Orthogonality
of Biphoton Polarization States
Abstract
Orthogonality of two-photon polarization states belonging to a single frequency and spatial mode is demonstrated experimentally, in a generalization of the well-known anti-correlation ’dip’ experiment.
pacs:
PACS Number: 42.50.-p, 42.50.Dv, 03.67.-aOrthogonality, one of the basic mathematical concepts, plays an important role in physics, especially quantum physics and, in particular, quantum optics. A well-known example including both classical and quantum cases is orthogonality of two polarization modes of electromagnetic radiation. Physically, orthogonality of two arbitrary polarization states means that if light is prepared in a certain (in the general case, elliptic) polarization state it will not pass through a filter selecting the orthogonal state. (A filter selecting an arbitrary polarization state can be made of a rotatable quarter-wave plate and a rotatable linear polarization filter Shurkliff .) Orthogonality of polarization states has an explicit representation on the Poincaré sphere where each polarization state is depicted by a point. The state orthogonal to a given one is shown by a point placed on the opposite side of the same diameter. Examples are states with vertical and horizontal linear polarization, right- and left- circularly polarized states, and any two elliptically polarized states with opposite directions of rotation and inverse axis ratios. This concept of orthogonality relates to both classical polarization states of light and single-photon quantum states of polarized light kljetp . Mathematically, orthogonality of two polarization states means that the scalar product of two corresponding Jones vectors Shurkliff is equal to zero. In quantum optics, this corresponds to zero scalar product of polarization state vectors, for instance, state vectors of single-photon states. This is a particular case of the general rule: orthogonality of quantum states means that their scalar product is equal to zero.
However, in addition to single-photon states there are other types of nonclassical light. In quantum optics, one of the central roles is played by two-photon states, which are most easily generated via spontaneous parametric down-conversion (SPDC) photons . In a two-photon state, radiation consists of photon pairs, often called biphotons, that are correlated in frequency, wavevector, moment of birth, and polarization. Focusing on the case of collinear frequency-degenerate SPDC, here we will discuss the so-called single-mode biphotons. Although even in the frequency-degenerate collinear case SPDC has a finite frequency and angular spectrum, under certain experimental conditions such biphotons can be treated as relating to a single frequency and angular mode.
One can show that the polarization state of a single-mode biphoton burkl can be described as a qutrit, a three- state quantum system. Using qutrits instead of qubits for the transmission of quantum information has been previously discussed trits , in particular, in connection with ternary cryptography protocols bechmann , bruss , peres . The general case of a qutrit represented by a biphoton in an arbitrary pure polarization state is given by the state vector
(1) |
where are complex amplitudes satisfying the normalization condition, , and denotes a two-photon () state with photons polarized horizontally and photons polarized vertically indistinguishable . In quarkobzor , it is shown that the state (1) allows an explicit representation on the Poincaré sphere. It can be written in the form of two arbitrarily polarized correlated single-photon states,
(2) |
Here, and are operators of photon creation in arbitrary polarization modes given by the coordinates , on the Poincaré sphere:
(3) |
where are photon creation operators in the horizontal and vertical linear polarization modes, and similarly for . Axial angles and azimuthal angles angles are in one-to-one correspondence with the four parameters describing the state (1), which are, for instance, , , , . Representation (2) means that a biphoton of arbitrary polarization can be shown as a pair of points on the Poincaré sphere. It turns out that the Stokes vector of a biphoton is simply a normalized sum of the Stokes vectors of photons forming it (’biphoton halves’), and the polarization degree of the pair is given by the angle at which the pair can be seen from the sphere center.
A question arises: what does orthogonality of two biphoton polarization states mean? As usually in quantum mechanics, it means that the product of their state vectors is equal to zero. For instance, orthogonality of two biphotons and , with the operators of photon creation in arbitrary polarization modes denoted now by , means that
(4) |
What does orthogonality of two biphotons mean from the viewpoint of physics? This question was answered in orthogonality where an operational criterion of orthogonality for arbitrarily polarized biphotons was formulated. Namely, orthogonality of two biphotons can be tested using a simple setup consisting of a non-polarizing beamsplitter, two detectors installed in its two output ports, with an arbitrary polarization filter inserted at the input of each detector, and a coincidence circuit. A biphoton is registered if there is a coincidence of photocounts from the two detectors. Let a biphoton be at the input and the filters in the output ports of the beamsplitter select polarization states corresponding to photon creation operators . Then orthogonality of and is equivalent to the absence of coincidences absence in such a setup. Note that orthogonality of any two polarization states among , , , is not required.
