Observation of geometric phases in quantum erasers
H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, M. Kitano
aa r X i v : . [ qu a n t - ph ] M a r Observation of Geometric Phases in Quantum Erasers
H. Kobayashi, S. Tamate, T. Nakanishi, K. Sugiyama, and M. Kitano Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510, Japan
In this study, we report the manifestation of geometric phases in the setup for quantum erasers.Our experiment includes a double-slit interferometer with the polarization as an internal state ofa photon. With regard to the visibility of the interference fringe, we can demonstrate the disap-pearance of fringes by which-path marking and the recovery of interference using quantum erasers,and the phase shift of the fringe due to the evolution of the polarization state is attributed to thegeometric phase or the Pancharatnam phase. For a certain arrangement, the geometric phase canbe very sensitive to a change in state and this is observed as a rapid displacement of the fringes.
I. INTRODUCTION
Wave-particle duality is one of the most intriguing fea-tures of quantum mechanics. This property manifestsitself prominently in the Young’s double slit experiment;each quanta creates a single spot on the observation planeaccording to the probability amplitude, and the spots cre-ated by thousands of quanta result in a clear fringe pat-tern due to the superposition of wavefunctions for thetwo possible paths followed by the quantum particle .Here, we assume that there exists a device to “mark”each particle according to the path followed by it. Thisoperation, called as which-path marking, enables us todistinguish the two states for the path so that the super-position of the path states has been collapsed and theinterference disappears.Surprisingly, although the which-path marking de-stroys the interference, we can recover the interferencefringe by erasing the which-path information. This ideacalled as the quantum eraser was first proposed by Scullyand Dr¨uhl , and it has been discussed extensively in con-nection with the wave-particle duality .A simple demonstration of the path marking and thequantum eraser using the internal states of a photon canbe demonstrated using a double-slit interferometer as fol-lows. A photon is marked with the right and left circularpolarization states according to the paths. Because wecould distinguish the path state by measuring the polar-ity of the circular polarization, no interference patternis observed. However, when a linear polarizer is placedbehind the double slit, the circular polarizations are pro-jected into the same linear polarization and which-pathinformation in the polarization state is completely erased.Therefore, the interference fringes are recovered .In addition to the recovery of interference, due to thechange in the polarization states, the quantum eraser alsoinduces an additional phase shift determined by threepolarization states: two states due to the which-pathmarking and one due to the linear polarizer used for thequantum eraser. This phase shift is called as the Pan-charatnam phase, which is proportional to the area ofthe spherical triangle connecting the three states on thePoincar´e sphere . From this geometric property, it isalso called as the geometric phase . It has been shownthat the Pancharatnam phase can be very sensitive to a FIG. 1: (Color online). (a) A typical Young’s double-slit in-terferometer using photons. The path states of the photon arerepresented as the state of the transverse wave numbers, | k A i and | k B i . (b) A double-slit interferometer with an internalstate. The which-path marker comprises of the linear polar-izer LP and the two quarter-wave plates QWP A and QWP B .The linear polarizer LP serves as the quantum eraser. Thestates | ψ i , | ψ i , | ψ A i , and | ψ B i are the polarization statesafter LP , LP , QWP A , and QWP B , respectively. The anglesof the transmission axes of LP and LP are θ and θ , andthose of the fast axes of QWP A and QWP B are 0 ◦ and 90 ◦ ,respectively. change in state for a certain arrangement .Recently, based on the interferometric point of view,Tamate and his co-workers revealed that the Pancharat-nam phase contributes to the weak measurements .In weak measurements, we can obtain unusual resultsthat lie well outside of the range of eigenvalues of anobservable. Owing to this property, the weak measure-ment is very useful for experimentally detecting minuteeffects . It has been shown that the high sensitivityof the Pancharatnam phase to a change in state plays anessential role in weak measurements .In this study, we report the manifestation of the Pan-charatnam phase in the setup for quantum erasers. Theloss of interference by which-path marking can be ex-plained by the fact that the interference pattern de-stroyed by the which-path marking contains two com-plete interference patterns that are shifted by differentamounts due to the Pancharatnam phase. This is demon-strated in our experiment for a double-slit interferometerwith internal states of a photon. With regard to thevisibility of the interference fringe, we can confirm thedisappearance of interference due to which-path markingand recovery of interference using the quantum eraser.On the other hand, we can observe the nonlinear varia-tion of the Pancharatnam phase with regard to the phaseshift of the fringe in the same setup. Although each phe-nomenon has already been described in previous works,in our experiment we can observe these phenomena us-ing a single setup. It may be useful for showing a uni-fied viewpoint of the quantum eraser and the geometricphase, and based on this viewpoint, our experiment canbe interpreted as the minimal setup of weak measurementfor the measured qubit (polarization states) coupled withthe qubit meter (path states) .The remainder of this paper is organized as follows. InSec. II, we introduce a theoretical model for a double-slitinterferometer with the internal states and analyze the in-terference pattern in the process of which-path markingand the quantum eraser. Moreover, we confirm the non-linear variation of the Pancharatnam phase in a certainarrangement. In Sec. III, we describe our experimentalsetup and results on the quantum eraser and the Pan-charatnam phase. A summary is presented in Sec. IV. II. THEORETICAL ANALYSIS OFDOUBLE-SLIT INTERFEROMETERS WITHINTERNAL STATES
An interferometer with internal states can be analyzedas a quantum system composed of the path state and theinternal state. In this section, we theoretically analyzethe interference patterns in our double-slit experimentwith regard to both the intensity and the phase.
