Rabi Oscillations, Entanglement and Teleportation in the Anti-Jaynes-Cummings Model ()
1. Introduction
The basic model of quantized light-matter interaction describing a two-level atom coupled to a single mode of quantized electromagnetic radiation is the quantum Rabi model (QRM) [1] [2] [3] [4] [5] initially introduced by Rabi, Isidor Isaac [6] [7] to discuss the phenomenon of nuclear magnetic resonance in a semi-classical way. The Jaynes-Cummings (JC) Hamiltonian [3] [4] [5] [8] and the anti-Jaynes-Cummings (AJC) Hamiltonian [3] [4] [5] are both generated from the QRM.
Exact analytical solutions of the eigenvalue equation for the QRM have been determined in [1] [2] [9] [10]. However, a major challenge in the QRM that remained an outstanding problem over the years is that while the JC component has a conserved excitation number operator and is invariant under the corresponding U (1) symmetry operation, a conserved excitation number and corresponding U (1) symmetry operators for AJC component had never been determined. Recently, it has been shown that the operator ordering principle distinguishes the JC and AJC Hamiltonians [3] [4] [5] as normal and anti-normal order components of the QRM. In this approach the JC interaction represents the coupling of a two-level atom to the rotating positive frequency component of the field mode while the AJC interaction represents the coupling of the two-level atom to the anti-rotating (anti-clockwise or counter-rotating [2] [3] [4] [5] [8] [11] ) negative frequency component of the field mode, because the electromagnetic field mode is composed of positive and negative frequency components [12]. The long-standing challenge of determining a conserved excitation number and corresponding U (1) symmetry operators for the AJC component was finally solved in [3]. The discovery and proof of a conserved excitation number operator of the AJC Hamiltonian [3] now means that dynamics generated by the AJC Hamiltonian is exactly solvable, as demonstrated in the polariton and anti-polariton qubit (photospin qubit) models in [4] [5].
Noting that fundamental features namely: collapses and revivals in the atomic inversion [13], generation of Schrdinger cat states of the quantized field [14] [15], transfer of atomic coherence to the quantized field [16], vacuum-field Rabi oscillations in a cavity [17] and many more have been extensively studied in the JC model in both theory and experiment in quantum optics, we now focus attention on the AJC model which has not received much attention over the years due to the erroneously assumed lack of a conserved excitation number operator.
We observe that the failure of the JC interaction component to account for some experimental features characterised by blue-sideband transitions has driven various workers to apply numerical methods to probe the full QRM into the ultrastrong coupling (USC) and deep strong coupling (DSC) regimes [18] [19] [20] [21] [22] to indirectly monitor the dynamical effects of the AJC interaction component. However, even such advanced approaches do not give explicitly the dynamical features generated solely by the AJC interaction. Fortunately, the reformulation developed in [3] [4] [5], drastically simplifies exact solutions of the AJC model, which we shall here apply.
In this paper, we are interested in analysis of quantum state configuration of the qubit states in the AJC model, entanglement of qubits in the AJC model and the application of the entangled qubit state vectors in teleportation of an entangled atomic quantum state.
The content of this paper is therefore summarized as follows. Section 2 presents an overview of the theoretical model. In Section 3, Rabi oscillations in the AJC model are studied. In Section 4, entanglement of AJC qubit state vectors is analysed. In Section 5, teleportation as an application of entanglement is presented. AJC state engineering and experimental implementation is briefly discussed in Section 6 and finally Section 7 presents the conclusion.
2. The Model
The quantum Rabi model of a quantized electromagnetic field mode interacting with a two-level atom is generated by the Hamiltonian [3]
(1)
noting that the free field mode Hamiltonian is expressed in normal and anti-normal order form
. Here,
are quantized field
mode angular frequency, annihilation and creation operators, while
are atomic state transition angular frequency and operators. The Rabi Hamiltonian in Equation (1) is expressed in a symmetrized two-component form [3] [4] [5]
(2)
where
is the standard JC Hamiltonian interpreted as a polariton qubit Hamiltonian expressed in the form [3]
(3)
while
is the AJC Hamiltonian interpreted as an anti-polariton qubit Hamiltonian in the form [3]
(4)
In Equations (3) and (4),
and
are the respective polariton and anti-polariton qubit conserved excitation numbers and state transition operators.
