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Zero Point Energy and Vacuum Fluctuations Effects,[object Object],Ana Trakhtman,[object Object],Department of Physics,[object Object],Ariel University Center of  Samaria,[object Object]
Table of Content,[object Object],[object Object]
 Zero Point Energy – a new friend or an old acquaintance?
 Vacuum Fluctuations – what is it?
 Spontaneous Emission and what it has to do with ZPE
 Casimir Effect or Van der Waals Attraction
 The Lamb Shift
 The Beam Splitter
 Science Fiction or is it?
 Conclusion,[object Object]
Zero Point Energy,[object Object],The origin of zero-point energy is the Heisenberg uncertainty principle.,[object Object],It is the lowest possible energy that a quantum mechanical system may have; it is the energy of its ground state. ,[object Object],The most famous such example of zero-point energy is 𝐸=12ℏ𝜔 associated with the ground state of the quantum harmonic oscillator.,[object Object],It is the expectation value of the Hamiltonian of the system in the ground state.,[object Object], ,[object Object]
Zero Point Energy,[object Object],𝐻=ℏ22𝑚 ∇2+𝑉(𝑟,𝑡),[object Object],𝑍𝑃𝐸=𝐻=Ψ𝑔𝑟𝑜𝑢𝑛𝑑𝐻Ψ𝑔𝑟𝑜𝑢𝑛𝑑,[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],In quantum field theory, the fabric of space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator. The zero-point energy is again the expectation value of the Hamiltonian; here, however, the phrase vacuum expectation value is more commonly used, and the energy is called the vacuum energy. Vacuum energy can also be thought of in terms of virtual particles (also known as vacuum fluctuations) which are created and destroyed out of the vacuum.,[object Object],The concept of vacuum energy was derived from energy-time uncertainty principle.,[object Object]
Vacuum Fluctuations,[object Object],The vacuum state |𝑣𝑎𝑐> of the field is the state of the lowest energy.,[object Object],The expectations values of both 𝑎𝑘𝑠 and 𝑎𝑘𝑠+ vanish in the vacuum state, because:,[object Object],𝑎𝑘𝑠|𝑣𝑎𝑐> =0=<𝑣𝑎𝑐|𝑎𝑘𝑠+,[object Object],Vector 𝐹(𝑟,𝑡), which may be the electric or magnetic or the vector potential, having a mode expansion of the general form:,[object Object],𝐹𝑟,𝑡=1𝐿32𝑘,𝑠𝑙𝜔𝑎𝑘𝑠𝜀𝑘𝑠𝑒𝑖𝑘∙𝑟−𝜔𝑡+h.𝑐,[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],Where 𝑙𝜔 is some slowly varying function of frequency which is different for each field vector.,[object Object],Expectation value of 𝐹𝑟,𝑡 in the vacuum state:,[object Object],<𝑣𝑎𝑐𝐹𝑟,𝑡𝑣𝑎𝑐> =0,[object Object],However, the expectation of the square of the field operator does not vanish, as we will show soon. This implies that there are fluctuations of the em field, even in its lowest energy.,[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],If we use the mode expansion and make use of the fact that:,[object Object],𝑣𝑎𝑐𝑎𝑘𝑠+𝑎𝑘′𝑠′𝑣𝑎𝑐=0,[object Object],𝑣𝑎𝑐𝑎𝑘𝑠+𝑎𝑘′𝑠′+𝑣𝑎𝑐=0,[object Object],𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′𝑣𝑎𝑐=0,[object Object],We find that:,[object Object],𝑣𝑎𝑐𝐹2(𝑟,𝑡)𝑣𝑎𝑐==1𝐿3𝑘𝑠𝑘′𝑠′𝑙𝜔𝑙∗(𝜔′)𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′+𝑣𝑎𝑐(𝜀𝑘𝑠∙𝜀𝑘′𝑠′∗)∙𝑒𝑖[𝑘−𝑘′𝑟−𝜔−𝜔′𝑡],[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],𝑎𝑘𝑠𝑡, 𝑎𝑘′𝑠′+(𝑡)=𝑎𝑘𝑠∙𝑎𝑘′𝑠′+−𝑎𝑘′𝑠′+∙𝑎𝑘𝑠=𝛿𝑘𝑘′3𝛿𝑠𝑠′,[object Object],With the help of the commutation relation we have:,[object Object],𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′+𝑣𝑎𝑐=𝑣𝑎𝑐(𝑎𝑘′𝑠′+∙𝑎𝑘𝑠+𝛿𝑘𝑘′3𝛿𝑠𝑠′)𝑣𝑎𝑐=𝛿𝑘𝑘′3𝛿𝑠𝑠′,[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],So that:,[object Object],𝑣𝑎𝑐𝐹2(𝑟,𝑡)𝑣𝑎𝑐=1𝐿3𝑘,𝑠𝑙𝜔2=2𝐿3𝑘𝑙𝜔2  ⟶ 22𝜋3𝑙𝜔2𝑑3𝑘,[object Object],This is clearly non-zero, and indeed is infinite for an unbounded set of modes. As it is know:,[object Object],𝑣𝑎𝑐∆𝐹2𝑣𝑎𝑐=𝑣𝑎𝑐𝐹2𝑣𝑎𝑐   ,    ∆𝐹=𝐹−𝐹,[object Object],∆𝐹 – the deviation from the mean,[object Object],This shows us that the field fluctuates in the vacuum state.,[object Object], ,[object Object]
Vacuum Fluctuations,[object Object],The effects of vacuum energy can be observed in various phenomena such as spontaneous emission, the Casimir effect and the Lamb shift, and are thought to influence the behavior of the Universe on cosmological scales.,[object Object]
