PHYSICAL REVIEW LETTERS

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1 PHYSICAL REVIEW LETTERS VOLUME DECEMBER 2000 NUMBER 24 Exerimental Demonstration of Three Mutually Orthogonal Polarization States of Entangled Photons Tedros Tsegaye, 1 Jonas Söderholm, 1 Mete Atatüre, 4 Alexei Trifonov, 1,2 Gunnar Björk, 1, * Alexander V. Sergienko, 3 Bahaa E. A. Saleh, 3 and Malvin C. Teich 3,4 1 Deartment of Electronics, Royal Institute of Technology (KTH), Electrum 229, SE Kista, Sweden 2 Ioffe Physical Technical Institute, 26 Polytekhnicheskaya, St. Petersburg, Russia 3 Quantum Imaging Laboratory, Deartment of Electrical and Comuter Engineering, Boston University, 8 Saint Mary s Street, Boston, Massachusetts Quantum Imaging Laboratory, Deartment of Physics, Boston University, 8 Saint Mary s Street, Boston, Massachusetts (Received 4 August 2000) Linearly olarized classical light can be exressed in a vertical and a horizontal comonent. Geometrically rotating vertically olarized light by 90 ± will convert it to the orthogonal horizontal olarization. We have exerimentally generated a two-hoton state of light which evolves into an orthogonal state uon geometrical rotation by 60 ±. Rotating this state by an additional 60 ± will yield a state which is mutually orthogonal to the first two states. Generalizing this rocedure, one can generate N 1 1 mutually orthogonal N-hoton states that cyclicly evolve from one to another uon a geometric rotation by 180 N 1 1 degrees. PACS numbers: Bz, a, Dv Polarized light is often described in terms of its Stokes arameters [1], which were introduced to classify the olarization roerties of classical light in the middle of the 19th century [2]. With the advent of lasers and hotoncounting technology, it was natural to introduce the Stokes oerators [3], describing the olarization roerties of quantized fields. Some time later it became evident that sensitive measurements of olarization were limited by the inherent olarization noise of the fields [4], and soon thereafter several roosals of olarization squeezed light were ut forth [5]. The Stokes oerators classify the second order correlations only between orthogonal field modes [6], but for quantized fields higher order correlations can lead to interesting results. Therefore the classification of olarization has been extended to fourth-order correlations [7]. Recently, oerational definitions and their exerimental imlementations of quantum olarization have been discussed [8]. In this aer we reort an alternative aroach to quantum olarization, where we find the eigenstates of the generator of geometrical rotation and then construct a comlementary oerator through the oerator s eigenstates. Our starting oint is to find states which are invariant under geometrical rotation. This is much in the sirit of Prakash and Chandra [9], and Agarwal [6] who defined unolarized quantum states of light as the states invariant under any geometrical rotation and any differential hase shift. (Later, and aarently indeendently, Lehner et al. defined unolarized states of light in the same manner [10].) We exand the electromagnetic field in two lane waves with horizontal-vertical linear olarization (indicated by the subscrit symbol 1) and exand the field state in the number basis jm, n 1 jm 1 jn 1. In this exansion the oerator generating geometric rotation is given by i â y ˆb 2 â ˆb y Ŝ y, (1) where â and ˆb are the annihilation oerators for the first and second mode, resectively, and Ŝ y is the Stokes oerator generating rotation around the direction of the wave vector of the field. The associated unitary oerator which rotates a field an angle u is ex iuŝ y. The oerator (1) can also be seen as the well-known Jaynes-Cummings interaction Hamiltonian used extensively in atomic hysics and (24) 5013(5)$ The American Physical Society 5013

2 in cavity quantum electrodynamics. It describes a linear couling between two oscillators, in this case, e.g., a vertically and a horizontally olarized electromagnetic field of the same frequency. We note that Ŝ y commutes with the total excitation oerator â y â 1 ˆb y ˆb (which is the Stokes oerator Ŝ 0 ) so the total excitation number is an invariant under geometrical rotation. It therefore makes sense to look at the field in each excitation manifold searately. Let us look at a state in excitation manifold N sanned by N 1 1 basis states j0, N 1, j1, N 2 1 1,...,jN,0 1.Itis well known from the theory of linear couling of bosonic oscillators with total excitation N that the eigenstates to the Jaynes-Cummings Hamiltonian (1) are dressed states j± N n differing in eigenvalue by DE 2. The states are nondegenerate and orthogonal, and therefore rovide an alternative basis for exanding any state