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Quantum Rabi oscillations in graphene Enamullah,* Vipin Kumar, Upendra Kumar, and Girish S. Setlur Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India *Corresponding author:
[email protected] Received August 23, 2013; revised January 2, 2014; accepted December 16, 2013; posted January 7, 2014 (Doc. ID 196252); published February 13, 2014 Graphene has been theoretically shown to exhibit anomalous Rabi oscillations (AROs) far from resonance in addition to conventional Rabi oscillations close to resonance (classical light frequency matches particle-hole frequency). The ARO has been attributed to the pseudospin degree of freedom that is unique to graphene-like systems. In this work, we show the same phenomenon also occurs in the single-photon limit or even in a vacuum. This is to be expected for conventional Rabi oscillations; however, the prediction that AROs also occur in the single-photon situation means that this notion of ARO is robust and not an artifact of approximations. We also study collapse and revival of both conventional and AROs in response to a coherent radiation field and extract the collapse and revival times in both cases. © 2014 Optical Society of America OCIS codes: (160.4236) Nanomaterials; (190.4400) Nonlinear optics, materials; (190.4720) Optical nonlinearities of condensed matter. http://dx.doi.org/10.1364/JOSAB.31.000484
1. INTRODUCTION Graphene [1–3] is a highly sought-after research topic in condensed matter in both theory and experiments due to its peculiar properties. It is made of carbon atoms arranged in a honeycomb lattice. The properties that distinguish it from other materials are its linear spectrum near the Dirac points and the presence of a pseudospin degree of freedom. Rabi oscillations [4,5] are among the most well-studied phenomena in two-level systems coupled to radiation. This is a periodic exchange of energy between the two-level system and the applied optical field. The frequency of such oscillations (Rabi frequency) depends upon the strength of the applied field rather than its frequency. The sinusoidal oscillations have maximum amplitude at resonance. Upon treating the radiation field quantum mechanically [6–8], some new phenomena are observed in Rabi oscillations. From a semiclassical perspective, an atom in the excited state cannot make a transition to a lower level in the absence of an external field, whereas in the quantum case, this transition is possible even in a vacuum because of spontaneous emission (zero-point quantum fluctuations of radiation). A solvable model of quantum radiation interacting with a two-level system has been given by Jaynes and Cummings [9,10]. The most interesting phenomena in this model are the collapse and revival oscillations that occur when the electromagnetic (EM) field is in a coherent state [11–13]. These phenomena may be intuitively understood as follows. The population density consists of a sum of oscillating terms, each corresponding to a well-defined photon number n, and each term oscillates with a particular Rabi p frequency proportional [10] to n 1. In this case, one may see that even without any relaxation, the distribution of Rabi frequencies produces an initial dephasing (“collapse”) in the sinusoidal Rabi oscillations [9]. The basic reason is, if two neighboring terms oscillate 180° out of phase, they destructively interfere, whereas if they are in phase, there is constructive interference [8]. In case of collapse, the various
0740-3224/14/030484-10$15.00/0
contributions destructively interfere, thereby producing the effect of decay or dephasing. Revival is purely a quantum phenomenon, however. Revival refers to the resurrection of previously extinguished oscillations due to destructive interference gradually diminishing, progressively revealing the original oscillations. It depends not only upon the coherent state of photons but also on the statistical distribution of photon numbers. A continuous photon distribution would give a collapse just like a classical field, but no revivals. The interaction between a two-level atom and a quantized EM field, viz. the phenomenon of collapse and revival of Rabi oscillations, has been observed in experiments such as cavity quantum electrodynamics (CQED) [14], performed with circular Rydberg atoms [15] (atoms excited to a high quantum number state), and the center-of-mass motion of a trapped ion. Rabi oscillations have long been studied in semiconductors [16] as well, where energy levels are replaced by bands. Rabi oscillations have also been studied in graphene single layer by Mishchenko [17] and Ishikawa [18] using the well-known rotating wave approximation [5] (RWA) close to resonance and also in Landau-quantized graphene by Dora et al. [19]. Recently, we predicted anomalous Rabi oscillations (AROs) in graphene that occur far from resonance using an alternative to RWA that we have been calling asymptotic rotating wave approximation [20] (ARWA). This is the regime in which the incident optical frequency is much higher than the frequency corresponding to particle-hole energy. The natural question that comes to mind is how can one detect AROs experimentally? In a forthcoming work (accepted for publication), we have suggested a technique, viz. the pump–probe method, by which one can detect AROs experimentally. We calculate the probe susceptibility as a function of the area of the pump pulse that has oscillatory behavior, and the frequency of oscillations is just anomalous Rabi frequency. It appears that this phenomenon has already been (unwittingly) experimentally verified in the paper of Breusing et al. [21], which we have © 2014 Optical Society of America
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analyzed in detail in this paper. In another work (to be published) we show that AROs are seen in few-layer graphene as well with some important differences. In our earlier work [20], we have treated the EM field as classical, whereas in this article we treat the EM field quantum mechanically. We wish to show that the phenomenon of AROs occurs even when the EM field is quantum, thereby establishing unequivocally that the AROs are due to pseudospin and not due to approximations or assumptions we have made.
