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Evolution processes of coupled thermal qubits
Physics Letters A 383 (2019) 125882
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Physics Letters A www.elsevier.com/locate/pla
Evolution processes of coupled thermal qubits Theerawat Chatchawaltheerat, Supitch Khemmani ∗ , Julian Poulter Department of Physics, Faculty of Science, Srinakharinwirot University, 114 Sukhumvit 23, Wattana, Bangkok 10110, Thailand
a r t i c l e
i n f o
Article history: Received 17 May 2019 Received in revised form 3 August 2019 Accepted 5 August 2019 Available online 9 August 2019 Communicated by M.G.A. Paris Keywords: Quantum speed limit time (QSLT) Born-Markov approximation Quantum speed-up and speed-down processes Thermal qubits Double cusp behaviour Markovian regime
a b s t r a c t We investigate the quantum speed limit time (QSLT) of quantum evolution before thermal equilibrium of two coupled qubits each of which is coupled to a separate thermal bath at the same temperature within the Born-Markov approximation. The evolution process in one particular initial state can change between speed-up and speed-down two times before reaching equilibrium. We call this double cusp behaviour. This behaviour is an anomalous phenomenon in evolution processes in the weak-coupling Markovian regime. We study QSLT corresponding to all pure initial energy eigenstates and categorise them. In addition, we also display the conditions for double cusp behaviour in terms of temperature, qubit interaction and frequency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The quantum speed limit (QSL) which describes the maximum speed of evolution between two distinguishable states is of much importance for many fields of applications such as quantum communication [1] and quantum computation [2]. QSL also provides limits to the computational capability of many physical devices [3] including, for example, quantum control [4–7] and the decay of unstable quantum systems [8]. Also, some exact results on QSL for qubit systems were derived [9]. Determining a QSL is equivalent to determining a minimal evolution time, called quantum speed limit time (QSLT), between two given states. The QSLT between two orthogonal pure states through unitary dynamics was first discussed by Mandelstam and Tamm (MT) [10]. An alternative bound was introduced by Margolus and Levitin (ML) [11]. This theory was generalized to the evolution between two arbitrary states not necessarily orthogonal and also to open systems and mixed states [12–14]. QSLT for open quantum systems have had much attention, and some valuable work has been done, in recent years [15]. Recently, some progress on the analysis of environmental effects on the quantum speed limit for open quantum systems have been made, for example [16–20] which found that non-Markovianity can speed up the quantum evolution. However, Markovianity can also speed up the evolution [21]. Moreover, there are some interest-
ing works on global and local master equations for two interacting systems coupled to two separate heat baths, for example [22,23]. In this paper, we shall derive the QSLT by using relative purity [24] as the distance measure [25] in the Born-Markov approximation. Within the Markovian regime, the evolution processes, including quantum speed-up and speed-down, corresponding to given initial states will be analysed for the system of two coupled qubits each of which is coupled to an environment which is, unlike most of the existing studies, a heat bath at finite temperature. This paper explains and categorises how this quantum system evolves in time, before reaching thermal equilibrium in terms of quantum speed-up and speed-down via the concept of QSLT. Our analysis contains controllable parameters such as a qubit-to-qubit coupling constant and energy gap for each qubit. In addition to a theoretical interest, our work will be useful for an understanding and realization of quantum evolution speed control at finite temperatures in devices for the future. 2. Evolution of two coupled thermal qubits We consider the system of two coupled qubits each of which is coupled to a separate bosonic bath. The total Hamiltonian of the system is given by [26]
H = H S + H BA + H BB + H IA + H IB ,
*
Corresponding author. E-mail addresses: [email protected] (T. Chatchawaltheerat), [email protected] (S. Khemmani), [email protected] (J. Poulter). https://doi.org/10.1016/j.physleta.2019.125882 0375-9601/© 2019 Elsevier B.V. All rights reserved.
(1)
where H S is the Hamiltonian of two coupled qubits (here, we set h¯ = k B = 1),
2
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
HS =
1 2
1
ω A σ AZ + ω B σ BZ + g σ A+ ⊗ σ B− + σ A− ⊗ σ B+ . 2
(2)
The Hamiltonian H S describes the qubit-to-qubit interaction, where σ jZ and σ j± are Pauli matrices. The parameter g represents the strength of the coupling between the two qubits. The Hamiltonian H B j of the reservoir for each qubit ( j = A , B) is given by
HBj =
ωk, j b† bk, j ,
(3)
k, j
k
which represents an infinite set of harmonic oscillators. The opera† tors b and b are bosonic creation and annihilation operators. k, j
| E 1 = |11 1 | E 2 = √ |10 + |01 2 1
|E 3 = √
2 | E 4 = |00
E 1 = ω, E2 = g,
|10 − |01
E3 = −g, E 4 = −ω.
(8)
Here, the transition frequencies (ωμ ) are
ω1 = ω − g ,
(9)
and
k, j
The interaction between a qubit and its bosonic bath is of the Jaynes-Cummings type and is represented by the Hamiltonian
σ j+ ⊗
HIj =
g b + σ j− ⊗ k, j ( j) k
=
k
g b k
k
∗( j ) †
k, j
k
† V j(
ωμ ) ⊗ f j + V j (ωμ ) ⊗ f j†(k) , (k)
(4)
μ
(k)
where the operators V j (ωμ ) and f j act on the two qubit system and the bosonic bath degrees of freedom, respectively. The μ index of ωμ labels all possible energy differences. There are two values of μ in this case, where V j (ωμ ) does not vanish. The master equation in the Born-Markov approximation for the reduced density operator of the qubits system in the interaction picture is [27]
d dt
∞
ρ (t ) = −
(I )
(I )
tr B H I (t ), H I
t − s , ρ (t ) ⊗ ρ B ds .
