Topological transition in measurement-induced geometric phases

Valentin Gebharta,b,1, Kyrylo Snizhko b,1 , Thomas Wellensa, Andreas Buchleitnera, Alessandro Romitoc, and Yuval Gefenb,2

aPhysikalisches Institut, Albert-Ludwigs-Universitat¨ Freiburg, 79104 Freiburg, Germany; bDepartment of , Weizmann Institute of Science, Rehovot 76100, Israel; and cDepartment of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

Edited by Yakir Aharonov, Chapman University, Orange, CA, and approved February 4, 2020 (received for review July 10, 2019)

The state of a quantum system, adiabatically driven in a cycle, may the Pancharatnam phase is equal to the Berry phase associated acquire a measurable phase depending only on the closed trajec- with the trajectory that connects the states |ψk i by the shortest tory in parameter space. Such geometric phases are ubiquitous geodesics in Hilbert space (6, 16). and also underline the physics of robust topological phenom- The Pancharatnam phase can quite naturally be interpreted ena such as the . Equivalently, a geometric as a result of a sequence of strong (projective) measurements phase may be induced through a cyclic sequence of quantum acting on the system and yielding specific measurement read- measurements. We show that the application of a sequence of outs (17). This interpretation is valid for optical experiments weak measurements renders the closed trajectories, hence the observing the Pancharatnam phase induced with sequences of geometric phase, stochastic. We study the concomitant probabil- polarizers (18). Such a phase can be consistently defined despite ity distribution and show that, when varying the measurement the fact that measurements (typically considered an incoher- strength, the mapping between the measurement sequence and ent process) are involved. A generic sequence of measure- the geometric phase undergoes a topological transition. Our find- ments is an inherently stochastic process. One thus expects a ing may impact measurement-induced control and manipulation distribution of measurement-induced geometric phases, deter- of quantum states—a promising approach to quantum informa- mined by the sequences of measurement readouts associated tion processing. It also has repercussions on understanding the with the corresponding probabilities. For a quasicontinuous foundations of quantum measurement. sequence of strong measurements (N → ∞ and k|ψk+1i − |ψk ik = O(1/N )), the induced evolution is fully deterministic quantum measurement | quantum trajectories | quantum feedback | due to the dynamical quantum Zeno effect (17), thus yielding Berry phase | topological phases of matter a unique Pancharatnam–Berry phase. Recently, geometric phases induced by weak measurements he overall phase of a system’s is an unmea- [that do not entirely collapse the system onto an eigenstate Tsurable quantity that can be freely assigned. However, when of the measured observable (19)] have been experimentally the system is driven slowly in a cycle, it undergoes an adiabatic observed (20). The setup of ref. 20 ensured that only one evolution which may bring its final state back to the initial one (1, 2); the accumulated phase then becomes gauge invariant and, Significance therefore, detectable. As noted by Berry (3), this is a geometric phase (GP) in the sense that it depends on features of the closed trajectory in parameter space and not on the dynamics of the We bring together two concepts at the forefront of current process. Geometric phases are key to understanding numerous research: measurement-induced back action (resulting in the physical effects (4–6), enabling the identification of topological steering of a quantum state) and topological transitions. It invariants for quantum Hall phases (7), topological insulators, has been widely believed that the transition from the limit and superconductors (8, 9); defining fractional statistics anyonic of a strong (projective) measurement to that of weak mea- quasiparticles (10, 11); and opening up applications to quantum surement involves a smooth crossover as a function of the information processing (12, 13). measurement strength. Here we find that varying the mea- Geometric phases are not necessarily a consequence of adi- surement strength continuously may involve a topological transition. Specifically, we address the measurement-induced abatic time evolution. For any pair of states |ψl i, |ψm i in geometric phase, which emerges following a cyclic sequence Hilbert space, it is possible to define a relative phase, χl,m ≡ of measurements. Our analysis suggests that measurement- arg [hψl |ψm i]. For a sequence of states (14, 15) |ψk i, k = based control and manipulations of quantum states (a promis- 0, ... , N , for which |ψN i ∝ |ψ0i, one can define the total phase accumulated through the sequence [the Pancharatnam ing class of quantum information protocols) may involve phase (14, 15)] subtle topological features that have not been previously appreciated. N −1 (P) X Author contributions: K.S., A.B., A.R., and Y.G. designed research; V.G., K.S., and T.W. χgeom = χk+1,k performed research; V.G., K.S., T.W., A.R., and Y.G. analyzed data; and V.G., K.S., T.W., k=0 A.B., A.R., and Y.G. wrote the paper.y = arg [hψ0|PN ... P2P1|ψ0i] = arghψ0 | ψ˜N i, [1] The authors declare no competing interest.y This article is a PNAS Direct Submission.y ˜ where |ψk i = Pk ... P2P1|ψ0i and Pk = |ψk i hψk | is the projec- Published under the PNAS license.y ˜ tor onto the kth state. Note that |ψk i ∝ |ψk i is not normalized, Data deposition: The code implementing the simulations and the relevant data can be which, however, does not undermine the definition of the phase found at GitHub, https://github.com/KyryloSnizhko/top-geom-meas.y (P) 1 (unless some |ψ˜k i = 0). Note also that χgeom is independent V.G. and K.S. contributed equally to this work.y of the gauge choice of phases of all |ψk i. For a quasicontin- 2 To whom correspondence may be addressed. Email: [email protected] uous sequence of states {|ψk i}, the Pancharatnam phase triv- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ ially coincides with the Berry phase under the corresponding doi:10.1073/pnas.1911620117/-/DCSupplemental.y continuous-state evolution. Moreover, for any sequence {|ψk i}, First published March 2, 2020.

