Floquet implementation - Minimal instance for Z 2 LGTs coupled to mattermatter

2. Minimal instance for Z 2 LGTs coupled to mattermatter

2.2. Floquet implementation

2. Minimal instance forZ2LGTs coupled to matter

strong influence on the oscillation amplitude and the oscillation frequency. The oscillation amplitude is J_a²/(J²_a +J²_f), i.e. it is 1 for J_f = 0 and vanishes for J_f → ∞. This behavior can be intuitively understood from theZ2ingredients: Gauss’s law forces a link, initially in the eigenstate of the electric field operator with eigenvalue τ^x = +1, to change its state to the eigenstate with eigenvalue τ^x = −1 when a matter particle traverses the link. According to Hamiltonian (2.7), this process has an energy cost of 2J_f. Thus, the tunneling of the matter particle is detuned and is suppressed for largeJ_f. For the same reason, the oscillation frequency is initially 2J_a/hforJ_f =0 and increases withJ_f, analog to Rabi-oscillations.

2.2. Floquet implementation

The Hamiltonian is written in the |Liand |Ribasis indicating that the particle occupies the left or right site of the double well, respectively:

Hˆ_1P(t) =−J⁰(|Li hR|+|Ri hL|) +{_∆+Acos(ωt+φ)} |Li hL|. (2.11) Here,J⁰is the bare single-particle tunneling rate,∆the energy offset between neighboring sites,Athe modulation amplitude,ωthe modulation frequency, andφthe modulation phase. Following references [69, 101], the infinite frequency limit of the effective Hamil-tonian is calculated. This limit is equivalent to the lowest (zeroth) order of the Floquet expansion. To this end, a unitary transformation |ψi → |ψ⁰i= R^ˆ(t)|ψiis chosen such that the new transformed Hamiltonian does not contain divergent terms in the high-fre-quency limit:

H(ˆ t) =R^ˆHˆ(t)R^ˆ^†−i ˆR∂tRˆ^† =

∑

k∈_Z

Hˆ⁽^k⁾e^ikωt. (2.12) Furthermore, the transformed Hamiltonian is expressed in a series of time-independent components ˆH⁽^k⁾. These components can be used to calculate the lowest orders of the Floquet Hamiltonian:

Hˆ_F= H^ˆ⁽⁰⁾+ ¹

¯ hω

∑

k>0

1 k

hHˆ⁽⁺^k⁾, ˆH⁽⁻^k⁾ⁱ+O 1

ω²

. (2.13)

A suitable transformation ˆRfor the time-dependent model (2.11) is Rˆ =exp

i∆t|Li hL|+ ^iA

hωsin(ωt+φ)|Li hL|

. (2.14)

In the high-frequency limit and for resonant driving∆ = ν¯hωwithν ∈ Z, the Fourier components of the Floquet expansion are

Hˆ⁽_1P^k⁾ =−J⁰n

J₋_k₋_ν(χ)eⁱ⁽^k⁺^ν⁾^φ|Ri hL|+J₊_k₋_ν(χ)eⁱ⁽^k⁻^ν⁾^φ|Li hR|^o (2.15) using the expression in Eq. (2.12) and the transformation (2.14). The symbolJ_ν(χ)refers to theνth order Bessel function of the first kind andχ= A/¯hωis the dimensionless driving strength. The zeroth order (k =0) of the Fourier components directly corresponds to the infinite-frequency Floquet Hamiltonian according to Eq. (2.13). The infinite-frequency Floquet Hamiltonian for the time-periodic model captured by Eq. (2.11) is then

Hˆ_F,1P= J^˜νeⁱ^ϕ|Ri hL|+h.c. (2.16) It is worth noting that the detuning∆between neighboring sites vanishes in the effective model. However, the ratio of the detuning to the driving frequencyν=_∆/¯hωhas a critical influence. The tunnel coupling is renormalized to ˜J_ν = J⁰|J_ν(χ)|and the tunneling process is associated with a tunneling phase¹ ϕ = ν φ(+π for odd positiveν). The tunneling

1A formal representation would beϕ= [(ν/|ν|)−1](−1)^νπ/2+ν φ.

2. Minimal instance forZ2LGTs coupled to matter

Floquet model

Figure 2.4.: Illustration of multiphoton processes in a periodically-driven double well.A dou-ble-well potential with energy offset∆ = ν∆ω, where∆ω = ¯hω, is periodically-driven at a frequencyω. The driving modulates the on-site energy of the left siteVL=Acos(ωt+φ)with an amplitudeAand a driving phaseφ. The different illustrated processes can be interpreted such that the particle absorbsνphotons from the drive during a hopping process to conserve energy.