(5)
0
dt
ρ (t ) = −i H S , ρ (t ) + L A (ρ ) + L B (ρ ),
(6)
where the operators L j describe dissipation to the corresponding reservoir,
L j (ρ ) =
2
† J ( j ) (−ωμ ) V j (ωμ )ρ (t ) V j (ωμ )
μ=1
1 † 1 † − V j (ωμ ) V j (ωμ )ρ (t ) − ρ (t ) V j (ωμ ) V j (ωμ ) 2 2 † + J ( j ) (ωμ ) V j (ωμ )ρ (t ) V j (ωμ ) 1 1 † † − V j (ωμ ) V j (ωμ )ρ (t ) − ρ (t ) V j (ωμ ) V j (ωμ ) . (7) 2
(10)
and the transition operators V j (ωμ ) are
1 V A (ω1 ) = √ | E 2 E 1 | + | E 4 E 3 | , 2 1 V A (ω2 ) = √ | E 4 E 2 | − | E 3 E 1 | , 2 1 V B (ω1 ) = √ | E 2 E 1 | − | E 4 E 3 | , 2 1 V B (ω2 ) = √ | E 3 E 1 | + | E 4 E 2 | . 2
(11)
Using eqs. (7)–(11), the analytical solution of eq. (6) in the ordered basis of eigenvectors {| E 1 , | E 2 , | E 3 , | E 4 } is given by [26]
⎛
a11 1 ⎜ a21 ⎜ ρii (t ) = Ω1 Ω2 ⎝ a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
⎞⎛
2
Here the spectral density J ( j ) (ωμ ) = γ j N j (ωμ ), where N j (ωμ ) = (e ωμ / T j − 1)−1 is the Planck distribution and J ( j ) (−ωμ ) = e ωμ / T j × J ( j ) (ωμ ). We choose the coupling constants to be equal and frequency independent, that is γ A = γ B = γ , which corresponds to a uniform spectral density as defined in much of the literature, for example [28] and [29]. By considering the resonance case, that is ω A = ω B = ω, chosen throughout this work, the eigenvectors | E n and the corresponding eigenvalues E n of H S of eq. (2) are
a11 = X 1+ + X 1− e −Ω1 t a31 = X 1+ + X 1− e −Ω1 t
Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t Y 2+ Y 2− + Y 2+ e −Ω2 t
a21 = 1 − e −Ω1 t X 1− Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t X 1− Y 2−
a22 = X 1− + X 1+ e −Ω1 t
a42 = X 1− + X 1+ e −Ω1 t
a23 = 1 − e −Ω1 t
1 − e −Ω2 t X 1+ Y 2+
a34 = 1 − e −Ω1 t X 1+ Y 2− + Y 2+ e −Ω2 t a41 = 1 − e −Ω1 t
a33 = X 1+ + X 1− e −Ω1 t
1 − e −Ω2 t X 1+ Y 2−
a13 = X 1+ + X 1− e −Ω1 t a14 = 1 − e −Ω1 t
(12)
1 − e −Ω2 t Y 2−
a12 = 1 − e −Ω1 t X 1+ Y 2+ + Y 2− e −Ω2 t a32 = 1 − e −Ω1 t
⎞
a14 ρ11 (0) ⎜ ρ22 (0) ⎟ a24 ⎟ ⎟⎜ ⎟, a34 ⎠ ⎝ ρ33 (0) ⎠ a44 ρ44 (0)
where the coefficients ai j are
In this paper, we consider thermal qubits which means that the operator ρ B is described by ρ B = e − H B A / T A /tr[e − H B A / T A ] ⊗ e − H B B / T B /tr[e − H B B / T B ], where T j ( j = A , B) is the absolute temperature of the j th bath for qubit. After performing the rotating wave approximation [27] in eq. (5), one arrives at the master equation in the Schrödinger picture
d
ω2 = ω + g ,
Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t Y 2−
1 − e −Ω2 t X 1− Y 2+
a43 = 1 − e −Ω1 t X 1− Y 2− + Y 2+ e −Ω2 t
a24 = X 1− + X 1+ e −Ω1 t a44 = X 1− + X 1+ e −Ω1 t
1 − e −Ω2 t Y 2+ Y 2− + Y 2+ e −Ω2 t
where
1 ( A) J (±ω1 ) + J ( B ) (±ω1 ) , 2 1 ( A) ± Y2 = J (±ω2 ) + J ( B ) (±ω2 ) , 2 Ω1 = X 1+ + X 1− , Ω2 = Y 2+ + Y 2− . X 1± =
(13) (14)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
3. Continuous evolution processes and some definitions In this section, the continuous evolution processes is studied by considering the evolution from ρ (t ) to ρ (t + τ D ), where the initial time t steps from 0, τ D , 2τ D , . . ., for given τ D . For an open quantum system, it is known that for fixed ρ (t ) and ρ (t + τ D ) (for each t), the evolution time between them is indeed not unique [30] with a lower bound denoted by τQSL called the quantum speed τ limit time. This means that we always have 0 ≤ τQSL ≤ 1, where D τQSL = 0 is a minimum lower bound, and if this ratio is decreased for each time step, then it means that there is a capability of quantum speeding up because there is more possibility for a system to evolve with time less than τ D for each time step. On the other hand, if this ratio is increased for each time step, then there is a capability of a quantum speeding down. At this point, it is clear that in order to investigate these speed behaviours instantaneously, the evolution processes should be considered continuously with small τ D . Moreover, if this ratio is indeed a proper measurement of quantum speed behaviour, it should be independent of τ D for small enough τ D so that, for any given τ D (but small enough), one will obtain the same behaviours of this ratio as a function of time. This fact can be demonstrated as follows. Starting with the relative purity [24]
f (t + τ D ) =
tr[ρ (t + τ D )ρ (t )] tr[ρ 2 (t )]
,
| f (t + τ D ) − 1|tr[ρ 2 (t )] τQSL = t +τ D n τD i =1 σi (t )ρi (t )dt t |tr[ρ (t + τ D )ρ (t )] − tr[ρ 2 (t )]| = t +τ D n , i =1 σi (t )ρi (t )dt t
γ 4 | f ii (t )| n , 4(Ω1 Ω2 )2 i =1 σi (t )ρi (t )
(19)
where
f 11 (t ) = α1 Ω1 (1 − α1 ) (1 − α2 )2 +
Ω22
γ2
Ω2 + α2 Ω2 (1 − α2 ) (1 − α1 )2 + 21 ,
γ
(20)
ρ (0) = | E 1 E 1 | or ρ11 (0) = 1,
Ω2 f 22 (t ) = −β1 Ω1 (1 + β1 ) (1 − α2 )2 + 22
for the initial condition
γ
(16)
γ
(21)
ρ (0) = | E 2 E 2 | or ρ22 (0) = 1,
Ω22 2 f 33 (t ) = α1 Ω1 (1 − α1 ) (1 + β2 ) + 2
for the initial condition
γ
Ω12 2 − β2 Ω2 (1 + β2 ) (1 − α1 ) + 2 ,
γ
(22)
ρ (0) = | E 3 E 3 | or ρ33 (0) = 1,
Ω22 2 f 44 (t ) = −β1 Ω1 (1 + β1 ) (1 + β2 ) + 2
for the initial condition
(17)
Notice that eq. (17) is independent of τ D and thus provides us with the well-defined speed function S (t ) as
|tr[ρ˙ (t )ρ (t )]| . S (t ) = n i =1 σi (t )ρi (t )
S (t ) =
Ω2 + α2 Ω2 (1 − α2 ) (1 + β1 )2 + 21 ,
where σi (t ) and ρi (t ) are the singular values of dρ (t )/dt and ρ (t ) respectively, and ordered as σ1 ≥ ... ≥ σn and ρ1 ≥ ... ≥ ρn . For small enough τ D , eq. (16) becomes
τQSL |tr[(ρ (t ) + τ D ρ˙ (t ))ρ (t )] − tr[ρ 2 (t )]| ≈ τD τ D ni=1 σi (t )ρi (t ) |tr[ρ˙ (t )ρ (t )]| = n . i =1 σi (t )ρi (t )
obtained in eq. (12) is derived from the Born-Markov master equation with the coarse-grained time axis. This means that the time step τ D must be larger than the time scale τ B over which the reservoir correlation function decays [27]. To show that such a small τ D satisfying this condition can always be chosen, we first notice from eq. (12) that the condition for small enough τ D described in the previous section is that Ωi τ D 1 or τ D 1/Ωi < 1/γ , and τ R = min{1/Ω1 , 1/Ω2 } which is the relaxation time of the reduced system. From the condition for obtaining the Markovian master equation, that is τ B τ R [27], we have τ B 1/γ . Since both τ D and τ B must be much less than 1/γ , one can always choose small enough τ D so that the speed function S (t ) can be defined while having τ D τ B . Now, substituting eq. (12) into eq. (17), the speed function corresponding to each initial energy eigenstate | E n can be expressed, after some algebra, in the form
(15)
the von Neumann trace inequality leads to the ML bound and thus τQSL of the form [25]
3
(18)
Now, the quantum speed behaviour of the evolution process can be defined as follows. The evolution process at time t is called “speed-up or acceleration process” if S˙ (t ) < 0, “speed-down or deceleration process” if S˙ (t ) > 0, and “stationary process” if S˙ (t ) = 0. Note that although the speed function S (t ) in eq. (18) is defined by treating the time as a positive continuous real number, S (t ) is also valid for a coarse-grained time axis of ρ (t ) satisfying the Born-Markov approximation too. This will be explained in more detail in the next section. 4. The speed of quantum evolution before thermal equilibrium To study the speed behaviour, we need to compute the speed function S (t ) in eq. (18) by using ρii (t ) in eq. (12). However, ρii (t )
γ
Ω2 − β2 Ω2 (1 + β2 ) (1 + β1 )2 + 21 ,
γ
(23)
for the initial condition ρ (0) = | E 4 E 4 | or ρ44 (0) = 1. Here, the parameters α1 , α2 , β1 and β2 are
α1 =
α2 = β1 = β2 =
Ω1
γ Ω2
γ Ω1
γ Ω2
γ
+ 1 e −Ω1 t , + 1 e −Ω2 t , − 1 e −Ω1 t , − 1 e −Ω2 t .
