Observing Topological Invariants Using Quantum Walks in Superconducting Circuits

Observing Topological Invariants Using Quantum Walks in Superconducting Circuits PHYSICAL REVIEW X 7, 031023 (2017) 1,2,* 1,2 1,2 1,2 1 1,2 E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi Department of Physics, University of California, Berkeley, California 94720, USA Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA (Received 6 December 2016; revised manuscript received 12 May 2017; published 3 August 2017) The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here, we motivate a toolbox based on time- dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schrödinger cat state and highlights a novel refocusing technique, which allows for the direct implementation of a digital version of Bloch oscillations. As the walk is performed in phase space, our scheme can be extended to higher synthetic dimensions by adding additional microwave cavities, whereby superconducting circuit-based simulations can probe topological phases ranging from the quantum-spin Hall effect to the Hopf insulator. DOI: 10.1103/PhysRevX.7.031023 Subject Areas: Quantum Physics, Quantum Information, Topological Insulators Topological phases elude the Landau-Ginzburg para- cavity’s phase space, as shown in Fig. 1(b). Its spin degrees digm of symmetry breaking [1]. Unlike conventional of freedom are formed by a superconducting transmon phases, they do not exhibit order parameters that can be qubit [14] with basis states fj↑i; j↓ig. To enable the qubit locally measured. Rather, their distinguishing features are state to control the direction of motion of the coherent state, hidden in quantized, nonlocal topological invariants, which we realize a strong dispersive coupling between the cavity are robust to all local perturbations [2,3]. While tremendous and qubit, theoretical progress has been made toward the full classi- fication of topological phases of matter [4,5], a general † † H=ℏ ¼ ω σ ˆ =2 þ ω a ˆ a ˆ − χ a ˆ a ˆσ ˆ =2; ð1Þ q z c qc z experimental platform for the direct measurement of topological invariants is lacking. Here, we demonstrate where ω are the qubit and cavity transition frequencies, q;c that time-dependent quantum walks comprise a powerful respectively, a ˆ (a ˆ ) is the lowering (raising) operator for the class of unitary protocols capable of digitally simulating cavity mode, σ is the Pauli z matrix for the qubit levels, single-particle topological band structures and directly and χ is the dispersive interaction strength [see Fig. 1(c)]. qc observing the associated nonlocal invariants. Dispersive coupling produces a qubit-dependent shift in the A quantum walk [6–11] describes the motion of a cavity oscillation frequency. Viewed in the rotating frame particle with internal (spin) degrees of freedom moving of the cavity at ω , the dispersive interaction causes the on a discrete lattice. Formally, the quantum walk is c coherent state to move clockwise (counterclockwise) at a comprised of two unitary operations [see Fig. 1(a)]: a coin ˆ rate χ =2 through phase space when the qubit is in the j↑i toss, denoted RðθÞ, which rotates the spin state, and a spin- qc (j↓i) state. Thus, free evolution under the dispersive dependent translation, denoted T , which translates the ↑↓ interaction precisely enables the spin-dependent translation particle’s position by a single lattice site in a direction needed for the quantum walk [15,16]. determined by the internal spin state. In our cavity quantum We realize a particular class of quantum algorithm known electrodynamics implementation of the quantum walk, the as the split-step quantum walk [17,18], which alternates particle is encoded as a coherent state of an electromagnetic two coin tosses (with rotation angles θ and θ ) between 1 2 cavity mode [12,13], where its position is defined in the two spin-dependent translations so that each step of the walk consists of the unitary operation U ðθ ; θ Þ¼ * W 1 2 emmanuel.flurin@berkeley.edu ˆ ˆ ˆ ˆ T Rðθ ÞT Rðθ Þ [see Fig. 1(a)]. The coin-toss operations ↑↓ 2 ↑↓ 1 iθσ ˆ =2 ˆ x Published by the American Physical Society under the terms of R ðθÞ¼ e are applied via short (7.5-ns) coherent the Creative Commons Attribution 4.0 International license. microwave pulses resonant with the qubit transition. By Further distribution of this work must maintain attribution to −1 waiting for a time interval t ¼ 2πð10χ Þ ¼ 124 ns qc the author(s) and the published article’s title, journal citation, and DOI. between successive coin tosses, we allow the dispersive 2160-3308=17=7(3)=031023(6) 031023-1 Published by the American Physical Society E. FLURIN et al. PHYS. REV. X 7, 031023 (2017) (a) We observe the expected ballistic expansion of the coherent (b) state in cavity phase space, consistent with theoretical predictions (population fidelities