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 geometrical 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 quantum 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 |↑i (black arrow). For finite mea- ing. Such geometrical dephasing (GD) depends on the surement strength, the state is only pulled towards |↑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 {rk}, different trajecto- Hamiltonian-generated geometrical phase, GD flips its (d) ries (red, cyan, purple) and different phases χ{r } 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 ϕk, sin θ sin ϕk, cos θ), ϕk = 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 |ψi, evolves after a mea- and the total phase, χ{r }, depend on the sequence of k surement that yields a specific readout r into ψ(r) = the readouts of the intermediate measurements, {rk}, cf. Fig.1(b). Different readout sequences result in dif- M(r) |ψi. The probability of each readout is given (r) (r) ferent trajectories, which leads to sequence-to-sequence by pr = hψ |ψ 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)† (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) {rk} P he i{rk} = e . (1) all readouts r pr = 1 independently of the system state |ψi, implying P M(r)†M(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 −iϕk cos 2 sin 2 e only induce a back-action on the system (the flying spin- R(nk) = , (2) sin θk − cos θk e−iϕk 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 {rk} {rk} 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 {rk} are identical for both −1 R (nk)SzR(nk). The back-action of this measurement, arms, hence, no “which path” measurement. The inter- when the measured observable is Sz, is described by op- ference pattern then corresponds to the relative phase erators M (r). 2iχ(d) e {rk} and is averaged over runs with different readout These Kraus operators imply the following character- sequences {rk}. istics of the measurement operation. If the spin points We find that, in general, the dephasing factor does in the direction of nk, the measurement will always yield not have a prescribed symmetry with respect to chang- r = 0 and keep the state unchanged. If the spin points to 3

(d) −nk, the measurement will yield r = 0 or 1 with prob- defines the measurement-induced phase χ and the {rk} abilities exp(−4C/N) and 1 − exp(−4C/N) respectively, (d) probability P of obtaining readout sequence {rk} (in- while the spin will remain pointing to −n independently {rk} k cluding r = 0 for the last projective measurement, bring- of r. For a general spin state |ψi, the probabilities of the ing the system to |ψ i). Considering all possible measure- outcomes are given by p above, and the measurement 0 r ment readout sequences {r }, the averaged phase χ¯(d) back-action will affect the system state. When r = 1, k and the dephasing parameter α(d) are given by the system state will be projected onto −nk, while for r = 0, the state will be pulled towards nk (to the ex- tent determined by C/N; hence, C controls the measurement strength) and rotated around n by angle −2A/N, 2 k (d) (d) X (r ) (r ) −1 2iχ¯ −α N 1 cf. Fig.1(a). The scaling ∼ N of the measurement e = hψ0| MN ...M1 |ψ0i parameters C and A is chosen such that the back-action {rk} N (0) X (d) 2iχ(d) M of N measurements performed along the same = P e {rk} . (5) axis would correspond to a single finite strength mea- {rk} {rk} surement [34]. The limit C → ∞ corresponds to projective measurements. The limiting case of C = 0 corresponds to no measurement, in which case the system state evolution is identical to that under Hamiltonian We note that both χ(d) and χ¯(d) can be measured in −1 {rk} H = 2A (nk · S − 1/2) for time ∆t = N ; here the interference experiments [29, 33]. (1) (0) −iH∆t Kraus operators are Mk = 0, Mk = e . (d) (d) The phase accumulated under a sequence of measure- We compute e2iχ¯ −α using the follow- ments can be calculated as follows. Denote the initial ing trick. Note that hψ | M(rN )...M(r1) |ψ i = θ θ 0 N 1 0 (r ) (r ) system state |ψ0i = cos 2 |↑i + sin 2 |↓i. After per- h↑| δRM N δR...δRM 1 δR |↑i, where δR = forming the sequence of N generalized measurements, −1 R(nk+1)R (nk) is a rotation matrix that for a given readout sequence {rk} = {r1, ..., rN }, the does not depend on k. In order to calcu- (rN ) (r2) (r1) system state becomes MN ...M2 M1 |ψ0i. The late the sum over {rk}, we define a matrix 0 0 last projective measurement makes the system state s1s2 P 0 (r) 0 (r) Ms1s2 = hs | M δR |s1i hs | M δR |s2i. Here (rN ) (r1) r 1 2 |ψ0i hψ0| M ...M |ψ0i. The matrix element 0 N 1 si (“before the measurement”) and si (“after the measurement”) take values ↑ / ↓ with i = 1, 2 being the q (d) (rN ) (r1) (d) iχ{r } hψ0| M ...M |ψ0i = P e k (4) replica index. Explicitly, N 1 {rk}

