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Weak-Measurement-Induced Asymmetric Dephasing: a Topological Transition

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Weak-Measurement-Induced Asymmetric Dephasing: a Topological Transition Weak-measurement-induced asymmetric dephasing: a topological transition Kyrylo Snizhko,1 Parveen Kumar,1 Nihal Rao,1, 2, 3 and Yuval Gefen1 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100 Israel 2Present affiliation: Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstr. 37, 80333 München, Germany 3Present affiliation: Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany Geometrical dephasing is distinct from dynamical dephasing in that it depends on the trajectory traversed, hence it reverses its sign upon flipping the direction in which the path is traced. Here we study sequences of generalized (weak) measurements that steer a system in a closed trajectory. The readout process is marked by fluctuations, giving rise to dephasing. The latter comprises a contribution which is invariant under reversal of the sequence ordering, and, in analogy with geo- metrical dephasing, one which flips its sign upon reversal of the winding direction, which may result in “coherency enhancement”. This asymmetric dephasing diverges at certain protocol parameters, marking topological transitions in the measurement-induced phase factor. Dephasing is a ubiquitous feature of open quantum systems [1,2]. Undermining coherency, it facilitates the crossover to classical behavior, and comprises a funda- mental facet of the dynamics of mesoscopic systems [3– 10]. Dephasing has to be taken into account when design- ing mesoscopic devices [11, 12], specifically those directed at quantum information processing [1, 13]. A particularly intriguing type of dephasing appears when geometrical phases [14, 15] emerge in open quan- tum systems [16–23]. On top of conventional dynamical Figure 1. Measurement-induced trajectories. (a)—The back- dephasing which arises due to the fluctuations of the sys- action of a generalized measurement. Under a projective mea- tem’s energy and is proportional to the evolution time, surement yielding readout r = 0, the system’s initial state Refs. [22–24] found a geometrical contribution to dephas- (red arrow) would become j"i (black arrow). For finite mea- ing. Such geometrical dephasing (GD) depends on the surement strength, the state is only pulled towards j"i and trajectory traversed, but not on the traversal time. Two also rotated around the z axis (green arrow). These two ef- facets underline GD that emerges due to the interplay fects of the back-action (illustrated by blue dashed lines) are respectively quantified by parameters C and A in Eq. (3). of Hamiltonian dynamics and dissipative environment. (b)—Schematic illustration of trajectories induced by a se- First, it can be expressed through an integral of the un- quence of measurements along some parallel (black line). De- derlying Berry curvature [25, 26]. Second, similarly to pending on the sequence of readouts frkg, different trajecto- Hamiltonian-generated geometrical phase, GD flips its (d) ries (red, cyan, purple) and different phases χfr g are induced. sign upon the reversal of the evolution protocol (the di- k rectionality in which the closed path is traversed). The existence of geometrical dephasing has been confirmed of geometrical phases does give rise to dephasing. (ii) experimentally [27]. Recent theoretical studies [25, 26] In similitude to Hamitonian dynamics of dissipative sys- have generalized GD to the case of non-Abelian phases. tems, leading to dynamical and geometrical components, On a seemingly unrelated front, measurement-induced here both the phase and the dephasing generated by mea- arXiv:2006.13244v1 [quant-ph] 23 Jun 2020 geometrical phases have recently become an object of surement protocols comprise a symmetric and an anti- both experimental [28] and theoretical [29] interest. symmetric (w.r.t. changing directionality) components. Notably, measurement in quantum mechanics involves (iii) The emergent dephasing in such measurement-based stochasticity. It is thus natural to ask whether dephasing steering protocols may diverge. These divergences are emerges in measurement-based protocols and to investi- associated with topological transitions underlying the gate its relation to Hamiltonian-induced dynamical and steering protocols. geometrical dephasing. Overview.