Indian J Phys DOI 10.1007/s12648-013-0419-9
ORIGINAL PAPER
Spontaneous emission spectrum in gravitational Jaynes–Cummings model with respect to counter-rotating terms M Mohammadi1* and M Keshavarz2 1
Department of Physics, Shahreza Branch, Islamic Azad University, Shahreza, Isfahan, Iran
2
Department of Chemistry, Dolatabad Branch, Islamic Azad University, Dolatabad, Isfahan, Iran Received: 26 July 2013 / Accepted: 30 October 2013
Abstract: Spontaneous emission spectrum in gravitational Jaynes–Cummings model (JCM) beyond rotating wave approximation is studied. First, in gravitational JCM with respect to counter-rotating terms, effective Hamiltonian of Schro¨dinger equation in interaction picture is obtained. Then, by transforming the Schro¨dinger equation to two differential equations, the amplitudes of probability are obtained. In this case, influence of certain parameters on spontaneous emission spectrum is investigated. Keywords:
Spontaneous emission spectrum; Jaynes–Cummings model; Gravity and counter-rotating terms
PACS Nos.: 42.50.Hz; 42.50.Dv
1. Introduction Spontaneous emission in a microcavity, where cavity length is of the order of wavelength is studied classically [1–3]. Although a good agreement between theory and experiment is obtained, the electric field amplitude for a single emission event is left as an unknown function. Atoms, effect of external electric field on the atom and lifetime of excited level for atom are studied [4–6]. Also, spontaneous emission spectrum from two-level system embedded in a continuum energy state is investigated within rotating wave approximation (RWA) when the counter-rotating terms (CRTs) are neglected [7–11]. However, RWA is questioned due to the recent progress in generation of intense femtosecond and attosecond optical pulses [12–19]. On the other hand, it is obvious that for atoms moving with a velocity of a few millimeters or centimeters per second for a time period of several milliseconds or more, influence of Earth’s acceleration becomes important and cannot be neglected [20]. Recently, within a quantum treatment of internal and external dynamics of the atom, we have investigated influence of a classical homogeneous gravitational field on atom-field properties in Jaynes–
Cummings model (JCM) [21–24] with RWA [25, 26]. In these works, it has been found that nonclassical properties are suppressed with increase of gravitational field influence. Moreover, we have used the weak coupling regime and off-resonance for detuning as other conditions. In this paper, influences of detuning, gravity and atomfield coupling parameter on the spontaneous emission spectrum in JCM beyond RWA are investigated. Without considering gravity and detuning, the sharp peaks in spontaneous emission spectrum appear when atom-field coupling is ultra-strong [22]. In this case, the model is used without RWA. Obtaining of optimum conditions for JCM in presence of gravity and detuning for using the model without RWA is motivation to calculations presented in this paper.
