Simplified algebraic description of weak measurements with Hermite–Gaussian and Laguerre–Gaussian pointer states

Optics Communications 331 (2014) 194–197

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Simplified algebraic description of weak measurements with Hermite–Gaussian and Laguerre–Gaussian pointer states Bertúlio de Lima Bernardo n, Sérgio Azevedo, Alexandre Rosas Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa, PB, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 23 January 2014 Received in revised form 5 June 2014 Accepted 6 June 2014 Available online 20 June 2014

Weak measurements are recognized as a very powerful tool in measuring tiny effects that are perpendicular to the propagation direction of a light beam. In this paper, we develop a simple algebraic description of the weak measurement protocol for both Laguerre–Gaussian and Hermite–Gaussian pointer states in the Schrödinger representation. Since a novel class of position and momentum expectation values could be derived, the present scenario appeared to be very efficient and insightful when compared to analytical methods. & 2014 Elsevier B.V. All rights reserved.

Keywords: Weak measurements Hermite–Gauss beams Laguerre–Gauss beams

1. Introduction Ideal measurements in quantum mechanics provide a well resolved distinction between the different eigenstates of the measured observable. Still, the price to pay is that the system is usually disturbed in an unpredictable way. Indeed, a standard quantum measurement necessarily projects the system onto one of the observable eigenstates. However, opposite to this usual approach, in 1988 a seminal paper of Aharonov, Albert, and Vaidman (AAV) introduced the notion of weak measurement, which is an experimental procedure to obtain information from a quantum system without greatly disturbing its state [1]. The characteristic trait of AAV's proposal is that the coupling between the measurement device M and the system S is extremely small such that their entanglement, which is an unavoidable feature in the standard measurement scheme [2], is prevented. Specifically, the authors considered the case where this weak coupling takes place between two strong measurements called pre-selection and postselection. The former is the preparation of the initial quantum state, jψ i 〉, whereas the latter amounts to the selection of a final state, jψ f 〉. The institution of the weak measurement scheme, similar to the standard measurement process, emerged from the von Neumann measurement model. In this formalism, S and M are coupled by an interaction depicted by the following Hamiltonian: H^ ¼ g A^  P^x ;

ð1Þ

where A^ is an Hermitian operator corresponding to the observable n

Corresponding author. E-mail address: bertulio@fisica.ufpb.br (B. de Lima Bernardo).

http://dx.doi.org/10.1016/j.optcom.2014.06.008 0030-4018/& 2014 Elsevier B.V. All rights reserved.

A of S, P^x is the momentum operator related to M, and g is the coupling constant, which differs from zero during the interaction time t. The main result of AAV's work is that, for a Gaussian probe, if the interaction is weak (so that the shift in the pointer position is much smaller than its initial uncertainty), after the post-selection, the position expectation value of the pointer state is given by 〈x〉f ¼ gt ReðAw Þ, where Re denotes the real part, and Aw is the socalled weak value, which is given by [1,3] Aw ¼

^ ψ〉 〈ψ f jAj i : 〈ψ f jψ i 〉

ð2Þ

Conversely, the momentum expectation value of the pointer depends on the imaginary part as 〈P x 〉f ¼ ðℏgt=2σ 2 Þ ImðAw Þ, with σ being the Gaussian uncertainty [4,5]. From the application point of view, weak measurements have been used to address foundational questions in quantum mechanics, such as Hardy's Paradox [6], the direct measure of the wave function [7], and the tunneling time [8]. Besides, it serves as a strong amplification method [9]. In fact, the possibility of the weak value becomes arbitrarily large when the overlap between the initial and final states, 〈ψ f jψ i 〉, is sufficiently small was used, for example, to produce superluminal and slow light propagation [10], to measure the angular deflection of a mirror to a precision of the order of 500 frad [11], and to detect the spin Hall effect of light [12]. In this paper we report a simplified study of the weak measurement protocol where the well known Hermite–Gauss (HG) and Laguerre–Gauss (LG) modes are used as pointer states. Recently, these two types of probes have been analytically studied [13–15], but constraints on the order of the modes were always necessary, so that a full description was never presented. Here we used the