A. Which-path marking
Because of the large distance between the double slitand the screen, the state of the photon through slits Aand B can be assumed to be an eigenstate of the trans-verse wave numbers on the screen, | k A i and | k B i , re-spectively, as shown in Fig. 1(a). These states satisfythe normalization condition h k | k ′ i = δ ( k − k ′ ), where δ ( · ) shows the Dirac delta function. In our setup, aphoton is marked with the polarization states | ψ A i and | ψ B i according to the paths using the quarter-wave platesQWP A and QWP B (see Fig. 1(b)). Assuming that thephoton has a 50:50 chance of passing through each slit, the total state vector for the composite system can berepresented as the following superposition: | Ψ m i = | ψ A i| k A i + | ψ B i| k B i . (1)Here, the path states and two polarization states are cor-related or entangled . We introduce the operator ˆ P x ≡ ˆ I ⊗ | x ih x | , which projects the path state into the positionstate | x i on the screen. With the position representationof the wave-number eigenfunction, h x | k i = e i kx / √ π , theprobability distribution P m ( x ) is given by P m ( x ) = h Ψ m | ˆ P x | Ψ m i∝ V m cos ( kx − δ m ) , (2)where k ≡ k B − k A and V m = |h ψ B | ψ A i| , (3) δ m = arg h ψ B | ψ A i . (4)For the double-slit apparatus, k is calculated as k =2 πd/λL , where λ is the wavelength of light; d , the dis-tance between two slits; and L , the distance between thedouble slit and the screen. The coefficient of the interfer-ence term, V m , can be experimentally obtained from thefringe pattern as the visibility V m = P max − P min P max + P min , (5)where P max and P min are the maximum and minimumvalues of P m ( x ), respectively.The degradation of visibility is related to the efficacy ofwhich-path marking, which depends on the inner product h ψ A | ψ B i . The lesser the value of |h ψ A | ψ B i| , the lesseris the visibility. In particular, when h ψ A | ψ B i = 0, twostates are perfectly distinguishable and the path followedby the photon is discriminated unambiguously. Then, theinterference is completely eliminated. B. Quantum eraser
Now, we erase the which-path information by the pro-jection of polarization using the linear polarizer LP thatprojects the polarization state into | ψ i (see Fig. 1(b)).The state vector for the composite system after LP iscalculated as follows: | Ψ f i = | ψ ih ψ | Ψ m i = | ψ i (cid:16) c A | k A i + c B | k B i (cid:17) , (6)where c A = h ψ | ψ A i and c B = h ψ | ψ B i . The probabilitydistribution P f ( x ) is given by P f ( x ) = h Ψ f | ˆ P x | Ψ f i∝ V f cos ( kx − δ f ) , (7)where the visibility V f and the phase shift δ f are given as V f = 2 | c A | · | c B || c A | + | c B | , (8) δ f = arg h ψ B | ψ ih ψ | ψ A i . (9) FIG. 2: (Color online). Separation of the partial interferencepattern. (a) The partial interference pattern can be separatedinto two fringes with 100% visibility shifted by γ ( ψ A , ψ B , ψ )and γ ( ψ A , ψ B , ψ ⊥ ). (b) The shifts are related to the solidangles of the spherical triangles on the Poincar´e sphere as γ ( ψ , ψ , ψ ) = − Ω( ψ , ψ , ψ ). Equation (8) shows that even when | ψ A i is orthogonal to | ψ B i , the visibility is recovered completely provided that | c A | = | c B | . In this case, the states of the which-pathmarker, | ψ A i and | ψ B i , are projected into the same polar-ization state | ψ i with the same probability, and it cannotbe determined whether the photon came from slit A or B.This implies that LP completely erases the which-pathinformation, and the interference is recovered. C. Quantum eraser and Pancharatnam phase