(24)
Next, we will divide our analysis into four cases corresponding to four initial states of the two qubit system. Here, in obtaining eqs. (19)–(23), we set the temperatures of the baths to be equal,
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T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
and use the rate change of this ratio, instead of the ratio itself, to identify the speed of quantum evolution in general. 4.2. Case 2 (ρ (0) = | E 2 E 2 |)
Fig. 1. S (t ) as a function of the initial time parameter t. Initial state ρ44 (0) = 1, g = 0.1; T = 0.1 (solid line), T = 0.5 (dashed line), T = 0.8 (dotted line). Other parameters are chosen as ω = 1, γ = 0.05.
that is T A = T B = T which corresponds logically to a single bath of temperature T . Therefore, Ωi (i = 1, 2) in eq. (14) becomes
Ωi = γ 2N (ωi ) + 1 , where we first assumed, without loss of generality, that ω > g.
(25)
ω1 > 0 or
4.1. Case 1 (ρ (0) = | E 4 E 4 |) Substituting eq. (23) into eq. (19), we numerically obtain an example graph as shown in Fig. 1. S (t ) ≈ 1 (solid line) indicates that the quantum evolution exhibits a stationary process, while for T = 0.5 (dashed line) and T = 0.8 (dotted line), the evolution exhibits, a speed-up or acceleration process and gradually goes to equilibrium. Note that in ref. [31], the ratio τQSL /τ D represents the potential capacity for quantum speed-up, that is for τQSL /τ D < 1, the quantum evolution has potential capacity for further speed-up. Here, however, we find that for T = 0.5 (dash line) or T = 0.8 (dotted line), the evolution reaches thermal equilibrium with equal τQSL /τ D < 1 in each time step, and the system in equilibrium should physically not possess any speed-up or speed-down processes. By this reason, we call this situation a stationary process
Fig. 2. S (t ) as a function of the initial time parameter t. (a) Initial state g = 0.8 (dotted line). Other parameters are chosen as, ω = 1, γ = 0.05.
Substituting eq. (21) into eq. (19), an example is shown in Fig. 2(a). Unlike Fig. 1 for the case ρ44 (0) = 1, there exists the cusp around t = t c (≈ 14 s) which indicates that just before t c the evolution exhibits a speed-up process but suddenly after t c the evolution instead exhibits a speed-down or deceleration process and then gradually goes to equilibrium. Moreover, one can also prove that there exists only one cusp in this case (see Appendix). At a cusp, the speed function and thus τQSL must be equal to zero because a cusp always appears at a time t c where f ii (t ) changes its sign under the absolute value in eq. (19). From eq. (18), the vanishing of τQSL at time t c comes from the fact that the state ρ (t c ) and its derivative ρ˙ (t c ) at this time are orthogonal. Since, as one can verify numerically from eq. (12), ρ˙ (t c ) = 0 or equivalently ρ (t c ) = ρ (t c + τ D ), the lower bound τQSL = 0 must be not tight otherwise the evolution speed will be infinite which would be impossible for the weak-coupling Markovian regime. Note that, unlike this case, the case 1 of initial state ρ44 (0) = 1 always contains no cusp because all terms in f 44 (t ) are negative. This means that the total evolution process is always smooth without a speed-down process. 4.3. Case 3 (ρ (0) = | E 1 E 1 |) An example graph is shown in Fig. 2(b). The quantum evolution process is similar to that of case 2 which contains only one cusp at time t c (not exactly the same value as t c in Fig. 2(a)) indicating the critical time where the evolution changes abruptly from speed-up to speed-down. Our extensive numerical investigations reveal that this case also has only one cusp. 4.4. Case 4 (ρ (0) = | E 3 E 3 |) The quantum evolution process in this case is divided into two subcases as follows. 4.4.1. Double cusp behaviour From our numerical survey, we found that, for example, for fixed g = 0.8 and T = 0.4 (see the dashed line of Fig. 3(a)), there exist two cusps at t c1 and t c2 which means that the abrupt change
ρ22 (0) = 1, T = 0.8; (b) initial state ρ11 (0) = 1, T = 0.5; g = 0.1 (solid line), g = 0.5 (dashed line)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
5
Fig. 3. S (t ) as a function of the initial time parameter t. Initial state ρ33 (0) = 1, (a) g = 0.8, T = 0.1 (solid line), T = 0.4 (dashed line), T = 0.6 (dotted line), T = 0.8 (dot dash line); (b) T = 0.7, g = 0.1 (line), g = 0.3 (dashed line), g = 0.7 (dotted line), g = 0.8 (dot dashed line). Other parameters are chosen as ω = 1, γ = 0.05.
Fig. 4. Grey zone corresponds to double cusp case; black zone corresponds to smoothed single minimum case. (a) chosen as γ = 0.05.
between speed-up and speed-down processes occurs twice before reaching equilibrium. Moreover, if T is decreased, the second critical time t c2 will be larger and tends to infinity for T approaching zero (see solid line in Fig. 3(a)) which implies that there exists only one cusp in this case at zero temperature. On the other hand, for fixed T = 0.7 and g = 0.3 (see the dash line of Fig. 3(b)), there exist also two cusps and if g is decreased, the second critical time t c2 will be larger (see solid line in Fig. 3(b)) and tends to infinity as g tends to zero. This means that there exists only one cusp at vanishing qubit coupling constant g, as in the case of a single qubit [16]. 4.4.2. Smoothed single minimum Starting with the dashed line in Fig. 3(a), if the temperature T is increased, we found that t c2 becomes closer to t c1 (which is almost fixed) and at a certain temperature t c2 becomes equal to t c1 which is the dotted line in the figure. Moreover, if T is still increased more, then the graph becomes like the dot dashed line. Since the minimum of both lines are smooth instead of a cusp, we call this case a smoothed single minimum. Physically, the evolution behaviour of this case is the same as the case of one cusp except that the change from speed-up to speed-down process occurs smoothly instead of abruptly like in the cusp cases. For Fig. 3(b),
ω = 1. (b) T = 0.5, g = ω (white line). Other parameter is
the explanation is the same but here the temperature is fixed while the coupling constant g is varied. From the above explanation, it is clear that the change from double cusps to smooth single minimum is started at the dotted lines in Fig. 3 when t c1 = t c2 corresponding to parameters that provide f 33 = 0 and ˙f 33 = 0 at this critical time. From this condition, we can find the diagram showing the zone of two cusps (grey) and of smoothed single minimum (black) separated by a common boundary satisfying this condition as shown in the Fig. 4(a). Furthermore, the value of the minimum of the dot dashed line graph in Fig. 4(a) becomes higher for higher temperature and at high enough temperature the graph then becomes like that in Fig. 1. This means that the speed-down process can be ignored at high enough temperature. In Fig. 4(b), the diagram shows both zones not only for ω > g but also for ω < g where the ω and g are interchanged in the formalism leaving the overall dynamics of evolution unchanged, as one can see the symmetry in Fig. 4(b) with respect to the g = ω line (white line). We note that the diagram in Fig. 4(a) is bounded by g- and T -axes which correspond to the case of a single cusp. Although the value of g presented in Fig. 4, can be any value, it is in fact bounded from below according to the condition for the secular or rotating wave approximation [27], that is τ S τ R or g γ /2,
6
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
as in our study τ S = |ω1 − ω2 |−1 = 1/2g and τ R = 1/Ω1 . Due to this condition, the value of the coupling constant γ should be appropriately set according to the range of qubit coupling strength of interest. For weaker γ , for example γ = 0.0005, Fig. 3 and Fig. 4 will be the same except that the value of t c1 and t c2 will be larger due to the fact that the system tends to equilibrium slower for weaker coupling constant. From Fig. 4(a), one can see, for fixed g (or T ), that the speed-up and speed-down processes are present at any temperature T (or coupling constant g) with a tendency for double cusp behaviour, where an abrupt change from speed-up and speed-down occurs twice, at low temperature T (or low coupling constant g). From Fig. 4(b), one also see, for fixed ω (or g) that the speed-up and speed-down processes present at any coupled constant g (or frequency ω ). These processes have a tendency for double cusp behaviour at low g (or high ω ) for ω > g (weak qubit coupling strength) and with a tendency for double cusp behaviour at high g (or low ω ) for ω < g (strong qubit coupling strength). By considering Fig. 4(a) and (b) together, the black zone on Fig. 4(b) will be extended as T is increased. This means that for both weak and strong qubit coupling strengths, speed-down can be ignored for high enough temperature. However, for fixed T (fixed black zone area), increasing g from weak (g < ω ) to strong (g > ω ) coupling strength cannot destroy the speed- down process. Instead, the processes swing from double cusp behaviours (grey) to single minimum (black) and then to double cusp again.
