# Simulating dynamical quantum Hall effect with superconducting qubits

###### Abstract

We propose an experimental scheme to simulate the dynamical quantum Hall effect and the related interaction-induced topological transition with a superconducting-qubit array. We show that a one-dimensional Heisenberg model with tunable parameters can be realized in an array of superconducting qubits. The quantized plateaus, which is a feature of the dynamical quantum Hall effect, will emerge in the Berry curvature of the superconducting qubits as a function of the coupling strength between nearest neighbor qubits. We numerically calculate the Berry curvatures of two-, four- and six-qubit arrays, and find that the interaction-induced topological transition can be easily observed with the simplest two-qubit array. Furthermore, we analyze some practical conditions in typical experiments for observing such dynamical quantum Hall effect.

###### pacs:

03.67.Ac, 03.65.Vf, 73.43.-f, 74.81.Fa## I introduction

The quantum Hall effect (QHE) is one of the most remarkable phenomena in condensed matter physics Klitzing1980 ; Tsui . The basic experimental fact characterizing QHE is that the non-diagonal conductivity is quantized in the form of with being an integer (the integer QHE) or a fractional number (the fractional QHE). The integer is a topological invariant which can be expressed as the integral of the Berry curvature Berry over the momentum space Thouless1982 ; Niu1985 . The Berry curvature and its associated Berry phase have many additional applications in condensed matter physics Xiao ; Zhu2006 and quantum computation Zanardi ; Sjoqvist ; Zhu2002 . Usually the Berry phase is measured with the interference experiments. Recently, it was proposed that the Berry curvature and hence the Berry phase in generic systems can be detected as a non-adiabatic response on physical observables to the rate of change of an external parameter Gritsev2012 ; Avron2011 . This phenomenon can be interpreted as a dynamical QHE in a parameter space, while the conventional QHE is a particular example of the general relation if one views the electric field as a rate of change of the vector potential Gritsev2012 . This work opens up the possibility to study the QHE in parameter space and to measure the Berry phase in many-body systems.

On the other hand, superconducting qubits have become one of the leading systems to study the Berry phase and to simulate some interesting phenomena emerged in condensed matter physics Georgescu2014 . The Berry phase Leek2007 , the non-Abelian non-adiabatic geometric gates Abdumalikov , and the geometric Landau-Zener interference Tan were experimentally demonstrated with superconducting qubits. Furthermore, a topological transition characterized by the change of the Chern number was also experimentally observed Schroer2014 ; Roushan2014 . These studies suggest that the superconducting qubit system can be a promising system for further exploring rich topological features of single-particle and many-body physics.

In this paper, we propose an experimental scheme to simulate the dynamical QHE and the related interaction-induced topological transition with a superconducting-qubit array. The one-dimensional (1D) Heisenberg spin chain was proposed to realize with superconducting qubits Pinto2010 ; Paraoanu2014 . We first extend this approach to show that an almost isotropic interaction (i.e., in Eq. (4)) between the nearest neighbor superconducting qubits can be achieved by coupling phase qubits with the Josephson junctions controlled with the bias current. One of the advantages of the system is that all parameters in this 1D Heisenberg model are controllable and tunable in experiments. We then show that the dynamical QHE and the related interaction-induced topological transition can be observed in the system. We numerically calculate the Berry curvatures of two-, four- and six-qubit arrays, and find that the interaction-induced topological transition can be easily observed with the simplest two-qubit array. We also discuss some practical conditions for observing the dynamical QHE in this system, such as the limit of ramp velocity, the control errors in experiments and the decoherence effects for the realistic open-system conditions.

The rest of this paper is organized as follows: Section II introduces our proposed superconducting phase qubit array and the realization of the required spin Hamiltonian. Section III presents our results for observing the dynamical QHE and the related interaction-induced topological transition in the proposed system. In Sec. IV, we analyze the ramp velocity limit, the robustness of our scheme against the control errors and the decoherence effects for realistic conditions, and finally present our conclusions.

## Ii system and Hamiltonian

It was demonstrated that the dynamical QHE can emerge in a D Heisenberg spin chain model with tunable parameters Gritsev2012 . We consider the D Heisenberg spin chain model with an external magnetic field:

(1) |

where stands for Pauli matrices, is the isotropic coupling constant between the nearest-neighbor spins, is the external magnetic field, and is the size of the spin chain. In the following, we will show that this Hamiltonian with tunable coupling constants can be realized in an array of superconducting phase qubits and the related dynamical QHE can be observed in this system.

