Fast-forward scaling theory for phase imprinting on a BEC: creation of a wave packet with uniform momentum density and loading to Bloch states without disturbance

We use the wave function ansatz given by

$$\begin{eqnarray}&&{\rm{\Psi }}(x,t)=\phi (x){{\rm{e}}}^{{\rm{i}}{R}(t)\eta (x)}{{\rm{e}}}^{-{\rm{i}}{E}{t}/{\hslash }},\end{eqnarray} \tag{ 8 }$$

where R changes monotonically from 0 to 1 with time. We typically choose the time dependence of R as [8]

$$\begin{eqnarray}R(t)=\left\{\begin{array}{ll}0 & \ t\lt 0,\\ \tfrac{t}{T}-\tfrac{1}{2\pi }\sin \left(\tfrac{2\pi }{T}t\right) & \ 0\leqslant t\leqslant T,\\ 1 & \ T\lt t,\end{array}\right.\end{eqnarray} \tag{ 9 }$$

so that the particle is in the ground state, ϕ, at t = 0 and becomes the target WPUM at t = T. The time dependence of R is arbitrary as long as it satisfies conditions $R(0)=0,R(T)=1,\dot{R}(0)=\dot{R}(T)=0$, although we use the particular form in equation (9) for the present study for concreteness. Conditions R(0) = 0 and R(T) = 1 guarantee that the state is in the given initial state at t = 0 and the target state at t = T, respectively. The reason of the conditions: $\dot{R}(0)=\dot{R}(T)=0$ is explained below in this section.

The time-dependent Schrödinger equation is represented as

$$\begin{eqnarray}&&{\rm{i}}{\hslash }\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{\rm{\Psi }}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{x}}^{2}}{\rm{\Psi }}+{V}_{\mathrm{FF}}{\rm{\Psi }}.\end{eqnarray} \tag{ 10 }$$

We divide equation (10) by Ψ and compare the real parts of the both sides to obtain the driving potential

$$\begin{eqnarray}&&{V}_{\mathrm{FF}}(x,t)=V(x,R)-{\hslash }\dot{R}(t)\eta (x),\end{eqnarray} \tag{ 11 }$$

where $\dot{R}$ is the time derivative of R, and $V(x,R)$ is defined by

$$\begin{eqnarray}&&V(x,R)={V}_{0}(x)-\displaystyle \frac{{{\hslash }}^{2}}{2m}{R}^{2}{\left(\displaystyle \frac{{\rm{d}}\eta }{{\rm{d}}{x}}\right)}^{2},\end{eqnarray} \tag{ 12 }$$

with V₀, the initial trapping potential, given by

$$\begin{eqnarray}&&{V}_{0}(x)=E+\displaystyle \frac{{{\hslash }}^{2}}{2m}\left(\displaystyle \frac{1}{\phi }\displaystyle \frac{{{\rm{d}}}^{2}\phi }{{{\rm{d}}{x}}^{2}}\right),\end{eqnarray} \tag{ 13 }$$

which is obtained by dividing equation (7) by $\phi (x)$. Note that the imaginary part vanishes because of equation (3). Here, V(x, R) is the potential which can maintain the WPUM in equation (8) for a fixed R. The second term in equation (11), which vanishes when $\dot{R}=0$, is for the non-adiabatic driving of the state. As seen in equation (9), $\dot{R}(t)=0$ for t < 0 and t > T. Therefore the driving potential in equation (11) continuously changes from V₀ to ${V}_{0}-\{{{\hslash }}^{2}/{(2m)\}({\rm{d}}\eta /{\rm{d}}{x})}^{2}$.

The results obtained in section 3.1 reveal the scaling property of the quantum dynamics. Once a pair of ϕ and η is found, we obtain the driving potential which realizes the state in equation (8) for any time dependence of R because the form of the driving potential in equation (11) is independent of the form of R. (Note that R does not have to start from 0 nor to become 1 at t = T for the validity of the scaling property.)

