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J. Opt. Soc. Am. A / Vol. 30, No. 8 / August 2013
F. De Zela
Secondary source of quantum or classical partially polarized states F. De Zela Departamento de Ciencias, Sección Física Pontificia Universidad Católica del Perú, Apartado 1761, Lima, Peru ([email protected]) Received May 14, 2013; accepted June 7, 2013; posted June 18, 2013 (Doc. ID 190428); published July 15, 2013 A simple device is presented that serves as a secondary source of light with prescribed polarization properties. The technique employed is based on the Schmidt purification of a mixed quantum state. Such a purification can be applied to quantum and to classical polarization states. The device presented here can be used with both classical and quantum primary sources of light. It allows controlling the degree of polarization as well as the Stokes vector that enters the decomposition of a light beam in a fully unpolarized and a fully polarized component. © 2013 Optical Society of America OCIS codes: (260.5430) Polarization; (260.3160) Interference. http://dx.doi.org/10.1364/JOSAA.30.001544
1. INTRODUCTION It is often desirable to have a source that produces light with prescribed statistical properties. This can be achieved by starting with a primary source of completely incoherent and unpolarized light, or else with a source of completely coherent and polarized light. A properly designed device should then transform this light into the desired one, constituting thereby a secondary source. Recently, such a secondary source has been proposed and experimentally tested [1]. It produces partially coherent and partially polarized light using a primary gas laser source, i.e., a source of almost completely coherent and polarized light. The optical components of such a source are commercially available, thereby improving previous proposals that achieved similar goals, but that required computer-controlled spatial light modulators with ad hoc characteristics [2–4]. The present work focuses on one of the aforementioned tasks, namely the task of producing light with prescribed polarization properties. We aim at controlling not only the (scalar) degree of polarization, but the full (threedimensional) Stokes vector that enters the decomposition of a light beam into a fully unpolarized and a fully polarized component. Furthermore, the secondary source should be usable with both classical and quantum primary sources. It could thus serve to experimentally test some issues that are of current interest in both the classical and the quantum domain. For example, some properties of Mueller matrices that have been theoretically analyzed [5] could be put under experimental test. This requires being able to realize a matrix that transforms an input polarization state, which is fixed by its Stokes parameters s0 and s, into different output states. The device presented here provides the required means to do this in a simple way. Another example is the study of geometric phases that are generated by a nonunitary evolution. Such a study can be performed in both the classical and the quantum domain. Furthermore, it is interesting to explore topological phases that arise when (polarization) states evolve ruled 1084-7529/13/081544-04$15.00/0
by, say, SL2; C transformations [6,7], instead of the much explored unitary evolutions that belong to the group SU2. In all these cases one has to go beyond controlling just the degree of polarization P jsj∕s0 ; it is also necessary to have control over the Stokes vector itself. This can be achieved with the help of a simple array, the design of which has been suggested by the so-called Schmidt decomposition [8] of a mixed quantum state ρˆ into a pure one jψi. The present work presents such a tool, which is equally applicable in the quantum and in the classical domain. Our results are thus valid when using either classical or quantum primary sources of light.
2. SCHMIDT DECOMPOSITION Let us first refer to classical light. The polarization properties of a classical, quasi-monochromatic, stochastic beam can be ˆ The components of described by the polarization matrix J. this matrix are hE α Eβ i, where angular brackets denote ensemble average. Being a two-dimensional matrix, Jˆ can be expressed in terms of the identity matrix 1 and the Pauli matrices: 1 Jˆ s0 1 s · σ: 2
(1)
Here, σ stands for the triple of Pauli matrices and ˆ Tr Jˆ · σ. When normalized to the beam’s intens0 ; s Tr J; sity, which is given by Tr Jˆ s0 , the matrix Jˆ can be written as 1 ρˆ 1 Pn · σ; 2
(2)