Following the physical property established in [5], that for the field mode in an initial vacuum state only an atom in an initial excited state
entering the cavity couples to the rotating positive frequency field component in the JC interaction mechanism, while only an atom in an initial ground state
entering the cavity couples to the anti-rotating negative frequency field component in an AJC interaction mechanism, we generally take the atom to be in an initial excited state
in the JC model and in an initial ground state
in the AJC model.
Considering the AJC dynamics, applying the state transition operator
from Equation (4) to the initial atom-field n-photon ground state vector
, the basic qubit state vectors
and
are determined in the form (
) [5]
(5)
with dimensionless interaction parameters
,
and Rabi frequency
defined as
(6)
where we have introduced sum frequency
to redefine
in Equation (4).
The qubit state vectors in Equation (5) satisfy the qubit state transition algebraic operations
(7)
In the AJC qubit subspace spanned by normalized but non-orthogonal basic qubit state vectors
,
the basic qubit state transition operator
and identity operator
are introduced according to the definitions [5]
(8)
which on substituting into Equation (7) generates the basic qubit state transition algebraic operations
(9)
The algebraic properties
and
easily gives the final property [5]
(10)
which is useful in evaluating time-evolution operators.
The AJC qubit Hamiltonian defined within the qubit subspace spanned by the basic qubit state vectors
,
is then expressed in terms of the basic qubit states transition operators
,
in the form [5]
(11)
We use this form of the AJC Hamiltonian to determine the general time-evolving state vector describing Rabi oscillations in the AJC dynamics in Section 3 below.
3. Rabi Oscillations
The general dynamics generated by the AJC Hamiltonian in Equation (11) is described by a time evolving AJC qubit state vector
obtained from the time-dependent Schrödinger equation in the form [5]
(12)
where
is the time evolution operator. Substituting
from Equation (11) into Equation (12) and applying appropriate algebraic properties [5], we use the relation in Equation (10) to express the time evolution operator in its final form
(13)
which we substitute into equation Equation (12) and use the qubit state transition operations in Equation (9) to obtain the time-evolving AJC qubit state vector in the form
(14)
This time evolving state vector describes Rabi oscillations between the basic qubit states
and
at Rabi frequency
.
In order to determine the length of the Bloch vector associated with the state vector in Equation (14), we introduce the density operator
(15a)
which we expand to obtain
(15b)
Defining the coefficients of the projectors in Equation (15b) as
(15c)
and interpreting the coefficients in Equation (15c) as elements of a
density matrix
, which we express in terms of standard Pauli operator matrices I,
,
and
as
(15d)
where
is the Pauli matrix vector and we have introduced the time-evolving Bloch vector
obtained in the form
(15e)
with components defined as
(15f)
The Bloch vector in Equation (15e) takes the explicit form
(15g)
which has unit length obtained easily as
(15h)
The property that the Bloch vector
is of unit length (the Bloch sphere has unit radius), clearly shows that the general time evolving state vector
in Equation (14) is a pure state.
We now proceed to demonstrate the time evolution of the Bloch vector
which in effect describes the geometric configuration of states. We have adopted class 4 Bloch-sphere entanglement of a quantum rank-2 bipartite state [23] [24] to bring a clear visualization of this interaction. In this respect, we consider the specific example (which also applies to the general n-photon case) of an atom initially in ground state
entering a cavity with the field mode starting off in an initial vacuum state
, such that the initial atom-field state is
. It is important to note that in the AJC interaction process the initial atom-field ground state
is an absolute ground state with both atom and field mode in the ground state
,
, in contrast to the commonly applied initial atom-field ground state
in the JC model where only the field mode
is in the ground state and the atom in the excited state
.
In the specific example starting with an atom in the ground state
and the field mode in the vacuum state
the basic qubit state vectors
and
, together with the corresponding entanglement parameters, are obtained by setting
in Equations (5) and (6) in the form
(16)
The corresponding Hamiltonian in Equation (11) becomes (
)
(17)
The time-evolving state vector in Equation (14) takes the form (
)
(18)
which describes Rabi oscillations at frequency
between the initial separable qubit state vector
and the entangled qubit state vector
.
The Rabi oscillation process is best described by the corresponding Bloch vector which follows from Equation (15g) in the form (
)
(19)
The time evolution of this Bloch vector reveals that the Rabi oscillations between the basic qubit state vectors
,
describe circles on which the states are distributed on the Bloch sphere as we demonstrate in Figure 1.
In Figure 1 we have plotted the AJC Rabi oscillation process with respective Rabi frequencies
determined according to Equation (16) for various values of sum frequency
. We have provided a comparison with plots of the corresponding JC process in Figure 2.
To facilitate the desired comparison of the AJC Rabi oscillation process with the standard JC Rabi oscillation process plotted in Figure 2, we substitute the redefinition
to express the Rabi frequency
in Equation (16) in the form
(20)
In the present work, we have chosen the field mode frequency
(
) such that for both AJC and JC processes we vary only the detuning frequency
. The resonance case
in the JC interaction now means
in the AJC interaction.
For various values of
, we use the general time evolving state vector in Equation (18), with
as defined in Equation (20) to determine the
coupled qubit state vectors
,
in Equation (16) by setting
,
describing half cycle of Rabi oscillation as presented below. In each case we have an accumulated global phase factor which does not affect measurement results [25] [26] [27], but we have maintained them here in Equations (21a)-(21c) to explain the continuous time evolution over one cycle.
(21a)
(21b)
(21c)
The AJC Rabi oscillations for cases
are plotted as red, black and blue circles in Figure 1, while the corresponding plots in the JC process are provided in Figure 2 as a comparison. Here, Figure 1 is a Bloch sphere entanglement [23] that corresponds to a 2-dimensional subspace of
with
and
while Figure 2 is a Bloch sphere entanglement corresponding to a 2-dimensional subspace of
with
and
, where we recall that, in the JC interaction the initial atom-field ground state with the field mode in the vacuum state is
.
Figure 1. Rabi oscillations in AJC interaction mechanism. The Rabi oscillations for values of sum frequencies are shown by red (
), black (
) and blue (
).
Figure 2. Rabi oscillations in JC interaction mechanism. Here, blue circle is at resonance with detuning
, red circle is for detuning
and black circle for detuning
.
In Figure 1 we observe:
1) that due to the larger sum frequency
in the AJC interaction process as compared to the detuning frequency
in the JC interaction process, the Rabi oscillation circles in the much faster AJC process are much smaller compared to the corresponding Rabi oscillation circles in the slower JC interaction process. This effect is in agreement with the assumption usually adopted to drop the AJC interaction components in the rotating wave approximation (RWA), noting that the fast oscillating AJC process averages out over time. We have demonstrated the physical property that the size of the Rabi oscillations curves decreases with increasing Rabi oscillation frequency by plotting the AJC oscillation curves for a considerably larger Rabi frequency
where we have set the field mode frequency
(
) in Figure 3. It is clear in Figure 3 that for this higher value of the Rabi frequency
the Rabi oscillation curves almost converge to a point-like form;
2) that Rabi oscillations in the AJC interaction process as demonstrated in Figure 1 occur in the left hemisphere of the Bloch sphere while in the JC interaction
Figure 3. Rabi oscillations in AJC interaction mechanism. The Rabi oscillations for values of sum frequencies are shown by red (
) and black (
).
process the oscillations occur in the right hemisphere as demonstrated in Figure 2. This demonstrates an important physical property that the AJC interaction process occurs in the reverse sense relative to the JC interaction process;
3) an interesting feature that appears at resonance specified by
. While in the JC model plotted in Figure 2 the Rabi oscillation at resonance
(blue circle) lies precisely on the yz-plane normal to the equatorial plane, the corresponding AJC Rabi oscillation (blue circle in Figure 1) is at an axis away from the yz-plane about the south pole of the Bloch sphere. This feature is due to the fact that the frequency detuning
takes a non-zero value under resonance
such that the AJC oscillations maintain their original forms even under resonance.