5706–5713 | PNAS | March 17, 2020 | vol. 117 | no. 11 www.pnas.org/cgi/doi/10.1073/pnas.1911620117 Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 (sin 1. Fig. by spanned is operators space the by as represented Hilbert are observables labeled whose measurements, qubit weak a is |↑i, system Our Quantum Strength Measurements Variable from their Phases Geometric facilitate Defining and detectors of geo- presence define the detection. consistently in to phases us inter- metric allow concrete number propose which Chern we protocols, the Finally, ferometry of surface. jump this a with through associated topolog- manifest our and is sphere, transition Bloch the surface ical on sequences: a area form certain measurement a sequences covers these of which by a induced family to trajectories related state a invariant. The is consider topological it integer-valued we that an Specifically, sense of the jump strength. in measurement discontinuous topological the is of function transition a The as place takes phase ric trajectory trajectories state possible all postselected over averaging single in of a effect the of consider mainly scenarios and We the strength. on measurement the focus on phases) crucially sequence the depend (and measurement trajectories here the the readouts, by measurement determined the and solely result- the are and phase trajectory projective postselecting state ing of the by case where the 18), out (17, to of measurements singled opposed phase As be sequences. the readout can analyze specific and which phases trajectories func- geometric distribution specific induced full the the of compute tion tunable We with strength. readouts detector measurement following and system measurements two-level of a sequence of a state quantum a by accrued phase quantum controlling successfully (23–26). been dynamically states has for which experimentally sys- measurements, employed the employing weak of context, through monitoring wider tem continuous a enables In measurements 22). weak (21, system measurements sequences the weak quasicontinuous sequences, for of even readout phase. stochastic remain possible observed dynamics the all to considering contributed When sequence readout possible ie(e o)t h ledtvatesots edsco h lc pee o ekmaueet,teqatmtaetr osntcicd ihthe with coincide not does the trajectory al. projects et quantum segment) Gebhart the (yellow measurements, measurement weak final The For lines). sphere. blue Bloch and geodesic. the (red shortest state the on initial via geodesic the state shortest at initial the terminate the not via may dot and blue sequence the measurement to dot) (red line all (c measurements strength (c measurements strong for at depicted sequence surement edsoe htatplgcltasto vis- transition topological a that discover We geometric the investigate and define we study, present the In IAppendix. SI r k θ h ytmudrosacrnlgclsqec of sequence chronological a undergoes system The |↓i. k + = cos esrmn eune n nue unu rjcois ( trajectories. quantum induced and sequences Measurement xetfrasingle a for except ϕ k sin , θ k sin ϕ k cos , = otdbakln)mtclul olw h esrmn eigenstates measurement the follows meticulously line) black dotted +∞, = ,rdln)dvae rmti ie ekmaueetidcdtaetr (c trajectory measurement-induced weak A line. this from deviates line) red 3, r k θ 0 k Left ,floigaprle ( parallel a following ), = − o ifrn esrmn tegh n edu eune.Tetaetr nue ythe by induced trajectory The sequences. readout and strengths measurement different for edu nue tt upfo t oiino h red the on position its from jump state a induces readout “−” The line. blue dotted-dashed the by depicted is readout k 1, = . . . , σ N n θ ` -i h geomet- the a-vis h measured The . k k = = and 4 π/ σ · n k where , ϕ k = Left 2π k esrmn eunesandb h directions the by spanned sequence measurement A ) /N probabilities described is mapping interaction-induced which the interaction by system–detector the by mediated is state initial before detector the the state while specify The so- is to (27–29). for measurement opt measurements the need we weak study we present null-type the analysis called For measurement. our the with of nature proceed further To de- be can phase geometric as fined measurement-induced weak The where space, {r strength {n measurement the Given Methods). and als where readout measurement obtained strength r its by acterized S orientations ment (sin σ le h tt ftesse;2 fteiiilsse tt is state system initial the if 2) system; the (a of is state state the alter system initial the b If 1) the properties: in following measured projectively is of detector basis the step, this Following k .(Right ). 2 0 = (σ = k k +, = 0 = } sdpce nFg 1, Fig. in depicted as χ } θ nue euneo states of sequence a induces geom fmaueetoinain ihcrepnigreadouts corresponding with orientations measurement of k nEq. in , M x cos b h oicto ftesse tt odtoa othe to conditional state system the of modification The −. , (a |±i unu rjcoiso h lc peeidcdb h mea- the by induced sphere Bloch the on trajectories Quantum ) σ 1 = k ( arg = r y → | ϕ k ni | , ) ψ ,i ie ihcranyreadout certainty with gives it 4), k tts oeta hsmaueetpooo a the has protocol measurement this that Note states. ˜ σ  p = sin , k + nEq. in z − a i ) M b h |ni = = ψ |−ni)|+ stevco fPuimtie and matrices Pauli of vector the is η θ | k 0 η PNAS k {n ψ + ˜ (n | |ψ sin ψ r and ˜ ,i ilsreadouts yields it 4), 1 k b k N i = ,...r } p | , = i ϕ + r )crepnigt h edu eunewith sequence readout the to corresponding 3) ensataetr nteui sphere unit the on trajectory a defines | arg = p k 1 ahwa esrmn schar- is measurement weak Each Left. k a n k ) + cos , ac 7 2020 17, March η − k i | r ru prtr 2,2)(Materi- 22) (21, operators Kraus are hl the while i, ni ∈ 1 = = η 0 1] [0, |−ni + M h θ | h esrmn rcs is process measurement The +i. ψ k − r b h euneo measure- of sequence The ). k ( k 0 where |−ni, r n η |M k  sgvnby given is epciey gi without again respectively, , ) | n w osbereadouts possible two and |+i ψ ˜ . . . k N ( {r r i | N k o + M {r ) +} = o.117 vol. k ntesse Hilbert system the in . . . b r 1 ( +} = √ r = r 1 M ) η + = σ rjcoyfrfinite- for trajectory − | |−ni|−i. | ψ ψ n 1 ( edu sequence readout |±ni n 0 r | M → i or 1 k i η n osnot does and ) N o 11 no. = . sequence a , |ψ r hsaeonto state th | (x ni = 0 + = k i. , ±|±ni, k ( y (a r k | k , |−ni n ) z with 1 = |ψ 5707 k k [2] [3] [4] ) = = i, ,

