Accurate effective Hamiltonians via unitary flow in Floquet space
We present a systematic construction of effective Hamiltonians of periodically driven quantum systems. Because of an equivalence between the time dependence of a Hamiltonian and an interaction in its Floquet operator, flow equations, that permit to decouple interacting quantum systems, allow us to identify time-independent Hamiltonians for driven systems. With this approach, we explain the experimentally observed deviation of expected suppression of tunneling in ultracold atoms.
The idea to use well-controllable quantum systems for simulations to explore physical phenomena has created big expectations to answer questions that exceed our computational and analytical means. In particular, strongly correlated many-body states, as they typically occur in flat-band systems Mielke (1991a); Yao et al. (2012); Möller and Cooper (2012), readily push us to our limitations of classical simulations.
Suitably driven systems can be used to mimic the dynamics of essentially any Hamiltonian, but the precise identification of such an effective Hamiltonian for given driving parameters is a big theoretical challenge. This can be exemplified by the dramatic increase of complexity in going from the static to the driven two-level system. The former is a standard textbook toy model; the latter is exactly solvable only in a few exceptional cases Barnes and Das Sarma (2012). The effective Hamiltonian of a given driven system is typically found in an approximate manner. The deviations between the actual and approximated dynamics accumulate in time and become significant for sufficiently long times. In order to perform precise quantum simulations it is therefore crucial to develop tools that allow one to systematically construct effective Hamiltonians with high accuracy.
As a prominent example of driving-induced effective dynamics, we highlight shaken optical lattices Eckardt et al. (2005); Lignier et al. (2007), which permit the engineering of the tunneling of ultracold bosonic atoms confined in an optical lattice by appropriately adjusting the driving parameters. This yields e.g. dynamical localization Lignier et al. (2007); Eckardt et al. (2009) and provides a promising route towards the simulation of artificial gauge fields Struck et al. (2012); Hauke et al. (2012). However, despite the proven success of the usually considered effective Hamiltonian for high driving frequencies, substantial deviations from theoretically predicted dynamical localization have been observed in many-body samples for moderate driving frequencies Lignier et al. (2007), which is in striking contrast to the single-particle case, where the exact dynamical localization occurs irrespectively of the driving frequency Kayanuma and Saito (2008); Dunlap and Kenkre (1986).
In this Letter, we introduce a new approach to derive effective Hamiltonians merging the concepts of Floquet theory Floquet (1883); Shirley (1965) and flow equations (FE) Wegner (1994). With this, we provide an explanation for the experimentally observed deviations Lignier et al. (2007) from the theoretically predicted suppression of tunneling Eckardt et al. (2005).
The starting point to arrive at an effective Hamiltonian is a periodically driven Hamiltonian . After full cycles of the driving, i.e. with integer , the time-evolution operator, ( denotes the time-ordering operator), can be written as , which defines the effective Hamiltonian 111The effective Hamiltonian is uniquely defined up to multiples of the driving frequency in its eigenvalues and some other subtleties discussed in Ref. Gesztesy and Mitter (1981).. The actual dynamics will follow the dynamics induced by the effective Hamiltonian only stroboscopically, but in the regime of fast driving, where exceeds the relevant scales of , the effective dynamics is a good approximation also for . The deviation between exact and effective dynamics defines the unitary
Since coincides with at , is periodic with period , and equals the identity at multiples of the period.
Given Eq. (1), the effective Hamiltonian can be obtained from the Schrödinger equation, (), and reads
Thus, the effective Hamiltonian is found after performing a periodic time-dependent unitary transformation, , such that the resulting transformed Hamiltonian is time independent and . In practice, finding this exact unitary transformation is an extremely difficult task. Here we present a method, using an unconventional approach, that allows one to systematically obtain the effective Hamiltonian up to a required accuracy.