B. de Lima Bernardo et al. / Optics Communications 331 (2014) 194–197

195

algebra of the quantum harmonic oscillator to circumvent this limitation. This framework allowed us to greatly simplify the calculations when compared to the analytical studies. Indeed, this is the first time that weak measurements could be described both in terms of the radial and azimuthal indices of the LG modes. A discussion about the benefits and validity conditions of the HG and LG probes is also provided.

creation and annihilation operators yield pffiffiffi a^ x jn; m〉 ¼ njn  1; m〉 pffiffiffiffiffiffiffiffiffiffiffi † a^ x jn; m〉 ¼ n þ 1jn þ 1; m〉 pffiffiffiffiffi a^ y jn; m〉 ¼ mjn; m  1〉 pffiffiffiffiffiffiffiffiffiffiffiffi † a^ y jn; m〉 ¼ m þ1jn; m þ 1〉:

2. Weak measurement with Hermite–Gaussian pointer states

Hence, by using (Eqs. (8) and 9) we can calculate the expectation value of the post-selected pointer state being

To begin with, let us consider the weak interaction between the pre-selected state jψ i 〉 of the system and the HG probe state jn; m〉, where n and m are the different mode indices of the HG beams [16]. The total initial state is then jΨ i 〉 ¼ jψ i 〉jn; m〉, whose evolution will be dictated by the von Neumann Hamiltonian in two dimensions: H^ ¼ g A A^ P^x þ g B B^ P^y :

ð3Þ

This equation describes the coupling between the system observable A^ with the x direction of the pointer state, and the observable B^ with the y direction. The observables P^x and P^y are the conjugate momenta associated with the position operators X^ and Y^ , respectively. The parameters gA and gB are the coupling constants. In the scenario of the Schrödinger representation, if we apply the evolution operator to the initial state by considering terms up to first order in the coupling constants, we can write ! ! 2 ^ ^  iHt iHt H^ t 2  jΨ ðtÞ〉 ¼ exp þ ⋯ jψ i 〉jn; m〉 jψ i 〉jn; m〉 ¼ 1  ℏ ℏ 2ℏ2  jψ i 〉jn; m〉 

ig A t ^ ig t ^ Ajψ i 〉P^x jn; m〉  B Bj ψ i 〉P^y jn; m〉: ℏ ℏ

ð4Þ

It is known that general HG modes can be generated from the fundamental Gaussian mode, j0; 0〉, by using the creation and annihilation operators: a^ x ¼ X^ =2σ þ iσ P^x =ℏ † a^ ¼ X^ =2σ  iσ P^x =ℏ x

† a^ y ¼ Y^ =2σ  iσ P^y =ℏ;

ð5Þ

in the following form [16]:   1 † † n; m〉 ¼ pffiffiffiffiffiffiffiffiffi ffiða^ x Þn ða^ y Þm j0; 0〉:  n!m! †

ig A t ^ ψ 〉P^x jn; m〉 〈ψ f jAj i ℏ

ig B t ^ ψ 〉P^y jn; m〉: 〈ψ f jBj i ℏ

ð7Þ

If a renormalization is performed, and we rewrite Eq. (7) in terms of the creation and annihilation operators, we obtain that †

jϕf 〉 ¼ jn; m〉 þ g A t〈A〉w ða^ x  a^ x Þjn; m〉 †

〈P x 〉f ¼

ℏg A t Im 〈A〉w ð2n þ 1Þ; 2σ 2

ð11Þ

ℏg B t Im 〈B〉w ð2m þ 1Þ: 2σ 2

ð12Þ

and 〈P y 〉f ¼

These results had been found in Ref. [13] without explicitly showing the modal dependence on n and m. As we can see, high-order HG probes provide an enhancement in the amplification of the momentum measurements which is linearly proportional to the order of the mode. Then, high-order HG probes could be an important experimental tool in measuring small effects by observing the conjugate variables. It is interesting now to calculate the joint expectation value of the position observables, 〈XY〉f . In doing so, if one considers terms only up to first order in the coupling constants (see Eq. (4)), the result is zero. However, if we proceed to calculate terms up to second order, we obtain