with n ≔ s∕jsj and P ≔ jsj∕s0 , so that 0 ≤ P ≤ 1. The above decompositions of Jˆ and ρˆ follow from Jˆ Jˆ† and ρˆ ρˆ† , as well as from the properties of 1 and the Pauli matrices. Equation (2) defines the degree of polarization, P, as a property of the polarization state ρˆ itself. That is, P does not © 2013 Optical Society of America
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Vol. 30, No. 8 / August 2013 / J. Opt. Soc. Am. A
ˆ is expressed. This can be depend on the basis in which ρˆ (or J) made clear by solving P from Eq. (2), thereby obtaining, on ˆ 0 , that account of ρˆ J∕s 4 Det Jˆ 1∕2 ˆ 1∕2 1 − : P 1 − 4 Det ρ ˆ2 Tr J
p p λ jX; n i λ− jY ; n− i:
BS
M
Dx
Q Q
(4)
We can now purify ρˆ by defining jΦi ∈ Ha ⊗ Hp as jΦi ≔
+
(3)
Equation (3) shows that P is invariant under unitary transformations, i.e., it is basis-independent. The above formulas and remarks hold true in the quantum domain as well, where ρˆ denotes a (generally) mixed polarization state. A pure state (i.e., a totally polarized one: P 1) can be written in the form ρˆ jψihψj. In the classical case, jψi denotes the Jones vector Ex ; E y T , while hψj ≔ Ex ; Ey . The Schmidt decomposition of ρˆ consists in writing it in the form Tra jΨihΨj. Here, jΨi is a state that belongs to the product space Ha ⊗ Hp , which is formed by the polarization space Hp and an auxiliary space Ha whose dimension is equal or larger than that of Hp , while Tra denotes the partial trace over the auxiliary space. That we can always express ρˆ in the form P ρˆ Tra jΨihΨj is easily ˆ i1;n λi jλi ihλi j and then conshown by first diagonalizing ρ P p structing jΨi i1;n λi jμi ia ⊗ jλi i, where the jμi ia are members of an arbitrary, orthonormal basis of Ha . We can exploit this result to design a secondary source that produces the states given by Eq. (2). Indeed, having a primary source that produces totally polarized, i.e., pure states, we can couple them to auxiliary ones, so as to have a state like jΨi. Tracing then jΨihΨj over the auxiliary space we can obtain the ˆ desired ρ. Thus, given the ρˆ of Eq. (2), we first diagonalize it: ˆ i λ jn i, with n · σjn i jn i and λ ρjn 1 P∕2. The prescribed parameters P and n fix the quantities jn i and λ as the input of our problem. The unit vector n can be specified by its polar coordinates on the Poincaré sphere: n cos 2χ; sin 2χ cos ϕ; sin 2χ sin ϕ. In that case, jn i cos χ; eiϕ sin χT and jn− i − sin χ; eiϕ cos χT . We use here the usual convention in optics for the Pauli matrices, i.e., the diagonal matrix in the triple σ is the matrix σ 1 . Using now that jn ihn j jn− ihn− j 1 and jn ihn j − jn− ihn− j n · σ, it follows that 1 jn ihn j 1 n · σ: 2
Dy
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Q
Y
Q
Xv
BS
X
M
Fig. 1. Mach–Zehnder array. BS, beam splitter; M, mirror; Q, quarterwave plate; and Dx Dy , detectors. The input state jXvi is a vertically polarized light-beam that propagates along the X direction. The array produces an output state carrying the desired characteristics of partial polarization. To extract them from the pure path-polarization productstate, one must trace over the path degree of freedom, thereby obtaining a mixed, i.e., partially polarized state. This can be done, e.g., by summing up the counts—or the registered intensities—of the two detectors.
Thus, in the case of an interferometric array, jXi and jY i denote propagation along one or the other arm of the interferometer, which we take, for the sake of concreteness, as one of the Mach–Zehnder type. Mounted on the arms of this interferometer there are optical components that transform the polarization state, see Fig. 1. As shown below, we will need two quarter-wave plates (QWPs) on each arm, to accomplish our specific task. The action of the Mach–Zehnder device on an incoming state jΨi i is to transform it into the output state U MZ jΨi i, where U MZ denotes the unitary transformation that is effected by the interferometer. We can write this transformation as U MZ U XX jXihXj U XY jXihY j U Y X jY ihXj U Y Y jY ihY j; (7) where the U ij represent operators acting on the polarization subspace Hp . These operators must be appropriately chosen, so as to realize the transformation that brings the input state jΨi i into the output state jΦi U MZ jΨi i. This is the task we address next.
3. MACH–ZEHNDER ARRAY (5)
Here, jXi and jY i mean two arbitrary, orthonormal vectors in Ha . It is then straightforward to see that ρˆ Tra jΦihΦj λ jn ihn j λ− jn− ihn− j. Hence, on account of Eq. (4), we obtain the desired result:
λ λ− λ − λ− 1 Tra jΦihΦj 1 n · σ 1 Pn · σ: 2 2 2 (6) Let us now specify jXi and jY i. The simplest choice is to take these vectors as representing additional degrees of freedom of the light-beam or photon; that is, we can take them as representing the path (or momentum) degree of freedom.