As shown in the previous section, the which-pathmarker can destroy the interference pattern effectively.However, using the quantum eraser, the interference canbe restored completely. This implies that the completeinterference pattern is buried under the destroyed inter-ference pattern. In this section, we will show that theinterference pattern destroyed by the which-path mark-ing contains two complete interference patterns that areshifted by different amounts according to the projection of the polarization state. These phase shifts can be inter-preted geometrically using the Poincar´e sphere, as shownbelow.Equation (2) can be separated into two terms using theprojected state | ψ i that satisfies |h ψ | ψ A i| = |h ψ | ψ B i| and its orthogonal state | ψ ⊥ i as follows: P m ( x ) = h Ψ | (cid:2)(cid:0) | ψ ih ψ | + | ψ ⊥ ih ψ ⊥ | (cid:1) ⊗ | x ih x | (cid:3) | Ψ i∝ | c A | P f1 ( x ) + (cid:0) − | c A | (cid:1) P f2 ( x ) , (10)with P f1 ( x ) ≡ (cid:2) kx − δ m − γ ( ψ A , ψ B , ψ ) (cid:3) , (11) P f2 ( x ) ≡ (cid:2) kx − δ m − γ ( ψ A , ψ B , ψ ⊥ ) (cid:3) , (12)where the additional phase shift γ is defined as γ ( ψ , ψ , ψ ) ≡ arg h ψ | ψ ih ψ | ψ ih ψ | ψ i . (13)Due to the difference in the phase shifts, γ ( ψ A , ψ B , ψ ) − γ ( ψ A , ψ B , ψ ⊥ ), even though both fringes P f1 ( x ) and P f2 ( x ) have 100% visibility, the sum of these patternshas reduced visibility [see Fig. 2(a)].The right-hand side of Eq. (13) is gauge-invariant, i.e.,independent of the choice of the phase factor of eachstate, because the bra and ket vectors for each stateappear in a pair. This phase shift γ is identified withthe Pancharatnam phase . It can be shown that thePancharatnam phase is proportional to the solid angleΩ( ψ , ψ , ψ ) of the spherical triangle connecting thestates | ψ i , | ψ i , and | ψ i with geodesic arcs on thePoincar´e sphere , i.e., γ ( ψ , ψ , ψ ) = −
12 Ω( ψ , ψ , ψ ) . (14)Therefore, each phase shift of two fringes, γ ( ψ A , ψ B , ψ )and γ ( ψ A , ψ B , ψ ⊥ ), can be represented geometrically onthe Poincar´e sphere. When the two marker states | ψ A i and | ψ B i are located along a meridian symmetrically withrespect to the equator, the projected states | ψ i and | ψ ⊥ i should be on the equator in order to satisfy the condition |h ψ | ψ A i| = |h ψ | ψ B i| , as shown in Fig. 2(b).In particular, if | ψ A i is perpendicular to | ψ B i , that is,a 100%-effective which-path marker is prepared, the fourstates | ψ A i , | ψ B i , | ψ i , and | ψ ⊥ i lie on the same greatcircle. Then, the following equation is satisfied: γ ( ψ A , ψ B , ψ ) − γ ( ψ A , ψ B , ψ ⊥ ) = π, | c A | = 12 . (15)The total pattern is composed of two fringes having thesame intensity but opposite phases, and the interferenceis completely washed out. A general case of partial era-sure is shown in Fig. 2. D. Nonlinear variation of Pancharatnam phase
From Eq. (14), we can calculate the Pancharatnamphase in our experiments. We modify the standard (a)(b)
FIG. 3: Variation of the Pancharatnam phase (a) with respectto θ and θ , and (b) with respect to θ for different θ . double-slit interferometer to include two linear polariz-ers and two quarter-wave plates, as shown in Fig. 1(b).First, we prepare the initial polarization state | ψ i us-ing the linear polarizer LP : | ψ i = cos θ | H i + sin θ | V i , (16)where θ is the angle between the horizontal line and thetransmission axis of LP ; | H i , the horizontal polarizationstate; and | V i , the vertical polarization state. In oursetup, the fast axes of two quarter-wave plates, QWP A and QWP B , are aligned to form angles of 0 ◦ and 90 ◦ ,respectively, from the horizontal line. Thus, they inducephase shifts of ± π/ | ψ A i = cos θ | H i + i sin θ | V i , (17) | ψ B i = i cos θ | H i + sin θ | V i . (18)Here, the pair of quarter-wave plates serves as the which-path marker in Eq. (1). The final state of the polarizationis expressed as | ψ i : | ψ i = cos θ | H i + sin θ | V i , (19)where θ is the angle between the horizontal line and thetransmission axis of LP . FIG. 4: (Color online). Geometrical interpretation of the non-linear variation of the Pancharatnam phase around ( θ , θ ) =(0 , π/ | ψ A i and | ψ B i are close to each other on thePoincar´e