 2iπd cos θ iπd sin θ iπd sin θ    1 + N − N − N 0 ↑↑ iπd sin θ C+iA iπd sin θ 1  − N 1 − 2 N 0 − N  ↑↓ M =  iπd sin θ C+iA iπd sin θ    + O 2 . (6)  − N 0 1 − 2 N − N  ↓↑ N iπd sin θ iπd sin θ 2iπd cos θ 4iA ↓↓ 0 − N − N 1 − N − N

In the limit of a quasicontinuous sequence of measure- scale with N, it is known that it admits a non-trivial ments (N → ∞), separation into the dynamical and geometrical components [35, 36]. At the same time these dynamical and ge- (d) (d) ↑↑ e2iχ¯ −α = lim MN . (7) ometrical components behave non-trivially with respect N→∞ ↑↑ to directionality reversal, d → −d, which hinders a sim- Characterization and classification of dephasing.— ple classification based on symmetry properties. Here Note that M = I + Λ/N + O(N −2), where I stands for we do not delve deeper into this classification issue [34] the identity matrix and Λ is a constant matrix. Hence, but rather focus on the behavior of dephasing and the N limN→∞ M = exp(Λ) does not depend on N. It measurement-induced phase. would thus be tempting to denote α(d) and χ¯(d) geomet- In order to understand the relation between α(d) and rical since they do not depend on the protocol duration α(−d), we note the following symmetries of M. Replac- (number of measurements). Such an identification would ing d → −d, A → −A, together with a complex conju- ∗ be erroneous, as can be easily seen from the following gation, leaves M invariant: Md→−d,A→−A = M . Us- argument: For C = 0, our measurement-induced evo- ing Eq. (7), this implies α(d)(C, A, θ) = α(−d)(C, −A, θ) lution is equivalent to non-adiabatic Hamiltonian evolu- and χ¯(d)(C, A, θ) = −χ¯(−d)(C, −A, θ). Consequently, the tion. While the accumulated phase in that case does not dephasing is only guaranteed to be symmetric (α(d) = 4