in excitation manifold N. To exemlify, we exlicitly exress these states in the N 2 excitation manifold: 1 j0, i 2 j1, j2, 0 1 2, (2) 2 j0, j2, 0 1 2, (3) 3 j0, i 2 j1, j2, (4) These states are eigenstates of Ŝ y with eigenvalues 2 n N and are circularly olarized since they are invariant under any geometrical rotation around the wave vector. If the field reresented by the left (right) index in the kets in Eqs. (2) (4) stands for the oscillation in the horizontal (vertical) direction, then we can identify 1 and 3 as left- and right-hand olarized states (assuming a harmonic time deendence e 2i vt2kr ). Exanded in a circularly olarized mode basis (left, right), indicated by the subscrit ±, the states are 1 j2, 0 ±, 2 j1, 1 ±, and 3 j0, 2 ±. The state 2 is symmetric with resect to the two oscillation modes. This was already noted in [10,11]. We suggest the state be called a neutrally olarized state (which should be distinguished from unolarized light [6,9,10]). We see that just like classically olarized light, the left-hand olarized state will be transformed into the right-hand olarized state under a relative hase shift of between the two modes, and vice versa. This can be accomlished with a l 2 late inserted at any angle. The neutrally olarized state is invariant under this transformation. (Similar rotational invariance, but in the context of two olarization-entangled states, has already been demonstrated [].) In classical otics the comlementary olarization to circular is linear. A linearly olarized field has the roerty that vertical (horizontal) olarization evolves to horizontal (vertical) olarization uon a geometrical rotation of 90 ±. A set of states with similar roerties can be constructed in manifold N as jz N k N11 1 X N 1 1 n 1 e i k21 n N11 j± N n, (5) where k 0, 1,..., N. The rincile of construction is identical to the construction of Pegg-Barnett hase states and relative hase states [13]. (A somewhat similar construction for excitation manifold 2 is roosed in [14] to achieve a ternary logic based on two-hoton states.) All the N 1 1 states in the set are mutually orthonormal. Furthermore, state jz N k evolves into state jz N k1l, where k 1 l should be taken modulus N 1 1, uon geometrical rotation by an angle l N 1 1 around the wave vector. In this sense the states are similar to classical linearly olarized fields. However, in contrast to their classical counterarts, the states do not have their electric-field oscillation confined to one satial direction. We note that any unitary rotation that leaves the circularly olarized states (2) (4) invariant will transform the states defined by (5) to states with identical roerties with the resect of rotation. Such a unitary transformation must have the form Û ex im 1 ± 2 1 j 1 3 j. The choice ex in 2 ± 2 2 j 1 ex ix 3 ± 2 m 0, n 2, x 0 (this articular choice will be motivated below) will generate the set of states i 2 j0, j1, i 2 j2, 0 1, (6) 2 1 i 6 j0, i 6 j2, 0 1, (7) i 2 j0, j1, i 2 j2, 0 1, (8) in the second excitation manifold. It is straightforward to show that geometrically rotating state 2 by 3 (2 3) will make it evolve into the orthogonal state ( 3 1 ). The reason we made the articular choice m 0, n 2, x 0 is that 2 can be generated from the state j1, 1 1 by the means of a hase late. The j1, 1 1 state can exerimentally be generated by sontaneous arametric down conversion (SPDC). In this rocess a um hoton is converted into a air of hotons with lower energy. SPDC is ossible in nonlinear, birefringent materials via the conservation of energy v v s 1v i and momentum k k s 1 k i, where the subscrits refer to the um () and the down-converted hotons, usually denoted signal (s) and idler (i). There are two tyes of SPDC; in tye-i SPDC the signal and idler have the same olarization, whereas in tye-ii SPDC they are orthogonally olarized. Our exerimental setu is outlined in Fig. 1. Similar setus have been used for other measurements of fringe visibility []. We use tye-ii SPDC so the state after 5014

3 FIG. 1. A hoton air roduced in BBO via tye-ii sontaneous arametric down-conversion iminges on a comensator late C (used to make the two hotons overla in time) and then on a filter F (used to narrow the sectral width of the air). A relative hase shift of h 2 arccos 1 3 is alied in a basis rotated by 45 ± from the vertical, so the outgoing state is the 2 3 state. A l 2 late inserted at 22.5 ± gives the same state in the horizontal and vertical basis. A second relative hase shift of 2h is then alied between the horizontal and vertical basis. A symmetric beam slitter is followed by two analyzers aligned at 45 ± and 135 ±. The two detectors are connected to a coincidence circuit that rojects out the state j1, 1 3. the nonlinear