Spooky Phenomenon Initiated by Two Laser Sources Ming-Chiang Li Abstract Spooky phenomena are physical basis for quantum computation and quantum encryption. Spooky phenomenon in atomic absorption initiated by two laser sources was discovered before 1980. Around 1990, there were a number of active experimental pursuits on spooky phenomena originated from crystal parametric down conversion. Present paper will review the similarity and difference between these two distinct processes. Introduction Understanding various spooky phenomena is the key to master quantum computation and quantum encryption. In 1975, an effort was initiated to investigate two- photon coherent atomic absorption1. The investigation revealed many interesting features associated with the absorption, which could not be explained classically or semi-quantum mechanically. These features were considered spooky. Random atomic motions could not prevent the coherence and the appearance of interference fringes. Not only one laser source, two difference laser sources still could lead to coherent absorption. The causes of these features are unique to quantum mechanics and deeply associated with its non-local nature. Around 1990, there were a number of active experimental pursuits2 on a spooky phenomenon initiated by a single laser source with the help of crystal parametric down conversion and classical correlation, which correlates photons received from two separated detectors. The objective of these activities was to investigate the non-local nature of quantum mechanics, and to examine entanglement of paired photons after down conversion. These activities led to current active pursuits in quantum computation and quantum encryption. Spooky phenomena arising from two-photon coherent atomic absorption and from crystal down conversion are similar despite their distinct difference. To understand their similarities and difference will lead to better understanding of the non-local nature of quantum mechanics as well as to successful quantum computation and quantum encryption. The basic theoretical principle on the two-photon coherent atomic absorption3 is discussed in the section of crossed beams. Each photon has to have two choices of paths in order for the absorption process to be coherent. Such absorption is a spooky phenomenon of quantum mechanics, and cannot occur classically. It will be pointed out in the section of overlapped beams that the overlapped experiment4 of two beams is a simplified version on the atomic absorption of the crossed beams5. Number of overlapped beams can be increased as discussed in the section of multiple overlapped beams6. The separated beam experiment7 of two-photon coherent atomic absorption is presented in the section of separated beams. As it is pointed out in the section of the beam width effect, a laser beam separation does not necessarily lead to a coherent absorption. Special experimental arrangement8 has to be made in order for interference fringes to appear. Two different laser sources would not lead to coherence in classical optics, but they would lead to coherence in two-photon atomic absorption9 as discussed in the 1 section of two different laser sources. In the section of parametric down conversion, an experiment on the delay “choice” quantum eraser10 is taken as an example to illustrate quantum mechanical spooky phenomena originated from crystal parametric down conversion. The similarity and difference between the two-photon processes of the atomic absorption and parametric down conversion are presented in the section on discussions. Crossed Beams Doppler free two-photon atomic absorption11 was demonstrated in 1974. An atom is able to absorb two photons, which travel in opposite directions without changing its momentum. In 1975, it was realized that such atomic absorption1 can be coherent and random motions of atoms will not M destroy interference fringes. The M Phase Laser shifter conceived experiment3 is depicted source r in Fig. 1. In the figure, laser beam r k1 − k2 is split by mirror SM. M denotes reflecting mirrors. Atoms are in the Atomic atomic cell, which is located in the SM cell r r intersection of laser beams. k2 − k1 Photons in the region of intersection r r r Detector have momentum hk1 , − hk1 , hk2 , r Fig. 1 M and − hk 2 respectively. Notation M h denotes Planck constant. r r r r r r Vectors k1 , − k1 , k2 , and − k2 are wave vectors with k1 = k2 = 2π / λ , where λ denotes the wave length of the laser beam. The energy diagram for the Doppler free two photon absorption is depicted in Fig. 2. The energy hω separation from the ground state to the excited state is ε ≡ hω0 . The laser is fine tuned to the atomic hω0 energy such that ω0 = 2ω = 2kc, where ω = kc is the circular frequency