and useful knowledge in explaining the speed of quantum computation or quantum information processes at finite temperature. Acknowledgements We thank Department of Physics, Faculty of Science, Srinakharinwirot University for support and facilities. Appendix A From
α2 and β1 in eq. (24), β1 can be written in terms of α2
as
β1 = G α2P ,
(A.1)
where p = Ω1 /Ω2 ≥ 1, and
( Ωγ1 − 1)
G= Ω < 1. ( γ2 + 1) P
(A.2)
Putting eq. (A.1) into eq. (21), we obtain f 22 (t ) in terms of only as
α2 (t )
f 22 (t ) = f 22
α2 (t ) = −G α2P −1 p 1 + G α2P (1 − α2 )2 +
Ω22
γ2
2 ( p Ω2 )2 . + (1 − α2 ) 1 + G α2P + 2
(A.3)
γ
5. Conclusion and discussion We have derived the QSLT in the Born-Markov approximation for the system of two coupled qubits each of which is coupled to a thermal bath at finite temperature with four initial states that are energy eigenstates of the system. By analysing the speed function defined via the QSLT through continuous processes, we found four types of evolution processes before reaching the equilibrium. First, the process that is composed of speed-up and speeddown processes where the change between them occurs in an abrupt way, called a cusp, at some critical time. This type of process contains only one cusp. Second, the totally smooth process without any cusp or minimum which occurs only for the initial ground state. Third, the process that has two cusps which indicates that, before reaching the equilibrium, the system of two coupled qubits exhibits a change from speed-up to speed-down twice. Our extensive numerical investigations reveal that the maximum number of cusps is two for the model we used. We believe that this double cusp behaviour is originated from the fact that we have two qubits in which their energy exchange mechanism for this model plays a crucial role. Note that if one does the same analysis in this paper for the same model but with a system of one qubit, then the number of existing cusp is at most one. However, multiple cusps can occur in some models (see for example, [28]). Fourth, the process that has one smooth single minimum which has the same behaviour as the first type except that the change from speed-up to speed-down process occurs smoothly instead of abruptly. This process can actually be viewed as a third process where two cusps merge together for appropriate parameters. This means that the speed-up and speed-down processes of any type except the second one can occur at any parameters, including temperature. We also found that low temperature or low (high) qubit coupling constant can stimulate double cusp behaviour for the weak (strong) qubit coupling strength. For high enough temperature the speed-down process can be ignored only for the fourth type. However, high qubit coupling constant cannot destroy the speed-down process. We expect our results to provide definite
Before proving that there exists only one cusp, we first need to show that f 22 (α2 ) is a monotonic decreasing function. Beginning from eq. (A.3), we have
d dα2
f 22 (α2 ) = −G ( p − 1)α2P −2 ( p ) 1 + G α2P
(1 − α2 )2 +
2
Ω22
γ2
Ω − G α2P −1 ( p ) pG α2P −1 (1 − α2 )2 + 22 p −1 1+ 2
γ
p G 2 (1 − 2 ) ( p Ω2 )2 P 2 . + 2 2
+ 4pG α
− 1 + Gα
α
α
(A.4)
γ
Notice that only the third term in eq. (A.4) can be positive. So, we claim that p −1 1+ 2
4pG α
p 2 (1 −
Gα
P 2 2
α2 ) −
1 + Gα
+
( p Ω2 )2
< 0.
γ2
First, notice that p −1
4pG α2
p
1 + G α2 (1 − α2 ) < 4pG (1 + G ) and
2 ( p Ω2 )2 ( p Ω2 )2 < − . − 1 + G α2P + 1 + 2 2
γ
γ
Then we have p −1
4pG α2
p
1 + G α2 (1 − α2 ) −
1 + G α2P
2
( p Ω2 )2 . < 4pG (1 + G ) − 1 + 2
+
( p Ω2 )2
γ2
γ
Now, for the right-hand side of the above inequality, suppose that
4Gp (1 + G ) − 1 + ( p Ω2 )2 /γ 2 > 0 or G>
1 2
1+
Ω2 Ω1
1+
Ω12
γ2
− 1 =: F .
(A.5)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882 ln G Since, for fixed Ω1 , one can verify from eq. (A.2) that ∂∂Ω >0 2 which implies that ∂ G /∂Ω2 > 0. Therefore,
( Ωγ1 − 1) . G max = G (Ω2 = Ω1 ) = Ω ( γ1 + 1) Next, since Ω2 ≥ γ ,
F min = F (Ω2 = γ ) =
1 2
1+
γ Ω1
1+
Ω12
γ2
−1 .
From these G max and F min , the inequality y 4 − 6 y 3 + 10 y 2 + 2 y + 1 > 0 implies that ( y = Ω1 /γ ) G max < F min which is the contradiction. Hence, we have the claim and, therefore df 22 (α2 )/dα2 < 0. Next, from α2 in eq. (24), we have 0 < α2 < Ωγ2 + 1. Since f 22 (α2 = 0) > 0 and f 22 (α2 = 1) < 0, it is clear from df 22 (α2 )/dα2 < 0 that f 22 (α2 ) has only one root. Because of α2 (t ) is a one-to-one function, f 22 (t ) has also only one root and thus there exists only one cusp. References [1] J.D. Bekenstein, Phys. Rev. Lett. 46 (1981) 623. [2] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England, 2000. [3] S. Lloyd, Nature (London) 406 (2000) 1047. [4] T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, G.E. Santoro, Phys. Rev. Lett. 103 (2009) 240501. [5] D. Sungny, C. Kontz, H.R. Jauslin, Phys. Rev. A 76 (2007) 023419.
