An experimental investigation of measurement-induced disturbance and time symmetry in quantum physics
Davor Curic, Magdalena C. Richardson, Guillaume S. Thekkadath, Jefferson Flórez, Lambert Giner, Jeff S. Lundeen
aa r X i v : . [ qu a n t - ph ] J a n An experimental investigation of measurement-induceddisturbance and time symmetry in quantum physics
D. Curic , M.C. Richardson , G.S. Thekkadath , J. Fl´orez , L. Giner , J.S. Lundeen Department of Physics and Centre for Research in Photonics,University of Ottawa, 25 Templeton Street,Ottawa, Ontario, K1N 6N5, Canada and Clarendon Laboratory, University of Oxford,Parks Road, Oxford, OX1 3PU, UK
Abstract
Unlike regular time evolution governed by the Schr¨odinger equation, standard quantum mea-surement appears to violate time-reversal symmetry. Measurement creates random disturbances(e.g., collapse) that prevents back-tracing the quantum state of the system. The effect of thesedisturbances is explicit in the results of subsequent measurements. In this way, the joint resultof sequences of measurements depends on the order in time in which those measurements areperformed. One might expect that if the disturbance could be eliminated this time-ordering de-pendence would vanish. Following a recent theoretical proposal [A. Bednorz et al 2013 New J.Phys. 151515 023043], we experimentally investigate this dependence for a kind of measurement thatcreates an arbitrarily small disturbance, weak measurement. We perform various sequences of aset of polarization weak measurements on photons. We experimentally demonstrate that, althoughthe weak measurements are minimally disturbing, their time-ordering affects the outcome of themeasurement sequence for quantum systems. P whose momentum ppp is coupled to measured observable AAA on the measured system S via unitary interaction UUU = exp ( − iδAAA ⊗ ppp ) = X a | a i h a | ⊗ TTT ( δa ) . (1)Here, δ denotes the interaction strength. On the right-side, we have rewritten UUU in theeigenbasis of
AAA , where | a i is an eigenstate of AAA with eigenvalue a. The translation operator,
TTT ( δa ) = exp( − iδappp ), shifts the pointer’s position, i.e., TTT ( δa ) | x i = | x − δa i , where | x i is aposition eigenstate. If the initial pointer state | ψ i has a position width σ < δa then the post-interaction pointer position x = δa unambiguously indicates the measurement result a . Thisis the case commonly found in conventional measurements, such as a polarizing beam splitter(PBS), or measuring the spin of a silver atom with a Stern-Gerlach apparatus. With thisexplicit model, one can consider reducing the interaction strength so that δa ≪ σ . Since thepointer now extends over multiple indicator marks, δa (where a is a particular value in the3pectrum of AAA ), a single trial’s measurement result will be ambiguous. However, averagedover many trials the mean measurement result is proportional to the average pointer position, δ h AAA i = h xxx i , regardless of measurement strength. It is this average result, the expectationvalue of AAA , that we will study.To extend the above formalism to include a sequence of measurements
AAA N · · · AAA AAA , onecomposes a product of unitaries UUU N . . . UUU UUU , where UUU i takes the form of Eq. 1. Here, A A A isthe first observable measured and AAA N is the last. Thus, each observable AAA i in the sequenceof measurements is independently coupled to a distinct pointer P i with state | ψ ( x i ) i . Thefinal result of the measurement is the expectation value h xxx ( AAA )1 xxx ( AAA )2 · · · xxx ( AAA N ) N i /δ N , where thesuperscript ( AAA i ) is the observable to which the P i pointer is coupled to [24]. With this, we canconsider the effect of reducing the interaction strength and testing different time-orderingsof the measurements. For example, how does h xxx ( AAA )1 xxx ( AAA )2 i compare to h xxx ( AAA )1 xxx ( AAA )2 i ?The experimental setup is shown in Fig. 1. It is technically challenging to use spatiallydistinct systems as our pointer P and measured system S . Instead, we use distinct degreesof freedom of a photon. The measured system S is the polarization degree of freedom,whereas the pointer P is the photon’s transverse position. A HeNe laser at 633 nm followedby a polarizing beam splitter (PBS) and a rotatable half-wave plate (HWP), set at anangle θ, prepares an ensemble of identically polarized photons as our system input state, | θ i = cos (2 θ ) | H i + sin (2 θ ) | V i .The coupling between S and P can be accomplished using polarization dependent walk-offin birefringent crystals. The walk-off transversely shifts the extraordinary polarized photonsby δ relative to ordinary polarized photons. A sequence of two weak measurements requirestwo independent pointers, which we take as the two transverse spatial degrees of freedom, x and y , of the photon, each with same initial wavefunction: h x, y | ψ x , ψ y i = ψ ( x ) ψ ( y ) =(2 πσ ) − exp ( − ( x + y ) / σ ). A