greater than 90%). As the walk unitary U directly couples the particle’s spin and position degrees of freedom, the resulting dynamics mimic those of spin-orbit interacting materials. More precisely, the unitary quantum-walk protocol simu- lates continuous evolution under an effective spin-orbit Hamiltonian H , which generates the same transformation −iH ˆ W as a single step of the walk when U ¼ e . Since the unitary is translation invariant, the effective Hamiltonian (c) exhibits Bloch bands of quasienergy ϵðkÞ, where the quasimomentum k lies in the Brillouin zone; Figs. 3(a) and 3(b) show, respectively, the band structures underlying ˆ ˆ the walks U and U . The corresponding eigenstates 0 1 consist of extended Bloch waves with spin polarization n⃗ ðkÞ [17]. Depending on symmetry, the band structure of such spin-orbit-coupled Hamiltonians can feature quan- FIG. 1. Quantum-walk implementation in cavity phase space. tized topological invariants. In the case of the split-step (a) Schematic representation of a split-step quantum walk on a −iπA·σ⃗ =2 quantum walk, Γ ¼ e plays the role of a so-called ˆ ˆ line, with rotations Rðθ Þ and Rðθ Þ and spin-dependent trans- 1 2 ˆ ˆ chiral symmetry [17,20], with Γ U Γ ¼ −U . This sym- lation T . Red (blue) lines show spin-up (-down) components W W ↑↓ metry constrains the spin polarization vector n⃗ ðkÞ to lie on a moving left (right). The opacity of each circle indicates the population on the corresponding lattice site. (b) Set of ten cavity ⃗ great circle of the Bloch sphere, perpendicular to A ¼ coherent states on which the walk takes place, in the phase space ( cosðθ =2Þ; 0; sinðθ =2Þ) [Figs. 3(c) and 3(d)]. Thus, the 1 1 of the TE cavity mode. (c) Cavity resonator and qubit. The number of times n⃗ ðkÞ wraps around the origin as k varies fundamental (TE , orange) mode at ω ¼ 2π × 6.77 GHz is 110 R through the Brillouin zone—known as the winding or used to measure the qubit state. This mode couples strongly Chern numberW—naturally defines the topological invari- [κ ¼ 2π × 600 kHz ¼ 1=ð260 nsÞ] to a 50-ohm transmission line ˆ ˆ ant [17] of the walk. While the energy spectra of U and U via the readout port at the center of the cavity. The TE cavity 0 1 mode (green) at ω ¼ 2π × 7.41 GHz is long lived with an are identical, they lie in topologically distinct phases, with inverse lifetime, κ ¼ 2π × 4 kHz ¼ 1=ð40 μsÞ. The transmon ˆ ˆ U having zero winding number and U a winding number 0 1 qubit (coin) has transition frequency ω ¼ 2π × 5.2 GHz, relax- of unity. Analogous to the number of twists in a closed ation times T ¼ 40 μs and T ¼ 5.19 μs, and is dispersively 1 2 ribbon, winding numbers are quantized and robust to local coupled to both cavity modes, with the dispersive shift of the perturbations [2]. walker mode, χ ¼ 2π × 1.61 MHz. qc The direct measurement of topological invariants in solid-state materials is an outstanding challenge [21–23], coupling to naturally implement the spin-dependent trans- owing to the nonlocal nature of the order parameter. Our lation. This time interval determines the lattice on which the method makes use of a time-dependent modification of the walk takes place; here, it is a circular lattice of ten sites in quantum walk, which, in the Hamiltonian picture, mimics cavity phase space [Fig. 1(b)]. an adiabatic translation of the underlying band structure We begin by performing a pair of topologically distinct across the Brillouin zone [24–28]. The resulting dynamics split-step quantum walks, the first (topologically trivial) effectively constitute digital Bloch oscillations, a phenome- ˆ ˆ with unitary U ¼ U ð3π=4; π=4Þ, and the second (topo- 0 W non whereby a particle on a lattice subjected to a constant ˆ ˆ logically nontrivial) with U ¼ U ðπ=4; 3π=4Þ. To dem- 1 W force undergoes oscillations [29] due to the periodicity of onstrate the robustness of the winding number, we also the Brillouin zone. In our system, these oscillations implement an additional pair of walks which are contin- manifest as a refocusing of the quantum walker to its ˆ ˆ uously connected to U and U (e.g., without closing the initial position, with a Berry phase—a signature of the 0 1 gap). The experimental sequence is shown in Fig. 2(a). band-structure topology (see Fig. 3)—imprinted during the The cavity mode is initialized (D ) in a coherent state jβi evolution. In practice, this refocusing depends on choosing with jβj ¼ 8 photons, after which the walk unitary is the number of steps in the walk such that the accrued repeatedly applied. To directly reconstruct the walker’s dynamical phase—which has opposite signs in either band quantum state on the phase-space lattice, we first projec- and thus impedes refocusing—effectively vanishes [19,24]. tively measure the qubit state and subsequently measure the In the general setting, one can experimentally determine the Q function of the cavity mode [19]. Figure 2(b) depicts the condition for dynamical phase refocusing by performing measured lattice site populations after each step of the walk. spectroscopy while varying the number of steps. 