Harnessing dark states: coherent control in coupled cavity-Rydberg-atom systems
Pith reviewed 2026-05-10 17:27 UTC · model grok-4.3
The pith
The arrowhead-matrix method identifies the explicit numbers and forms of dark states in excitation subspaces of N Rydberg atoms coupled to a cavity.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Using the arrowhead-matrix method, the work obtains the numbers and forms of the dark states in certain excitation-number subspaces for the two-, three-, and four-atom cases, as well as in the single-excitation subspace for a general N-atom case. It further suggests characterizing these states by inspecting the populations of specific quantum states detectable in experiments and analyzes the effect when both dipole-dipole and atom-cavity couplings vary with atomic position.
What carries the argument
The arrowhead-matrix method applied to the Hamiltonian restricted to fixed excitation-number subspaces, which isolates the zero-energy eigenstates arising from destructive interference between cavity and atomic excitations.
If this is right
- Dark states appear in definite numbers and explicit superpositions inside each relevant excitation subspace for two to four atoms.
- In the single-excitation manifold the dark-state structure generalizes to arbitrary atom number N.
- Dark states can be identified experimentally by measuring populations in a small set of basis states.
- Position dependence in the interaction strengths does not eliminate the dark states but modifies their concrete form.
- The cavity-Rydberg platform therefore supports coherent control protocols based on these states.
Where Pith is reading between the lines
- The same subspace-by-subspace counting could be used to design protected subspaces for quantum memory or gates that avoid cavity loss.
- Extending the arrowhead construction to multi-mode cavities or inhomogeneous fields would test how robust the dark-state count remains under more complex geometries.
- Population-based characterization offers a direct experimental route to verify dark-state presence without full state tomography.
Load-bearing premise
The system stays confined to isolated excitation-number subspaces and the arrowhead matrix fully captures the eigenstates without decoherence or higher-order corrections.
What would settle it
Prepare a two-atom system in the single-excitation manifold, let it evolve under the cavity and dipole-dipole couplings, and check whether the measured populations of the predicted dark-state basis vectors remain constant while other states oscillate.
Figures
read the original abstract
The dark-state effect, caused by destructive interference, not only is an important fundamental research topic in atomic physics and quantum optics, but also has wide potential application in quantum physics and quantum information science. Using the arrowhead-matrix method, here we study the dark-state effect in a coupled cavity-Rydberg-atom system, in which $N$ Rydberg atoms with the dipole-dipole interactions are coupled to a single-mode cavity field. We obtain the numbers and form of the dark states in certain excitation-number subspaces for the two-, three-, and four-atom cases, as well as in the single-excitation subspace for a general $N$-atom case. We also suggest to characterize the dark states by inspecting the populations of some specific quantum states, which can be detected in experiments. Furthermore, we analyze the dark-state effect in a realistic case, where both the atomic dipole-dipole interaction strengths and the atom-cavity-field coupling strengths depend on the position of the atoms. Our findings pave the way for studying dark-state physics and applications in the cavity-Rydberg-atom platform.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript applies the arrowhead-matrix method to a Hamiltonian describing N Rydberg atoms with dipole-dipole interactions coupled to a single-mode cavity. It derives the number and explicit forms of dark states in selected excitation-number subspaces for the N=2, 3, and 4 cases, as well as in the single-excitation subspace for arbitrary N. The authors further propose characterizing these states via measurable populations of specific basis states and extend the analysis to position-dependent atom-cavity and dipole-dipole couplings.
Significance. If the derivations hold, the work supplies exact, parameter-free expressions for dark states in conserved excitation subspaces of a hybrid cavity-Rydberg platform. This is valuable for coherent control and quantum-information applications, as the arrowhead structure arises directly from the uniform cavity coupling and is solved exactly without additional approximations. The inclusion of position-dependent parameters addresses experimental realism and strengthens the practical utility of the results.
minor comments (3)
- [Abstract] The abstract and introduction refer to 'certain excitation-number subspaces' for the two-, three-, and four-atom cases without explicitly listing the subspaces (e.g., single-excitation, double-excitation) considered in each; adding this detail would improve clarity.