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[6] V. Mukherjee, A. Carlini, A. Mari, T. Caneva, S. Motangero, T. Calarco, R. Fazio, V. Giovannetti, Phys. Rev. A 88 (2013) 062326. [7] S. Deffner, S. Cambell, J. Phys. A, Math. Theor. 50 (2017) 453001. [8] A. Chenu, M. Beau, J. Cao, A. del Campo, Phys. Rev. Lett. 118 (2017) 140403. [9] L. Zhang, Y. Sun, S. Luo, Phys. Lett. A 382 (2018) 2599. [10] L. Mandelstam, I. Tamm, J. Phys. USSR 9 (1945) 249–254. [11] N. Margolus, L.B. Levitin, Phys. D 120 (1998) 188–195. [12] J. Anandan, Y. Aharonov, Phys. Rev. Lett. 65 (1990) 1697–1700. [13] A. Uhlmann, Phys. Lett. A 161 (1992) 329–331. [14] D. Mondal, A.K. Pati, Phys. Lett. A 380 (2016) 1395. [15] A. del Campo, I.L. Egusquiza, M.B. Plenio, S.F. Huelga, Phys. Rev. Lett. 110 (2013) 050403. [16] S. Deffner, E. Lutz, Phys. Rev. Lett. 111 (2013) 010402. [17] Z.Y. Xu, S. Luo, W.L. Yang, C. Liu, S. Zhu, Phys. Rev. A 89 (2014) 012307. [18] Y.J. Zhang, W. Han, Y.J. Xia, J.P. Cao, H. Fan, Phys. Rev. A 91 (2015) 032112. [19] M.M. Taddei, B.M. Escher, L. Davidovich, R.L. de Matos Filho, Phys. Rev. Lett. 110 (2013) 050402. [20] I. Marvian, D.A. Lidar, Phys. Rev. Lett. 115 (2015) 210402. [21] N. Mirkin, F. Toscano, D.A. Winiacki, Phys. Rev. A 94 (2016) 052125. [22] A. Levy, R. Kosloff, Europhys. Lett. 107 (2014) 20004. [23] P.P. Hofer, M. Perarnau-Llobet, L. David, M. Miranda, G. Haack, R. Silva, J.B. Brask, N. Brunner, New J. Phys. 19 (2017) 123037. [24] K.M.R. Audenaert, Quantum Inf. Comput. 14 (2014) 31–38. [25] Y.J. Zhang, W. Han, Y.J. Xia, J.P. Cao, H. Fan, Sci. Rep. 4 (2014) 4890. [26] I. Sinaysky, F. Petruccione, D. Burgarth, Phys. Rev. A 78 (2008) 062301. In our notation, this reference used {| E 4 , | E 1 , | E 2 , | E 3 } as the ordered basis. [27] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002. [28] J.-Q. Cheng, G.-Q. Zhang, J.-B. Xu, J. Opt. Soc. Am. B 35 (9) (2018) 2192–2201. [29] M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer, Berlin, 2007. [30] X. Meng, C. Wu, H. Guo, Sci. Rep. 5 (2015) 16357. [31] W.-J. Wu, K. Yan, Y. Wu, X. Hao, Mod. Phys. Lett. B 30 (2016) 1650389.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Evolution processes of coupled thermal qubits Theerawat Chatchawaltheerat, Supitch Khemmani ∗ , Julian Poulter Department of Physics, Faculty of Science, Srinakharinwirot University, 114 Sukhumvit 23, Wattana, Bangkok 10110, Thailand
a r t i c l e
i n f o
Article history: Received 17 May 2019 Received in revised form 3 August 2019 Accepted 5 August 2019 Available online 9 August 2019 Communicated by M.G.A. Paris Keywords: Quantum speed limit time (QSLT) Born-Markov approximation Quantum speed-up and speed-down processes Thermal qubits Double cusp behaviour Markovian regime
a b s t r a c t We investigate the quantum speed limit time (QSLT) of quantum evolution before thermal equilibrium of two coupled qubits each of which is coupled to a separate thermal bath at the same temperature within the Born-Markov approximation. The evolution process in one particular initial state can change between speed-up and speed-down two times before reaching equilibrium. We call this double cusp behaviour. This behaviour is an anomalous phenomenon in evolution processes in the weak-coupling Markovian regime. We study QSLT corresponding to all pure initial energy eigenstates and categorise them. In addition, we also display the conditions for double cusp behaviour in terms of temperature, qubit interaction and frequency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The quantum speed limit (QSL) which describes the maximum speed of evolution between two distinguishable states is of much importance for many fields of applications such as quantum communication [1] and quantum computation [2]. QSL also provides limits to the computational capability of many physical devices [3] including, for example, quantum control [4–7] and the decay of unstable quantum systems [8]. Also, some exact results on QSL for qubit systems were derived [9]. Determining a QSL is equivalent to determining a minimal evolution time, called quantum speed limit time (QSLT), between two given states. The QSLT between two orthogonal pure states through unitary dynamics was first discussed by Mandelstam and Tamm (MT) [10]. An alternative bound was introduced by Margolus and Levitin (ML) [11]. This theory was generalized to the evolution between two arbitrary states not necessarily orthogonal and also to open systems and mixed states [12–14]. QSLT for open quantum systems have had much attention, and some valuable work has been done, in recent years [15]. Recently, some progress on the analysis of environmental effects on the quantum speed limit for open quantum systems have been made, for example [16–20] which found that non-Markovianity can speed up the quantum evolution. However, Markovianity can also speed up the evolution [21]. Moreover, there are some interest-
ing works on global and local master equations for two interacting systems coupled to two separate heat baths, for example [22,23]. In this paper, we shall derive the QSLT by using relative purity [24] as the distance measure [25] in the Born-Markov approximation. Within the Markovian regime, the evolution processes, including quantum speed-up and speed-down, corresponding to given initial states will be analysed for the system of two coupled qubits each of which is coupled to an environment which is, unlike most of the existing studies, a heat bath at finite temperature. This paper explains and categorises how this quantum system evolves in time, before reaching thermal equilibrium in terms of quantum speed-up and speed-down via the concept of QSLT. Our analysis contains controllable parameters such as a qubit-to-qubit coupling constant and energy gap for each qubit. In addition to a theoretical interest, our work will be useful for an understanding and realization of quantum evolution speed control at finite temperatures in devices for the future. 2. Evolution of two coupled thermal qubits We consider the system of two coupled qubits each of which is coupled to a separate bosonic bath. The total Hamiltonian of the system is given by [26]
H = H S + H BA + H BB + H IA + H IB ,
*
Corresponding author. E-mail addresses: [email protected] (T. Chatchawaltheerat), [email protected] (S. Khemmani), [email protected] (J. Poulter). https://doi.org/10.1016/j.physleta.2019.125882 0375-9601/© 2019 Elsevier B.V. All rights reserved.
(1)
where H S is the Hamiltonian of two coupled qubits (here, we set h¯ = k B = 1),
2
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
HS =
1 2
1
ω A σ AZ + ω B σ BZ + g σ A+ ⊗ σ B− + σ A− ⊗ σ B+ . 2
(2)
The Hamiltonian H S describes the qubit-to-qubit interaction, where σ jZ and σ j± are Pauli matrices. The parameter g represents the strength of the coupling between the two qubits. The Hamiltonian H B j of the reservoir for each qubit ( j = A , B) is given by
HBj =
ωk, j b† bk, j ,
(3)
k, j
k
which represents an infinite set of harmonic oscillators. The opera† tors b and b are bosonic creation and annihilation operators. k, j
| E 1 = |11 1 | E 2 = √ |10 + |01 2 1
|E 3 = √
2 | E 4 = |00
E 1 = ω, E2 = g,
|10 − |01
E3 = −g, E 4 = −ω.
(8)
Here, the transition frequencies (ωμ ) are
ω1 = ω − g ,
(9)
and
k, j
The interaction between a qubit and its bosonic bath is of the Jaynes-Cummings type and is represented by the Hamiltonian
σ j+ ⊗
HIj =
g b + σ j− ⊗ k, j ( j) k
=
k
g b k
k
∗( j ) †
k, j
k
† V j(
ωμ ) ⊗ f j + V j (ωμ ) ⊗ f j†(k) , (k)
(4)
μ
(k)
where the operators V j (ωμ ) and f j act on the two qubit system and the bosonic bath degrees of freedom, respectively. The μ index of ωμ labels all possible energy differences. There are two values of μ in this case, where V j (ωμ ) does not vanish. The master equation in the Born-Markov approximation for the reduced density operator of the qubits system in the interaction picture is [27]
d dt
∞
ρ (t ) = −
(I )
(I )
tr B H I (t ), H I
t − s , ρ (t ) ⊗ ρ B ds .
