Measurement of orbital angular momentum of a single photon: a noninterferrometric

Transcription

1 Measurement of orbital angular momentum of a single photon: a noninterferrometric approach R. Dasgupta and P. K. Gupta * Biomedical Applications Section, Centre for Advanced technology, Indore, Madhya Pradesh, India ABSTRACT We report observation of a shift, in the frequency of light, having non-zero orbital angular momentum (OAM), when it undergoes multiple total internal reflections from a liquid-glass interface. The frequency shift was found to be proportional to the OAM state of the light and thus provides a non interferrometric means to determine OAM state of the photon. Keywords: Orbital angular momentum, single photon, rotational Doppler effect There is considerable current interest in the use of light angular momentum for quantum information processing. The spin angular momentum (polarization) of photon has already been used to demonstrate physical realization of a single qubit. 1 The use of orbital angular momentum (OAM) of light would allow the generation and manipulation of qunits, that is quantum states in an N-dimensional space. Such multidimensional quantum states will allow enhanced capacity and more secure key distribution in quantum cryptography 3. For retrieval of information coded in qunits, means to detect the orbital angular momentum state of a single photon are needed. This has motivated considerable efforts on development of approaches that can meet this objective. Computer generated holograms 4 that transform photon with a particular OAM state into l=0 state have been used to test photons for that particular OAM value by exploiting the fact that l=0 state has an on-axis intensity maximum, and therefore couples efficiently through a pin hole or single-mode fiber on focusing. 5 However, use of this approach to test photons for N values of l will require that the beam be split into N different parts, each of which is then tested for one of the l values of interest. Such an approach will automatically limit the quantum efficiency of an N-state measurement to 1/N. A major recent development was to make use of cascaded stages of Mach-Zehnder interferometers where by use of suitably oriented dove prisms, beam in one arm is rotated with respect to the other to result in an l-dependent phase difference in the two optical paths. 6, 7 The rotation angle could be chosen such that photons with specific values of l interfere constructively at, and exit through one output port, while those with other values interfere constructively at, and exit through the second port. It was shown that a cascade of such interferometers, in conjunction with, either the use of holograms to add appropriate values of l to the passing photons 6 or by slight adjustment of the length of one arm of some of the interferometers 7, could sort OAM states of photons. Two serious drawbacks of this approach are; the inherent susceptibility of this method to mechanical vibrations, and that the uncertainty of OAM determination increases with increasing l value because the photon needs to pass through increasing number of cascaded interferometers. In this letter we report a non-interferrometric approach for sorting OAM states of photons that will be free from these drawbacks. This approach exploits our observation of a shift, in the frequency of light, having non-zero OAM, when it undergoes multiple total internal reflections from a liquid-glass interface. The frequency shift was found to be proportional to the OAM state of the light and thus provides a non interferrometric means to determine OAM state of the photon. The use of this approach to sort photons with OAM states of l = 0, ± 1,,± 5 is discussed. In Fig. 1 we show a Laguerre-Gaussian (LG) beam with non-zero OAM undergoing total internal reflection at the liquidglass boundary. Reflection reverses the sign of the OAM state of the beam and should therefore result in transfer of OAM to the interface.

2 N lħ θ θ lħ l Glass N' 1-Bromonapthalene Figure 1. LG beam with non-zero OAM undergoing total internal reflection at 1-bromonapthalene-glass interface. For a linearly polarized photon the net orbital angular momentum transferred ( l) at the interface is given as, l = l ħ (1-cosθ) 0.5 (1) Where, θ is the angle between the direction of the incident beam and the surface normal NN'. A transfer of OAM to freely rotating liquid molecules 8, 9, 10 at the interface will change the angular momentum state of the molecule from L 1 to L. It has also been shown that since electronic motions do not contribute to exchange of OAM in case of dipole transition 8 the internal energy of the molecule remains the same. Therefore, making use of the fact that the rotational kinetic energy is given by E = L /, (I being the moment of inertia of the molecule), one can write the following equation for conservation of energy L1 L = + h ν () Conservation of angular momentum will require that L = l. Therefore, ν the shift in the frequency of the light arising due to transfer of OAM can be written as 1 L L 1 L( L1 + L ) l. L ν = h = (3) h Ih Putting the value of l from equation (1) and making use of the fact that L = πiω, where Ω is the rotational frequency of the liquid molecule we get