Let us choose as input state jΨi i jXvi, a vertically polarized state that propagates along the X direction toward the Mach– Zehnder device, as shown in Fig. 1. The output state is then given by jΨf i U MZ jXvi U XX jXvi U Y X jY vi. Tracing jΨf ihΨf j over the auxiliary space we get Tra jΨf ihΨf j U XX jvihvjU †XX U Y X jvihvjU †Y X :
(8)
ˆ with ρˆ given by We should achieve that Tra jΨf ihΨf j ρ, Eq. (2). By following techniques that are similar to those explained in [9], we first define the operators V X eiϕX jtX ihhj jsX ihvj; V Y eiϕY jtY ihhj jsY ihvj; (9) with the phases ϕX , ϕY being arbitrary and fjti i; jsi ig, i X, Y , denoting two orthonormal bases in Hp , yet to be fixed. Setting
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F. De Zela
U XX V X V Y ∕2 and U Y X V X − V Y ∕2, Eq. (8) gives, on account of Eq. (9), 1 Tra jΨf ihΨf j V X jvihvjV †X V Y jvihvjV †Y 2 1 jsX ihsX j jsY ihsY j: 2
xc3 t3 2 1 xc3 t3 2 xc1 t1 2 xc2 t2 2 : (13) (10)
4. Calculate αi tan−1 xi c2 t2 ∕xi c1 t1 and βi t3 for all the roots xi ∈ R of Eq. (13). 5. Choose the angles αi , βi that satisfy the two conditions: sin αi ∕ sin βi xi c2 t2 ∕xi c3 t3 and cos αi ∕ cos βi xi c1 t1 ∕xi c3 t3 . This fixes the angles to which the QWPs should be oriented. These angles enter the QQ-gadget in the form Qαi − βi ∕2Qαi βi ∕2. cot−1 xi c3
At firstpsight, our task would be accomplished by setting p jsX i λ jn i and jsY i λ− jn− i. However, jsX i and jsY i must be unit vectors for V X , V Y to be unitary. This is a necessary condition for these transformations to be implementable with retarders. A possible choice that satisfies all our requirements reads as follows: p p jtX i − λ jn− i i λ− jn i; p p jtY i − λ jn− i − i λ− jn i;
2. Construct the auxiliary vectors c si sf and t si × sf ∕1 si · sf . 3. Solve, for x ∈ R, the fourth-order algebraic equation
p p jsX i λ− jn− i i λ jn i; p p jsY i λ− jn− i − i λ jn i: (11)
With this choice, we obtain from Eq. (10) that 1 Tra jΨf ihΨf j λ jn ihn j λ− jn− ihn− j 1 Pn · σ; (12) 2 as desired. To complete our task we have to implement the operators V X and V Y of Eq. (9) with the help of retarders. V X , for instance, transforms the orthonormal basis fjhi; jvig into the orthonormal basis fjtX i; jsX ig, up to the phase factor eiϕX . However, Eq. (12) is independent of the value taken by eiϕX . Similar observations hold for V Y and eiϕY . Although V X and V Y could each be implemented with a gadget formed by two quarter-wave (Q) plates and one half-wave (H) plate, see [9], in the present case we can use a minimal array of two Qs [10]. This is because all we need to do is to transform the projectors fjhihhj; jvihvjg into the projectors fjtX ihtX j; jsX ihsX jg, so that both transformations jhi → jtX i and jhi → eiϕX jtX i, say, will do. In other words, we need to transform the (normalized) Stokes vector that represents jhi on the Poincaré sphere, into the Stokes vector representing jtX i. A device that accomplishes this task will automatically transform into one another the Stokes vectors representing jvi and jsX i. This device is a QQ-gadget. Our present task reads then as follows: given some initial and final Stokes vectors, si and sf , respectively, we seek for the angles α and β to which two Q-plates must be oriented, in order to transform si into sf . In the present case, in order to implement V X we set si Trσ · jhihhj and sf Trσ · jtX ihtX j, while for implementing V Y we set si Trσ · jhihhj and sf Trσ · jtY ihtY j. In [10] it is shown how the Q-plates must be oriented in cases like these. The procedure can be given in the form of an algorithm, which for the sake of completeness we reproduce here. Algorithm: 1. Calculate the Stokes parameters si and sf that correspond to the input and output polarization states, respectively. In the case of jhi and jtX i, for example, we obtain si 1; 0; 0 and p sf −P cos 2χ; − 1 − P 2 sin ϕ p − P cos ϕ sin 2χ; 1 − P 2 cos ϕ − P sin ϕ sin 2χ