2. MODELS OF GRAPHENE AND PSEUDOSPINLESS GRAPHENE In this section, we introduce models that describe two species of fermionic oscillators coupled to a single-mode photon. This is meant to isolate and highlight the role played by pseudospin in determining the nature of AROs far from resonance in the presence of a quantized single-mode photon field. We demonstrate through an exact solution and otherwise the central result viz. that anomalous oscillations far from resonance occur only in graphene-like systems that possess pseudospin. In addition, conventional Rabi oscillations seen in pseudospinless systems [e.g., the Jaynes–Cummings (JC) model] are, of course, also present. Consider first a graphene-like system. We suppress the momentum label to bring out the role played by other degrees of freedom such as pseudospin. The Hamiltonian for a graphenelike system we suggest is H G c†A ⃗σ AB · ϵ⃗ cB c†B ⃗σ BA · ϵ⃗ cA λ c†A cB b† eiωt λbc†B cA e−iωt :
(1)
Here cA , cB are the A and B sublattice electrons, σ represents the pseudospin matrices, and b; b† 1 are the photon operators. Here we wish to show that in the ARWA limit, the amplitude of the part of current density that oscillates with the frequency of the external radiation in the case of graphene oscillates slowly with the anomalous Rabi frequency, whereas in the case of pseudospinless “graphene” this amplitude is constant in time. The Hamiltonian for a pseudospinless system that possesses all attributes of graphene barring pseudospin is obtained by making the correspondence cA → d† , c†B → c† , but now the kinetic energy is chosen to be purely diagonal. The subscript PSL stands for pseudospinless system. H PSL ϵc† c d† d λc† d† be−iωt λ b† dceiωt :
(2)
These Hamiltonians are designed so that when the fields are p treated classically, b n (a constant), the results, both for graphene and for the peudospinless system are identical to the ones found in our earlier work [20]. Hence the reason for this particular choice of Hamiltonians. Furthermore, the Hamiltonian in Eq. (2), which we have been calling [20] PSL graphene, deserves such an epithet only because we choose to think of ϵ as being linearly related to the momentum. Since the kinetic term is purely diagonal, it is also “pseudospinless”. We make use of the observation that this model is exactly solvable even when b is an operator. This is not surprising since this model may be mapped to the well-known JC model through the following identifications:
σ c† d† ;
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σ − dc;
together with a unitary transformation on the photons, which means we replace b with be−iωt .
3. EQUATIONS FOR THE PROBABILITY AMPLITUDE In order to study the differences that emerge when radiation is treated classically or quantum mechanically, it is best to keep the matter sector as simple as possible. To this end, we assert that all processes in the case of graphene involve just one electron either on site A or on site B, and for the pseudospinless system each state has either (i) no electron or hole or (ii) one electron and one hole. This means the only nonzero amplitudes are h0; 1; njΦtiG and h1; 0; njΦtiG in the case of graphene and h0; 0; njΦtiPSL and h1; 1; njΦtiPSL in the case of the PSL system. Using Schrodinger/Dirac equation ˆ iℏ∂∕∂tjΦti HjΦti, the general equations for the amplitudes may be written down (putting ℏ 1). For the graphene system they are i∂t h0; 1; njΦtiG ⃗σ BA · ϵ⃗ h1; 0; njΦtiG p n 1λe−iωt h1; 0; n 1jΦtiG ;
(3)
i∂t h1; 0; n 1jΦtiG ⃗σ AB · ϵ⃗ h0; 1; n 1jΦtiG p n 1λ eiωt h0; 1; njΦtiG :
(4)
In the case of the PSL system they are p iωt nλ e h1; 1; n − 1jΦtiPSL ;
(5)
i∂t h1; 1; n − 1jΦtiPSL 2ϵh1; 1; n − 1jΦtiPSL p nλe−iωt h0; 0; njΦtiPSL :
(6)
i∂t h0; 0; njΦtiPSL
Unlike the graphene equations (3) and (4) that require domain-specific approximate methods for their solution (to be discussed subsequently), the equations for the pseudospinless model, viz. Eqs. (5) and (6), are exactly solvable by virtue of being the same as the equations for the JC model. The current density in the case of graphene is defined by j⃗ G ⃗σ AB c†A cB ⃗σ BA c†B cA , whereas in the case of the pseudospinless system it is j⃗ PSL − ⃗pvc c† d† dc. We wish to evaluate the expectation value of current density in both cases. In the case of graphene’s toy model, we assume that there is one electron so that we have two possible states nA ; nB 0; 1; 1; 0: hj⃗ G it ⃗σ BA
X n
⃗σ AB
hΦtj0; 1; nih1; 0; njΦti
X hΦtj1; 0; nih0; 1; njΦti; n
whereas in the case of the pseudospinless system, we assert that nc ; nd 0; 0; 1; 1:
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hj⃗ PSL it − ⃗pvc
X
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hΦtj1; 1; nih0; 0; njΦti
n
− ⃗pvc
X hΦtj0; 0; nih1; 1; njΦti: n
In the case of graphene, the polarization function in our earlier work was defined by the following expression: pG t hΦtjc†A cB jΦti