We use the framework of Floquet theory which asserts that the Schrödinger equation with a time-periodic Hamiltonian has a complete set of solutions that decompose into a phase factor and a time-periodic state vector, i.e. with . Because of their periodicity, the state vectors can be expanded in a discrete set of periodic functions , which are vectors in the space of time-dependent functions defined in the interval . In the following we will use the functions as basis and associate with each such function a state vector in a Hilbert space .
Any T-periodic operator can now be mapped to an operator in ‘Floquet space’
where the Fourier components act on the Hilbert space of the actual system, and the acting on are defined by 222 Applying the ‘operator’ on a ‘wave function’ , yields , i.e. raises the quantum number by .. Similarly, the time derivative is associated with
with the number operator , so that the Floquet operator is mapped to
Formally, this is equivalent to a time-independent Hamiltonian of a composite system with a Hamiltonian for the individual components and an interaction . Because of this analogy, techniques to treat interactions in time-independent Hamiltonians are applicable.
If was time independent all Fourier components but the static would vanish, so that the interaction term would vanish as well. Consequently, a static Hamiltonian is equivalent to a noninteracting system in the present framework. Our goal is, therefore, to find an operator that corresponds to a periodic unitary transformation according to Eq. (3), such that the transformed Floquet operator describes two noninteracting systems, . Once such a transformation is found, the sought transformation reads , and the effective Hamiltonian is given by .
We target the required block diagonalization of the Floquet operator with the method of flow equations Wegner (1994); Głazek and Wilson (1993), which is considered a generalization of conventional scaling approaches and is based on a unitary flow that makes the Hamiltonian increasingly diagonal Wegner (2006); Kehrein (2006). The method defines a family of unitarily equivalent time-independent Hamiltonians related to each other by a continuous parameter
where is the anti-Hermitian generator, , of a unitary transformation. The boundary conditions are such that coincides with the given Hamiltonian, and needs to be chosen such that is in the desired form, i.e. typically diagonal or block diagonal. The canonical approach Wegner (1994) to eliminate an interaction of a Hamiltonian is to define the flowing Hamiltonian and the corresponding generator . The main advantage of the FE method is that it permits an equal treatment of different energy scales in a renormalization formalism, and a focus on a special regime, e.g. low lying excitations, is not necessary. This is important if one wants to study dynamical properties in nonequilibrium situations Kehrein (2005); Moeckel and Kehrein (2008).
The FE method is typically used to decouple an interacting many-body system, e.g. the spin from the bosonic bath in the spin-boson model Kehrein and Mielke (1998). Here, on the other hand, we will use it to remove the interaction in Eq. (6). For this purpose, we will define a flowing Floquet operator and apply Eq. (7) analogously. Additional care is however necessary in the choice of generator to ensure that the unitary transformation corresponds indeed to a periodic time-dependent transformation. This is the case exactly if is invariant under the symmetry transformation , i.e. , and exactly generators of the form preserve this property.
The generator of interest for our purposes reads . This generator will induce a flow (dynamics with the flowing parameter ) that comes to an end if the interaction commutes with . This, in turn, implies that the interaction is trivial in , i.e. it is of the form and the decoupling has been achieved. Eq. (7) defines an infinite set of non-linear differential equations, so that an exact solution can be found only in very exceptional cases. We will therefore strive for a high-frequency expansion, where this set of equations is truncated at a given power in . This requires a modification of the generator Kehrein (2006) as discussed in section I in the supplementary material. In section II we also discuss the driven two-level system for explanatory purposes and reproduce Rahav et al. (2003) the energy shift up to fourth order in . Here, however, we will focus on the shaken optical lattice in order to address the above mentioned question of suppressed tunneling.
The Hamiltonian of the one-dimensional shaken optical lattice can be written, in the co-moving reference frame Eckardt et al. (2005), as , with the Bose-Hubbard model (with periodic boundary conditions) and an additional driving term that describes the shaking. denotes the hopping matrix element between nearest-neighbor sites and is the on-site interaction energy. The operators are the usual bosonic annihilation (creation) operators satisfying and . In lowest order in we find the desired unitary to read , so that the transformed Hamiltonian reads
with the previously known Eckardt et al. (2005) effective Hamiltonian and . The effective hopping matrix element reads in terms of the zeroth order Bessel function and is a small deviation. This effective Hamiltonian, , is a good approximation for sufficiently large driving, Eckardt et al. (2005). Experimentally, however, deviations from this limiting case have been observed and, as we shall see, a systematic improvement of for a finite driving frequency permits to explain these deviations very well.
