# Floquet dynamical quantum phase transitions

###### Abstract

Dynamical quantum phase transitions (DQPTs) are manifested by time-domain nonanalytic behaviors of many-body systems. Introducing a quench is so far understood as a typical scenario to induce DQPTs. In this work, we discover a novel type of DQPTs, termed “Floquet DQPTs”, as intrinsic features of systems with periodic time modulation. Floquet DQPTs occur within each period of continuous driving, without the need for any quenches. In particular, in a harmonically driven spin chain model, we find analytically the existence of Floquet DQPTs in and only in a parameter regime hosting a certain nontrivial Floquet topological phase. The Floquet DQPTs are further characterized by a dynamical topological invariant defined as the winding number of the Pancharatnam geometric phase versus quasimomentum. These findings are experimentally demonstrated with a single spin in diamond. This work thus opens a door for future studies of DQPTs in connection with topological matter.

DQPTs are often associated with quantum quenches, a protocol in which parameters of a Hamiltonian are suddenly changed HeylPRL2013 (); PollmannPRE2010 (). A quantum quench across an equilibrium quantum critical point may induce a DQPT. If pre-quench and post-quench systems are in topologically distinct phases, DQPTs may also be characterized by dynamical topological invariants DTOP1 (); DTOP2 (); DTOP3 (). As a promising approach to classify quantum states of matter in nonequilibrium situations, DQPTs have been theoretically explored in both closed and open quantum systems at different physical dimensions DQPTRev1 (); DQPTRev2 (); ZhouArXiv2017 (). Experimentally, DQPTs have been observed in trapped ions DQPTExp1 (); DQPTExp2 (), cold atoms DQPTExp3 (); DQPTExp4 (), superconducting qubits DQPTExp5 (), nanomechanical oscillators DQPTExp6 (), and photonic quantum walks DQPTExp7 (); DQPTExp8 ().

To date, in most studies of DQPTs, a quantum quench acts as a trigger for initiating nonequilibrium dynamics and then exposing the underlying topological features. However, DQPTs under more general nonequilibrium manipulations are still largely unexplored SharmaPRB2016 (); PuskarovSPP2016 (); BhattacharyaPRB2017 (). In particular, because the dynamics of systems under time-periodic modulations has led to fascinating discoveries like Floquet topological states Lindner2011 (); FTIRev (); LW2014 (); HoPRL2012 (); LiPRL2018 () and discrete time crystals TCRev (); Yao2018 (); BomantaraPRL (), it is urgent to investigate how DQPTs may occur in such Floquet systems. Along this avenue, there have been scattered studies, but still with the notion that DQPTs are best aroused by a quench to some system parameters FDQPT1 (); FDQPT2 (). Here we introduce a novel class of DQPTs, termed Floquet DQPTs, which can be regarded as intrinsic features of systems with time-periodic modulations. As schematically shown in Fig. 1, the Floquet DQPTs we discovered occur within each period of a continuous driving field, without the need for any quenches. In the model investigated below, the occurrence of Floquet DQPTs and the emergence of Floquet topological phases are in the same parameter regime. We further perform a quantum simulation experiment using a single electron spin in diamond to verify our main theoretical findings.

For a periodically modulated system with the modulation period and the frequency, the time-dependent Schrödinger equation yields solutions , with the Floquet mode and the quasienergy HolthausFloTut (). Though a Floquet mode by definition becomes parallel to itself at multiple ’s, it still evolves nontrivially within each driving period (hence the micromotion dynamics). More specifically, consider a periodically driven system prepared initially at a many-particle Floquet state , which is the product state of Floquet modes filling the quasienergy band . The return amplitude of the system at is then . In the thermodynamic limit, the rate function is defined as

(1) |

where is the return probability, and is the number of degrees of freedom, such as the number of lattice sites in a chain. The rate function plays the role of a dynamical free energy HeylPRL2013 (); SM (), which is periodic in time with period . Now if there exists a critical time , at which is orthogonal to , one finds . Then as defined in Eq. (1) or its time derivatives becomes nonanalytic at .

