Optimal Shortcuts to Adiabaticity for a Quantum Piston
aa r X i v : . [ m a t h . O C ] J a n Optimal Shortcuts to Adiabaticity for aQuantum Piston ⋆ Dionisis Stefanatos Abstract
In this paper we use optimal control to design minimum-time adiabatic-like pathsfor the expansion of a quantum piston. Under realistic experimental constraints, wecalculate the minimum expansion time and compare it with that obtained from astate of the art inverse engineering method. We use this result to rederive the knownupper bound for the cooling rate of a refrigerator, which provides a quantitativedescription for the unattainability of absolute zero, the third law of thermodynamics.We finally point out the relation of the present work to the fast adiabatic-likeexpansion of an accordion optical lattice, a system which can be used to magnifythe initial quantum state (quantum microscope).
Key words:
Quantum control; optimal control.
Quantum science promises computational power unattainable by any classi-cal computer [16], as well as unprecedented precision measurement of variousphysical phenomena [24]. At the heart of these important applications lies theproblem to accurately control and manipulate the states of quantum systems[11]. For many cases of interest, this control is achieved using adiabatic pro-cesses, where the system parameters are changed slowly from the initial to thedesired final value [12]. If the change is slow enough, the system follows theinstantaneous eigenvalues and eigenstates of the time-dependent Hamiltonianand obtains the desired final state with a good approximation. The inherentdrawback of adiabatic processes is that they require long times which may ⋆ The material of this paper has not been presented at any conference.
Email address: [email protected] (Dionisis Stefanatos). Tel.: +30-697-4364682; Fax: +30-26740-23266.
Preprint submitted to Elsevier 1 March 2018 ender them impractical, since most of the systems are not isolated but areexposed to undesirable interactions with the surrounding environment (deco-herence) that lead to dissipation [4].Several methods have been proposed to speed up adiabatic quantum dynam-ics. Their common characteristic is that they prepare a similar final state asthe adiabatic process at a given final time (which in principle can be madearbitrarily short), without necessarily following the adiabatic path at eachmoment. The corresponding adiabatic-like trajectories are successfully charac-terized as shortcuts to adiabaticity. In the method of counter-diabatic control[10], the applied electromagnetic field restores the adiabatic dynamics of thesystem by suppressing diabatic effects as they are generated. In the method oftransitionless quantum driving [3], an appropriate auxiliary time-dependentinteraction is added such that the augmented system arrives in finite time ata similar quantum state with the adiabatic trajectory of the unperturbed sys-tem. In a closely related method, the trajectory of the shortcut is picked firstand then the time-dependent interaction generating the corresponding evolu-tion is inversely engineered [5], using the theory of Lewis-Riesenfeld invariants[15]. The above methods have been tested experimentally [20,2] and provenquite robust to various types of noise [7]. They also share another interestingfeature: they do not specify a unique shortcut but rather provide entire fam-ilies of them [6]. This ample freedom can be exploited using optimal controlmethods to find the shortcuts which minimize relevant physical criteria, liketime, under realistic experimental constraints [23].In this paper we use optimal control theory to design the shortest adiabatic-like shortcut for a potential well with a moving boundary, in the presenceof restrictions suggested by the experimental setup. This system provides amodel for a quantum piston and has some interesting applications. Note thatthe shortcuts to adiabaticity for this quantum piston have been studied in [9],as well as their potential applications to the control of many-body quantumdynamics. Here we mostly concentrate on the control aspects of the problem,but we also highlight some nice applications. In Section 2 we summarize theimportant points of the above work which are essential for the current anal-ysis and also formulate the time-optimal control problem for the expansionof the piston. We solve this problem in Section 3 and obtain the minimumexpansion time as a function of the expansion factor of the piston. In Sec-tion 4 we compare our optimal results with those obtained using the inverseengineering method. Using the fact that the minimum expansion time has alogarithmic dependence on the expansion factor for large values of the latter,we reobtain the cooling rate of a refrigerator [13], a result which quantifies theunattainability of absolute zero implied by the third law of thermodynamics[18]. Finally, we discuss how the control problem studied here is related to thefast adiabatic-like expansion of an optical lattice with dynamically variablespacing [25]. Section 5 concludes the paper.2