Spontaneous Emission,[object Object],Quantum electrodynamics shows that spontaneous emission takes place because there is always some electromagnetic field present in the vicinity of an atom, even when a field is not applied. Like any other system with discretely quantized energy, the electromagnetic field has a zero-point energy. Quantum electrodynamics shows that there will always be some electromagnetic field vibrations present, of whatever frequency is required to induce the charge oscillations that cause the atom to radiate 'spontaneously'.,[object Object]
The Casimir Effect,[object Object],One of the more striking examples is the attractive force between a pair of parallel, uncharged, conducting plates in vacuum. This force is also referred to as a Van der Waals attraction and has been ,[object Object],   calculated by Dutch physicists,[object Object],Hendrik B. G. Casimir and,[object Object],    Dirk Polder (1948).,[object Object]
The Casimir Effect,[object Object],One can account for this force (also known as Casimir force), and obtain an approximate value of its magnitude, by assuming that the force is a consequence of the separation-dependent vacuum field energy trapped between the two plates. If the plats are squares of side L and are separated by a distance z, we may suppose that the system constitutes a “cavity” that supports modes with wave number k down to about 1/z. the vacuum field energy trapped between the plates may therefore be written approximately as:,[object Object],𝑈=𝑘,𝑠12ℏ𝜔≈𝐿2𝑧1𝑧𝐾ℏ𝑐𝑘 𝑘2𝑑𝑘≈14𝐿2ℏ𝑐𝑧𝐾4−1𝑧3=𝑈𝑢𝑝𝑝𝑒𝑟−𝑈𝑙𝑜𝑤𝑒𝑟,[object Object], ,[object Object]
The Casimir Effect,[object Object],we have introduced a high frequency cut-off K to make the energy finite.,[object Object],We can think of the negative rate of change of the lower cut-off energy 𝑈𝑙𝑜𝑤𝑒𝑟 with separation z as constituting a force of attraction, whose magnitude F per unit are is given by:,[object Object],𝐹=−1𝐿2𝑑𝑈𝑙𝑜𝑤𝑒𝑟𝑑𝑧~ℏ𝑐𝑧4,[object Object], ,[object Object]
The Casimir Effect,[object Object],It is interesting to note from the structure of F that the force is proportional to ℏ and is therefore quantum mechanical.,[object Object],Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small. ,[object Object],On a submicrometre scale, this force becomes so strong that it becomes the dominant force between uncharged conductors. ,[object Object], ,[object Object]
The Casimir Effect,[object Object],At separations of 10 nm—about 100 times the typical size of an atom—the Casimir effect produces the equivalent of 1 atmosphere of pressure (101.325 kPa), the precise value depending on surface geometry and other factors.,[object Object],In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon; and in applied physics, it is significant in some aspects of emerging micro technologies and nanotechnologies.,[object Object]
The Lamb Shift,[object Object],The Lamb shift, named after Willis Lamb (1913–2008), is a small difference in energy between two energy levels 2S1 / 2 and 2P1 / 2 of the hydrogen atom in quantum electrodynamics. According to Dirac, the 2S1 / 2 and 2P1 / 2 orbitals should have the same energies. However, the interaction between the electron and the vacuum causes a tiny energy shift on 2S1 / 2. Lamb and Robert Retherford measured this shift in 1947. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.,[object Object]
The Lamb Shift,[object Object]
The Lamb Shift,[object Object],In 1948 Welton succeeded in accounting for the Lamb shift between the s and p energy levels of atomic hydrogen in terms of the perturbation of the electronic orbit brought about by vacuum fluctuations. ,[object Object],A perturbation 𝛿𝑟 in electronic position in general causes a change of potential energy 𝛿𝑉 given by:,[object Object],𝛿𝑉=𝑉𝑟+𝛿𝑟−𝑉𝑟=∇𝑉∙𝛿𝑟+12𝜕𝜕𝑟𝑖𝜕𝜕𝑟𝑗𝑉𝛿𝑟𝑖𝛿𝑟𝑗+⋯,[object Object], ,[object Object]