We now focus on a harmonically driven spin chain described by , where are quantum spin- variables localized at site of the chain; , , and are system parameters. The associated Bloch Hamiltonian is found to be SM ()

(2) |

where is the quasimomentum. The dynamics governed by is analytically solvable and has been already experimentally realized in other quantum simulation studies GTPExp (). The bulk dynamics of the system can be solved because each occupied component of the system evolves independently. The Floquet state at each is given by . Here are quasienergy dispersions of two Floquet bands with the Floquet modes , is responsible for the micromotion dynamics, and are the eigenstates of the static Hamiltonian SM ().

The micromotion dynamics arising from is the rotation around the axis (with frequency ) on the Bloch sphere SM (). If there exists such that lies in the - plane, then the micromotion dynamics can always rotate this initial state to a state pointing at exactly the opposite direction at for . That is, at the time-evolving state at this quasimomentum becomes orthogonal to its initial state. Assuming that the collective initial state of the system fills one of the Floquet bands with band indices or . At a later time , the return amplitude of the system is given by , with . It can be shown that, under the condition

(3) |

there exists a critical momentum with , such that lies in the - plane SM () and consequently . Under the condition (3), the rate function becomes non-analytic at each critical time . It can be also shown that there exists a topological order parameter given by the winding number of the geometric phase over the Brillouin zone, namely,

(4) |

where is the Pancharatnam geometric phase acquired during the evolution of a Floquet state at quasimomentum SM (). When time passes the critical time of a Floquet DQPT, makes a quantized jump. The winding number as a dynamical topological invariant thus reflects the topological nature of Floquet DQPTs identified in the spin chain model here.

Before proceeding to experiment, three traits of Floquet DQPTs deserve to be highlighted. Firstly, no quantum quench from one equilibrium phase to another is required in our proposal, as the continuous driving field suffices to generate non-analyticities in the time domain. Secondly, the Floquet DQPTs introduced in this work repeat periodically in time, whereas the DQPTs following a single quench are usually visible only in transient time windows due to the decay of the rate function. Floquet DQPTs are therefore more accessible to experiments. Thirdly, precisely under the same condition as in (3), our spin chain model is found to reside in a nontrivial “chiral-symmetric” Floquet topological phase SM (), as featured by two winding numbers defined with the one-period evolution operator in two chiral-symmetric time frames AsbothPRB2013 (). These two winding numbers can be used to predict topological edge states under open boundary conditions. As such, the Floquet DQPTs reported here not only provide an indirect probe to study these intriguing Floquet topological phases, but also suggest a remarkable connection between jumps in the dynamical topological invariant and the emergence of a Floquet topological phase SM ().

A negatively charged nitrogen-vacancy (NV) center in diamond is used in our experiment to simulate GTPExp (), the many-body Hamiltonian in its quasimomentum representation. As shown in Fig. 2(a), the NV center is composed of one substitutional nitrogen atom and an adjacent vacancy Doherty (); Schirhagl (); Prawer (); Wrachtrup (). In our experiment, an external static magnetic field around G is parallel to the NV symmetry axis, which enables both the NV electron spin and the host N nuclear spin to be polarized by optical excitation Jacques (); Sar (). The Hamiltonian of the electronic ground state of the NV center with a static magnetic field applied along the NV axis (also the axis) is , where is the angular momentum operator for spin-1, MHz is the zero-field splitting, and MHz/G is the gyromagnetic ratio of the NV electron. As illustrated in Fig. 2(b), microwaves generated by an arbitrary waveform generator drives the transition between the electronic levels and which compose a qubit. The level remains idle due to large detuning. The probability in can be read out via fluorescence detection during optical excitation. All the optical procedures are performed on a home-built confocal microscope, and a solid immersion lens is etched on the diamond above the NV center to enhance the fluorescence collection Robledo (); Rong ().