2. Evolution of system with respect to CRTs Total Hamiltonian for atom-field system with gravity and in absence of RWA with atomic motion along the position vector ~ x^ is given by [21] ^¼H ^free þ H ^RWA þ H ^CRT ; H
ð1Þ
where
*Corresponding author, E-mail:
[email protected]
2 1 1 ^free ¼ p^ Mg ~ ~þ x^ hx a^y a^ þ H þ hxeg r^z ; 2 2 2M
ð2Þ
2013 IACS
M Mohammadi and M Keshavarz
h i ^RWA ¼ ~~ ~~ H hk exp iq x^ a^y r^ þ exp iq x^ r^þ a^ ; h i ^CRT ¼ ~ ~ ~~ H hk exp iq x^ a^y r^þ þ exp iq x^ r^ a^ ;
ð3Þ ð4Þ
Here a^ and a^y denote, the annihilation and creation operators respectively, of a single-mode traveling wave with frequency x; ~ q is wave vector of the running wave; r^ denote raising and lowering operators of two-level atom with electronic levels jei; jgi and Bohr transition frequency xeg. The atom-field coupling is given by parameter k; ~; p^ ~ x^ denote, respectively, momentum and position operators of the atomic center of mass motion and g is Earth’s gravitational acceleration. Schro¨dinger equation is given by i h
ojuðtÞi ^ ¼ HjuðtÞi: ot
ih
oju1e ðtÞi ^1e ju1e ðtÞi: ¼H ot
It is not easy to solve Eq. (12) because of presence of CRTs in Eq. (10). Thus, the unitary squeezing operator is defined as ^ ¼ exp g a^2 g a^y2 : SðgÞ ð13Þ 2 2 The Hamiltonian corresponding to state vector j/2e ðtÞi ¼ S^y ðgÞju1e ðtÞi is obtained ^2e ¼ S^y H ^1e S^ ihS^y S^_ H h ^ ay r^ ¼ hk AðtÞ^ a2 þ A ðtÞ^ ay2 þ fðtÞ^
ð5Þ
It is convenient to consider evolution of state vector juðtÞi of the total system in an interaction picture induced by unitary operator R ^ ^ ^ ¼ exp itH0 exp i Hp ðtÞd t ; TðtÞ ð6Þ h h where 2 2 q h y ^ ^0 ¼ p^ þ Mg ~ ~ a^ a^ þ jeij x hr^z d; x h H 8M 2M ð7Þ "
~ p^ 3 d^ ¼ 2x xeg ~: q ~þ x^ M 2
!# ;
ð8Þ
ð12Þ
i
ð14Þ
^ r^ a^ þ Cg ðtÞ ; þ f^ ðtÞr^þ a^ þ n^ ðtÞ^ ay r^þ þ nðtÞ where pffiffiffiffiffiffiffiffiffiffi _ g_ gÞ g sinh 4g g iðgg pffiffiffiffiffiffiffiffiffi ffi AðtÞ ¼ x 4g g 4g g _ g_ gÞ ig_ iðgg ; 4g g 2 pffiffiffiffiffiffiffi sinh g g g c ^ ¼ c^ ðtÞ cosh pffiffiffiffiffiffiffi ^ ffiffiffiffiffiffiffi p fðtÞ g ðtÞ ; þ g g pffiffiffiffiffiffiffi sinh g g g c ^ ¼ c^ ðtÞ cosh pffiffiffiffiffiffiffi ^ ffiffiffiffiffiffiffi p g ðtÞg nðtÞ ; þ gg pffiffiffiffiffiffiffiffiffiffi _ g_ gÞ x iðgg g 1 ; cosh 4g Cg ðtÞ ¼ 2 8g g
ð15Þ
ð16Þ ð17Þ ð18Þ
with
with 1 ^p ðtÞ ¼ ~ H p^ þ ~ q ~ gt þ Mg2 t2 : 2
ð9Þ
h i ^ ð~; c^ ðtÞ ¼ exp itM p^ ~ g; tÞ :
ð19Þ
In this scenario, evolution of the transformed state vector ^ is governed by Hamiltonian ju1e ðtÞi ¼ TðtÞjuðtÞi
By considering two constraints A = 0 and n = 0, timedependent function g(t) is obtained
^1e ¼ T^y H ^ T^ ihT^y T^_ H i h ^ ð~;g p^ ~;tÞ a^y r^ ¼ hk exp itM h i i h ^^ ^ ð~;g ^þ ð~;g þexp itM p ~;tÞ r a^ þ hk exp itM p^ ~;tÞ a^y r^þ þ h i ^ ^ ^ þ ð~;g þexp itM p ~;tÞ r a^ ; ð10Þ
1 gðtÞ ¼ expðivðtÞÞ 2 2xeg expðivðtÞ=2Þ þ ixf1 ðtÞ expðivðtÞ=2Þ ln ; 2xeg expðivðtÞ=2Þ ixf1 ðtÞ expðivðtÞ=2Þ
