IOP PUBLISHING
JOURNAL OF OPTICS
J. Opt. 13 (2011) 064017 (6pp)
doi:10.1088/2040-8978/13/6/064017
Demonstration of the angular uncertainty principle for single photons B Jack1, P Aursand1, S Franke-Arnold1, D G Ireland1 , J Leach1 , S M Barnett2 and M J Padgett1 1 Department of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, UK 2 Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, UK
E-mail:
[email protected]
Received 20 December 2010, accepted for publication 1 March 2011 Published 28 April 2011 Online at stacks.iop.org/JOpt/13/064017 Abstract We present an experimental demonstration of a form of the angular uncertainty principle for single photons. Producing light from type I down-conversion, we use spatial light modulators to perform measurements on signal and idler photons. By measuring states in the angle and orbital angular momentum basis, we demonstrate the uncertainty relation of Franke-Arnold et al (2004 New J. Phys. 6 103). We consider two manifestations of the uncertainty relation. In the first we herald the presence of a photon by detection of its paired partner and demonstrate the uncertainty relation on this single photon. In the second, we perform orbital angular momentum measurements on one photon and angular measurements on its correlated partner exploring, in this way, the uncertainty relation through non-local measurements. Keywords: spatial light modulator, uncertainty relations, quantum entanglement
(Some figures in this article are in colour only in the electronic version)
1. Introduction
periodic nature of the angular variable means that care must be applied when considering its validity within quantum systems. A convenient method for defining the standard deviation of an angular distribution is to restrict the integral to a single cycle, say between θ and θ + 2π . The uncertainty relation that we investigate arises from studies of phase and angle operators [8–10], and has been tested experimentally with classical light [11]. It relates the standard deviation of the OAM distribution, !L z = !$h¯ (where z is the axis of propagation), to the standard deviation of the angular distribution !φθ by
The Heisenberg uncertainty principle imposes a fundamental limit upon the precision with which we can specify conjugate properties of a system [1]. The original form of Heisenberg’s uncertainty principle relates linear position and linear momentum, with the familiar form !x!p x ! h¯ . Beyond linear position and linear momentum, there are 2 uncertainty relations between any two incompatible variables which have a non-zero commutator [2]. In most cases the resulting lower bound is itself state dependent, which complicates the problem of finding the states that minimize the uncertainties [3]. The uncertainty relations also play a crucial role in demonstrations of non-local phenomena as manifested in the EPR paradox [4–7]. The key idea is that a local and realistic perspective allows us to construct a chain of inference, for entangled states, that brings us into conflict with the uncertainty principle. In this paper we investigate the uncertainty relation between the conjugate variables of angle and orbital angular momentum (OAM), measured on photon pairs produced in spontaneous parametric down-conversion (SPDC). The 2π 2040-8978/11/064017+06$33.00
!L z !φθ !
h¯ |1 − 2π P(θ )|, 2
(1)
where P(θ ) is the probability density at the boundary of the chosen angular range [10]. Without loss of generality, we set the boundary as θ = ±π . This angular form of the uncertainty relation is notably different from the linear case as the right-hand side of the inequality depends on the state under consideration. The intelligent state (i.e. state that saturates the inequality such that !$!φ = 12 |1 − 2π P(π)|) for the φ variable is a truncated Gaussian distribution wrapped in 2π , 1
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basis [17], where the probability amplitude for each mode being produced is determined by parameters such as the geometry of the crystal and the size of the pump beam [18]. As angular momentum is conserved in the down-conversion process and our pump beam is in the fundamental $ = 0 mode, the process is symmetric about $ = 0, i.e. $s = −$i where $s is the OAM state of signal photons, and $i is the OAM state of the idler photons. The probability amplitudes are also symmetric about $ = 0 (c$ = c−$ ). The two-photon field produced in this configuration can then be described by