In our experiment, the evolution at different is performed separately in different experimental runs. The pulse sequence for each is sketched in Fig. 2(c). At first, the qubit is polarized to the state by a laser pulse [green bar in the preparation section in Fig. 2(c)]. A resonant microwave pulse [orange bar in Fig. 2(c)] is then applied to prepared the initial state as , which occupies the lower quasienergy band. The Bloch vector of this initial state is along the direction . The parameters adopted in our experiment are MHz, MHz, and MHz. For MHz, the prepared initial state lies in the - plane for . This is not the case for any if MHz, thus excluding DQPTs.

Upon initial-state preparation, the qubit is left to evolve under , namely, in the presence of a field with the transverse component and the longitudinal component . The field is rotated around the axis with the angular frequency , which is implemented by applying a microwave pulse starting from , as depicted by the blue bar in Fig. 2(c).

The evolution governed by lasts for some duration , and then the return probability needs to be measured. The measurement procedure begins with a resonant microwave pulse [the magenta bar in Fig. 2(c)], which steers the direction of to the direction. It is followed by a laser pulse [green bar in the measurement section of Fig. 2(c)] together with fluorescence detection. The later is collected via two counting windows represented by the red bars in Fig. 2(c), with the first recording the signal while the second recording the reference SM (). The fluorescence collection amounts to the measurement of the probability in , and this effect combined with the last microwave pulse is equivalent to the measurement of .

The above sequence is performed for a series of within 0.6 s, and is iterated five hundred thousand times to obtain the expectation value. One can then get as a function of . This procedure is repeated for different values of . The experimental data with MHz and MHz are illustrated in the upper panels of Fig. 3(a) and Fig. 3(b), respectively. The experimental results agree with the theoretical ones which are shown in the lower panels. The results in Fig. 3(a) confirm that, in the case of MHz where the DQPT condition in Eq. (3) is satisfied, the return probability at the critical momentum vanishes at the critical times such as , , and . The DQPT condition in Eq. (3) is not satisfied in the case of MHz. The return probability never vanishes in this case as confirmed by the results in Fig. 3(b).

Numerical integration over based on the negative logarithm of these experimental data yields the experimental values of the rate function . Under the condition in Eq. (3) for Floquet DQPTs, the rate function behaves non-analytically at critical times, as shown by the kinks in Fig. 3(c) noerrorbar (). By contrast, the rate function stretches smoothly over time without non-analyticity when the DQPT condition is not satisfied, as shown in Fig. 3(d). This further demonstrates that Floquet DQPTs can be observed through the rate function.

In order to investigate the topological character of Floquet DQPTs, we next extract the Pancharatnam geometric phase of the time-evolving state from the expectation values of , , and measured in our experiment SM (). The pulse sequence for measuring these expectation values is similar to that in Fig. 2(c) except for the last microwave pulse. This resonant microwave pulse rotate the direction of or to for the measurement of or , respectively. The pulse is not needed for the measurement of . The experimental and theoretical values of the geometric phases for MHz and MHz are illustrated in Fig. 4(a) and (b). Geometric phases with difference are equivalent and we have thus set them in the range from to . The net discontinuous jump of the geometric phase along the dimension is a signature of the winding of the geometric phase with quasimomentum. The number of winding then yields defined in Eq. (4) as a dynamical topological order parameter. In the case of Fig. 4(a) with Floquet DQPTs, the measured geometric phases manifest a jump from to at the first critical time . The number of net jumps increases by one once a new critical time is passed, in particular at and . Therefore, increases from one at to three at . By contrast, in the case of Fig. 4(b) where the DQPT condition in Eq. (3) is not satisfied and hence there are no DQPTs, the measured geometric phases has no net jumps along the dimension. Even when there is a discontinuous jump from to , it is cancelled by a reversed jump from to , resulting in no change in . Some noisy but insignificant patterns in the experimental results are mainly due to the imperfection of the microwave pulses. Nevertheless, even in the presence of such experimental errors, all the principal characteristics of Floquet DQPTs in connection with jumps in the dynamical topological order parameter have been clearly demonstrated.