crystal is j1, 1 1. In a linear basis rotated by 45 ± around the wave vector (indicated by the subscrit 3), this state can be exressed j0, j2, Inserting a variable birefringence comonent (in our case a birefringent rism air) with its rincial axes at 45 ± and 135 ±, a state of the form j0, ex ih j2, will be generated. Secifically the state 2 3 is generated if the relative hase shift (indicated with the letters PS in the figure) is chosen to be h 2 arccos ±. The generated state can subsequently be geometrically rotated by a l 2-late. If the late s fast axis is set at an angle a 2 from the 45 ± direction, the state is geometrically rotated by a degrees, e.g., if the fast axis is set at 22.5 ± from the vertical, the state 2 is rotated so that the bases refer to the horizontal and vertical directions. (It should be noted that a l 2 late does not strictly corresond to a geometrical rotation. For a general state exressed in some linear basis it corresonds to a ure geometrical rotation of the basis, followed by a relative hase shift of in the rotated basis. However, for the state 2 which only has comonents with an even number of hotons, 0 or 2, this articular hase shift is inconsequential as ointed out in the aragrah following Eq. (4). In other cases it can be canceled with the hel of a second l 2-late, succeeding the first, oriented with its rincial axes in the rotated basis directions.) To detect the state we transform it to a j1, 1 state which subsequently can be detected using coincidence counting. To this end a second relative hase shift of 2h is alied in the horizontal-vertical basis. If the state in this basis rior to the second relative hase shift is 2 1, the state after the hase shift is j0, j2, Following the second relative hase shift is a nonolarizing beam slitter (BS) and two olarization analyzers (A 1 and A 2 ), one for each arm. The analyzer in the transmitted arm is oriented at 45 ± and the analyzer in the reflected arm is oriented at 135 ±. When the l 2 late is inserted at 22.5 ± we see a maximum in the coincidences since at this angle we roject the state 2 onto itself. By rotating the l 2 late to b 2 the coincidence rate is roortional to j z 2 2 j ex ibŝ y 2 j 2 sin2 3b 9 sin 2 b. (9) At b 60 (b 260) degrees this rojection robability is zero since the state 2 evolves into the orthogonal state 3 ( 1 ). In the exeriment our light source is a single-mode cw argon-ion laser with a wavelength of nm. The SPDC crystal is b-bab 2 O 4 (BBO) with a length 0.5 mm. The crystal is aligned so that collinear orthogonally olarized hoton airs with equal energy are roduced. The um is searated from the hoton airs by a disersion rism. Pinholes and one interference filter (F), with a bandwidth of 10 nm and centered at nm, further select hoton airs that have the same energy and that travel in the same satiotemoral mode. Because of the linear disersion of the crystal the horizontally and vertically olarized hotons are searated in time. A birefringent crystalline quartz late (C) is used to comensate for the linear disersion. The detectors (D 1 and D 2 ) are actively quenched single-hoton counting modules (EG&G SPCM). The exerimental data resented in Fig. 2 is just as collected (raw data) without any background subtraction. The achieved visibility is 90%. The minima are achieved for rotations of aroximately 45 ± 6 60 ± from the vertical as redicted (corresonding to a rotation of the l 2 late by 22.5 ± 6 30 ± from the vertical). The nonerfect visibility is exlained by the fact that the measurement requires careful mode matching and alignment of 7 birefringent and 4 nonbirefringent otical comonents. Furthermore, the Coincidence counts er 500 s ζ 1 ζ 2 ζ Rotation angle (deg) Background level FIG. 2. Exerimentally measured values of the coincidences as a function of twice the rotated angle of the l 2 late from its original osition at 22.5 ±.Atb 660 ±, the coincidence rate is reduced to the background level. The dotted line indicates the behavior of a classical state under the same geometrical rotation. 5015