A is shown in Fig.2(b). Remarkably, the phase makes a winding around each of the dephasing singularity points, cf. Fig.2(a). Exactly at the singularity points, the phase (+1) (+1) is undefined since e2iχ¯ −α = 0. The windings of the phase are of size π, and not 2π. However, since the (+1) measurable quantity is e2iχ¯ , there is no physical dis- continuity as χ¯(+1) → χ¯(+1) +π. The windings cannot be eliminated by a continuous deformation of the phase, and thus constitute topological features. Such phase windings Figure 2. Dephasing α(+1) (a) and phase χ¯(+1) (b), cf. Eq. (1), at θ = 3π/4 color-coded as functions of the measurement accompany all the divergences we found. strength (C) and asymmetry (A) parameters. Note the two Another way of viewing the divergences as topolog- singularities at C ≈ 2, where α(+1) diverges. The phase makes ical features arises when considering the set of all di- π-windings around the points of divergent α(+1). vergences. The divergences of α(+1) form a critical line (Ccrit,Acrit) in the (C,A) plane, shown in Figure3. For each (Ccrit,Acrit), there is a value of θcrit ∈ [0; π] at (−d) α ) when A = 0. Away from A = 0 there may be an which α(+1) diverges. The critical line separates the additional antisymmetric component. We therefore de- plane into three regions. The θ-dependence of the phase note A as the asymmetry parameter. Using the above χ¯(+1)(C, A, θ) is topologically different in each of these re- symmetry relations, we write down the symmetric and gions. To see this, consider the dependence on the polar antisymmetric dephasing components angle, θ, of χ¯(+1)(θ) for a given value of measurement pa- (+1) 1 rameters (C,A). For each given θ, χ¯ is defined mod- αs = α(+1)(C, A, θ) + α(+1)(C, −A, θ) , (8) ulo π. However, taking the whole dependence on θ into 2 1 account, we unfold the phase to form a continuous func- αa = α(+1)(C, A, θ) − α(+1)(C, −A, θ) . (9) tion χ¯(+1)(θ) which is not confined to the interval [0; π). 2 (+1) (+1) Furthermore, note that e2iχ¯ (θ=0) = e2iχ¯ (θ=π) = 1. Next, defining a diagonal matrix U = This implies diag(1, −1, −1, 1), one shows that Md→−d,θ→π−θ = (d) (−d) π (+1) UMU, which implies α (C, A, θ) = α (C, A, π − θ) (+1) (+1) (+1) dχ¯ (θ) (d) (−d) χ¯ (π) =χ ¯ (π) − χ¯ (0) = dθ = πn,¯ and χ¯ (C, A, θ) =χ ¯ (C, A, π − θ). This symmetry ˆ0 dθ can be understood from a simple consideration: a (10) clockwise (d = −1) protocol in southern hemisphere where n¯ ∈ Z and we have used the freedom to becomes a counterclockwise (d = 1) protocol in the fix χ¯(+1)(0) = 0. No transition between differ- northern hemisphere upon exchanging the roles of the ent values of integer n¯ can happen when χ¯(+1)(θ) is south and the north poles. smoothly deformed, making n¯ a topological index. How- We next calculate numerically and analyze the behav- ever, n¯(C,A) can jump when χ¯(+1)(θ) is not a well- ior of α(+1) (the behavior of α(−1) can be inferred by defined smooth function. This happens at the diver- (d) swapping θ → π − θ). Figure2(a) shows the dependence gence points when α (Ccrit,Acrit, θcrit) = +∞, where (+1) (+1) of α on the measurement parameters C and A at χ¯ (Ccrit,Acrit, θcrit) is undefined. Therefore, the θ = 3π/4. Note that α(+1)(C, A, θ) 6= α(+1)(C, −A, θ) = (C,A) plane can be divided into regions, each with a dis- α(−1)(C, A, θ), revealing that the antisymmetric compo- tinct value of n¯. In the present example, n¯ = 0 in region nent αa is indeed generically present. Note also the I, n¯ = −1 inside II, and n¯ = −2 inside III, as illustrated (+1) (−1) two divergences, α → ∞, at (Ccrit ≈ 2,Acrit > 0). in Fig.3(inset). The behavior of χ¯ (θ) is recovered via There are no corresponding divergences at A < 0, im- relation χ¯(d)(C, A, θ) =χ ¯(−d)(C, A, π−θ)(mod π), imply- (−1) plying that α (Ccrit,Acrit, θ) is non-singular. This im- ing that for d = −1 similar topological transitions happen a s plies that both α (Ccrit,Acrit, θ) and α (Ccrit,Acrit, θ) di- at the same (Ccrit,Acrit) but at different θcrit. verge. Moreover, the strength of divergence is identical Summary.—We have presented here a protocol com- as α(−1) = αs − αa is finite. Contrast this to the case prising a set of generalized measurements, which steers of dephasing in Hamiltonian-induced dynamics, where a spin-1/2 system along a closed trajectory on the Bloch α(d) = βET + γd with ET 1 being the adiabaticity sphere. Fluctuations in the readout sequences are re- parameter (E is the energy gap and T is the protocol sponsible for dephasing, which is not symmetric under execution time) [22, 23, 27]. There the symmetric com- changing of path directionality, d → −d. Rather it component, αs = βET , associated with dynamical dephasing, prises two components: symmetric and antisymmetric. always dominates over the antisymmetric geometrical de- Such measurement-induced dynamics bears similitude to phasing αa = γd so that αs/αa ∼ ET 1. adiabatic Hamiltonian dynamics of open quantum sys- Divergences as topological features.—Consider the tems, where symmetric (dynamical) and antisymmetric phase, χ¯(+1)(θ = 3π/4), whose dependence on C and (geometrical) dephasing components have been predicted 5

exhibit dephasing for an arbitrary number of measurements, N < ∞, and for arbitrary Kraus operators, M(r). The dephasing will, in general, be asymmetric w.r.t. reversal of the protocol directionality, and may diverge under certain conditions. For N < ∞, the dephasing and the induced phase will depend on N. We thank V. Gebhart for useful discussions. We ac- knowledge funding by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) – Projekt- nummer 277101999 – TRR 183 (project C01) and Pro- jektnummer EG 96/13-1, and by the Israel Science Foun- dation (ISF). Figure 3. Topological transition in the measurement- induced phase χ¯(+1). Main panel—The critical line of points (Ccrit,Acrit) for which there exists θcrit such that the dephas- (+1) ing α (Ccrit,Acrit, θcrit) diverges. The values of θcrit ∈ [0; π] are shown with color. The averaged phase χ¯(+1) exhibits three [1] Dieter Suter and Gonzalo A. 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