3 ν =1.411 lω (1-cosθ) 0.5 (4) Since each transfer of OAM will impart a frequency shift, to ensure easily measurable frequency shift, the LG beam can be made to undergo a large number of total internal reflections at the interface. The experimental arrangement used to look for the expected shift in frequency of LG beam undergoing a large number of total internal reflections at a glass liquid interface is shown in Fig.. A 1 mw He:Ne laser (63.8 nm) was used to generate a Laguerre-Gaussian beam with azimuthal index varying from l = -5 to +5. We used computergenerated holograms for generating LG beam with desired l value. A beam splitter (B) was used to combine the He:Ne laser beam with another LG beam with l=1, generated from a frequency doubled Nd:YAG laser ( 53 nm, 10 Hz, ~ 7 ns, mj per pulse). The combined beams were coupled to a 1.5 m long, 4 mm inner diameter glass tube, filled with 1- Bromonapthalene at an angle of ~70 with respect to the normal to the interface. Since the index of refraction for 1- Bromonapthalene and glass are 1.65 and 1.51 respectively leading to a critical angle of ~ 66, the two beams propagated through the tube via total internal reflections inside the glass tube. The approximate number of total internal reflections suffered by the two beams before coming out from the other end of the tube was estimated to be around 130. The He:Ne laser LG beam emerging from the tube was coupled to a scanning grating monochromator (Oriel ¼ m, Model 7700, having 100 lines/mm grating) to measure the expected frequency shift. With 0.1 mm slits at both input and output sides the resolution of the system is 0.1 nm. M3 Glass Tube M1 1-Bromonapthalene 53 nm, l=1 M4 Dispersive element 633 nm l=1 l= l=n 53 nm M B 633 nm, l =0, 1,,., N Figure. Experimental set-up for measuring orbital angular momentum state of a light. In Fig. 3 we show the measured shift in the wavelength of the emerging He-Ne laser LG beam with different values of l. For OAM state with l = 1 the observed shift was ~ 0.4 nm which could be conveniently measured by the monochromator. The shift is seen to be proportional to the OAM state of the light with light having OAM ranging from 1 to 5 showing a up shift and light with negative values of OAM showing down shift of wavelength. This implies that in the energy exchange process between rotating liquid molecules and photons, photons with positive l values have lost part of their energy to the liquid molecules and those with negative l values have gained energy from rotating liquid molecules. It is important to emphasize that no shift in the frequency of the He-Ne laser LG beam was observed when the frequency doubled Nd:YAG laser LG beam was blocked. Further, the sign of frequency shift was observed to change if the azimuthal index of the pulsed 53 nm LG beam was changed from +1 to 1. In such condition the sense of rotation of the liquid molecules will be reversed, so the energy exchange procedure will get changed accordingly. We assume that the function of the high power pulses is to overcome the surface tension existing at the glass- bromonapthalene interface and render the liquid molecules as freely rotating. The energy of 53 nm pulses (~ mj) is sufficient to overcome the force of adhesion. This follows because the surface energy at the liquid surface (πr S.secθ, where r is the radius of beam spot at the liquid-glass interface, S the surface energy of 1-bromonapthalene (45 mj/m ) and θ is the angle of the surface normal and the beam) has a maximum value of 4.13 x 10-4 mj for θ ~ 70 and r ~ 1mm.

4 Wavelength Shift (nm) Orbital Quantum Number -1 - Figure 3. Observed rotational frequency shift of LG photons according to their orbital angular momentum states. A multi-channel detection of frequency of the LG beam can be used for a spatial discrimination of the photons with different OAM states. The efficiency of this method will only be limited by the first-order diffraction efficiency of the grating which can be ~ 90% for a blazed grating. Further a combination of prisms providing the required resolution 11 can also be used to have a completely deterministic system. The use of a longer glass tube will further enhance the shift and may facilitate use of a single prism to spatially discriminate the wavelength shifted beams. Very low absorption coefficient of 1-bromonapthalene at the He:Ne laser wavelength implies that longer tubes can be used with no significant loss of photons. However, with increasing length of glass tube, ensuring spatial overlap of the two beams will becomes more demanding. To check for the validity of the scheme at single photon level, we followed the approach used in Ref. 8. Output of the 1mW He:Ne laser was attenuated down to ~ 1nW using a series of neutral-density filters. This intensity is so low that we can expect only one photon to be present inside the monochromator at one time. The photons coming out of the output slit of the monochromator were detected using a photomultiplier tube. The wavelength shifts measured was in proportion with the orbital angular momentum number and clearly demonstrate the validity of our scheme for determination of orbital angular momentum state at single photon level. To conclude a shift has been observed in the frequency of light having non-zero orbital angular momentum when it undergoes multiple total internal reflections from a liquid-glass interface in presence of an overlapping pulsed laser LG beam. The frequency shift was found to be proportional to the OAM state of the light and provides a convenient, non interferrometric means for determination of orbital angular momentum state at single photon level. The use of liquid light guides in place of the glass tube used in present experiments needs to be explored, as that would make the approach particularly easy to use. ACKNOWLEDGMENTS Authors would like to thank S. K. Majumder, S. K. Mohanty and K. Das for their valuable suggestions and technical input.

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