sphere, the area of the spherical triangle blows upvery rapidly with the movement of | ψ i around θ = π/ From Eqs. (17), (18), and (19), we can obtain the Pan-charatnam phase as γ ( ψ A , ψ B , ψ )= ( − (cid:0) tan θ tan θ (cid:1) (cos 2 θ ≥ , − (cid:0) tan θ tan θ (cid:1) + π (cos 2 θ < . (20)Figure 3(a) shows the variation of the first term ofEq. (20) with respect to θ and θ . It is noteworthythat this variation exhibits strong nonlinearity around( θ , θ ) = (0 , π/ π/ , π/ , π ), and ( π, π/ θ is plotted for different values of θ . The figure showsthat the smaller the value of θ , the faster is the changein the Pancharatnam phase with respect to θ near θ = π/
2. We can observe this nonlinear variation asa rapid displacement of the fringes when we change θ by rotating LP .The nonlinear variation of the Pancharatnam phasecan be explained by the spherical geometry on thePoincar´e sphere, as shown in Fig. 4. In our experiment, | ψ A i and | ψ B i , given by Eqs. (17) and (18), respectively,can be depicted at a latitude of ± θ on the prime merid-ian, and the final state | ψ i , given by Eq. (19), can bedepicted on the equator at a longitude of 2 θ . We as-sume that | ψ A i and | ψ B i are located near | H i , that is,0 < θ ≪ π/ | ψ i moves on the equa-tor from | H i . When the distance between | ψ i and | V i is greater than 2 θ , the area of the spherical trianglespanned by | ψ A i , | ψ B i , and | ψ i remains small. How-ever, when | ψ i approaches | V i and the distance betweenthem becomes lesser than 2 θ , the area of the sphericaltriangle increases very rapidly, and after traversing | V i ,the triangle covers most of the Poincar´e sphere. This is FIG. 5: (Color online). Experimental setup for double-slitquantum eraser. Light passing through the right and left ofthe wire interferes. Each path is marked by two film-typequarter-wave plates, QWP A and QWP B , whose fast axes Fmake angles of 0 ◦ and 90 ◦ , respectively. The interferencefringe is captured using a CCD camera. the geometrical reasoning why the Pancharatnam phasechanges rapidly in certain conditions. III. EXPERIMENTS
The experimental setup is shown in Fig. 5. The lightsource is a 532-nm green laser with a 3-mm beam diam-eter (model DPGL-2200, SUWTECH). A thin opaquewire crossing the beam works as the double slit; the lightpassing through the right- and left-hand sides of the wireinterferes due to diffraction. We attached two film-typequarter-wave plates having the orthogonal fast axes, 0 ◦ and 90 ◦ , with a thin piece of double-sided adhesive tapethat works as a wire. A double slit having a similar de-sign has been introduced by Hilmer and Kwiat .Two film-type linear polarizers, LP and LP , are at-tached to the rotatable mounts with graduated scales foradjusting the angles θ and θ . At a distance of approx-imately 1 m from the double slit, the recombined beamis captured using a charge-coupled device (CCD) camera(model LBP-2-USB, Newport) connected to a personalcomputer (PC). The CCD camera has a resolution of640 ×
480 pixels, each having a size of 9 µ m × µ m, andit is equipped with a gain controller. A. Experimental results of quantum erasers
First, by setting θ = π/ , the initial state of polarization | D i isevolved into two orthogonal states through the quarter-wave plates, right circular polarization, and left circu-lar polarization according to the paths. Because we candetermine which slit the photon has passed through bymeasuring the polarity of the circular polarization of thephoton, no interference pattern is obtained. (This ismathematically confirmed from Eq. (3), which vanishes FIG. 6: (Color online). Interference patterns captured usinga CCD camera. (a) By setting θ = π/ ,a typical diffraction pattern is observed. (b) By setting θ = π/ θ = 0, the fringe pattern reappears. (c) By setting θ = π/ θ = π/
