Abstract
Noise is ubiquitous in real quantum systems, leading to nonHermitian quantum dynamics, and may affect the fundamental states of matter. Here we report in an experiment a quantum simulation of the twodimensional nonHermitian quantum anomalous Hall (QAH) model using the nuclear magnetic resonance processor. Unlike the usual experiments using auxiliary qubits, we develop a stochastic average approach based on the stochastic Schrödinger equation to realize the nonHermitian dissipative quantum dynamics, which has advantages in saving the quantum simulation sources and simplifying the implementation of quantum gates. We demonstrate the stability of dynamical topology against weak noise and observe two types of dynamical topological transitions driven by strong noise. Moreover, a region where the emergent topology is always robust regardless of the noise strength is observed. Our work shows a feasible quantum simulation approach for dissipative quantum dynamics with stochastic Schrödinger equation and opens a route to investigate nonHermitian dynamical topological physics.
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Introduction
As a fundamental notion beyond the celebrated LandauGinzburgWilson framework^{1}, the topological quantum matter has stimulated extensive studies in recent years, with tremendous progress having been achieved in searching for various types of topological states^{2,3,4,5,6,7}. A most important feature of topological matter is the bulksurface correspondence^{2,3,4}, which relates the bulk topology to boundary states and provides the foundation of most experimental characterizations and observations of topological quantum phases, such as via transport measurements^{8,9,10} and angleresolved photoemission spectroscopy^{11,12,13}.
Despite the fact that topological phases are defined at the ground state at equilibrium, quantum quenches in recent studies provide a nonequilibrium way to investigate topological physics^{14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}. Particularly, as a momentumspace counterpart of the bulkboundary correspondence, the dynamical bulksurface correspondence was proposed^{29,30,31,32,33,34}, which relates the bulk topology of an equilibrium phase to a nontrivial dynamical topological phase emerging on certain momentum subspaces called bandinversion surfaces (BISs) when quenching the system across topological transitions. This dynamical topology enables a broadly applicable way to characterize and detect topological phases by quantum dynamics and has triggered many experimental studies in quantum simulations, such as in ultracold atoms^{35,36}, nitrogenvacancy defects in diamond^{37,38,39}, nuclear magnetic resonance (NMR)^{40}, and superconducting circuits^{41}.
The quenchinduced dynamical topological phase has been mainly studied in Hermitian systems, while the system is generally nonHermitian when coupled to the environment^{42}. Recently, the interplay between nonHermiticity and topology has attracted considerable attention^{43,44}, with rich phenomena being uncovered, such as the exotic topological phases driven by exceptional points^{45,46,47,48}, the anomalous bulkboundary correspondence^{46,49,50,51}, and the nonHermitian skin effect^{52}. Experimental observations of the nonHermitian topological physics have been reported in classical systems with gain and loss, like the photonic systems^{53,54}, the active mechanical metamaterial^{55}, as well as topolectrical circuits^{56}, and in quantum simulators, like the nitrogenvacancy center^{57,58}, where the nonHermitian effects are engineered by coupling to auxiliary qubits.
As an important source of dissipation and nonHermiticity, the dynamical noise is ubiquitous and inevitable in the real quantum simulations, especially for the quantum quench dynamics, and can be described by the stochastic Schrödinger equation^{59,60}. Without the necessity of applying auxiliary qubits, the quantum simulation using the stochastic Schrödinger equation may enable a direct and more efficient way to explore nonHermitian dynamical phases, hence facilitating the discovery of nonHermitian topological physics with minimal quantum simulation sources. In particular, the controllable noise can provide a fundamental scheme to explore nonHermitian dissipative quantum dynamics, and the noise effects on the quenchinduced dynamical topological phase give rise to rich nonequilibrium topological physics^{61}. However, the experimental study is currently lacking.
In this article, we report the experimental observation of quenchinduced nonHermitian dynamical topological states by simulating a noising twodimensional (2D) quantum anomalous Hall (QAH) model on an NMR quantum simulator. Unlike previous experiments using auxiliary qubits^{57,58}, we achieve with advantages the nonHermitian quench dynamics via simulating the stochastic Schrödinger equation and by averaged measurements over different noise configurations^{59,60,61}. We observe the dynamical topology emerging in the nonHermitian dissipative quench dynamics on BISs by measuring the timeaveraged spin textures in momentum space, and identify two types of dynamical topological transitions classified by distinct dynamical exceptional points by varying the noise strength. Moreover, the existence of a sweet spot region with the emergent topology being robust under arbitrarily strong noise is experimentally verified. Our experiment demonstrates a feasible technique in simulating dynamical topological physics with minimal sources.