4 comensator late gives imerfect disersion cancellation between the vertically and horizontally olarized hoton. The state is then in a mixture between the desired state and the states j0, 1 1 and j1, 0 1. These latter states slightly deteriorate the quality of the measurement. The exeriment described above is an exerimental demonstration of Heisenberg-limited olarimetry, although not the first []. The rotation sensitivity of the states defined by Eq. (5) scales as N 1 1, clearly exceeding the standard quantum limit ~ 1 N. A roblem with the scheme is that the states are highly entangled. Therefore, in higher excitation manifolds, highly nonlinear otical comonents will be needed to generate the states from, e.g., two-mode number states. This is a difficulty this scheme shares with many other alications where entanglement is used to otimize the device erformance, such as quantum comuting. Let us now briefly discuss the difference between our formalism and the traditional Stokes oerator formalism. Our starting oint was finding the eigenstates of the generator of geometrical rotation Ŝ y. From these, we constructed the eigenstates of a comlementary oerator through Eq. (5). It is clear that the set of states is associated with a Hermitian oerator since in each excitation manifold the set is comlete. The comlementary nature of the oerator follows from the fact that j ± N k jz N l j 2 1 N 1 1 ; k, l 1,2,...,N 1 1. Comlete knowledge of the circular olarization state of a field recludes any knowledge of its comlementary olarization state (all outcomes are equally robable), and vice versa. It is also clear that the latter is not true for the eigenstates of the Stokes oerators, excet in manifold N 1. The state j0, 2 1 is a linearly olarized state and is an eigenstate of Ŝ z â y â 2 ˆb y ˆb. However, the fact that j ± 2 2 j0, 2 1 j fi 1 3 shows that Ŝ y and Ŝ z are not comlementary in site of the fact that they do not commute. Similarly, j z 2 2 j0, 2 1 j 2 1 2, showing that Ŝ z is not comlementary to the olarization oerator that can be defined through Eqs. (6) (8) either. There are three noncommuting Stokes oerators Ŝ x, Ŝ y, and Ŝ z, and one Stokes oerator Ŝ 0 which commutes with all the other three, defining a SU(2) algebra. So far, in our theory of olarization rotation, we have talked about only three Hermitian oerators, Ŝ y, Ŝ 0, and the oerator defined by its set of eigenstates (5). However, as shown in [15], in every manifold there exists (at least) one more oerator, defined through the set of states (11), (17), and (36) in [15], which commutes with Ŝ 0. In excitation manifold N 1 the three noncommuting oerators exhaust the set of mutually comlementary oerators, they define a SU(2) algebra and therefore a choice can be made so that they coincide with the Stokes oerators. However, in a manifold N. 1, defining a Hilbert sace of dimension N 1 1, there exist at least three, and sometimes (when N 1 1 is a rime or a ower of a rime [15]) as many as N 1 2, mutually comlementary oerators, all commuting with Ŝ 0. Therefore this algebra is richer than the SU(2) grou. However, the oerators detailed roerties, e.g., their commutation relations, will have to be left out of this aer due to sace limitations. What is imortant to retain is that, excet for one hoton state, linearly olarized states and circularly olarized states are not eigenstates of comlementary oerators. The set of comlementary olarization oerators may include either Ŝ y or Ŝ z, but not both. In this aer we made the choice to start from Ŝ y, but we could just as well have started from Ŝ z. With the latter choice, the states corresonding to Eqs. (6) (8) would have been the twomode relative hase states [13]. Let us now briefly touch uon the difference between the classical and a quantum mechanical definition of olarized light. In classical hysics, olarization is a manifestation of correlations between the two transverse field modes defined in an Euclidian sace. Therefore only second order correlations are needed to fully classify the state of olarization. A quantized transverse field with a fixed number of hotons N, on the other hand, is defined in a Hilbert sace of dimension N 1 1. In this case, the olarization is a manifestation of correlations between N 1 1 quantum states. Therefore, if N. 1, the quantum descrition offers more degrees of freedom than the classical descrition. Just as it is ossible to use quantized fields to imrove the resolution of interferometers, it is ossible to use quantized fields to increase the resolution of olarimetry. Finally, it may be noteworthy to oint out that the quantized olarization states derived in this aer defy the corresondence rincile. They will be increasingly nonclassical as the excitation N increases. They share this feature with many modern alications of quantum mechanics such as quantum comuters. The reason is that the states are maximally entangled irresectively of their excitation, and therefore lack classical counterarts. This work was funded by the Swedish Foundation for Strategic Research, INTAS through Grant No , the Swedish Royal Academy of Sciences, John och Karin Engbloms Stiendiefond, the Boston University Photonics Center, and the U.S. National Science Foundation. *Electronic address: gunnarb@ele.kth.se [1] A short oerational introduction to the Stokes arameters is given in E. Hecht, Am. J. 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