X
hΦtj1; 0; nih0; 1; njΦti:
n
In the PSL case it was X hΦtj0; 0; nih1; 1; njΦti: (7) pPSL t hΦtjdcjΦti n
We now make the observation that Eqs. (5) and (6) for the PSL system (essentially the JC model) may be solved through the following substitution: h0; 0; njΦtiPSL h0; 0; njΦtiPSL;r eiωt :
(8)
We have observed earlier [20] that this is tied to the global symmetry, h1; 1; njΦtiPSL → h1; 1; njΦtiPSL eiθ , h0; 0; njΦtiPSL → h0; 0; njΦtiPSL , λ → eiθ λ enjoyed by the PSL equations. With this substitution, the solution for the PSL equations may be written as follows. Let q z 4njλj2 −2ϵ ω2 :
in graphene occur at resonance when the external frequency matches the particle-hole frequency—an unsurprising result shared by many systems such as two-level atoms and semiconductors. But we also found that graphene due to the pseudospin degree of freedom also exhibits AROs for frequencies much higher than the frequency associated with particle-hole creation. We find below that both types of oscillations survive in the ultra quantum limit as well. Indeed there are even zeropoint AROs in a vacuum. This means that the notion of ARO is robust and not an artifact of any approximations. A. Conventional Oscillations: RWA Here we attempt to solve the graphene equations using the p conventional RWA. In this regime, Δ ≡ ω − 2ϵ ∼ njλj2 ≪ ϵ. For this we use the following substitution on all the probability amplitudes: hi hi e−iϵt hi−;r eiϵt eiΔt : Then we substitute into the full equations and then write ω 2ϵ Δ and finally take the limit ϵ → ∞ keeping Δ fixed: i∂t h0; 1; njΦtiG; ϵh0; 1; njΦtiG; p n 1λh1; 0; n 1jΦtiG;−;r ⃗σ BA · ϵ⃗ h1; 0; njΦtiG; ; i∂t h1; 0; n 1jΦtiG; ϵh1; 0; n 1jΦtiG; ⃗σ AB · ϵ⃗ h0; 1; n 1jΦtiG; ;
(9)
i∂t h0; 1; njΦtiG;−;r − Δh0; 1; njΦtiG;−;r − ϵh0; 1; njΦtiG;−;r
Then,
⃗σ BA · ϵ⃗ h1; 0; njΦtiG;−;r ;
1
h0; 0; njΦtiPSL;r
ie−2t2iϵiωiz − 2iϵ−1 eitz 2z iω iz eitz −iω iz;
and i∂t h1; 0; n 1jΦtiG;−;r − Δh1; 0; n 1jΦtiG;−;r
1
h1; 1; n − 1jΦtiPSL
e−2t2iϵiωiz −1 eitz λ p n: − z
(10)
Substituting the above equations into Eq. (2) allows us to conclude that there are no AROs in this situation. The only Rabi oscillations occur close to resonance, viz. ω ∼ 2ϵ. However, the graphene system may not be solved by a simple substitution. But there are some limits in which the graphene system may also be studied. One is the semiclassical radiation limit. In this case n n0 ≫ 1, a fixed quantity. Since now we may set n − 1 ≈ n, Eqs. (3) and (4), instead of being an infinite tower (in the variable n) of coupled equations, just become two coupled equations. One may then proceed to study them using either the conventional RWA or the newly introduced ARWA [20] valid far from resonance where we encountered the phenomenon of ARO. The purpose of this article is to show that the same effect is also present in the single-photon limit, making this phenomenon robust and not an artifact of approximations used.
4. RABI OSCILLATIONS IN GRAPHENE In this section, we explore the two main regions where Rabi oscillations occur according to our earlier work [20] and see whether they still survive in the extreme quantum limit of the radiation field. In that work, we showed that Rabi oscillations
− ϵh1; 0; n 1jΦtiG;−;r ⃗σ AB · ϵ⃗ h0; 1; n 1jΦtiG;−;r
p n 1λ h0; 1; njΦtiG; : (11)
In the semiclassical limit, we set n 1 ≈ n n0 . This together with the identification i∂t → ΩR gives a secular equation for ΩR , the amplitude Rabi frequency. ΩR; 1∕2Δ p Δ2 n0 jλj2 (ψ ∼ eiΩR; t ). Since the current is a bilinear in the amplitude (j ∼ ψ † ψ − ), the Rabi frequency associated with the current comes out as shown in textbooks as p ΩRabi Δ2 n0 jλj2 . The Rabi frequency at zero detuning p is ωR n0 jλj2 . Now we consider the other extreme limit, viz. when there is only one photon. In this case, in an amplitude such as h0; 1; njΦtiG the value of n 0, 1. Now there are a large number of amplitudes since there is a distinction between n and n 1. If we are interested, say, in the slow part of the current, j G;s t ∼ h1; 0; 0jΦtiG;−;r h0; 1; 0jΦtiG;−;r h1; 0; 0jΦtiG; h0; 1; 0jΦtiG; h1; 0; 1jΦtiG; h0; 1; 1jΦtiG; h1; 0; 1jΦtiG;−;r h0; 1; 1jΦtiG;−;r ;
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then it is sufficient to focus on the following set of mutually coupled equations for the amplitudes that appear in the expression for j G;s t:
and i∂t h1; 0; njΦtiG;s ⃗σ AB · ϵ⃗ h0; 1; njΦtiG;s
i∂t h0; 1; 0jΦtiG; ϵh0; 1; 0jΦtiG; ⃗σ BA · ϵ⃗ h1; 0; 0jΦtiG; λh1; 0; 1jΦtiG;−;r ;
(12)
jλj4 n1 n − jλj2 ωΩ ω2 ϵ − Ωϵ Ω 0:
⃗σ AB · ϵ⃗ h0; 1; 1jΦtiG;−;r λ h0; 1; 0jΦtiG; ;
(14) and
Ω n
−jλj2
(15) ΩARWA n 2