Such an experiment is the most general case of the well-known anticorrelation experiment hong . Earlier, a particular case of this polarization version of anticorrelation experiment has been performed for type-II SPDC typeiianti . The absence of coincidences in the anticorrelation experiment typeiianti can be interpreted as orthogonality of the states and , a pair of photons polarized linearly at angles to the vertical axis. Similarly, in ternary , orthogonality of the states and to the state of right- and left polarized photons, , has been demonstrated. In both cases, orthogonal biphotons had zero polarization degree, which means that they were pairs of orthogonally polarized photons. Another example of a basis formed by three mutually orthogonal biphotons with zero polarization degree was demonstrated in trifonov .
Several examples of orthogonal biphotons of other polarization degrees are shown in Fig.1. Fig.1a shows three mutually orthogonal states , , of biphotons with zero polarization degree studied in typeiianti and ternary . All three states are biphotons consisting of two orthogonal photons; at the same time, no photon forming a biphoton is orthogonal to any photon of the other two biphotons. Three biphotons shown in Fig.1a form an orthogonal basis.
Whenever a biphoton is fixed (two points are fixed on the Poincaré sphere), there are infinitely many biphotons orthogonal to it. Replacing the Poincaré sphere by the globe, one can pick two spots denoting a biphoton to be, for instance, Moscow (Russia) and Turin (Italy). Then, one should make a choice for the third point. Let it be Baltimore(MD, USA). Then the fourth point is found from Eq. (4), and it turns out to be near New Zealand and the Bounty isles. So, the biphoton ’Moscow - Turin’ is orthogonal to the biphoton ’Baltimore - Bounty’!
The idea of our experiment was to prepare some arbitrary input biphoton state, to make the registration part select the state orthogonal to it, and to demonstrate orthogonality by scanning the parameters of the input and registered states around the minimum of coincidence counting rate. We chose the input biphoton state to be a pair of photons polarized linearly at the opposite angles to the horizontal direction, the polarization degree being . On the Poincaré sphere, this is shown as two points on the equator placed symmetrically at the angles with respect to the ’H’ axis, which corresponds to photons polarized linearly at the angles to the horizontal axis. For the reasons that we will explain later, it is convenient to make one of the polarization filters in the registration part select the state polarized linearly at the angle to the horizontal axis. The other filter, as one can easily find, should then select linear polarization at the angle to the horizontal axis. This configuration, which is used in one of our experiments, is shown in Fig.1b. Finally, Fig.1c gives an example of two orthogonal biphotons with polarization degree in a ’non-plane’ configuration. This time, the points denoting the biphoton ’ab’ are placed on the ’Greenwich meridian’, if we follow the globe terminology. Again, one of the polarizers selects the state polarized linearly at . However, the position of the other polarizer is changed: now, it should select linear polarization at the angle to the horizontal axis. Under each Poincaré sphere in Fig.1, the corresponding polarization states are shown schematically. In all these examples, there is no orthogonality between separate photons, or ’halves of biphotons’.
The experimental setup is shown in Fig.2. Collinear frequency-degenerate SPDC is generated in two similar type-I lithium iodate crystals of length 1 cm, cw radiation of argon laser at wavelength 351 nm used as the pump. The optic axis of the first crystal is in the vertical plane while the optic axis of the other crystal is in the horizontal plane. The two-photon state generated after the crystals is of the form (1), with , the amplitudes and can be varied by rotating the plate in the pump beam, and the phase can be varied by tilting the two quartz plates QP, whose optic axes are oriented in the vertical plane. The pump after the crystals is cut off by a UV mirror UVM. Spatial and frequency filtering of the SPDC radiation is performed by a pinhole P and an interference filter IF with nm central wavelength and nm bandwidth. The right-hand side of the setup shows the registration part (the Brown-Twiss interferometer). It includes a non-polarizing beamsplitter BS and two detectors (photomultiplier tubes) D1, D2 inserted in its output ports. At the input of each detector, there is a polarization filter consisting of a rotatable quarter-wave plate QWP1,2 qp and a rotatable polarizer P1,2. Coincidences between the photocounts of the detectors are registered using a coincidence circuit with a resolution ns.
The measurements were performed for two configurations shown in Figs 1b,c. The plates QP were tilted so that the phase was equal to . Then the angle of the plate was scanned from to . As a result, the two points corresponding to the produced biphoton state travelled on the Poincaré sphere: first, from the ’VV’ point () to the ’HH’ point () symmetrically along the opposite sides of the equator, then again to the ’VV’ point () but this time, along the opposite sides of the ’Greenwich meridian’. This way, both cases shown in Fig.1b,c were realized. In the first run, the polarizer P1,2 orientations were fixed as in Fig.1b: and . In the dependence of coincidence counting rate on (Fig.3a), the minimum was at , which corresponded to . For this point, the ratio was measured using the tomography procedure developed in tomography ; this ratio turned out to be . The orientation of P1 is convenient since in this case, rotation of the half-wave plate in the pump beam does not lead to the variation of D1 single counting rate .