Results
NonHermitian QAH model
We consider the nonHermitian 2D QAH model with the magnetic dynamical white noise described by the Hamiltonian
where \({{{{\mathcal{H}}}}}_{{{{\rm{QAH}}}}}({{{\bf{k}}}})={{{\bf{h}}}}({{{\bf{k}}}})\cdot {{{\boldsymbol{\sigma }}}}\) describes the QAH phase^{62,63}, with Bloch vector \({{{\bf{h}}}}=({\xi }_{\rm{so}}\sin {k}_{x},{\xi }_{\rm{so}}\sin {k}_{y},{m}_{z}{\xi }_{0}\cos {k}_{x}{\xi }_{0}\cos {k}_{y})\). Here ξ_{0} (or ξ_{so}) simulates the spinconserved (spinflipped) hopping coefficient, and m_{z} is the magnetic field. The white noise w_{i}(k, t) of strength \(\sqrt{{w}_{i}}\) couples to the Pauli matrix σ_{i} and satisfies \({\langle \langle {w}_{i}({{{\bf{k}}}},t)\rangle \rangle }_{{{{\rm{noise}}}}}=0\) and \({\langle \langle {w}_{i}({{{\bf{k}}}},t){w}_{j}({{{\bf{k}}}},t^{\prime} )\rangle \rangle }_{{{{\rm{noise}}}}}={w}_{i}{\delta }_{ij}\delta (tt^{\prime} )\), where 〈〈⋅〉〉_{noise} is the stochastic average over different noise configurations. Without noise, the Hamiltonian \({{{{\mathcal{H}}}}}_{{{{\rm{QAH}}}}}\) hosts nontrivial QAH phase for 0 < ∣m_{z}∣ < ∣ξ_{0}∣ with Chern number C_{1} = sgn(m_{z}), and the phase is trivial for ∣m_{z}∣ > ∣ξ_{0}∣ or m_{z} = 0^{62}. The random noise can change the topology of the QAH model, and plays a vital role on the quantum dynamics induced in the present system. We start with the simple situation with a single noise configuration. In this case the quantum dynamics governed by the stochastic Schrödinger equation \({{{\rm{i}}}}{\partial }_{t}\left\psi ({{{\bf{k}}}},t)\right\rangle ={{{\mathcal{H}}}}({{{\bf{k}}}},t)\left\psi ({{{\bf{k}}}},t)\right\rangle\) describes a random unitary evolution, which can be further converted into the socalled Itô form^{59,60} in simulation (see “Methods” for details)
Here \({{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}={{{{\mathcal{H}}}}}_{{{{\rm{QAH}}}}}({{{\rm{i}}}}/2){\sum }_{i}{w}_{i}\) is the effective nonHermitian Hamiltonian, such that the increment of a Wiener process \({W}_{i}({{{\bf{k}}}},t)\equiv (1/\sqrt{{w}_{i}})\int\nolimits_{0}^{t}{{{\rm{d}}}}s\,{w}_{i}({{{\bf{k}}}},s)\) is independent from the wavefunction function \(\left\psi (t)\right\rangle\), for which we have the Itô rules dtdW_{i}(t) = 0 and dW_{i}(t)dW_{j}(t) = δ_{ij}dt, and the corresponding expectation value is zero. The formal solution of the above equation reads \(\left\psi (t)\right\rangle =U(t)\left\psi (0)\right\rangle\) with
where \({{{\mathcal{T}}}}\) denotes the time ordering. Note that while the equation (3) describes a random unitary evolution in the regime with single noise configuration, after the noise configuration averaging the nonHermitian dissipative quantum dynamics emerges and is captured by the master equation
where \(\rho ({{{\bf{k}}}},t)\equiv {\langle \langle \left\psi ({{{\bf{k}}}},t)\right\rangle \left\langle \psi ({{{\bf{k}}}},t)\right\rangle \rangle }_{{{{\rm{noise}}}}}\) is the stochastic averaged density matrix; see “Methods” for details. The configuration averaging is a key point for the present quantum simulation of nonHermitian dynamical topological phases.
Quantum simulation approach
We next develop the quantum simulation approach by introducing the discrete Stochastic Schrödinger equation for the nonHermitian dissipative quantum dynamics, since the continuous evolution cannot be directly emulated with digital quantum simulators. Specifically, we discretize the continuous time as t_{n} = nτ with smalltime step τ, where the integer n ranges from zero to the total number of time steps M. The increment of the Wiener process can be simulated by random numbers \({{\Delta }}{W}_{i}({t}_{n})={N}_{i}({t}_{n})\sqrt{\tau }\) for each noise configuration, and we obtain the discretized stochastic Schrödinger equation
with \(\tilde{{{{\mathcal{H}}}}}({{{\bf{k}}}},{t}_{n})={{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}({{{\bf{k}}}})+{\sum }_{i}\sqrt{{w}_{i}}{\sigma }_{i}{N}_{i}({{{\bf{k}}}},{t}_{n})/\sqrt{\tau }\). Here N_{i}(t_{n}) is sampled from the standard normal distribution to match the expectation and variance of dW_{i}, and the wavefunction is normalized in each time step. The corresponding unitary evolution operator from time t_{n} to t_{n+1} reads
leading to the discrete equation of motion
in the linear order of τ after stochastic average, which describes the desired nonHermitian quantum dynamics. We shall analyse the quality of this discretization versus time step τ in the experiment. The stochastic average of a physical operator \(\hat{{{{\bf{O}}}}}\) at time t_{n} can now be obtained by
This formalism can be directly simulated in experiment.
