Critical properties of two-level atom systems interacting with a

PHYSICAL REVIEW A 70, 033808 (2004)

Critical properties of two-level atom systems interacting with a radiation field Giuseppe Liberti* and Rosa Letizia Zaffino Dipartimento di Fisica, Università della Calabria, INFN—Gruppo Collegato di Cosenza, 87036 Rende (CS), Italy (Received 30 April 2004; published 22 September 2004) The thermodynamic properties of N two-level atoms coupled to radiation field are analyzed taking into account the diamagnetic term by a slight modification to the usual approach to quantizing the electromagnetic field. Following this procedure, we are able to solve the discrepancy originating in the use of different states as a basis for expressing the state of radiation field. We develop a perturbative expansion of partition function and a simple analytic expression is found for a high coupling constant. We show that, in this limit, the interaction of independent atomic spins with a single photon mode can be interpreted as an effective spin-spin interaction of long range nature. Even in this approach, if the diamagnetic term is taken into account, a phase transition at nonzero temperature cannot occur due to sum-rule arguments. DOI: 10.1103/PhysRevA.70.033808

PACS number(s): 42.50.Ar, 05.70.Fh, 64.90.⫹b

I. INTRODUCTION

The possibility of superradiant phase transition (SPT) in condensed matter systems has received a renewed interest in the last three years by different authors [1–3] and has been investigated with different methods. The possibility of a phase transition involving N two-level atomic systems interacting with the radiation field, first calculated in an early paper by Hepp and Lieb [4], is connected to the problem as to what form of interaction Hamiltonian should be used to describe the electromagnetic interactions (more specifically in the role of the counterrotating and of the diamagnetic terms) and of the exact treatment of the thermodynamical properties of the system, which is sufficiently complicated and is useful to study some limiting cases. Wang and Hioe [5] proposed a computational method based on the use of Glauber’s coherent states [6] for the radiation field and on the assumption that, in the thermodynamic limit (N → ⬁, V → ⬁ but ␳ = N / V finite) the field operators can be treated as c-number functions. For the simple Dicke model Hamiltonian [7], in which both the counterrotating terms and the diamagnetic term are truncated, above a certain critical density of the atoms and below a critical temperature, the formation of a coherent condensed phase is found. The critical properties do not change qualitatively when the counterrotating terms are taken into account [8] but, when the diamagnetic term is included, the SPT is forbidden to occur [9]. On the contrary, Gilmore and Bowden [10], from a coupled order-parameter treatment of a model Hamiltonian in which the counterrotating term are truncated but the diamagnetic term is retained, and Orszag [11], using the Fock states 兩n典 as a basis for the field, shows that the diamagnetic term does not affect the phase transition. Yamanoi [12] shows that the presence of the counterrotating term in the presence of the diamagnetic term plays a crucial role in the instability property of the system and that the rotating wave approximation (RWA) as well as the neglect of the diamagnetic term should not be made for the study of the thermodynamic phase tran-

*Corresponding author; email address: [email protected] 1050-2947/2004/70(3)/033808(5)/$22.50

sition and instability properties of the ground state of the system. A more realistic model of atomic system is adopted by Bialynicki-Birula and Rzazewski [13]. They proved that the SPT cannot occur in a general system of atoms if only charges (not spin magnetic moments) of the particles interact with a finite number of radiation modes in the dipole approximation. This result was shown to hold without the dipole approximation by Gawedzski and Rzazewski [14]. They showed that the ground-state energy of the atomic system interacting with an arbitrary number of modes of the transverse radiation field is bound from below by the sum of the ground-state energies of atoms noninteracting with radiation thus excluding any SPT. More recently Sivasubramanian, Widom, and Shrivastava [3] argued that, in a gauge invariant formulation of the atomic electric dipole-photon interaction, the sequence in which the canonical transformation and the Wang and Hioe procedure are performed is not irrelevant. Performing the transition to the classical limit and the application of the Wang and Hioe method at the end of the calculations, after all canonical transformation, they show that the SPT remains intact. In all of these arguments the classical limit is taken only for the radiation field but not for the atomic system that is described as a system of quantum spin. Emary and Brandes [1] discuss the phase transition and the chaotic properties for the Dicke Hamiltonian, in the thermodynamic limit, without the RWA and without the diamagnetic term, using the Fock states as a basis for the radiation field and the Dicke states as a basis for the atomic system and, in addition, discuss the status of the two systems (atomic plus radiation) obtained by taking the classical limit for one or both of them. In the next section we show that, taking into account the counter rotating and the diamagnetic terms by a slight modification to the usual approach to quantizing the electromagnetic field, the thermodynamical properties of the system are independent from the choice of different states as a basis for expressing the state of radiation field. In Sec. II we develop an approximation scheme with a perturbative expansion of the partition function in order to perform thermodynamical calculations without to have recourse to the Wang and Hioe computational method. A simple analytic solution is found for a high coupling constant. The existence of this soluble

70 033808-1

©2004 The American Physical Society

PHYSICAL REVIEW A 70, 033808 (2004)

G. LIBERTI AND R. L. ZAFFINO

N

冋

册

model allows us to show that, in this limit, the interaction of independent atomic spins with a single photon mode can be interpreted as an effective spin-spin interaction of long range nature. Despite this simple physical interpretation we obtain that, if the diamagnetic term is taken into account, a phase transition at nonzero temperature cannot occur due to sumrule arguments.

All thermodynamic properties of the system are derived from the partition function

II. CLASSICAL LIMIT

Z共N,T兲 = TrATrF兵exp共− ␤H兲其

Throughout this paper, we will consider the Hamiltonian N

H = ⍀a†a +

兺 i=1

冋

⑀ z ⌳ † ␴ + 共a + a兲共␴+i + ␴−i 兲 2 i 冑N

册

共1兲

that describes a system of N two-level atoms (with energy difference between the two levels ⑀) interacting with a single quantized mode of radiation. ␴zi , ␴+i , and ␴−i are, respectively, the z component, the raising, and the lowering operators of the Pauli matrices used to describe the ith atom. The diamagnetic term is included in the field Hamiltonian by use of a new field-mode of frequency ⍀ = 冑␻共␻ + 4k兲

共2兲

instead of frequency ␻. In this model the atomic parameters are ⌳ = ⑀d

冑

2␲␳ , ⍀

e2 ␲␳ k= , m ␻

a† = 共⍀Q − iP兲/冑2⍀,

⑀⍀ 4⌳2

We also obtain ⌳ ⯝ ␭ = ⑀d

冑

2␲␳ , ␻

kⰆ␻

and, in this limit, the Crisp Hamiltonian (1) becomes

冑

1+

冉 冑

16⌳2z2 ␤⑀ = tanh 2 ⑀ 2

1+

冊

16⌳2z2 , ⑀2

共10兲

where z = 共Re ␣兲2 / N, using the coherent states 兩␣典, and z2 = n / N using the Fock states 兩n典 for the field. Equation (10) holds for

⑀⍀ . 4

共11兲

Therefore, we obtain a phase transition only for

共4兲

d2⑀ ⬎ e2/2m

共12兲

in contradiction with the following inequality derived from the Thomas-Reiche-Kuhn sum rule [9]: d2⑀ ⬍ e2/2m.

共13兲

The inclusion of the diamagnetic term in the field Hamiltonian through the “shifted” frequency (2) tells us that, in the classical limit, the result of the calculations are independent of the choice for the basis for expressing the state of radiation field. III. PERTURBATION EXPANSION OF PARTITION FUNCTION

共5兲

k Ⰶ ␻.

共9兲

and requires summation over atomic and field variables. Except for some special cases, the trace (9) cannot be computed in closed form and, when that happens, only approximate results can be obtained. Assuming that, for the radiation field, the thermodynamic limit (N → ⬁, V → ⬁ but ␳ = N / V finite) is equivalent to the classical limit, with the additional constraints of Ref. [5], a phase transition can be predicted if exist a nonzero solution of equation

共3兲

When the full Hamiltonian is written out with these new definitions, the a2 and a†2 terms are no longer present and their physical consequences are contained in the new mode frequency (2). This frequency change reduces to a usual shift equal to 2k value when k Ⰶ ␻ (where ␻ is the “free-field frequency”), i.e., ⍀ ⯝ ␻ + 2k,

⑀ z ␭ † 共a + a兲共␴+i + ␴−i 兲 . ␴ + 2 i 冑N 共8兲

where Q and P are the conjugate coordinates and momenta subjecting to the usual commutation relation 关Q, P兴 = i.