We note that the qubit state transitions described by the Bloch vector in the AJC process (Figure 1) are blue-side band transitions characterized by the sum frequency
according to the definition of the Rabi frequency
in Equation (20).
The geometric configuration of the state space demonstrated on the Bloch-sphere in Figure 2 determined using the approach in [5] agrees precisely with that determined using the semi-classical approach in [28] corresponding to a 2-dimensional subspace of
Span
. In the approach [28], at resonance where detuning
the atomic population is inverted from
to
and the Bloch-vector
describes a path along the yz-plane on the Bloch-sphere. For other values of detuning, the atom evolves from
to a linear superposition of
and
and back to
and the Bloch-vector
describes a circle about the north pole of the Bloch-sphere.
4. Entanglement Properties
In quantum information, it is of interest to measure or quantify the entanglement of states. In this paper we apply the von Neumann entropy as a measure of entanglement. The von Neumann entropy [29] [30] [31] [32] [33] of a quantum state
is defined as
(22)
where the logarithm is taken to base d, d being the dimension of the Hilbert space containing
and
‘s are the eigenvalues of
. It follows that
, where
if and only if
is a pure state.
Further, the von Neumann entropy of the reduced density matrices of a bipartite pure state
is a good and convenient entanglement measure
. The entanglement measure defined as the entropy of either of the quantum subsystem is obtained as
(23)
where for all states we have
. Here the limit 0 is achieved if the pure state is a product
and 1 is achieved for maximally entangled states, noting that the reduced density matrices are maximally mixed states.
In this section we analyse the entanglement properties of the qubit state vectors and the dynamical evolution of entanglement generated in the AJC interaction.
4.1. Entanglement Analysis of Basic Qubit State Vectors
and
Let us start by considering the entanglement properties of the initial state
which according to the definition in Equation (16) is a separable pure state. The density operator of the qubit state vector
is obtained as
(24a)
Using the definition
, we take the partial trace of
in Equation (24a) with respect to the field mode and atom states respectively, to obtain the respective atom and field reduced density operators
,
in the form (subscripts
and
)
(24b)
which take explicit
matrix forms
(24c)
The trace of
,
and
,
of the matrices in Equation (24c) are
(24d)
The unit trace determined in Equation (24d) proves that the initial qubit state vector
is a pure state.
Next, we substitute the matrix form of
and
from Equation (24c) into Equation (23) to obtain equal von Neumann entanglement entropies
(24e)
which together with the property in Equation (24d) quantifies the initial qubit state vector
as a pure separable state, agreeing with the definition in Equation (16).
We proceed to determine the entanglement properties of the (transition) qubit state vector
defined in Equation (16). For parameter values
we ignore the phase factor in Equation (21a), to write the transition qubit state vector in the form
(25a)
The corresponding density operator of the state in Equation (25a) is
(25b)
which takes the explicit
matrix form
(25c)
with eigenvalues
,
,
,
. Applying Equation (22), its von Neumann entropy
(25d)
quantifying the state
in Equation (25a) as a bipartite pure state.
Taking the partial trace of
in Equation (25b) with respect to the field mode and atom states respectively, we obtain the respective atom and field reduced density operators
together with their squares in the form
(25e)
The trace of
and
in Equation (25e) gives
(25f)
demonstrating that
and
are mixed states, satisfying the general property
for a mixed state
.
To quantify the mixedness we determine the length of the Bloch vector along the z-axis as follows
(25g)
which shows that the reduced density operators
are non-maximally mixed states.
The eigenvalues
of
and
are
and
respectively, which on substituting into Equation (22), gives equal von Neumann entanglement entropies
(25h)
Taking the properties in Equations (25d), (25f)-(25h) together clearly characterizes the qubit state
in Equation (25a) as an entangled bipartite pure state. However, since
the state is not maximally entangled.