The above presents the essential idea for simulating the nonHermitian systems based on the stochastic Schrödinger equation. This method is fundamentally different from that applied in the previous experiments^{57,58} using auxiliary qubits, where the nonHermiticity is obtained from a Hermitian Hamiltonian in the extended Hilbert space by tracing the auxiliary degrees of freedom and careful designs of the quantum circuit with complex unitary operations are required^{64,65}. In contrast, our temporal average approach based on the stochastic Schrödinger equation saves the resources of qubits and avoids the implementation of complex gates, which benefits the experimental platforms in various scenarios. Moreover, this quantum simulation approach can be directly extended to exploring higher dimensional nonHermitian topological phases and phase transitions.
NonHermitian dynamical topological phases
Before presenting the experiment, in this section, we briefly introduce the nonHermitian dynamical topological phases emerging in the quench dynamics described by Eq. (4) and to be studied in this work.
The system is initially prepared at the fully polarized ground state \({\rho }_{0}=\left\downarrow \right\rangle \left\langle \downarrow \right\) of a deep trivial Hamiltonian with ∣m_{z}∣ ≫ ∣ξ_{0}∣. After quenching m_{z} to a nontrivial value at time t_{n} = 0, the system starts to evolve under the postquench Hamiltonian \({{{\mathcal{H}}}}({{{\bf{k}}}},{t}_{n})\); see Fig. 1a. Without noise, the spin polarization \(\langle {{{\boldsymbol{\sigma }}}}({{{\bf{k}}}},{t}_{n})\rangle \equiv \rm{Tr}[{{{\boldsymbol{\sigma }}}}\mathop{\prod }\nolimits_{i = 0}^{n1}U({t}_{ni},{t}_{ni1}){\rho }_{0}\mathop{\prod }\nolimits_{i = 0}^{n1}{U}^{{\dagger} }({t}_{i+1},{t}_{i})]\) precesses with respect to the Hamiltonian vector h; see Fig. 1a. The postquench QAH phase can be determined by the dynamical topology emerging on BISs^{29}, identified as the momentum subspaces with h_{z} = 0, where the initial state is perpendicular to the SO field h_{so} ≡ (h_{x}, h_{y}), leading to vanishing timeaveraged spin polarizations.
In the presence of nonHermiticity, the precession axis for each noise configuration is distorted, leading to the deformation for the BISs and dissipative effect. To characterize the noise effect, the spin polarization needs to be stochastically averaged as
over different noise configurations (Fig. 1b). Compared to the spin polarization 〈σ(k, t_{n})〉 without noise, the stochastic averaged s(k, t_{n}) follows the nonHermitian dynamics and exhibits dephasing and amplitude decaying effects. We compensate the amplitude decay by rescaling s(k, t_{n}), leading to the rescaled spin polarization \(\tilde{{{{\bf{s}}}}}({{{\bf{k}}}},{t}_{n})\equiv {{{{\bf{s}}}}}_{0}({{{\bf{k}}}})+{{{{\bf{s}}}}}_{+}({{{\bf{k}}}}){{{{\rm{e}}}}}^{{{{\rm{i}}}}\omega ({{{\bf{k}}}}){t}_{n}}+{{{{\bf{s}}}}}_{}({{{\bf{k}}}}){{{{\rm{e}}}}}^{+{{{\rm{i}}}}\omega ({{{\bf{k}}}}){t}_{n}}\), where the coefficients s_{0,±} and oscillation frequency ω are extracted from the experimental data by fitting; see “Methods”. Similar to the noiseless case, the time average
vanishes on the deformed BISs (dubbed as dBISs)^{61}, with the number of steps M being large enough to minimize the error. The nonHermitian dynamical topological phase is captured by the dynamical invariant \({{{\mathcal{W}}}}\equiv \frac{1}{2\pi }{\oint }_{{{{\rm{dBIS}}}}}{{{\bf{g}}}}({{{\bf{k}}}}){{{\rm{d}}}}{{{\bf{g}}}}({{{\bf{k}}}})\), which describes the winding of dynamical field \({{{\bf{g}}}}({{{\bf{k}}}})=(1/{{{{\mathcal{N}}}}}_{{{{\bf{k}}}}}){\partial }_{{k}_{\perp }}(\overline{{\tilde{s}}_{x}({{{\bf{k}}}})},\overline{{\tilde{s}}_{y}({{{\bf{k}}}})})\) on the dBISs. Here k_{⊥} is perpendicular to the dBISs and \({{{{\mathcal{N}}}}}_{{{{\bf{k}}}}}\) is a normalization factor. Under the dynamical noise, the nonHermitian dynamical topological phases and phase transitions may be induced, as studied in the experiment presented below.