In conclusion, we have discovered and experimentally demonstrated a new class of DQPTs as intrinsic features of quantum systems subject to smooth and periodic time modulations. Floquet DQPTs here are found to have two-fold topological nature. Firstly, their occurrence leads to jumps in a dynamical topological invariant. Secondly, the system parameters yielding Floquet DQPTs lie in a regime accommodating topologically nontrivial Floquet phases. Our work thus opens a door for future studies of DQPTs in connection with topological matter.

###### Acknowledgements.

The authors at University of Science and Technology of China are supported by the National Key RD Program of China (Grant No. 2018YFA0306600, No. 2016YFA0502400), the National Natural Science Foundation of China (Grants No. 81788101, No. 11227901, No. 31470835, No. 91636217, and No. 11722544), the CAS (Grants No. GJJSTD20170001, No. QYZDY-SSW-SLH004, and No. YIPA2015370), Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000), the CEBioM, and the Fundamental Research Funds for the Central Universities (WK2340000064). J.G. acknowledges support from the Singapore NRF Grant No. NRF-NRFI2017-04 (WBS No. R-144-000-378-281) and by the Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112). L.Z. acknowledges support from the Young Talents Project at Ocean University of China (Grant No. 861801013196). K.Y., L.Z., W.M., and X.K. contributed equally to this work.## References

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Supplemental Material

## Appendix A I. Theory

In this supplementary note, we present in detail our theory of Floquet dynamical quantum phase transitions (DQPTs). We first briefly review the basics of conventional DQPTs and Floquet theory. Following that, we introduce our definition of Floquet DQPTs and discuss its general features. To demonstrate these features, we study the periodically driven spin chain model whose Hamiltonian in thermodynamic limit can be mapped to a qubit in a rotating field. We further solve this driven qubit model analytically and describe its Floquet DQPTs by (i) the Fisher zeros of the Loschmidt (return) amplitude, (ii) the rate function of Loschmidt echo (return probability), (iii) the pattern of a non-adiabatic, non-cyclic geometric phase, and (iv) a dynamical topological winding number. We also reveal the relation between Floquet DQPTs found in our model and its underlying Floquet topological phases. Finally, we illustrate numerically our theoretical predictions in three examples.

### a.1 A. Elements of DQPTs

In this section, we briefly review the definition of DQPTs for quenched evolutions. According to Ref. HeylPRL2013 (), a DQPT means that the time derivative of certain observable at a given order shows a jump or a singularity in time. It is therefore a real-time non-analytic signature in dynamics. The central quantity in the description of DQPTs is the return amplitude, defined as

(S1) |

where is the initial state of the system, usually chosen to be the ground state of some many-body Hamiltonian , and is the Hamiltonian governing the evolution of the system after the initial time . A nonequilibrium dynamical process is generated if , which can be realized by performing a quantum quench from to at .

Formally, mimics the so-called boundary partition function in equilibrium statistical mechanics, which can be seen by rotating the time to the complex plane (), yielding DQPTRev1 (). The zeros of are called Fisher zeros. They form a dense set in thermodynamic limit. A real-time Fisher zero of appears when this set crosses the imaginary time axis at some critical time , where the rate function of return probability,

(S2) |

or its time derivative becomes non-analytic as a function of time. Here is the number of degrees of freedom of the system. Mathematically, this is closely related to the microscopic Lee-Yang theory of phase transitions. Therefore and are sometimes called dynamical partition function and dynamical free energy, respectively, whereas is called the critical time of a DQPT, analogous to the critical temperature of a thermal phase transition. Theoretical studies have shown that DQPTs usually happen in dynamics following a quench across the equilibrium quantum phase transition point of the prequench Hamiltonian DQPTRev1 (). Experimentally, DQPTs characterized by kinks in return rates at critical times were first observed in a one-dimensional chain of trapped ions DQPTExp1 ().