ð20Þ
with vðtÞ ¼ i ln½ xf ðtÞ;
ð21Þ
where where ! 2 ^ ~ ~ q p h q ^ ~; þ~ q ~ gt þ 3 M p^ ~ g; t ¼ x xeg ; M 2M
f ðtÞ ¼ xf1 ðtÞ ð11Þ
has been introduced as Doppler shift detuning at time t. The Schro¨dinger equation governing JCM beyond the RWA is
i ln f2 ðtÞ; 2x
ð22Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 þ if1 ðtÞ ;
f2 ðtÞ ¼
ixf1 ðtÞh 2 þ if1 ðtÞ þ 2 2xeg
f1 ðtÞ ¼
2ixeg expð2ixeg tÞ: x
ð23Þ ð24Þ
Spontaneous emission spectrum in gravitational Jaynes–Cummings model
By using Eq. (20), Hamiltonian given in Eq. (14) is rewritten as h i ^ ay r^ þ f^ ðtÞr^þ a^ þ Cg ðtÞ : ^2e ¼ H hk fðtÞ^ ð25Þ Schro¨dinger equation with regard to Eq. (25) is oju ðtÞi ^2e ju2e ðtÞi: i h 2e ¼H ot Finally, by use of Te ¼ expði Hamiltonian is given by
and state vector for center-of-mass degree of freedom is R ~Þjp ~i: The initial state in Eq. (32) reads jwc:m ð0Þi ¼ d3 p/ðp as Z X ~Þje; ni jp ~i jwðt ¼ 0Þi ¼ d 3 p ðwn ce /ðp n¼0
~Þjg; n þ 1i jp ~i þwnþ1 cg /ðp Z ~Þcg jg; 0i jp ~i: þ d3 pw0 /ðp
ð26Þ Rt 0
Cg ðt0 Þdt0 Þ; the effective
^eff ¼ Tey H ^2e Te ihT ye T_e ¼ H ^2e Cg ðtÞ H ^ ay r^ þ f^ ðtÞr^þ a^; ¼ hk½fðtÞ^
When we compare Eq. (29) in t = 0 with Eq. (33), we find following initial conditions ð27Þ
^ and Cg(t) in Eqs. (16) and (18), where we have defined fðtÞ respectively. Schro¨dinger equation is considered as ojwðtÞi ^eff jwðtÞi; ¼H i h ot
ð28Þ
n¼0
~Þ; w2 ðt ¼ 0Þ ¼ wnþ1 cg /ðp ~Þ; w1 ðt ¼ 0Þ ¼ wn ce /ðp ~Þ: wg;0 ðt ¼ 0Þ ¼ w0 cg /ðp
ð34Þ
We can solve two coupled first order differential equations [Eqs. (30) and (31)] in a straightforward way. We have o2 w 1 ow a ðtÞ 1 þ bn ðtÞw1 ¼ 0; 2 ot ot
for the state vector by following form Z X ~; ~ ~i jwðtÞi ¼ d3 p we;n ðp g; tÞje; ni jp ~; ~ ~i g; tÞjg; n þ 1i jp þwg;nþ1 ðp Z ~; tÞjg; 0i jp ~i: þ d3 pwg;0 ðp
ð33Þ
ð35Þ
and ð29Þ
o2 w 2 ow aðtÞ 2 þ bn ðtÞw2 ¼ 0; ot2 ot
ð36Þ
where
^ ~ eff jg; 0i ¼ 0 which means, vacuum cannot excite We find H an atom initially in ground state and therefore, the state jg; 0i decouples from rest of the states. Equations of motion ~; ~ for the time-dependent probability amplitudes we;n ðp g; tÞ ~; ~ w1 ; wg;nþ1 ðp g; tÞ w2 by substituting Eqs. (27) and (29) into Eq. (28) is found to be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w_ 1 ¼ ik ðn þ 1Þf ðtÞw2 ; ð30Þ
aðtÞ ¼
_ fðtÞ ;n ðtÞ ¼ k2 ðn þ 1Þf ðtÞfðtÞ; fðtÞ
ð37Þ
are time-dependent. The exact solutions of Eqs. (35) and (36) read as, respectively, w1 ðtÞ ¼ expðtbn ðtÞ=aðtÞÞ½Cn ð1ÞD1n ðtÞ þ Cn ð2ÞE1n ðtÞ; ð38Þ and
and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w_ 2 ¼ ik ðn þ 1ÞfðtÞw1 :
ð31Þ
At time t = 0, atom is uncorrelated with single-mode cavity-field and the state vector of system can be written as a direct product jwðt ¼ 0Þi ¼ jwc:m ð0Þi jwatom ð0Þi jwfield ð0Þi Z 3 ~Þjp ~i ce jei þ cg jgi ¼ d p/ðp ! X wn jni ;
ð32Þ
w2 ðtÞ ¼ expðtbn ðtÞ=aðtÞÞ½Cn ð1ÞD2n ðtÞ þ Cn ð2ÞE2n ðtÞ; ð39Þ where "