A solution to this yields the amplitude Rabi frequency as p ΩR 1∕2Δ Δ2 jλj2 . This means that the slow part p 2 2 of the current oscillates as j G;s t ∼ ei Δ jλj t . Thus conventional Rabi oscillations are also present in the singlephoton limit in graphene. Now we wish to ascertain whether AROs predicted in our earlier work [20] are also present in the single-photon limit. B. Anomalous Oscillations: ARWA In this section, we study the other extreme where ω ≫ ϵ, p njλj2 . This regime has been termed the ARWA regime. Unlike the earlier discussion, which was near resonance, we now consider the other extreme, viz ω ≫ ϵ, njλj2 . In this situation we set hi his hi e−iωt hi− eiωt : Inserting this into the full amplitude Eqs. (3) and (4) and making use of the observation ji∂t gs; j ≪ jωgs; j, we obtain p n 1λ h1; 0; n 1jΦtiG;s ; h0; 1; njΦtiG; ω ⃗σ · ϵ⃗ h0; 1; n 1jΦtiG; h1; 0; n 1jΦtiG; AB ω ⃗σ · ϵ⃗ h1; 0; njΦtiG;− ; h0; 1; njΦtiG;− − BA ω p n 1λ h0; 1; njΦtiG;s : h1; 0; n 1jΦtiG;− − ω
(18)
p jλj4 1 2n2 4ϵ2 ω2 : 2ω
(19)
This is the Rabi frequency associated with the probability amplitude: ψ ∼ eiΩ t . But the current is j ∼ ψ † ψ − ; hence the actual Rabi frequency in the ARWA regime associated with the current is
i∂t h1; 0; 0jΦtiG; ϵh1; 0; 0jΦtiG; ⃗σ AB · ϵ⃗ h0; 1; 0jΦtiG; :
(17)
Therefore, the amplitude Rabi frequency is
i∂t h0; 1; 1jΦtiG;−;r − Δh0; 1; 1jΦtiG;−;r − ϵh0; 1; 1jΦtiG;−;r ⃗σ BA · ϵ⃗ h1; 0; 1jΦtiG;−;r ;
njλj2 h1; 0; njΦtiG;s : ω
To find the amplitude Rabi frequency we set i∂t ≡ Ω and obtain a secular equation for Ω:
i∂t h1; 0; 1jΦtiG;−;r − Δh1; 0; 1jΦtiG;−;r − ϵh1; 0; 1jΦtiG;−;r (13)
487
r 2 jλj4 n 12 ϵ2 ω2 ω
:
(20)
In the semiclassical limit, when n n0 ≫ 1, we obtain the forq mula of our earlier work, viz. ΩARWA n0 2 ω4R ϵ2 ω2 ∕ω, p where ωR n0 jλj. We may see that the same effect also survives in the single (or even zero) photon limit. The smallest anomalous Rabi frequency is one for which n 0. We see that zero-point fluctuations of the photon field lead to AROs even in a vacuum, which is detectable only when jλj2 ∕2 ≫ ϵω. C. Interpolation As in the earlier work [20], we wish to find a single formula for the Rabi frequency that interpolates between the two regimes discussed above. The best way to solve the graphene equations is by use of Fourier transforms. This method yielded [20] one formula for the Rabi frequency valid both close to resonance and far from resonance, which matches with the results obtained by solving these equations separately in those limits. To this end, we make the following identifications: Z
dω0 −iω0 t f 01 n; ω0 ; e −∞ 2π Z∞ dω0 −iω0 t f 10 n; ω0 : e h1; 0; njΦtiG −∞ 2π h0; 1; njΦtiG
∞
(21)
Substituting these into the graphene equations above yields the following set of coupled double recursion relations: ω0 f 01 n; ω0 ⃗σ BA · ϵ⃗ f 10 n; ω0
Inserting these into the equations for the slow part of the amplitude, we get
p n 1λf 10 n 1; ω0 − ω (22)
and i∂t h0; 1; njΦtiG;s ⃗σ BA · ϵ⃗ h1; 0; njΦtiG;s n 1jλj2 − h0; 1; njΦtiG;s ω
(16)
ω0 f 10 n 1; ω0 ⃗σ AB · ϵ⃗ f 01 n 1; ω0 p n 1λ f 01 n; ω0 ω:
(23)
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We may also combine these two into one equation,
C 2 ω0 ; n ≡ f 2 ω0 ; ω; ϵ ω0 −
C 1 ω0 ; nf 01 n; ω0 a1 ω0 ; nf 01 n − 1; ω0 ω 0
0
b1 ω ; nf 01 n 1; ω − ω
−
(24)
by defining ϵ2 n 1jλj2 C 1 ω0 ; n ω0 − 0 − ; a1 ω0 ; n 0 ω ω −ω p p ⃗σ · ϵ⃗ nλ ⃗σ BA · ϵ⃗ ; b1 ω0 ; n n 1λ AB : ω0 − ω ω0 Since Eq. (24) is a homogeneous system of equations, it defines an eigenvalue problem. The eigenvalues are precisely the variable ω0 . An iterative solution of this eigenvalue problem yields a continued fraction description of the eigenvalue equation. The continued fraction method of solution of the Rabi problem has been introduced in the context of the study of Bloch–Siegert shift by Swain [22]: C 2n ω0 ; nf 01 n; ω0 a2n ω0 ; nf 01 n − 2n ; ω0 2n ω
−
C 2n ω0 ; n C 2n−1 ω0 ; n a n−1 ω0 ; nb2n−1 ω0 2n−1 ω; n − 2n−1 − 2 C 2n−1 ω0 2n−1 ω; n − 2n−1
a2n−1 ω0 ; na2n−1 ω0 2n−1 ω; n − 2n−1 ; C 2n−1 ω0 2n−1 ω; n − 2n−1
b2n ω0 ; n
b2n−1 ω0 ; nb2n−1 ω0 − 2n−1 ω; n 2n−1 : C 2n−1 ω0 − 2n−1 ω; n 2n−1 (25)
The Rabi frequency ω0 is determined by the solution of C 2n ω0 ; n 0; where n is the level of iteration. We saw in our earlier work [20] that choosing n 0 amounts to ignoring all harmonics (even the first harmonic). This leads to an erroneous result for the Rabi frequency when ω ∼ 2ϵ (the conventional RWA), but is quite adequate for obtaining the ARWA result when ϵ → 0. Here too we find that when ϵ 0, C 1 ω0 ; n 0 gives n 1jλj2 C 1 ω0 ; n ≈ ω0 − 0. −ω Thus the Rabi frequency associated with the probability amplitude comes out as n 1jλj2 ∕ω, which is ω2R ∕ω for n large compared to unity. The Rabi frequency associated with polarization and current is twice this value—a result that matches with the one obtained in the earlier work [20]. We need to do better of course. We need one equation that correctly describes both regimes—the RWA and the ARWA regimes. Again, the earlier work shows the way. Just iterating once more gets the job done.