Experimental setup
The demonstration is performed on the NMR quantum simulator. The sample is the ^{13}Clabeled chloroform dissolved in acetoned6, with ^{13}C and ^{1}H nuclei denoted as two qubits. The 2D QAH model is simulated by the qubit ^{13}C, while the other qubit ^{1}H enhances the signal by Overhauser effect (see Fig. 1c and “Methods”). In the doublerotating frame, the total Hamiltonian of this sample is
where J = 215 Hz is the coupling strength, B_{i} is the amplitude of the control pulse, and ϕ_{i} is the phase. We firstly initialize the system into the fully polarized state \(\left\downarrow \right\rangle\) using the nuclear Overhauser effect^{66}. Then we quench m_{z} to the nontrivial region with ∣m_{z}∣ < 2ξ_{0} and allow the system to evolve under the effective Hamiltonian \(\tilde{{{{\mathcal{H}}}}}\), in which the nonHermitian constant term i∑_{i}w_{i} can be ignored. The evolution is realized by the Trotter approximation combined with control pulse optimizations as follows.
We study the nonHermitian dissipative quantum dynamics from time t = 0–30 ms. For each noise configuration, numerical results show that the discrete evolution approximates the continuous evolution of the stochastic Schrödinger equation quite well, when the total number of time steps is greater than 100; see Fig. 2. In the experiment, we discretize the time into 300 segments, such that the Hamiltonian in each interval is approximately timeindependent. As the interval τ is sufficiently small, the evolution in the nth step can be realized using the firstorder Trotter decomposition:
with \({\eta }_{x,y}={\xi }_{\rm{so}}\sin {k}_{x,y}+\sqrt{{w}_{x,y}}{N}_{x,y}({t}_{n})/\sqrt{\tau }\) and \({\eta }_{z}={m}_{z}{\xi }_{0}\cos {k}_{x}{\xi }_{0}\cos {k}_{y}+\sqrt{{w}_{z}}{N}_{z}({t}_{n})/\sqrt{\tau }\). Here ξ_{0} is set to 1 kHz. Each term on the righthand side represents a singlequbit rotation with a rotating angle 2η_{i}τ along axis σ_{i}, which can be experimentally realized by tuning the amplitude and phase of the control pulse in Eq. (11) (zrotation can be indirectly realized via x and yrotations), with further pulse optimization techniques to reduce control errors; see Fig. 1c.
We measure the spin polarization 〈σ(k, t)〉 for single noise configuration at every 20τ interval. After averaging over all noise configurations, we obtain the stochastic averaged spin polarization s(k, t), from which the rescaled spin polarization \(\tilde{{{{\bf{s}}}}}({{{\bf{k}}}},t)\) can be constructed by fitting. We repeat the above procedures for the whole momentum space to obtain the timeaveraged spin textures \(\overline{\tilde{{{{\bf{s}}}}}({{{\bf{k}}}})}\).
Experimental results
We start from the weak noise regime, where the noise strength is chosen as w_{x} = 0.05ξ_{0}, w_{y} = 0, and w_{z} = 0.01ξ_{0} with ξ_{so} = 0.2ξ_{0}. The system is quenched to the topological phase with m_{z} = 1.2ξ_{0}. In Fig. 3a, we plot the spin polarization 〈σ(t)〉 at the momentum k = (1.286, −0.257) for four different noise configurations. For each noise configuration, no notable decay exists in the spin polarization, manifesting the unitary evolution. However, after averaged over all noise configurations, the system clearly exhibits the nonHermitian dissipative quantum dynamics; see Fig. 3b.
Figure 3c shows the measured timeaveraged spin textures \(\overline{{\tilde{s}}_{i}({{{\bf{k}}}})}\) with fixed k_{y} = −0.257 and k_{x} ∈ [−1.8, 1.8], obtained by rescaling the stochastic averaged spin polarization s(k, t). The momenta with vanishing values represent dBIS points. To obtain the 2D timeaveraged spin texture, we discretize the whole momentum space k_{x}, k_{y} ∈ [−1.8, 1.8] into a 15 × 15 lattice and repeat the above measurements. The results are shown in Fig. 3d, from which the dBIS momenta can be identified. Although the corresponding shape is slightly deformed from the ideal BIS with h_{z} = 0 in the absence of noise (see “Methods”), it is obvious that under weak noise, the dynamical field g(k) can be defined everywhere on dBIS and characterizes the nontrivial nonHermitian dynamical topological phase (Fig. 3e). Indeed, this emergent dynamical topology is robust against the weak noise and is protected by the finite minimal oscillation frequency on the dBISs, serving as a bulk gap for the dynamical topological phase. The experimental minimum oscillation frequency on dBISs is given by \({\omega }_{\min }=0.4175\) kHz, close to the theoretical value 0.4063 kHz (Fig. 3b). Further, this nonHermitian dynamical topological phase may break down under strong noise, with two types of dynamical transition being observed below.