A pair composed of the dynamics of the system and the corresponding driving potential, {Ψ[R], V_FF[R]} , are characterized by R, a function of time. We can consider the ensemble composed of such infinite number of pairs. The ensemble can be represented by one of Ψ[R]. Thus, infinite number of the pairs of the dynamics and the potential corresponding to different R are related to each other. The scaling property is explained in detail for a more general case in the following section.

Due to the scaling property, the final state of an adiabatic dynamics with $\dot{R}\simeq 0$ can be realized for any finite time T. As mentioned, this fast-forwarding of the dynamics is important for the creation of the target WPUM because WPUMs are short-lived under a finite potential.

As an example we apply the driving potential in equation (11) to a BEC of ⁸⁷Rb in a harmonic trap to create a WPUM. The dynamics is governed by the GP equation

$$\begin{eqnarray}&&{\rm{i}}{\hslash }\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{\rm{\Psi }}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{x}}^{2}}{\rm{\Psi }}+{V}_{\mathrm{FF}}{\rm{\Psi }}+g| {\rm{\Psi }}{| }^{2}{\rm{\Psi }},\end{eqnarray} \tag{ 14 }$$

where m is the mass of the atom, g the coupling constant and Ψ the condensate order parameter. For simplicity we consider the case of g = 0. (The form of the driving potential for finite g is the same as equation (11) because $g| {\rm{\Psi }}{| }^{2}$ can be regarded as a part of the potential, although the driving potential differs from the one for g = 0 because of the dependence of ϕ on g [8].) We assume that the initial state is the ground state of the harmonic trap, of which the condensate order parameter is ϕ. We take the initial harmonic potential as

$$\begin{eqnarray}&&{V}_{0}(x)=\displaystyle \frac{m{\omega }^{2}}{2}{x}^{2}.\end{eqnarray} \tag{ 15 }$$

The target state is given in equation (1) with the phase in equation (4).

Figure 1 shows the phase η(x) of the target WPUM and the final potential V in equation (12) for a parameter set: ω/(2π) = 500 Hz and η₀ = 2.1 × 10⁶ (m⁻²). The real part of the order parameter oscillates with respect to x due to the phase change. The potential diverges to negative infinity where the amplitude of the order parameter is small.

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Now we simulate evolution of the state driven from the ground state in the harmonic potential in equation (15) to a WPUM for the same ω and η₀ as in figure 1. In the following simulation we use the approximate potential of which $\tfrac{{\rm{d}}\eta }{{\rm{d}}{x}}(x)$ is set to $\tfrac{{\rm{d}}\eta }{{\rm{d}}{x}}(\pm {x}_{0})$ for x > x₀ (=0.72 μm) and x < −x₀, respectively, as exhibited in figure 1(b) to avoid drastic change in potential with respect to x and t. Figure 2(a) shows approximate V_FF during the control for T = 32 μs. (Hereafter we simply refer the approximate potential as V_FF instead of approximate V_FF.) The potential is tilted to imprint a phase η, which is asymmetric, generating positive momentum (see the potential for t = T/4 and T/2). At the end of the control, t = T, the potential coincides with the one represented by the dashed line in figure 1(b).

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Figure 2(b) shows the fidelity, as a function of T, defined as the amplitude of the overlap between the state at t = T and the target state in equation (1). The fidelity decreases with the increase of T because of the imperfection of the approximate potential. Therefore, fast control is essential for creation of the target state.