beam expander magnifies the HeNe’s transverse Gaussianmode to set the a width of σ = 600 µ m. With these two pointers, the measurement couplingis implemented with two walk-off crystals, one of which displaces the horizontal polarizationalong the x -axis, followed by an identical crystal rotated by 90 ◦ so that it shifts the verticalpolarization along the y -axis. Both crystals impart a shift of δ = 160 µ m ensuring that weare in the weak measurement regime, δ/σ ≈ . . In order to change which observable each crystal implements we add waveplates that ef-fectively rotate the basis of the measurement. With this, we use either the xxx = xxx or xxx = yyy y FIG. 1. Setup to measure the effect of time-ordering of weak measurements. (a)(a)(a). State preparation:a beam expander is used to decrease the interaction strength by increasing the width σ of thepointer, i.e., the photon transverse distribution. The input system polarization state | θ i is preparedby a polarizing beam splitter (PBS) and a half-wave plate (HWP). Using quarter-wave plates(QWP) and a HWP (QWP at 45 ◦ → HWP → QWP at 90 ◦ ) incoherently polarized light can begenerated. The HWP is attached to a motor that spins at a rate faster than the collection time ofthe camera. (b)(b)(b) Weak Measurements: Each weak measurement is implemented by a HWP followedby a walk-off crystal (xtal). The first effects the πππ I projector by shifting | I i polarized light by δ < σ in the x direction. Likewise, the second implements the πππ J projector by shifting | J i polarized lightby δ in the y direction. (c)(c)(c) Strong measurement: The combination of a HWP and a PBS realizesthe third measurement of πππ H when desired, and is taken out of the setup otherwise. (d)(d)(d) A 4 f system images the shifted beam onto a imaging camera. positions to read out measurements of the AAA = | I i h I | = πππ I , and AAA = | J i h J | = πππ J polar-ization projectors, depending on the ordering. We demonstrate the effect of time-ordering onweak measurements by comparing the result of a sequential measurement h xxx ( πππ I ) yyy ( πππ J ) i withthe reversed sequence h xxx ( πππ J ) yyy ( πππ I ) i . In all cases, the expectation values h xyxyxy i of the photon5ransverse two-dimensional distribution are found by imaging onto a camera.We begin the experiment by placing the first HWP in Fig. 1(b) at 0 ◦ and the secondat 67 . ◦ so that the two crystals implement a measurement of πππ H followed by πππ D , twoincompatible observables. By switching the first HWP to 22 . ◦ and leaving the second as itis, the crystals now implement the reverse sequence, πππ D followed by πππ H . In both cases, werecord the joint result of the sequence h xyxyxy i . In Fig. 2 we plot this joint result as a functionof the input system state angle, θ . The two orderings agree within errors. Thus, as expectedsince the measurement disturbance is now minimized, the joint result does not depend onthe time-ordering of the measurements. Π H π D π H Π H π H π D System Input State |θ(cid:0), Polarization Angle θ (deg) M e a s u r e m e n t R e s u l t (cid:1) x y (cid:0) / δ FIG. 2. The measurement result h xyxyxy i /δ for a sequence of two measurements. In one, the sequence πππ H πππ D is measured (red triangles), and in the other, πππ D πππ H is measured (blue squares). Since thepoints agree within error, the plot shows that the results do not depend on the order in which themeasurements are performed. The red solid and blue dashed line are the respective theoreticalcurves. Error bars are the standard error obtained by averaging over four experimental runs.Imperfections in the HWP birefringence likely introduces the differences between the experimentalpoints and the theoretical curve, as they can create systematic errors not only when preparing theinput polarization state | θ i , but also when aligning the walk-off crystals. So far, nothing surprising has been revealed: when the measurement disturbance is mini-6ized, the result of a sequence of quantum measurements is indeed time-ordering invariant.However, there are fundamental phenomena in quantum physics that only appear in se-quences of three or more measurements. An example is the violation of the Leggett-Garginequality [25, 26]. Hence, we extend the sequence to three measurements, where the thirdmeasurement is a conventional (i.e., ‘strong’) measurement of ΠΠΠ K = | K i h K | as implementedby a HWP and PBS (here the capital pi indicates a strong measurement). Our goal is againto test the role of measurement-order when the disturbance is not a factor. Since this addedconventional measurement will substantially disturb the system and, thus, any subsequentmeasurements, we always perform it last.The final joint-result of the three measurement sequence is h πππ K πππ J πππ I i = 1 δ h ΠΠΠ K xxx ( πππ J ) yyy ( πππ I ) i = 1 δ Z xy Prob( x, y, K )d x d y (2)where Prob( x, y, K ) is the probability that a given input photon is transmitted