We now increase the noise strength to a strong regime with w_{x} = 0.1ξ_{0}, w_{y} = 0.05ξ_{0}, and w_{z} = 0.45ξ_{0}. The averaged spin polarization is measured in the same way as in the weak noise regime. However, the quench dynamics are essentially different, where the spin polarization s(t) at certain momenta, for instance k_{x} = −1.286 and k_{y} = −0.257, displays pure decay without oscillation; see Fig. 4a, b. For these momenta, the dynamical field g vanishes. In Fig. 4c, we show the corresponding spin textures. From the result for \(\overline{{\tilde{s}}_{z}}\), we find that singularities emerges on the dBISs and interrupt their continuity. Thus the dBIS breaks down, while the deformation of the shape of dBIS is small, and the nonHermitian dynamical topological phase transition occurs. In Fig. 4d, we increase the noise strength to w_{x} = 1.6ξ_{0}, w_{y} = 0, w_{z} = 0.8ξ_{0} and set a strong SO coupling coefficient with ξ_{so} = 2ξ_{0}. A qualitatively different dynamical transition is uncovered, where the dBISs are dramatically deformed by the noise and are connected to the topological charge at k = 0. Due to this singularity, the dynamical topology also breaks down. The above two qualitatively different phenomena are referred to as typeI and typeII dynamical transitions, respectively, which we examine below in more detail.
We notice that the equilibrium topological phase transition usually corresponds to the close of energy gap. In the nonequilibrium regime, the analogous quantity is the oscillation frequency. Here we observe the corresponding momentum distribution in Fig. 5a. One can see that the oscillation frequency is in general nonzero but may vanish on certain dBISs momenta when these two types of dynamical transition occur, i.e., \({\omega }_{\min }({{{{\bf{k}}}}}_{c})\to 0\). Indeed, the momenta (k_{c}) with just vanishing oscillation frequency are exceptional points of the Liouvillian superoperator, on which the eigenvectors \({{{{\bf{s}}}}}_{\pm }^{L(R)}\) coalesce^{61}. Thus the dynamical transitions are driven by exceptional points with vanishing oscillation frequency on dBISs. To further distinguish these two types of dynamical transition and the corresponding exceptional points, we treat the Liouvillian superoperator as a threelevel system; see “Methods”. The coefficient s_{+} of rescaled dynamical spin polarization \(\tilde{{{{\bf{s}}}}}({{{\bf{k}}}},t)\) contains the information of corresponding eigenvectors \({{{{\bf{s}}}}}_{\pm }^{L(R)}\). Like the spin1 system, we measure the Liouvillian polarization \(\langle {L}_{\alpha }\rangle \equiv {{{{\bf{s}}}}}_{+}^{{\dagger} }{L}_{\alpha }{{{{\bf{s}}}}}_{+}\) to characterize the Liouvillian superoperator. Here the operator L_{α} is defined as
and L_{z} = i[L_{y}, L_{x}], which satisfies [L_{α}, L_{β}] = iϵ^{αβγ}L_{γ}. The measured momentum distribution of these quantities in the experiment is shown in Fig. 5b, c, from which an important feature of exceptional points is observed that the component 〈L_{x}〉 ≈ 0 and 〈L_{y}〉 ≈ 0 vanish on these points while 〈L_{z}〉 is in general nonzero (e.g., see Fig. 5c). Therefore, the exceptional points are actually the singularities in the twocomponent vector field (〈L_{x}〉, 〈L_{y}〉).
With this observation and to characterize the exceptional points, we consider the Liouvillian polarization on a small loop \({{{\mathcal{S}}}}\) enclosing the exceptional points, as shown in Fig. 6a, b. Although the component 〈L_{z}〉 is nonzero on this loop, the trajectory projected on the 〈L_{x}〉〈L_{y}〉 plane indeed defines a winding number^{61}
which distinguishes the two types of dynamical transitions. We observe that for typeI transition, the winding number N_{E} = 0 is trivial, while the winding N_{E} = 1 is nontrivial for the typeII dynamical transition. Consequently, these distinct exceptional points on dBISs shows the fundamental difference between the typeI and typeII dynamical transitions. Moreover, regardless of the shape and size of the loop \({{{\mathcal{S}}}}\), the winding number N_{E} only depends on the topological properties of the enclosed exceptional points as long as the loop does not cross any other singular points; see Fig. 6a, b. Here we note that the topological charges are always singularities of the field (〈L_{x}〉, 〈L_{y}〉) and have nontrivial winding number^{61} (Fig. 6c, d). The loop \({{{\mathcal{S}}}}\) should be introduced without enclosing any nonexceptional charge momentum in characterizing the dynamical transitions and corresponding exceptional points. This also tells that the typeII dynamical transition is similar to the equilibrium topological phase transition, in which the topological charges serve as singular points and the transition occurs when they pass through the BISs^{29,30}. On the other hand, the typeI transition is a peculiar feature of the quenchinduced nonHermitian dynamical topological phase transition.