- [Section on general N] In the general-N single-excitation analysis, the explicit form of the dark-state coefficients is given but the normalization step is only sketched; providing the closed-form normalization constant would aid reproducibility.
- [Characterization proposal] The suggestion to characterize dark states by inspecting populations of specific states is useful, but the manuscript does not discuss how finite detection efficiency or measurement back-action would affect this protocol; a brief remark on experimental feasibility would be helpful.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately captures the scope of our work on dark states in the cavity-Rydberg system using the arrowhead-matrix approach. We provide responses below to the points raised in the report.
Circularity Check
No significant circularity detected
full rationale
The derivation applies the arrowhead-matrix method directly to the block-diagonal Hamiltonian in conserved excitation-number subspaces (single-excitation for general N, and specific subspaces for N=2,3,4). These subspaces follow rigorously from the excitation-conserving structure of the cavity-Rydberg Hamiltonian with no decay terms; the arrowhead form is generated by the uniform atom-cavity coupling term and is diagonalized exactly by the cited method without any fitted parameters, self-citation load-bearing premises, or ansatz smuggling. Position-dependent dipole-dipole and coupling strengths enter only as numerical entries inside the same blocks and do not mix subspaces or alter the algebraic counting of dark states. No step reduces the claimed dark-state counts or forms to a re-labeling or re-fitting of the input Hamiltonian.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The system Hamiltonian is block-diagonal in excitation-number subspaces
- domain assumption The arrowhead-matrix method yields the exact eigenstates of the system
Reference graph
Works this paper leans on
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[1]
C(1) [2] C(1) [2] † L(1) [2] = −∆a g1 g2 g1 0V dd g2 Vdd 0 ,(9) whereU (1)
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[2]
are the submatrices related to the upper- and lower-state components in the single-excitation subspace, respectively, andC (1)
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[3]
is the coupling matrix describing the couplings between these two components. It should be noted that the superscript “ (1)” and the subscript “ [2]” in Eq. (9) are introduced to denote the single-excitation subspace and two Rydberg atoms coupled to the cavity field, respectively. We consider that the coupling strengthsg 1 andg 2, as well as the dipole-di...
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[4]
with the unitary matrix Sl = 1/ √ 2 1/ √ 2 −1/ √ 2 1/ √ 2 ! ,(10) the Hamiltonian can be transformed into an arrowhead matrix ˜H(1)
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[5]
= U(1) [2] ˜C(1) [2] ˜C(1) [2] † ˜L(1) [2] = −∆a G1 G2 G1 Vdd 0 G2 0−V dd ,(11) where the coupling strengths are introduced asG 1 = (g1 +g 2)/ √ 2 andG 2 =(−g 1 +g 2)/ √
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[6]
Here, these three new basis states for the Hamiltonian ˜H(1)
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[7]
are given by |u1⟩= |1,g,g ⟩ ,(12a) |L(1)
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[8]
(1)⟩= 1√ 2 |0⟩ (|e,g ⟩ + |g,e ⟩) ,(12b) |L(1)
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[9]
(2)⟩= 1√ 2 |0⟩ (− |e,g ⟩ + |g,e ⟩) .(12c) The two states|L (1)
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[10]
The dark states in this case can be obtained by analyzing Eq
(2)⟩are the eigenstates of the lower-state submatrix ˜L(1) [2], with the corresponding eigenval- uesV dd and−V dd. The dark states in this case can be obtained by analyzing Eq. (11) with the arrowhead-matrix method [39, 81]. (1) Consider the case of zero coupling column vector: (i) Wheng 1 =−g 2, the corresponding coupling strengthG 1 be- tween the lower ...
