenoccur at room temperature for some dielectric substrates such as sapphire and graphite which ismuch lower than the temperature for the transition to take place in the thermal equilibrium case,thus making the experimental demonstration of such force transitions easier.
PACS numbers: 31.30.jh, 12.20.-m, 34.35.+a, 42.50.Nn z − with z being the distance betweenthe atom and the wall, and it dominates over the contribution of thermal fluctuations, whileat distances much larger than the wavelength of thermal photons, z ≫ ¯ hck B T , the contributionof thermal fluctuations is proportional to T /z , which provides the leading contribution tothe total force.However, this behavior of the CP force changes when the situation comes to that ofout of thermal equilibrium [15–21]. In this regard, Antezza et al found that for an atomat very large distances from the surface of a half-space dielectric substrate, the CP forceexhibits, when the temperature is low, a new, stronger asymptotic behavior out of thermalequilibrium as compared to that in the equilibrium case [17, 18], which has been verifiedin experiment [19] , providing the first experimental observation of the thermal effect inthe CP force. Later, Dedkov and Kyasov extended the analysis to different nonequilibriumconfigurations [20] and Ellingsen et al showed that Antezza et al’s result can also be obtainedin the framework of the Keldysh Green function method [21]. This has spurred a great deal ofinterest in the Casimir effect out of thermal equilibrium [23–35] as well as other effects suchas manipulation of atomic populations [36, 37] and entanglement [38] in situations out ofthermal equilibrium. Recently, by generalizing the formalism proposed by Dalibard, Dupont- Let us note here that the thermal CP force at thermal equilibrium has also been experimentally observedrecently [22]. z → ∞ ) in [17, 18] by giving a concrete region where this behavior holds [41].In the present paper, we investigate the CP force of a typical polarizable molecule withlong-wavelength transitions near the surface of a dielectric substrate out of thermal equilib-rium. At this point, let us note that Ellingsen, et. al demonstrated that for a typical moleculewith long-wavelength transitions placed near a plane metal surface at thermal equilibrium,the CP potentials can be entirely independent of temperature even when the thermal pho-ton number is large [42], while we showed that when the molecular transition wavelengthsare comparable to the molecule-surface separation, the CP force can be dependent on theambient temperature and the molecular polarization, and it can even change from attractiveto repulsive at room temperature [43]. What we are particularly interested in here are thenew features coming from being out of thermal equilibrium with a real dielectric substrateas opposed to thermal equilibrium and a metal surface.We assume, for simplicity, that the dielectric substrate is non-dispersive and the moleculeis modeled by a two-level system with two stationary states represented by | + i and |−i ,and energy spacing between the two levels being ¯ hω . The left half-space with z < ǫ and temperature T s withthe surface of the substrate coinciding with the plane z = 0, and the right half-space with z > T e . Generally, thetemperatures of the substrate and the environment do not coincide and each half-space isassumed to be in local thermal equilibrium. The molecule is placed at a distance z > | a i as α = X i α i = X i,b |h a | µ i (0) | b i| hω (1)where µ i is the i − th spatial component of the dipole moment of the particle, and thesummation over b extends over its complete set of states.3he position-dependent particle-surface potential for the state | a i can be expressed in asum of three parts as [41],( U a ) bndtot ( z ) = ( U a ) bndvac ( z ) + ( U a ) bndeq ( z, β e ) + ( U a ) bndneq ( z, β s , β e ) (2)with β e = ¯ hck B T e and β s = ¯ hck B T s being the wavelength of thermal photons, ( U a ) bndvac ( z ) repre-senting the contribution of zero-point fluctuations, ( U a ) bndeq ( z, β e ) the contribution of thermalfluctuations at thermal equilibrium at a temperature T e , and ( U a ) bndneq ( z, β s , β e ) the contribu-tion of non-thermal equilibrium. The explicit expressions for the above three parts