With Eq. (8) as starting point, our approach yields the effective Hamiltonian with
including effects , as described in more detail in section III of the supplementary material. As a qualitative change as compared to the lowest order effective Hamiltonian , there is tunneling interaction dependent on site occupation, whose rate is given in terms of the th order Bessel functions, . In contrast to the tunneling term or the on-site interaction term of the Bose-Hubbard Hamiltonian, cannot be diagonalized through a suitable choice of single-particle basis, which makes it difficult to develop a simple physical interpretation. The effect of is probably best demonstrated by its action on a Fock state with well-defined particle number on each lattice site. For such a state, the operators reduce to scalars (), such that reduces to a regular tunneling term with site-dependent tunnel rates. For a Fock state with site-independent particle numbers, this rate is even site independent , but only depends on the direction of tunneling. That is, enhances the tunneling in one direction and suppresses it in the other direction. The directionality is determined by the sign of the driving amplitude and the interaction energy as well as by the specific value of . On the other hand, for a Fock state with a large particle difference between adjacent sites the tunneling rate depends on the particle gradient, . In fact, similarly as above the driving parameters can be tuned such that the rate for tunneling events towards highly populated sites is enhanced, which is impossible for the usual kinetic term of the Bose-Hubbard model.
An advantage of the flow equation method over other methods like perturbation theory is its systematics. In section IV and V of the online supplementary material we explicitly show how this method permits, with little extra effort, the identification of the effective Hamiltonian in first order in one of the parameters but in all orders in the other one. In the large interaction energy regime, , we obtain the effective Hamiltonian
and in the large tunneling regime, ,
and the operators and are defined recursively via the relations
For non-interacting particles, and all higher order terms vanish and our result confirms Kayanuma and Saito (2008); Dunlap and Kenkre (1986) that is the exact effective Hamiltonian independently of the value of . Thus, for noninteracting particles, the exact suppression of tunneling is expected at multiples of the driving period whenever coincides with the zeros of even for slow driving.
With interacting particles, however, the exact suppression of the tunneling is not possible in general, since and all higher order terms do not necessarily vanish, and only an approximate suppression in the large-frequency regime can be obtained. For moderate driving frequencies Lignier et al. (2007), is of comparable magnitude as and, if , describes the dominant tunneling mechanism that can no longer be considered a small higher order correction. In particular, as shown in Fig. 1, the tunneling rate proportional to is close to maximal for driving amplitudes at which the rate vanishes. This explains the recently experimentally observed deviations from predictions based on that are particularly pronounced when the effective tunneling is expected to be vanishing. For slower drivings, also higher order terms get more and more important. Eq. (12) predicts that not only is maximal when vanishes, but all show the same property. Also the even coefficients are finite (though nonmaximal) when vanishes, but rapidly tend to for increasing .
An enhancement of the tunneling rate would also appear in a more accurate description of the optical lattice system than the Bose-Hubbard model, where next-nearest-neighbor tunneling is neglected assuming that the trapping potential is sufficiently deep Jaksch et al. (1998); Eckardt et al. (2009). , however, appears also for deep lattices and, as argued above, its signatures can be experimentally observed even with a deep trapping potential, where the Bose-Hubbard model is an excellent approximation. This shows that particular care is required in the identification of effective Hamiltonians and that supposedly high-order contributions can take over a dominant role.
The importance of accurate effective Hamiltonians and their derivation with the present method are by no means limited to the suppression of tunneling discussed above. For example, in flat band systems a small interaction term results in the emergence of strongly correlated states Mielke (1991b), and only the correct identification of such a seemingly small correction to the effective Hamiltonian will permit the correct prediction of such states.