-
[11]
(1)⟩and the upper state|u 1⟩is zero, 5 then|L (1)
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[12]
(ii) Wheng 1 =g 2, we haveG 2 =0, then the state|L (1)
(1)⟩becomes a dark state. (ii) Wheng 1 =g 2, we haveG 2 =0, then the state|L (1)
-
[13]
(2)⟩is decoupled from the upper state|u 1⟩and becomes a dark state. We point out that these two states|L (1) [2](1)⟩and|L (1) [2](2)⟩are the Bell states [86], which are maximally entangled states involving two atoms. (2) Consider the case of degenerate lower-state subspace: For avoiding change of the coupling structure of the system, we consider the case ...
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[14]
(1)⟩is coupled to the upper state|1,g,g⟩, while the other dressed lower state|L (1)
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[15]
(2)⟩is the dark state and here it is decoupled from the upper state|1,g,g⟩. In addition, the upper state|1,g,g⟩is connected to the ground state|0,g,g⟩ through the cavity-field dissipation. Since the dark state is de- coupled from the cavity, it can be witness by selecting proper initial states. For example, the state|0,e,g⟩is the superposi- tion of the br...
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[16]
(1)⟩and dark state|L (1)
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[17]
(2)⟩, i.e., |0,e,g⟩=(|L (1)
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[18]
Under the dissipation of the cavity, the bright state|L (1)
-
[19]
(1)⟩coupled to the upper state |1,g,g⟩will be finally dissipated into the ground state|0,g,g⟩; while the dark state|L (1)
-
[20]
There- fore, the steady-state population of the dark state|L(1)
(2)⟩will remain unchanged. There- fore, the steady-state population of the dark state|L(1)
-
[21]
(2)⟩will be a signature for the existence of the dark state, and this fea- ture can be used to characterize the dark-state effect. In partic- ular, the dark-state population can be distinguished from the ground state|0,g,g⟩by detecting the excited-state probability of the two atoms. Hence, the present dark-state characteriza- tion can be realized in exper...
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[22]
(2)⟩, and|0,g,g⟩ as functions of the scaled timeg 1tin Fig. 2(b). It can be seen that the dressed lower states|L (1)
-
[23]
As time evolves, the populations of the states|1,g,g⟩and the dressed lower state|L (1)
(2)⟩each pos- sess population of 1/2 at the initial moment. As time evolves, the populations of the states|1,g,g⟩and the dressed lower state|L (1)
-
[24]
(1)⟩exhibit coherent oscillations, then decay to the ground state|0,g,g⟩, while the population of the dark state |L(1)
-
[25]
Therefore, we can observe the dark state|L (1)
(2)⟩remains unchanged all the time. Therefore, we can observe the dark state|L (1)
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[26]
(2)⟩by measuring the atomic pop- ulations, and there are populations in these two atoms. We čaĎ čbĎ 1g t Probability 0.5 0 0 10 20 30 (1) [2]| (1)L 〉 (1) [2]| (2)L 〉 | 0, ,g g〉 |1, ,g g〉κ 1G (1) [2]| (1)L 〉 (1) [2]| (2)L 〉 | 0, ,g g〉 |1, ,g g〉 FIG. 2. (a) Energy-level diagram of the coupled cavity-two-atom system confined in the zero- and single-excitatio...
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[27]
(b) The occupation probabilities of the states|1,g,g⟩(green),|L (1)
(2)⟩is the dark state in this case. (b) The occupation probabilities of the states|1,g,g⟩(green),|L (1)
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[28]
(1)⟩(yel- low),|L (1)
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[29]
(2)⟩(purple), and|0,g,g⟩(black) as functions of time in the open-system case. The initial state is|0,e,g⟩. Other parameters used are∆ a/g1 =0,g 2/g1 =1,V dd/g1 =0.5, andκ/g 1 =0.3. should point out that the initial state|0,e,g⟩can be prepared by only driving the first atom with the Hamiltonian in Eq. (8b). In particular, by driving the second atom and set...