1 nϵ2 jλj2 2
ϵ ω0 − ω2 ω0 − jλjω02n −2ω − ω0 −ω − ω
02
ω
2
nϵ2 jλj2
0. (26)
2 2 − jλjω0n ω0 ω − ω0ϵω
jλj2 n 1 1 Limϵ→∞ f 2 ϵ x; 2ϵ Δ; ϵ 2x − O 0: 2x − Δ ϵ (27) The conventional Rabi frequency associated with the probability amplitude is ψ ∼ eiΩRWA; t , q 1 Δ jλj2 1 n Δ2 : 2
(28)
Thus the conventional Rabi frequency associated with the current j ∼ ψ † ψ − is
b n−1 ω0 ; na2n−1 ω0 − 2n−1 ω; n 2n−1 − 2 ; C 2n−1 ω0 − 2n−1 ω; n 2n−1 a2n ω0 ; n
As we did in our earlier work we consider the two regions p separately: (i) RWA, Δ ≡ ω − 2ϵ ≪ ω and ϵ ≫ ωR ≡ jλj2 n, and (ii) ARWA, ω ≫ ϵ, ωR but ωϵ b2 ∼ ω2R . In order to investigate case (i) we have to examine the limit,
Ω n ϵ
b2n ω0 ; nf 01 n 2n ; ω0 − 2n ω;
ϵ2 jλj2 1 n − ω0 − ω ω0
ΩRWA n
q jλj2 1 n Δ2 :
(29)
This result matches the ones obtained in our earlier work [20] when n ≫ 1. In order to investigate case (ii) we have to examine the limit Limω→∞ f 2
x r 1 r2 ; ω; jλj2 n 1 2 x ω ω ω jλj n − x 1 O 2 0. ω
(30)
Therefore, the anomalous Rabi frequency associated with the probability amplitude is Ω n
q 1 −jλj2 jλj4 1 2n2 4r 2 : 2ω
(31)
The anomalous Rabi frequency matches with the one obtained earlier [Eq. (20)], by a series solution of the differential equation. The ARWA frequency for the current is
ΩARWA n 2
r 2 jλj4 n 12 ϵ2 ω2 ω
:
(32)
These results are identical to what we obtained earlier. It is now important to ascertain that these values are retained when we do a further iteration. We have remarked that iterating the recursion Eq. (25) once (i.e., set n 1) is sufficient to reproduce both the RWA and ARWA frequencies correctly. Now we must make sure that these expectations continue to hold even at the next level of iteration, i.e., n 2. This is indeed the case since
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Limϵ→∞ f 4 ϵ x; 2ϵ Δ; ϵ − f 2 ϵ x; 2ϵ Δ; ϵ jλj4 n 1n 2 1 : O 2 96x − Δ ϵ ϵ2
(33)
Therefore, differences show up only at strong enough fields. In the ARWA case, the agreement is even more striking: x r x r jλj4 n − 1nr 4 ; ω; − f2 ; ω; − : Limω→∞ f 4 ω ω ω ω 2jλj2 n − x2 ω7 (34)
¯n w 1 1¯ p h1; 0; njΦtiG;s −e−2ww n! ΩARWA n n 1jλj2 −iΩ nt × −Ω n e ω n 1jλj2 −iΩ− nt ; − −Ω− n e ω (36) wn1 ϵ2 1¯ p h0; 1; n 1jΦtiG;s −e−2ww n 1! ϵAB ΩARWA n 1 × eiΩ n1t − eiΩ− n1t ;
5. COLLAPSE AND REVIVAL In this section, we study the phenomenon of collapse and revival oscillations. These are well explained in various textbooks on quantum optics [5,8]. The phenomenon of collapse is the extinguishing of Rabi oscillations after a large number of cycles. This could be a purely classical effect brought about by dephasing, or more interestingly it could be due to the quantum nature of light that causes destructive interference that progressively builds up and finally destroys the oscillations. This is in fact the case in the exactly solvable JC model, which describes a two-level system coupled to a single-mode photon. While collapse could be due to any one of these causes, the reverse phenomenon, viz. revival, is a purely quantum effect. Dephasing is irreversible and therefore cannot be the reason for revival. Revivals are caused when the destructive interference due to the quantum nature of light progressively diminishes revealing the original Rabi oscillations. We wish to study these phenomena both in the exactly solvable pseudospinless model (JC model) and also in graphene. Close to resonance ω ∼ 2ϵ, both graphene and the JC model show identical behavior. Hence the collapse and revival oscillations in this case are identical to what may be found in textbooks. We therefore focus on the other extreme case, namely far from resonance, where only graphene is expected to show Rabi-like oscillations. Collapse and revival phenomena occur when the radiation is in a coherent state. In the textbooks, the photon number states are used as a basis, where the calculations involve a saddle point approximation around a classical average number of photons n hni δn, which then leads to a closed formula for the dynamical variables, which then is seen to exhibit collapse and revival oscillations. To see this, we first explicitly solve Eqs. (16) and (17). Since the terms h…iG; are smaller than h…iG;s by factors of λ∕ω or ϵ∕ω, we may apply the initial conditions on the slow parts without introducing much error. We assert jΦtiG;s j1; 0ijwi, where jwi is the boson coherent state † ¯ jwi e−1∕2ww ewb j0i. As we have seen, the amplitude Rabi frequencies come out to be
