1 Complementarity and the Quantum Eraser

Size: px

Start display at page:

Download "1 Complementarity and the Quantum Eraser"

Francine Henry
6 years ago
Views:

1 Complementarity and the Quantum Eraser In this little text I will first give a short review of the concept of Complementarity, Which Path Detectors reversibility and how this can be used to

1 1 Complementarity and the Quantum Eraser In this little text I will first give a short review of the concept of Complementarity, Which Path Detectors reversibility and how this can be used to introduce the quantum eraser and connected to this the possibilitty of delayed choice. 1.1 Complementarity Historically complementarity is closely linked to Albert Einstein and Niels Bohrs discussion on wave particle duality in the beginning of this century. Their discussion led to the introduction of the complementarity princple, which excludes the simultaniuous manifestation of particle- and wavelike behavior. Today we can look on complementarity in a broader sence and state Figure 1: Recoiling slit apparatus that any two observables for which precise knowlegde of the one of them implies total uncertainity in the other are complementary. Whith other word we might say that any noncommuting or non compatible observables are complementary. To illustrate this we might think of an electrons spin. If we measure measure the z component and get spin up z = 1/ 2( x + x ) then any possible outcome of the measurment spin in the x direction will be equally probable. But let us now return to the original idea of Niels Bohr and Albert Einstein, which can be most easely understood by considering Young s two slit experiment. Here the wavelike behaviro is manifested, if we see interference, but if we on the other hand gain which way (Welcher Weg) 1

2 information the interference is washed out and the particlelike behavior is manifested. The peculiar thing here is that we do not even have to read out the Which Path Detector as long as the mere possibility of obtaining this information excists in principle. 1.2 Which Path Detectors Through out history people have thought of many different kinds of Which Path Detectors where the first approaches where characterised by irreversible interactions between the interfering system and the detectors, so that the dissapearence of interference could be explained by Heisenbergs Uncertainity Principle. As an example we could mention Albert Einsteins recoiling slit Gedanken experiment. The basic idea is illustrated in figure 1. We send light trough a double slit which can recoil. Here Einstein argues if the mass of the double slit is small enough we can measure the recoil of the slit and by conservation of momentum deduce which way the electron went. Bohr then answers that we also have to treat the double slit quantum mechanically, so we can only know the position of the recoiling slit to within an uncertainty of x due to the uncertainty principle, but this will introduce an uncertainty in the phase of our light beam and the interference pattern will be washed out. A similar example goes back to Feynman. He is sending electrons through a double slit and is wachting the other side by a light source. See figure 3 and the interference disapears as opposed to the unwatched case see figure 2 where interference persists. This can be understood intuitively by Feynmans rules for calculating probabilities: Indisinguishable paths Probability amplitudes are summed, then absolute squared - leading to interference terms Distinguishable path Probabilities are summed, yielding no interference And again one can argue that a random phase shift is introduced due to Heisenbergs uncertainty relation. Indeed Feynman himself concludes that no one ever has found or even thought a way around this principle. But as we shall see there have been proposed more subtle Which Path Detectors in which the loss of coherence can be considered as due to the entanglement of the systems wave function to the Which Path detectors. In this case the loss of coherence is reversible and interference might be regain if one by a subsequent measurment manages to erase the which path information. Scully and co-workers put the following idea forward. A Micromaser Which Path Detector. For an illustration see page 564 in Quantum Optics by M.O.Scully. Here he considers a variant of youngs double slit experiment, 2

Figure 2: Physics Wavelike behavior of electrons from the Feynman lectures on Figure 3: Particlelike behavior of electrons from The Feynmans lectures of physics Vol

3 Figure 2: Physics Wavelike behavior of electrons from the Feynman lectures on Figure 3: Particlelike behavior of electrons from The Feynmans lectures of physics Vol III. y d/2 (d separation of slits), p y θ/2p x (θ angle to first maximun), Debroglie wave length λ = h/p, d sin θ dθ = λ implies y p y h/4 contrary to Heisenberg. 3

4 where a beam of two level atoms is excited by an laser from b to a and passing two high q cavities before hitting the double slit. If no cavities and laser are present the system can be described by. ψ = 1 2 (ψ 1 (r) + ψ 2 (r)) b (1) where r is the center of mass coordinate of the atom, hence the probability denstity for particles on the screen at R is given by P (R) = ψ ψ (ψ 1ψ 2 + ψ 2ψ 1 ) b b. (2) In the presens of the lase-cavity system we first excite the atom to a long lifed Rydberg state a and then prepare the cavity length such that the atom will decay whith near unit probability. If we further assume the initial absence of photons in the cavity we get the following state after the cavity ψ = 1 2 (ψ 1 (r) ψ 2 (r) ) b (3) where denotes one photon in the first cavity and no in the second cavity. Now the probability density at the screen is given by P (R) = ψ ψ 2 2 (4) where the interference terms have vanished, but how can we tell whether the this is due to Heisenberg Uncertaintyrelation or not? Scully shows in his book that phase factors introduced due to the interaction with the field are not random but just the Rabi factor. 2 Different realisations of the Q-eraser In 1994 T.J.Herzog, J.G.Rarity, Harald Weinfurter and Anton Zeilinger published a paper on Frustrated Two-Photon Creation via Interference. Here they present a way to get interference between two possible ways of creating a entangled photon pair produced in a non-linear LiIO 3 crystal by parametric downconversion. The results represented here are the foundation of the experiment on Complementatity and the Quantum Eraser by Thomas Herzog et al. in 1995, so if one is interessted in experimental details one should consult the former article. In the article on the q-eraser they put forward three different q-eraser experiments. In firgure FIG.1. the basic setup is shown. In the non linear crystal a UV pump photon (351.1nm) is split 4