PHYSICS altering the state of the system. In general, when the system For a general choice of η the final state |ψ˜N i may not be state is in a superposition of |ni and |−ni, the measurement proportional to the initial state |ψ0i, meaning the trajectory does alter the system state. For η  1, the detector remains |ψ1i → ... → |ψN i is not closed. For simplicity, we take the practically always in its initially prepared state (r = +, i.e., null last measurement to be strong (ηN = 1) and postselect it to outcome), modifying the system state only slightly; yet with prob- yield rN = + (i.e., discard those experimental runs that yield 2 (rN ) (+) ability η|b| the readout is r = −, inducing a jump in the system rN = −); hence MN = MN = |ψ0i hψ0| = P0. The probabil- state to |−ni. Considering only the experimental runs resulting in ity of getting a specific sequence of readouts {rk } can then be r = + allows one to define “null weak values” (27, 28). For arbi- expressed as trary η, such postselected measurements may be implemented, with imperfect polarizers, as depicted in Fig. 2: A photon of one D E 2 D E 2 P = ψ ψ˜ = ψ ψ˜ [5] polarization is always transmitted (r = +), while a photon of the {rk ,rN =+} 0 N 0 N −1 orthogonal polarization has finite probability to be transmitted ˜ ˜ (r = +) or absorbed (r = −). Here, we do not restrict ourselves with |ψN i = |ψ{rk }i as defined in Eq. 2. Thus, to postselected measurements and to the η  1 limit. Below (Materials and Methods) we address a Hamiltonian implemen- (rN −1) (r1) p iχgeom hψ0|M ... M |ψ0i = P{r ,r =+}e . [6] tation of such measurements in the spirit of the von Neumann N −1 1 k N

(30) measurement model. We next study χgeom as a function of the measurement strength. q N → ∞ Define the normalized state |ψk i = |ψ˜k i/ hψ˜k |ψ˜k i. With We consider measurements with measurement orienta- {n } (θ , ϕ ) = iαk tions k that follow a given parallel on the sphere, k k the standard parameterization, |ψk i = e (cos Θk /2|↑i + (θ, 2πk/N ). The initial state is |ψ0i = cos θ/2|↑i + sin θ/2|↓i; iΦk e sin Θk /2|↓i), the sequence of states is mapped onto a cf. Fig. 1, Right. The measurement strength of each individual discrete trajectory on the Bloch sphere with spherical coordi- measurement is ηk = η = 4c/N → 0 with c being a nonnegative Θ Φ Right nates k and k . Fig. 1, depicts state trajectories that constant (except for ηN = 1). The sequence of N − 1 weak mea- correspond to measurement orientation sequences (Fig. 1, surements can be characterized by an effective measurement Left) of various measurement strengths. The particular type −4c strength ηeff = 1 − e , 0 6 ηeff 6 1. of measurement we employ guarantees that hψ˜k |ψ˜k−1i = ∗ (rk )†  (rk )  Probability Distribution of the Measurement-Induced hψ˜k−1|M |ψ˜k−1i = hψ˜k−1|M |ψ˜k−1i > 0, enabling us k k Geometric Phase to express the above geometric phase in the same form as the The probability distribution of the geometric phases is reported Pancharatnam phase (Eq. 1). It thus follows that χgeom = −Ω/2 in Fig. 3 as a function of the parameter c quantifying the effec- can be expressed via the Ω subtended by a piecewise tive measurement strength. For c → 0, the distribution is peaked trajectory on the Bloch sphere that connects the neighboring around χgeom = 0, corresponding to a vanishing back action states (|ψ i and |ψ i; here we imply |ψN +1i := |ψ0i) along k k+1 from the measurement process. With increasing measurement shortest geodesics. Note the difference between weak and strength, the distribution develops a main peak which contin- projective measurements. In the latter, the system states |ψ i k uously evolves toward the Pancharatnam phase in the strong are fully determined by the measurement orientation and the measurement limit. This peak corresponds to the most proba- measurement readout r . By contrast, the system state following k ble trajectory associated with a specific readout sequence, r = + a weak measurement also depends on its strength η < 1 and k for all k; cf. Fig. 1 (red solid line). A secondary peak develops on the state before the measurement. Furthermore, for a for intermediate measurement strengths due to the nonvanishing quasicontinuous sequence of strong measurements (N → ∞ and probability of obtaining r = − for some k. kn − n k = O(1/N )), the readout r = − is impossible due to k k+1 k k We first turn our attention to the geometric phase associated the dynamical quantum Zeno effect (17), rendering all readouts with a specific readout sequence, r = + for all k (this means r = + and the measurement-induced trajectory deterministic. k k that if any r = −, that particular experimental run should be For a quasicontinuous sequence of weak measurements, the k discarded). For a generic measurement strength, it is the most trajectory is, instead, stochastic, manifested in a variety of probable measurement outcome, and hence the corresponding possible readout sequences {r }; cf. Fig. 1, Right. The probability k GP is the most likely one. We parameterize the Hilbert space of obtaining a specific sequence of readouts {rk } is given trajectory as |ψ(t)i, t ∈ [0, 2π], so that |ψ(t = πk/N )i = |ψk i for P = hψ˜ |ψ˜ i by {rk } N N . k = 1, ... , N − 1 and |ψ(t ∈ [π, 2π])i is the shortest Bloch sphere geodesic between |ψN −1i and |ψN i = |ψ0i; cf. Fig. 1, Right. This parameterization results in a quasicontinuous trajectory since k|ψk+1i − |ψk ik = O(1/N ) for k < N − 1. We investigate the behavior of χgeom and link it to the behavior of |ψ(t)i as a func- tion of the measurement strength η. Since the measurements are not projective (measurement strength η → 0), the state after each

measurement is not necessarily the ↑ eigenstate of σnk . The state trajectory on the Bloch sphere for θ = π/4 and c = 3 is shown in Fig. 1 (red solid line). The probability P = P of measur- Fig. 2. An imperfect polarizer implementing a null-type weak measure- {rk =+} ing the desired readouts and the corresponding geometric phase ment. Each polarizer transmits a certain polarization (here, ez), with cer- tainty. An impinging beam with generic polarization (blue arrows) is either χgeom (Eq. 6) can be calculated analytically for N → ∞ and are transmitted, resulting in a null readout, r = +, or absorbed, r = −. (Left)A given by vertically polarized photon (|↑i) is transmitted without altering its polariza- √ tion. (Center) A horizontally polarized photon (|↓i) is transmitted with a Peiχgeom = −e−c (cosh(τ) + z sinh(τ)/τ), [7] probability 1 − η < 1 (pale blue arrow). (Right) A photon of generic polar- ization |ψi = a|↑i + b|↓i is transmitted with probability |a|2 + (1 − η)|b|2 p 2 2 2 (+) with τ = z − π sin θ and z = c + iπ cos θ; cf. Fig. 3 (red and a modified polarization state, |ψi → Mη (ez, r = +)|ψi = M |ψi. By rotating the polarizers and adding phase plates (e.g., quarter- plates), solid lines). it is possible to engineer a fully transmitted polarization direction | + ni = We note three qualitatively different regimes depending cos θ/2|↑i + eiϕ sin θ/2|↓i. on the parameter c controlling the effective measurement