(5)
0
dt
ρ (t ) = −i H S , ρ (t ) + L A (ρ ) + L B (ρ ),
(6)
where the operators L j describe dissipation to the corresponding reservoir,
L j (ρ ) =
2
† J ( j ) (−ωμ ) V j (ωμ )ρ (t ) V j (ωμ )
μ=1
1 † 1 † − V j (ωμ ) V j (ωμ )ρ (t ) − ρ (t ) V j (ωμ ) V j (ωμ ) 2 2 † + J ( j ) (ωμ ) V j (ωμ )ρ (t ) V j (ωμ ) 1 1 † † − V j (ωμ ) V j (ωμ )ρ (t ) − ρ (t ) V j (ωμ ) V j (ωμ ) . (7) 2
(10)
and the transition operators V j (ωμ ) are
1 V A (ω1 ) = √ | E 2 E 1 | + | E 4 E 3 | , 2 1 V A (ω2 ) = √ | E 4 E 2 | − | E 3 E 1 | , 2 1 V B (ω1 ) = √ | E 2 E 1 | − | E 4 E 3 | , 2 1 V B (ω2 ) = √ | E 3 E 1 | + | E 4 E 2 | . 2
(11)
Using eqs. (7)–(11), the analytical solution of eq. (6) in the ordered basis of eigenvectors {| E 1 , | E 2 , | E 3 , | E 4 } is given by [26]
⎛
a11 1 ⎜ a21 ⎜ ρii (t ) = Ω1 Ω2 ⎝ a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
⎞⎛
2
Here the spectral density J ( j ) (ωμ ) = γ j N j (ωμ ), where N j (ωμ ) = (e ωμ / T j − 1)−1 is the Planck distribution and J ( j ) (−ωμ ) = e ωμ / T j × J ( j ) (ωμ ). We choose the coupling constants to be equal and frequency independent, that is γ A = γ B = γ , which corresponds to a uniform spectral density as defined in much of the literature, for example [28] and [29]. By considering the resonance case, that is ω A = ω B = ω, chosen throughout this work, the eigenvectors | E n and the corresponding eigenvalues E n of H S of eq. (2) are
a11 = X 1+ + X 1− e −Ω1 t a31 = X 1+ + X 1− e −Ω1 t
Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t Y 2+ Y 2− + Y 2+ e −Ω2 t
a21 = 1 − e −Ω1 t X 1− Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t X 1− Y 2−
a22 = X 1− + X 1+ e −Ω1 t
a42 = X 1− + X 1+ e −Ω1 t
a23 = 1 − e −Ω1 t
1 − e −Ω2 t X 1+ Y 2+
a34 = 1 − e −Ω1 t X 1+ Y 2− + Y 2+ e −Ω2 t a41 = 1 − e −Ω1 t
a33 = X 1+ + X 1− e −Ω1 t
1 − e −Ω2 t X 1+ Y 2−
a13 = X 1+ + X 1− e −Ω1 t a14 = 1 − e −Ω1 t
(12)
1 − e −Ω2 t Y 2−
a12 = 1 − e −Ω1 t X 1+ Y 2+ + Y 2− e −Ω2 t a32 = 1 − e −Ω1 t
⎞
a14 ρ11 (0) ⎜ ρ22 (0) ⎟ a24 ⎟ ⎟⎜ ⎟, a34 ⎠ ⎝ ρ33 (0) ⎠ a44 ρ44 (0)
where the coefficients ai j are
In this paper, we consider thermal qubits which means that the operator ρ B is described by ρ B = e − H B A / T A /tr[e − H B A / T A ] ⊗ e − H B B / T B /tr[e − H B B / T B ], where T j ( j = A , B) is the absolute temperature of the j th bath for qubit. After performing the rotating wave approximation [27] in eq. (5), one arrives at the master equation in the Schrödinger picture
d
ω2 = ω + g ,
Y 2+ + Y 2− e −Ω2 t
1 − e −Ω2 t Y 2−
1 − e −Ω2 t X 1− Y 2+
a43 = 1 − e −Ω1 t X 1− Y 2− + Y 2+ e −Ω2 t
a24 = X 1− + X 1+ e −Ω1 t a44 = X 1− + X 1+ e −Ω1 t
1 − e −Ω2 t Y 2+ Y 2− + Y 2+ e −Ω2 t
where
1 ( A) J (±ω1 ) + J ( B ) (±ω1 ) , 2 1 ( A) ± Y2 = J (±ω2 ) + J ( B ) (±ω2 ) , 2 Ω1 = X 1+ + X 1− , Ω2 = Y 2+ + Y 2− . X 1± =
(13) (14)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
3. Continuous evolution processes and some definitions In this section, the continuous evolution processes is studied by considering the evolution from ρ (t ) to ρ (t + τ D ), where the initial time t steps from 0, τ D , 2τ D , . . ., for given τ D . For an open quantum system, it is known that for fixed ρ (t ) and ρ (t + τ D ) (for each t), the evolution time between them is indeed not unique [30] with a lower bound denoted by τQSL called the quantum speed τ limit time. This means that we always have 0 ≤ τQSL ≤ 1, where D τQSL = 0 is a minimum lower bound, and if this ratio is decreased for each time step, then it means that there is a capability of quantum speeding up because there is more possibility for a system to evolve with time less than τ D for each time step. On the other hand, if this ratio is increased for each time step, then there is a capability of a quantum speeding down. At this point, it is clear that in order to investigate these speed behaviours instantaneously, the evolution processes should be considered continuously with small τ D . Moreover, if this ratio is indeed a proper measurement of quantum speed behaviour, it should be independent of τ D for small enough τ D so that, for any given τ D (but small enough), one will obtain the same behaviours of this ratio as a function of time. This fact can be demonstrated as follows. Starting with the relative purity [24]
f (t + τ D ) =
tr[ρ (t + τ D )ρ (t )] tr[ρ 2 (t )]
,
| f (t + τ D ) − 1|tr[ρ 2 (t )] τQSL = t +τ D n τD i =1 σi (t )ρi (t )dt t |tr[ρ (t + τ D )ρ (t )] − tr[ρ 2 (t )]| = t +τ D n , i =1 σi (t )ρi (t )dt t
γ 4 | f ii (t )| n , 4(Ω1 Ω2 )2 i =1 σi (t )ρi (t )
(19)
where
f 11 (t ) = α1 Ω1 (1 − α1 ) (1 − α2 )2 +
Ω22
γ2
Ω2 + α2 Ω2 (1 − α2 ) (1 − α1 )2 + 21 ,
γ
(20)
ρ (0) = | E 1 E 1 | or ρ11 (0) = 1,
Ω2 f 22 (t ) = −β1 Ω1 (1 + β1 ) (1 − α2 )2 + 22
for the initial condition
γ
(16)
γ
(21)
ρ (0) = | E 2 E 2 | or ρ22 (0) = 1,
Ω22 2 f 33 (t ) = α1 Ω1 (1 − α1 ) (1 + β2 ) + 2
for the initial condition
γ
Ω12 2 − β2 Ω2 (1 + β2 ) (1 − α1 ) + 2 ,
γ
(22)
ρ (0) = | E 3 E 3 | or ρ33 (0) = 1,
Ω22 2 f 44 (t ) = −β1 Ω1 (1 + β1 ) (1 + β2 ) + 2
for the initial condition
(17)
Notice that eq. (17) is independent of τ D and thus provides us with the well-defined speed function S (t ) as
|tr[ρ˙ (t )ρ (t )]| . S (t ) = n i =1 σi (t )ρi (t )
S (t ) =
Ω2 + α2 Ω2 (1 − α2 ) (1 + β1 )2 + 21 ,
where σi (t ) and ρi (t ) are the singular values of dρ (t )/dt and ρ (t ) respectively, and ordered as σ1 ≥ ... ≥ σn and ρ1 ≥ ... ≥ ρn . For small enough τ D , eq. (16) becomes
τQSL |tr[(ρ (t ) + τ D ρ˙ (t ))ρ (t )] − tr[ρ 2 (t )]| ≈ τD τ D ni=1 σi (t )ρi (t ) |tr[ρ˙ (t )ρ (t )]| = n . i =1 σi (t )ρi (t )
obtained in eq. (12) is derived from the Born-Markov master equation with the coarse-grained time axis. This means that the time step τ D must be larger than the time scale τ B over which the reservoir correlation function decays [27]. To show that such a small τ D satisfying this condition can always be chosen, we first notice from eq. (12) that the condition for small enough τ D described in the previous section is that Ωi τ D 1 or τ D 1/Ωi < 1/γ , and τ R = min{1/Ω1 , 1/Ω2 } which is the relaxation time of the reduced system. From the condition for obtaining the Markovian master equation, that is τ B τ R [27], we have τ B 1/γ . Since both τ D and τ B must be much less than 1/γ , one can always choose small enough τ D so that the speed function S (t ) can be defined while having τ D τ B . Now, substituting eq. (12) into eq. (17), the speed function corresponding to each initial energy eigenstate | E n can be expressed, after some algebra, in the form