031023-2 OBSERVING TOPOLOGICAL INVARIANTS USING … PHYS. REV. X 7, 031023 (2017) (a) (b) (c) FIG. 2. Quantum-walk protocol and resulting populations. (a) Protocol used to perform the quantum walk, showing cavity state preparation (blue), quantum walk (green), qubit state measurement (blue), and Q function measurement (pink). The dashed boxes with σ gates are performed to implement the Bloch oscillation. (b) Cavity Q functions after each step of the quantum walk without Bloch ˆ ˆ oscillations, U (top strip) and U (bottom strip). Spin-up (red) and spin-down (blue) Q functions are superimposed. The average 0 1 ˆ ˆ fidelities of the populations compared to theoretical predictions are 0.97 and 0.96 for U and U , respectively. (c) Cavity Q functions 0 1 after each step of the refocusing quantum walk with Bloch oscillations. The state refocuses after ten steps, as shown in the final frame for ˆ ˆ ˆ ˆ both U and U . Refocusing fidelities (to the initial state) for U and U are 0.83 and 0.87, respectively. 0 1 0 1 In addition to the dynamical phase, upon traversing The additional steps used in performing the time- the Brillouin zone, the particle’s spin winds around the dependent walks are shown in dashed boxes in Fig. 2(a). ˆ ˆ Bloch sphere, encoding its path in the accumulated Berry Beginning with either U or U , we insert rotations by Δk 0 1 ˆ ˆ phase [30], about σ ˆ after each coin toss rotation Rðθ Þ and Rðθ Þ.In z 1 2 ˆ ˆ contrast to the original operations comprising U and U , 0 1 the rotation angle Δk varies in time. Since a σ ˆ rotation is ϕ ¼ i hk; n⃗ ðkÞj∂ jk; n⃗ ðkÞidk ¼ π × W; ð2Þ B k equivalent to a translation of the underlying Hamiltonian in BZ quasimomentum space [24], this time-varying rotation which thus becomes an observable manifestation of the angle implements a digital Bloch oscillation. We choose winding number W—the Hamiltonian’s topological invari- Δk to vary in steps of π=10 from 0 to π, traversing the ant. As one cannot directly observe the quantum mechani- Brillouin zone exactly once. cal phase of a wave function, measuring this Berry phase Populations resulting from the time-dependent walks requires an interferometric approach. To this end, we (with the system initialized in a single coherent state) are shown in Fig. 2(c). Unlike the ballistic dynamics resulting perform the time-dependent walk with the cavity-qubit from the original walks, the Bloch oscillation (traversal of system initialized in a Schrödinger cat superposition of two the Brillouin zone) causes the walker wave function to coherent-state components: One component undergoes the walk, while the other is unaffected by the unitaries. The refocus [25–28] to both its initial position and spin state. Berry phase thus appears as the relative phase between The intuition underlying this refocusing is that both the the two components and is observable via direct Wigner dynamical and Berry phases accumulated by each quasi- momentum component of the walker is identical upon full tomography. 031023-3 E. FLURIN et al. PHYS. REV. X 7, 031023 (2017) (a) (b) (a) (e) (b) (c) (d) (c) (d) FIG. 3. Topological classes of split-step quantum walks. Calculated band structures, quasienergy ϵ versus quasimomen- tum k, corresponding to the two walks we perform in the ˆ ˆ ˆ ˆ ˆ ˆ experiment, U ¼ T Rðπ=4ÞT Rð3π=4Þ (a) and U ¼ 0 ↑↓ ↑↓ 1 ˆ ˆ ˆ ˆ T Rð3π=4ÞT Rðπ=4Þ (b). Though the energy bands of the ↑↓ ↑↓ two walks are identical, they are topologically distinct, with FIG. 4. Winding number measurement via direct Wigner the topology given by the winding of n⃗ ðkÞ as k varies through the tomography of refocused Schrödinger cat states. (a) Protocol Brillouin zone, shown in diagrams (c) and (d). In diagram (c), the for measuring topology via a time-dependent walk (Bloch trivial case U , n⃗ ðkÞ does not complete a full revolution around 0 oscillations). The Schrödinger cat state is first prepared (blue), the Bloch sphere, while in the topological case U diagram (d), it after which the ten-step refocusing quantum walk is performed does perform a full revolution. This also provides a direct (green). The qubit and cavity state are then disentangled, the qubit connection to the Berry phase, as for a spin-1=2 system the state is purified (blue), and direct Wigner tomography on the Berry phase is simply half the subtended solid angle of the Bloch cavity state is performed (pink). Wigner tomography of (b) the cat sphere path. A schematic representation of the variation of n⃗ ðkÞ is undergoing no quantum walk, (c) the cat after undergoing the ˆ ˆ shown by the ribbons below the Bloch spheres. The arrows on