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Squeezed states in the Jaynes-Cummings model
Volume 89A, number 8
PHYSICS LETTERS
7 June 1982
SQUEEZED STATES IN THE JAYNES—CUMMINGS MODEL P. MEYSTRE and M.S. ZUBAIRY 1 Max-Planck-In srirut für Quanrenoptik, D-8046 Garching, Fed.
Rep. Germany
Received 7 April 1982
It is shown that squeezed states of the electromagnetic field are obtained in the Jaynes—Cummings model of an atomic transition coupled to a single mode, quantized electromagnetic field.
Recently, a substantial interest has centered around the generation of so-called “squeezed states” of the electromagnetic field, in view of their potential use in gravitational wave detection [1,2]. A number of nonlinear optical systems susceptible to produce squeezed states have been analyzed theoretically. These include degenerate parametric oscillators [3,4], resonance fluorescence [5], degenerate four-wave mixing [6], optical bistabiity [7], and free-electron lasers [8]. Denoting by at and a the creation and annihilation operators of the electromagnetic field, with [a, all = 1, we introduce the hermitian amplitude operators a1 and
The simplest nontrivial model of quantum optics exhibiting nonclassical behaviour is the Jaynes— Cummings model [10], describing the interaction between a two-level atom and a single mode of the radiation field. This model has been studied by a number of authors [11,12]. Early results showed in particular the appearance of the so-called “Cummings-collapse”. Very recently, it was shown that this is followed by “revivals” and “recorrelations” [13—15].None of these effects can be described classically. In view of these results and of the interest in understanding how
a2, which are defined (in the interaction picture) as
squeezing comes about, it is instructive to analyze whether this simple model also leads to squeezed
a = a1 + ia2
states. Our task heremodel Jaynes—Cummings is particularly is amenable easy, to since an exact the solu-
It follows that [a1,a2] uncertainty relation is z~a1z~a2 1/4. ~‘
(P “ ‘
-
=
i/2, and the corresponding
tion. The Jaynes—.Cummings hamiltonian is
(2)
H= hc~xj~ + h~2ata+g(a+a+(rat),
A state of the field is said to be squeezed when one of the amplitudes a1 and a2 satisfies the relation 2< 1/4 i= 1 2 (3) (i~a.) Squeezed states can be shown to be purely quantummechanical states, i.e., their corresponding Glauber Frepresentation [9] is non-positive-defmite. Thus, they can be generated only via a purely quantum-mechanical process.
(4)
where (rand a~are the Pauli spin matrices, w and &2 are the frequency of the two-level atom and the field, respectively, and g approximation is a coupling constant. usual, the rotating wave has beenAs per~
formed, and we take for simplicity the resonant case = &7. In the interaction picture, the hamiltonian reduces to H = g(a~a+ rat) (5) I
For an atom initially in the excited state, the rePermanent address: Institute for Modern Optics, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA.
390
duced density operator for the field is given by
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 89A, number 8 ~
PHYSICS LEUERS I
2
(Aa
I
7 June 1982
(a)
1)
Fig. 1. (~a 2as a function of the dimensionless time At for = 0. The various 1) curves are labeled according to their conesponding initial average photon number al2 = 1,4,8, and 10. Form = ir/2, (~a 2becomes (~a 2.(a) 0 ~ At ~ 5;(b) 0 ‘~At’~20. 1) 2)
_____________________________________ I I I 0
pat)
1
=
2
3
0
5
Tratom u(t, 0) (Pf(O) 0
0) Ut(t, 0), 0
10 At
(6)
pf(t) = cos(AtV’~)pf(0)cos(AtV’~) sin(At~J~) sin(At’~J~) pf(O) a, ~[aat =—
=
=
X [Ia ~
+
(13
(7)
lpt(Olm)
exp[—i(n 2
—
m)&2t] [eHaI2 &l—1 (a*)m_lis,/n!,n!1
cos(AtV~~i) cos(AtV’m+l) sin(At’~/~) sin(At~~)],
operators, we find readily that (~a 2= ((a 2> 1) 1 (a1>) = ~ (2(ata) + 1 + (az> +
(8)
where al2 is the average number of photons in the field at t = 0. In terms of the creation and annihilation
+
1
(~a2) In
where A = gill. For a field initially in a coherent state, the matrix elements of the density-matrix, in the number-state representation, become Pnm(t)
20
—
where pf(0) is the density matrix of the field at t = 0 and U(t, 0) = exp(—iH 1 t/h) is the time-evolution operator. A straightforward calculation gives
+ at
15
—
(az)
—
—
~(a+ at>2,
(9a)
(at 2>) + ~2. (9b) —
2((~a 2)as a function of 1) 2) time, for 0 = 0 (ir/2), where 0 is the phase of a, i.e. a = al exp(iØ), for a field initially in a coherent state of various intensities. For short times, we observe oscillations, and the value of (~a 2((~a 2)does indeed reach nonclassical values1) (smaller2) than 1/4). As time goes on, (~~a ((~a 2 2)increases monoto1) This 2) corresponds to the nically, and almost linearly. Cummings collapse region [12,16]. This increase of (~a 2((&z 2) continues until the appearance of 1) [13],2)see fig. ib, where it starts oscillating revivals again, and can again reach values less than 1/4. Thus, we see that squeezing takes place both in the shorttime regime and during the revivals. In fig. 2, we plot (i.~a 2((~a 2)versus time At 1)figures that the for 0 = 0 (ir/2). It is clear 2) from the product of the uncertainties (i~a 1)(~a2)is always fig.