(15)
the von Neumann trace inequality leads to the ML bound and thus τQSL of the form [25]
3
(18)
Now, the quantum speed behaviour of the evolution process can be defined as follows. The evolution process at time t is called “speed-up or acceleration process” if S˙ (t ) < 0, “speed-down or deceleration process” if S˙ (t ) > 0, and “stationary process” if S˙ (t ) = 0. Note that although the speed function S (t ) in eq. (18) is defined by treating the time as a positive continuous real number, S (t ) is also valid for a coarse-grained time axis of ρ (t ) satisfying the Born-Markov approximation too. This will be explained in more detail in the next section. 4. The speed of quantum evolution before thermal equilibrium To study the speed behaviour, we need to compute the speed function S (t ) in eq. (18) by using ρii (t ) in eq. (12). However, ρii (t )
γ
Ω2 − β2 Ω2 (1 + β2 ) (1 + β1 )2 + 21 ,
γ
(23)
for the initial condition ρ (0) = | E 4 E 4 | or ρ44 (0) = 1. Here, the parameters α1 , α2 , β1 and β2 are
α1 =
α2 = β1 = β2 =
Ω1
γ Ω2
γ Ω1
γ Ω2
γ
+ 1 e −Ω1 t , + 1 e −Ω2 t , − 1 e −Ω1 t , − 1 e −Ω2 t .
(24)
Next, we will divide our analysis into four cases corresponding to four initial states of the two qubit system. Here, in obtaining eqs. (19)–(23), we set the temperatures of the baths to be equal,
4
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
and use the rate change of this ratio, instead of the ratio itself, to identify the speed of quantum evolution in general. 4.2. Case 2 (ρ (0) = | E 2 E 2 |)
Fig. 1. S (t ) as a function of the initial time parameter t. Initial state ρ44 (0) = 1, g = 0.1; T = 0.1 (solid line), T = 0.5 (dashed line), T = 0.8 (dotted line). Other parameters are chosen as ω = 1, γ = 0.05.
that is T A = T B = T which corresponds logically to a single bath of temperature T . Therefore, Ωi (i = 1, 2) in eq. (14) becomes
Ωi = γ 2N (ωi ) + 1 , where we first assumed, without loss of generality, that ω > g.
(25)
ω1 > 0 or
4.1. Case 1 (ρ (0) = | E 4 E 4 |) Substituting eq. (23) into eq. (19), we numerically obtain an example graph as shown in Fig. 1. S (t ) ≈ 1 (solid line) indicates that the quantum evolution exhibits a stationary process, while for T = 0.5 (dashed line) and T = 0.8 (dotted line), the evolution exhibits, a speed-up or acceleration process and gradually goes to equilibrium. Note that in ref. [31], the ratio τQSL /τ D represents the potential capacity for quantum speed-up, that is for τQSL /τ D < 1, the quantum evolution has potential capacity for further speed-up. Here, however, we find that for T = 0.5 (dash line) or T = 0.8 (dotted line), the evolution reaches thermal equilibrium with equal τQSL /τ D < 1 in each time step, and the system in equilibrium should physically not possess any speed-up or speed-down processes. By this reason, we call this situation a stationary process
Fig. 2. S (t ) as a function of the initial time parameter t. (a) Initial state g = 0.8 (dotted line). Other parameters are chosen as, ω = 1, γ = 0.05.
Substituting eq. (21) into eq. (19), an example is shown in Fig. 2(a). Unlike Fig. 1 for the case ρ44 (0) = 1, there exists the cusp around t = t c (≈ 14 s) which indicates that just before t c the evolution exhibits a speed-up process but suddenly after t c the evolution instead exhibits a speed-down or deceleration process and then gradually goes to equilibrium. Moreover, one can also prove that there exists only one cusp in this case (see Appendix). At a cusp, the speed function and thus τQSL must be equal to zero because a cusp always appears at a time t c where f ii (t ) changes its sign under the absolute value in eq. (19). From eq. (18), the vanishing of τQSL at time t c comes from the fact that the state ρ (t c ) and its derivative ρ˙ (t c ) at this time are orthogonal. Since, as one can verify numerically from eq. (12), ρ˙ (t c ) = 0 or equivalently ρ (t c ) = ρ (t c + τ D ), the lower bound τQSL = 0 must be not tight otherwise the evolution speed will be infinite which would be impossible for the weak-coupling Markovian regime. Note that, unlike this case, the case 1 of initial state ρ44 (0) = 1 always contains no cusp because all terms in f 44 (t ) are negative. This means that the total evolution process is always smooth without a speed-down process. 4.3. Case 3 (ρ (0) = | E 1 E 1 |) An example graph is shown in Fig. 2(b). The quantum evolution process is similar to that of case 2 which contains only one cusp at time t c (not exactly the same value as t c in Fig. 2(a)) indicating the critical time where the evolution changes abruptly from speed-up to speed-down. Our extensive numerical investigations reveal that this case also has only one cusp. 4.4. Case 4 (ρ (0) = | E 3 E 3 |) The quantum evolution process in this case is divided into two subcases as follows. 4.4.1. Double cusp behaviour From our numerical survey, we found that, for example, for fixed g = 0.8 and T = 0.4 (see the dashed line of Fig. 3(a)), there exist two cusps at t c1 and t c2 which means that the abrupt change
ρ22 (0) = 1, T = 0.8; (b) initial state ρ11 (0) = 1, T = 0.5; g = 0.1 (solid line), g = 0.5 (dashed line)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
5
Fig. 3. S (t ) as a function of the initial time parameter t. Initial state ρ33 (0) = 1, (a) g = 0.8, T = 0.1 (solid line), T = 0.4 (dashed line), T = 0.6 (dotted line), T = 0.8 (dot dash line); (b) T = 0.7, g = 0.1 (line), g = 0.3 (dashed line), g = 0.7 (dotted line), g = 0.8 (dot dashed line). Other parameters are chosen as ω = 1, γ = 0.05.