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[30]
(2)⟩. IV . DARK STA TES IN THE THREE-A TOM CASE We now turn to the three-atom case in which the system is described by the Hamiltonian ˆH[3] [N=3 for Eq. (2)]. Similarly, we consider a simplified case where the dipole- dipole interaction strengths are identical, namelyV j j′ =V dd forj,j ′ =1,2,3 andj,j ′. We study the dark states in both the single- and ...
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[31]
Single-excitation subspace In the single-excitation subspace, there are four ba- sis states {|1,g,g,g ⟩ , |0,e,g,g ⟩ , |0,g,e,g ⟩ , |0,g,g,e ⟩} for the three-atom system. According to the involved cavity pho- ton number, there is one upper state|u 1⟩= |1,g,g,g ⟩ and three lower states|l 1⟩= |0,e,g,g ⟩,|l 2⟩= |0,g,e,g ⟩, and |l3⟩= |0,g,g,e ⟩ .We can divide...
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[32]
= − 3 2∆a g1 g2 g3 g1 − 1 2∆a Vdd Vdd g2 Vdd − 1 2∆a Vdd g3 Vdd Vdd − 1 2∆a ,(13) where we consider non-zerog j (forj=1,2,3) andV dd to en- sure the coupling structure unchanged for the three-atom case. By diagonalizing the lower-state submatrix with the unitary matrix Sl = 1/ √ 3 1/ √ 3 1/ √ 3 −1/ √ 2 1/ √ 2 0 ...
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[33]
is transformed into an arrowhead matrix ˜H(1)
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[34]
= − 3∆a 2 G1 G2 G3 G1 −∆a+4Vdd 2 0 0 G2 0− ∆a+2Vdd 2 0 G3 0 0− ∆a+2Vdd 2 ,(15) where the coupling strengths are introduced asG 1 = (g1 +g 2 +g 3)/ √ 3,G 2 =(−g 1 +g 2)/ √ 2, andG 3 = −(g1 +g 2 −2g 3)/ √
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[35]
We point out that the matrix ˜H(1)
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[36]
is ex- pressed with the following basis states |u1⟩= |1,g,g,g ⟩ ,(16a) |L(1)
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[37]
(1)⟩= 1√ 3 |0⟩ (|e,g,g ⟩ + |g,e,g ⟩ + |g,g,e ⟩) ,(16b) |L(1)
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[38]
(2)⟩= 1√ 2 |0⟩ (− |e,g ⟩ + |g,e ⟩) |g⟩ ,(16c) |L(1)
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[39]
(15) with the arrowhead-matrix method
(3)⟩= 1√ 6 |0⟩ (− |e,g,g ⟩ − |g,e,g ⟩ +2 |g,g,e ⟩) .(16d) The dark states in this case can be obtained by analyzing Eq. (15) with the arrowhead-matrix method. (1) Consider the case of zero coupling column vector: (i) Wheng 1 +g 2 +g 3 =0, we haveG 1 =0, then the state|L (1)
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[40]
(1)⟩ is decoupled from the upper state|u 1⟩and becomes a dark state. This state|L (1)
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[41]
(ii) Wheng 1 =g 2, the coupling strengthG 2 =0, then the state|L (1)
(1)⟩is aWstate [87] involving three atoms. (ii) Wheng 1 =g 2, the coupling strengthG 2 =0, then the state|L (1)
- [42]
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[43]
(iii) When g1 +g 2 =2g 3, we getG 3 =0, then the state|L (1)
(2)⟩is a Bell state of the first and second atoms, and the third atom is decoupled from other subsystems. (iii) When g1 +g 2 =2g 3, we getG 3 =0, then the state|L (1)
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[44]
(2) Consider the case of degenerate lower-state subspace: It can be seen from Eq