—As a concrete (but generalizable) example The challenge of the present paper is two-fold. We we consider a “system” made up of spin 1=2, represented first ask whether weak-measurement-induced geometrical by the operator S. Our detector has two possible read- phases go hand-in-hand with emergent dephasing. Sec- outs, r = 0 and r = 1; the back-action of a quantum ondly, provided that dephasing is part of such protocols, measurement on the system state is encoded in the re- does this dephasing have a term similar to GD? Our main spective Kraus operators M(r) operating on the system findings are: (i) Indeed, measurement-induced generation [30–32]. In the case of a strong (projective) measurement, 2 the back-action projects the system’s state onto the ±1=2 ing the directionality of the trajectory, d ! −d, i.e., eigenstates of the measured spin projection n · S. For α(d) 6= ±α(−d). Rather, it comprises a symmetric and non-projective (generalized) measurements, the effect of an antisymmetric term, α(d) = αs + αad. The total de- back-action is more subtle, cf. Fig.1(a). phasing must represent a suppression factor of any co- Below, we study the following protocol that leads herent contribution, hence αs ≥ αa. Remarkably, we to the generation of a trajectory-related phase and the find that for certain protocol parameters both the sym- emergence of dephasing. We prepare the system in the metric and the antisymmetric terms diverge. We study +1=2 eigenstate of n0 · S, where n0 = (sin θ; 0; cos θ). this divergence and link it to a topological transition in We then consider a sequence of N generalized measure- the behavior of the averaged phase χ¯(d). ments corresponding to measurement directions nk = For the case of adiabatic Hamitonian dynamics, the (sin θ cos 'k; sin θ sin 'k; cos θ), 'k = 2πkd=(N +1). Here symmetric component of dephasing arises due to fluc- d = ±1 defines the directionality of the trajectory. Fol- tuations of the dynamical phase and is called dynami- lowing the sequence of generalized measurements, we per- cal dephasing; the antisymmetric component is the GD, form a projective measurement corresponding to nN+1 = which arises from cross-correlations of the fluctuations n0 and postselect it on r = 0 readout (corresponding to of the dynamical and the geometrical phase components n0 ·S = +1=2), which guarantees that the final and initial [25, 26]. This association of symmetric/antisymmetric states coincide. with dynamical/geometrical does not apply here [34]. Be- The back-action due to the measurements dictates low we characterize and analyze the symmetric and an- a closed trajectory in the Hilbert space, along which tisymmetric parts of the dephasing factor. the system’s state accumulates a geometrical phase and Derivation of the dephasing factor.—Kraus operators some dynamical phase on top of that. The trajectory [30–32], M(r), account for the measurement back-action. (d) The system, initially in state j i, evolves after a mea- and the total phase, χfr g, depend on the sequence of k surement that yields a specific readout r into (r) = the readouts of the intermediate measurements, frkg, cf. Fig.1(b). Different readout sequences result in dif- M(r) j i. The probability of each readout is given (r) (r) ferent trajectories, which leads to sequence-to-sequence by pr = h j i. The case of standard projective fluctuations. To obtain the dephasing, one averages over measurement corresponds to the Kraus operators being the readout sequences. This amounts to not registering projectors onto the measurement eigenspaces: M(r) = (r) (r)y (r) (r0) (r) the detectors’ readouts (“blind measurements”), i.e., trac- P = P , P P = δr;r0 P . In general, the ing out over the detectors’ states. One may then define Kraus operators are arbitrary matrices acting in the sys- the averaged phase, χ¯(d), and the dephasing parameter, tem Hilbert space and the readouts r do not necessarily α(d), through correspond to the system being projected to a particular (d) state. The only restriction is that the total probability of 2iχ 2iχ¯(d)−α(d) frkg P he ifrkg = e : (1) all readouts r pr = 1 independently of the system state j i, implying P M(r)yM(r) = , where represents the The factor of 2 in the exponent of Eq. (1) is related r I I identity operator. to the phase observation protocol [33]: A phase could be We focus on a particular type of generalized measure- measured by an interference experiment, where a “flying ments. Namely, for the kth measurement in the protocol spin 1/2”, represented by an impinging electron, is split above (1 ≤ k ≤ N) yielding readout r , the respective between two arms and subjected to measurements in one k (rk) −1 (rk) of them. Such a protocol, however, presents the follow- Kraus operator is Mk = R (nk)M R(nk) with ing problem: the detector changing its state would not θk θk −i'k cos 2 sin 2 e only induce a back-action on the system (the flying spin- R(nk) = ; (2) sin θk − cos θk e−i'k 1/2), but would also constitute a “which-path” measure- 2 2 ment, undermining the interference. Instead, we resort to a measurement setup where each detector is coupled ! (0) 1 0 (1) 0 0 to respective points on both arms. The detector–system M = −2 C+iA ;M = p − 4C : 0 e N 0 1 − e N couplings are engineered such that the phases accumu- (3) lated in the respective arms are χ(d) and −χ(d) . With frkg frkg In essence, these describe the same measurement proce- these designed couplings, the probabilities of obtaining dure applied to measure different observables nk · S = a specific readout sequence frkg are identical for both −1 R (nk)SzR(nk).
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