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For continuous.) is < geom = { 0, ∈ θ , c c 7 χ crit crit o ie esrmn tegh osdrthe Consider strength. measurement given a for −α geom . sra,ipyn that implying real, is , , −π χ | := R ψ PNAS χ = χ −π geom (θ geom π/ geom }. h χ ydmnigthat demanding by {r X 0 = ) r e 2 geom hs h neplto of interpolation the Thus, . k k 2i (π )= 2) (π/ } | + = oeas that also Note 0. = 2) (π/ c χ 2) (π/  M r )i geom ac 7 2020 17, March > efidta h eie finfinitely of regimes the that find We . k h (θ η ψ + = a o ece h qao smid- ’s equator the reached not has c [0, : ) (e o all for c 0 crit i |M  z realizations sildfie steprobability the as ill-defined is θ , −π 2π = Ω ntemaueetsequence measurement the on r = n snnoooi for nonmonotonic is and 1 N ( ) π r c θ N (i.e., h xrsini the in expression the 2, π/ −1 k sra (Eq. real is oaciia strength, critical a to ] > = −1 → ar elgbeprobabili- negligible carry χ c 2 π/ ) | geom crit and , Nt htalthough that (Note R. o 2π mod . . . η θ χ o.117 vol. eff 2 =π/ χ geom h rjcoyafter trajectory the , isetrl nthe on entirely lies M geom (θ → χ χ 1 ( = (t θ 0 0 = (0) geom geom r ,i eae as behaves it 1), ,wihguar- which 17), 1 (θ = eufl it unfold we , 2) π/ ∈ ) | |ψ χ = ) 2 π/ [0, o 11 no. (θ (θ χ geom 0 θ i geom π π = =  a take can = n that and 2 Fg 5). (Fig. tra- ])i) (cos r (θ , )= 2) π/ )= 2) π/ k 2 π/ c | 2) (π/ crit ) + = 5709 c θ sa is [8] ≈ − < is )

PHYSICS B˜ (θ, t) = Im (∂t hψθ(t)|∂θ|ψθ(t)i − ∂θ hψθ(t)|∂t |ψθ(t)i). [10]

Transitions in quantum dynamics as a function of the measure- ment strength have been known for single qubits (29) and more recently discovered for many-body systems (31). Notably, the topological nature of the transition we report here is of signifi- cance. Importantly, it implies that the transition is robust against perturbing the protocol. For example, if one considers sequences of measurements that follow generic closed curves different from the parallels considered above, it would not be possible to define χgeom(π/2) and determine the transition from its discrete set of values. However, as long as the family of measurement sequences wraps the sphere, there is a Chern number which assumes a discrete set of values (characterizing a global property of the set of measurement-induced trajectories) controlled by the mea- surement strength. Importantly, the Chern number is different Fig. 4. Nonmonotonicity of geometric phases. Shown is dependence of the in the limits of weak and strong measurements. This guaran- geometric phase on the polar angle θ for the postselected trajectory (red tees the existence of a critical measurement strength, ccrit, at solid lines) for different values of the integrated measurement strength which the Chern number changes abruptly, and a concurrent (key). The ideal strong measurement dependence for c → ∞ is presented jump of the phase χgeom associated with a critical measurement θ as a gray dashed line. The asymptotic dependence of the GP on dis- sequence. Unlike the transition, whose existence is protected plays an abrupt transition from monotonic to nonmonotonic behavior in the by the change of the topological invariant, the value ccrit at vicinity of c = 2.15. The behavior is underlined by the fact that χgeom(π/2) can assume only discrete values, 0 or −π. Inset shows the probability of which it takes place and the corresponding critical measure- observing the most probable trajectory with postselected readout sequence ment sequence are nonuniversal and depend on the specifics of

{rk = +} at c = 2.1; the gray dashed line indicates P = 1 for c → ∞, showing the protocol. the dynamical quantum Zeno effect. Experimental Implementations This picture, in fact, extends beyond the trajectories on the To detect the postselected GP in an experiment, we design a equator. Consider the manifold formed by all state trajecto- protocol based on a Mach–Zehnder interferometer incorporat- ries, which is obtained via measurement sequences at θ ∈ [0, π] ing detectors in one of its arms (Fig. 6A). An impinging particle (Fig. 5). For c > ccrit, this manifold covers the Bloch sphere, with an internal degree of freedom (spin for electrons, polar- while for subcritical c it does not. The GP transition then ization for photons) in state |ψ0i is split into two modes in corresponds to a change in the topology of the set of state the two interferometer arms. The compound system-detector√ trajectories—thence the designation “topological transition.” state is then |Ψi i = |ψ0i ⊗ (|a = 1i + |a = −1i) ⊗ |+ ··· +i/ 2, While this transition can be intuitively understood from the where a = ±1 describes the particle being in the upper or lower behavior of the trajectory on the equator, we prove it formally arm, respectively, and |+ ··· +i is the initial state of the detec- below (Materials and Methods) by considering the Chern number tors. In the upper arm, the particle is subsequently measured by all of the detectors; in addition it acquires an extra dynam- 1 Z π Z 2π ical phase γ controlling the interference. Running through the C ≡ dθ dt B˜ (θ, t) lower arm, the state is left untouched. Traversing the inter- 2π 0 0 √ |Ψ i = |ψ i|a = −1i|+ ··· +i/ 2 + 1 ferometer, the state is then f 0 √ = (χgeom(π) − χgeom(0))∈ {0, −1}, [9] iγ P QN (rk ) e M |ψ0i|a = 1i|{rk }i/ 2, where rk is the 2π {rk } k=1 k readout of the kth measurement, and |{r }i is the correspond- ˜ k where B(θ, t) is the Berry curvature ing collective state of all of the detectors. The state with all