trivial U walk, and (d) the cat after undergoing the topological U 0 1 these strips point in the direction of n⃗ ðkÞ. Analogous to the walk. Fidelities of these resulting cat states compared to pure cat number of twists in closed ribbons, winding numbers are states are 0.68, 0.69, and 0.67, respectively. (e) A cut of the quantized and robust to local perturbations. Wigner function, showing the fringes that encode the relative phase between the two cat components for no walk (black), a trivial walk (red), and a topological walk (blue). The relative phase traversal [19,24]. In practice, we observe refocusing fidel- corresponds to the phase of the measured interference fringes pffiffiffi ities greater than 80%, limited by incomplete adiabaticity following the relation A exp½−2jImðαÞj  cos½2 n¯ImðαÞþ ϕ, and experimental imperfections. where A, ϕ are the amplitude and phase of the fringes. The Having verified the refocusing behavior of the time- Berry phase—captured by the phase difference between the dependent quantum walks, we initialize the cavity-qubit topological and the trivial walks—is ϕ ¼ 1.05π  0.06π in experiment, consistent with the theoretical expectations of π. system in a Schrödinger cat state to measure the accumu- lated Berry phase [19]. One component of the cat is walking component of the cat refocuses, we disentangle the precisely the initial state of the previous walks, jβ;↑i. qubit from the cavity with the operation j0;fi → j0;↑i. The other component is j0;fi, where the cavity is in its ground (vacuum) state and the transmon is in its second This leaves the oscillator in the state excited state [14], jfi. Shelving the vacuum component of iϕ the cat in the jfi state renders it immune to the coin-toss jψi¼j0i − e jβi; ð3Þ rotations, as the jfi ↔ j↓i transition is far detuned (225 MHz) from the j↑i ↔ j↓i transition. Thus, this where ϕ is the Berry phase. component of the cat lies dormant during the walk, acting While Q tomography lends itself well to measuring as a phase reference for the observation of the Berry phase. coherent-state occupations, coherences between these states Our method of preparing the cat, a modification of the are largely invisible in this representation. To measure the protocol introduced in Ref. [31], is shown in Fig. 4(a). With Berry phase, we therefore apply direct Wigner tomography the cat initialized, we perform the time-dependent walk to the final cavity state [19,31,32]. As Figs. 4(b)–4(d) show, over a full Bloch oscillation, applying the same set of the Wigner functions of two-component cat states display pulses that resulted in the final frames of Fig. 2(c). After the interference fringes, whose phase directly encodes the 031023-4 OBSERVING TOPOLOGICAL INVARIANTS USING … PHYS. REV. X 7, 031023 (2017) A single step of these walks consists of multiple spin- relative phase between the dormant (j0;fi) and walking (jβ;↑i) components of the cat. Figures 4(c) and 4(d) display dependent translation steps in different directions, neces- the measured Wigner functions for both split-step walks. sitating the modification in time of the dispersive shifts. In the topologically trivial phase [Fig. 4(c)], the interference Using the current Bloch-oscillating protocol, the 2D fringes do not acquire any phase shift after the walk, besides a Brillouin zone can be swept out in stripes, whereby a small offset due to technical imperfections. For the topo- measurement of the Berry phase acquired along each stripe logically nontrivial walk [Fig. 4(d)], however, the fringes allows the extraction of the Chern number [35]. visibly shift [Fig. 4(e)], corresponding to an acquired phase The generalization of quantum-walk-based protocols to measurements of many-body topological invariants repre- of ϕ ¼ 1.05π  0.06π. The topologies of the Hamiltonians that generate the walks are thereby clearly imprinted on the sents an exciting frontier at the interface of topology, Wigner functions of the refocused states. A key feature of interactions, and quantum simulation [36,37]. such topology is its robustness to all perturbations that do not The authors acknowledge discussions with David Toyli, close the spectral gap. To this end, we have performed an Chris Macklin, Kevin Fischer, Mark Rudner, Eugene ˆ ˆ additional pair of quantum walks, U ¼ U ð0.64π; 0.28πÞ 0 W Demler, and Carlos Navarette-Benloch for motivating the ˆ ˆ and U ¼ U ð0.28π; 0.64πÞ, which are continuously 1 W use of a refocusing quantum walk. V. V. R. and L. S. M. deformable from the original walks. In this case, line cuts acknowledge funding via the National Science Foundation. of the two Wigner functions yield an extracted Berry phase N. Y. Y. acknowledges support from the Miller Institute for difference of Δϕ ¼ 1.07π  0.09π [19]. Thus, we have Basic Research in Science. 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Observing Topological Invariants Using Quantum Walks in Superconducting Circuits