Figure 3 shows the initial and the final profiles of the square of the amplitude of the order parameter for T = 32 μs and T = 191 μs. For longer control the order parameter moves to the x-direction because of the imperfect potential. In the numerical simulation there are hard walls at x = ±4.8 μm, which reflect the order parameter. The oscillation in the order parameter for T = 191 μs is due to the interference of the order parameter with the one reflected by the hard walls, while for T = 32 μs, the order parameter approximately keeps its initial form. It might not be very obvious why the distortion of the order parameter is reduced at the final time of the creation, t = T, when T is sufficiently short and the approximated potential is sufficiently closer to the exact one in spite of the asymmetry of the potential. It is because that the target WPUM has the spatially uniform current of the amplitude of order parameter. Because the current is independent of x, the amplitude of the order parameter is stationary in the ideal case. Due to the current flowing to the x-direction, the order parameter is reflected back by the hard wall at x = 4.8 μm in the both cases for T = 32 and 192 μs. However t = 32 μs is so short that the major part of the order parameter has not been reflected yet at that time. This is the reason why the oscillation due to interference is not clearly seen at t = 32 μs.

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Suppose we release the order parameter in equation (1) by putting V = 0 at t = 0. The pattern of the order parameter is distinct from that without phase η. Thus, observation of the amplitude of the BEC after releasing can be used for an indirect measurement of the WPUMs.

Evolution of $| {\rm{\Psi }}{| }^{2}$ in free space is shown in figure 4(a) as a function of t and x. The spatial oscillation of $| {\rm{\Psi }}{| }^{2}$ is clearly seen. The fringe pattern is moving toward positive direction of x, and the intervals between the peaks increase. Such fringe pattern is not observed with the ground state order parameter which is simply broaden. This pattern of the order parameter which resembles an expanded wing of birds is a manifestation of the spatially changing phase. Figure 4(b) shows the amplitude of the Fourier transform of Ψ at the initial time. Note that there is a fringe pattern in the Fourier transform of the order parameter in k > 0, and this initial distribution of the order parameter in the k-space is attributed to the pattern of the released order parameter in figure 4(a). Detailed study on this pattern such as the correlation between η₀ and the peak intervals is beyond the scope of the present paper and will be conducted elsewhere.

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Now we derive the driving potential which generates the target state. We assume that the wave function during the creation of the target state is represented as

$$\begin{eqnarray}&&{{\rm{\Psi }}}^{\mathrm{FF}}(x,t)=\phi (x,R){{\rm{e}}}^{{\rm{i}}\eta (x,R)}{{\rm{e}}}^{{\rm{i}}{f}(x,t)}{{\rm{e}}}^{-({\rm{i}}/{\hslash }){\int }_{0}^{t}{\rm{d}}{t}^{\prime} E(R(t^{\prime} ))},\end{eqnarray} \tag{ 18 }$$

where R is a time-dependent parameter, which starts from 0 and becomes $R^{\prime}$ at t = T. In the intermediate wave function in equation (18), an additional phase f is introduced. The time-dependent Schrödinger equation is represented as

$$\begin{eqnarray}&&{\rm{i}}{\hslash }\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{{\rm{\Psi }}}_{{\rm{FF}}}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{x}}^{2}}{{\rm{\Psi }}}_{{\rm{FF}}}+{V}_{{\rm{FF}}}{{\rm{\Psi }}}_{{\rm{FF}}},\end{eqnarray} \tag{ 19 }$$

with the driving potential V_FF.

We divide equation (19) by the phase factor of Ψ_FF and substitute equation (18), and compare the imaginary and the real parts of the both sides of the equation. Using the phase-amplitude relation in equation (3), the imaginary part leads to the equation for additional phase f as

$$\begin{eqnarray}&&\phi (x,R)\displaystyle \frac{{\partial }^{2}f}{\partial {x}^{2}}(x,t)+2\displaystyle \frac{\partial f}{\partial x}(x,t)\displaystyle \frac{\partial \phi }{\partial x}(x,R)+\displaystyle \frac{2m}{{\hslash }}\dot{R}\displaystyle \frac{\partial \phi }{\partial R}(x,R)=0.\end{eqnarray} \tag{ 20 }$$

Equation (20) is to be solved to obtain f(x, t). A solution of ∂ f/∂ x is given as

$$\begin{eqnarray}&&\displaystyle \frac{\partial f}{\partial x}=-\displaystyle \frac{2m\dot{R}}{{\hslash }{\phi }^{2}(x,R)}{\int }_{0}^{x}\ {\rm{d}}{x}^{\prime} \phi (x^{\prime} ,R)\displaystyle \frac{\partial \phi }{\partial R}(x^{\prime} ,R).\end{eqnarray} \tag{ 21 }$$