are( U a ) bndvac ( z ) = − hω π ε c Z ∞ dω ω ω − ω ab X σ α σ f σ ( z, ω ) , (3)( U a ) bndeq ( z, β e ) = 3¯ hω π ε c Z ∞ dω (cid:18) ω ω + ω ab − ω ω − ω ab (cid:19) e β e ω/c − X σ α σ f σ ( z, ω ) , (4)( U a ) bndneq ( z, β s , β e ) = 3¯ hω π ε c Z ∞ dω (cid:18) ω ω + ω ab − ω ω − ω ab (cid:19)(cid:18) e β s ω/c − − e β e ω/c − (cid:19) × X σ α σ Z dt A σ ( t ) e − z √ ǫ − ωt/c (5)where σ = k , ⊥ , ε is the permittivity of vacuum, α k = α x + α y , α ⊥ = α z and f σ ( z, ω ) = Z dt [ A σ ( t ) e − z √ ǫ − ωt/c + T σ ( t ) cos(2 zωt/c )] . (6)In the above equations, we have defined A k ( t ) = 12 √ ǫ − ǫ + 1)( ǫ − t + 1( ǫ − t + 1 t √ − t , (7) A ⊥ ( t ) = ǫ √ ǫ − ǫ − t + 1( ǫ − t + 1 t √ − t , (8) T k ( t ) = 14 (cid:18) t − √ ǫ − t t + √ ǫ − t − t ǫt − √ ǫ − t ǫt + √ ǫ − t (cid:19) , (9) T ⊥ ( t ) = 12 (1 − t ) ǫt − √ ǫ − t ǫt + √ ǫ − t . (10)The CP force on the particle can be obtained by taking the derivative of z on the particle-surface potential, Eq. (2), F a = − ∂∂z ( U a ) bndtot ( z ) . Let us first study the temperature dependence of the CP force for a typical moleculewhose transition wavelength is much larger than the typical experimental molecule-surfaceseparation placed near the surface of a substrate with a real relative permittivity ǫ . For4his purpose, we can define a geometric temperature, T z = ¯ hczk B , i.e., the temperature ofradiation whose wavelength is of order z , and a spectroscopic temperature, T ω = ¯ hω k B , whichis roughly the temperature required to noticeably populate the upper level. For the case weare considering, we have { zω c , z √ ǫ − ω c } ≪ , i.e., { T z , T z √ ǫ − } ≫ T ω , as long as ǫ is not verylarge. Then the analysis of the temperature dependence of the CP force can be divided intothe following three typical regions: the low temperature region where both T s and T e aremuch smaller than the spectroscopic temperature, i.e., { T s , T e } ≪ T ω ≪ { T z , T z √ ǫ − } , theintermediate temperature region where both T s and T e are much smaller than the geometrictemperature and much larger than the spectroscopic temperature, i.e., T ω ≪ { T s , T e } ≪{ T z , T z √ ǫ − } , and the high temperature region where both T s and T e are much larger than thegeometric temperature, i.e., T ω ≪ { T z , T z √ ǫ − } ≪ { T s , T e } .Generally, the thermal radiation that originates both from the substrate and the envi-ronment contributes to the molecular CP force. In the low and intermediate temperatureregions, though the contribution of thermal radiation that originates from the substrate ismuch larger than the contribution of radiation from the environment if T s /T e is not ex-tremely small, it is much smaller than the contribution of zero-point fluctuations, and thusin these two regions, the CP force behaves like z − , i.e., it obeys van-der Waals law. So,the CP force here is essentially temperature-independent just as the CP force of a moleculelocated near a perfect conducting plate [43].However, if we go to the high temperature region, T ω ≪ { T z , T z √ ǫ − } ≪ { T s , T e } , i.e., { β s , β e } ≪ { z, z √ ǫ − } ≪ λ where λ = cω is the wavelength of the molecule, then depen-dence shows up and the CP force of the ground and excited molecules near the substrateout of thermal equilibrium can now be written respectively as F − ≈ − ¯ h πε (cid:26) ǫ − ǫ + 1 9 ω ( α k + 2 α z )32 z + (cid:18) α k α z (cid:19) ω z cβ e − (cid:20) α k g ( ǫ ) + α z g ( ǫ ) (cid:21) ω z cβ s (cid:27) , (11) F + ≈ − ¯ h πε (cid:26) ǫ − ǫ + 1 9 ω ( α k + 2 α z )32 z − (cid:18) α k α z (cid:19) ω z cβ e + (cid:20) α k g ( ǫ ) + α z g ( ǫ ) (cid:21) ω z cβ s (cid:27) (12) Hereafter, { a, b } ≫ c means a ≫ c and b ≫ c . Similarly, { a, b } ≪ c means a ≪ c and b ≪ c . g ( ǫ ) = 2 ǫ + ǫ + 1( ǫ + 1) , g ( ǫ ) = (3 ǫ + 1) ǫ ( ǫ + 1) . (13)Let us note here that although Eqs. (11) and (12) have well-defined perfect-reflector limits( ǫ → ∞ ), they are not valid for the case of a molecule located near a perfect reflector as thecondition { z, z √ ǫ − } ≪ λ breaks down. Just as has been pointed out previously [41], forthe case of a perfect conducting plate, the limit ǫ → ∞ should be taken in all expressionsbefore analyzing the asymptotic behaviors.We now first examine the case of thermal equilibrium and see how it differs