(3)⟩becomes a dark state, which is also an entangled state involving these three atoms. (2) Consider the case of degenerate lower-state subspace: It can be seen from Eq. (15) that the second and third eigenval- ues are identical, then there is a two-dimensional degenerate lower-state subspace{|L (1)
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[45]
(3)⟩}. As a result, there exists one dark state |D(1) [3]⟩= 1 N (1) [3] G2|L(1) [3](3)⟩ −G 3|L(1) [3](2)⟩ = 1 N (1) [3] |0⟩ " − G2 √ 6 + G3 √ 2 ! |e,g,g⟩ − G2 √ 6 + G3 √ 2 ! |g,e,g⟩+ G2 √ 6 |g,g,e⟩ # ,(17) where the constantN (1)
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[46]
The dark state|D (1) [3]⟩is aWstate [87], which is an entangled state involving three atoms
=(G 2 2 +G 2 3)1/2 is introduced. The dark state|D (1) [3]⟩is aWstate [87], which is an entangled state involving three atoms. It should be pointed out that, when G2 =0 orG 3 =0, the state|D (1) [3]⟩will be reduced to|L (1)
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[47]
Based on the above discussions, we know that, whenG 1 = 0, there are two dark states|L (1)
(3)⟩, respectively. Based on the above discussions, we know that, whenG 1 = 0, there are two dark states|L (1)
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[48]
(1)⟩and|D (1) [3]⟩in the single- excitation subspace; whenG 1 ,0, then there is one dark state |D(1) [3]⟩
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[49]
Double-excitation subspace In the double-excitation subspace, there are seven basis states{ |2,g,g,g ⟩ , |1,e,g,g ⟩ , |1,g,e,g ⟩ , |1,g,g,e ⟩ , |0,e,e,g ⟩ , |0,e,g,e ⟩ , |0,g,e,e ⟩}. According to the involved cavity pho- ton number, these seven states can be divided into the upper- and lower-state components. Concretely, there are four upper states{|u 1⟩=...
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[50]
C(2) [3] C(2) [3] † L(2) [3] = −3∆a 2 √ 2g1 √ 2g2 √ 2g3 0 0 0 √ 2g1 −∆a 2 Vdd Vdd g2 g3 0 √ 2g2 Vdd −∆a 2 Vdd g1 0g 3 √ 2g3 Vdd Vdd −∆a 2 0g 1 g2 0g 2 g1 0 ∆a 2 Vdd Vdd 0g 3 0g 1 Vdd ∆a 2 Vdd 0 0g 3 g2 Vdd Vdd ∆a 2 . (18) Similarly, h...
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[51]
and the corresponding cou- pling submatrix ˜C(2)
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[52]
=diag ∆a 2 +2V dd, ∆a 2 −V dd, ∆a 2 −V dd ! ,(19a) ˜C(2)
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[53]
=(G 1,G 2,G 3)= 0 0 0 g2+g3√ 3 −g2+g3√ 2 −g2−g3√ 6g1+g3√ 3 −g1√ 2 −g1+2g3√ 6g1+g2√ 3 g1√ 2 −g1+2g2√ 6 .(19b) Note that the dressed lower states of the submatrix ˜L(2)
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[54]
(1)⟩= 1√ 3 |0⟩ (|e,e,g ⟩ + |e,g,e ⟩ + |g,e,e ⟩) ,(20a) |L(2)
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[55]
(2)⟩= 1√ 2 |0⟩ |e⟩ (− |e,g ⟩ + |g,e ⟩) ,(20b) |L(2)
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[56]
(19), we can analyze the dark states with the arrowhead-matrix method
(3)⟩= 1√ 6 |0⟩ (− |e,e,g ⟩ − |e,g,e ⟩ +2 |g,e,e ⟩) .(20c) Based on Eqs. (19), we can analyze the dark states with the arrowhead-matrix method. (1) Consider the case of zero coupling column vector: We find that there are no proper non-zero parametersg 1,g 2, and g3 satisfyingG i =0fori=1,2,3. As a result, no dark states corresponding to the zero coupling c...