Although the nonHermitian dynamical topological phase may typically be destroyed in the strong noise regime, a quite interesting feature of the present system is the existence of a sweet spot region satisfying^{61}
in which regime the dynamical topology is always robust at any finite noise strength, as characterized by the tapertype region in Fig. 7. In particular, for the central line with w_{x} = w_{y} = w_{z}, we experimentally increase the noise strength w_{i} in each direction from 0.5ξ_{0} to a very large value w_{i} ≃ 10ξ_{0} (points O_{1,2,3}) and measure the corresponding timeaveraged dynamical spin textures. We observe that although the noise strength is much large compared with all other energy scales, the dBIS in \(\overline{{\tilde{s}}_{z}}\) remains stable, without suffering singularities. Inside the tapertype region the dynamical topology is welldefined on the dBIS, in sharp contrast to outside points (P_{1,2}). The experimental confirmation of this sweet spot region may offer guidance in designing noisetolerant topological devices.
Discussion
We have experimentally reported the quantum simulation of nonHermitian quantum dynamics for a 2D QAH model coupled to dynamical noise based on a stochastic average approach of the stochastic Schrödinger equation, and simulated nonHermitian dynamical topological phases and phase transitions. Our method does not require the ancillary qubits and careful designs of complex unitary gates, hence saving the simulation sources and avoiding the implementation of complex gates in the experiment. The dynamical topological physics driven by dynamical noise has been observed, including the stability of nonHermitian dynamical topological states protected by the minimal oscillation frequency of quench dynamics under weak noise and two basic types of dynamical topological transitions driven by strong noise and classified by distinct exceptional points. Moreover, a sweet spot region is observed, where the nonHermitian dynamical topological phase survives at arbitrarily strong noise.
Our experiment has shown an advantageous quantum simulation approach to explore the nonHermitian dynamical topological physics, in which only a minimal number of qubits are used. This approach is directly applicable to high dimensions by taking into account more, but still minimal number of qubits, in which the rich phenomena are expected, and also to other digital quantum simulators.
Methods
Stratonovich stochastic Schrödinger equation
We consider the nonHermitian 2D QAH model (1) with the magnetic dynamical white noise w_{i}(k, t). Since the dynamical white noise is in some sense infinite, the dynamical equation \({\partial }_{t}\left\psi ({{{\bf{k}}}},t)\right\rangle ={{{\rm{i}}}}{{{\mathcal{H}}}}({{{\bf{k}}}},t)\left\psi ({{{\bf{k}}}},t)\right\rangle\) cannot be considered as an ordinary differential equation. Instead, it should be regarded as an integral equation
where \({W}_{i}({{{\bf{k}}}},t)=(1/\sqrt{{w}_{i}})\int\nolimits_{0}^{t}{{{\rm{d}}}}s\,{w}_{i}({{{\bf{k}}}},s)\) is a Wiener process. For brevity, the symbols of integration are usually dropped, leading to the stochastic Schrödinger equation
In general, there are two definitions of stochastic integration, i.e., the Stratonovich form
and the Itô form
The basic difference is that the integrand f(t) and the increment dW(t) are independent of each other in the Itô form, namely 〈〈f(t)dW(t)〉〉_{noise} = f(t)〈〈dW(t)〉〉_{noise} = 0, while they are not independent in the Stratonovich form. The Schrödinger equation (17) must be interpreted as a Stratonovich stochastic differential equation^{59,60}, such that the quantum mechanical probability is preserved, i.e., d〈ψ(t)∣ψ(t)〉 = 0.