Thus, after having determined α and β by the above algorithm, we get the transformation QαX QβX , such that QαX QβX jhi eiϕX jtX i and QαX QβX jvi eiϕX jsX i, with αX α − β∕2, βX α β∕2. The phase ϕX could be determined by calculating, e.g., arghtX jV X jhi; but knowledge of this phase is unnecessary for the present purposes. Proceeding similarly with V Y we obtain the orientation of the gadget QαY QβY that is placed on the Y -arm of the Mach–Zehnder array. In this way we can produce the partially polarized states ρˆ of Eq. (2). In order to experimentally verify that we have produced the ˆ we can submit each of the two output intensities of our state ρ, Mach–Zehnder array to the standard procedure that is used to characterize a polarization state [11]. In our case, we have to sum up the measured intensities (or photon counts) of two detectors set at the output of the interferometric array, see Fig. 1. With this information at hand we can construct the coherency matrix, as we explain next. The polarization states that are available at the two output ports are given by ρX Tra jΨf ihΨf j⊗jXihXjλ jn ihn j and ρY Tra jΨf ihΨf j⊗ jY ihY jλ− jn− ihn− j. Each of these states is submitted to the standard procedure for determining the coherency matrix. That is, with the help of a polarizer and a QWP we measure a series of intensities I iX;Y θ; ϵ. Here, θ is the polarizer’s orientation and ϵ is the phase retardation: ϵ 0 means that there is no QWP, and ϵ π∕2 that we insert a QWP with its fast axis set along the horizontal direction. We need [11] Iθ; ϵ ≔ I X θ; ϵ I Y θ; ϵ for the cases I0; 0, Iπ∕4; 0, Iπ∕2; 0, I3π∕4; 0, Iπ∕4; π∕2, I3π∕4; π∕2. With the help of these values we can construct the coherency ˆ whose components are given by J 11 I0; 0, matrix J, J 22 Iπ∕2; 0, J 12 J 21 Iπ∕4;0−I3π∕4;0iIπ∕4;π∕2− I3π∕4;π∕2∕2. Analytically, these values can be calculated from the general expression I i θ; ϵ Trp P θ Qε ρi Q†ε P θ ; with i X, Y . Here, cos2 θ Pθ cos θ sin θ
cos θ sin θ sin2 θ
(14) 15
describes the action of the polarizer set at angle θ, while Qπ∕2
e−iπ∕4 0 0 eiπ∕4
16
and Q0 1 correspond, respectively, to placing or not a QWP. A straightforward calculation gives
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Vol. 30, No. 8 / August 2013 / J. Opt. Soc. Am. A
1 Jˆ 2
1 P cos 2χ Pe−iϕ sin 2χ ; iϕ Pe sin 2χ 1 − P cos 2χ
(17)
which is in accordance with Eq. (2). After having made sure that our device produces the desired polarization states, we can use it to manipulate these states. For example, we can arrange a cascade of two or more devices like the one shown in Fig. 1, with detectors set at the output of the last device. Each device transforms an input state ρˆi into an output state ρˆo . These transformations could be designed to test, e.g., topological phases that arise from SL2; C transformations, as already mentioned. Other tasks can be similarly addressed. An experimental realization that illustrates the potential capabilities of the present approach has been recently reported [12]. The approach in [12] differs from ours in that polarization is not coupled to the path degree of freedom, but to “spatial-parity.” The latter is also a binary degree of freedom. Although the experimental techniques in [12] present some restrictions as compared to our bulk setting, they have the advantage of being realized with integrated optics. Several results of the present work should thus be realizable with integrated optics, by exploiting the isomorphism that exists between spatial-parity and the path degree of freedom.
4. SUMMARY AND OUTLOOK We have proposed a simple array that serves as a secondary source to produce partially polarized states. These states are given as a superposition of a totally unpolarized and a totally polarized state, as per Eq. (2). The proposed array gives us the possibility to control both the degree of polarization, P, and the direction n on the Poincaré sphere that corresponds to the polarized component. The array is a Mach–Zehnder interferometer having a gadget placed on each of its arms. This gadget is made of two QWPs. The orientation of the wave plates is fixed by the input parameters, P and n, with the help of an algorithm that is easily implemented for numerical calculation. In this way, by just changing the orientation of the wave plates, one can scan a whole range of values of P and n. Furthermore, by concatenation of several secondary sources like the present one, it is possible to evolve a state under nonunitary transformations. This serves, for instance, to exhibit some topological phases. In quantum computation it is
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interesting to study the robustness of geometrical phases under decoherence. The latter can be simulated by a nonunitary evolution, which can be implemented with the help of arrays whose building blocks could be like the present one. This array has the great advantage of requiring only commercially available equipment, in contrast to other arrays that require ad hoc components, for instance custom designed dichroic elements and the like.
ACKNOWLEDGMENTS The author gratefully acknowledges DGI-PUCP for financial support under Grant No. 2013-0130.