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[57]
Therefore, there is no dark state
(3)⟩}, the corresponding coupling submatrix re- lated to the degenerate subspace is full rank for non-zerog j (forj=1,2,3). Therefore, there is no dark state. Based on the above analyses, we know that there is no dark state in the double-excitation subspace for the three-atom case. B. Characterization of the dark states In this section, we study the chara...
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[58]
(1)⟩and|D (1) [3]⟩, and only the bright state |B(1) [3]⟩=(G 2|L(1)
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[59]
is coupled to the up- per state|1,g,g,g⟩. The upper state|1,g,g,g⟩is connected to a b Probability d 1g t Probability 0.70 0.35 0 (1) [3]| (1)L 〉 (1) [3]| B 〉 (1) [3]| D 〉 | 0, , ,g g g〉 |1, , ,g g g〉 c 0 10 20 30 0.70 0.35 0 (1) [3]| (1)L 〉 (1) [3]| B 〉 (1) [3]| D 〉 | 0, , ,g g g〉|1, , ,g g g〉 κ 2 3,G G (1) [3]| (1)L 〉 (1) [3]| B 〉 (1) [3]| D 〉 |1, , ,g g...
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[60]
The initial states are: (c)|0,g,g,e⟩, (d) |0,e,g,g⟩
(1)⟩(blue),|B (1) [3]⟩ (yellow),|D (1) [3]⟩(purple), and|0,g,g,g⟩(black) as functions of time in the open-system case. The initial states are: (c)|0,g,g,e⟩, (d) |0,e,g,g⟩. The used parameters are (c)g 2/g1 =0.9 andg 3/g1 = −1.9, (d)g 2/g1 =0.8 andg 3/g1 =1.5. Other parameters used are Vdd/g1 =0.5,∆ a/g1 =0, andκ/g 1 =0.3. the ground state|0,g,g,g⟩through ...
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[61]
This is because the state |0,g,g,e⟩and the dark state|D (1) [3]⟩are almost orthogonal in this case
(1)⟩can be dis- tinguished from the dark state|D (1) [3]⟩. This is because the state |0,g,g,e⟩and the dark state|D (1) [3]⟩are almost orthogonal in this case. Therefore, the presence of the dark state|L (1)
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[62]
(1)⟩ is signaled by its steady-state population, which provides a useful means for characterizing the dark-state effect. Notably, one can differentiate the dark-state population from that of the ground state|0,g,g,g⟩via detection of the excited-state prob- ability of the three atoms. As a result, the proposed scheme for dark-state characterization is real...
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[63]
Similarly, the dark-state effect can be identified from the population of the system
(1)⟩and the bright state|B (1) [3]⟩are coupled to the upper state|1,g,g,g⟩ which is coupled to the ground state|0,g,g,g⟩via cavity-field dissipation, while the dark state|D (1) [3]⟩is decoupled from the upper state|1,g,g,g⟩. Similarly, the dark-state effect can be identified from the population of the system. When the initial state is|0,e,g,g⟩, which is t...
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[64]
(1)⟩,|B (1) [3]⟩, and|D (1) [3]⟩, the steady state of the system will be the superposition of the dark state|D (1) [3]⟩and the ground state|0,g,g,g⟩. Except the dark state, all states eventually de- cay to the ground state|0,g,g,g⟩because of the dissipation of the cavity. Hence, the existence of the dark state|D (1) [3]⟩can be revealed through its finite ...
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[65]
(1)⟩,|B (1) [3]⟩, |D(1) [3]⟩, and|0,g,g,g⟩as functions of the scaled timeg 1twhen the initial state is|0,g,g,e⟩in Fig. 3(c). At the initial time, populations are present for these three states|L (1)
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[66]
(1)⟩and |B(1) [3]⟩while the population of the dark state|D (1) [3]⟩is approach zero. As time evolves, we can find that the populations of states|1,g,g,g⟩and|B (1) [3]⟩exhibit oscillations, then decay to the ground state|0,g,g,g⟩, while the population of the dark state|L (1)
-
[67]
(1)⟩remains unchanged all the time. Therefore, the system eventually relaxes to a steady state with population only in the dark state|L (1)
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[68]
(1)⟩and the ground state|0,g,g,g⟩, and these two states can be distinguished by measuring the atomic populations. In addition, the initial state|0,g,g,e⟩can be prepared by only driving the third atom with the Hamilto- nian in Eq. (8b). All these features increase the probability for experimental implementation of this system. For case (2), in Fig. 3(d) we...