1, we show (~a
391
Volume 89A, number 8
PHYSICS LETTERS
7 June 1982
8 (Aa~
0
5
10
15
At 2,and for times 0
Fig. 2. Same as fig. 1, but for (~a
~ At
~ 20. For ~
=
7r/2, this becomes (~aj)2.
2)
larger than 1/4. Note also the relative lack of temporal structure in (~a 2((~a 2),fig. 2, as compared to 2) 1) the conjugate component shown in fig. 1. In conclusion, we have shown that even the simplest mcxdel of quantum optics follows a dynamics leading to squeezed states of the electromagnetic field. However, this squeezing is rather weak, (~a 2~ 0.2. Fur1)academic thermore, this model is obviously only of interest. References [1] C.M. Caves, Phys. Rev. D23 (1981) 1693. [2] For a recent review of this problem, see: P. Meystre and M.D. Scully, eds., Quantum optics, experimental gravitation, and measurement theory (Plenum, New York, 1982), to be published. [3] D. Stoler, Phys. Rev. Lett. 33 (1974) 1397.
392
[4] G. Milburn and D.F. Walls, Opt. Commun. 39 (1981) 401. [5] D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (1981) 709. [6] H.P. Yuen and J.H. Shapiro, Opt. Lett. 4 (1979) 334. [7] L.A. Lugiato and G. Strini, to be published. [8] W. Becker, M.O. Scully and M.S. Zubairy, Phys. Rev. Lett. 48 (1982) 475. [9] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [10] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51(1963) 89. [111S. Stenholm, Phys. Rep. 6C (1973) 1. [12J T. von Foerster, J. Phys. A8 (1975) 95. [13] J.H. Eberly, N.B. Narozhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44 (1980) 1323. [14] S. Singh, Phys. Rev. A, to be published. [15] P.L. Knight and P.M. Radmore, Phys. Rev. A, to be published. [16] P. Meystre, E. Geneux, A. Quattropani and A. Faist, Nuovo Cimento 25 (1975) 521.
PHYSICS LETTERS
7 June 1982
SQUEEZED STATES IN THE JAYNES—CUMMINGS MODEL P. MEYSTRE and M.S. ZUBAIRY 1 Max-Planck-In srirut für Quanrenoptik, D-8046 Garching, Fed.
Rep. Germany
Received 7 April 1982
It is shown that squeezed states of the electromagnetic field are obtained in the Jaynes—Cummings model of an atomic transition coupled to a single mode, quantized electromagnetic field.