through thePBS and is detected at transverse coordinate ( x, y ) on the imaging camera. Since the lastis a strong measurement, in Eq. 2 we directly evaluate the measurement outcome h ΠΠΠ K i ,rather than use the Von Neumann formalism. We set K = H for all of the followingmeasurements. Figure 3(a) shows experimental results for two orderings of a sequence ofthree measurements, ΠΠΠ H πππ D πππ H and ΠΠΠ H πππ H πππ D . The joint result of one ordering substantiallydisagrees with the other ordering. Theoretically, the difference is maximum at θ =45 ◦ and θ =135 ◦ . The nearest experimental points, 48 ◦ and 138 ◦ , differ by 0.4 ± . ± . N pointer positions h xxx ( AAA )1 xxx ( AAA )2 · · · xxx ( AAA N ) N i was found to be proportional to the recursively nested anti-commutatorstructure {{ . . . {{ AAA N , AAA N − } , AAA N − } , . . . } , AAA } . While all the anti-commutators are symmet-7
30 60 90 120 150 1800.00.51.0 (a) Π H π D π H Π H π H π D (b) Π H π D π H Π H π H π D System Input State |θ(cid:0), Polarization Angle θ (deg) M e a s u r e m e n t R e s u l t (cid:1) x y Π H (cid:0) / δ FIG. 3. The measurement result h xy Π xy Π xy Π H i /δ for a sequence of three measurements. In one, thesequence ΠΠΠ H πππ D πππ H is performed (blue squares), and in the other, ΠΠΠ H πππ H πππ D is performed (redtriangles). The blue dashed and red solid lines are the respective theoretical curves. ( a )( a )( a ) The inputsystem state is | θ i . The two distinct curves show that the order in which the measurements aremade changes the measurement result. ( b )( b )( b ) An incoherently polarized system state ρ ( θ ) is usedinstead. Now the measurement result does not depend on the ordering of the measurements. ric under interchange of their two arguments, the only anti-commutator that is invariantunder interchange of the measurements is the innermost one, which is non-nested. As such,it, and thus the expectation value, is invariant to the ordering of the last two measurements, AAA N and AAA N − . This explains why a sequence of two measurements always exhibits orderinginvariance; the entire sequence is the last two measurements. The result of a sequenceof three or more measurements will be invariant solely to the ordering of the last twomeasurements.While we now have a mathematical reason for the time-ordering dependence of mini-mally invasive measurements, a physical explanation is still absent. A distinguishing featureof weak measurements is that they preserve the coherence of the measured system andthe coherence of the pointer. This can allow a disturbance to propagate in unexpectedways, as in measurement back-action [28, 29]. Since in a classical system this coherence8ould be absent, we test what happens to the time-ordering dependence as we decrease thecoherence in the H-V basis of the initial system state. To generate these reduced coher-ence (i.e., mixed) states, we send our polarized input state | θ i through a rapidly spinningHWP that is sandwiched between two quarter-waveplates (QWP). Since this spinning isfaster than the imaging camera acquisition time, the resulting state is effectively mixed: ρ ( θ ) = sin ( θ ) | H i h H | + cos ( θ ) | V i h V | . We test the same pair of three-measurement se-quences with these input states. The experiment results, shown in Fig. 3(b), show that thejoint result of the sequence does not depend on the order in which the observables weremeasured. Quantum coherence indeed appears to play an important role in time-orderingsymmetry.To summarize our experimental findings, in the case of two measurements, the orderin which weak measurements are made does not impact the end result. But, in the caseof three or more measurements, the order of the measurements matter. In short, we haveshown that reducing the disturbance induced by measurements does not restore the timesymmetry of quantum evolution, as exhibited by the Schrodinger equation. Our findingsconfirm a recent mathematical description of sequential weak measurements [27]. Whilethe physical mechanism for this time-ordering invariance is still not clear, we have shownthat coherence plays a role. We expect these results will guide the development of closelyrelated areas, such as whether different times of system can be considered separate Hilbertspaces [3], and how cause and effect can be identified in quantum systems [30].We would like to thank E. Giese for his thoughtful discussions. This work was supportedby the Natural Sciences and Engineering Research Council (NSERC), the Canada Excel-lence Research Chairs (CERC) Program, and the Canada First Research Excellence Fund(CFREF). [1] G. Muga, R. S. Mayato, and I. Egusquiza, Time in quantum mechanics , Vol. 734 (SpringerScience & Business Media, 2007).[2] J. Hilgevoord, Stud. Hist. Philos. Mod. Phys. , 29 (2005).[3] D. Horsman, C. Heunen, M. F. Pusey, J. Barrett, and R. W. Spekkens, Proc. R. Soc. A (2017).
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