Fig. 4. Grey zone corresponds to double cusp case; black zone corresponds to smoothed single minimum case. (a) chosen as γ = 0.05.
between speed-up and speed-down processes occurs twice before reaching equilibrium. Moreover, if T is decreased, the second critical time t c2 will be larger and tends to infinity for T approaching zero (see solid line in Fig. 3(a)) which implies that there exists only one cusp in this case at zero temperature. On the other hand, for fixed T = 0.7 and g = 0.3 (see the dash line of Fig. 3(b)), there exist also two cusps and if g is decreased, the second critical time t c2 will be larger (see solid line in Fig. 3(b)) and tends to infinity as g tends to zero. This means that there exists only one cusp at vanishing qubit coupling constant g, as in the case of a single qubit [16]. 4.4.2. Smoothed single minimum Starting with the dashed line in Fig. 3(a), if the temperature T is increased, we found that t c2 becomes closer to t c1 (which is almost fixed) and at a certain temperature t c2 becomes equal to t c1 which is the dotted line in the figure. Moreover, if T is still increased more, then the graph becomes like the dot dashed line. Since the minimum of both lines are smooth instead of a cusp, we call this case a smoothed single minimum. Physically, the evolution behaviour of this case is the same as the case of one cusp except that the change from speed-up to speed-down process occurs smoothly instead of abruptly like in the cusp cases. For Fig. 3(b),
ω = 1. (b) T = 0.5, g = ω (white line). Other parameter is
the explanation is the same but here the temperature is fixed while the coupling constant g is varied. From the above explanation, it is clear that the change from double cusps to smooth single minimum is started at the dotted lines in Fig. 3 when t c1 = t c2 corresponding to parameters that provide f 33 = 0 and ˙f 33 = 0 at this critical time. From this condition, we can find the diagram showing the zone of two cusps (grey) and of smoothed single minimum (black) separated by a common boundary satisfying this condition as shown in the Fig. 4(a). Furthermore, the value of the minimum of the dot dashed line graph in Fig. 4(a) becomes higher for higher temperature and at high enough temperature the graph then becomes like that in Fig. 1. This means that the speed-down process can be ignored at high enough temperature. In Fig. 4(b), the diagram shows both zones not only for ω > g but also for ω < g where the ω and g are interchanged in the formalism leaving the overall dynamics of evolution unchanged, as one can see the symmetry in Fig. 4(b) with respect to the g = ω line (white line). We note that the diagram in Fig. 4(a) is bounded by g- and T -axes which correspond to the case of a single cusp. Although the value of g presented in Fig. 4, can be any value, it is in fact bounded from below according to the condition for the secular or rotating wave approximation [27], that is τ S τ R or g γ /2,
6
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882
as in our study τ S = |ω1 − ω2 |−1 = 1/2g and τ R = 1/Ω1 . Due to this condition, the value of the coupling constant γ should be appropriately set according to the range of qubit coupling strength of interest. For weaker γ , for example γ = 0.0005, Fig. 3 and Fig. 4 will be the same except that the value of t c1 and t c2 will be larger due to the fact that the system tends to equilibrium slower for weaker coupling constant. From Fig. 4(a), one can see, for fixed g (or T ), that the speed-up and speed-down processes are present at any temperature T (or coupling constant g) with a tendency for double cusp behaviour, where an abrupt change from speed-up and speed-down occurs twice, at low temperature T (or low coupling constant g). From Fig. 4(b), one also see, for fixed ω (or g) that the speed-up and speed-down processes present at any coupled constant g (or frequency ω ). These processes have a tendency for double cusp behaviour at low g (or high ω ) for ω > g (weak qubit coupling strength) and with a tendency for double cusp behaviour at high g (or low ω ) for ω < g (strong qubit coupling strength). By considering Fig. 4(a) and (b) together, the black zone on Fig. 4(b) will be extended as T is increased. This means that for both weak and strong qubit coupling strengths, speed-down can be ignored for high enough temperature. However, for fixed T (fixed black zone area), increasing g from weak (g < ω ) to strong (g > ω ) coupling strength cannot destroy the speed- down process. Instead, the processes swing from double cusp behaviours (grey) to single minimum (black) and then to double cusp again.
and useful knowledge in explaining the speed of quantum computation or quantum information processes at finite temperature. Acknowledgements We thank Department of Physics, Faculty of Science, Srinakharinwirot University for support and facilities. Appendix A From
α2 and β1 in eq. (24), β1 can be written in terms of α2
as
β1 = G α2P ,
(A.1)
where p = Ω1 /Ω2 ≥ 1, and
( Ωγ1 − 1)
G= Ω < 1. ( γ2 + 1) P
(A.2)
Putting eq. (A.1) into eq. (21), we obtain f 22 (t ) in terms of only as
α2 (t )
f 22 (t ) = f 22
α2 (t ) = −G α2P −1 p 1 + G α2P (1 − α2 )2 +
Ω22
γ2
2 ( p Ω2 )2 . + (1 − α2 ) 1 + G α2P + 2
(A.3)
γ
5. Conclusion and discussion We have derived the QSLT in the Born-Markov approximation for the system of two coupled qubits each of which is coupled to a thermal bath at finite temperature with four initial states that are energy eigenstates of the system. By analysing the speed function defined via the QSLT through continuous processes, we found four types of evolution processes before reaching the equilibrium. First, the process that is composed of speed-up and speeddown processes where the change between them occurs in an abrupt way, called a cusp, at some critical time. This type of process contains only one cusp. Second, the totally smooth process without any cusp or minimum which occurs only for the initial ground state. Third, the process that has two cusps which indicates that, before reaching the equilibrium, the system of two coupled qubits exhibits a change from speed-up to speed-down twice. Our extensive numerical investigations reveal that the maximum number of cusps is two for the model we used. We believe that this double cusp behaviour is originated from the fact that we have two qubits in which their energy exchange mechanism for this model plays a crucial role. Note that if one does the same analysis in this paper for the same model but with a system of one qubit, then the number of existing cusp is at most one. However, multiple cusps can occur in some models (see for example, [28]). Fourth, the process that has one smooth single minimum which has the same behaviour as the first type except that the change from speed-up to speed-down process occurs smoothly instead of abruptly. This process can actually be viewed as a third process where two cusps merge together for appropriate parameters. This means that the speed-up and speed-down processes of any type except the second one can occur at any parameters, including temperature. We also found that low temperature or low (high) qubit coupling constant can stimulate double cusp behaviour for the weak (strong) qubit coupling strength. For high enough temperature the speed-down process can be ignored only for the fourth type. However, high qubit coupling constant cannot destroy the speed-down process. We expect our results to provide definite
Before proving that there exists only one cusp, we first need to show that f 22 (α2 ) is a monotonic decreasing function. Beginning from eq. (A.3), we have
d dα2
f 22 (α2 ) = −G ( p − 1)α2P −2 ( p ) 1 + G α2P
(1 − α2 )2 +
2
Ω22
γ2
Ω − G α2P −1 ( p ) pG α2P −1 (1 − α2 )2 + 22 p −1 1+ 2
γ
p G 2 (1 − 2 ) ( p Ω2 )2 P 2 . + 2 2
+ 4pG α
− 1 + Gα
α
α
(A.4)
γ
Notice that only the third term in eq. (A.4) can be positive. So, we claim that p −1 1+ 2
4pG α
p 2 (1 −
Gα
P 2 2
α2 ) −
1 + Gα
+
( p Ω2 )2
< 0.
γ2
First, notice that p −1
4pG α2
p
1 + G α2 (1 − α2 ) < 4pG (1 + G ) and
2 ( p Ω2 )2 ( p Ω2 )2 < − . − 1 + G α2P + 1 + 2 2
γ
γ
Then we have p −1
4pG α2
p
1 + G α2 (1 − α2 ) −
1 + G α2P
2
( p Ω2 )2 . < 4pG (1 + G ) − 1 + 2
+
( p Ω2 )2
γ2
γ
Now, for the right-hand side of the above inequality, suppose that
4Gp (1 + G ) − 1 + ( p Ω2 )2 /γ 2 > 0 or G>
1 2
1+
Ω2 Ω1
1+
Ω12
γ2
− 1 =: F .
(A.5)
T. Chatchawaltheerat et al. / Physics Letters A 383 (2019) 125882 ln G Since, for fixed Ω1 , one can verify from eq. (A.2) that ∂∂Ω >0 2 which implies that ∂ G /∂Ω2 > 0. Therefore,
( Ωγ1 − 1) . G max = G (Ω2 = Ω1 ) = Ω ( γ1 + 1) Next, since Ω2 ≥ γ ,
F min = F (Ω2 = γ ) =
1 2
1+
γ Ω1
1+
Ω12
γ2
−1 .
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