REFERENCES 1. A. S. Ostrovsky, G. Rodríguez-Zurita, C. Meneses-Fabián, M. A. Olvera-Santamaría, and C. Rickenstorff-Parrao, “Experimental generating the partially coherent and partially polarized electromagnetic source,” Opt. Express 18, 12864–12871 (2010). 2. A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. MartínezNiconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010). 3. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004). 4. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005). 5. S.-Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998). 6. S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004). 7. R. Simon, S. Chaturvedi, V. Srinivasan, and N. Mukunda, “Hamilton’s turns for the Lorentz group,” Int. J. Theor. Phys. 45, 2051–2070 (2006). 8. J. Audretsch, Entangled Systems: New Directions in Quantum Physics (Wiley-VCH, 2007). 9. B.-G. Englert, C. Kurtsiefer, and H. Weinfurter, “Universal unitary gate for single-photon two-qubit states,” Phys. Rev. A 63, 032303 (2001). 10. F. De Zela, “Two-component gadget for transforming any two nonorthogonal polarization states into one another,” Phys. Lett. A 376, 1664–1668 (2012). 11. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999). 12. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
J. Opt. Soc. Am. A / Vol. 30, No. 8 / August 2013
F. De Zela
Secondary source of quantum or classical partially polarized states F. De Zela Departamento de Ciencias, Sección Física Pontificia Universidad Católica del Perú, Apartado 1761, Lima, Peru ([email protected]) Received May 14, 2013; accepted June 7, 2013; posted June 18, 2013 (Doc. ID 190428); published July 15, 2013 A simple device is presented that serves as a secondary source of light with prescribed polarization properties. The technique employed is based on the Schmidt purification of a mixed quantum state. Such a purification can be applied to quantum and to classical polarization states. The device presented here can be used with both classical and quantum primary sources of light. It allows controlling the degree of polarization as well as the Stokes vector that enters the decomposition of a light beam in a fully unpolarized and a fully polarized component. © 2013 Optical Society of America OCIS codes: (260.5430) Polarization; (260.3160) Interference. http://dx.doi.org/10.1364/JOSAA.30.001544
1. INTRODUCTION It is often desirable to have a source that produces light with prescribed statistical properties. This can be achieved by starting with a primary source of completely incoherent and unpolarized light, or else with a source of completely coherent and polarized light. A properly designed device should then transform this light into the desired one, constituting thereby a secondary source. Recently, such a secondary source has been proposed and experimentally tested [1]. It produces partially coherent and partially polarized light using a primary gas laser source, i.e., a source of almost completely coherent and polarized light. The optical components of such a source are commercially available, thereby improving previous proposals that achieved similar goals, but that required computer-controlled spatial light modulators with ad hoc characteristics [2–4]. The present work focuses on one of the aforementioned tasks, namely the task of producing light with prescribed polarization properties. We aim at controlling not only the (scalar) degree of polarization, but the full (threedimensional) Stokes vector that enters the decomposition of a light beam into a fully unpolarized and a fully polarized component. Furthermore, the secondary source should be usable with both classical and quantum primary sources. It could thus serve to experimentally test some issues that are of current interest in both the classical and the quantum domain. For example, some properties of Mueller matrices that have been theoretically analyzed [5] could be put under experimental test. This requires being able to realize a matrix that transforms an input polarization state, which is fixed by its Stokes parameters s0 and s, into different output states. The device presented here provides the required means to do this in a simple way. Another example is the study of geometric phases that are generated by a nonunitary evolution. Such a study can be performed in both the classical and the quantum domain. Furthermore, it is interesting to explore topological phases that arise when (polarization) states evolve ruled 1084-7529/13/081544-04$15.00/0
by, say, SL2; C transformations [6,7], instead of the much explored unitary evolutions that belong to the group SU2. In all these cases one has to go beyond controlling just the degree of polarization P jsj∕s0 ; it is also necessary to have control over the Stokes vector itself. This can be achieved with the help of a simple array, the design of which has been suggested by the so-called Schmidt decomposition [8] of a mixed quantum state ρˆ into a pure one jψi. The present work presents such a tool, which is equally applicable in the quantum and in the classical domain. Our results are thus valid when using either classical or quantum primary sources of light.