Similarly, the transition qubit state vector
obtained for
in Equation (21b) is an entangled bipartite pure state, but not maximally entangled.
Finally, we consider the resonance case
, characterized by
in the AJC model. Ignoring the phase factor in Equation (21c) the transition qubit state vector
takes the form
(26a)
The corresponding density operator of the state in Equation (26a) is
(26b)
which takes the explicit
matrix form
(26c)
with eigenvalues
. Applying Equation (22) its von Neumann entropy
(26d)
quantifying the state in Equation (26a) as a bipartite pure state.
Taking the partial trace of
in Equation (26b) with respect to the field mode and atom states respectively, we obtain the respective atom and field reduced density operators
together with their squares in the form
(26e)
The trace of
and
in Equation (26e) is
(26f)
which reveals that the reduced density operators
are mixed states. To quantify the mixedness, we determine the length of the Bloch vector along the z-axis as follows
(26g)
showing that the reduced density operators
and
are maximally mixed states.
The eigenvalues
of
and
are
respectively which on substituting into Equation (22), gives equal von Neumann entanglement entropies
(26h)
The unit entropy determined in Equation (26h) together with the properties in Equations (26d)-(26g) quantifies the transition qubit state determined at resonance
in Equation (26a) (or Equation (21c)) as a maximally entangled bipartite pure state. Due to this maximal entanglement property, we shall use the resonance transition qubit state
in Equation (26a) to implement teleportation by an entanglement swapping protocol in Section 5 below.
Similar proof of entanglement of the AJC qubit states is easily achieved for all possible values of sum frequency parameter
, confirming that in the initial vacuum-field AJC interaction, reversible transitions occur only between a pure initial separable qubit state vector
and a pure entangled qubit state vector
. This property of Rabi oscillations between an initial separable state and an entangled transition qubit state occurs in the general AJC interaction described by the general time evolving state vector
in Equation (14).
4.2. Entanglement Evolution
Let us consider the general dynamics of AJC interaction described by the general time-evolving qubit state vector
in Equation (14). Substituting
from Equation (14) into Equation (15a) and using the definitions of
,
in Equation (5) the density operator takes the form
(27)
The reduced density operator of the atom is determined by tracing over the field states, thus taking the form
(28)
after introducing the general time evolving atomic state probabilities
,
obtained as
(29)
where the dimensionless interaction parameters
,
are defined in Equation (6) and the Rabi frequency takes the form
(30)
Expressing
in Equation (28) in
matrix form
(31)
We determine the quantum system entanglement degree
defined in Equation (23) as
(32)
which takes the final form
(33)
Using the definitions of the dimensionless parameters
,
and the Rabi frequency
in Equations (6), (30), we evaluate the probabilities in Equation (29) and plot the quantum system entanglement degree
in Equation (33) against scaled time
for arbitrarily chosen values of sum frequency
and photon number
in Figures 4-6.
The graphs in Figures 4-6 show the effect of photon number n and sum frequency
on the dynamical behavior of quantum entanglement measured by the von Neumann entropy
(
;
).
Figure 4. Degree of entanglement against scaled time for sum frequency
when
and
.
Figure 5. Degree of entanglement against scaled time for sum frequency
and
when
.
Figure 6. Degree of entanglement against scaled time for sum frequency
when
,
,
and
.
In the three figures, the phenomenon of entanglement sudden birth (ESB) and sudden death (ESD) is observed during the time evolution of entanglement similar to that observed in the JC model [34] [35] [36]. In ESB there is an observed creation of entanglement where the initially un-entangled qubits are entangled after a very short time interval. For fairly low values of photon numbers n and sum frequency
as demonstrated in Figure 4 for
plotted when
,
, the degree of entanglement rises sharply to a maximum value of unity (
) at an entangled state, stays at the maximum level for a reasonably short duration, decreases to a local minimum, then rises back to the maximum value before falling sharply to zero (
) at the separable state. The local minimum disappears for larger values of sum frequency
at low photon number n and re-emerge at high photon number
(see Figure 5 and Figure 6) as examples. However, in comparison to the resonance case
in the JC model [36] we notice a long-lived entanglement at
in the cases of
plotted when
in Figure 5 and
plotted when
in Figure 6. The process of ESB and ESD then repeats periodically, consistent with Rabi oscillations between the qubit states.
In Figure 4 and Figure 6 sum frequencies are kept constant at
and
respectively and photon number n is varied in each case. We clearly see that the frequency of oscillation of
increases with an increase in photon number n. This phenomenon in which the frequency of oscillation of
increases with an increase in photon number n is also observed in the JC model [35] [36].
To visualize the effect of sum frequency parameter
on the dynamics of
, we considered values of sum frequency set at
and
for photon number
in Figure 5. It is clear that the frequency of oscillation of
increases with an increase in sum frequency
. In the JC model when detuning
is set at off resonance
results into a decrease in the frequency of oscillation of
as seen in [35] [36] [37] in comparison to the resonance case
.
Finally, for
plotted when
in Figure 5 and in Figure 6 in comparison to
plotted when
in Figure 5, it is clear in Figure 5 that the degree of entanglement
decreases at a high value of sum frequency a phenomenon similar to the JC model in [37]. The observed decrease in degree of entanglement is due to the property that the system loses its purity and the entropy decreases when the effect of sum frequency is considered for small number of photons n. This is remedied when the effect of sum frequency is considered for higher photon numbers n as shown in Figure 6.
5. Teleportation
In the present work we consider an interesting case of quantum teleportation by applying entanglement swapping protocol (teleportation of entanglement) [38] [39] [40] [41] where the teleported state is itself entangled. The state we want to teleport is a two-atom maximally entangled state in which we have assigned subscripts to distinguish the atomic qubit states in the form [42]
(34)
and it is in Alice’s possession. In another location Bob is in possession of a maximally entangled qubit state
generated in the AJC interaction in Equation (21c) and expressed as
(35)
where we have also assigned subscripts to the qubits in Equation (35) to clearly distinguish them.
An observer, Charlie, receives qubit-1 from Alice and qubit-x from Bob. The entire state of the system
(36a)
which on substituting
and
from Equations (34), (35) and reorganizing takes the form
(36b)
after introducing the emerging Bell states obtained as
(37)
Charlie performs Bell state projection between qubit-1 and qubit-x (Bell state measurement (BSM)) and communicates his results to Bob which we have presented in Section 5.1 below.
5.1. Bell State Measurement
BSM is realized at Charlie’s end. Projection of a state
onto
is defined as [43]
(38)
Using
from Equation (36b) and applying Equation (38) we obtain a Bell state projection outcome communicated to Bob in the form
(39a)
The Bell state
in Equation (39a) is in the form of Alice’s qubit in Equation (34). Alice and Bob now have a Bell pair between qubit-2 and qubit-3. Similarly the other three Bell projections take the forms
(39b)
(39c)
(39d)
For these cases of Bell state projections in Equations (39b)-(39d) it will be necessary for Bob to perform local corrections to qubit-3 by Pauli operators as shown in Table 1. We also see that the probability of measuring states
in
Equations (39a)-(39d) in Charlie’s lab is
. In general, by application of the
entanglement swapping protocol (teleportation of entanglement), qubit-2 belonging to Alice and qubit-3 belonging to Bob despite never having interacted before became entangled. Further, we see that a maximally entangled anti-symmetric atom-field transition state
(in Equation (21c)) easily generated in the AJC interaction, can be used in quantum information processing (QIP) protocols like entanglement swapping (teleportation of entanglement) which we have demonstrated in this work. We note that it is not possible to generate such an entangled anti-symmetric state in the JC interaction starting with the atom initially in the ground state and the field mode in the vacuum state [5]. Recall that the JC interaction produces a meaningful physical effect, namely, spontaneous emission only when the atom is initially in the excited state and the field mode in the vacuum state.