The schematic diagram of the whole system we consider is shown in Fig 1. It is an array of superconducting phase qubits coupled with Josephson junctions. The phase qubit constitutes an ”atom-like” two-level system. The truncated Hamiltonian of the lowest two levels in the energy bases is , where represents the energy difference between and and is the Pauli operator in the direction Orlando1999 ; Tan . For simplicity, we assume the same parameters for all the phase qubits (i.e., ). Moreover, the state of each qubit can be controlled by microwaves. In the rotating frame of an applied microwave with the frequency , the Hamiltonian for the qubits can be written as Georgescu2014 ; Paraoanu2014

(2) |

Here the interacting part of the Hamiltonian will be addressed later, and is an effective magnetic field induced by the microwave and can be parameterized as Schroer2014 ; Roushan2014 ; Tan

(3) |

Here the parameter represents the phase of the applied microwave, is the Rabi oscillation frequency proportional to the amplitude of the microwave, and is the detuning with being the frequency of the microwave and being the mixing angle. The mixing angle will be used as the quench parameter for observing the dynamical QHE in the following.

As shown in Fig. 1, the interaction between nearest-neighbor qubits and is realized by the inductances and and the Josephson junction characterized by capacitance . The two qubits also contain the capacitances and , while the circuit has a negative mutual inductance and a tunable bias current . Thus in this system, the coupling strengths can be tuned via the bias current of the coupled Josephson junctions and the Hamiltonian of the interacting part can be written as Pinto2010 ; Paraoanu2014

(4) |

where the coupling strengths along the three spin directions are respectively given by Pinto2010

(5) | ||||

(6) |

Here with and being the critical current of inter-Josephson junction, the renormalization parameters (the mutual and coupling inductances) are and

From Eqs. (5,6), we calculate the coupling coefficients and for the typical homogenous parameters GHz, pF, nH, nH, nH, and A Pinto2010 . In this case, we get homogenous (qubit-independent) coupling strengths and thus the qubit label in the coupling strengths () will be omitted for simplicity hereafter. The results are plotted in Fig. 2, showing that if we adjust the bias current from 0 to , the coupling strength will continuously and monotonically decrease from about MHz to zero. Furthermore, when the bias current is less than the critical current the ratio remains in the region about . As we will show in the following, this parameter region already allows the observation of the dynamical QHE.

## Iii Simulating dynamical quantum Hall effect

The topological features of the superconducting qubit system can be probed by measuring the Berry curvature, while Fig. 3 depicts a typical sequence used to measure the Berry curvature. To demonstrate the dynamical QHE in this system, we follow the proposal in Ref. Gritsev2012 to analyze the quantized response of the system to a rotating magnetic field. We consider all the superconducting qubits initially in the ground state with , and then ramp the system with fixed to undergo a quasi-adiabatic evolution by varying the mixing angle for a ramp time , where denotes the ramp velocity. At the end of such a ramp, the velocity of the -component of the magnetic field is exactly , and we can measure the Berry curvature of the system. We note that this choice of ramping field guarantees that the angular velocity is turned on smoothly and the system is not excited at the beginning of the evolution Gritsev2012 .

During the ramping process, the three components of the effective magnetic field are depicted in Fig. 3 (a). The generalized force for the full Hamiltonian , which is measured at , is along the latitude direction (at the point of measurement it is along -axis) and given by

(7) |

while the quench velocity is along to the longitude direction. Then we can obtain the Berry curvature within the linear response approximation Gritsev2012 ; Avron2011 ,

(8) |

In experiments, one can measure the of each superconducting qubit, and then the Berry curvature can be derived by substituting the results into Eq. (8). In other words, the qubits system in the described evolution progress is initially prepared in the ground state and then is slowly and smoothly driven along the direction, as shown in Fig. 3(b). The generalized force along the orthogonal direction is measured as a linear response to the ramping magnetic field. We will see the quantization of this response in the following, and in this sense it is called dynamical QHE Gritsev2012 .

The simplest system to observe the dynamical QHE and its related interaction-induced topological transition should be a two-qubit system. Therefore we first address whether one can observe this phenomenon in an array with two superconducting qubits. To this end, we numerically calculate the Berry curvature in a two-qubit array as a function of the bias current for the described ramp process by time-dependent exact diagonalization ED . The results are plotted in Fig. 4(a), where the parameters are the same with those in Fig. 2. The red solid line in Fig. 4(a) shows that although the interaction strengths are not exactly isotropic in our superconducting qubit system, the plateaus in the Berry curvature are strictly stable at and , and the transition between the two plateaus is very sharp. For comparison, we also calculate the Berry curvature for the isotropic coupling case [the blue dashed line in Fig. 4(a)], where we choose an isotropic coupling strength determined by (here depend on the bias current as shown in Fig. 2). From Fig. 4(a), it is clear that for the chosen typical parameters the difference of the Berry curvatures between the isotropic and anisotropic cases can be neglected.