5 into a photon pair, where one of the created photons conventionally is called signal (632.8nm) and the other idler (788.7nm). Note ν p = ν s + ν i. Due to dispersion (n(ω i ) n(ω s )) the signal and idler will normaly get out of fase, however in a birefringent crystal the setup can be allign, such that the phases are mached. This is done here and therefore the idler and the signal share the same polarisation. The state of the pair after the crystal is ψ = α 1 sr 1 ir + O( α 2 ) (5) wheres α is the probability amplitude for emitting the photons into the spatial modes sr ir and α 1. As seen on FIG.1. the pump is retroreflected back into the crystal so that another signal idler pair can be created. ψ = α 1 sr 1 ir + exp(iφ p )α 1 sd 1 id (6) where φ p is the total phaseshift that the pump accumulated between the to emissons. Now also the signal and idler modes are reflected back so that they over lap with the second signal idler pair. Thus we may set sr = sd s and ir = id s and the final state of the indistiguishable pairs is ψ = α (exp(iφ p ) + exp(i(φ s + φ i ))) 1 s 1 i (7) where φ s and φ s are the phasehifts of the reflected signal and idler respectivly. Thus we get oscilatory count rates I i = I s = 2I 0 (1 + cos( φ)) (8) 5

6 where I 0 α 2 is the intensity of the respective modes when no mirrors are present and φ = φ s + φ i φ p. So it is seen that both the idler and the signal intensity vary when any of the mirrors is moved and the period will be given by half the wave lenght of the light who s mirror is moved. In the first experiment they extend there basic setup by putting a quarter wave plate (QWP) into the path of the reflected idler and upun correct orientation this will rotate the polarisation from vertical (V) to horizontal (H). Thus they are able to the distinguish the direct idler (V) from the reflected idler (H), buy putting a polariser in front of the idler detector. This will wash out the interference of both idler and signal, since which path information of the one implies which path information of the other. However it is possible to erase the which path information carried by the idler completly, if we measure its polarisation at an angle of 45 degree. In this case the authors only consider the cases of complet which path information or complet erasure, one could also have chosen to vary the orientation of the polariser continuously and one would have seen a continous loss of interference, as shown by Paul G. Kwiat, Aephraim M Steinberg and Raymond Y. Chiao in In the last two experiment the inseperability and nonlocality of the photon pair becomes much more apparent. Here they imprint the which path information of the idler photon in the signal photon, by two different methods - polarisation and timing. In the second experiment they also insert a QWP into the reflected signal path and employ a half wave plate (HWP) and a polarising beam splitter to measure the polarisation of the signal along different directions. The two entangled two photon state right after the crystal is know given by ψ = 1 2 ( V s V i + exp(i φ) H i H s ). (9) Note that this state also can be used to test bells inequality and that a vertical orientated signal corresponds to a direct idler and a horizontal signal to a reflected idler. Again the which path information can be erased but this time by measuring the signal polarisation at an angle of ±45. This is however not enough to restore 1. order coherences - it it would be enough we could send superluminal signal so only the second order coherrences are restored, as it can be seen on FIG 3. in the coincidence rates. 6

7 7

8 In the last experiment they introduce the possibility to obtain which path information by increasing the signal mirror to crystal distance by more than the coherncelenght (l c 260µm). Thus they can tell wether the idler was reflected or direct by the relative arrivel times of the photons. It can be seen in FIG.4 that the interference dissapears even though they in reality could not resolve the time delay (450µm). As a quantum eraser they this time used a interference filter, which increased the coherencelenght of to (800µm) and the second order coherences reappeared. For the last two experiments it is in principle possible to delay the choice as whether to erase or contain the which path information, until after the photons left the crystal. In principal it is even possible to delay this choice until after detection of the idler photon, eg. be increasing the opical path of the signal by inserting a optical fiber into the path. So the article strongly corroborates Niels Bohrs notion of complementatity between total which path information and interference and they propose a highly feasable delayed choice experiment. 8

arxiv:quant-ph/ v1 13 Jun 2001

arxiv:quant-ph/ v1 13 Jun 2001 A double-slit quantum eraser arxiv:quant-ph/1678v1 13 Jun 1 S. P. Walborn 1, M. O. Terra Cunha 1,, S. Pádua 1, and C. H. Monken 1 1 Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal