1. Introduction
Although the standard approach based on time-ordered exponentials is extremely useful [1] [2] [3] , it may occasionally turn out to be challenging, particularly, in the case of revealing nonlinear quantum dynamics [4] [5] that requires rigorous numerical simulations [6] [7] [8] . Quantum dynamics for arbitrary system are traditionally realized by time evolutions of wave functions in Hilbert space, which can also be expressed in terms of density operators in the Liouville space [2] [3] . In this work, we introduce a new nonstandard iterative technique formulated as follows. 1) Finite time interval is divided into a large number of discrete subintervals with an ultrashort width. 2) The Liouville space is expanded with an additional (i.e., virtual) space for this ultrashort time duration. The system’s original Hamiltonian is, then, modified for the system’s space plus virtual space, where the force terms are replaced with the virtual quantum operators. 3) The density operator for the system is extracted by tracing over the virtual operator space. In principle, various virtual operators can be chosen depending on the specific quantum system. Here we choose two-state spin raising and lowering operators because of their simple algebra. In the next section, we present the standard approach using S-operator defined as time-ordered exponentials in Hilbert, and then, in the Liouville space. In Section 3, we introduce our technique and implement it to the well-known example of a charged particle in a harmonic potential. The temporal evolutions of the probability for the particle being in the ground state are obtained by our technique and compared to the analytical solutions obtained using the standard S-operator. By extending this example, we perform numerical simulations for temporal evolutions for the ground state probability for the generalized systems governed by time-dependent nonlinear Hamiltonians. We further discuss the physics insight of this technique based on a thought-experiment, in which a large number of polarized atoms successively interact with a lossless cavity field. The last section is a conclusion.
2. Standard Approach
In this section, the standard approach for quantum dynamics both in Hilbert space and the Liouville space is presented. We consider the system with the Hamiltonian given by
(1)
here
is the unperturbed (free) and
interaction Hamiltonians and we set
.
2.1. Quantum Dynamics in Hilbert Space
We begin with the approach for the Hilbert space. In the interaction representation, the rapid state evolution due to
is removed as
,
where
and
are wave functions in the interaction and Schrödinger representations, respectively. Unitary transformation of initial state in the interaction picture is given as
, here unitary operator
satisfies
and is expressed as
. Time evolution of
can be derived from
. The Hamiltonian is in the interaction representation as
. Choosing time interval between
and
(
), unitary transformation is expressed as [1]
(2)
with S-operator
. We divide time interval
into N sub-intervals with a width of
. At mid-time
in the jth interval, the S-operator is written as [1]
(3)
where
and
but
is finite. Equation (3) leads to the traditional time-ordered exponential given as [1]
(4)
where time ordering for boson operators is defined as
with
.
2.2. Examples
As an example, we consider a driven harmonic oscillator. For that temporal evolutions of the ground state using S-operator are given in Equation (4). Let a particle of a charge
, mass
be in a harmonic potential (
). The driving electric field is
, if
, and otherwise, it is zero and
is the frequency of the oscillator. In the interaction picture, the Hamiltonian is written as
(5)
where time-dependent operators are
and
and the force terms are
. The probability
for the particle to remain in the ground state
after time T is written as
(6)
The probability amplitude is given by S-operator from Equation (4) as
(7)
where
and
is the Green’s function. For this simple example, the Green’s function is well known
(8)
Therefore, for the particle, its probability to remain in the ground state after time T is analytically found to be as [1]
(9)
with pulse area
.
This example is the simplest case when a linear Hamiltonian is considered. That conveniently ensures to use the well known Green function in Equation (8). However, in general, the Green’s functions are mostly unknown and a laborious numerical method is often needed. Next, we consider two more examples that use nonlinear Hamiltonians. The first example for the nonlinear Hamiltonian is a driven anharmonic oscillator. The Hamiltonian is given in the form in Equation (5) [2] where degenerate two-boson nonlinear operators
and
and the force terms
are assumed to have a similar form as in the harmonic oscillator case. The second example for the nonlinear Hamiltonian is a driven intensity-dependent oscillator. The Hamiltonian is given in the form in Equation (5) [9] with the intensity-dependent nonlinear boson operators
and
and
is assumed to be the same
as before. In Section 3, we numerically solve for the probability time evolutions for these nonlinear systems and compare with the approximate analytical results.
3. Nonstandard Approach
3.1. Quantum Dynamics in Liouville Space
Before introducing our technique, let us first replace the wave functions in Hilbert space with density operators in Liouville space [2] [3] . We recall that
and
from Equation (2). Using S-operator in the jth interval from Equation (3), we rewrite Equation (2) in terms of density operators rather than wave functions as
(10)
here
and
. In the traditional approach, to obtain
at later time
for any given initial state
at
Equation (10) is repeatedly evaluated, where
and
but
is finite. It is also important to note that Equation (10) is the formal solution of the Liouville-von Neumann equation [2] .
3.2. Quantum Dynamics in Liouville Space Restructured with a Virtual Space
From this point, we implement our new nonstandard approach, rather directly evaluating Equation (10). As before, finite time interval
is divided into N discrete subintervals with an ultrashort width of
. The Liouville space is expanded with a two-state spin operator space for duration of
. The system’s original Hamiltonian is, then, modified for the system’s space plus spin space, where the force terms are replaced with the spin operators. The density operator for the system is extracted by tracing over the spin operator space. In the jth interval with an infinitesimally short width of
, it is an acceptable ansatz where we replace the original Hamiltonian
by new Hamiltonian expanding it with an additional virtual space
as
(11)
where
. For the sake of simplicity,
can be chosen to be a two-state spin operator defined as
(12)
with
. For this choice, the force terms are replaced with the raising
and lowering
operators specifically as
(13)
Thus, in the jth interval with
width, this original Hamiltonian Equation (5) can be replaced with a new Hamiltonian
, also known as the unified Jaynes-Cummings Hamiltonian [10] [11] as
(14)
Instead of the original approach given by Equation (10) for
, we introduce an iterative relation for new density operator
using the modified Hamiltonian given in Equation (14) as
(15)
Therefore, our goal is to demonstrate that the two density operators converge
(16)
for the same pure initial state.
3.3. A Thought-Experiment
The essentials of our iterative technique are explained by the following thought-experiment. As sketched in Figure 1, let us consider a monokinetic beam consisting of individual two-level atoms. Each atom is prepared in arbitrary coherent superposition of the upper and lower states [8] [12] [13] . The atoms are then injected into a lossless cavity in a well controlled rate where only one atom at a time is present inside the cavity for duration
. At the exit from the cavity
Figure 1. A monokinetic beam of N two-level atoms passing through a lossless cavity. Each atom interacts with the cavity field for a duration of
. The cavity field builds up to its final state
(or
) from initial state
(or
) after a finite time
.
the individual atoms are not intended to be measured. Total number of atoms is N and the jth atom-field coupling constant is
. Although the present model can be generalized to multi-level atoms [8] [9] [14] , for the sake of simplicity, we consider only two-level atoms, where
and
are upper and lower atomic states, respectively. Correspondingly,
and
are probability amplitudes for the jth atomic upper and lower states. Thus, as a result of numerically solving Equation (15), the final cavity field state is evaluated from the existing initial quantum state in the cavity after time
. For example, when atoms are prepared in the same phase then the cavity field evolves to the so-called superradiant state [5] [8] [15] [16] [17] [18] . The mean number of photons created in the cavity (i.e., field intensity) is proportional to N2 rather than N. On the other hand, when each successive pair of atoms are prepared in perfectly out-of-phase, then the cavity field evolves to the sub-radiant state [5] [8] . Moreover, we justify that time evolutions involve pure states after tracing over the virtual space operator. As demonstrated in our earlier work [8] , an initial coherent state given as
evolves into
, with
at later time t. Therefore, the above statement that our technique maintains time evolutions for pure states is justified not only for infinitesimally short
interval, but also for finite time t.
3.4. Examples
Here we employ our technique for the previous example for a charged particle in harmonic potential. We evaluate Equation (15) using the Hamiltonian given in Equation (14) both numerically and analytically, however, for the sake of simplicity, only for initial vacuum state. Thus, the modified Hamiltonian
in Equation (14) is written in terms of
. Comparing the Hamiltonian in Equation (5) with the ansatz in Equation (13), we obtain
. The parameter
stands for a coherence between spin states. For example, for parameters chosen to be as
,
, it is given by
with
. In the Fock state representation, the jth density matrix elements are
. We numerically evaluate
from Equation (15), to obtain
, at
with
to compare the probability
given in Equation (9). Equation (9) is rewritten in terms of time-independent parameters
,
associated to the virtual operator space as
(17)
In Figure 2, the density matrix elements for
(red circles) and
(black curves) are plotted as functions of pulse area per
,
. For the plots in Figure 2(a) and Figure 3(a), the parameters include coherence
, total number
, width of the subintervals
, time
frequency
. For the plot in Figure 2(a’), except for the larger total
Figure 2. Analytical (black curves) and numerical (red circles) results for temporal evolutions of the probability
for a charged particle driving by the external field being in the ground state as functions of
. Left column: The system with a linear Hamiltonian. Middle column: The system with the degenerate two-boson Hamiltonian. Right column: The system with the intensity-dependent Hamiltonian.
