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Fast-Forward of Adiabatic Dynamics in Quantum Mechanics

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Fast-Forward of Adiabatic Dynamics in Quantum Mechanics Fast-forward of adiabatic dynamics in quantum mechanics By Shumpei Masuda1 and Katsuhiro Nakamura2,3 1 Department of Physics, Kwansei Gakuin University, Gakuen, Sanda, Hyogo 669-1337, Japan 2 Heat Physics Department, Uzbek Academy of Sciences, 28 Katartal Str., 100135 Tashkent, Uzbekistan 3 Department of Applied Physics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan We propose a way to accelerate adiabatic dynamics of wave functions in quantum mechanics to obtain a final adiabatic state except for the spatially uniform phase in any desired short time. We develop the previous theory of fast-forward (Masuda & Nakamura 2008) so as to derive a driving potential for the fast-forward of the adi- abatic dynamics. A typical example is the fast-forward of adiabatic transport of a wave function which is the ideal transport in the sense that a stationary wave func- tion is transported to an aimed position in any desired short time without leaving any disturbance at the final time of the fast-forward. As other important exam- ples we show accelerated manipulations of wave functions such as their splitting and squeezing. The theory is also applicable to macroscopic quantum mechanics described by the nonlinear Schr¨odinger equation. Keywords: atom manipulation, mechanical control of atoms, quantum transport 1. Introduction The adiabatic process occurs when the external parameter of Hamiltonian of the system is adiabatically changed. Quantum adiabatic theorem (Born & Fock 1928; Kato 1950; Messiah 1962), which states that, if the system is initially in an eigen- state of the instantaneous Hamiltonian, it remains so during the process, has been studied in various contexts (Berry 1984; Aharonov & Anandan 1987; Samuel & Bhandari 1988; Shapere & Wilczek 1989; Nakamura & Rice 1994; Bouwmeester et al. 1996; Farhi et al. 2001; Roland & Cerf 2002; Sarandy & Lidar 2005; Du et al. 2008). The rate of a change in the parameter of Hamiltonian is infinitesimal, so that it takes infinite time to achieve the final result in the process. In our previous paper (Masuda & Nakamura 2008), we investigated a way to arXiv:0910.0301v1 [cond-mat.mes-hall] 2 Oct 2009 accelerate quantum dynamics with use of an additional phase of a wave function (WF). We can accelerate a given quantum dynamics to obtain a target state in any desired short time. This kind of acceleration is called the fast-forward of quantum dynamics. The idea of the adiabatic process seems to be incompatible with that of fast- forward. But here we combine these two ideas, that is, we propose a theory to accelerate the adiabatic dynamics in quantum mechanics and obtain, in any desired Article submitted to Royal Society TEX Paper 2 Shumpei Masuda & Katsuhiro Nakamura short time, the target state originally accessible after infinite time through the adiabatic dynamics. By using this theory we can find a driving potential to generate the target state exactly except for a spatially-uniform time-dependent phase like dynamical and adiabatic phases (Berry 1984). The fast-forward of the adiabatic dynamics makes an ideal transport of quantum states possible: a stationary wave packet is moved to an aimed position without leaving any disturbance at the end of transport. After the transport, the wave packet becomes stationary again, and is in the same energy level as the initial one. Before embarking upon the main part of the text, we briefly summarize the previous theory of the fast-forward of quantum dynamics (Masuda & Nakamura 2008). In Schr¨odinger equation with a given potential V0 = V0(x,t) and nonlinearity constant c0 (appearing in macroscopic quantum dynamics) ~2 dΨ0 2 2 i~ = Ψ0 + V0(x,t)Ψ0 c0 Ψ0 Ψ0, (1.1) dt −2m0 ∇ − | | Ψ0(x,t) is supposed to be a known function of space (x) and time (t) and is called a standard state. For any long time T called a standard final time, we choose Ψ0(t = T ) as a target state that we are going to generate. Let Ψα(x,t) be a virtually fast-forwarded state of Ψ0(x,t) defined by Ψ (t) >= Ψ (αt) >, (1.2) | α | 0 where α is a time-independent magnification factor of the fast-forward. The time- evolution of the WF is speeded up for α> 1 and slowed down for 0 <α< 1 like a slow-motion. A rewind can occur for α< 0, and the WF pauses when α = 0. In general, the