Shortcuts to Adiabaticity for a Quantum Piston
Consider the potential V p ( x, t ) = , < x < a ( t ) ∞ , otherwise (1)which describes the infinite square well with a moving boundary at x = a ( t ).The evolution of the wavefunction ψ ( x, t ) of a particle trapped in this potentialis given by the following Schr¨odinger equation i ¯ h ∂ψ∂t = " − ¯ h m ∂ ∂x + V p ( x, t ) ψ, (2)where m is the particle mass and ¯ h is Planck’s constant. The wavefunction ψ is square-integrable on the interval [ o, a ], with | ψ ( x, t ) | dx expressing theprobability to find the particle between x and x + dx at time t . The abovephysical system can serve as a model for a quantum piston.We are interested in the expansion of the piston from a (0) = a > a ( T ) = a T > a , at some final time t = T . If the expansion is slow enough(adiabatic) then, according to the adiabatic theorem, the solution of (2) isgiven approximately by ψ ( x, t ) ≈ ∞ X n =1 c n exp [ iφ n ( t )]Ψ n ( x, t ) , (3)whereΨ n ( x, t ) = s a ( t ) sin " nπxa ( t ) (4)are the instantaneous eigenstates of the right hand side in (2) [14], φ n are theso-called adiabatic phases, and c n are constant coefficients determined by theinitial condition. The slower is the expansion, the better is the approximationin (3). For fast expansion the adiabatic approximation breaks down.A method has been proposed recently to accelerate adiabatic quantum dy-namics, according to which an auxiliary potential is designed such that the3ystem acquires a similar quantum state with the reference adiabatic path (3)in arbitrarily short time T [3]. For the case of an expanding piston, it has beenshown in [9] that if the following potential is added in (2) V a ( x, t ) = 12 k ( t ) x , (5) k ( t ) = − m ¨ a ( t ) a ( t ) , (6)then the exact solution of the resulting Schr¨odinger equation can be expressedas ψ ( x, t ) = ∞ X n =1 c n exp − i t Z E n ( t ′ ) dt ′ / ¯ h × (7)exp " i m ˙ a ( t )2¯ ha ( t ) x Ψ n ( x, t ) , where E n ( t ) = n π ¯ h ma ( t ) (8)are the instantaneous energy eigenvalues corresponding to the eigenstates (4)[14]. The boundary conditions a (0) = a , a ( T ) = a T , ˙ a (0) = ˙ a ( T ) = 0 , (9)ensure the expansion of the piston from a to a T in the interval 0 < t < T ,and that the exact solution (7) is similar to the adiabatic one (3) at t = 0 and t = T . The additional conditions k (0) = k ( T ) = 0 (10)guarantee that the auxiliary potential is active only within 0 < t < T . The auxiliary potential can be designed using an elegant inverse engineeringmethod [5]. The final time T is fixed and the condition (10) is translatedthrough (6) to¨ a (0) = ¨ a ( T ) = 0 . (11)4 polynomial ansatz satisfying (9) and (11) has been found in [9], a ( τ ) /a =1 + ( γ − τ (6 τ − τ + 10), where τ = t/T and γ = a T /a is the expansionfactor. Then, the stiffness k ( t ) of the auxiliary potential can be determinedfrom (6) k ( τ ) = − mT γ − τ (2 τ − τ + 1)1 + ( γ − τ (6 τ − τ + 10) , (12)where again τ = t/T . Note that there is no mathematical limitation on thesize of T , which can be chosen arbitrarily small in theory. In practise, there are always experimental constraints, for example − k ≤ k ( t ) ≤ k , (13)which restrict T to some finite value. In such cases, finding the shortestadiabatic-like path can be expressed as an optimal control problem. If weset x = a ( t ) a , x = s mk ˙ a ( t ) a ( t ) , u ( t ) = k ( t ) k , (14)and normalize time according to t new = t old /T , where T = q m/k , we obtainthe following system, equivalent to equation (6)˙ x = x , (15)˙ x = − ux . (16)The minimum time adiabatic shortcut, under the constraint (13), can be foundby solving the following optimal control problem: Problem 1
Find u ( t ) with − ≤ u ( t ) ≤ such that starting from ( x (0) , x (0)) =(1 , , the above system with x ( t ) > reaches the final point ( x ( T ) , x ( T )) =( γ, , γ > , in minimum time T . The boundary conditions on x and x correspond to those for a and ˙ a from(9), while the additional constraint x ( t ) > a ( t ) >