兺 i=1

⌳2 ⬎

where d is the projection of the transition dipole moment on the polarization vector of the field mode, ␳ is the density of the atoms, and e and m, respectively, are the charge and the mass of electron. The Hamiltonian (1) is the generalization to the N two-level atom system of the Crisp Hamiltonian [15] used to describe the interaction of a single atomic system with a single mode of a quantized electromagnetic field in the dipole approximation with a different quantization procedure of annihilation and creation field operators. The new annihilation and creation field operators are given by a = 共⍀Q + iP兲/冑2⍀,

H = 共␻ + 2k兲a a + †

A convenient starting point to calculate Z共N , T兲 is to split the Hamiltonian (1) as 共14兲

H = H0 + HI , where

冉

冊

1 H 0 = ⍀ a †a + S z , 2

共6兲

HI =

共7兲

and where 033808-2

⌬ z ⌳ † + − S + 冑N 共a + a兲共S + S 兲, 2

⌬ = ⑀ − ⍀,

共15兲

PHYSICAL REVIEW A 70, 033808 (2004)

CRITICAL PROPERTIES OF TWO-LEVEL ATOM…

关H0,HI兴 = 0,

N

S

共z,±兲

=

␴共z,±兲 兺 i i=1

共16兲

are the collective atomic operators. The exponential operator of Eq. (9) can be disentangled into a product of an infinite series of exponential operators as

where Ci is a homogeneous polynomial of degree n in H0 and HI. All the Ci⬘s contain the commutator 关HI , H0兴 and using the method given by Wilcox [16] they can be determined as

共27兲

IV. PARTITION FUNCTION

exp关共− ␤兲iCi兴, 兿 i=2 共17兲

Ci = 0 共i 艌 2兲

that implies that the difference of the two terms of Eq. (18), usually neglected in RWA, is zero. In this sense condition (27) is a weaker condition than the RWA, in which only conserving energy terms are considered.

⬁

exp关− ␤共H0 + HI兲兴 = exp共− ␤H0兲exp共− ␤HI兲

i.e.,

A. Coherent-state representation

Let us calculate the partition function. Using 兩␣典 for the photon field, we have

再

冉

⍀ Z共N,T兲 = TrA exp − ␤ Sz 2

冋 冉

冊冕

d 2␣ 具␣兩exp共− ␤⍀a†a兲 ␲

⌬ z ⌳ + − † S + 冑N 共S + S 兲共a + a兲 2

冊册

冎

C2 = 21 关HI,H0兴,

共18兲

C3 = − 61 †H0,关HI,H0兴 ‡ − 31 †HI,关HI,H0兴 ‡

共19兲

共28兲

with increasing complexity for higher i. By applying the harmonic-oscillator commutation relations of the field mode operators a†, a:

For the case of resonant interaction (⌬ = 0) the coherent integral is exactly resoluble. Introducing the overcompleteness of the field coherent states

关a,a†兴 = 1,

关a†a,a†兴 = a†,

关a†a,a兴 = − a

⫻exp − ␤

冕

共20兲

and that for the atomic operators 关S+,S−兴 = Sz,

关Sz,S±兴 = ± 2S± .

共21兲

C3 =

再

⌳⍀

冑N

冕 冕 d 2␣ ␲

冋

共aS− − a†S+兲,

冎

2 ⌳⍀ ⌳ − + 2 † 2 z ⑀共aS− + a†S+兲 + 冑N 关共S + S 兲 − 共a + a 兲 S 兴 . 3 冑N

具HIC2典 = ⌳2⍀共a + a†兲2,

and

exp关− ␤共H0 + HI兲兴 = exp关− ␤H0兴exp关− ␤HI兴

冑N 共S

冋

⫻exp

共30兲

冋

+

册

+ S−兲共a† + a兲 兩␣典

␤⌳

冑N 共␥

*

+ ␣兲共S+ + S−兲

册

册

␤ 2⌳ 2 + − 2 共S + S 兲 . 2N

共32兲

Integral (30) can be now easily performed and the partition function (28) becomes

共25兲

is correct up to the second order in ␤ or rather for

␤3⌳2⍀ ⬍ 1.

␤⌳

= 具␥兩␣典exp −

具C3典 = 34 ⌳2⍀,

where 具Ci典 = 兺S1=±1 . . . 兺SN=±1 具S1 . . . SN兩Ci兩S1 . . . SN典. From Eqs. (23) and (24) we conclude that the approximation

冋

具␥兩exp −

共23兲

共24兲

册

+ S−兲共a† + a兲 兩␣典.

共31兲

Taking the expectation value over the atomic states we obtain 具C2典 = 具H0C2典 = 0,

+

具␣兩exp共− ␤⑀a†a兲兩␥典 = 具␣兩␥典exp兵− ␣*␥关1 − exp共− ␤⑀兲兴其

3

+ HI兲C2 + C3兴 − ¯其.