þ g B t〈B〉w ða^ y  a^ y Þjn; m〉:

   gA gB t2 ½ Reð〈A〉nw 〈B〉w Þ þ Re ðAB þ BAÞ=2 w : 2

ð13Þ

This same result had been found for Gaussian states in order to extract joint weak values using only local, single-particle Hamiltonians [17]. Here we demonstrate that it is independent of the modes n and m. That is, high-order HG probes cannot improve this kind of measurement when compared to Gaussian pointer states.

ð6Þ

These operators obey the standard commutation rule ½a^ i ; a^ j  ¼ δij , and allow us to write the position and momentum operators as † † † X^ ¼ σ ða^ x þ a^ x Þ, Y^ ¼ σ ða^ y þ a^ y Þ, and P j ¼ ðiℏ=2σ Þða^ j  a^ j Þ, with j ¼ x; y. Accordingly, after the weak interaction, if we post-select (project) the quantum system onto the state jψ f 〉, the pointer state is left in the following final state: jϕf 〉 ¼ 〈ψ f jΨ ðtÞ〉 ¼ 〈ψ f jψ i 〉jn; m〉 

ð10Þ

Likewise, one can find that 〈Y〉f ¼ g B t Re〈B〉w . Notice that the expectation values of the position operators are equal to those of the Gaussian mode [4], that is, there exists no modal dependence for these observables. On the other hand, if we calculate the expectation values of the conjugate momenta, we obtain that

〈XY〉f ¼

a^ y ¼ Y^ =2σ þ iσ P^y =ℏ



〈X〉f ¼ 〈ϕf jX^ jϕf 〉 ¼ g A t Re〈A〉w :

ð9Þ

ð8Þ

^ ψ 〉=〈ψ jψ 〉 and 〈B〉w ¼ 〈ψ jBj ^ ψ 〉=〈ψ jψ 〉, The factors 〈A〉w ¼ 〈ψ f jAj i f i f i f i according to Eq. (2), are the weak values of the observables A^ and ^ respectively. The HG modes jn; m〉 when operated by the B,

3. Weak measurement with Laguerre–Gaussian pointer states Now, let us turn to the analysis of weak measurements with LG probes. Here, we extend previous works [14,15] by providing a full description in terms of both the radial index p, and the azimuthal index l. Like HG modes, LG modes also define a possible set of basis vectors for paraxial light beams. It was predicted [18] and experimentally observed [19] that this class of modes, besides the intrinsic spin angular momentum 7 ℏ, related to the light polarization, each photon carries a quantized orbital angular momentum (OAM) along the propagation direction given by 7 lℏ. This extra degree of freedom is related to the jlj intertwined helical wave fronts, whose handedness is given by the sign of l. Moreover, these beams present a topological phase singularity (optical vortex) at the beam axis. Herein, similar to the previous section, we apply a quantum mechanical operator formalism to describe the behavior of general LG modes in a pre- and a post-selected weak interaction. The interaction Hamiltonian is the same as in Eq. (1), however, instead of the standard creation and annihilation operators, which rule HG

196

B. de Lima Bernardo et al. / Optics Communications 331 (2014) 194–197

modes, we introduce the following ladder operators: pffiffiffi a^ 7 ¼ 1= 2½a^ x 8 ia^ y  pffiffiffi † † † a^ 7 ¼ 1= 2½a^ x 7 ia^ y ;

Next, we calculate the momentum expectation values being

ð14Þ

which generate high-order LG modes in the following way [16]: 1 † † js; q〉 ¼ pffiffiffiffiffiffiffiffiffiða^ þ Þα ða^  Þβ j0; 0〉: α !β !