### a.2 B. Elements of Floquet theory

In this section, we recap some basics of Floquet theory HolthausFloTut () that will be used in this study. A Floquet system is described by a time periodic Hamiltonian , where is the driving period and is the driving frequency. A complete set of solutions of the time-dependent Schrödinger equation is given by the Floquet states , with each of them satisfying (take )

(S3) |

The Floquet state can be further decomposed as , where is called the quasienergy and is called the Floquet mode. A Floquet mode is time periodic, i.e., . All such Floquet modes form an orthonormal, complete set at any time , i.e.,

(S4) |

In terms of the quasienergies and Floquet modes, the time evolution operator of the system from an initial time to a later time can be expressed as

(S5) |

It is not hard to verify that and

(S6) |

as expected. One may further write a Floquet mode as , where is called the micromotion operator.

### a.3 C. Floquet DQPTs — definition and properties

We will introduce our definition of Floquet DQPTs in this section. Consider a system described by a time-periodic Hamiltonian , which is prepared initially () at an -particle Floquet state , given by Floquet modes filling the quasienergy band . At a later time , the return amplitude of the system to its initial state is given by

(S7) |

The time evolution operator , where denotes the time ordering operator. In terms of quasienergies and Floquet modes, can be expressed as Eq. (S5). Then we can write the return amplitude for each Floquet mode involved in the dynamcis as

(S8) |

The corresponding return probability to the initial state is given by

(S9) |

with the micromotion operator introduced in the previous section. In thermodynamic limit, the rate function of return probability reads

(S10) |

where is the number of degrees of freedom (e.g., number of particles) of the system and the summation is taken over the occupied states. In our construction of Floquet DQPTs, plays the role of a dynamical free energy. If there exists a critical time , at which is orthogonal to the initial state , will vanishes. Then as defined in Eq. (S10) or its time derivatives will become non-analytic as a function of time . We refer to this as a Floquet DQPT. From Eq. (S9), one may interpret a Floquet DQPT as originated from the interplay between the stroboscopic nature of the system encoded in its Floquet mode , and the micromotion within a driving period described by .

Compared with the DQPTs following a sudden quench HeylPRL2013 (), a notable difference of Floquet DQPTs is that they happen in time periodically. This is not hard to see from Eq. (S10), since

(S11) |

Therefore, if is a critical time, then is also a critical time for any , with taking equal values at these critical times. Practically, this means that Floquet DQPTs will not just happen in a transient time scale as the DQPTs following a single quench, but will be observable with equal strength in a much wider time window thanks to the periodic drivings applied to the system. Furthermore, the driving fields also yield more freedom to control Floquet DQPTs in the system.

In the following sections of this theoretical part, we will study an analytically solvable driven spin chain model as described in the main text, which possesses the defining features of Floquet DQPTs. We will further map the Hamiltonian of this driven spin chain to the parameter space of a qubit in a rotating magnetic field, which allows us to develop a systematic description of Floquet DQPTs in this system (see Table 1 for an outline).

Concepts | Expressions | Annotations |
---|---|---|

Spin chain | ||

Hamiltonian | ||

Lattice | Obtained from the spin chain Hamiltonian through | |

Hamiltonian | Jordan-Wigner and Fourier transforms, | |

Bloch Hamiltonian | Rotation | |

in rotating frame | ||

Quasienergies and | and are eigenvalues | |

Floquet eigenstates | and eigenstates of | |

Initial | Fill the Floquet band or | |

state | ||

Return | “Dynamical partition function” | |

amplitude | ||

Rate | “Dynamical free energy” | |

function | ||

Critical time and | Floquet DQPTs happen at each | |

critical momentum | if there exists such a | |

Pancharatnam | Total phase minus dynamical phase | |

geometric phase | ||

Dynamical | Topological order parameter | |

winding number | of the Floquet DQPTs |

### a.4 D. Floquet DQPTs in a periodically driven spin chain

We start with a periodically driven spin chain with the Hamiltonian

(S12) | ||||

(S13) |

where , and are real parameters, is the driving frequency with being the driving period, and are quantum spin- variables located at site of the spin chain. The terms on the right hand side of Eq. (S12) describes the usual XY model with time-dependent nearest-neighbor couplings, whereas the terms in Eq. (S13) are some anomalous coupling terms. In terms of Pauli spin variables , the spin chain Hamiltonian can be expressed as