Fig. 5. Mapping of measurement sequences onto the system’s quantum trajectories: a topological transition. Shown is a θ-dependent family of trajectories on the Bloch sphere for c < ccrit (Left) and c > ccrit (Right). Yellow segments represent final projective measurements, ascertaining closed trajectories. For stronger measurements (Right), the S2 space of the measurement orientations n is mapped through the measurement process onto the whole S2 Bloch sphere of quantum trajectories. For weaker measurement strengths, the S2 space of measurement orientations is mapped onto a subset of the Bloch sphere. The corresponding Chern numbers are −1 and 0, respectively.

5710 | www.pnas.org/cgi/doi/10.1073/pnas.1911620117 Gebhart et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 nefrmtrotusD n 2are D2 the at and signals D1 obtained outputs postselected The interferometer the pattern. detect interference the to through us GP the allow with would traversing state arms interferometer’s beam postselected the a readouts reproduces of all polarizers state of polarization sequence The is. polarization of surement orienta- photon the a control absorb to possible is tion it plates), quarter-wave (e.g., considered (Eq. measurement above null-type postselected a implementing ih.Mr pcfial,ec oaie ul rnmt n given − (| one transmits fully (| polarizer polarization each polarizer specifically, imperfect More light. 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Fig. ehr tal. et Gebhart † iet nefrne hrfr,i h nlitreec,w e nytegeometric the only see we interference, giving final one the only the the in of is phase Therefore, it information, interference. which-path to no yields rise sequence this only since but oeta nti ceew culyd o oteetthe postselect not do actually we scheme this in that Note geom npatc,ti rtclcnb mlmne yemploying by implemented be can protocol this practice, In n | | + k + sdrcl eae oteitreec hs.Nt htthe that Note phase. interference the to related directly is .Atasitdba orsod oa to corresponds beam transmitted A i). xeietlstp o bevn esrmn-nue P.(A) GPs. measurement-induced observing for setups Experimental · · · n k e r i k i {r hrfr rdcn interference producing therefore +i, γ I .B oaigteplrzr n digpaeplates phase adding and polarizers the rotating By 4). N 1,2 + = hti ul rnmte.Telre h rbblt to probability the larger The transmitted. fully is that k rdcdb en te hnmeasurements. than other means by produced r r + } = k k ∞, → = = + + = + = onie ihteiiilsaeo l ftedetec- the of all of state initial the with coincides n I I 2 2 trajectory. 0 0 k edus o“hc-ah nomto simplied, is information “which-path” No readouts. ntligasto oaiesi n fthe of one in polarizers of set a Installing . n atal bob h rhgnlone orthogonal the absorbs partially and i)  √ 1 1 Pe rasr (readout absorb or +) ± ± P √ i Re χ = P geom P e Re i {r γ k e h sgvnb Eq. by given is =+} − | i ψ χ 0 geom | n k hsdtrie h inter- the determines thus Y k N =1 h togrtemea- the stronger the i, +i M γ  {r k (+) , k +} = h impinging the −) † |ψ + h intensities The . 7 0 edu,thus readout, i and edu sequence readout ! {r I 0 k +} = sthe is [11] is netgtdaoei rtce ytehriiiyo h Kraus the of hermiticity the by operators, protected partic- transition is the is of above This nature polarizer. investigated topological accu- a the is because through important passing polarizations ularly light two the the by (15, addi- between mulated no phase difference that such Pancharatnam phase designed the is carefully tional be of polarization must polarizers realization orthogonal The a 18). the is the corre- while blocked) of transmitted polarizers fully polarization fully of one is (i.e., limit beam measurements the strong that to sponding Note con- polarizers. pattern imperfect interference the from 11 extracted by be trolled can GP The adsrcue n yaia vlto)oesa intriguing an opens evolution) dynamical horizon. and structure, quasiparticles, non-Abelian band e.g., sys- with, the (associated of measured structure tem topological possible and measurement the of any for place on depend take will will strength transition the number measurements, weak Fur- (N of Hermitian). quasicontinuous still investigated (yet we used while we ther, ones Kraus the by than place characterized other types, take operators different infinitely will of of transition measurements limit for the the also different, in are and numbers measurements measurements Chern strong weak the of Since limit the nature. in topological its by guaranteed transition. measurement strong-to-weak a of nature aver- the the in in of phases phase analysis An geometric geometric one. aged the nonmonotonous a of angle, to dependence monotonous polar the the of on through change manifest also abrupt is transition an This number. a Chern through a classified of is jump transition mea- This the varied. as is to strength transition sequences surement topological measurement a of undergoes trajectory). mapping phases postselected the geometric specific that shown a have of We readout words, postselected other specific (in a sequence with have associated the measuring phase We for geometric protocols stochastic. experimental inherently schematic forward are put the directly and on depend strength be adiabatic here measurement an obtained by phases can induced the phases evolution, state Hamiltonian geometric to quantum measure- system opposed of sequence As the the the ments. during words, accrued of by phase other the onto state In traced mapped phase. the trajectory geometric of purely the mea- phase a quantum the inducing generalized measured, modify of sequences may how surements the shown have by We unaffected is that Discussion level extra an by measurements. replaced should interferometer then the form of be arm (which reference junction lower low- Josephson The two qubit). superconducting the the a by of replaced levels is est freedom the of implementation, degree an internal such particle’s In (23–26). hardware qubit ducting ‡ prtr r o-emta ilb efre elsewhere. performed be will non-Hermitian are operators nivsiaino h ekmaueetidcdgoercpaewe h Kraus the when phase geometric measurement-induced weak the of investigation An ebleeta h nepa ewe h oooia nature topological the between interplay the that believe We is which context, broader much a in prevails transition The supercon- using implemented be also can protocol above The consfrtels u otelgtasrto ythe by absorption light the to due loss the for accounts θ N eedne u nlssudrcrstetopological the underscores analysis Our dependence. γ M ≥ I h ifrnei h neste oprdt Eq. to compared intensities the in difference The . 1,2 3 k ( r fmaueet abi h rtclmeasurement critical the (albeit measurements of k = = ) . ‡ I I 4 PNAS 0 0  |1 θ + 1 ftemaueetsqec rma from sequence measurement the of , ± N | 4 √ ). P ac 7 2020 17, March IAppendix SI Pe ± i χ 2 1 geom √ P +i Re γ | | e hw iia feature similar a shows 2 i . o.117 vol. χ geom sequences ∞) → +i γ |  o 11 no. | [12] 5711