冑N 共S

By direct calculation, we obtain

Z共N,T兲 = TrATrF兵exp关− ␤H0兴exp关− ␤HI兴 − ␤ C2 − ␤ 关共H0 2

共29兲

d 2␥ 具␣兩exp共− ␤⑀a†a兲兩␥典具␥兩 ␲

␤⌳

⫻exp −

共22兲

The basic step in the construction of approximants to Eq. (17) is to find a product of exponential operators which is correct up to a certain power of ␤. Straightforward algebra yields the result

d 2␥ 兩␥典具␥兩 = I ␲

the integral in Eq. (28) can be written as

It is easy to show that C2 =

兩␣典 .

共26兲

This result is similar to that obtained by Orszag in Ref. [17]. He assume that H0 is conserved or that 033808-3

再 冉 冊 冋

⑀ ␤ 2⌳ 2 + − 2 共S + S 兲 Z共N,T兲 = TrA exp − ␤ Sz exp 2 2N ⫻

冕

册

d 2␣ exp关− 兩␣兩2共1 − exp共− ␤⑀兲兲兴 ␲

冋冑

⫻exp −

␤⌳ N

共S+ + S−兲共␣*exp共− ␤⑀兲 + ␣兲

册冎

PHYSICAL REVIEW A 70, 033808 (2004)

G. LIBERTI AND R. L. ZAFFINO

=

再 冉 冊

1 ⑀ TrA exp − ␤ Sz 2 1 − exp共− ␤⑀兲

冋

⫻exp

冉 冊

␤ 2⌳ 2 ␤⑀ coth 共S+ + S−兲2 2N 2

册冎

Z共N,T兲 =

Using the Fock states 兩n典 for the photon field, we have Z共N,T兲 = TrA

再

冉

冋 冉

⫻exp − ␤

冊兺

⫻exp −

⬁

exp共− ␤⍀n兲具n兩

f共x兲 = −

n=0

⌬ z ⌳ + − † S + 冑N 共S + S 兲共a + a兲 2

冊册

冎

冋

⬁

␤⌳

兺 exp共− ␤⑀n兲具n兩exp − 冑N 共S+ + S−兲共a† + a兲 n=0

冋

= exp

冋

⫻ −

␤ 2⌳ 2 + − 2 共S + S 兲 2N

册兺 册

册

冋冑

tanh

共35兲

⬁

兺

兺 e−␤⑀nLn n=0 =

冋

−

␤⌳ + −2 共S + S 兲 N

冋

册

共36兲

册

共37兲

V. PHASE TRANSITION

In order to disentangle the contribution of the various spins we introduce a new variable z and we make use of the identity

冋

⫻exp −

dz

册

共41兲

.

冉 冊

␤ c⑀ = ␤2c ⌳2 . 2

共42兲

冋 冉 冊册

1 ␤⑀ 2 cosh 1 − exp共− ␤⑀兲 2

N

共43兲

2N 共4␲兲1/2␤⑀

冕

冉 冊 冉冑 冑 冊

+⬁

dz exp −

−⬁

z2 coshN 4

␤ ⌳z ⑀ N

and the minimum condition becomes ˜x =⌳ 2

册

冑

冋冑 册

␤ tanh ⌳ ⑀

␤ ˜x . ⑀

共45兲

In this limit the leading term of Hamiltonian (1) becomes an Ising term with the spins interacting equally with each other [18]. The interaction of independent spins with a photon mode induce an effective spin-spin interaction of long range nature that may be described by the Hamiltonian HIsing = ⑀a†a +

⌳2 † 共a + a兲共S+ + S−兲2 . N⑀

共46兲

In the Ising model limit, we get

z2 + z␩共S+ + S−兲 . 共38兲 4

Indeed, the partition function can be written as

2 tanh 共␤⑀/2兲

corresponding to the expected partition function of N twolevel atomic systems plus radiation field in the noninteracting regime. For ␤ ⬎ ␤c, Eq. (41) has a solution of ˜x ⫽ 0. In the limit ␤c⑀ / 2 Ⰶ 1 the partition function is given by

+⬁

−⬁

˜ ␤⌳x

共40兲

共44兲

The partition function can be calculated substituting this result into Eq. (35) and found to be identical to Eq. (33).