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Abstract

PHYSICAL REVIEW X 7, 031023 (2017) 1,2,* 1,2 1,2 1,2 1 1,2 E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi Department of Physics, University of California, Berkeley, California 94720, USA Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA (Received 6 December 2016; revised manuscript received 12 May 2017; published 3 August 2017) The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here, we motivate a toolbox based on time- dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schrödinger cat state and highlights a novel refocusing technique, which allows for the direct implementation of a digital version of Bloch oscillations. As the walk is performed in phase space, our scheme can be extended to higher synthetic dimensions by adding additional microwave cavities, whereby superconducting circuit-based simulations can probe topological phases ranging from the quantum-spin Hall effect to the Hopf insulator. DOI: 10.1103/PhysRevX.7.031023 Subject Areas: Quantum Physics, Quantum Information, Topological Insulators Topological phases elude the Landau-Ginzburg para- cavity’s phase space, as shown in Fig. 1(b). Its spin degrees digm of symmetry breaking [1]. Unlike conventional of freedom are formed by a superconducting transmon phases, they do not exhibit order parameters that can be qubit [14] with basis states fj↑i; j↓ig. To enable the qubit locally measured. Rather, their distinguishing features are state to control the direction of motion of the coherent state, hidden in quantized, nonlocal topological invariants, which we realize a strong dispersive coupling between the cavity are robust to all local perturbations [2,3]. While tremendous and qubit, theoretical progress has been made toward the full classi- fication of topological phases of matter [4,5], a general † † H=ℏ ¼ ω σ ˆ =2 þ ω a ˆ a ˆ − χ a ˆ a ˆσ ˆ =2; ð1Þ q z c qc z experimental platform for the direct measurement of topological invariants is lacking. Here, we demonstrate where ω are the qubit and cavity transition frequencies, q;c that time-dependent quantum walks comprise a powerful respectively, a ˆ (a ˆ ) is the lowering (raising) operator for the class of unitary protocols capable of digitally simulating cavity mode, σ is the Pauli z matrix for the qubit levels, single-particle topological band structures and directly and χ is the dispersive interaction strength [see Fig. 1(c)]. qc observing the associated nonlocal invariants. Dispersive coupling produces a qubit-dependent shift in the A quantum walk [6–11] describes the motion of a cavity oscillation frequency. Viewed in the rotating frame particle with internal (spin) degrees of freedom moving of the cavity at ω , the dispersive interaction causes the on a discrete lattice. Formally, the quantum walk is c coherent state to move clockwise (counterclockwise) at a comprised of two unitary operations [see Fig. 1(a)]: a coin ˆ rate χ =2 through phase space when the qubit is in the j↑i toss, denoted RðθÞ, which rotates the spin state, and a spin- qc (j↓i) state. Thus, free evolution under the dispersive dependent translation, denoted T , which translates the ↑↓ interaction precisely enables the spin-dependent translation particle’s position by a single lattice site in a direction needed for the quantum walk [15,16]. determined by the internal spin state. In our cavity quantum We realize a particular class of quantum algorithm known electrodynamics implementation of the quantum walk, the as the split-step quantum walk [17,18], which alternates particle is encoded as a coherent state of an electromagnetic two coin tosses (with rotation angles θ and θ ) between 1 2 cavity mode [12,13], where its position is defined in the two spin-dependent translations so that each step of the walk consists of the unitary operation U ðθ ; θ Þ¼ * W 1 2 emmanuel.flurin@berkeley.edu ˆ ˆ ˆ ˆ T Rðθ ÞT Rðθ Þ [see Fig. 1(a)]. The coin-toss operations ↑↓ 2 ↑↓ 1 iθσ ˆ =2 ˆ x Published by the American Physical Society under the terms of R ðθÞ¼ e are applied via short (7.5-ns) coherent the Creative Commons Attribution 4.0 International license. microwave pulses resonant with the qubit transition. By Further distribution of this work must maintain attribution to −1 waiting for a time interval t ¼ 2πð10χ Þ ¼ 124 ns qc the author(s) and the published article’s title, journal citation, and DOI. between successive coin tosses, we allow the dispersive 2160-3308=17=7(3)=031023(6) 031023-1 Published by the American Physical Society E. FLURIN et al. PHYS. REV. X 7, 031023 (2017) (a) We observe the expected ballistic expansion of the coherent (b) state in cavity phase space, consistent with theoretical predictions (population fidelities greater