ð15Þ

The integers α and β are defined by α ¼ ðs þ qÞ=2 and β ¼ ðs  qÞ=2, whereas the parameters s and q are related, respectively, to the radial and azimuthal indices of the LG modes by the relations p ¼ ðs  jljÞ=2 and l¼ q. Notably, the operators in Eq. (14) comprise the algebra of LG modes; nevertheless, they were originally derived by Schwinger to describe the angular momentum of the two dimensional harmonic oscillator [20,21]. The appearance of these operators here can be explained by the fact that the Helmholtz paraxial equation of light has the same form of the Schrödinger equation [22], and LG modes, as well as the two dimensional harmonic oscillator, deliver the same set of solutions in polar coordinates. These ladder operators obey the Bosonic commutation rules ^ ^ ½a ^7 ; a†7  ¼ 1 and ½a ^7 ; a†8  ¼ 0, which can be used to rewrite the position and momentum operators as follows: pffiffiffi † † X^ ¼ σ = 2½a^ þ þ a^  þ a^ þ þ a^   pffiffiffi † † Y^ ¼ iσ = 2½  a^ þ þ a^  þ a^ þ  a^   pffiffiffi † † P^x ¼ iℏ=2 2σ ½a^ þ þ a^   a^ þ  a^   pffiffiffi † † ð16Þ P^y ¼ ℏ=2 2σ ½a^ þ  a^  þ a^ þ  a^  : The application of the ladder operators in the state js; q〉 is given by [16] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a^ þ js; q〉 ¼ ðs þ qÞ=2js 1; q  1〉 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † a^ þ js; q〉 ¼ ðs þ q þ 2Þ=2js þ 1; q þ1〉 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a^  js; q〉 ¼ ðs  qÞ=2js 1; q þ 1〉 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † a^  js; q〉 ¼ ðs  q þ 2Þ=2js þ 1; q 1〉: ð17Þ In this manner, we can express the system-probe state after the weak interaction as ! ! 2 ^ ^ H^ t 2  iHt iHt 0  jΨ ðtÞ〉 ¼ exp þ ⋯ jψ i 〉js; q〉 jψ i 〉js; q〉 ¼ 1  ℏ ℏ 2ℏ2  jψ i 〉js; q〉 

ig A t ^ ig t ^ ψ i 〉P^y js; q〉: Ajψ i 〉P^x js; q〉  B Bj ℏ ℏ

ð18Þ

Therefore, if we substitute Eqs. (16) into Eq. (18), and apply the algebra of Eqs. (17), by post-selecting the state jψ f 〉, similar to Eq. (7), after some calculations, the expectation values of the position operators are found to be 〈X〉f ¼ 〈ϕ

0 ^ f jX j

ϕ

0 f〉¼

g A t Re〈A〉w þ l½g B t Im 〈B〉w :

ð19Þ

0 0 〈Y〉f ¼ 〈ϕf jY^ jϕf 〉 ¼ g B t Re〈B〉w  l½g A t Im 〈A〉w ;

ð20Þ

and

0 f〉

where jϕ is the probe state after the post-selection. As we can see, unlike Gaussian states, the spatial displacements along x and y for high-order LG modes depend not only on the real part of the weak value, but also on the imaginary part. This is due to the fact that LG modes, contrary to HG modes, are not factorable in functions which depend only on x and y. It causes a coupling between the observables A^ and B^ with the x and y dimensions of the pointer state. These results were found in Ref. [15] with the constraint p ¼0. Here, we demonstrate that the position expectation values are independent of the radial index p.