Converting into the Itô form
Since the wavefunction \(\left\psi (t)\right\rangle\) and the increment dW_{i}(t) are not independent in the Stratonovich form, it is usually convenient to convert the Stratonovich stochastic Schrödinger equation (17) into the Itô form, which takes the form
Due to \((1/2)(\left\psi (t+{{{\rm{d}}}}t)\right\rangle +\left\psi (t)\right\rangle )=[{{{\bf{1}}}}({{{\rm{i}}}}/2)({{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}{{{\rm{d}}}}t+{\sum }_{i}{\alpha }_{i}{{{\rm{d}}}}{W}_{i}(t))]\left\psi (t)\right\rangle\), we have the following relation between the Stratonovich integral and the Itô integral
where we have used the Itô rules dtdW_{i}(t) = 0 and dW_{i}(t)dW_{j}(t) = δ_{ij}dt for the increment of a Wiener process. Substituting this into the Itô stochastic Schrödinger equation (20), we obtain
Compared with the original Stratonovich stochastic Schrödinger equation (17), it is easy to find
In the main text, we have shown that the formal solution of the Itô stochastic Schrödinger equation (20) is given by a unitary evolution U(t) [see Eq. (3)]. To prove that U(t) is indeed the solution of the Itô equation, we shall note that
where the terms other than dt and dW_{i}dW_{i} = dt vanish according to the Itô rules. Thus we recover the Itô stochastic Schrödinger equation, i.e., \({{{\rm{d}}}}U(t)=U(t+{{{\rm{d}}}}t)U(t)={{{\rm{i}}}}[{{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}{{{\rm{d}}}}t+{\sum }_{i}\sqrt{{w}_{i}}{\sigma }_{i}{{{\rm{d}}}}{W}_{i}(t)]U(t)\).
NonHermitian dissipative quantum dynamics
We now consider the equation of motion for the stochastic density operator \(\varrho (t)=\left\psi (t)\right\rangle \left\langle \psi (t)\right\), namely
Since the increments dW_{i}(t) are independent of ϱ(t) in the Itô form, after average over different noise configurations the last term vanishes and we arrive at the Lindblad master equation (4) for the stochastic averaged density matrix ρ(t) ≡ 〈〈ϱ(t)〉〉_{noise}, which describes the nonHermitian dissipative quantum dynamics.
Stochastically averaged spin dynamics
In this section, we show the stochastically averaged spin dynamics. According to the master equation (4), the stochastically averaged spin polarization s(k, t) is governed by the equation of motion
with the Liouvillian superoperator
The solution to this dissipative quantum dynamics can be written as
with the coefficients \({{{{\bf{s}}}}}_{\alpha }({{{\bf{k}}}})=[{{{{\bf{s}}}}}_{\alpha }^{L}({{{\bf{k}}}})\cdot {{{\bf{s}}}}({{{\bf{k}}}},0)]{{{{\bf{s}}}}}_{\alpha }^{R}\) for α = 0, ±. Here \({{{{\bf{s}}}}}_{\alpha }^{L(R)}\) satisfying \({{{{\bf{s}}}}}_{\alpha }^{L}({{{\bf{k}}}})\cdot {{{{\bf{s}}}}}_{\beta }^{R}({{{\bf{k}}}})={\delta }_{\alpha \beta }\) are the left (right) eigenvectors of the Liouvillian superoperator \({{{{\mathcal{L}}}}}^{T}{{{{\bf{s}}}}}_{\alpha }^{L}={\lambda }_{\alpha }{{{{\bf{s}}}}}_{\alpha }^{L}\), \({{{\mathcal{L}}}}{{{{\bf{s}}}}}_{\alpha }^{R}={\lambda }_{\alpha }{{{{\bf{s}}}}}_{\alpha }^{R}\) with eigenvalues λ_{0} and λ_{±} = λ_{1} ± iω, respectively. The oscillation frequency is denoted as ω.
In experiments, the coefficients s_{α}, decay rates λ_{0,1}, and oscillation frequency can be extracted by fitting the experimental data. By ignoring λ_{0,1}, we obtain the rescaled spin polarization \(\tilde{{{{\bf{s}}}}}({{{\bf{k}}}},t)\).
NMR sample
The experiment is performed on the nuclear magnetic resonance processor (NMR). The sample we used is the ^{13}Clabeled chloroform dissolved in rmacetone—d6. The ^{13}C spin is used as the working qubit, which is controlled by radiofrequency (RF) fields. The ^{1}H is decoupled throughout the experiment by Overhauser effect which can enhance the signal strength of ^{13}C.
Overhauser effect
Applying a weak RF field at the Larmor frequency of one nuclear spin for a sufficient duration may enhance the longitudinal magnetization of the others, this is the steadystate nuclear Overhauser effect (NOE). In modern NMR, the steadystate NOE is mainly exploited in heteronuclear spin systems, where the enhancement of magnetization is useful and dramatic.
For an ensemble of heteronuclear systems made up with a nuclei I with gyromagnetic ratio γ_{I} and a nuclei S with gyromagnetic ratio γ_{S}, with ∣γ_{I}∣ > ∣γ_{S}∣, the thermal equilibrium state of the heteronuclear system is
where β_{I}/β_{S} = γ_{I}/γ_{S}, \(\frac{1}{4}{\hat{{{{\rm{I}}}}}}_{z}=\hat{{\sigma }_{z}}\otimes \hat{1}\), \(\frac{1}{4}{\hat{{{{\rm{S}}}}}}_{z}=\hat{1}\otimes \hat{{\sigma }_{z}}\). Assume that a continuous RF field is applied at the Ispin Larmor frequency, inducing transitions across two pairs of energy levels. After sufficient time, the RF field equalizes the populations across the irradiated transitions. At that time, the populations settle into steadystate values, which do not change any more, as long as the RF field is left on. The steadystate spin density operator is
By comparing with thermal equilibrium Eq. (29), the Sspin magnetization is enhanced by factor ϵ_{NOE}. For our experiment I = ^{1}H and S = ^{13}C.