PHYSICS Materials and Methods In the quasicontinuous limit, the Pancharatnam phase in Eq. 1 reduces Measurement Model. The measurement sequence leading to the geometric to the Berry phase accumulated by |Ψi and can be computed by standard phase in Eq. 3 consists of positive-operator valued measurements (POVMs) methods (3) to express it as an integral of the Berry curvature. In fact, for R θ0 R 2π (rk) (rk) θ ∈ [ π] χ θ = θ ˜ θ defined by the Kraus operators M = M (n , r ), |ψi → M |ψi, as any 0 0, , we have geom( 0) 0 0 d dt B( , t) (Eq. 21): k ηk k k k described in the main text. Such POVMs can be implemented with a detec- tion apparatus consisting of a second qubit, whose Hilbert space is spanned χgeom(θ0) = χgeom(θ0) − χgeom(0) by |+i and |−i and which is coupled to the system via the Hamiltonian 2π−dt Y = arg hΨ(Θ(θ0, t + dt), Φ(θ0, t + dt))|Ψ(Θ(θ0, t), Φ(θ0, t))i (s) (d) t=0 Hn(t) = λ(t)(1 − σn )σy /2. [13] Z 2π = i dt hΨ(Θ(θ0, t), Φ(θ0, t))|∂t |Ψ(Θ(θ0, t), Φ(θ0, t))i / Here, σ(s d) denote the Pauli matrices acting on the system and detector, 0 Z θ Z 2π respectively, σn = n · σ and n = (sin θ cos ϕ, sin θ sin ϕ, cos θ), 0 6 θ 6 π, 0 6 0 = −Im dθdt (∂θ hΨ(Θ, Φ)|∂t |Ψ(Θ, Φ)i ϕ < 2π, defines the measurement direction. The system and detector are 0 0 (in) initially (t = 0) decoupled in the state |ψs i ⊗ |+i, where −∂t hΨ(Θ, Φ)|∂θ |Ψ(Θ, Φ)i) Z θ Z 2π a 0 ˜ |ψ(in)i = a|↑i + b|↓i = . [14] = dθdt B(θ, t), [21] s b 0 0

˜ The measurement coupling λ(t) 6= 0 is then switched on for t ∈ [0, T] to where B(θ, t) is the Berry curvature introduced in Eq. 10. Alternatively, using obtain the entangled state: the mapping F : (θ, t) 7→ (Θ, Φ), the geometric phase can be expressed in terms of a curvature on the Bloch sphere as h (s) (d) i (in) |ψenti = exp −ig(1 − σn )σy /2 |ψs i |+i Z θ Z 2π 0 ˜ χgeom(θ ) − χgeom(0) = dθdt B(θ, t) (in) (in) 0 0 0 = Mη (n, +)|ψs i |+i + Mη (n, −)|ψs i |−i . [15] Z θ0 Z 2π ∂(Θ, Φ) = dθdt B(Θ, Φ), [22] R T 2 ∂(θ, t) Here, g = 0 dtλ(t) determines the measurement strength η = sin g. The 0 0 matrices Mη (n, +) and Mη (n, −) are defined by with M (n, r) = R−1(n)M (e , r)R(n), [16] η η z ∂(θ, t) B(Θ, Φ) = B˜(θ(Θ, Φ), t(Θ, φ)) ∂(Θ, Φ) where 1 0  0 0  = −Im (∂Θ hΨ(Θ, Φ)|∂Φ|Ψ(Θ, Φ)i Mη (ez, +) = p , Mη (ez, −) = √ [17] 0 1 − η 0 η − ∂Φ hΨ(Θ, Φ)|∂Θ|Ψ(Θ, Φ)i) are the Kraus operators for the measurement orientation along the z axis 1 = − sin Θ(θ, t), [23] (n = ez) and 2 ! cos θ/2 e−iϕ sin θ/2 R(n) = −iϕ [18] and where sin θ/2 −e cos θ/2 ∂(Θ, Φ) ∂Θ ∂Φ ∂Θ ∂Φ = − [24] is a unitary matrix corresponding to the rotation of the measurement ori- ∂(θ, t) ∂θ ∂t ∂t ∂θ −1 −1 entation | ± ni = R (n)| ± ezi = R (n)| ↑ / ↓i. This implements a null-type is the Jacobian of F. weak measurement as defined in the main text. The right-hand side of Eq. 22 admits a simple interpretation: χgeom(θ0) − χ = −A / A geom(0) θ0 2, where θ0 is the oriented area of the Bloch sphere Geometric Phase from a Quasicontinuous Measurement Sequence and Posts- covered by the measurement-induced trajectories |Ψ(Θ(θ, t), Φ(θ, t))i with election. The geometric phase χgeom obtained from the quasicontinuous θ ∈ [0, θ0] (here the orientation of each infinitesimal contribution is given trajectory with all outcomes rk = + is given in Eq. 7. This result is obtained by the sign of the Jacobian). In particular, for θ0 = π, Aπ = 4π if the surface −1 starting from Eq. 3. By setting |ψ0i = R (n0)|↑i, the readouts rk = + and generated by all of the trajectories wraps the Bloch sphere once, and Aπ = 0 the measurement orientations (θk, ϕk) = (θ, 2πk/N), and using the explicit if it does not wrap around the Bloch sphere (Fig. 5). This provides the two form of Kraus operators in Eq. 16, one rewrites possible values for the Chern number (9), C ∈ {0, −1}. Formally, this can be proved by explicitly using the degree of the map F (32). The degree of (+) (+) N−1 the map F, deg F, is the number of points (θ, t) that map to a given point hψ0|MN−1 ... M1 |ψ0i = h↑|δR Mη=4c/N(ez, +)δR |↑i, [19] (Θ, Φ) (provided that (Θ, Φ) is a regular point of F), taking the orientation where into account. The degree does not depend on the specific point (Θ, Φ) and can be expressed as −1 δR = R(nk+1)R (nk) X ∂(Θ, Φ) ! deg F = sgn , [25] cos2 θ + e−2πi/N sin2 θ 1 (1 − e−2πi/N) sin θ ∂(θ, t) 2 2 2 −1 = 1 −2πi/N 2 θ −2πi/N 2 θ [20] (θ,t)∈F [(Θ,Φ)] 2 (1 − e ) sin θ sin 2 + e cos 2 where F −1[(Θ, Φ)] is the set of points (θ, t) that are mapped by F into is a matrix independent of k. The quasicontinuous limit is obtained by (Θ, Φ), and sgn is the sign function. Considering the integral as the sum of diagonalizing the 2 × 2 matrix M (e , +)δR and calculating the matrix η=4c/N z infinitesimal contributions and grouping those by the image points (Θ, Φ), elements in Eq. 19 in the limit N → ∞. This yields Eq. 7. one then shows that