whose Fourier transform gives the discrete OAM distribution with a near-Gaussian envelope. For small values of !φ , the probability P(π) is small, and the relationship simplifies to !$!φ = 1/2. However, for large values of !φ the value of the uncertainty product falls, and for a uniform angular √ distribution P(π) = 1/2π is a constant while !φ = π/ 3. In this case the uncertainty product becomes !$!φ = 0, i.e. !$ itself falls to zero. This is a direct consequence of the cyclic nature of angle; the Fourier transform of a flat distribution of angular position is a delta function in OAM, i.e. a pure OAM eigenstate. It is worth noting that in the case of the angular uncertainty relationship, the intelligent states are not the same as the minimum uncertainty states [12] (for the linear position– momentum case they are the same) because the right-hand side of the inequality depends on the angle. We note that the presence of P(θ ) in the uncertainty relation introduces the possibility of intelligent and minimum uncertainty states with large values of !φ [13, 14]. We test this angular uncertainty principle using correlated pairs of photons and use spatial light modulators (SLM) to measure the angular position of each photon, as well as to analyse the light in terms of its OAM distribution. We first show that this uncertainty relation holds at the single-photon level, using photons heralded by the coincident detection of the paired partner. We then investigate the uncertainty relation for correlated photon pairs using non-local measurements, by performing the angle state measurement and the OAM measurement on spatially separated optical paths. These single-photon results build upon the previous classical work of [11] and, in doing so, verify the uncertainty relationship at the single-photon level. The non-local measurements also allow us to explore the same uncertainty relation but are complicated by the fact that the generation of photon pairs has a restricted angular/OAM bandwidth.
|ψ$ =
c$ |$$s | − $$i ,
(4)
3. The experiment Our down-converted photons are generated by optically pumping a type I BBO crystal with a quasi-CW, mode locked, frequency tripled Nd:YAG laser at 355 nm and a pulse frequency of 100 MHz. The phase matching angle is such that the down-converted signal and idler photons are collinear with respect to the pump. Separation of signal and idler photons takes place at a 50/50 beam splitter, where the signal and idler beams are each imaged onto spatially separated SLMs. The light is then imaged into single-mode fibres with 100× objective lenses. In this configuration both the SLM and the fibre facet are positioned in image planes of the crystal. The magnification from fibre to SLM is approximately 600, and the demagnification from SLM to crystal face is approximately 10. This means that the back-projected signal and idler beams are much smaller than the 1 mm diameter pump beam, giving an overlap for a large range of OAM modes. Our coincidence counting device (Ortec) has a time resolution of !t = 10 ns, and we ensure frequency degeneracy of the measured photons by filtering with 2 nm wide band-pass filters centred around 710 nm. This experimental configuration is shown in figure 1. The use of diffractive optics and SLMs to couple only specific modes, or their superpositions, into single-mode optical fibre is now an established technique [20]. An important aspect of this technique is that when using offaxis holograms, where the desired modes are produced in the first diffracted order, the technique is robust to residual phase
Angular distributions and OAM are Fourier related, with the probability amplitudes of the angular states given by
and angular momentum amplitudes by " π 1 ψ$ = √ &(φ)e−i$φ dφ. 2π −π
$=−∞
where we have restricted our attention to modes whose radial mode index, p , is zero. For c$ = const. this is an infinite dimensional OAM state space. However, due to the finite size of the pump beam and phase matching conditions within the crystal, c$ decreases with increasing |$|. When this is combined with the measurement sensitivity which is limited by experimental noise, the result is that we can experimentally access around 20 OAM states. The Shannon dimensionality is related to the number of modes produced and their relative weightings in a given measurement basis [19]. The width of distribution of the OAM modes measured in a system is referred to as the spiral bandwidth [18], and is affected both by the distribution of modes produced (generation bandwidth) and by the experimental limitations on measurements (detection bandwidth).
2. Theory
+∞ 1 ! &(φ) = √ ψ$ ei$φ . 2π $=−∞
$=∞ !