magnification factor can be time-dependent, α = α(t). The time-evolution of a WF is accelerated and decelerated when α(t) is increasing and decreasing, respectively. In this case, the virtually fast-forwarded state is defined as, Ψ (t) >= Ψ (Λ(t)) >, (1.3) | α | 0 where t ′ ′ Λ(t)= α(t )dt . (1.4) Z0 Since the generation of Ψα requires an anomalous mass reduction, we can not generate Ψα (Masuda & Nakamura 2008). But we can obtain the target state by considering a fast-forwarded state ΨF F = ΨF F (x,t) which differs from Ψα by an additional space-dependent phase, f = f(x,t), as if if ΨF F (t)= e Ψα(t)= e Ψ0(Λ(t)). (1.5) The Schr¨odinger equation for ΨF F is given by ~2 dΨF F 2 2 i~ = ΨF F + VF F (x,t)ΨF F c0 ΨF F ΨF F , (1.6) dt −2m0 ∇ − | | where VF F (x,t) is called a driving potential. If we appropriately tune the initial and final behaviours of the time dependence of α (the detail is shown later), the Article submitted to Royal Society Fast-forward of adiabatic dynamics 3 additional phase can vanish at the final time of the fast-forward TF , and we can obtain the exact target state, that is, ΨF F (TF )=Ψ0(T ), (1.7) where TF is the final time of the fast-forward defined by TF T = α(t)dt. (1.8) Z0 (In the case of a constant α, TF = T/α.) From equations (1.1), (1.4), (1.5) and (1.6) we obtain an equation for the addi- tional phase f (Masuda & Nakamura 2008) 2 2 ∗ Ψα f + 2Re[Ψα Ψα] f | | ∇ ∇ · ∇ ∗ +(α 1)Im[Ψ 2Ψ ]=0, (1.9) − α∇ α and the driving potential of the fast-forward VF F ~2 df 2 VF F = αV0 ~ ( f) − dt − 2m0 ∇ ~2 ~2 2 +Re[ (α 1) Ψα/Ψα + i f Ψα/Ψα] − − 2m0 ∇ m0 ∇ · ∇ (α 1)c Ψ 2. (1.10) − − 0| α| With use of the phase η = η(x,t) of the standard state Ψ0, f is given by (Masuda & Nakamura 2008) f(x,t) = (α(t) 1)η(x, Λ(t)), (1.11) − which satisfies equation (1.9) and determines VF F in equation (1.10). We impose the initial and final conditions for α(t): α must start from 1, increase for a while and decrease back to 1 with dα/dt = 0 at the final time of the fast-forward. Then the additional phase f vanishes in the initial and final time of the fast-forward. Once we have a standard state, we can obtain a target state in any desired short time by applying the driving potential with suitably tuned α(t). However, in the fast-forward of the adiabatic dynamics we shall use infinitely- large α. Then the expression of VF F in equation (1.10) and f in equation (1.11) should diverge. This difficulty will be overcome by regularization of the standard potential and states which will be described in Section 2. In Section 3 we apply the theory to the ideal transport by accelerating the adiabatic transport of a wave function in a moving confining potential. Ideal manipulations such as wave packet splitting and squeezing are shown as other examples of the fast-forward of adiabatic dynamics. Section 4 is devoted to summary and discussions. 2. Regularization of standard state and driving potential for fast-forward (a) Difficulty involved in standard adiabatic dynamics We consider a dynamics of a WF, Ψ0, under the potential: V0 = V0(x, R(t)) which is adiabatically varies, where R = R(t) is a parameter in the potential which Article submitted to Royal Society 4 Shumpei Masuda & Katsuhiro Nakamura is adiabatically changed from a constant R0 as R(t)= R0 + εt. (2.1) The constant value ε is the rate of adiabatic change of R(t) with respect to time and is infinitesimal, that is, dR(t) = ε, (2.2) dt ε 1. (2.3) ≪ The Hamiltonian is represented as p2 H0 = + V0(x, R(t)), (2.4) 2m0 and Schr¨odinger equation for Ψ0 is given as ~2 dΨ0 2 2 i~ = Ψ0 + V0(x, R(t))Ψ0 c0 Ψ0 Ψ0, (2.5) dt −2m0 ∇ − | | where c0 is a nonlinearity constant. If a system is in the n-th energy eigenstate at the initial time, the adiabatic theorem guarantees that, in the limit ε 0, Ψ → 0 remains in the n-th energy eigenstate of the instantaneous Hamiltonian. Then Ψ0 is represented as − i t ′ ′ ~ R0 En(R(t ))dt iΓ(t) Ψ0(x,t,R(t)) = φn(x, R(t))e e , (2.6) where En = En(R) and φn = φn(x, R) are the n-th energy eigenvalue and eigenstate corresponding to the parameter R, respectively, and Γ = Γ(t) is the adiabatic phase defined by t ∞ ∗ d Γ(t)= i dxdtφn φn, (2.7) Z0 Z−∞ dt which is independent of space coordinates. φn fulfills ∂φ n = 0, (2.8) ∂t ~2 2 2 φn + V0(x, R)φn c0 φn φn = En(R)φn. (2.9) −2m0 ∇ − | | The second factor in the right hand side of equation (2.6) is called a dynamical phase factor, which is also space-independent.
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