Since the FE have proven very valuable in the treatment of nonperturbative effects, they bear great potential for situations in which a clear separation of scales is no longer valid. That is, the method presented here can deal with driven strongly interacting systems and identify driving schemes that simulate e.g. three-particle interactions.
As a final remark, we would like to mention that our approach shows an equivalence between the time-independent and the time-dependent FE Tomaras and Kehrein (2011). In our treatment, we use the framework of time-independent FE to treat time-dependent systems. Translating our analysis from Floquet space back to the framework of time-dependent operators, one neatly reproduces the formalism of time-dependent FE. This equivalence is completely general and not restricted to periodic Hamiltonians, since one can always treat the time window of interest as the fundamental period of driving. That is, our approach also permits to translate all the existing expertise on generators of time-independent FE to the much less mature field of time-dependent FE. In particular, the time-dependent canonical generator Tomaras and Kehrein (2011) appears as a natural extension of the time-independent canonical one.
Financial support by the European Research Council within the project ODYCQUENT is gratefully acknowledged.
- Mielke (1991a) A. Mielke, J. Phys. A 24, 3311 (1991a).
- Yao et al. (2012) N. Y. Yao, C. R. Laumann, A. V. Gorshkov, S. D. Bennett, E. Demler, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 109, 266804 (2012).
- Möller and Cooper (2012) G. Möller and N. R. Cooper, Phys. Rev. Lett. 108, 045306 (2012).
- Barnes and Das Sarma (2012) E. Barnes and S. Das Sarma, Phys. Rev. Lett. 109, 060401 (2012).
- Eckardt et al. (2005) A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. 95, 260404 (2005).
- Lignier et al. (2007) H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403 (2007).
- Eckardt et al. (2009) A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. A 79, 013611 (2009).
- Struck et al. (2012) J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Phys. Rev. Lett. 108, 225304 (2012).
- Hauke et al. (2012) P. Hauke, O. Tieleman, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, Phys. Rev. Lett. 109, 145301 (2012).
- Kayanuma and Saito (2008) Y. Kayanuma and K. Saito, Phys. Rev. A 77, 010101 (2008).
- Dunlap and Kenkre (1986) D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
- Floquet (1883) Floquet, Ann. École Norm. Sup. 12, 47 (1883).
- Shirley (1965) J. H. Shirley, Phys. Rev. 138, B979 (1965).
- Wegner (1994) F. Wegner, Annalen der Physik 506, 77 (1994).
- (15) The effective Hamiltonian is uniquely defined up to multiples of the driving frequency in its eigenvalues and some other subtleties discussed in Ref. Gesztesy and Mitter (1981).
- (16) Applying the ‘operator’ on a ‘wave function’ , yields , i.e. raises the quantum number by .
- Głazek and Wilson (1993) S. D. Głazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993).
- Wegner (2006) F. Wegner, J. Phys. A 39, 8221 (2006).
- Kehrein (2006) S. Kehrein, The Flow Equation Approach to Many-Particle Systems (Springer, 2006).
- Kehrein (2005) S. Kehrein, Phys. Rev. Lett. 95, 056602 (2005).
- Moeckel and Kehrein (2008) M. Moeckel and S. Kehrein, Phys. Rev. Lett. 100, 175702 (2008).
- Kehrein and Mielke (1998) S. Kehrein and A. Mielke, J. Stat. Phys. 90, 889 (1998).
- Rahav et al. (2003) S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. A 68, 013820 (2003).
- Jaksch et al. (1998) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
- Mielke (1991b) A. Mielke, J. Phys. A 24, L73 (1991b).
- Tomaras and Kehrein (2011) C. Tomaras and S. Kehrein, Europhys. Lett. 93, 47011 (2011).
- Gesztesy and Mitter (1981) F. Gesztesy and H. Mitter, J. Phys. A 14, L79 (1981).