than 90%). As the walk unitary U directly couples the particle’s spin and position degrees of freedom, the resulting dynamics mimic those of spin-orbit interacting materials. More precisely, the unitary quantum-walk protocol simu- lates continuous evolution under an effective spin-orbit Hamiltonian H , which generates the same transformation −iH ˆ W as a single step of the walk when U ¼ e . Since the unitary is translation invariant, the effective Hamiltonian (c) exhibits Bloch bands of quasienergy ϵðkÞ, where the quasimomentum k lies in the Brillouin zone; Figs. 3(a) and 3(b) show, respectively, the band structures underlying ˆ ˆ the walks U and U . The corresponding eigenstates 0 1 consist of extended Bloch waves with spin polarization n⃗ ðkÞ [17]. Depending on symmetry, the band structure of such spin-orbit-coupled Hamiltonians can feature quan- FIG. 1. Quantum-walk implementation in cavity phase space. tized topological invariants. In the case of the split-step (a) Schematic representation of a split-step quantum walk on a −iπA·σ⃗ =2 quantum walk, Γ ¼ e plays the role of a so-called ˆ ˆ line, with rotations Rðθ Þ and Rðθ Þ and spin-dependent trans- 1 2 ˆ ˆ chiral symmetry [17,20], with Γ U Γ ¼ −U . This sym- lation T . Red (blue) lines show spin-up (-down) components W W ↑↓ metry constrains the spin polarization vector n⃗ ðkÞ to lie on a moving left (right). The opacity of each circle indicates the population on the corresponding lattice site. (b) Set of ten cavity ⃗ great circle of the Bloch sphere, perpendicular to A ¼ coherent states on which the walk takes place, in the phase space ( cosðθ =2Þ; 0; sinðθ =2Þ) [Figs. 3(c) and 3(d)]. Thus, the 1 1 of the TE cavity mode. (c) Cavity resonator and qubit. The number of times n⃗ ðkÞ wraps around the origin as k varies fundamental (TE , orange) mode at ω ¼ 2π × 6.77 GHz is 110 R through the Brillouin zone—known as the winding or used to measure the qubit state. This mode couples strongly Chern numberW—naturally defines the topological invari- [κ ¼ 2π × 600 kHz ¼ 1=ð260 nsÞ] to a 50-ohm transmission line ˆ ˆ ant [17] of the walk. While the energy spectra of U and U via the readout port at the center of the cavity. The TE cavity 0 1 mode (green) at ω ¼ 2π × 7.41 GHz is long lived with an are identical, they lie in topologically distinct phases, with inverse lifetime, κ ¼ 2π × 4 kHz ¼ 1=ð40 μsÞ. The transmon ˆ ˆ U having zero winding number and U a winding number 0 1 qubit (coin) has transition frequency ω ¼ 2π × 5.2 GHz, relax- of unity. Analogous to the number of twists in a closed ation times T ¼ 40 μs and T ¼ 5.19 μs, and is dispersively 1 2 ribbon, winding numbers are quantized and robust to local coupled to both cavity modes, with the dispersive shift of the perturbations [2]. walker mode, χ ¼ 2π × 1.61 MHz. qc The direct measurement of topological invariants in solid-state materials is an outstanding challenge [21–23], coupling to naturally implement the spin-dependent trans- owing to the nonlocal nature of the order parameter. Our lation. This time interval determines the lattice on which the method makes use of a time-dependent modification of the walk takes place; here, it is a circular lattice of ten sites in quantum walk, which, in the Hamiltonian picture, mimics cavity phase space [Fig. 1(b)]. an adiabatic translation of the underlying band structure We begin by performing a pair of topologically distinct across the Brillouin zone [24–28]. The resulting dynamics split-step quantum walks, the first (topologically trivial) effectively constitute digital Bloch oscillations, a phenome- ˆ ˆ with unitary U ¼ U ð3π=4; π=4Þ, and the second (topo- 0 W non whereby a particle on a lattice subjected to a constant ˆ ˆ logically nontrivial) with U ¼ U ðπ=4; 3π=4Þ. To dem- 1 W force undergoes oscillations [29] due to the periodicity of onstrate the robustness of the winding number, we also the Brillouin zone. In our system, these oscillations implement an additional pair of walks which are contin- manifest as a refocusing of the quantum walker to its ˆ ˆ uously connected to U and U (e.g., without closing the initial position, with a Berry phase—a signature of the 0 1 gap). The experimental sequence is shown in Fig. 2(a). band-structure topology (see Fig. 3)—imprinted during the The cavity mode is initialized (D ) in a coherent state jβi evolution. In practice, this refocusing depends on choosing with jβj ¼ 8 photons, after which the walk unitary is the number of steps in the walk such that the accrued repeatedly applied. To directly reconstruct the walker’s dynamical phase—which has opposite signs in either band quantum state on the phase-space lattice, we first projec- and thus impedes refocusing—effectively vanishes [19,24]. tively measure the qubit state and subsequently measure the In the general setting, one can experimentally determine the Q function of the cavity mode [19]. Figure 2(b) depicts the condition for dynamical phase refocusing by performing measured lattice site populations after each step of the walk. spectroscopy while varying the number of steps. 