The Lamb Shift,[object Object],When we average this over the random displacements 𝑟, the term in 𝛿𝑟2 is the leading non-zero term and we find that:,[object Object],𝛿𝑉=16∇2𝑉 𝛿𝑟2,[object Object],In order to calculate the value of 𝛿𝑟2 resulting from the fluctuations of the vacuum field, we observe that, under the influence of an electric field 𝐸𝜔 of frequency 𝜔, the electronic position r obeys the equation of motion:,[object Object],𝑚𝑟=−𝑒𝐸𝜔cos𝜔𝑡,[object Object], ,[object Object]
The Lamb Shift,[object Object],And this results in a mean squared displacement about its equilibrium value of:,[object Object],𝛿𝑟𝜔2=12𝑒2𝑚2𝜔4 𝐸𝜔2𝑣𝑎𝑐=ℏ𝑒22𝜋3𝜀0𝑚2𝑑3𝑘𝜔3=ℏ𝑒22𝜋2𝜀0𝑚2𝑐3𝜔0Ω𝑑𝜔𝜔,[object Object], ,[object Object]
The Lamb Shift,[object Object],The integral diverges logarithmically at the upper end, and had to be provided with a cut-off Ω, which is usually chosen to be of order 𝑚𝑐2/ℏ. ,[object Object],When this expression for 𝛿𝑟2 is inserted in 𝛿𝑉, and we average ∇2𝑉(𝑟) over the electronic orbit with the help of the wave function 𝜓(𝑟), we obtain finally for the perturbation of the atomic energy level:,[object Object],𝛿𝑉=ℏ𝑒212𝜋2𝜀0𝑚2𝑐3 𝑑3𝑟 ∇2𝑉𝑟𝜓𝑟2𝜔0Ω𝑑𝜔𝜔,[object Object], ,[object Object]
The Lamb Shift,[object Object],If we take the potential energy 𝑉(𝑟) to be:,[object Object],𝑉𝑟=−𝑒24𝜋𝜀0𝑟 ,[object Object], then:			∇2𝑉𝑟=𝑒2𝜀0𝛿3(𝑟),[object Object],and the volume integral reduces to:,[object Object],𝑒2𝜀0𝜓02,[object Object], ,[object Object]
The Lamb Shift,[object Object],This vanishes for a p-state but gives a finite value for an s-state. ,[object Object],The difference between the s and p energy levels is therefore:,[object Object],∆𝐸=ℏ𝑒412𝜋2𝜀02𝑚2𝑐3𝜓𝑠02ln𝑚𝑐2ℏ𝜔0,[object Object], ,[object Object]
The Lamb Shift,[object Object],This leads to:,[object Object],∆𝐸ℏ~1040 𝑀𝐻𝑧,[object Object],For the 2s-state of hydrogen, and is in reasonable agreement with measurements by Lamb and Retherford (1947).,[object Object], ,[object Object]
The Beam Splitter,[object Object]
The Beam Splitter,[object Object],After decomposing all fields into plane-wave modes in the usual way, we consider a single incident mode labeled 1, which gives rise to a reflected mode 2 and a transmitted mode 3. ,[object Object],r, t are the complex amplitude reflectivity and transmissivity for light incident from one side.,[object Object],𝑟′, 𝑡′ for light coming from the other side,[object Object],there are no losses in the beam splitter,[object Object], ,[object Object]
The Beam Splitter,[object Object],Then these parameters must obey the following reciprocity relations (due to Stokes, 1849):,[object Object],𝑟=𝑟′  ,  𝑡=𝑡′,[object Object],𝑟2+𝑡2=1,[object Object],𝑟𝑡∗+𝑟∗𝑡=0,[object Object], ,[object Object]
The Beam Splitter,[object Object],It follows that an incoming classical wave of complex amplitude 𝜐1 gives rise to a reflected wave 𝜐2, and a transmitted wave 𝜐3 such that:,[object Object],𝑣2=𝑟𝑣1   ,   𝑣3=𝑡𝑣1,[object Object],From these relations it follows immediately that:,[object Object],𝑣22+𝑣32=𝑡2+𝑟2𝑣12,[object Object],So that the incoming energy is conserved.,[object Object], ,[object Object]
The Beam Splitter,[object Object],Now suppose that we wish to apply a similar argument to the treatment of a quantum field. ,[object Object],Then 𝑣1, 𝑣2, 𝑣3 have to be replaced by the complex amplitude operators 𝑎1, 𝑎2 , 𝑎3 , which obey the commutation relations:,[object Object],𝑎𝑗, 𝑎𝑗+=1,  𝑗=1, 2, 3,[object Object],𝑎2, 𝑎3+=0,[object Object], ,[object Object]
The Beam Splitter,[object Object],if we simply replace 𝑣1, 𝑣2, 𝑣3 by the operators 𝑎1, 𝑎2 , 𝑎3 , we readily find that the commutation equations do not hold for 𝑎2 , 𝑎3. Instead we obtain:,[object Object],𝑎2 , 𝑎3+=𝑟2𝑎1 , 𝑎1+=𝑟2,[object Object],𝑎3 , 𝑎3+=𝑡2𝑎1 , 𝑎1+=𝑡2,[object Object],𝑎2 , 𝑎3+=𝑟𝑡∗𝑎1 , 𝑎1+=𝑟𝑡∗,[object Object], ,[object Object]
The Beam Splitter,[object Object],The reason for the discrepancy is that we have ignored the fourth beam splitter input port, which is justifiably ignored in the classical treatment because no light enter that way.,[object Object],However, even if no energy is flowing through the mode labeled 0, in a quantized field treatment there is a vacuum field that enters here and contributes to the two output modes.,[object Object]

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