Figure 3. Realizations of dynamics of quantum states from initial vacuum. Numerical calculations of the temporal evolutions displayed by the contour plots of the Husimi Q-functions accompanied with the trajectories (red curves) of the centers of these contours. Left column: The system with a linear Hamiltonian. Middle column: The system with the degenerate two-boson Hamiltonian. Right column: The system with the intensity-dependent Hamiltonian. All parameters are the same as used in Figure 2.
number
and later time
, the rest of parameters remain the same as that given in Figure 1(a). In Figure 3, the quasi-distributions given by the Husimi Q-functions are plotted. The Husimi Q-function [2] [4] [5] is defined as
, here
. Because of coherent state representations, the Q-functions conveniently illustrate the coherent state as a displaced vacuum state with a perfect ring shape [8] , preserved for entire time. In Figure 3, the red curves indicate the trajectories of displacements of the initial coherent state over time. These trajectories are the centers of single selected contour plots with the fixed value of the Q-functions at any given time t. For example, in Figure 3(a), this trajectory follows a circle but is not yet complete circle opposite to that case in Figure 3(a’). The parameters used for the plots in Figures 3(a)-(c) and Figures 3(a’)-(c’) are the same as those used in Figures 2(a)-(c) and Figures 2(a’)-(c’), respectively. In Figure 2(b) and Figure 2(b’) and Figure 2(c) and Figure 2(c’), the realizations of quantum dynamics for nonlinear Hamiltonians with
in (b, b’) representing two-boson processes and
in (c, c’) representing intensity-dependent processes are demonstrated. Similar to Figure 2, the numerical results for
are compared to approximate analytical expressions for time evolutions for the probabilities being in the ground state
after time T in Figure 3(b) and Figure 3(c). In the case of two-boson transition processes, the approximate analytical expressions are obtained to be
(18)
For Figure 2(b) and Figure 2(c), the parameters are given as
,
,
,
and
, while for Figure 2(b’) and Figure 2(c’), the parameters are the same as in (b) except for frequency
and, thus,
. In Figure 3(b), the Q-functions display how the initial vacuum state with a ring shape is transformed to the significantly squeezed states with its signature oval shape [2] [8] [19] [20] for a slower process with a frequency of
. However, for the fast process with
, the state remains merely in vacuum state without observable squeezing. Lastly, Figure 2(c) and Figure 2(c’) and Figure 3(c) and Figure 3(c’) represent the temporal evolutions for the Holstein-Primakoff
transformed states [9] [21] . Similarly, the probabilities for slow (Figure 2(c)) versus fast (Figure 2(c’)) processes are compared. It is important to note that the analytical formula for the probability for these processes is identical to Equation (17). However, the deviation (i.e., displacement) is not as much pronounced as for coherent states (see, Figure 3(c) and Figure 3(c’)).
4. Conclusions
In the standard approach, quantum dynamics for arbitrary system are realized by the time evolutions of wave functions in Hilbert space, which can also be expressed in terms of density operators in Liouville space. However, the standard quantum simulations may occasionally turn out to be challenging, particularly, for nonlinear dynamical systems.
In this work, we introduce a new nonstandard iterative technique, formulated as follows. 1) A finite time interval is divided into a large number of discrete subintervals with an ultrashort width. 2) The Liouville space is synthesized with an additional virtual space for ultrashort time duration and the quantum system’s original Hamiltonian is modified accordingly. In particular, the force terms are replaced with virtual quantum operators. 3) The density operator for the system is extracted by tracing over the virtual operator space. In principle, various virtual operators can be chosen depending on specific quantum system. For example, the simple algebra of using two-state spin raising and lowering operators reduces the cost of time-consuming calculations. After introducing our technique, we implement it to the well-known example of a charged particle in a harmonic potential. Temporal evolutions of the probability for the particle being in the ground state are obtained by the present technique and compared to the analytical solutions given by the standard approach. We further discuss the physics insight of this technique based on a thought-experiment. Lastly, we perform numerical simulations for temporal evolutions for the ground state probability for generalized systems governed by the time-dependent nonlinear Hamiltonians. The quantum dynamics are realized by using the quasi-distributions.
Successive processes implicitly “hitchhiking” via virtual space for discrete ultrashort time duration, are the hallmark of our technique. We believe that this novel technique has potential for solving numerous problems otherwise challenging to address using the standard approach based on time-ordered exponentials.