031023-2 OBSERVING TOPOLOGICAL INVARIANTS USING … PHYS. REV. X 7, 031023 (2017) (a) (b) (c) FIG. 2. Quantum-walk protocol and resulting populations. (a) Protocol used to perform the quantum walk, showing cavity state preparation (blue), quantum walk (green), qubit state measurement (blue), and Q function measurement (pink). The dashed boxes with σ gates are performed to implement the Bloch oscillation. (b) Cavity Q functions after each step of the quantum walk without Bloch ˆ ˆ oscillations, U (top strip) and U (bottom strip). Spin-up (red) and spin-down (blue) Q functions are superimposed. The average 0 1 ˆ ˆ fidelities of the populations compared to theoretical predictions are 0.97 and 0.96 for U and U , respectively. (c) Cavity Q functions 0 1 after each step of the refocusing quantum walk with Bloch oscillations. The state refocuses after ten steps, as shown in the final frame for ˆ ˆ ˆ ˆ both U and U . Refocusing fidelities (to the initial state) for U and U are 0.83 and 0.87, respectively. 0 1 0 1 In addition to the dynamical phase, upon traversing The additional steps used in performing the time- the Brillouin zone, the particle’s spin winds around the dependent walks are shown in dashed boxes in Fig. 2(a). ˆ ˆ Bloch sphere, encoding its path in the accumulated Berry Beginning with either U or U , we insert rotations by Δk 0 1 ˆ ˆ phase [30], about σ ˆ after each coin toss rotation Rðθ Þ and Rðθ Þ.In z 1 2 ˆ ˆ contrast to the original operations comprising U and U , 0 1 the rotation angle Δk varies in time. Since a σ ˆ rotation is ϕ ¼ i hk; n⃗ ðkÞj∂ jk; n⃗ ðkÞidk ¼ π × W; ð2Þ B k equivalent to a translation of the underlying Hamiltonian in BZ quasimomentum space [24], this time-varying rotation which thus becomes an observable manifestation of the angle implements a digital Bloch oscillation. We choose winding number W—the Hamiltonian’s topological invari- Δk to vary in steps of π=10 from 0 to π, traversing the ant. As one cannot directly observe the quantum mechani- Brillouin zone exactly once. cal phase of a wave function, measuring this Berry phase Populations resulting from the time-dependent walks requires an interferometric approach. To this end, we (with the system initialized in a single coherent state) are shown in Fig. 2(c). Unlike the ballistic dynamics resulting perform the time-dependent walk with the cavity-qubit from the original walks, the Bloch oscillation (traversal of system initialized in a Schrödinger cat superposition of two the Brillouin zone) causes the walker wave function to coherent-state components: One component undergoes the walk, while the other is unaffected by the unitaries. The refocus [25–28] to both its initial position and spin state. Berry phase thus appears as the relative phase between The intuition underlying this refocusing is that both the the two components and is observable via direct Wigner dynamical and Berry phases accumulated by each quasi- momentum component of the walker is identical upon full tomography. 031023-3 E. FLURIN et al. PHYS. REV. X 7, 031023 (2017) (a) (b) (a) (e) (b) (c) (d) (c) (d) FIG. 3. Topological classes of split-step quantum walks. Calculated band structures, quasienergy ϵ versus quasimomen- tum k, corresponding to the two walks we perform in the ˆ ˆ ˆ ˆ ˆ ˆ experiment, U ¼ T Rðπ=4ÞT Rð3π=4Þ (a) and U ¼ 0 ↑↓ ↑↓ 1 ˆ ˆ ˆ ˆ T Rð3π=4ÞT Rðπ=4Þ (b). Though the energy bands of the ↑↓ ↑↓ two walks are identical, they are topologically distinct, with FIG. 4. Winding number measurement via direct Wigner the topology given by the winding of n⃗ ðkÞ as k varies through the tomography of refocused Schrödinger cat states. (a) Protocol Brillouin zone, shown in diagrams (c) and (d). In diagram (c), the for measuring topology via a time-dependent walk (Bloch trivial case U , n⃗ ðkÞ does not complete a full revolution around 0 oscillations). The Schrödinger cat state is first prepared (blue), the Bloch sphere, while in the topological case U diagram (d), it after which the ten-step refocusing quantum walk is performed does perform a full revolution. This also provides a direct (green). The qubit and cavity state are then disentangled, the qubit connection to the Berry phase, as for a spin-1=2 system the state is purified (blue), and direct Wigner tomography on the Berry phase is simply half the subtended solid angle of the Bloch cavity state is performed (pink). Wigner tomography of (b) the cat sphere path. A schematic representation of the variation of n⃗ ðkÞ is undergoing no quantum walk, (c) the cat after undergoing the ˆ ˆ shown by the ribbons below the Bloch spheres. The arrows on trivial U