In the next run, the orientations of P1, P2 were fixed as in Fig.2c: and . In this case, the minimum was achieved for (Fig.3b), corresponding to .
Finally, orthogonality of two biphoton states was checked by fixing all parameters (, , ) in the configuration shown in Fig.1b and then scanning them around their optimal values. The plot in Fig.4 shows the dependence of the coincidence counting rate on the orientation of polarizer P1, with the other polarizer fixed at and the half-wave plate in the pump beam fixed at . Similar dependencies were obtained for scanning and .
Two important notes should be made about the measurement procedure. First, when the angle is scanned with fixed positions of the two polarizers (Fig.3a), a considerable modulation in the singles counting rate of detector D2 is observed (Fig.5). This is quite natural since in the course of variation the state of biphoton light changes from being vertically polarized through completely non-polarized state (at ) to being horizontally polarized. With the polarizer P1 oriented at , the intensity of transmitted light should vary three times, as it indeed does in Fig.5. However, the minimum of the coincidence counting rate does not correspond to the point where is minimal; according to the calculation, it occurs at . The second remark is that in all kinds of anticorrelation experiments, the coincidence counting rate in the minimum is given by the level of accidental coincidence counting rate, which corresponds to the normalized second-order Glauber’s correlation function . To demonstrate this, instead of the coincidence counting rate, in all plots we present g2 instead of .
To find the dependence of the coincidence counting rate on all parameters, one can write
(5) |
The operators , , , are substituted in (5) in the form (3), with the angles taking the values , respectively. The axial angles for can be expressed as functions of via the relations given in orthogonality and the formulas , . The axial angles for can be calculated as functions of , the angles of P1,P2 orientations. Then we obtain the coincidence counting rate,
(6) |
and the single-photon counting rate in detector 2,
(7) |
Equations (6,7) were used to plot the theoretical dependencies shown in Figs.3,4.
To conclude, we have experimentally demonstrated orthogonality of two biphotons having polarization degree between and . Our experiment is a generalization of the ’anti-correlation dip’ experiment to the case of arbitrarily polarized photon pairs. The observed effect can find applications in ternary quantum cryptography protocols cryptography .
This work was supported in part by the Russian Foundation for Basic Research grants #02-02-16664,#03-02-16444, INTAS grant #01-2122, and the Russian program of scientific school support (#166.2003.02)
References
- (1) W.A.Shurkliff, Polarized Light (Harvard University Press, Cambridge, Massachusetts, 1962).
- (2) D. N. Klyshko, JETP 84 (6), 1065 (1996).
- (3) D.N.Klyshko, Photons and Nonlinear Optics (Gordon & Breach, New York, 1988).
- (4) A. V. Burlakov and D. N. Klyshko, JETP Lett. 69, 839 (1999).
- (5) K. Mattle et.al., Phys. Rev. Lett. 76, 4656 (1996).
- (6) H. Bechmann-Pasquinnucci and W. Tittel, Phys.Rev.A, 61, 062308 (2000).
- (7) D. Bruss and C. Macchiavello, Phys.Rev.Lett., 88, 127901 (2002).
- (8) H. Bechmann-Pasquinnucci and A. Peres, Phys.Rev.Lett., 85, 3313 (2002).
- (9) Here, it is important that the two photons may differ only in polarization. In the case of distinguishable photons, the state is replaced by two Bell states .
- (10) A. V. Burlakov and M. V. Chekhova, JETP Lett. 75, 505 (2002).
- (11) is measured with respect to the ’H’ axis, giving the state of horizontal polarization and is measured with respect to the ’D’ axis giving linear polarization.
- (12) A.A.Zhukov, G.A.Maslennikov, and M.V.Chekhova, JETP Lett. 76, 696 (2002).
- (13) Coincidences are not completely absent but their rate corresponds to the Poissonian (accidental) level.
- (14) C.K.Hong, Z.Y.Ou, and L.Mandel, Phys.Rev.Lett., 59, 2044 (1987).
- (15) Y.H. Shih and A.V. Sergienko, Phys.Rev.Lett. A 186, 29 (1994); ibid 191, 201 (1994).
- (16) A. V. Burlakov et. al., Phys.Rev.A, 60, R4209 (1999).
- (17) T.Tsegaye et. al., Phys.Rev.Lett., 85, 5013 (2000).
- (18) Although quarter-wave plates are necessary in the general case, in the measurements performed here they could be omitted.
- (19) L. A. Krivitsky et. al., JETP 124 (2003).
- (20) To calculate the normalized second-order correlation function in experiment, one divides the coincidence counting rate by the product of two single-photon counting rates and the coincidence circuit resolution time .
- (21) G. A. Maslennikov et. al., Journal of Optics B: Quantum Semiclass.Opt.5,530 (2003).