Noise configurations
For the stochastic average, it is clear that the more noise configurations are considered, the more reasonable result we obtain, as shown in Fig. 8. On the other hand, the large number of noise configurations takes a lot of time. We have performed numerical simulations, and found that the average of 5000 noise configurations can precisely approximate the nonHermitian dissipative quantum dynamics; see Fig. 8d. However, in NMR experiments, as the relaxation time is in the magnitude of seconds, a complete implementation of all 5000 noise configurations requires an extremely longrunning time that we cannot afford.
An alternative method to solve the issue is to reduce the number of noise configurations by numerical simulation prior to the implementation of experiments. We test different number of noise configurations, and plot their average dynamics in comparison with the ideal dynamics of the nonHermitian Hamiltonian; see Fig. 8ad. The simulated results show that with the increase of the number of noise configurations, the stochastic averaged spin polarization 〈〈〈σ(k, t)〉〉〉_{noise} would eventually approach to the spin polarization s(k, t) solved by the Lindblad master equation^{61}. The opposite is that with the decrease of the number of noise configurations, the performance of the approximation becomes more fluctuating (Fig. 8e). But the 〈〈〈σ(k, t)〉〉〉_{noise} always fluctuates above and below the theoretical spin polarization s(k, t). After a sufficient number of averaging, the stochastic averaged spin polarization that in the opposite side of theoretical value will be offset by each other. We randomly generated 5000 noise configurations N(t_{n}) that satisfy the normal distribution and separate these noise configurations into two subgroups in which the noise has an opposite effect on 〈〈〈σ(k, t)〉〉〉_{noise}. Then we use numerical simulations to select two noise configurations from these two subgroups respectively such that the 〈〈〈σ(k, t)〉〉〉_{noise} obtained from these four noise configurations precisely approximate the one obtained from the 5000 configurations (Fig. 8f). From experimental results and the corresponding fidelities, it can be concluded that the experiment is in excellent accordance with the simulations. And the theory and experiment results of each group of noise are in good agreement (Fig. 8g) So, it is somehow reasonable to utilize four noise configurations to replace a full description of the nonHermitian dynamics under 5000 noise configurations. We would like to emphasize that the above numerical simulations to reduce the number of noise configurations does not affect the applicability of the method. In many other quantum systems such as the superconducting circuits or nitrogenvacancy centers in diamond, the implementation of experiments takes a much shorter time, so they can realize the stochastic average with a larger number of noise configurations.
Experimental results vs. theoretical results
In this section, we show the agreement of our experimental results with the theoretical calculations. In Fig. 9, we compare the experimental spin textures with theoretical ones. Although the resolution of experimental data is lower than that of numerical calculations, the experimental results and the theoretical simulations reach the same conclusion. In Fig. 10, we show the numerical calculations for exceptional points and the corresponding winding numbers, which are consistent with our experimental results (Fig. 7).
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work is supported by the National Key Research and Development Program of China (2019YFA0308100, 2021YFA1400900), the National Natural Science Foundation of China (12075110, 11825401, 11975117, 11905099, 11875159, and U1801661), the Guangdong Basic and Applied Basic Research Foundation (2019A1515011383, 2021B1515020070), the Guangdong International Collaboration Program (2020A0505100001), the Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217, KQTD20190929173815000, JCYJ20200109140803865, JCYJ20170412152620376, RCYX20200714114522109, and JCYJ20180302174036418), the Open Project of Shenzhen Institute of Quantum Science and Engineering (Grant No.SIQSE202003), the Pengcheng Scholars, the Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044), and the Guangdong Provincial Key Laboratory (2019B121203002). L.Z. also acknowledges support from Agencia Estatal de Investigación (the R&D project CEX2019000910S, funded by MCIN/AEI/10.13039/501100011033, Plan National FIDEUA PID2019106901GBI00, FPI), Fundació Privada Cellex, Fundació MirPuig, and Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA program).
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D.L. and X.L. supervised the experiments. L.Z. and X.L. elaborated the theoretical framework. Z.L. and X.L. wrote the computer code and accomplished the NMR experiments. All authors analyzed the data, discussed the results, and wrote the manuscript. Z.L., L.Z., and X.L. who made equal contributions to this work are considered “cofirst authors".
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Lin, Z., Zhang, L., Long, X. et al. Experimental quantum simulation of nonHermitian dynamical topological states using stochastic Schrödinger equation. npj Quantum Inf 8, 77 (2022). https://doi.org/10.1038/s41534022005873
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DOI: https://doi.org/10.1038/s41534022005873
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