Thus, we obtain the additional phase by integrating equation (21). $f/\dot{R}$ is explicitly given by

$$\begin{eqnarray}&&\displaystyle \frac{f}{\dot{R}}={c}_{0}-\displaystyle \frac{2m}{{\hslash }}{\int }_{0}^{x}{\rm{d}}{x}^{\prime} \displaystyle \frac{1}{{\phi }^{2}(x^{\prime} ,R)}{\int }_{0}^{x^{\prime} }{\rm{d}}{x}^{\prime\prime} \phi (x^{\prime\prime} ,R)\displaystyle \frac{\partial \phi }{\partial R}(x^{\prime\prime} ,R),\end{eqnarray} \tag{ 22 }$$

where c₀ is a constant. On the other hand, the real part gives rise to the driving potential

$$\begin{eqnarray}{V}_{\mathrm{FF}}(x,t) & = & V(x,R)-{\hslash }\dot{R}\displaystyle \frac{\partial \eta }{\partial R}(x,R)-{\hslash }\displaystyle \frac{\partial f}{\partial t}(x,t)\\ & & -\displaystyle \frac{{{\hslash }}^{2}}{2m}\left\{2\displaystyle \frac{\partial f}{\partial x}(x,t)\displaystyle \frac{\partial \eta }{\partial x}(x,R)+{\left[\displaystyle \frac{\partial f}{\partial x}(x,t)\right]}^{2}\right\}.\end{eqnarray} \tag{ 23 }$$

Once we set the time dependence of R, the additional phase and the driving potential are calculated using equations (22) and (23). Importantly, the additional phase vanishes at the initial time and the final time by imposing $\dot{R}=0$ at those times as seen from equation (22). Thus we are guaranteed to obtain the target state from the initial state at predetermined time. The form of the driving potential for finite g is the same as equation (23), although the driving potential differs from the one for g = 0 because of the dependence of ϕ on g.

It is seen, from equation (20) or (22), that $f/\dot{R}$ depends on R and x but not explicitly on t because ϕ depends explicitly only on R and x. This reveals that if we have the additional phase for a particular function for R, say R₁(t), the additional phase for a different function R₂(t) is obtained using a simple relation between the phases. (Note that we do not have an analytical form of f for higher dimensions in general.)

For example, we assume that the additional phase is f₁ for R₁ and f₂ for R₂. Then we have the relation

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{1}(x,{t}_{1})}{{\dot{R}}_{1}({t}_{1})}=\displaystyle \frac{{f}_{2}(x,{t}_{2})}{{\dot{R}}_{2}({t}_{2})}\end{eqnarray} \tag{ 24 }$$

if R₁(t₁) = R₂(t₂) because the both sides of the above equation do not depend on t explicitly. (We assume that there exists t₂ such that R₂(t₂) = R₁(t₁) for any t₁.) Thus we obtain

$$\begin{eqnarray}&&{f}_{2}(x,{t}_{2})=\displaystyle \frac{{\dot{R}}_{2}({t}_{2})}{{\dot{R}}_{1}({t}_{1})}{f}_{1}(x,{t}_{1}).\end{eqnarray} \tag{ 25 }$$

The driving potential in equation (23) is a functional of f and R. Therefore the driving potential for R₂ is easily obtained if we have additional phase f₁ for R₁.

As in the previous section, this theory can be regarded as providing a shortcut to adiabaticity protocol because it shows the connection between the adiabatic dynamics corresponding to the $\dot{R}=0$ limit and accelerated dynamics with finite $\dot{R}$. However, we emphasize that this theory covers wider concept because it uncovers the connection between infinite number of non-adiabatic dynamics characterized by different R.