of the laser beam. Constant c hω denotes the velocity of light. Due to fine tuning and Fig. 2 energy conservation, only the Doppler free two photon absorption can occur during the atomic transition. Otherwise, an atom will gain or lose the kinetic energy and the transition will not occur. Atoms are in the intersection region of two laser beams as depicted in Fig. 1, then every atom has two possible ways to receive two photons without alter its momentum, namely r r r r r r r P + hk1 + (−hk1 ) = P + hk2 + (−hk2 ) = P r where P is the momentum of an atom before and after absorption. It means that the r r atom in two-photon absorption can either receive photon pair (hk1 , − hk1 ) or photon 2  r r pair (hk2 , − hk2 ) . According to the rule of coherence in quantum mechanics, two possible choices dictate that the absorption be coherent. The above situation is very similar to that of Young’s double slits experiment in Fig. 3. As the detector in Source Fig. 3 is concerned, it cannot tell which Detector path a photon comes from. If a phase shifter is inserted into one of the path in Fig. 3, then by varying the phase shifter Fig. 3 the light intensity received by the detector will vary in Young’s double slits experiment. Similarly, if a phase shifter is inserted into one of the crossed beams in Fig. 1, then by varying the phase shifter the number of excited atoms will vary. The excited atoms will decay through florescence, which can be measured by the detector. A variation of the phase shift will vary the detected intensity of florescence. The observed intensity variation is the interference fringe in the coherent two-photon atomic absorption. If the above experiment is realized, then important physics emerges from it. One of them is the direct phase measurement of two-photon transition amplitude. The other is a new quantum phenomenon of coherence despite atoms in random motion. In the view of quantum mechanics, Young’s double slit experiment is the coherence phenomenon of a single photon, where a photon has two choices through slits in reaching a detector to create the interference. In the above experiment, two photons are involved where one photon moves from right directions toward the atom and the other moves in opposite directions. Each of them has to have two choices in order for interference to appear. If one photon has the choices and other does not have, then the interference would not appear in the two-photon atomic absorption. In physics, coherence is also synonymous to high precision measurement. The above experiment leads to high precision measurements of two photon atomic absorption spectrum. Common approaches to reveal coherence of a physical system is to eliminate its random motion by either cooling it to low temperature or preparing it especially as in molecular beam experiments. However atoms in the atomic cell of Fig. 1 are in random motion. Conventionally, random motion will destroy coherence. However in two photon absorption as in Fig.1, random motions are allowed. It is a spooky nature of quantum mechanical coherence. The new quantum phenomenon of coherence does not arise from the semi- quantum mechanical treatment, in which the atomic energy level is quantized and the atom is considered as a potential center. In other words, the atom as a whole is not quantized. The whole atom has to be quantized in order for the new quantum phenomenon to appear. Hence the process in Fig. 1 should be treated according to quantum field theory. According to the theory of representation, a quantum mechanical particle should be either described as in momentum representation or in coordinate representation, not both. Classically, a particle should be described in both coordinate and momentum representations. More variables are needed to describe a classical particle than a quantum mechanical particle. In view of coherence, an extra variable will lead to an extra random phase. Hence a quantum mechanical particle is more coherent than a classical particle. 3 This is the main reason that, despite random motions, the experimental setup in Fig.1 still leads to coherence in two-photon atomic absorption. A quantum mechanical particle with definite momentum is a quantum mechanical wave, which is non-local in nature and leads to coherence. If the atom is a classical particle and only its energy level is quantized, than such treatment leads to interference fringes which contradict with experiments as being discussed later. Hence the interference as appeared in the coherence two-photon atomic absorption reflects the non- local characteristic of the atom and demonstrates the atom as a quantum mechanical wave. Overlapped Beams During the course of OW M the experiment, it was realized SM SM that the continuous wave laser