2. SCHMIDT DECOMPOSITION Let us first refer to classical light. The polarization properties of a classical, quasi-monochromatic, stochastic beam can be ˆ The components of described by the polarization matrix J. this matrix are hE α Eβ i, where angular brackets denote ensemble average. Being a two-dimensional matrix, Jˆ can be expressed in terms of the identity matrix 1 and the Pauli matrices: 1 Jˆ s0 1 s · σ: 2
(1)
Here, σ stands for the triple of Pauli matrices and ˆ Tr Jˆ · σ. When normalized to the beam’s intens0 ; s Tr J; sity, which is given by Tr Jˆ s0 , the matrix Jˆ can be written as 1 ρˆ 1 Pn · σ; 2
(2)
with n ≔ s∕jsj and P ≔ jsj∕s0 , so that 0 ≤ P ≤ 1. The above decompositions of Jˆ and ρˆ follow from Jˆ Jˆ† and ρˆ ρˆ† , as well as from the properties of 1 and the Pauli matrices. Equation (2) defines the degree of polarization, P, as a property of the polarization state ρˆ itself. That is, P does not © 2013 Optical Society of America
F. De Zela
Vol. 30, No. 8 / August 2013 / J. Opt. Soc. Am. A
ˆ is expressed. This can be depend on the basis in which ρˆ (or J) made clear by solving P from Eq. (2), thereby obtaining, on ˆ 0 , that account of ρˆ J∕s 4 Det Jˆ 1∕2 ˆ 1∕2 1 − : P 1 − 4 Det ρ ˆ2 Tr J
p p λ jX; n i λ− jY ; n− i:
BS
M
Dx
Q Q
(4)
We can now purify ρˆ by defining jΦi ∈ Ha ⊗ Hp as jΦi ≔
+
(3)
Equation (3) shows that P is invariant under unitary transformations, i.e., it is basis-independent. The above formulas and remarks hold true in the quantum domain as well, where ρˆ denotes a (generally) mixed polarization state. A pure state (i.e., a totally polarized one: P 1) can be written in the form ρˆ jψihψj. In the classical case, jψi denotes the Jones vector Ex ; E y T , while hψj ≔ Ex ; Ey . The Schmidt decomposition of ρˆ consists in writing it in the form Tra jΨihΨj. Here, jΨi is a state that belongs to the product space Ha ⊗ Hp , which is formed by the polarization space Hp and an auxiliary space Ha whose dimension is equal or larger than that of Hp , while Tra denotes the partial trace over the auxiliary space. That we can always express ρˆ in the form P ρˆ Tra jΨihΨj is easily ˆ i1;n λi jλi ihλi j and then conshown by first diagonalizing ρ P p structing jΨi i1;n λi jμi ia ⊗ jλi i, where the jμi ia are members of an arbitrary, orthonormal basis of Ha . We can exploit this result to design a secondary source that produces the states given by Eq. (2). Indeed, having a primary source that produces totally polarized, i.e., pure states, we can couple them to auxiliary ones, so as to have a state like jΨi. Tracing then jΨihΨj over the auxiliary space we can obtain the ˆ desired ρ. Thus, given the ρˆ of Eq. (2), we first diagonalize it: ˆ i λ jn i, with n · σjn i jn i and λ ρjn 1 P∕2. The prescribed parameters P and n fix the quantities jn i and λ as the input of our problem. The unit vector n can be specified by its polar coordinates on the Poincaré sphere: n cos 2χ; sin 2χ cos ϕ; sin 2χ sin ϕ. In that case, jn i cos χ; eiϕ sin χT and jn− i − sin χ; eiϕ cos χT . We use here the usual convention in optics for the Pauli matrices, i.e., the diagonal matrix in the triple σ is the matrix σ 1 . Using now that jn ihn j jn− ihn− j 1 and jn ihn j − jn− ihn− j n · σ, it follows that 1 jn ihn j 1 n · σ: 2
Dy
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Q
Y
Q
Xv
BS
X
M
Fig. 1. Mach–Zehnder array. BS, beam splitter; M, mirror; Q, quarterwave plate; and Dx Dy , detectors. The input state jXvi is a vertically polarized light-beam that propagates along the X direction. The array produces an output state carrying the desired characteristics of partial polarization. To extract them from the pure path-polarization productstate, one must trace over the path degree of freedom, thereby obtaining a mixed, i.e., partially polarized state. This can be done, e.g., by summing up the counts—or the registered intensities—of the two detectors.
Thus, in the case of an interferometric array, jXi and jY i denote propagation along one or the other arm of the interferometer, which we take, for the sake of concreteness, as one of the Mach–Zehnder type. Mounted on the arms of this interferometer there are optical components that transform the polarization state, see Fig. 1. As shown below, we will need two quarter-wave plates (QWPs) on each arm, to accomplish our specific task. The action of the Mach–Zehnder device on an incoming state jΨi i is to transform it into the output state U MZ jΨi i, where U MZ denotes the unitary transformation that is effected by the interferometer. We can write this transformation as U MZ U XX jXihXj U XY jXihY j U Y X jY ihXj U Y Y jY ihY j; (7) where the U ij represent operators acting on the polarization subspace Hp . These operators must be appropriately chosen, so as to realize the transformation that brings the input state jΨi i into the output state jΦi U MZ jΨi i. This is the task we address next.
3. MACH–ZEHNDER ARRAY (5)
Here, jXi and jY i mean two arbitrary, orthonormal vectors in Ha . It is then straightforward to see that ρˆ Tra jΦihΦj λ jn ihn j λ− jn− ihn− j. Hence, on account of Eq. (4), we obtain the desired result:
λ λ− λ − λ− 1 Tra jΦihΦj 1 n · σ 1 Pn · σ: 2 2 2 (6) Let us now specify jXi and jY i. The simplest choice is to take these vectors as representing additional degrees of freedom of the light-beam or photon; that is, we can take them as representing the path (or momentum) degree of freedom.