Equation (25) leads to

$$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{2}/\partial x(x,{t}_{2})}{\partial {f}_{1}/\partial x(x,{t}_{1})}=\displaystyle \frac{{\dot{R}}_{2}({t}_{2})}{{\dot{R}}_{1}({t}_{1})},\end{eqnarray} \tag{ 26 }$$

which can be interpreted as follows. The left hand side is the ratio of the currents of the probability density at x for the dynamics corresponding to R₁ to that for R₂ because the current is proportional to ${\phi }^{2}\tfrac{\partial f}{\partial x}$, and we have ϕ(R₁(t₁), x) = ϕ(R₂(t₂), x). This ratio is the same as the ratio between $\dot{R}{\rm{s}}$ (right hand side) because it determines the ratio between the rates of the change in the probability density.

Study on the motion of a particle in a spatially periodic potential has revealed interesting features of the system such as acceleration theorem formulated by Bloch [57]. The dynamics of a particle under a periodic potential and an additional linear potential has been extensively studied, e.g., asymptotic expression of Bloch oscillators in the limit of weak electric field [58] and numerical study of the dynamics of Bloch oscillation [59]. Dynamics of a particle in a periodically driven lattice potential has been also studied [60, 61]. However, to the best of our knowledge, no explicit form of the driving potential, which can load a particle into a predetermined Bloch state, has been presented.

We apply the result obtained in the above subsection to derive the driving potential without tight binding approximation, which generates a predetermined Bloch state in the lowest band from the ground state as depicted in figure 5(a).

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We consider a BEC in an optical lattice potential

$$\begin{eqnarray}&&{V}_{p}(x)={V}_{0}{\sin }^{2}\left(\displaystyle \frac{\pi }{L}x\right),\end{eqnarray} \tag{ 27 }$$

with period L and potential height V₀. The order parameter of a Bloch state in V_p is represented as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{k}(x,t)=\phi (x,k){{\rm{e}}}^{{\rm{i}}\eta (x,k)}{{\rm{e}}}^{-{\rm{i}}{E}(k)t/{\hslash }},\end{eqnarray} \tag{ 28 }$$

with the amplitude of the order parameter ϕ(x, k) and the energy E(k). Here, k is the wavenumber. Phase η can be separated as

$$\begin{eqnarray}&&\eta (x,k)={\rm{\Delta }}\eta (x,k)+{kx},\end{eqnarray} \tag{ 29 }$$

where Δη(x, k) is periodic with respect to x with period L as well as ϕ, that is, Δη(x + L) = Δη(x) and ϕ(x + L) = ϕ(x). Note that the profile of ϕ(x, k) depends on k as shown in figure 5(b). Note also that η satisfies the phase-amplitude relation in equation (6) because Ψ_k is an instantaneous energy eigenstate.

As in the previous subsections, the order parameter ansatz is written as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{\mathrm{FF}}(x,t)=\phi (x,k(t)){{\rm{e}}}^{{\rm{i}}\eta (x,k(t))}{{\rm{e}}}^{{\rm{i}}{f}(x,t)}{{\rm{e}}}^{-({\rm{i}}/{\hslash }){\int }_{0}^{t}{\rm{d}}{t}^{\prime} E(k(t^{\prime} ))}\end{eqnarray} \tag{ 30 }$$

with the additional phase f. The additional phase is given by

$$\begin{eqnarray}&&f=-\dot{k}{\int }_{0}^{x}{\rm{d}}{x}^{\prime} \displaystyle \frac{2m}{{\hslash }{\phi }^{2}(x^{\prime} ,k)}{\int }_{0}^{x^{\prime} }{\rm{d}}{x}^{\prime\prime} \phi (x^{\prime\prime} ,k)\displaystyle \frac{\partial \phi }{\partial k}(x^{\prime\prime} ,k),\end{eqnarray} \tag{ 31 }$$

where we used equation (22) with c₀ = 0 without loss of generality.