冕

Z共N,T兲 ⯝

Z共N,T兲 =

1 ␤2⌳2 共S+ + S−兲2 exp . 1 − exp共− ␤⑀兲 N exp共␤⑀兲 − 1

1 exp关␩2共S+ + S−兲2兴 = 共4␲兲1/2

册冎

For ␤ ⬍ ␤c we obtain ˜x = 0 and the partition function takes the simple form

exp共− ␤⑀n兲Ln

n=0

2

␤⌳x

冑2 tanh 共␤⑀/2兲

˜x ␤⌳ tanh = 2 冑2 tanh共␤⑀/2兲

⬁

e−xt/共1−t兲 = tnLn共x兲. 1−t n=0

2

再 冋

x2 + ln cosh 4

兩n典

where Ln共x兲 is a Laguerre polynomial. Using the formula for the generating function of a Laguerre polynomial

⬁

册

␤⌳z 冑 z2 coshN coth共␤⑀/2兲 . 共39兲 4 2N

Through this result we can easily compute a critical temperature and we find

␤ 2⌳ 2 + − 2 共S + S 兲 , N

We obtain

dz

−⬁

is minimized. The minimum condition implies

兩n典 . 共34兲

The sum, for ⍀ = ⑀, is given by

冕

+⬁

z is an order parameter and the integral may be evaluated in the limit N → ⬁ by the steepest descent method. Writing the integral in term of a new variable x = z / 冑N we search the value ˜x for which

B. Fock-state representation

⍀ exp − ␤ Sz 2

冉 冊 冋冑

共33兲

.

2N coshN共␤⑀/2兲 共4␲兲1/2 1 − exp共− ␤⑀兲

␤c ⯝

⑀ 2⌳2

and the order parameter near the critical temperature is 033808-4

共47兲

PHYSICAL REVIEW A 70, 033808 (2004)

CRITICAL PROPERTIES OF TWO-LEVEL ATOM…

˜x ⯝ 冑6

␤c ␤

冑

␤ − ␤c . ␤c

␤3 ⬍ ␤3c ⯝

共48兲

Since ␤c⌳ 艋 1 we obtain

1 1 3 ⬍ ⌳ ⌳ 2⑀

共52兲

and for the superradiant phase only for

⑀ ⌳⬎ . 2

1 1 3 3 , 2 ⬎ ␤ 艌 ␤c ⯝ ⌳⑀ ⌳3

共49兲

The Ising model limit is then a strong long range interaction limit. Applying the inequality derived from the ThomasReiche-Kuhn sum rule (13), condition (49) cannot be satisfied. In this limit we find again the same conclusions found in the context of the Wang and Hioe method [9,14]. In the limit ␤c⑀ / 2 Ⰷ 1 we obtain

共53兲

i.e., for ⌳ ⬎ ⑀ in both cases and no conclusions can be reached for this regime in our approximate scheme. VI. CONCLUSION

In this “weak coupling regime” the direct interactions between different atoms are negligible and the phase transition, if it exists, is not restricted by the TKR inequality (13). Condition (26) permits us to derive accurate results for the normal phase only for

We have discussed the analytic evaluation of the thermodynamical properties of two-level atoms in interaction with the radiation field whether in the framework of the method developed by Wang and Hioe or in the strong coupling limit, through the construction of an approximate partition function. We found that in this limit, the interaction of independent atomic spins with a single photon mode can be interpreted as an effective spin-spin interaction of long range nature. Our results depends to the order of approximation but the one adopted is sufficient to illustrate the salient feature of the approach. It is possible to derive higher-order approximation and some of the methods discussed in this paper may also be applied to the calculation of thermodynamical properties of other atomic system for which the phase transition is possible [19]. These are the subjects of ongoing research.

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[11] M. Orszag, J. Phys. A 10, L21 (1977); 10, 1995 (1977). [12] M. Yamanoi, J. Phys. A 9, 573 (1976). [13] I. Bialynicki-Birula and K. Rzazewski, Phys. Rev. A 19, 301 (1979). [14] K. Gawedzki and K. Rzazewski, Phys. Rev. A 23, 2134 (1981). [15] M. D. Crisp, Phys. Rev. A 44, 563 (1991). [16] R. M. Wilcox, J. Math. Phys. 8, 962 (1967). [17] M. Orszag, J. Phys. A 12, 2205 (1979). [18] G. Parisi, Statistical Field Theory (Addison-Wesley, Redwood City, CA, 1988). [19] J. M. Knight, Y. Aharonov, and G. T. C. Hsieh, Phys. Rev. A 17, 1454 (1978).

1 ⌳

共50兲

⑀ ⌳⬍ . 2

共51兲

␤c ⯝ that is valid for

033808-5