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[69]
Similarly, at the initial time, populations are present for these three states|L (1)
(1)⟩,|B (1) [3]⟩,|D (1) [3]⟩, and|0,g,g,g⟩as func- tions of the scaled timeg 1twhen the initial state is|0,e,g,g⟩. Similarly, at the initial time, populations are present for these three states|L (1)
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[70]
As time evolves, the populations of these states|1,g,g,g⟩,|L (1)
(1)⟩,|B (1) [3]⟩, and|D (1) [3]⟩. As time evolves, the populations of these states|1,g,g,g⟩,|L (1)
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[71]
(1)⟩, and|B (1) [3]⟩ex- hibit oscillations, and then decay to the ground state|0,g,g,g⟩, while the population of the dark state|D (1) [3]⟩stays the same throughout the dynamics. Therefore, the system eventually relaxes to a steady state with population only in the dark state |D(1) [3]⟩and the ground state|0,g,g,g⟩. The dark state|D (1) [3]⟩can be identifi...
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[72]
Single-excitation subspace In the single-excitation subspace, the basis states are given by{ |1,g,g,g,g ⟩ , |0,e,g,g,g ⟩ , |0,g,e,g,g ⟩ , |0,g,g,e,g ⟩ , |0,g,g,g,e ⟩}for the four-atom system, and there is one up- per state|u 1⟩= |1,g,g,g,g ⟩ and four lower states{|l 1⟩= |0,e,g,g,g ⟩ ,|l 2⟩= |0,g,e,g,g ⟩ ,|l 3⟩= |0,g,g,e,g ⟩ ,|l 4⟩= |0,g,g,g,e ⟩}.We define...
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[73]
= −2∆a g1 g2 g3 g4 g1 −∆a Vdd Vdd Vdd g2 Vdd −∆a Vdd Vdd g3 Vdd Vdd −∆a Vdd g4 Vdd Vdd Vdd −∆a .(21) Similarly, we consider that both the coupling strengthg j (for j=1-4) andV dd are non-zero to prevent change of the cou- pling configuration for the system. By diagonalizing the lower-state submatrix with the unitary...
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[74]
becomes an arrowhead matrix ˜H(1)
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[75]
= U(1) [4] ˜C(1) [4] ˜C(1) [4] † ˜L(1) [4] ,(23) where these submatrices are given by U(1)
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[76]
=diag (−∆a +3V dd,−∆ a −V dd,−∆ a −V dd,−∆ a −V dd) , (24b) ˜C(1)
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We point out that the five new basis states of the Hamiltonian 9 ˜H(1)
=(G1,G 2,G 3,G 4).(24c) Here, we introduce the coupling strengthsG 1 = (g1 +g 2 +g 3 +g 4) /2,G 2 = (−g1 +g 2) / √ 2,G 3 = (−g1 −g 2 +2g 3) / √ 6, andG 4 = (−g1 −g 2 −g 3 +3g 4) /2 √ 3. We point out that the five new basis states of the Hamiltonian 9 ˜H(1)
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[78]
are given by |u1⟩=|1,g,g,g,g⟩,(25a) |L(1)
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[79]
(1)⟩= 1 2 |0⟩(|e,g,g,g⟩+|g,e,g,g⟩+|g,g,e,g⟩ +|g,g,g,e⟩),(25b) |L(1)
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[80]
(2)⟩= 1√ 2 |0⟩(−|e,g⟩+|g,e⟩)|g,g⟩,(25c) |L(1)
discussion (0)
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