Now we turn to address the dynamical QHE and the interaction-induced topological transition in an -qubit array. For this 1D Heisenberg spin chain, the plateaus in the Berry curvature should appear in an integer for an even Gritsev2012 . We have numerically confirmed this phenomenon for a four- and six-qubit array, with for typical parameters shown as the blue dashed lines in Figs. 4(b) and 4(d), respectively. We then further check whether these multi-plateaus can be observed in this superconducting qubit system. For the same corresponding parameters, we calculate the Berry curvature as a function of in the region , which corresponds to the coupling strength in the region [34, 40] MHz as shown in Fig. 2. The results are given by the red lines in Figs. 4(b) and 4(d), where we find that only two quantized plateaus with a topological transition appear and other quantized plateaus could not be observed. One simple approach to solve this problem is to simultaneously change the magnetic field strength and the bias current . For instance, we can choose MHz in simulations, and then the obtained Berry curvature as a function of (and ) for four-qubit and six-qubit are shown in Figs. 4(c) and 4(e), respectively. It is clear that all quantized plateaus can be observed in this approach. In the above calculation, the amplitude of given by the relation equation is yet to be optimized and it is about MHz at . However, the required amplitude of the effective magnetic field to observe all the quantized plateaus can be much smaller after the optimization.

## Iv Discussions and conclusions

In the previous calculations, the Berry curvature is considered to be a linear response to the ramp velocity . In general, the magnetization (the generalized force) is determined by Gritsev2012 ; Avron2011 , where the constant term gives the value of the magnetization in the adiabatic limit and in our cases. The linear response theory breaks down when the velocity is too large to neglect the term related to . To check the velocity limit in this linear response theory, we numerically calculate the Berry curvature as a function of the ramp time for a two-qubit array, with the results for parameters and MHz being plotted in Fig. 5. We can see that the Berry curvature saturates to nearly one when ns and becomes very stable when ns. In addition, the magnetization is plotted in the inset of Fig. 5 as a function of the finial ramp velocity , which further shows the linear response approximation works well within . Therefore, to observe the quantized plateaus in Fig. 4(a), the ramp velocity should be slower than , corresponding to the ramp time longer than ns. We also simulate the same procedures for the four-qubit and six-qubit arrays and find that the results are similar to those in Fig. 5. Thus the velocity limit to observe the quantized plateaus does not change much for arrays with different number of qubits.

Then we further study the robustness of the quantized plateaus of the Berry curvature against the control errors which stem from the fluctuations of the parameters in the Hamiltonian (2). We assume and , with and randomly distributing in the region (here describes the fluctuation strength). For a single realization with randomly chosen and , we calculate the corresponding as that in Fig. 4(a) and then we can obtain the averaged Berry curvature , where denotes the sampling number. The averaged Berry curvature for the two-qubit case as a function of is plotted in Fig. 6(a). We find that the plateaus are still stable when the parameter fluctuation strength is less than about , even though their transition is slightly smoothed by the fluctuation. Furthermore, we calculate the corresponding Chern number by integrating the Berry curvature in the sphere in Fig. 6(b). As we expected, the Chern number is more robust and the quantized plateaus there are more significant due to the averaging over different runs of -ramping with the parameter fluctuations. In addition, we also plot the energy gap between the ground state and the first excited state of the two-qubit array in Fig. 6(b). We can see that the gap closes at the topological transition point.

We now turn to discuss the decoherence effects in our system for the realistic open-system conditions. For simplicity, we assume that each superconducting qubit of the system interacts independently with the environment, which is commonly modeled as a bath of oscillators. The quantum dynamics of the system is thus described by the master equation book

(9) |

where the density matrix is spanned by the -qubit basis and the Lindblad superoperator describes the decoherence due to the independent interaction between each qubit and the bath. We further assume the weak qubit-bath interaction and the Markovian limit and thus the Lindblad superoperator can be written as book . Here the first two terms describe the energy relaxation progress with parameter and the third term describe the pure dephasing progress with parameter , and the effective boson number on each qubit depends on the temperature of the bath with (here is the Boltzmann constant). In superconducting qubit system, we have because for mK and is on the order of gigahertz in practical experiments Tan ; Schroer2014 ; Roushan2014 . Then the usually measured relaxation time and dephasing time of each qubit are determined by and book , respectively.