2, the fringe pattern is out of phasewith the case of θ = 0. po w e r / a r b . un it position / mm q = 0 q = p /2 FIG. 7: Recovered interference fringes for the quantum eraserwith θ = 0 and θ = π/ when | ψ A i is orthogonal to | ψ B i .) We observed a typicaldiffraction pattern that only has broad peaks, as shownin Fig. 6(a).By inserting LP , the right and left circular polar-izations are projected into the same linear polarizationwith the same probability, and therefore, the polariza-tion provides no which-path information. As a result,the interference fringe reappears. (Mathematically, thiscorresponds to the fact that Eq. (8) becomes unity when | c A | = | c B | .) Figure 6(b) shows the recovered interfer-ence fringe for θ = 0. Similarly, for θ = π/
2, we canobtain the corresponding interference fringe, as shown inFig. 6(c), which is out of phase with that observed for θ = 0 (see Fig. 7). This phase difference is attributed tothe Pancharatnam phase. The sum of these interferencepatterns reproduces the broad peak pattern, as shown inFig. 6(a), that is obtained in the absence of LP . There-fore, the quantum eraser actually filters out one of these FIG. 8: (Color online). The shift of fringes induced by thePancharatnam phase with respect to θ (a) when θ = 45 ◦ and (b) when θ = 9 ◦ . The light intensity of each frame isnormalized individually. When θ = 9 ◦ , the fringe exhibits arapid displacement around θ = 90 ◦ . x / D x q / ˚ q = 45˚ q = 30˚ q = 18˚ q = 9˚ FIG. 9: Experimental results of Pancharatnam phase withrespect to θ for different θ . The vertical axis shows thedisplacement of the fringe normalized by the spatial period ofthe fringe. fringes and perfectly recovers the visibility. B. Observation of Pancharatnam phase and itsnonlinearity
In order to observe the variation of the Pancharatnamphase with respect to θ and θ , we measured the dis- placement of the fringes. Figure 8 shows the fringe shiftwith respect to θ for fixed θ . The light intensity ofeach fringe in Fig. 8 is normalized individually. When θ = 45 ◦ , the fringe moves linearly with respect to thechange in θ , as shown in Fig. 8(a). However, setting θ = 9 ◦ , the fringe exhibits a quick displacement around θ = 90 ◦ , as shown in Fig. 8(b).Figure 9 shows the variation of the Pancharatnamphase with respect to θ for θ = 45 ◦ , 30 ◦ , 18 ◦ , and 9 ◦ .The points in Fig. 9 indicate the experimental results andthe solid lines indicate the theoretical lines calculated us-ing Eq. (20). The vertical axis is the displacement of thefringe x normalized by the spatial period of the fringe ∆ x .The origin of the vertical axis is determined by the po-sition of the fringes when θ = 0 ◦ . All the experimentalresults agree well with the theoretical ones. The gradientof the variation of the shift around θ = 90 ◦ increases as θ is decreased. This implies that the variation of theshift becomes more sensitive to the variation of the lastpolarization state. IV. SUMMARY
We have shown that the Pancharatnam phase mani-fests in the setup for quantum erasers. In our experiment,we have introduced a double-slit interferometer with in-ternal states of a photon and demonstrated which-pathmarking, quantum erasers, and the variation of the ge-ometric phases. The visibility of the interference fringeis related to the which-path marking and the quantumeraser, and the phase shift of the interference shows themanifestation of the Pancharatnam phase. Moreover, wehave demonstrated that the Pancharatnam phase couldbecome sensitive to a change in the polarization state.This fact can be utilized for high-precision measurementof the polarization.Even though our experiment is performed with classi-cal light, it serves the purpose of showing the quantum-mechanical meaning of which-path marking, quantumerasers, and geometric phases since photons are nonin-teracting Bose particles and our tests can be straightfor-wardly extended to experiments with single photons . Acknowledgments
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