Ω n
r 2 −jλj2 2 jλj4 n 12 ϵ2 ω2 2ω
:
(35)
Using the initial conditions just mentioned we may write
489
(37)
p n 1λ −1ww wn1 1 e 2 ¯ p ω n 1! ΩARWA n 1 n 2jλj2 iΩ n1t × −Ω n 1 e ω n 2jλj2 iΩ− n1t ; − −Ω− n 1 e ω (38)
h0; 1; njΦtiG; −
and h1; 0; n 1jΦtiG;−
p ¯n n 1λ −1ww w ϵ2 e 2 ¯ p ω n! ϵBA ΩARWA n × e−iΩ nt − e−iΩ− nt :
(39)
The polarization function is given by pG t hΦtjc†A cB jΦti
∞ X hΦtj1; 0; nih0; 1; njΦti: n0
The amplitude of the part of that oscillates as e−iωt is pG; t
∞ X
h1; 0; njΦtiG;s h0; 1; njΦtiG;
n0
∞ X
h1; 0; n 1jΦtiG;− h0; 1; n 1jΦtiG;s :
n0
After substitution and some simplification we obtain (for n¯ ≫ 1) pG; t ≈ ×
∞ X λ n0
2ϵ2
ω
we−n¯
n¯ n n!
1 0 ¯ 2 ΩARWA n
2
− 2jλj ω
1 ¯ 2 ΩARWA n
2
¯ n3∕2jλj ω
¯ Ω2ARWA n ¯ 0ARWA nt ¯ in−nΩ
¯ e × eiΩARWA nt
:
(40)
This expression shows that there is a threshold behavior in the current density [20] in the frequency domain with the threshold frequency (with ϵ 0) given by ΩARWA;0 2n¯ 1∕2jλj2 ∕ω. Thus we shall be content with examining the expressions in this situation. The induced current P density is given by jt j te−iωt , where j t k pG; t,
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where ϵ vF jkj. Therefore, the envelope of the current density is j t
∞ ¯ 1 2 1 jλj2 X λ −n¯ n¯ n i2jλj2 n 2 ¯ 2jλj ω t: we e ω t ein−n 2 2π 4ivF ωt n0 ω n!
(41)
We see that the current density in the time domain has an oscillatory behavior governed by the threshold anomalous Rabi frequency ΩARWA;0 and an envelope that has a power law decay (∼t−1 ) with exponent −1. This exponent is characteristic of the linear dispersion of the single layer graphene quasiparticles; e.g., this exponent is different in bilayer graphene. The method for evaluating this sum over the integer n is described by Eberly et al. [11] and also in various textbooks [5,8]. It involves noting that the Poisson distribution is sharply peaked at n n ≡ ww for ww ≫ 1 so that we may write n ¯ 2 and higher. n n − n and ignore terms of order n − n However, it is important to retain the discrete summation over n (rather than replace by integration, for example) since this leads to the phenomenon of revivals. From this expression, it is possible to extract the collapse and revival times. Revival occurs when successive terms interfere constructively. Thus 0 ¯ rev if trev is the revival time, eiΩARWA nt 1. This means the oscillations revive roughly (because we used the saddle point approximation) after every trev such that trev
2πm ; ¯ Ω0ARWA n
where m 1; 2; …. To study collapse, we perform the sum
which means it is sufficient to study any one of these times as a function of other parameters. Consider the two revival times: the revival time of conventional Rabi oscillation ¯ trev;RWA 2π∕Ω0RWA n and the revival time of ARO ¯ trev;ARWA 2π∕Ω0ARWA n. But we know from the earlier p ¯ n¯ jλj and ΩARWA n ¯ discussion that ΩRWA n p ¯ ¯ 2 ∕ω so that Ω0RWA n ¯ 1∕2 n¯ jλj 2Ω2RWA n∕ω ≈ 2njλj ¯ ≈ 2jλj2 ∕ω. Therefore, and Ω0ARWA n ωπ ; jλj2 p 4π n¯ ; jλj
trev;ARWA trev;RWA
r 2 Log10 ωπ ; n¯ jλj2 1 p 4π 2 Log10 : 2π jλj
tcol;ARWA tcol;RWA
1 2π
6. RESULTS AND DISCUSSION In this section, we plot the graphs obtained from above calculations. In Fig. 1, the real part of the polarization pG; t is plotted versus tjλj2 ∕ω and n 15. This figure depicts AROs followed by collapse and revival with this sequence repeating indefinitely (in the saddle point scheme). In Fig. 2 we see the collapse time associated with both conventional Rabi oscillations and AROs in arbitrary units with a choice ω 100
pG 150
∞ X
n¯ n iΩARWA nt ¯ 0ARWA nt ¯ ¯ e−n¯ ein−nΩ I e n! n0
100 50
to obtain
t 0
Collapse occurs on a time scale small compared to trev so that we may expand 1 ¯ ≈ 1 itΩ0ARWA n ¯ itΩ0ARWA n ¯ 2… ExpitΩ0ARWA n 2 1 2 02 ¯ ¯ I ≈ eiΩARWA nt e−n¯ 2t ΩARWA n : We define collapse time tcol to be 1/10 of the original. This means 1 2
4
0
¯ ARWA nt ¯ ¯ ¯ ¯ n ARWA e−inΩ e−n¯ enExpitΩ : I eiΩARWA nt
02
¯ e−n¯ 2tcol ΩARWA n
1 10
6
8
10
2
12
50 100
Fig. 1. Collapse and revival phenomenon. The real part of the polarization is plotted versus time (tjλj2 ∕ω). To plot, we have taken n 10.