The additional timescale for measurement is another issue one should consider for finite decoherence time. We assume that each phase qubit in the array can be manipulated and measured independently Roushan2014 ; Lucero2012 . Since the qubits can only be naturally read out in the basis (i.e. the measurement), an additional spin rotation

for each qubit (effectively an operation in experiments Schroer2014 ; Roushan2014 ) have to be inserted in order to measure after the ramp. This rotation can be achieved by additional microwave pulses Tan ; Roushan2014 with the duration time ns for the cases with MHz in Figs. 4 and 5. Finally, the measurement of each qubit requires a duration time , which is typically several nanoseconds Tan ; Lucero2012 . So the total time required for measurement is around ns. Since the measurement fidelity for each phase qubit in the coupled system can be more than Lucero2012 , we do not further consider the measurement errors.

To see the decoherence effects in the dynamical QHE in our proposed system, we take the two-qubit array for example and numerically simulate the whole progress with the ramp and measurement sequences by calculate the master equation (see the Appendix for details). For simplicity in our simulations, we treat the evolution of the qubits in the whole measurement progress with time ns as free evolution under decoherence. In addition, the relaxation time and dephasing time of single phase qubit are usually longer than those of multi-qubit in the coupled system; this effect is somehow contained in the master equation, which includes the increases of decoherence channels and decoherence rates (see Eq. (A8) in the Appendix). We estimate that the effective times and of two qubits are about times smaller than those of a single qubit in the master equation ( and ), thus we will choose typical decoherence times in simulations from single phase qubit experiments.

We first take the decoherence times ns and ns of each qubit in experiments of phase qubits Whittaker as a typical example. From Fig. 5, we know that the linear response condition satisfies when the ramp time ns. So we numerically calculate the Berry curvature with ns and the result is plotted as the green triangles in Fig. 4(a). In this case, the two plateaus in the Berry curvature are respectively near and (the difference is about and the transition point remains), as expected. However, we find that the two plateaus in the Berry curvature are gradually shifted from and when the ramp time becomes longer. For instance, the difference between the two plateaus decreases to about for the ramp time ns, which is shown as yellow squares in Fig. 4(a). This is due to the fact that the total evolution time (i.e., 110 ns) of the system is now comparable with the effective decoherence times (the effective times ns and ns) and then the Chern number is no longer a well-defined topological index note . Therefore, the observation of the dynamical QHE is crucially dependent on the long decoherence time since the Berry curvature (which associates with the Berry phase factor) has no classical correspondence. To demonstrate the topological features of the dynamical QHE more clearly (or in a longer ramp time), we should make improvements in coherence time for superconducting qubits in experiments. In current technology, the relaxation time of the phase qubit can be as long as s Whittaker ; Patel . One can use dynamical decoupling to increase the dephasing time up to the limit DD . Thus, we also numerically calculate the result for s, and the result is plotted as black circles in Fig. 4(a). It clearly shows that the decoherence effects are almost negligible in this case.

In conclusion, we have proposed an experimental scheme to simulate the dynamical QHE and the related interaction-induced topological transition with a superconducting-qubit array. We find that the typical topological features can even be observed in the simplest two-qubit array under practical experimental conditions.

## V Acknowledgements

We thank Z.-Y. Xue and C.-J. Shan for helpful discussions. This work was supported by the NSFC (Grants No. 11125417 and No. 11474153), the SKPBR of China (Grants No. 2011CB922104), and the PCSIRT (Grant No. IRT1243). D.W.Z. acknowledges support from the postdoctoral fellowship of HKU.

Appendix: The master equation for the two-qubit case

In this Appendix, we derive the master equation for two-qubit array with the Hamiltonian

(A1) |

where the components of the effective magnetic field are given by Eq. (3) in the text. In the two-qubit basis , the Hamiltonian matrix can be written as

(A2) |

where the matrix elements are given by

(A3) |

The quantum dynamics of the system is described by the master equation

(A4) |

where the density matrix is denoted by

(A5) |

We consider the system in the Markovian and low temperature limit, and thus the Lindblad superoperator can be written as

(A6) |

Here the relaxation rate and pure dephsing rate are determined by the measured decoherence times: and . Using the expansions and , one can obtain the Lindblad superoperators:

(A7) |

By substituting Eq. (A7) into Eq. (A4), one can obtain the master equation as

(A8) |

After the evolution of the system with the ramp time and the total measurement time , one can obtain the final polarization along the direction at by tracing the final density matrix governed by Eqs. (A8) as

(A9) |

For simplicity in our simulations, we treat the evolution of the qubits in the whole measurement progress as free evolution under decoherence.

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