For concreteness we consider k defined by

$$\begin{eqnarray}k(t)=\left\{\begin{array}{ll}0 & \ t\lt 0,\\ {k}_{{\rm{fin}}}\left[\tfrac{t}{T}-\tfrac{1}{2\pi }\sin \left(\tfrac{2\pi }{T}t\right)\right]\hspace{0.3cm} & \ 0\leqslant t\leqslant T,\\ {k}_{{\rm{fin}}} & \ T\lt t,\end{array}\right.\end{eqnarray} \tag{ 32 }$$

where k_fin is the final (target) wavenumber. We use the parameter set: L = 0.8 μm, k_fin = 0.96π/L and T = 67.5 μs. m is the mass of a ⁸⁷Rb atom. V₀ = 0.14 E_R, where E_R is defined by ${E}_{R}={({\hslash }{k}_{L})}^{2}/(2m)$ with k_L = π/L. Figure 5(c) shows ϕ²(x, k) as a function of t and x for k(t) defined in equation (32).

The driving potential is given by

$$\begin{eqnarray}{V}_{\mathrm{FF}}(x,t) & = & {V}_{p}(x)-{\hslash }\displaystyle \frac{{\rm{d}}{k}}{{\rm{d}}{t}}x-{\hslash }\displaystyle \frac{{\rm{d}}{k}}{{\rm{d}}{t}}\displaystyle \frac{\partial {\rm{\Delta }}\eta }{\partial k}(x,k)-{\hslash }\displaystyle \frac{\partial f}{\partial t}(x,t)\\ & & -\displaystyle \frac{{{\hslash }}^{2}}{2m}\left\{2\displaystyle \frac{\partial f}{\partial x}(x,t)\displaystyle \frac{\partial \eta }{\partial x}(x,k)+{\left[\displaystyle \frac{\partial f}{\partial x}(x,t)\right]}^{2}\right\},\end{eqnarray} \tag{ 33 }$$

where we used equations (23) and (29). Since dk/dt = 0 for t ≤ 0 and t ≥ T the additional phase f disappears and the potential V_FF coincides with the original potential V_p because f and ∂f/∂t vanish as seen from equation (31). The first and the second terms in equation (33) are the original periodic potential and the time-dependent linear potential, respectively. The linear potential corresponds to imprinting the linear phase kx. The third term corresponds to the periodic phase Δη. Other terms are attributed to the change of the profile of the amplitude of the order parameter with respect to k. There is no divergence in phase η and driving potential V_FF because the amplitude of the order parameter of Bloch states in the lowest band has no node.

The driving potential in equation (33) is shown as a function of x and t in figure 6(a). In figure 6(b), simpler potential composed of the original and the linear potentials, ${V}_{p}-{\hslash }\dot{k}x$, is plotted for comparison. We refer this potential as the simple potential. The left panels in figures 6(a) and (b) are for 0 ≤ t ≤ T/2, while the right panels are for T/2 < t ≤ T. The difference between the two cases is apparent for t ≥ T/2 when ϕ(x, k) for $| x| \gt 0.2\,\mu {\rm{m}}$ changes rapidly as shown in figure 5(c). In figures 6(c) and (d) the evolution of $| {\rm{\Psi }}{| }^{2}$ is shown for the potentials in figures 6(a) and (b), respectively. The fidelity, which is defined as the amplitude of the overlap between the state at t = T and the target state in equation (28), is 0.99 for the control with V_FF and 0.95 for the control with the simple potential. Infidelity (decrease of the fidelity) of the control with V_FF is attributed to numerical error. The evolutions of $| {\rm{\Psi }}{| }^{2}$ driven by the potentials exhibit clear difference although the difference of the fidelities is not significant. The evolution of $| {\rm{\Psi }}{| }^{2}$ is almost the same as the ideal one shown in figure 5(c). On the other hand, the evolution of $| {\rm{\Psi }}{| }^{2}$ in figure 6(d) clearly differs from the one in figure 5(c) due to the lack of the potential which supports the change in the amplitude of the order parameter.

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