from the caseof a metal plate. When T s coincides with T e , i.e., T s = T e = T , the above results reduce tothe total CP force for the molecule at thermal equilibrium as F − ≈ − ¯ h πε (cid:26) ǫ − ǫ + 1 9 ω ( α k + 2 α z )32 z − (cid:20) α k g ( ǫ ) + α z g ( ǫ ) (cid:21) ω z cβ (cid:27) , (14) F + ≈ − ¯ h πε (cid:26) ǫ − ǫ + 1 9 ω ( α k + 2 α z )32 z + (cid:20) α k g ( ǫ ) + α z g ( ǫ ) (cid:21) ω z cβ (cid:27) (15)with g ( ǫ ) = ǫ ( ǫ − ǫ + 1) , g ( ǫ ) = (2 ǫ + 1)( ǫ − ǫ + 1) (16)and β = ¯ hck B T . For the ground-state molecule, there exists a threshold temperature, T = 3¯ hc k B z ω α k + 2 α z α k ǫǫ +1 + 2 α z ǫ +1 ǫ +1 , (17)which depends on both the molecular polarization and the dielectric permittivity and beyondwhich the attractive-to-repulsive transition of the CP force occurs, while for the excitedmolecule, the CP force is always attractive. This transition happens for molecules with anypolarization. This is in sharp contrast to the case of a metal surface where the behavior ofthe CP force depends crucially on the sign of α k − α z and the transition of attractive-to-repulsive is possible for molecules both in the ground and excited states [43].When T s does not coincide with T e , Eqs. (11) and (12) describe the CP forces of a moleculenear the substrate out of thermal equilibrium. If T e ≪ T s , the second term in Eqs. (11) and(12) is much smaller than the third term, and this means that the contribution of thermalfluctuations originating from the environment is negligible as compared to that from thesubstrate. For the molecule in the ground state, there exists a threshold temperature of the6ubstrate at which the first term balances the third term in Eqs. (11) , T s = 3¯ hc k B z ω ǫ − ǫ + 1 α k + 2 α z α k g ( ǫ ) + 2 α z g ( ǫ ) . (18)Obviously, this threshold depends again on both the molecular polarization and the permit-tivity of the substrate. When the temperature of the substrate is lower than that of thethreshold, the CP force is attractive as a result of the fact that the first term in Eq. (11)overtakes the third one so that F − <
0, and repulsive otherwise, while for the molecule inthe excited state, the CP force is always attractive since F + is always negative as can beseen from Eq. (12). This behavior is qualitatively similar to that in the case of thermalequilibrium. However, if T e ≫ T s , the second term in Eqs. (11) and (12) is much largerthan the third term, so that the contribution of thermal fluctuations that originate from thesubstrate is negligible as compared to that from the environment. Now for the molecule inthe excited state, there exists a threshold temperature of the environment at which the firstterm balances the third term in Eqs. (12) , T e = 3¯ hc k B z ω ǫ − ǫ + 1 . (19)When the temperature of the environment is lower than that of the threshold, the CP forcethe excited molecule feels is attractive, and repulsive otherwise, while for the molecule in theground state, the CP force is always attractive. This indicates that the molecular CP forceout of thermal equilibrium near a real dielectric substrate can be dramatically manipulatedby varying the relative magnitude of the temperatures of the substrate and the environment,and it displays behaviors distinctive from that at thermal equilibrium near a metal surface.Interestingly, this threshold, unlike T s , is independent of the molecular polarization.The above analysis shows that, in the near-zone and in the high temperature limit,( { β s , β e } ≪ { z, z √ ǫ − } ≪ λ ), the attractive-to-repulsive transition of the CP force canoccur for molecules in both the ground and excited states in an environment out of thermalequilibrium under certain conditions, while for a thermal equilibrium environment, such atransition can only occur for ground-state molecules as the CP force on the excited moleculeis always attractive. Now a few comments are in order. First, the contribution of the thermalradiation from the environment to the CP force is dependent on the molecular polarization(refer to the second terms in Eqs. (11) and (12)), while the contribution of the thermalradiation from the substrate depends on both the molecular polarization and the dielectric7roperties of the substrate (see the third terms in Eqs. (11) and (12)). Moreover, they