Atomic Cell strength was too weak and Laser intersection region was too Source small. Pulse laser was a better choice. Instead of crossed beams, overlapping beams Detector would increase interaction region between the laser and atoms. A detour has to be introduced to mimic two paths. M M An experiment was setup as in Fig. 4 Fig. 4. This is accomplished through the use of splitters SM and reflection mirrors M. Optical isolator OW is used to reduce disturbance to laser source. A detector is mounted above the atomic cell to detect the fluorescence from the excited atoms. The simplicity eliminates the direct phase measurement of two-photon transition amplitude, but still retains the main feature of coherence. An atom in the atomic cell has two possible choices to absorb two photons. In one choice, two photons both take the short direct path and one of them is reflected by the mirror in the back of the atomic cell. In the other, two photons both take the longer detour path and one of them is also reflected by the mirror in the back of the atomic cell. Two choices lead to interference. Such an experiment was carried out by Salour and Cohen-Tannoudji4. A main characteristic in interference is the interference fringe. They claimed that they observed Ramsey fringe Cos[T (ω 0 − 2ω )], where T is the time difference for a photon to pass through either the short direct path or the longer detour path. However their claim was not valid. Because they based their consideration on the second order perturbation theory and took the atom as a potential center. This is to say that the atom was treated as a classical particle in their consideration, but not as a quantum mechanical wave. Hence, their consideration is semi-quantum mechanical. How to calculate the observed interference fringe? The rule is the same as Young’s double slits experiment in calculating the phase difference from path differences. Let L ≡ T / c denotes the path difference from possible direct and detour paths. Young’s double slits experiment is a single photon interference experiment and leads to the phase 4 difference kL , where the wave number k has been defined before. The Doppler free two photon absorption needs two photons. Phase differences from these two photons have to be added together. Then the interference fringe is Cos (2kL) = Cos (2Tω ) . Pulse is not monochromatic and contains many components. The atomic energy level has a line width as shown in Fig.5. Total energy for two photons falls within the line width will be absorbed. An hω average has to be taken in order to take hω hω0 hω0 care of the absorbed components. After averaging, the interference fringe becomes hω hω as Cos(2Tω ) ⇒ Cos(Tω0 ) . This was what Salour and Cohen-Tannoudji had observed. In making the above interference fringe to Fig. 5 appear as the Ramsey fringe, Salour and Cohen-Tannoudji locked the center frequency of the laser pulse to the path difference with the condition Tω = nπ , where n is an integer. Ramsey fringe comes from molecular beam experiments, where molecules in the beam passing through two spatial separated regions of standing RF waves. The experimental setup in molecular beam experiments is completely different as in comparison with that of Fig. 4. First, RF wave does not associate with photons. Second, pulsed laser beams in Fig. 4 do not form standing waves. Third, atoms before and after excitations remain in a single spatial region as defined by the atomic cell, and do not pass through two spatial regions. Furthermore, there are no similarities on the observed interference fringes. Hence, the overlapped beam experiment with pulsed laser beams is completely different from the classical separated molecular beam experiment5. Multiple Overlapped Beams An alternative to the overlapped beams as described in the previous section is to place the atomic cell inside an SM M optical resonator cavity as in Fig. 6. M denotes a mirror and SM a Atomic cell partially transmitting mirror. These Pulsed Laser two mirrors form the resonator cavity. A laser pulse is injected into the cavity. Many pulses are Fig. 6 Detector created inside the cavity and provide many choices for an atom inside the atomic cell to catch a two-photon pair instead of two choices as in Fig. 4. The difference between experimental setups in Figs. 4 and 6 is the same difference as that in double and multiple slits diffractions. The setup of Fig. 6 will lead to shaper interference peak than that of Fig. 4. The experiment in Fig. 6 was carried out by Hansch et al6. There were problems in explaining the observed interference fringe. Hansch et al. , like Salour and Cohen-Tannoudji , also based their consideration on the second order perturbation theory of quantum mechanics and took the atom as