Chern Number for the Mapping of Measurement Orientations onto State Tra- Z π Z 2π ∂(Θ, Φ) jectories. The mapping of quasicontinuous measurement orientations onto dθdt sin Θ(θ, t) = 4π deg F. [26] 0 0 ∂(θ, t) state trajectories is topologically classified by the Chern number in Eq. 9. The

discrete values of the Chern number are in correspondence with the differ- This topological feature is reflected in the discrete value of χgeom(π/2) ent regimes of the θ dependence of χgeom(θ). To prove this, we parameterize and hence in the θ dependence of χgeom. To show this, note that Eq. iα(θ,t) Θ(θ,t) each state |ψθ (t)i, θ ∈ [0, π], t ∈ [0, 2π], as |ψθ (t)i = e (cos 2 |↑i + 7 is symmetric under complex conjugation supplemented by θ → π − θ; Θ(θ,t) iΦ(θ,t) iα(θ,t) sin 2 e |↓i) = e |Ψ(Θ, Φ)i with (Θ, Φ) being coordinates on the i.e., χgeom(π − θ) = −χgeom(θ) mod 2π. Using the continuity of χgeom(θ), Bloch sphere. Since |ψθ (2π)i = |ψθ (0)i and |ψθ=0,π (t)i = |ψθ=0,π (0)i, the we obtain χgeom(π − θ) = −χgeom(θ) + 2χgeom(π/2), and hence (for θ = parameters t and θ can be regarded as a parameterization of a sphere, and 0), χgeom(π/2) = (χgeom(π) + χgeom(0))/2 = χgeom(π)/2 = πC. Therefore, we the map (θ, t) 7→ (Θ, Φ) is equivalent to the mapping of a sphere to a sphere: have χgeom(π/2) = πC = −π for c > ccrit and χgeom(π/2) = πC = 0 for c < 2 2 F : S 3 (θ, t) 7→ (Θ, Φ) ∈ S . ccrit as shown in Fig. 4.

