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 Condensed Matter Physics, 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 j k 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 quantum Hall effect. 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 ! 1 and kj k+1i − j k ik = O(1=N )), the induced evolution is fully deterministic quantum measurement j quantum trajectories j quantum feedback j due to the dynamical quantum Zeno effect (17), thus yielding Berry phase j topological phases of matter a unique Pancharatnam–Berry phase. Recently, geometric phases induced by weak measurements he overall phase of a system’s quantum state 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 j l i, j m 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 [h l j m i]. For a sequence of states (14, 15) j k i, k = based control and manipulations of quantum states (a promis- 0, ::: , N , for which j N i / j 0i, 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 [h 0jPN ::: P2P1j 0i] = argh 0 j ~N i, [1] The authors declare no competing interest.y This article is a PNAS Direct Submission.y ~ where j k i = Pk ::: P2P1j 0i and Pk = j k i h k j is the projec- Published under the PNAS license.y ~ tor onto the kth state. Note that j k i / j k 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 j ~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 j k i. For a quasicontin- 2 To whom correspondence may be addressed. Email: [email protected] uous sequence of states fj k ig, 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 fj k ig, First published March 2, 2020. 5706–5713 j PNAS j March 17, 2020 j vol. 117 j no. 11 www.pnas.org/cgi/doi/10.1073/pnas.1911620117 Downloaded by guest on September 29, 2021 possible readout sequence contributed to the observed phase. σ = (σx , σy , σz ) is the vector of Pauli matrices and nk = When considering all possible readout sequences, the system (sin θk cos 'k , sin θk sin 'k , cos θk ). The sequence of measure- dynamics remain stochastic even for quasicontinuous sequences ment orientations fnk g defines a trajectory on the unit sphere of weak measurements (21, 22). In a wider context, employing S 2 as depicted in Fig. 1, Left. Each weak measurement is char- weak measurements enables continuous monitoring of the sys- acterized by its strength η 2 [0, 1] and two possible readouts tem through weak measurements, which has been successfully rk = +, −. The modification of the system state conditional to the employed experimentally for dynamically controlling quantum (rk ) obtained measurement readout rk is given by j i ! Mk j i, states (23–26). (rk ) where M = Mη (n , r ) are Kraus operators (21, 22) (Materi- In the present study, we define and investigate the geometric k k k k als and Methods). Given the measurement strength η, a sequence phase accrued by a quantum state of a two-level system following fn g of measurement orientations with corresponding readouts a sequence of measurements and detector readouts with tunable k n ~ o measurement strength. We compute the full distribution func- frk g induces a sequence of states j k i in the system Hilbert tion of the induced geometric phases and analyze the phase of space, where specific trajectories which can be singled out by postselecting specific readout sequences. As opposed to the case of projective j ~ i = j ~ i = M(rk ) ::: M(r1) j i : [2] measurements (17, 18), where the state trajectory and the result- k r1,:::,rk k 1 0 ing phase are solely determined by the measurement sequence The weak measurement-induced geometric phase can be de- and the measurement readouts, the trajectories (and the phases) fined as here depend crucially on the measurement strength. We mainly focus on the scenarios of a single postselected state trajectory (r ) (r ) χ = arg h j ~ i = arg h jM N ::: M 1 j i: [3] and consider the effect of averaging over all possible trajectories geom 0 N 0 N 1 0 in SI Appendix. We discover that a topological transition vis-`a-vis the geomet- To further proceed with our analysis we need to specify the ric phase takes place as a function of the measurement strength. nature of the measurement. For the present study we opt for so- The transition is topological in the sense that it is related to a called null-type weak measurements (27–29). The state before j i = ajni + b|−ni σ |±ni = ±|±ni discontinuous jump of an integer-valued topological invariant. the measurement is , where n , j+i Specifically, we consider a family of measurement sequences: while the detector initial state is . The measurement process is The state trajectories induced by these sequences form a surface mediated by the system–detector interaction which is described which covers a certain area on the Bloch sphere, and our topolog- by the interaction-induced mapping PHYSICS ical transition is manifest through a jump of the Chern number associated with this surface.