0. In the following section we solve the above problem on theinterval 0 < t < T . In order to much the boundary conditions u (0) = u ( T ) = 0,5orresponding to (10), the optimal control may be complemented with instan-taneous jumps at the initial and final times which do not affect the cost (time),see Fig. 1 in Section 4. This approach is similar to that used in our recentwork [23,21], as well as in [13]. Note that the problem of finding the control u ( t ) with 0 < u min ≤ u ≤ u max which drives in minimum time the system(15) and (16) (the parametric oscillator) from the point ( x , x ) to the ellipse x + u T x = 2 E T , with specified oscillator energy E T for stiffness u T , has beenstudied thoroughly in [1]. In Problem 1 the control is allowed to take negativevalues, while the target is a point and not a curve of the state space. The system described by (15) and (16) can be expressed in compact form as˙ x = f ( x ) + ug ( x ) , (17)where the vector fields are given by f = x , g = − x (18)and x ∈ D = { ( x , x ) ∈ R : x > } , u ∈ U = [ − , U .Given an admissible control u defined over an interval [0 , T ], the solution x of the system (17) corresponding to the control u is called the correspondingtrajectory and we call the pair ( x, u ) a controlled trajectory.For a constant λ and a row vector λ ∈ ( R ) ∗ the control Hamiltonian forsystem (17) is defined as H = H ( λ , λ, x, u ) = λ + λ [ f ( x ) + ug ( x )]. Pon-tryagin’s Maximum Principle [17] provides the following necessary optimalityconditions: Theorem 2 (Maximum principle)
Let ( x ∗ ( t ) , u ∗ ( t )) be a time-optimal con-trolled trajectory that transfers the initial condition x (0) = x of system (17)into the terminal state x ( T ) = x T . Then it is a necessary condition for opti-mality that there exists a constant λ ≤ and nonzero, absolutely continuousrow vector function λ ( t ) such that:(1) λ satisfies the adjoint equation ˙ λ = − ∂H/∂x .(2) For ≤ t ≤ T the function u H ( λ , λ ( t ) , x ∗ ( t ) , u ) attains its maximumover the control set U at u = u ∗ ( t ) . H ( λ , λ ( t ) , x ∗ ( t ) , u ∗ ( t )) ≡ . In the following we use maximum principle to solve Problem 1.
Definition 3
We denote the vector fields corresponding to the constant bangcontrols u = − and u = 1 by X = f − g and Y = f + g , respectively, and callthe corresponding trajectories X - and Y -trajectories. A concatenation of an X -trajectory followed by a Y -trajectory is denoted by XY while the concatenationin the inverse order is denoted by Y X . Theorem 4 (Optimal solution)
The optimal trajectory for Problem 1 hasthe one-switching form XY . The optimal control is u ( t ) = − , < t < T X , T X < t < T X + T Y , (19) where T X = sinh − s γ − , (20) T Y = sin − γ s γ − . (21) The minimum expansion time is T = T X + T Y (22) Proof.
We show first that the optimal control is bang-bang, i.e., alternatesbetween the boundary values u = ± H ( λ , λ, x, u ) = λ + λ x − λ x u, (23)and thus˙ λ = uλ , (24)˙ λ = − λ . (25)Observe that H is a linear function of the bounded control variable u . Thecoefficient at u in H is − λ x and, since x >
0, its sign is determined by Φ = − λ , the so-called switching function . According to the maximum principle,7oint 2 above, the optimal control is given by u = sign Φ, if Φ = 0. Themaximum principle provides a priori no information about the control at times t when the switching function Φ vanishes. Now observe that whenever Φ( t ) = − λ ( t ) = 0 at some time t , then ˙Φ( t ) = − ˙ λ ( t ) = λ ( t ) = 0 since the maximumprinciple requires that it is always λ = ( λ , λ ) = 0. Hence, when Φ( t ) = 0it is also ˙Φ( t ) = 0 and there is a switch between the control boundary valuesat time t . The optimal trajectory consists of a concatenation of X - and Y -trajectories.We now move to narrow the candidate sequences for optimality. We showfirst that the concatenation XY X cannot be part of the optimal trajectory.Without loss of generality assume that the XY switch takes place at t = 0,so λ (0) = 0, while it is also x (0) >
0. Along the Y -trajectory it is u = 1, sofrom (24) and (25) we find λ ( t ) = − λ (0) sin t . The subsequent Y X switchshould take place at t = π , where the switching function Φ = − λ becomeszero again. But observe that for u = 1 the state equations (15) and (16)correspond to a rotation around the origin with period 2 π , so at t = π thestate x has been rotated by half circle and consequently x ( π ) <
0, whichis forbidden. We next show similarly that the sequence
Y XY cannot also bepart of the optimal trajectory. We assume that the
Y X switch takes place at t = 0, thus λ (0) = 0. Along the X -trajectory it is u = −
1, so from (24) and(25) we obtain λ ( t ) = − λ (0) sinh t . Observe that Φ = − λ = 0 for t > XY switch is not allowed.We conclude that the only candidates for optimality left are the XY and Y X trajectories. Furthermore, it is not hard to see that only the former correspondsto expansion (final γ > X -trajectory starting from (1 ,
0) and of the Y -trajectory ending at ( γ,