ℏg t 0 0 〈P x 〉f ¼ 〈ϕf jP^x jϕf 〉 ¼ A2 Im〈A〉w ð2p þ jlj þ1Þ; 2σ

ð21Þ

and ℏg t 0 0 〈P y 〉f ¼ 〈ϕf jP^y jϕf 〉 ¼ B2 Im〈B〉w ð2p þ jlj þ 1Þ: 2σ

ð22Þ

Observe that these expressions depend both on the radial and azimuthal indices, and that the higher the order of the mode, the stronger the amplification properties. However, the shift in the momentum expectation value depends more strongly on the radial index p. As expected, (Eqs. (21) and 22) reduce to the momentum expectation value of the Gaussian probe (p ¼ l ¼ 0) [4]. In the specific case of LG probes, it is interesting to verify which is the final OAM in the propagation direction, Lz. In terms of the position and momentum operators we have that L^ z ¼ X^ P^y  Y^ P^x , † † which renders L^ z ¼ ℏða^ þ a^ þ  a^  a^  Þ. Thence, it can be shown that the final OAM expectation value is given by 0 0 〈Lz 〉f ¼ 〈ϕf jL^z jϕf 〉 ¼ ℏl:

ð23Þ

That is, the weak measurement process does not change the average OAM of the photons. It was expected, since the Hamiltonian couples the observables only with the transverse linear momenta of the probe. For the sake of comparison with the HG pointer states, we now calculate the joint expectation value of the position operators, 〈XY〉f . Once again, if we consider terms only up to first order in the coupling constants, a null expectation value is obtained. However, when second order terms are taken into account, after some calculations (See Eq. (18)), we derive the following expression: gA gB t2 Reð〈A〉nw 〈B〉w Þ½f ðp; lÞ þ 2 4 g g t2  A B Reð〈ðAB þ BAÞ=2〉w Þ½f ðp; lÞ  2 4 l 2 2  ½g A t Im〈A2 〉w  g 2B t 2 Im〈B2 〉w ; 2

〈XY〉f ¼

ð24Þ

2

with f ðp; lÞ ¼ 2ðp2 þpjlj þ pÞ l þ jlj. Notice that Eq. (24) coincides with Eq. (13) for p ¼ l ¼ 0 (Gaussian probe) [17]. For p ¼0 this result is reduced to those found in Refs. [14,15]. Actually, these references have shown that the joint expectation value of LG modes with p¼ 0 and l a 0 has modal dependence on l. This fact could be used to extract joint weak values of single-particle operators. Here we showed that the joint expectation value also depends on the radial index p. This extra degree of freedom can also be handled in order to obtain joint weak values.

4. Validity conditions As a last point, we address the validity conditions for weak measurements with both HG and LG pointer states. For the general case, it was obtained in Ref. [23] that weak measurements obeying the first-order (linear in gt) approximation are limited by the following condition: gt j〈A〉w jΔp{1; ℏ

ð25Þ

with Δp being the initial momentum uncertainty of the pointer (See Ref. [3] for review). Then, by using the relation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δp ¼ 〈p2 〉 〈p〉2 , it is possible to demonstrate that the validity conditions for weak measurements with HG pointer states, according to the Hamiltonian of Eq. (3), for the x and y directions are given, respectively, by pffiffiffiffiffiffiffiffiffiffiffiffiffi gA t j〈A〉w j 2n þ 1{1 ð26Þ 2σ

B. de Lima Bernardo et al. / Optics Communications 331 (2014) 194–197

and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gB t j〈A〉w j 2m þ 1{1: 2σ

ð27Þ

Similarly, if we use the algebra presented in the last section, we find the validity relations for LG pointer states with respect to the x and y directions as given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gA t j〈A〉w j 2p þ jlj þ 1{1 ð28Þ 2σ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gB t j〈A〉w j 2p þ jlj þ 1{1; 2σ

197

calculate for the first time how the LG expectation values depend on the radial index p. As a matter of fact, the present results are consistent with recent literature [4,14,15,17]. In what concerns the benefits of high-order HG and LG probes in weak measurements, we can conclude by saying that, in general, LG modes take advantage over HG modes both in the amplification properties and experimental implementation for the extraction of joint weak values. Also, in a recent paper [24], weak measurements with LG probes were shown to be useful in the realization of quantum tomography of light polarization states.