Let us choose as input state jΨi i jXvi, a vertically polarized state that propagates along the X direction toward the Mach– Zehnder device, as shown in Fig. 1. The output state is then given by jΨf i U MZ jXvi U XX jXvi U Y X jY vi. Tracing jΨf ihΨf j over the auxiliary space we get Tra jΨf ihΨf j U XX jvihvjU †XX U Y X jvihvjU †Y X :
(8)
ˆ with ρˆ given by We should achieve that Tra jΨf ihΨf j ρ, Eq. (2). By following techniques that are similar to those explained in [9], we first define the operators V X eiϕX jtX ihhj jsX ihvj; V Y eiϕY jtY ihhj jsY ihvj; (9) with the phases ϕX , ϕY being arbitrary and fjti i; jsi ig, i X, Y , denoting two orthonormal bases in Hp , yet to be fixed. Setting
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U XX V X V Y ∕2 and U Y X V X − V Y ∕2, Eq. (8) gives, on account of Eq. (9), 1 Tra jΨf ihΨf j V X jvihvjV †X V Y jvihvjV †Y 2 1 jsX ihsX j jsY ihsY j: 2
xc3 t3 2 1 xc3 t3 2 xc1 t1 2 xc2 t2 2 : (13) (10)
4. Calculate αi tan−1 xi c2 t2 ∕xi c1 t1 and βi t3 for all the roots xi ∈ R of Eq. (13). 5. Choose the angles αi , βi that satisfy the two conditions: sin αi ∕ sin βi xi c2 t2 ∕xi c3 t3 and cos αi ∕ cos βi xi c1 t1 ∕xi c3 t3 . This fixes the angles to which the QWPs should be oriented. These angles enter the QQ-gadget in the form Qαi − βi ∕2Qαi βi ∕2. cot−1 xi c3
At firstpsight, our task would be accomplished by setting p jsX i λ jn i and jsY i λ− jn− i. However, jsX i and jsY i must be unit vectors for V X , V Y to be unitary. This is a necessary condition for these transformations to be implementable with retarders. A possible choice that satisfies all our requirements reads as follows: p p jtX i − λ jn− i i λ− jn i; p p jtY i − λ jn− i − i λ− jn i;
2. Construct the auxiliary vectors c si sf and t si × sf ∕1 si · sf . 3. Solve, for x ∈ R, the fourth-order algebraic equation
p p jsX i λ− jn− i i λ jn i; p p jsY i λ− jn− i − i λ jn i: (11)
With this choice, we obtain from Eq. (10) that 1 Tra jΨf ihΨf j λ jn ihn j λ− jn− ihn− j 1 Pn · σ; (12) 2 as desired. To complete our task we have to implement the operators V X and V Y of Eq. (9) with the help of retarders. V X , for instance, transforms the orthonormal basis fjhi; jvig into the orthonormal basis fjtX i; jsX ig, up to the phase factor eiϕX . However, Eq. (12) is independent of the value taken by eiϕX . Similar observations hold for V Y and eiϕY . Although V X and V Y could each be implemented with a gadget formed by two quarter-wave (Q) plates and one half-wave (H) plate, see [9], in the present case we can use a minimal array of two Qs [10]. This is because all we need to do is to transform the projectors fjhihhj; jvihvjg into the projectors fjtX ihtX j; jsX ihsX jg, so that both transformations jhi → jtX i and jhi → eiϕX jtX i, say, will do. In other words, we need to transform the (normalized) Stokes vector that represents jhi on the Poincaré sphere, into the Stokes vector representing jtX i. A device that accomplishes this task will automatically transform into one another the Stokes vectors representing jvi and jsX i. This device is a QQ-gadget. Our present task reads then as follows: given some initial and final Stokes vectors, si and sf , respectively, we seek for the angles α and β to which two Q-plates must be oriented, in order to transform si into sf . In the present case, in order to implement V X we set si Trσ · jhihhj and sf Trσ · jtX ihtX j, while for implementing V Y we set si Trσ · jhihhj and sf Trσ · jtY ihtY j. In [10] it is shown how the Q-plates must be oriented in cases like these. The procedure can be given in the form of an algorithm, which for the sake of completeness we reproduce here. Algorithm: 1. Calculate the Stokes parameters si and sf that correspond to the input and output polarization states, respectively. In the case of jhi and jtX i, for example, we obtain si 1; 0; 0 and p sf −P cos 2χ; − 1 − P 2 sin ϕ p − P cos ϕ sin 2χ; 1 − P 2 cos ϕ − P sin ϕ sin 2χ