tcollapse
30 25
so that tcol
s 2 Log10 : ¯nΩ02 ¯ ARWA n
We note that
20 Anomalous 15 10
trev 2π tcol
s n¯ ; 2 Log10
Conventional 40
n 60
80
100
Fig. 2. Collapse times versus the average number of photons n.
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trevival
491
The polarization and populations are defined as
300
Anomalous
pt ψ †A tψ B t;
250
p t ψ †B tψ A t;
nt ψ †A tψ A t − ψ †B tψ B t:
200
Thus the equation of motion, 150
e⃗ ⃗p − At pt − c:c:; c e i∂t pt v ⃗σ BA · ⃗p − A⃗ t nt; c e⃗ i∂t p t −v ⃗σ AB · ⃗p − At nt: c
i∂t nt 2v ⃗σ AB ·
Conventional
100 50
40
60
80
100
n
Fig. 3. Revival times versus the average number of photons n.
Choices: and jλj 1 versus n. In Fig. 3 we see the corresponding plot for the revival times. In Appendix A, the differences in the formulas for the anomalous and conventional Rabi frequencies when the photon field is treated as a complex exponential (as we do here) versus when the fields are treated as real (as done by Mishchenko [17]) are brought out.
7. CONCLUSIONS In this paper, we have conclusively established that the phenomenon of AROs that occur far from resonance, shown to be unique to graphene-like systems in an earlier work and attributable to pseudospin, survives even when the radiation is treated quantum mechanically. New phenomena such as zero-point or vacuum AROs in graphene have now been predicted. In the presence of a coherent photon field, the phenomenon of collapse and revival is seen in both conventional Rabi oscillations and AROs. The anomalous collapse and revival times of graphene are extracted and contrasted with the corresponding forms seen in the JC model.
APPENDIX A Here we show that treating the vector potential as a real quantity leads to a formula for the Rabi frequency found by Mishchenko [17], whereas treating as complex with one branch only viz. e−iωt leads to our formula [20]. We favor treating the vector potential as complex since it simplifies the introduction of higher harmonics while paying the small price of getting a trignometric factor wrong in the final formula for the Rabi frequency. A superposition of linearly polarized waves leads to a different picture of Rabi oscillations than the one found in the case of a single wave. This is a small price to pay since the discrepancy between the two results is significant only when the electric field and the momentum are in the same direction. While summing over momenta to find the current density, all directions are included and thus the discrepancy is mitigated. The graphene Hamiltonian is H
ψ †A v ⃗σ AB
·
e⃗ e⃗ † ⃗p − At ψ B ψ B v ⃗σ BA · ⃗p − A t ψ A : c c
ˆ sinωt. (i) Mishchenko: At A t −xcE∕ω ˆ −iωt , (ii) Enamullah et al.: At Ax 0xˆ Ay 0ye iωt ˆ A t Ax 0xˆ Ay 0ye . First Mishchenko’s Theorem: If At is real and linearly polarized along p, there are no Rabi oscillations. Proof: In this case we may write e∕cAt pat so that ⃗σ AB · p px − ipy p− . This means i∂t pt vp 1−atnt, i∂t nt2vp− 1−atpt−2vp 1−atp t, and i∂t p t −vp− 1 − atnt. Set p− pt − p p t ≡ zt. This gives the following simplified coupled equations: i∂t zt 2vjpj2 1 − atnt and i∂t nt 2v1 − atzt. There is a bilinear conserved quantity associated with these equations that may be exploited to generate a solution. z2 t − jpj2 n2 t −C 2 , where C is a constant. This means we may write zt iC cosθt; jpjnt C sinθt. R Therefore, θt θ0 2vjpjt − 0t dt0 at0 . Setting at a0 0∕ω sinωt we conclude that at resonance, ω 2vjpj, the quantities zt and nt have no terms that have a frequency other than the harmonics of ω, i.e., no Rabi frequency. Corollary: If e∕cAt pa0e−iωt is complex but linearly polarized along p, there are Rabi oscillations with the frequency the same as what is found in textbooks on two-level systems. Here we have three coupled equations: i∂t pt vp 1− a 0eiωt nt, i∂t nt 2vp− 1−a0e−iωt pt−2vp 1− iωt a 0e p t, and i∂t p t−vp− 1−a0e−iωt nt. In this situation the three coupled equations do not reduce to two. To solve the above equations we employ the conventional RWA, where we set ω 2ϵ Δ and ignore terms that are varying with frequency 2ω, 4ω, etc. This yields the following conventional Rabi frequency: Ω2 Δ2 ja0j2 pv2 . Since e∕cAt pa0e−iωt , it follows that ve∕c ⃗σ BA · A0 vp a0. According to our earlier work [20], ωR jve∕c ⃗σ BA · A0j vjpjja0j. This means the Rabi frequency is what is found in the context of two-level atoms, i.e., q Ω Δ2 ω2R . Thus we see there are qualitative differences in treating the field as having a real sinusoidal time dependence versus a complex exponential time dependence. We prefer the complex exponential since allows us to maintain contact with the two-level system and is also easier to handle than the trignometric functions when higher harmonics are