pffiffiffiffiffiffiffiffiffiffi# pffiffiffi b2n ðtÞ 2bn ðtÞ t a ðtÞ D1n ðtÞ ¼ H 3 ; 3=2 þ pffiffiffi ; ð40Þ a ðtÞ a ðtÞ 2 2 pffiffiffiffiffiffiffiffiffiffi!2 3 pffiffiffi 2 t a ðtÞ 5 b ðtÞ 2 b ðtÞ n ; E1n ðtÞ ¼ 1 F1 4 3n ; 1=2; þ pffiffiffi 3=2 2a ðtÞ a ðtÞ 2
n¼0
where, we have assumed that initially the field is in a single-mode coherent superposition of Fock states. Atom is in a coherent superposition of its excited and ground states
ð41Þ "
pffiffiffiffiffiffiffiffi# pffiffiffi b2n ðtÞ 2bn ðtÞ t aðtÞ D2n ðtÞ ¼ H 3 ; 3=2 þ pffiffiffi ; a ðtÞ a ðtÞ 2
ð42Þ
M Mohammadi and M Keshavarz
2 pffiffiffiffiffiffiffiffi!2 3 pffiffiffi 2 t aðtÞ 5 b ðtÞ 2 b ðtÞ n ; E2n ðtÞ ¼ 1 F1 4 3n ; 1=2; þ pffiffiffi 2a ðtÞ a3=2 ðtÞ 2 ð43Þ with Cn ð1Þ ¼
E2n ð0Þw1 ð0Þ E1n ð0Þw2 ð0Þ ; E2n ð0ÞD1nð0Þ E1n ð0ÞD2n ð0Þ
ð44Þ
Cn ð2Þ ¼
D2n ð0Þw1 ð0Þ D1n ð0Þw2 ð0Þ ; E1n ð0ÞD2nð0Þ E2n ð0ÞD1n ð0Þ
ð45Þ
and H, 1F1 denote Hermite and confluent hypergeometric functions respectively.
coupling in atom-field interaction. We assume at t = 0, two-level atom is in a coherent superposition of excited state and ground state with cg ð0Þ ¼ p1ffiffi2; ce ð0Þ ¼ p1ffiffi2 and cavity-field is prepared in a Glauber coherent state wn ð0Þ ¼ jaj2 expð 2 Þan pffiffiffi : In this figure and all subsequent figures we set n! 2 q ¼ 107 m1 ; M ¼ 1026 Kg; g ¼ 9:8sm2 ; xrec ¼ 2M ¼ :5 7rad 7rad 106rad s ; xeg ¼ 8:822 10 s ; x ¼ 9 10 s ; a ¼ 2 and p 1 ~Þ ¼ pffiffiffiffiffiffiffi /ðp expð r2 Þ with r0 = 1 [25, 26]. Here, it is 2pr0 0 necessary to point out that relevant time scale introduced by gravitational influence is t ¼ sa ¼ p1ffiffiffiffiffi [26]. Therefore ~g q~
(a)
3. Spontaneous emission spectrum with respect to CRTs In this section, by calculation of spontaneous emission spectrum, influences some of certain parameters on its evolution are studied. The spontaneous emission spectrum is expressed by following form [23, 24] Z PðxÞ ¼ dt expðixtÞCðtÞ; ð46Þ with CðtÞ ¼ hwðtÞjr^þ r^ jwðtÞi:
ð47Þ
We obtain Z Z 1 X PðxÞ ¼ dt d3 p jwn j2 expðixtÞjw1 ðtÞj2 :
ð48Þ
(b)
n¼0
By using jwðtÞi and w1(t) given by Eqs. (29) and (38) respectively we obtain Z Z 1 X PðxÞ ¼ dt d 3 p jwn j2 expðixtÞ n¼0
j expðtbn ðtÞ=aðtÞÞ½Cn ð1ÞD1n ðtÞ þ Cn ð2ÞE1n ðtÞj2 : ð49Þ where we have defined a(t), bn(t), D1n(t), E1n(t), Cn(1) and Cn(2) in Eqs. (37), (40), (41), (44) and (45), respectively. According to Eqs. (16) and (19), fðtÞ and c± (t) in a(t) and bn(t) are functions of ~: p Spontaneous emission spectrum in gravitational JCM with respect to CRTs for three values of coupling parameter k with weak gravity ~ q ~ g ¼ 0:01 107 s12 and the atom-field detuning M ¼ x xeg ¼ 0 [25] is shown in Fig. 1. Weak gravity means very small ~ q ~ g; i.e., momentum transfer from laser beam to the atom is only slightly altered by gravitational acceleration, because the latter is very small or nearly perpendicular to laser beam. Atom-field detuning with M ¼ x xeg ¼ 0 means on-resonance and coupling parameter k is degree of