areof opposite signs, thus when the system is out of thermal equilibrium, a disparity betweenthe contribution of the thermal radiation from the substrate and that from the environmentappears when T e ≪ T s or T e ≫ T s , and this results in a threshold temperature for eitherthe substrate or the environment across which the attractive-to-repulsive transition of theCP force can occur for either the ground-state or excited-state molecules. However, for thethermal equilibrium case, i.e., when the temperatures of the substrate and the environmentcoincide, the contributions of the thermal radiation from the substrate and the environmentcan be incorporated into one term which gives a repulsive force component for the CPforce of the ground-state molecules and an attractive one for the excited-state molecules(see Eqs. (14) and (15)). As a result, at thermal equilibrium, the attractive-to-repulsivetransition of the CP force can occur only for ground-state molecules while the CP force onthe excited-state molecules is always attractive in the near zone. Second, the main interestof the present paper is the temperature dependence of the CP force on the typical long wave-length molecules. However, one can show that the attractive-to-repulsive transition of theCP force for the excited-state molecules can also happen in the far-zone ( { z, z √ ǫ − } ≫ λ )as the molecule-substrate distance varies and in fact the CP force oscillates as a functionof the distance just as what happens for the CP force on an atom in excited states in thefar-zone. It is worth noting however that for a typical molecule with long wave-length, thefar-zone is too far for the molecular CP force to have any experimental significance.Now we turn our attention to the CP force of the molecule at a distance comparable toits transition wave-length both in and out of thermal equilibrium near a dielectric substrate.We start with the equilibrium case and take the LiH molecule, whose vibrational transitionfrequency is ω ∼ . × Hz ( λ ∼ . µm ), as an example. Assume that the moleculeis in the ground state and located at a distance z = 6 µm from the surface of a dielectricsubstrate. Consider first the case in which the temperatures of the environment and thesubstrate coincide, T s = T e , i.e., the molecule-substrate system is at thermal equilibrium.For the ground-state molecule polarizable along the z -direction and the substrate with agiven value of ǫ , the attractive-to-repulsive changeover of the CP force can occur at a cer-tain temperature, and with the increasing of the value of ǫ , the change occurs at highertemperature, as is shown in Fig. 1(a). While for the molecule polarizable parallel to the8urface of the substrate, the CP force of the ground-state molecule is always attractive, asis shown in Fig. 1(b). For small ǫ , the CP force varies very slowly with the temperaturein the T < ∼ K region and is effectively temperature-independent. And in fact this tem-perature independence of the CP force does not change appreciably as ǫ increases. Thisis similar to that of the CP force of a molecule located near a metal surface at thermalequilibrium which can be viewed as independent of the ambient temperature though thenumber of photons may be large [42, 43]. However, in the region T > ∼ K , the CP forcebecomes temperature-dependent as long as ǫ is not very small, and the dependence becomesmore obvious with larger ǫ . This is consistent with the result that the CP force of the samemolecule near the surface of a perfect conducting plate varies dramatically with the tem-perature when T > ∼ K [43]. For an isotropically polarizable molecule, Fig. 1(c) showsthat the attractive-to-repulsive changeover of the CP force can also happen, depending onthe value of the relative permittivity of the substrate ǫ . For the CP force of the moleculenear a half-space granite substrate with ǫ ∼
5, the change occurs at about 320 K . However,for the same molecule located at the same distance near the surface of a perfect conductingplane, the CP force is always attractive and no attractive-to-repulsive change occurs [43].This demonstrates again that the behaviors of the CP force for typical long wave-lengthmolecules near a dielectric substrate are sharply different from those near a metal surface.