Thus, after having determined α and β by the above algorithm, we get the transformation QαX QβX , such that QαX QβX jhi eiϕX jtX i and QαX QβX jvi eiϕX jsX i, with αX α − β∕2, βX α β∕2. The phase ϕX could be determined by calculating, e.g., arghtX jV X jhi; but knowledge of this phase is unnecessary for the present purposes. Proceeding similarly with V Y we obtain the orientation of the gadget QαY QβY that is placed on the Y -arm of the Mach–Zehnder array. In this way we can produce the partially polarized states ρˆ of Eq. (2). In order to experimentally verify that we have produced the ˆ we can submit each of the two output intensities of our state ρ, Mach–Zehnder array to the standard procedure that is used to characterize a polarization state [11]. In our case, we have to sum up the measured intensities (or photon counts) of two detectors set at the output of the interferometric array, see Fig. 1. With this information at hand we can construct the coherency matrix, as we explain next. The polarization states that are available at the two output ports are given by ρX Tra jΨf ihΨf j⊗jXihXjλ jn ihn j and ρY Tra jΨf ihΨf j⊗ jY ihY jλ− jn− ihn− j. Each of these states is submitted to the standard procedure for determining the coherency matrix. That is, with the help of a polarizer and a QWP we measure a series of intensities I iX;Y θ; ϵ. Here, θ is the polarizer’s orientation and ϵ is the phase retardation: ϵ 0 means that there is no QWP, and ϵ π∕2 that we insert a QWP with its fast axis set along the horizontal direction. We need [11] Iθ; ϵ ≔ I X θ; ϵ I Y θ; ϵ for the cases I0; 0, Iπ∕4; 0, Iπ∕2; 0, I3π∕4; 0, Iπ∕4; π∕2, I3π∕4; π∕2. With the help of these values we can construct the coherency ˆ whose components are given by J 11 I0; 0, matrix J, J 22 Iπ∕2; 0, J 12 J 21 Iπ∕4;0−I3π∕4;0iIπ∕4;π∕2− I3π∕4;π∕2∕2. Analytically, these values can be calculated from the general expression I i θ; ϵ Trp P θ Qε ρi Q†ε P θ ; with i X, Y . Here, cos2 θ Pθ cos θ sin θ
cos θ sin θ sin2 θ
(14) 15
describes the action of the polarizer set at angle θ, while Qπ∕2
e−iπ∕4 0 0 eiπ∕4
16
and Q0 1 correspond, respectively, to placing or not a QWP. A straightforward calculation gives
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1 Jˆ 2
1 P cos 2χ Pe−iϕ sin 2χ ; iϕ Pe sin 2χ 1 − P cos 2χ
(17)
which is in accordance with Eq. (2). After having made sure that our device produces the desired polarization states, we can use it to manipulate these states. For example, we can arrange a cascade of two or more devices like the one shown in Fig. 1, with detectors set at the output of the last device. Each device transforms an input state ρˆi into an output state ρˆo . These transformations could be designed to test, e.g., topological phases that arise from SL2; C transformations, as already mentioned. Other tasks can be similarly addressed. An experimental realization that illustrates the potential capabilities of the present approach has been recently reported [12]. The approach in [12] differs from ours in that polarization is not coupled to the path degree of freedom, but to “spatial-parity.” The latter is also a binary degree of freedom. Although the experimental techniques in [12] present some restrictions as compared to our bulk setting, they have the advantage of being realized with integrated optics. Several results of the present work should thus be realizable with integrated optics, by exploiting the isomorphism that exists between spatial-parity and the path degree of freedom.
4. SUMMARY AND OUTLOOK We have proposed a simple array that serves as a secondary source to produce partially polarized states. These states are given as a superposition of a totally unpolarized and a totally polarized state, as per Eq. (2). The proposed array gives us the possibility to control both the degree of polarization, P, and the direction n on the Poincaré sphere that corresponds to the polarized component. The array is a Mach–Zehnder interferometer having a gadget placed on each of its arms. This gadget is made of two QWPs. The orientation of the wave plates is fixed by the input parameters, P and n, with the help of an algorithm that is easily implemented for numerical calculation. In this way, by just changing the orientation of the wave plates, one can scan a whole range of values of P and n. Furthermore, by concatenation of several secondary sources like the present one, it is possible to evolve a state under nonunitary transformations. This serves, for instance, to exhibit some topological phases. In quantum computation it is
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interesting to study the robustness of geometrical phases under decoherence. The latter can be simulated by a nonunitary evolution, which can be implemented with the help of arrays whose building blocks could be like the present one. This array has the great advantage of requiring only commercially available equipment, in contrast to other arrays that require ad hoc components, for instance custom designed dichroic elements and the like.
ACKNOWLEDGMENTS The author gratefully acknowledges DGI-PUCP for financial support under Grant No. 2013-0130.
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