(S14) | ||||

(S15) |

Pauli spins are anticommute at the same site, i.e. , but commute at different sites, i.e. , with . In the following, we first fermionize by applying the Jordan-Wigner transformation, and then apply the Fourier transform to the resulting free fermionic lattice model to obtain a single qubit Hamiltonian in momentum representation under periodic boundary conditions.

#### a.4.1 1. Fermionization of the spin chain

The Jordan-Wigner transformation is a non-local transformation. It is often used to exactly solve 1D spin chains such as the Ising and XY models by transforming the spin operators to fermionic operators, and then doing diagonalizations in the fermionic basis FranchiniBook ().

We first express and in terms of spin raising and lowering operators as

(S16) |

In terms of and , the spin chain Hamiltonian reads

(S17) |

The Jordan-Wigner transformation (JWT) is defined as

(S18) |

where the fermionic creation and annihilation operators and satisfy the anti-commutation relations

(S19) |

Using the JWT, we find

(S20) | ||||

(S21) |

(S22) | ||||

and

(S23) |

Using these relations, we find the fermionized spin chain model to be

(S24) |

Dropping the constant term , which only introduces a global shift to the energy, we arrive at the following quadratic fermionic Hamiltonian

(S25) |

The first term on the RHS of Eq. (S25) describes a nearest neighbor (NN) hopping of fermions with hopping amplitude . The second term describes an onsite potential with strength . The third term describes superconducting pairing interactions between NN fermions, with a complex pairing strength modulated periodically in time with the period .

#### a.4.2 2. Mapping a fermionic chain to a qubit

Under the anti-periodic boundary condition FranchiniBook (), the fermionic Hamiltonian Eq. (S25) can be further simplified by performing a Fourier transform, where is the total number of lattice sites.

The Fourier transform is given by

(S26) |

where the quasimomentum is in the first Brillouin zone (BZ), and the lattice constant has be set to . Using this transform, we find

(S27) |

Plugging Eq. (S27) into Eq. (S25), we find

(S28) |

In terms of the spinor basis , we can further express as (up to a constant)

(S29) |

where the Hamiltonian is given by

(S30) |

with

(S31) |

Notably, the Hamiltonian is exactly the Hamiltonian realized in the single-qubit simulation of generalized Thouless pump GTPExp (), where the quasimomentum is mapped to angles of the qubit on the Bloch sphere. Furthermore, as will be shown in the next section, the dynamics described by is analytically solvable. Therefore this model serves as an ideal playground to explore Floquet DQPTs.

### a.5 E. Simulation of Floquet DQPTs by a driven qubit

The Hamiltonian in Eq. (S30) also describe a qubit in a rotating field. The dynamics of the qubit obeys the Schrödinger equation

(S32) |

In the rotating frame defined by

(S33) |

the Hamiltonian in Eq. (S30) is transformed to

(S34) |

where is a identity matrix. with the rotated state

(S35) |

The eigenvalues of the Hamiltonian in Eq. (S34) are

(S36) |

The eigenstates are

(S37) |

with the energy gap .

The solution of the Schrödinger equation (S32) can be written as

(S38) |

The Floquet states and the Floquet modes are

(S39) |

With these results, we are ready to explore Floquet DQPTs through this driven qubit model. Following our definitions in Sec. A.3, we choose the initial state of the system to fill one of the Floquet band, i.e.,