5712 | www.pnas.org/cgi/doi/10.1073/pnas.1911620117 Gebhart et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 7 .Fch,A .Ken .Psai,L .Shla,Brypaefo unu Zeno quantum a from phase Berry Schulman, S. L. Pascazio, S. Klein, G. A. phase. Facchi, P. Berry’s for 17. setting General Bhandari, R. Samuel, J. 16. tool a as interferometer Mach-Zehnder Electronic Gefen, Y. Feldman, E. D. Law, T. K. 11. 8 .V er,S li,Goercpae rmsak fcytlplates. crystal of stacks from phases Geometric Klein, S. Berry, V. M. 18. applications. its and interference, of theory Generalized Pancharatnam, S. 15. and anyons Non-Abelian Sarma, Das S. Freedman, stochastic M. against Stern, A. robust Simon, H. are S. that Nayak, C. gates quantum 13. Geometric Zanardi, P. Zhu, S.-L. 12. Wilczek, F. 10. 4 .Aaoo,J nna,Paecag uigacci unu evolution. quantum cyclic a during change Phase Anandan, J. Aharonov, Y. 14. ie.Atrrpaigti simulation this repeating After mined. ehr tal. et Gebhart ueetwssmltd fe simulating After z simulated. was surement represented the was For measurement). trajectory postselected quasicontinuous strong the Namely, by account. to |h probabilities different their over taking sum readouts the surement of simulation Carlo Monte readouts measurement a using obtained been Simulations. Numerical Carlo Monte state readout measurement ties the mined | culated .F ice,A Shapere, A. Wilczek, F. changes. adiabatic 4. accompanying factors phase Quantal Berry, V. M. 3. Messiah, adiabatensatzes. A. des Beweis 2. Fock, V. Born, M. 1. .J .Asb K. J. 9. con- Hall Quantized Bernevig, Nijs, A. den B. M. 8. Nightingale, P. M. Kohmoto, M. Thouless, J. D. 7. properties. Jamiolkowsk, electronic A. on Chruscinski, effects D. phase Berry 6. Niu, Q. Chang, C. M. Xiao, D. 5. ψ { ˜ ψ r k k effect. (1988). 2342 A Sect. Sci. Acad. Indian statistics. fractional probe to 1990). 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(r (r oooia nuaosadTplgclSuperconductors Topological and Insulators Topological = M k k )|ψ M )i/ 3–4 (1999). 232–240 257, 1 (r 55 (1984). 45–57 392, η n,A P A. any, ( = ´ 1 k 0 p (n ) emti hssi Physics in Phases Geometric (r |ψ {r 4–6 (1956). 247–262 44, k k p(r h hn o h selected the for Then, )i. , 0 k ψ 231(2005). 020301 72, r i| } k k 0 hs e.B Rev. Phys. k 2 )|ψ a acltd fe hc h etmea- next the which after calculated, was ) |ψ alyi, =1,...,N ´ emti hssi lsia n unu Mechanics Quantum and Classical in Phases Geometric = N DvrPbiain,1961). Publications, (Dover k −1 |h − −1 hr oreo oooia Insulators Topological on Course Short A ψ e.Md Phys. Mod. Rev. 1 i) | ∝ i 0 −1 = |M 2 h eut o h vrgdG have GP averaged the for results The = esmltdtesqecso mea- of sequences the simulated We . 9 ekmaueet n one and measurements weak 499 439(2006). 045319 74, r η ψ k ˜ (n P +/− = k { i N .Phys. Z. r N N −1 (|ψ =+ realizations , P N WrdSinic 1989). Scientific, (World r k k 0315 (2008). 1083–1159 80, { } N i hmaueet ecal- we measurement, th − e r −1 6–8 (1928). 165–180 51, k hs e.Lett. Rev. Phys. 2i codn oprobabili- to according n admydeter- randomly and 2) ,r stenraiainof normalization the is ekmeasurements, weak 1 χ N ) geom =+ . . . hs e.Lett. Rev. Phys. times, r M } k ( h normalized the , { = η r k (n |h } WrdScientific (World .Md Optic. Mod. J. ) 1 ψ e , rc .Sc A Soc. R. Proc. i 0 r 2 a deter- was 1 | χ ¯ 405–408 49, ψ )|ψ ˜ geom N (Princeton e.Mod. Rev. hs Rev. Phys. 2339– 60, (Springer −1 0 i| −α i| 2 Proc. 2 in- 43, = = 9 .Aaoo,D .Abr,L ada,Hwtersl famaueeto com- a of Cho measurement Y.-W. a 20. of result the How Vaidman, L. Albert, Z. D. Aharonov, Y. 19. 1 .Syizwk,A oio .Shmrs nageettasto rmvariable- from transition Entanglement Schomerus, H. Romito, A. the Szyniszewski, M. in 31. dynamics qubit in mechanics” phase quantum in measurement of of model Crossover Neumann von “The Korotkov, Mello, A. P. N. A. 30. Mizel, A. Ruskov, in values” R. weak 29. null and “Standard Gefen, Y. Romito, A. Zilberberg, O. 28. trajectories quantum single Observing Siddiqi, I. Macklin, C. Weber, J. S. Murch, W. K. Milburn, 23. J. G. Wiseman, M. H. 22. Jacobs, K. 21. 3 .Gbat .Wles oeaddt sdi h ae Tplgcltasto in transition “Topological paper the in used data and Code Wellens, T. Gebhart, V. 33. Dancer, N. Ambrosio, L. 32. Zilberberg O. 27. Minev K. Z. 26. Naghiloo M. 25. Weber J. S. 24. otfo h niern n hsclSine eerhCuclvaGrant via Council Research Sciences Physical sup- and EP/P010180/1. acknowledges Engineering A.R. the Foundation), C01). from the port funding (project Research 183 under acknowledge (German 277101999–TRR Y.G. Forschungsgemeinschaft Foundation Projektnummer and Deutsche Minerva K.S. the program. the by Grant by Research support Short-Term financial acknowledges ACKNOWLEDGMENTS. be can data (33). relevant GitHub the at and described found simulations algorithm the data the implementing code the to and The on according above. postselection based simulations plotted Carlo of been Monte absence have by 3) Eq. produced the (Fig. formula, in phase analytical geometric phases averaged an the geometric of the means of by distribution produced been have phase Availability. Data using obtained was 3 Fig. h e 2i back-action. (1988). 1351–1354 teghwa measurements. weak strength 136–165. pp. 1575, vol. 2014), York, New Publishing, (AIP Casta Eds. O. Bijker, R. Proceedings, Conference measurement. weak negative-result a of presence 377–387. pp. 2014), Italy, Milan, Story Success Two-Time A Theory (2013). 170405 110, bit. quantum superconducting a of 2010). UK, Cambridge, Press, Univerity 2014). UK, Cambridge, Press, o-emma.Dpstd3 aur 2020. January 31 Deposited top-geom-meas. https://github.com/KyryloSnizhko/ GitHub. phases.” geometric measurement-induced 2000). Germany, Berlin/Heidelberg, (Springer, (2019). 204 2019). April (29 arXiv:1703.05885 (2014). 570–573 511, oeto h pno pn12pril a unott e100. be to out turn can particle spin-1/2 a of spin the of ponent χ geom i realizations unu esrmn hoyadisApplications its and Theory Measurement Quantum mrec ftegoercpaefo unu measurement quantum from phase geometric the of Emergence al., et apn h pia ot ewe w unu states. quantum two between route optimal the Mapping al., et octhadrvreaqatmjm mid-flight. jump quantum a reverse and catch To al., et a.Phys. Nat. etadwr ln niiultaetre faqatmbit. quantum a of trajectories individual along work and Heat al., et ulvle n unu tt discrimination. state quantum and values Null al., et l ftedt eadn h oteetdgeometric postselected the regarding data the of All = N PNAS 6–7 (2019). 665–670 15, realizations −1 ..tak .Hle o epu icsin.V.G. discussions. helpful for Holder T. thanks V.G. aclso aitosadPrilDfeeta Equations Differential Partial and Variations of Calculus N realizations | hs e.B Rev. Phys. .C tup,J .Tlasn d.(pigrMilan, (Springer Eds. Tollaksen, M. J. Struppa, C. D. , ac 7 2020 17, March unu esrmn n Control and Measurement Quantum P Nature realizations = 4,000. 1–1 (2013). 211–214 502, 624(2019). 064204 100, o,R J R. nos, ˜ z raiain a calculated. was (realization) hs e.B Rev. Phys. ueu,R eu,O Rosas-Ortiz, O. Lemus, R. auregui, ´ | o.117 vol. CmrdeUniversity (Cambridge 251(2007). 220501 75, | hs e.Lett. Rev. Phys. Nature o 11 no. hs e.Lett. Rev. Phys. (Cambridge Quantum 200– 570, | The 7. 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PHYSICS