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involved. We are more interested in establishing the general results that graphene exhibits peudospin-linked AROs rather than obtaining an accurate and realistic formula for the Rabi frequency in terms of the components of the fields. With this limited goal in mind, a mathematically elegant method that can bring out these ideas is more desirable than one that is tedious but more realistic. Second Mishchenko’s Theorem: If At is real and linearly polarized making an angle χ p with the vector p, conventional Rabi oscillations exist with Rabi frequency ωR evE∕ω sinχ p . Set e∕cAt −eE∕ωxˆ sinωt. Without loss of generality we have chosen the x axis to coincide with the direction of the linearly polarized field. Proof: Proved in Mishchenko’s paper [17]. Existence Theorem of ARO: Even when the vector potential is chosen real and sinusoidal, AROs are seen even though the dependence of the anomalous Rabi frequency on the components of the electric field depends on whether the fields are treated as (i) complex and exponential or (ii) real and sinusoidal. In our earlier work [20] we proved (i). To prove (ii) we set At ReA0 e−iωt ; ˆ where θ, δ, ω, and A0 where A0 A0 cosθxˆ sinθeiδ y, are real. The real vector potential is given by At A0 cosθxˆ cosωt A0 sinθyˆ cosωt − δ, where θ is the angle made by the electric field and the x axis, δ is the phase difference between the x and y components, and A0 is the amplitude, all real quantities. We follow the ARWA procedure outlined in the earlier work [20] to obtain nt ns t nf te−iωt nf teiωt and pt ps t p te−iωt p− teiωt with ωnf t −2ve∕cA0 cosθ1∕2ps t − ps t i2ve∕cA0 sinθ1∕2ps t ps teiδ , ωp t −ve∕cA0 cosθ1∕2 ns t − ve∕ciA0 sinθ1∕2eiδ ns t, and −ωp− t −ve∕c A0 cosθ1∕2ns t − ve∕ciA0 sinθ1∕2e−iδ ns t, together with ps t ve∕cA0 2 ∕ω sin2θSin δ1∕4p− vns t 1∕4p− vi∂t ns t and ns t ns 0CosωARWA t. The anomalous Rabi frequency is given by 1 2 v e A0 4 ωARWA 4v2 p2 c 2 sin2 2θSin2 δ : ω
(A1)
We see that when the fields are real, the anomalous Rabi frequency is zero when the electric field is linearly polarized. We saw in the earlier work that the anomalous Rabi frequency is simply related to the conventional one through the relation ωARWA p 0 2ω2RWA Δ 0∕ω when we treat the fields as complex exponential time varying fields. We now evaluate the conventional Rabi frequency and see whether the same relationship between conventional and anomalous Rabi frequency holds when the fields are real and sinusoidal. One q finds the result, ΩRWA ω2RWA Δ2 , where the zero detuning conventional Rabi frequency is ωRWA
A0 ev q 1 − cos2θ cos2χ p − cosδ sin2θ sin2χ p ; c (A2)
where χ p tan−1 py ∕px . We see that this relationship is not strictly obeyed now. However, magnitude wise, it is still true
that ωARWA ∼ ω2RWA ∕ω. The main point we are making is that no matter what, there is such a thing as AROs in graphene, which is absent in conventional (pseudospinless) systems whose frequency is smaller compared to the zero detuned conventional Rabi frequency (ωRWA ) by a factor of ωRWA ∕ω. These two approaches (real versus complex fields) are equivalent when considering linear equations, but for nonlinear systems such as the one we consider here, they are different. Mathematically, this may understood as follows: while Imeiωt sinωt so that the two approaches are the same for linear fields, from the observation that Ime2iωt ≠ sinωt2 we may conclude that it does not work in nonlinear (Rabi-like) situations. While Mishchenko’s [17] approach has the appeal that it uses a physical real electric field, our approach has the appeal that the model with complex vector potentials (the Hamiltonians are real of course) is able to map the graphene system at resonance to the two-level system with the Rabi frequency given by the textbook value, which depends on the electric field alone (it does not vanish in some situations unlike Mishchenko’s). Furthermore, our approach of using time varying complex exponentials renders calculations involving higher harmonics (which is the norm in few-layer graphene, e.g., or while considering higher-order effects such as Bloch Siegert shift) quite easy and elegant, unlike what would be the case had we insisted on using real fields.
ACKNOWLEDGMENT We are deeply grateful to Prof. J. H. Eberly for a critical reading of the manuscript and for his valuable suggestions. We are also grateful to Dr. T. N. Dey for his fruitful input.
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