In an effective Floquet picture, the energy difference between neighboring sites is therefore zero.

However, the tunneling is renormalized by theνth order Bessel function of the first kindJ_ν(χ)in dependence on the dimensionless driving strengthχ= A/¯hω.

phase depends on the phase of the modulation and has aπ-shift depending on the sign of the tilt ∆ for oddν. This π-shift has its origin in the anti-symmetry of the Bessel-function with odd order: J₋_ν(χ) = (−1)^νJ_ν(χ). Fig. 2.4 illustrates the corresponding time-dependent and effective models. In Sec. 3.1.3, a measurement of the renormalization of the tunneling rate is presented forν={0, 1, 2}.

2.2.3. Matter–gauge coupling

The multi-photon processes introduced in the previous section are the basis of the im-plementation of the minimal instance for Z2 LGTs. In this minimal instance, a matter particle is coupled to a Z2 gauge-field degree-of-freedom. In this implementation the gauge field is realized using exactly one additional particle f per link, which can occupy either the left or right site of the minimal instance. These two configurations directly represent the two values of the gauge-fieldτ^z. First, a static value of the gauge field or simply a localized f-particle is assumed. This is sufficient to understand the effective matter–gauge coupling. For the matter particlesa, a lattice potential without an energy offset between neighboring sites is considered. Thea- and f-particles interact via a strong on-site Hubbard interactionU. In addition, a periodic modulation of the left site’s on-site energy is applied, which is experienced equally by both species. Then, the time-dependent

2.2. Floquet implementation

Hamiltonian is

Hˆ^a(t) =−J ˆ

a^†_Raˆ_L+aˆ_L^†aˆ_R

∑

j={L,R}

n^a_jnˆ_j^f +Acos(ωt+φ)nˆ^a_L, (2.17)

whereJis the tunneling rate, ˆa^†_j creates ana-particle on sitej, ˆn^a_j is the number operator,A is the modulation amplitude, ¯hωis the resonant driving frequency, andφthe modulation phase. Depending on the f-particle’s position, the relative on-site energy difference between neighboring sites for thea-particles becomes either±U(Fig. 2.5). This density-dependent sign change is the essential ingredient leading to the coupling between the matter and the gauge field. In the limitU Jand A=0, tunneling of thea-particles is suppressed. However, it can be restored by resonant driving with ¯hω≈U. Analog to the multiphoton processes described above, the tunneling rate is renormalized in the effective Floquet Hamiltonian and gains a tunneling phase of ϕ. The zeroth order effective model is

Hˆ_eff^a =−J^˜ae^iϕaˆ^†_Raˆ_L+h.c. (2.18) with the renormalized tunneling rate ˜Ja = J|J_±₁(χ)|, which is independent of the f -particle’s position. However, the phaseϕdepends explicitly on the f-particle’s position and is

ϕ= ^π

2 +nˆ_L^f −nˆ_R^f φ−^π 2

. (2.19)

When defining the link variable ˆτ^z ≡nˆ_R^f −nˆ_L^f and choosing the modulation phaseφ=0, the gauge–matter coupling becomes explicit:

Hˆ_eff^a = −J_aτˆ^zaˆ^†_Raˆ_L+h.c. (2.20)

2.2.4. Gauge field dynamics

In order to analyze the gauge field dynamics, the matter particleais first kept localized, while the f-particle is allowed to tunnel. If identical potentials for thea- and f-particles were chosen, for symmetry reasons the result would be the same effective Hamiltonian.

However, to realize the minimal instance, it is required that noa-particle-dependent phase is associated with tunneling of the f-particle. Note that in general the motion of the f-particle should be fully independent of the position of thea-particles. It is therefore not possible to use the same potentials for a- and f-particles as the f-particle would acquire a π-phase depending on the a-particle’s position. To break this symmetry, a

2. Minimal instance forZ2LGTs coupled to matter

Figure 2.5.: Implementation of the matter–gauge coupling.The gauge field is implemented by a second particle species, thef-particles (red). The gauge field can have two possible valuesτ^z=±1.