5.2. Maximal Teleportation Fidelity
For any two-qubit state
the maximal fidelity is given by [44]
(40)
where
is the fully entangled fraction defined in the form [32]
(41)
From Table 1
(42)
Table 1. Showing how Bob applies an appropriate gate to his qubit based on BSM from Charlie.
(43)
Substituting the results in Equation (42) and Equation (43) into the fully entangled fraction Equation (41) we obtain
(44)
Substituting the value of the fully entangled fraction into Equation (40) we get
(45)
a maximal teleportation fidelity of unity, showing that the state was fully recovered,i.e. Alice’s qubit in Equation (34) was successfully teleported to Bob. We obtain an equal outcome to all the other measured states. We have thus achieved teleportation using a maximally entangled qubit state generated in an AJC interaction, using the case where the atom and field are initially in the absolute ground state
,
as an example.
6. AJC State Engineering and Experimental Implementation
In order to systematically implement the AJC Hamiltonian with a single tuned blue-sideband interaction, the simulation process will involve AJC state preparation followed by unitary transformation and measurement.
The state of the whole system as an interaction of a two-level atom and one photon where both the atom and photon are in ground state
,
will take the form of Equation (18). In a field mode that keeps the cavity field with upto one photon, the main focus should be to determine the experimental values of the probability amplitudes
(46a)
(46b)
for the initial states
and
respectively in Equation (18) and show their variation with time that has a direct correspondence to Rabi frequency
, which is of the form
(46c)
The measurement procedure can be easily implemented using efficient experimental schemes for manipulating quantum entanglement with atoms and photons in a cavity strictly in the AJC model, during which process difficulties can be determined as appropriate. The most common scheme being cavity quantum electrodynamics [45].
Entanglement swapping is realised in an experimental set-up through Bell state measurement. Initially, the two sets of entangled states in Equations (34) and (35) are prepared. The entire state of the system then takes the form of Equation (36b). The required Bell state measurement is achieved in this case by first applying a quantum controlled-NOT(C-NOT) gate operation followed by a quantum Hadamard gate operation to qubit 1, which we now explain with examples below. In order to realise a C-NOT quantum gate operation in this case, we note that state evolution operator in the AJC interaction is generated by the time evolution operator in Equation (12), which on substituting the Hamiltonian
from Equation (11) and dropping the factorizable global phase factor
, we define a C-NOT gate operator in the AJC model in the general form in Equation (10), which we rewrite here for ease of reference
(10')
The C-NOT gate process consists of a two-level atom as the control qubit, which constitutes a two dimensional Hilbert space spanned by the atomic excited and ground states
,
as basis vectors. Two non-degenerate and orthogonal polarized cavity modes
and
make the target qubit. The target qubit is defined in two-dimensional Hilbert space spanned by the state vector
, which expresses the presence of one photon in mode A, when there is no photon in mode B, and the state vector
, which indicates that mode A is in the vacuum state and one photon is present in mode B.
Let us consider the case when qubit 1 (in Charlie’s possession) in ground state
enters an electromagnetic cavity with mode A in vacuum state and a single photon in mode B. The atom couples to the anti-rotating negative frequency component of the field mode undergoing an AJC qubit state transition. After the
atom interacts with mode A for a time
, equal to half Rabi oscillation time, the driving field is modulated such that
(47)
Redefining [5]
(48)
and considering a resonance case where
with the coupling strength
far much greater than the quantized field mode angular frequency
, that is
in the deep strong coupling regime of the AJC model,
in Equation (48) becomes very small thus
(49)
since
in Equation (47) determined from the general form in Equation (6). The evolution of this interaction determined by applying the AJC qubit state transition operation in Equation (10) noting the definition of
and
[5] in Equation (8) is of the form
(50a)
which reduces to
(50b)
We observe that the atom interacted with mode A and completed half of the Rabi oscillation, as a result, it contributed a photon to mode A and evolved to excited state
. Now, after the interaction time, it enters mode B containing a single photon, interacting with the cavity mode as follows
(50c)
After an interaction with mode B for a time
such that
, the driving field is modulated such that
with
since
and
since
. Therefore,
. The form of Equation (50c) results into the evolution
(50d)
The results in Equation (50d) show that the atom evolves to ground state and absorbs a photon initially in mode B. Therefore the atom clearly performs a swapping of the electromagnetic field between the two field modes by controlled interaction.