a potential center. They again did not account for the atom as a quantum mechanical wave. Hence their 5 consideration could not lead to the observed interference fringe. To remedy the problem, Hansch et al. superficially adopted the observed interference fringe. Such an action masked the true spooky nature of quantum mechanics as embedded in the two-photon coherent atomic absorption. It only takes few more steps to calculate the interference fringe from overlapped beams in Fig.4 to that of multiple overlapped beams in Fig. 6. These steps are very similar to that from double slit diffraction to multiple silt diffraction. The important matter to remember is that the phase differences in interference come from counting path length differences of choices for two absorbing photons. Separated Beams The separated beam M M experiment of two-photon coherent Doppler free atomic absorption Atomic was suggested out by Chebotayev beam A A* et al. The experimental setup is depicted in Fig. 7, where the laser Laser source is continuous and not source Detector pulsed as in overlapped beam M M Fig. 7 experiments. An atomic beam intersects two coherent and spatially separated laser beams. They stated that the separation of laser beam is able to overcome the width broadening of laser beam in two-photon atomic absorption. In their theory, the continuous laser beams were treated as standing waves. Such a theory is very similar to that of classical molecule beam experiment as discussed above, and leads to Ramsey fringes. Standing wave is needed in order to support the theory of Chebotayev et al. However, the experimental arrangement in Fig. 7 does not lead to standing waves of laser beams. Furthermore, the physical parameter on the beam width never appeared in their theoretical derivation. As it will be shown later, the beam width has drastic effect on the observed interference fringe, and the experimental setup in Fig. 7 does not lead to Ramsey interference fringe. In the following consideration, atoms and photons are totally quantized according to quantum field theory. When the atomic beam enters the first intersection region, some of the atoms absorb a pair of opposite-momentum photons without changing their momentum and become excited. The excited and unexcited atoms move together at same velocity. Upon entering the second intersection region some of the atoms again absorb a pair of opposite-momentum photons and become excited too. These later excited atoms mix with the previously excited atoms and move ahead with the rest of atoms. The fluorescence from the excited atoms is measured by the detector. Due to two possible choices of excitations, the detector cannot distinguish whether the fluorescence is emitted by excited atoms from the first intersection or by that from the second intersection region. According to the rule of coherence in quantum mechanic, two choices means interference and the detector should observe such an interference phenomenon. The next step is on how to calculate these fringes. Atoms and their excited counter parts are staying at the same spatial location in Figs.1 and 4. But atoms and their 6 excited counter parts are at different spatial locations in Fig. 7. A simple field theory calculation8 leads to the interfering Ramsey fringe Cos[T (ω0 − 2ω )], where T is the time difference for excitations at first or second intersection regions. An intuitive interpretation might be viewed as follows. In the overlapped experiment only phase contribution from two photons is counted and lead to the interference fringe dependence of 2Tω . In the separated beam experiment, since atoms are excited at different regions, the phase contribution from the atom before and after excitation has to be accounted too, that leads to the phase dependence (E + ε )T / h − ET / h = ω0T , where E denotes the kinetic energy of atom before and after absorption. The minus sign in the dependence is due to the fact that the final atomic state is conjugated to its initial state. Ramsey fringe comes from counting these two dependences together. Hence Ramsey interference fringe only appears in the separated beams experiment, but not in cross beam or overlapped beam experiments. Beam Width Effect The above discussion did not account for the finite-width effect of the laser beam. The finite-width leads to uncertainty in the transverse momentum of photons in the laser beam, which will destroy or modify the Ramsey fringe8. A simplified discussion is presented here. Let a small arrow denotes the uncertain transverse momentum in the incident laser beam. The uncertainty leads to uncertainties in all other beams as well. These uncertainties are indicated M M by respective small arrows as in Fig. 8. The transverse momentum uncertainties