0) can be found from (15) and (16) for u = ∓ x − x = 1 , (26) x + x = γ . (27)The switching point satisfies both equations and it is ( √ γ + 1 / √ , √ γ − / √ T X and T Y spent on each segment of theoptimal trajectory can be easily derived as in (20) and (21). ✷ In Fig. 1 we plot the time-optimal control u ( t ) from (19) (solid line, T =3 . T ), as well as the control k ( t ) /k obtained from the inverse engineering8 u ( t ) Fig. 1. Optimal control (solid line, T = 3 . T ) and inverse engineering control(dashed line, T = 6 . T ), under the constraint − ≤ u ( t ) ≤ γ = 10. Observe that both controls are activeonly within 0 < t < T . method (12) and with minimum duration T under the constraint (13) (dashedline, T = 6 . T ), both for the final expansion factor a T /a = γ = 10. Ob-serve that the control derived from the inverse engineering method is actuallylimited by the lower bound u = − x and velocity x . The acceleration (force) acting on the particle is − ux . As we can observefrom Fig. 2, in both trajectories the particle traverses the same distance x but along the time-optimal trajectory its speed ˙ x = x is always higher.In Fig. 3 we plot the expansion time T in units of T = q m/k as a function ofthe expansion factor γ for both control policies. This graph provides the speedlimits of the adiabatic-like expansion under condition (13). Even if the optimalbang-bang controls are not experimentally exactly realizable, knowledge ofthe time-optimal solutions is a useful guide for the design of more realisticcontrols. For example, the abrupt changes of the optimal control shown in Fig.1 can be approximated by ramps of finite duration [22,13]. In general, morecomplicated constraints on the control or even the state can be incorporated inthe current formalism using a powerful numerical optimization method basedon pseudospectral approximations [22].The above results have an interesting thermodynamic application. The quan-tum piston using noninteracting particles as the working medium and execut-ing the Otto cycle provides a model for a refrigerator, similar to that consideredin [19]. Following the procedure described in this work, the cooling rate R ofthe refrigerator can be calculated, as the temperature of the cold reservoir τ c approaches absolute zero. This rate is defined as R = Q/T , the ratio of the9 x Fig. 2. Corresponding trajectories for the control inputs of Fig. 1. heat Q extracted from the cold reservoir on each cycle to the duration T ofthe cycle. We find Q first. When the working medium is in contact with thehot reservoir of temperature τ h , its internal energy is τ h / a to a T . At the end of this process the populations ofthe energy levels are preserved (3), while the energies are reduced by a factorof γ = a T /a (8), thus the internal energy of the working medium becomes τ h / γ . After the expansion the working medium is brought in contact withthe cold reservoir and its internal energy is raised to τ c /
2. The heat extractedfrom the cold reservoir is Q = τ c / − τ h / γ . A necessary condition for op-eration of the refrigerator is Q > ⇒ γ > q τ h /τ c . For τ c → γ → ∞ , and in this limit the duration of the cycle is dominated by theduration of the adiabatic expansion. From (20) and (21) we find that for thefastest adiabatic-like expansion it is T = T X + T Y → ln γ for γ → ∞ , thus thecooling rate is restricted as R = Q/T < − τ c / ln τ c for τ c →
0, and the result of[13] is reobtained. As τ c → ω of the external harmonicpotential, necessary to keep the atoms trapped during the expansion, is relatedto the lattice scale parameter Λ by the relation ω = − ¨Λ / Λ, which is similar to(6). As a consequence, the present analysis applies also to this context. Notethat such accordion lattices are useful since the final lattice spacing can bemade large enough to be resolved experimentally, for example by imaging ofthe atoms at individual sites. The final quantum state is a scaled-up versionof the initial state due to the adiabatic-like evolution, thus the optical latticeacts like a quantum dynamical microscope [8].10 γ T / T Fig. 3. Expansion time T (in units of T = p m/k ) as a function of the expan-sion factor γ , for the optimal (solid line) and the inverse engineering (dashed line)strategies. In this paper we formulated and solved the problem of minimum-time adiabatic-like expansion for a quantum piston, in the presence of experimental con-straints. As a result, we obtained the speed limit for this fast quantum driv-ing, and used it to rederive an interesting result related to the third law ofthermodynamics. We also highlighted the possible application of the presentwork to the adiabatic-like expansion of an optical lattice.
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