walk, and (d) the cat after undergoing the topological U 0 1 these strips point in the direction of n⃗ ðkÞ. Analogous to the walk. Fidelities of these resulting cat states compared to pure cat number of twists in closed ribbons, winding numbers are states are 0.68, 0.69, and 0.67, respectively. (e) A cut of the quantized and robust to local perturbations. Wigner function, showing the fringes that encode the relative phase between the two cat components for no walk (black), a trivial walk (red), and a topological walk (blue). The relative phase traversal [19,24]. In practice, we observe refocusing fidel- corresponds to the phase of the measured interference fringes pffiffiffi ities greater than 80%, limited by incomplete adiabaticity following the relation A exp½−2jImðαÞj  cos½2 n¯ImðαÞþ ϕ, and experimental imperfections. where A, ϕ are the amplitude and phase of the fringes. The Having verified the refocusing behavior of the time- Berry phase—captured by the phase difference between the dependent quantum walks, we initialize the cavity-qubit topological and the trivial walks—is ϕ ¼ 1.05π  0.06π in experiment, consistent with the theoretical expectations of π. system in a Schrödinger cat state to measure the accumu- lated Berry phase [19]. One component of the cat is walking component of the cat refocuses, we disentangle the precisely the initial state of the previous walks, jβ;↑i. qubit from the cavity with the operation j0;fi → j0;↑i. The other component is j0;fi, where the cavity is in its ground (vacuum) state and the transmon is in its second This leaves the oscillator in the state excited state [14], jfi. Shelving the vacuum component of iϕ the cat in the jfi state renders it immune to the coin-toss jψi¼j0i − e jβi; ð3Þ rotations, as the jfi ↔ j↓i transition is far detuned (225 MHz) from the j↑i ↔ j↓i transition. Thus, this where ϕ is the Berry phase. component of the cat lies dormant during the walk, acting While Q tomography lends itself well to measuring as a phase reference for the observation of the Berry phase. coherent-state occupations, coherences between these states Our method of preparing the cat, a modification of the are largely invisible in this representation. To measure the protocol introduced in Ref. [31], is shown in Fig. 4(a). With Berry phase, we therefore apply direct Wigner tomography the cat initialized, we perform the time-dependent walk to the final cavity state [19,31,32]. As Figs. 4(b)–4(d) show, over a full Bloch oscillation, applying the same set of the Wigner functions of two-component cat states display pulses that resulted in the final frames of Fig. 2(c). After the interference fringes, whose phase directly encodes the 031023-4 OBSERVING TOPOLOGICAL INVARIANTS USING … PHYS. REV. X 7, 031023 (2017) A single step of these walks consists of multiple spin- relative phase between the dormant (j0;fi) and walking (jβ;↑i) components of the cat. Figures 4(c) and 4(d) display dependent translation steps in different directions, neces- the measured Wigner functions for both split-step walks. sitating the modification in time of the dispersive shifts. In the topologically trivial phase [Fig. 4(c)], the interference Using the current Bloch-oscillating protocol, the 2D fringes do not acquire any phase shift after the walk, besides a Brillouin zone can be swept out in stripes, whereby a small offset due to technical imperfections. For the topo- measurement of the Berry phase acquired along each stripe logically nontrivial walk [Fig. 4(d)], however, the fringes allows the extraction of the Chern number [35]. visibly shift [Fig. 4(e)], corresponding to an acquired phase The generalization of quantum-walk-based protocols to measurements of many-body topological invariants repre- of ϕ ¼ 1.05π  0.06π. The topologies of the Hamiltonians that generate the walks are thereby clearly imprinted on the sents an exciting frontier at the interface of topology, Wigner functions of the refocused states. A key feature of interactions, and quantum simulation [36,37]. such topology is its robustness to all perturbations that do not The authors acknowledge discussions with David Toyli, close the spectral gap. To this end, we have performed an Chris Macklin, Kevin Fischer, Mark Rudner, Eugene ˆ ˆ additional pair of quantum walks, U ¼ U ð0.64π; 0.28πÞ 0 W Demler, and Carlos Navarette-Benloch for motivating the ˆ ˆ and U ¼ U ð0.28π; 0.64πÞ, which are continuously 1 W use of a refocusing quantum walk. V. V. R. and L. S. M. deformable from the original walks. In this case, line cuts acknowledge funding via the National Science Foundation. of the two Wigner functions yield an extracted Berry phase N. Y. Y. acknowledges support from the Miller Institute for difference of Δϕ ¼ 1.07π  0.09π [19]. Thus, we have Basic Research in Science. 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