(2)
(3)
This Fourier relationship shows that a restriction of angular width will result in a spread of the OAM distribution, with an envelope determined by the Fourier transform of the aperture [15]. When the transmission varies azimuthally as a Gaussian function, the resulting OAM spectrum is also approximately Gaussian in form. In analogy to the spreading of linear momentum from the restriction of linear position (diffraction), we refer to this effect as angular diffraction [16]. The light field produced in type I parametric downconversion can be decomposed in the Laguerre–Gaussian 2
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Figure 1. Experimental set-up and holograms used. Insets for signal and idler SLMs show example holograms used in the two different configurations to test the uncertainty relation—those for heralded local measurements of !$ and !φ (red inset) and those for the non-local measurements (blue inset).
modulation of the SLM (because the phase error only degrades the diffraction efficiency and not the mode purity). Also, by adjusting the blazing function over the aperture of the SLM it is possible to shape both the phase and intensity of the diffracted beam. In previous experiments, we have used photons generated in SPDC and measurements with SLMs to quantify entanglement in the OAM degree of freedom [21–23], through measurement of the non-local correlations at the detectors. Here we are not seeking to explore entanglement; rather we are using the correlated photon pairs to study the nature of the angular uncertainty relation. Thanks to the reprogrammability of the SLMs, we can test the uncertainty relation in two different experimental configurations. We vary the width of an angular mask (setting !φi ) in the idler arm to introduce uncertainty in the OAM of the photons, which in the first experiment is measured in the same arm as the mask (!$i ) and in the second is in the signal arm (!$s ), non-local with respect to the angular mask. In the first configuration, figure 1 (red inset), we are interested in testing the angular uncertainty relationship for measurements on the same single photon. Experimentally we realize this by detecting a photon in the signal arm ($s ) to herald the presence of a photon in the idler arm (−$i ). The spread in OAM introduced by the angular mask can then be measured by scanning through a range of spiral phases implemented on the same SLM as the angular mask itself. This measurement forms the basis from which we calculate !$i . In the second configuration, figure 1 (blue inset), we measure the uncertainty in the OAM state of the signal photon arising from the presence of the angular mask in the idler arm. In the signal arm we scan $s to obtain the OAM spectrum and hence the uncertainty !$s .
To summarize, in the first configuration we measure both !φ and !$ for the same photon, whereas in the second configuration we measure !φ for one photon and !$ for its correlated partner. In both of these configurations we vary the width of the angular mask, !φ , and measure the width of the associated OAM distribution, !$. By comparing the product !$!φ with the aforementioned inequality, we can then explore the uncertainty relation for both local and nonlocal measurement systems.
4. Measuring the uncertainty !" via maximum likelihood estimation We calculate !$ from the width of the measured OAM However, distribution for a given aperture width !φ . the accuracy of the measurements of this distribution is compromised in the two extreme cases of very wide and very narrow apertures. In the case of a very narrow aperture, the low number of correlated photons that reach the detectors means that any measurement is subject to a high noise component. This can be partially improved by increasing the time over which the data are collected. However, the presence of a comparatively large noise floor means the overestimation of !$ is inevitable. In the other extreme of no angular restriction we anticipate a value of !$ = 0. In this situation, any noise in the measurements results in an overestimation of !$. To improve our estimation of !$, we use a maximum likelihood estimate (MLE) technique. Our MLE model assumes that the OAM distribution is Gaussian in form, giving a predicted coincidence count of # $ ($i − $s )2 C = Ss × η × exp − (5) 2(!$m )2 3
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Figure 2. Test of the uncertainty principle for heralded single photons in the idler arm (by measuring $s = 0 photons in the signal arm). (a) Width of the measured OAM distribution !$i for width of angular distribution !φi . (b) Uncertainty product !$i !φi against !φi . The insets shown are examples of the angular transmission functions for different !φi .
where !$m is the modelled width of the OAM distribution, Ss and Si are the single-channel count rates of signal and idler photons, and η is the quantum efficiency of signal and idler detection apparatus. There is an accidental coincidence rate— given by A = Ss × Si × !t , where Ss , Si are the singlechannel count rates and !t is the gate time of the coincidence detector. This gives a total predicted number of coincidence counts λ = A + C . We assume a Poisson distribution of anticipated counts and therefore use a log-likelihood function of the form
˜ = ln( L)
N ! j =1
(n j ln(λ j ) − λ j ),
(6) Figure 3. Measured OAM distribution shown with and without spiral bandwidth compensation for an angular aperture width !φs = 0.2.
where n j is the actual number of counts measured. By comparing the measured counts n j against each λ j for a range of possible !$m values, we extract the most likely value of !$. From this maximum likelihood function, we can separate the coincidence signal from the accidental ones. From this corrected OAM spectrum we obtain a best estimate of the width !$.
few accidental coincidences results again in an overestimation of !$i .