will change the Atomic velocity of excited atoms. Atoms beam A A* excited at first intersection region gain the uncertain transverse Laser momentum, hence move slower. source Detector Those excited at second M intersection region move faster. M Fig. 8 Then there is a way to tell in which M intersection region the excited M atoms are excited. According to the rule of coherence in quantum mechanics, any distinction Atomic destroys interference. The beam A A* experimental setup in Fig. 7 will not lead to coherent atomic Laser absorption and Ramsey Detecto interference could not be observed. M r However, such a rule does not apply to the saturated atomic M absorption8 of separated beams. In Fig. 9 saturated absorption, two photons are absorbed in sequence that is different from two-photon coherent absorption. 7  Next consider another case which is depicted in Fig. 9. A lens is inserted before the end mirror, which is placed on the focal plane of the lens. Then the uncertain momentum of the incident laser beam is reversed after reflection by the end mirror. The uncertain momentum in respective beams is indicated by small arrows. When an atom absorbs two photons in intersection regions, the uncertain momentum of one photon cancels that of the other photon in the pair moving from opposite direction. Then there is no distinction in which one of the intersection regions the atom gets the excitation. The excitation becomes coherent and Ramsey interference fringe appears. A realization on the reverse of uncertain momentum will improve outcomes on the two photon coherent atomic absorption of separated beams. This is also a spooky nature of quantum mechanical coherence. If a distinction can be identified, then the choice is eliminated and interference is no longer possible. The beam separation alone is not enough for Ramsey fringe to appear. Special arrangement has to be made in order to observe Ramsey fringe. Various experimental setups are possible. Setups in Figs. 8 and 9 are just two examples. Experimental realizations of these various setups will lead to more understanding on the two-photon spooky phenomenon of quantum mechanical coherence. Since the importance of the atomic quantum mechanical wave has not been recognized by most experimental groups, the separated beam experiment has not reached the benefit of its full potential in achieving high precision atomic spectroscopy. Two Different Laser Sources Two laser sources OW OW with different frequencies SM SM Atomic SM SM can also be used to excited Cell atoms in two-photon Laser Laser coherent absorption, and Source Source the absorption needs not being Doppler free9. Detector Consider the experimental setup as depicted in Fig. 10 with two different pulsed laser sources of Fig. 10 frequencies M M M M ω1 and ω2 respectively. The atomic energy levels hω10 and hω20 for the above transition are depicted in Fig. 11. It is a ε 2 = hω20 hω 2 three-level atomic transition. Laser sources are tuned to these levels. In general the absorption is not coherent. ε 1 = hω10 However when the following alignment condition9 hω1 is satisfied, the absorption becomes coherent Fig. 11 ω10T1 = ω20T2 where T1 and T2 are the time differences for photons in the pair to pass through their respective direct short or long detour paths. The 8  interference fringe has a form Cos(ω10T1 + ω20T2 ) . It is clear in comparing interference fringes that the overlapped experiment in Fig. 4 is a special case of that in Fig. 10, when ω10 = ω20 and T1 = T2 . In the above overlapped beam experiment, each of photons in two-photon absorption has to have two choices in order for interference to appear. If one photon has the choices and other does not have, then the alignment condition is violated and the interference would not appear. This is a spooky phenomenon of two-photon coherence in quantum mechanics. Parametric Down Conversion Two-photon spooky phenomena associated with crystal parametric down conversion have many variations. The case cited here has been referred to as the delay “choice” quantum eraser. Its experimental setup is depicted in Fig. 12. a laser beam is divided by a double slit and directed onto a nonlinear crystal BBO located at regions A and B. A pair of down converted signal-idler photons is generated either from the A or B region. The signal photon (coming either from A or B region) propagates through the lens to detector D0, which is on the focal plane of the lens. Detector D0 can be scanned along its x axis by a step motor. The idler photon (coming either from A or B region) is sent to detectors D1, D2, D3, or D4 with the help of a prism, beam splitters BSA, BSB, and BS, and mirrors MA and MB. According to the setup, the idler photon from either A or B region can reach