These values are encoded by the f-particle’s position ˆτ^z=nˆ_R^f −nˆ_L^f, where left and right correspond to+1 and−1, respectively. The matter–gauge coupling requires that the matter tunneling is associated with a phase of{0,π}in dependence of thef-particle’s position ˆτ^zaˆ^†_Laˆ_R+_{h.c. Thus,} the tunnel couplings for the matter particle (a-particle, blue) are analyzed. The double well has no on-site energy shift between neighboring sites anda- and f-particles interact via strong on-site interactions. Therefore, the f-particle introduces an effective on-site energy difference between neighboring sites. Resonant driving with a frequency of ¯hω = Uleads to an effective Floquet model in the high-frequency limit realizing the matter–gauge coupling.

species-dependent energy-offset between the sites is introduced for the f-particles. The time-dependent Hamiltonian is then

Hˆ^f(t) =−J⁰

fˆ_R^†fˆ_L+ f^ˆ_L^†fˆ_R

∑

j={L,R}

n^a_jnˆ_j^f +_∆_fnˆ_L^f +Acos(ωt+φ)nˆ_L^f, (2.21) where f_j^† creates an f-particle on site j. The relative on-site energy difference between neighboring sites for the f-particle becomes now ∆f ±U (Fig. 2.6). For ∆f ≡ U both processes are resonant in the high-frequency limit for a driving with frequencyω. The tunnel coupling is renormalized by the zeroth-J₀(χ)and second-order Bessel function J₂(χ)according to the derived multiphoton processes because the energy offset is either zero or twice the driving frequency. It is important to recognize that the tunneling is not associated with a density-dependentπ-shift becauseνhas the same sign and is even for both processes. Nevertheless, the two-photon processν =2 inherits twice the modulation phase ϕ=2φ. Thus, a modulation phase ofφ=0 orπis required and the effective model in the high-frequency limit reduces to

Hˆ_eff^f =−J^ˆ_f

fˆ_R^†fˆ_L+ f^ˆ_L^†fˆ_R

=−J^ˆ_fτˆ^x, (2.22) where ˆJ_f is operator-valued and depends on thea-particle configuration

Jˆ_f = JJ₀(χ)nˆ^a_L+JJ₂(χ)nˆ_R^a. (2.23) Simply combining the two processes described by ˆH_eff^a and ˆH_eff^f , already constitutes aZ2 -symmetric minimal instance. However, the f-particle tunneling rate depends in general on the position of thea-particle. This complication can be avoided by choosing the driving strengthχ₀'1.84 such thatJ₀(χ₀) =J₂(χ₀).

2.2. Floquet implementation

Figure 2.6.: Implementation of the gauge-field dynamics.The gauge-field particle dynamics (red) is presented for fixed positions of the matter particle (blue). In contrast to the configuration for the matter particle, an energy offset between neighboring sites of sizeUis introduced. This additional species-dependent tilt breaks the symmetry between thea- and thef-particles. Again looking at the effect of the energy offsets between neighboring sites taking the on-site interaction into account reveals two possible coupling processes: a zero and a two-photon transition. Here, the tunneling rate is renormalized differently depending on the twoa-particle configurations but the tunneling phase isϕ=0 forφ=0 (Eq. 2.15).

2.2.5. Time-dependent and effective Hamiltonian

So far, the processes fora- and f-particles are treated separately. Here, the processes are combined and the time-dependent and effective Hamiltonian of the scheme for realizing a minimal instance ofZ2LGTs are summarized. A detailed derivation including higher order terms can be found in Sec. A. The assumptions are strong inter-species on-site interactionsU Jand a resonant periodic driving ¯hω≈U. The resulting time-dependent Hamiltonian is

Hˆ(t) =−J ˆ

a_R^†aˆ_L+ f^ˆ_R^†fˆ_L+h.c.

∑

j={L,R}

n^a_jnˆ_j^f +_∆_fnˆ_L^f +Acos(ωt+φ)nˆ_L^a +nˆ_L^f

(2.24)

In the high-frequency limit U/J 1 and forφ = 0, the lowest order of the effective Floquet Hamiltonian takes the form

Hˆ_eff =−Jaτˆ^z

a^†_Raˆ_L+aˆ^†_Laˆ_R

−J^ˆ_fτˆ^x, (2.25) where the a-particle position dependence vanishes for a specific driving strength, e.g.

χ0=1.84. In conclusion, this scheme realizes a minimal instance forZ2LGTs.

3. Experimental realization and

Im Dokument Minimal instances for ℤ₂ lattice gauge theories and spin pumps (Seite 44-51)