6. Testing the uncertainty relationship for !" and !φ for separated photon pairs
5. Testing the uncertainty relationship for !" and !φ for the same photon
In the second configuration the spread !φi is measured for the idler photon and !$s is measured for its correlated photon in the signal beam. The entangled nature of the photon pairs suggests that the OAM spectrum should be the same whether the angle and angular momentum measurements are spatially separated or not. However, the measurement process which gives rise to these non-local results is different from that in the previous configuration. In this non-local configuration, the measured OAM distribution is also dependent on the number of modes which can be measured and, as discussed, this bandwidth needs to be taken into account. One way to interpret the non-local influence of the angular mask is through ghost imaging using coincident photon pairs [24]. By introducing an angular mask in the signal arm,
Shown in figure 2 are the results for the first experimental configuration, where !φi and !$i are measured for the same photon. For a given angular width, !φi (yet, with a randomly chosen mean azimuthal angle φi ), the spread in the OAM, !$i , is compared to the theoretical value of [11]. We see a close match of experimental results with the theoretical predictions. It can be seen that for both small and large values of !φi , !$i , and thus !$i !φi , are consistently overestimated. For small values of !φi , the number of coincidences measured is significantly reduced, which in turn reduces the effectiveness of the MLE fit. For large values of !φi , we would anticipate √ a small value of !$i . For no angular restriction (!φi = π/ 3), we predict !$i = 0. However, even the presence of a 4
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Figure 4. Non-local uncertainty measurements. (a) !$s measurements for a given !φi with and without spiral bandwidth compensation compared with theory. (b) Uncertainty product with and without spiral bandwidth compensation.
These measurements contain within them information about the OAM generation bandwidth of our source. A high dimensional, maximally entangled state would be an ideal case on which to perform these non-local measurements.
we also produce a ghost aperture in the idler arm. However, our source of photon pairs both has a finite aperture and produces photon pairs with a finite mode spectrum. This means that the ghost aperture in the signal arm is not a perfect reconstruction of the aperture in the idler arm—its spectral decomposition is restricted by the number of modes produced by the crystal. The narrower the angular width !φi , the greater the number of modes which are present in its OAM spectrum. This means that the effect of a limited OAM mode spectrum is more pronounced for narrow apertures. Given that we know the number of modes which are produced in our experiment, we can use the measured spiral bandwidth to compensate for the underestimation of !$i . Multiplying the measured OAM spectrum by the reciprocal of the spiral bandwidth increases the amplitude of the higher order OAM components, and compensates for the bandwidth limitations of our system (figure 3). Conversely, the value of the uncertainty product !$i !φs as it deviates away from the predicted value indicates the actual spiral bandwidth of the experiment. Shown in figure 4 are the results for non-local measurements !$s and !φi . As described, we see that the uncertainty product falls for small values of !φi . This is because as the width !φi tends to zero, !$s tends to a constant (equal to the width of the OAM spectrum of the system).
Acknowledgments This work was supported by the UK EPSRC. SMB and MJP thank the Royal Society and the Wolfson Foundation. We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under the FET Open grant agreement HIDEAS number FP7-ICT-221906 and Phorbitech. We would like to thank Hamamatsu for their support of this work.
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7. Conclusions The uncertainty relation for angle and angular momentum has been tested previously in the regime of classical optics [11]. In this paper we report a test in the manifestly quantum regime of single photons. Our results are fully consistent with the theoretically expected uncertainty relation: !$!φ ! 12 |1 − 2π P(θ )|. In a second experiment we measure !φ and !$ for different but correlated photons. There is, of course, no restriction analogous to an uncertainty relation on the product of these uncertainties. The fact that they are related is a consequence of the correlations between the angular momenta of the two photons. By producing a ghost aperture in the signal arm, we measure the OAM spread of signal photons. 5
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