detectors D1 and D2. Detectors D3 only can detect the idler photon from B region, and Detectors D4 from A region respectively. Coincidences between detectors D0 and Di (i =1, 2, 3, 4) are recorded. The optical distance between regions A, B and detector D0 is much shorter than f x D0 D1 D4 Laser B A BBO MA BSB Coincidence BSA BS Circuit MB D2 D3 Fig. 12 the optical distance between regions A, B and beam splitters BSA or BSB. This means that any information one can infer from the idler photon must come later than the 9 registration of the signal photon by the detector D0 based on the view of the delay “choice” quantum eraser. The experimental results confirmed the interference pattern as detector D0 shifting along its x axis in coincidence correlation measurements between detectors D0 and D1, or between detectors D0 and D2. It means that the detector D0 can not tell which region A or B the signal photon is from. No interference pattern was observed in coincidence correlation measurements between detectors D0 and D3, or between detectors D0 and D4. The latter implies that the detector D0 can tell which region the signal photon is from. The idea of the delay “choice” is from the double slit diffraction of a single photon. It is the interference phenomenon of a single photon. The delay “choice” quantum eraser as in Fig. 13 involves two photons. The rule of the two-photon interference should be applied to such interference. Coincidence circuit with detectors D0 and Di (i =1, 2, 3, 4), similar to the atom in two-photon absorption, performs a correlation measurement of two photons. The signal photon and idler photon should both have two choices of paths in order to observe the interference as in the coincidence correlation measurements between detectors D0 and D1, or between detectors D0 and D2. When the signal photon has two choices and the idler photon does not have, the interference is no longer possible. No interference pattern will be observed as in the coincidence correlation measurements between detectors D0 and D3, or between detectors D0 and D4. The rule of the two-photon interference is different from that of the single photon interference. It is better to consider the delay “choice” quantum eraser in Fig. 12 as a two-photon interference phenomenon, instead of the single photon interference phenomenon. Discussions Two photons in the parametric down conversion come from a single laser source. Their correlation is accomplished with the help of two spatially separated detectors. It is a classical correlation. On the other hand, two photons in the coherent atomic absorption can be from two different laser sources. It is the atom, which performs the two-photon quantum mechanical correlation. Hence the coherent process of the parametric down conversion and that of the atomic absorption are distinctively different, however, their rule of coherence is the same. Each photon has to have two choices in its path. Two photons are entangled in the parametric down conversion. Entanglement is a relative concept. The concept of entanglement at the present is only applied to the classical correlation. In the case of Doppler free two photon atomic absorption, two photons are entangled as in the classical sense. In the non-free case, even though two photons might not be entangled as in the classical sense, they can be arranged to entangle through alignment in view of the absorbing atom. Without such entanglement, the two- photon coherent atomic absorption will not occur when atoms are in random motion and the two photons are emitted from two different laser sources. Hence the concept of entanglement should be enlarged in view of the quantum mechanical correlation. Although two-photon coherent atomic absorption was first suggested about thirty years ago, the importance of the absorbing atom as a quantum mechanical wave has not been properly recognized by most experimentalists. The present paper should put the importance in proper perspective and clear some of its theoretical misunderstandings. 10 Reference: 1. M. C. Li, Bull. Am. Phys. Soc. 20 (1975) 654. 2. J. Horgan, Scientific American 267 (1992) 94. 3. M. C. Li, Nuovo Cimento 39B (1977) 165. 4. M. M. Salour, and C. Cohen-Tannoudji, Phys. Rev. Lett. 38 (1977) 757. 5. M. C. Li, Phys. Rev. A 16 (1977) 2480. 6. R. Teets, J. N. Eckstein, and T. W. Hansch, Phys. Rev. Lett. 38 (1977) 760. 7. Ye. V. Baklanov, B. Ya. Dubetsky, and V. P. Chebotayev, Appl. Phys. 9 (1976) 171. 8. M. C. Li, Phys. Rev. A 23 (1981) 2995. 9. M. C. Li, Phys. Rev. A 22 (1980) 1323. 10. Y. H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84 (2000) 1. 11. F. Biraben, B. Cagnac, and G. Grynberg, Phys. Rev. Lett. 32 (1974) 643; M. D. Levenson, and N. Bloembergen, Phys. Rev. Lett. 32 (1974) 645. 11