When the atom in ground state
, enters the electromagnetic cavity containing a single photon in mode A and mode B in vacuum state, the atom and the field interact as follows
(50e)
After an interaction with field mode B for a time
equal to half Rabi oscillation time, the driving field is modulated such that
, with
since
. Therefore
. The form of Equation (50e) results in the evolution
(50f)
The atom then enters mode A containing one photon and interacts as follows
(50g)
After an interaction with the cavity mode for a time
such that
we obtain a driving field modulation
, with
since
and
since
. Therefore
. The form of Equation (50g) results into the evolution
(50h)
This shows that the atom evolves to ground state and performs a field swapping by absorbing a photon in mode A.
When the qubit 1, a two-level atom in excited state
enters mode A in vacuum state, that is target qubit
, the atom propagates as a free wave without coupling to the field mode in vacuum state
[5], leaving the cavity without altering the state of the cavity-field mode. A similar observation is made when the atom in excited state
enters cavity B in vacuum state for the case of target qubit
.
The Hadamard gate operation then follows. Noting the qubit state transition algebraic operations in Equation (9), we identify the normalized qubit state transition operator
defined in Equation (8) as the AJC Hadamard gate operator which we use Equation (4) to express in the general form
(51a)
where
is defined in Equation (6). For the specific example where atom begins in the ground state
and the field mode in the vacuum state
, we set
and take
in Equation (6) to define the corresponding Hadamard gate operator in the form
(51b)
Applying this Hadamard gate operator, rotates the initial atomic ground state
to
(51c)
On the other hand, if the atom starts from an initial excited state
, the appropriate Hadamard gate operator for such a process follows from the definition of the relevant normalised qubit state transition operator
in [5], which on setting
and
takes the form
(52a)
which rotates the initial atomic excited state
to
(52b)
Application of the C-NOT and Hadamard gate operations using the respective operators defined in Equation (10') or earlier (10) and Equations (51b), (52a) as briefly explained in the above example, provides a practical platform for experimental implementation of the AJC quantum teleportation process described in Sec. 5. Here, results of the Bell state measurement are communicated to Bob (by Charlie) who applies appropriate single-qubit rotation to qubit 3 in his possession. Details of experimental design, procedures and difficulties can be provided as appropriate, noting that the quantum Rabi interaction is generally achieved in cavity or circuit quantum electrodynamics, quantum dots or ion traps, etc.
7. Conclusion
In this paper we have analysed entanglement of a two-level atom and a quantized electromagnetic field mode in an AJC qubit formed in the AJC interaction mechanism. The effect of sum-frequency parameter and photon number on the dynamical behavior of entanglement measured by von Neumann entropy was studied which brought a clear visualization of this interaction similar to the graphical representation on Bloch sphere. The graphical representation of Rabi oscillations on the Bloch sphere demonstrated an important physical property that the AJC interaction process occurs in the reverse sense relative to the JC interaction process. We further generated an entangled AJC qubit state in the AJC interaction mechanism which we used in the entanglement swapping protocol as Bob’s qubit. We obtained an impressive maximal teleportation fidelity
showing that the state was fully recovered. This impressive result of fidelity, opens all possible directions for future research in teleportation strictly within the AJC model. In conclusion we observe that the operator ordering that distinguishes the rotating (JC) component and anti-rotating component (AJC) has an important physical foundation with reference to the rotating positive and anti-rotating negative frequency components of the field mode which dictates the coupling of the degenerate states of a two-level atom to the frequency components of the field mode, an important basis for realizing the workings in the AJC interaction mechanism and JC interaction mechanism.
Acknowledgements
We thank Maseno University Department of Physics and Materials Science for providing a conducive environment to do this work.