(c)
Fig. 1 Spontaneous emission of system versus Dþ for (a) k = 0.01x in the off-resonance D ¼ 0, (b) k = 1.11x in the off-resonance D ¼ 0, and (c) k = 10.1x in the off-resonance D ¼ 0
Spontaneous emission spectrum in gravitational Jaynes–Cummings model
(a)
(b)
appear because of CRTs. Therefore, this interaction model is valid without RWA. Moreover, with increasing value of coupling parameter k, amplitude of P(x) decreases and peaks of spontaneous emission spectrum present obvious height and position asymmetry. In Fig. 2(a) and 2(b), sharp peaks in spontaneous emission spectrum decrease due to M ¼ 0:18 107rad s (off-resonance). This means that in offresonance (D = 0), for three values of the atom-field coupling the degree of atom-field interaction decreases which is agreeable with [25] and this model is valid with RWA. By comparing Figs. 1 and 2 by considering three conditions: (i) the ultra-strong coupling, (ii) the weak gravity and (iii) the proper detuning (on-resonance), the sharp peaks in spontaneous emission spectrum are seen which is agreeable with [22]. Therefore, we can use calculations without RWA when we have three above conditions. Moreover, by considering strong gravity, offresonance and weak coupling this interaction model is valid with RWA.
4. Conclusions
(c)
Spontaneous emission spectrum in gravitational JCM beyond RWA is studied. First, in gravitational JCM with respect to CRTs the effective Hamiltonian of Schro¨dinger equation in interaction picture is obtained. Then, by transforming the Schro¨dinger equation to two differential equations, amplitudes of probability are obtained. In this case, influence of certain parameters on spontaneous emission spectrum is investigated. With increase of coupling parameter in our system without RWA, the sharp peaks in spontaneous emission spectrum are analyzed by considering three conditions: (i) ultra-strong coupling, (ii) weak gravity and (iii) proper detuning (on-resonance). Acknowledgments Authors wish to thank the Office of Research of Shahreza and Dolatabad Branches, Islamic Azad University for their support.
Fig. 2 Spontaneous emission of system versus Dþ for (a) k = 0.01x in the on-resonance M ¼ 0:18 107rad s , (b) k = 1.11x in the onresonance M ¼ 0:18 107rad s , and (c) k = 10.1x in the on-resonance M ¼ 0:18 107rad s
~j ¼ 107 m1 ; t ¼ sa is about for an optical field with jq -4 10 s. In Fig. 1(a), one can not see the sharp peaks in spontaneous emission spectrum P(x) in weak coupling regime with k = 0.01x [27, 28] which is agreeable with [22]. Because in atom-field system beyond RWA, value of coupling parameter is small even when weak gravity and atom-field detuning M ¼ 0 [29] are considered. By comparing Fig. 1(b) and 1(c) for strong coupling k = 1.11x [30] and ultra-strong coupling k = 11.1x [31] respectively, sharp peaks in spontaneous emission spectrum
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