100 200 300 400 T H K L - F - , z (a)
100 200 300 400 T H K L - - - - - - F - , ÈÈ (b)
100 200 300 400 T H K L - - - - F - (c) FIG. 1: Casimir-Polder force for the ground-state molecule at thermal equilibrium near a realdielectric substrate when z = 6 um . The solid, dashed, dotted, dot-dashed lines correspond to ǫ = 2 , , ,
15 respectively. The force is in the unit of ¯ hcω α/ (128 πε ). (a) The molecule ispolarized along the z -direction; (b) parallel to the surface of the substrate; (c) isotropically. When the temperatures of the environment and the substrate do not coincide, T s = T e ,i.e., the system is out of thermal equilibrium. Figure 2 shows how the CP force of theground-state molecule varies with the temperature of the substrate when the temperature9f the environment is kept at room temperature, T e ∼ K . For transversely polarizablemolecules, Fig. 2(a) shows that the CP force is similar to that in the case of thermal equi-librium (see Fig. 1(a)). In both cases, the attractive-to-repulsive changeover of the CP forceoccurs around T s ∼ K . In contrast, for longitudinally polarizable molecules, the CP forceof the molecule changes dramatically as opposed to that in the case of thermal equilibrium(see Fig. 1(b)). As shown in Fig. 2(b), the attractive-to-repulsive transition of the CP forcewhich doesn’t exist in the case of thermal equilibrium (see Fig. 1(b)) can occur in the outof thermal equilibrium case. Especially, for a substrate with small permittivity, a cementsubstrate with ǫ ∼ T s ∼ K . For an isotropicallypolarizable molecule, Fig. 2(c) shows that the attractive-to-repulsive changeover of the CPforce can occur when the system is out of thermal equilibrium. Especially, for a substratewith a relatively large relative permittivity, a sapphire substrate with ǫ ∼
10 and a graphitesubstrate with ǫ ∼
15 for example, the changes occur at T s ∼ K and 380 K respectivelywhich are much lower than the temperatures for the changeover to occur in the case ofthermal equilibrium (Fig. 1(c)).
100 200 300 400 Ts H K L F - , z
100 200 300 400 Ts H K L - - - - - F - , ÈÈ (b)
100 200 300 400 Ts H K L - - - - F - (c) FIG. 2: Same as Fig.1, but for the ground-state molecule out of thermal equilibrium.
In summary, we have studied the CP force of a typical molecule with long wave-lengthtransitions near a real dielectric substrate both in and out of thermal equilibrium. In thecase of thermal equilibrium, we find that, when the molecule-substrate separation is muchsmaller than the molecular transition wave-length, the CP force, in the high temperaturelimit, for the molecules in excited states is always attractive while that for the groundstate can change from attractive to repulsive beyond some threshold temperature and thisattractive-to-repulsive transition can happen for any molecular polarization. This is in clearcontrast to the CP force of the same molecule placed near a metal plate where the behaviorof the CP force depends crucially on the molecular polarization, on the sign of α k − α z ,10o be specific, and the transition of attractive-to-repulsive is possible for molecules both inthe ground and excited states [43]. The behaviors of the CP force are also sharply differentwhen the molecule-substrate separation is comparable to the transition wave-length and themost outstanding feature is that the CP force for isotropically polarizable molecules canchange from attractive to repulsive beyond a certain threshold temperature, which is about320K for a granite substrate. However, for the same molecule near a metal surface, the CPforce is always attractive and this kind of changeover never happens.For the case of being out of thermal equilibrium but in a stationary regime, when themolecule-substrate separation is much less than the molecular transition wave-length, theCP force in the high temperature limit can be dramatically manipulated by varying therelative magnitude of the temperatures of the substrate and the environment. In particular,the attractive-to-repulsive transition can occur beyond a certain threshold temperature ofeither the substrate or the environment for molecules both in the ground and excited states.If the temperature of the substrate is much higher, then there exists a threshold for thesubstrate temperature beyond which the transition happens for the molecules in the groundstate while the CP force for the excited states is always attractive. And it is just theother way around if the environment is of a much higher temperature, i.e., the CP force forthe ground state is attractive while that for the excited state can change from attractiveto repulsive beyond some threshold temperature of the environment. On the other hand,when the separation is comparable to the wave-length, the attractive-to-repulsive transitionwhich is impossible when the substrate and the environment are at thermal equilibrium, mayhappen for molecules polarizable along the surface of the substrate with a small permittivity,while for isotropically polarizable ones the transitions can occur even at room temperaturefor some dielectric substrate such as sapphire and graphite, and the transition temperatureis much lower than the temperature for the same transition to take place in the thermalequilibrium case, thus making the experimental demonstration of such force transitionseasier. 11 cknowledgments This work was supported in part by the NSFC under Grants No. 11375092, No. 11435006and No. 11405091, the SRFDP under Grant No. 20124306110001, the Zhejiang ProvincialNatural Science Foundation of China under Grant No. LQ14A050001, the Research Programof Ningbo University under Grants No. E00829134702, No. xkzwl10 and No. XYL14029,and the K.C. 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