ð29Þ

respectively. From these relations, we see that the validity conditions for the fundamental Gaussian state are

Acknowledgments

gA t j〈A〉w j{1 2σ

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

ð30Þ

and gB t j〈A〉w j{1; 2σ

ð31Þ

for the x and y directions, respectively. Note that, in general, whether a measurement is weak or not dependent on the mode and uncertainty of the pointer, as well as on the weak value, 〈A〉w . By investigating these relations, we can see that for high-order HG and LG probes, the coupling constants gA and gB (or the weak value) must be smaller than for the Gaussian probe in order to satisfy the general validity condition, Eq. (25). However, these relations depend on the square root of the mode indices, whereas the amplification of the average momenta of the pointers depends linearly on the indices. See (11), (12), (21) and (22). This ensures that the amplification properties of both HG and LG probes outperform the Gaussian pointer in what concerns the enhancement of the shift of the final momentum distribution. 5. Conclusion In this work, we have provided a simple description of weak measurements with both HG and LG pointer states. In doing so, contrary to the usual analytical methods [14,15], the operator algebra of the quantum harmonic oscillator was applied in the Schrödinger representation. This procedure allowed us to calculate the position and momentum expectation values (single and joint) with more simplicity and efficiency. Notably, we were able to

References [1] Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev Lett. 60 (1988) 1351. [2] J. von Neuman, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955. [3] A.G. Kofman, S. Ashhab, F. Nori, Phys. Rep. 520 (2012) 43. [4] J.S. Lundeen, K.J. Resch, Phys. Lett. A 334 (2005) 337. [5] R. Jozsa, Phys. Rev. A 76 (2007) 044103. [6] J.S. Lundeen, A.M. Steinberg, Phys. Rev. Lett. 102 (2009) 020404. [7] J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, Nature (London) 474 (2011) 188. [8] A.M. Steinberg, Phys. Rev. Lett. 74 (1995) 2405. [9] K.J. Resch, Science 319 (2008) 733. [10] D.R. Solli, C.F. McCormick, R.Y. Chiao, S. Popescu, J.M. Hickmann, Phys. Rev. Lett. 92 (2004) 043601. [11] P.B. Dixon, D.J. Starling, A.N. Jordan, J.C. Howell, Phys. Rev. Lett. 102 (2009) 173601. [12] O. Hosten, P. Kwiat, Science 319 (2008) 787. [13] J. Dressel, A.N. Jordan, Phys. Rev. Lett. 109 (2012) 230402. [14] G. Puentes, N. Hermosa, J.P. Torres, Phys. Rev. Lett. 109 (2012) 040401. [15] H. Kobayashi, G. Puentes, Y. Shikano, Phys. Rev. A 86 (2012) 053805. [16] G. Nienhuis, L. Allen, Phys. Rev. A 48 (1993) 656. [17] K.J. Resch, A.M. Steinberg, Phys. Rev. Lett. 92 (2004) 130402. [18] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. [19] G. Molina-Terriza, J.P. Torres, L. Torner, Nat. Phys. 3 (2007) 305. [20] J. Schwinger, Quantum Theory of Angular Momentum, Academic Press, New York, 1965. [21] J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Redwood City, 1985. [22] S.J. van Enk, G. Nienhuis, Opt. Commun. 94 (1992) 147. [23] I.M. Duck, P.M. Stevenson, E.C.G. Sudarshan, Phys. Rev. D 40 (1989) 2112. [24] H. Kobayashi, K. Nonaka, Y. Shikano, Phys. Rev. A 89 (2014) 053816.