Recently, a substantial interest has centered around the generation of so-called “squeezed states” of the electromagnetic field, in view of their potential use in gravitational wave detection [1,2]. A number of nonlinear optical systems susceptible to produce squeezed states have been analyzed theoretically. These include degenerate parametric oscillators [3,4], resonance fluorescence [5], degenerate four-wave mixing [6], optical bistabiity [7], and free-electron lasers [8]. Denoting by at and a the creation and annihilation operators of the electromagnetic field, with [a, all = 1, we introduce the hermitian amplitude operators a1 and
The simplest nontrivial model of quantum optics exhibiting nonclassical behaviour is the Jaynes— Cummings model [10], describing the interaction between a two-level atom and a single mode of the radiation field. This model has been studied by a number of authors [11,12]. Early results showed in particular the appearance of the so-called “Cummings-collapse”. Very recently, it was shown that this is followed by “revivals” and “recorrelations” [13—15].None of these effects can be described classically. In view of these results and of the interest in understanding how
a2, which are defined (in the interaction picture) as
squeezing comes about, it is instructive to analyze whether this simple model also leads to squeezed
a = a1 + ia2
states. Our task heremodel Jaynes—Cummings is particularly is amenable easy, to since an exact the solu-
It follows that [a1,a2] uncertainty relation is z~a1z~a2 1/4. ~‘
(P “ ‘
-
=
i/2, and the corresponding
tion. The Jaynes—.Cummings hamiltonian is
(2)
H= hc~xj~ + h~2ata+g(a+a+(rat),
A state of the field is said to be squeezed when one of the amplitudes a1 and a2 satisfies the relation 2< 1/4 i= 1 2 (3) (i~a.) Squeezed states can be shown to be purely quantummechanical states, i.e., their corresponding Glauber Frepresentation [9] is non-positive-defmite. Thus, they can be generated only via a purely quantum-mechanical process.
(4)
where (rand a~are the Pauli spin matrices, w and &2 are the frequency of the two-level atom and the field, respectively, and g approximation is a coupling constant. usual, the rotating wave has beenAs per~
formed, and we take for simplicity the resonant case = &7. In the interaction picture, the hamiltonian reduces to H = g(a~a+ rat) (5) I
For an atom initially in the excited state, the rePermanent address: Institute for Modern Optics, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA.
390
duced density operator for the field is given by
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 89A, number 8 ~
PHYSICS LEUERS I
2
(Aa
I
7 June 1982
(a)
1)
Fig. 1. (~a 2as a function of the dimensionless time At for = 0. The various 1) curves are labeled according to their conesponding initial average photon number al2 = 1,4,8, and 10. Form = ir/2, (~a 2becomes (~a 2.(a) 0 ~ At ~ 5;(b) 0 ‘~At’~20. 1) 2)
_____________________________________ I I I 0
pat)
1
=
2
3
0
5
Tratom u(t, 0) (Pf(O) 0
0) Ut(t, 0), 0
10 At
(6)
pf(t) = cos(AtV’~)pf(0)cos(AtV’~) sin(At~J~) sin(At’~J~) pf(O) a, ~[aat =—
=
=
X [Ia ~
+
(13
(7)
lpt(Olm)
exp[—i(n 2
—
m)&2t] [eHaI2 &l—1 (a*)m_lis,/n!,n!1
cos(AtV~~i) cos(AtV’m+l) sin(At’~/~) sin(At~~)],
operators, we find readily that (~a 2= ((a 2> 1) 1 (a1>) = ~ (2(ata) + 1 + (az> +
(8)
where al2 is the average number of photons in the field at t = 0. In terms of the creation and annihilation
+
1
(~a2) In
where A = gill. For a field initially in a coherent state, the matrix elements of the density-matrix, in the number-state representation, become Pnm(t)
20
—
where pf(0) is the density matrix of the field at t = 0 and U(t, 0) = exp(—iH 1 t/h) is the time-evolution operator. A straightforward calculation gives
+ at
15
—
(az)
—
—
~(a+ at>2,
(9a)
(at 2>) + ~2. (9b) —
2((~a 2)as a function of 1) 2) time, for 0 = 0 (ir/2), where 0 is the phase of a, i.e. a = al exp(iØ), for a field initially in a coherent state of various intensities. For short times, we observe oscillations, and the value of (~a 2((~a 2)does indeed reach nonclassical values1) (smaller2) than 1/4). As time goes on, (~~a ((~a 2 2)increases monoto1) This 2) corresponds to the nically, and almost linearly. Cummings collapse region [12,16]. This increase of (~a 2((&z 2) continues until the appearance of 1) [13],2)see fig. ib, where it starts oscillating revivals again, and can again reach values less than 1/4. Thus, we see that squeezing takes place both in the shorttime regime and during the revivals. In fig. 2, we plot (i.~a 2((~a 2)versus time At 1)figures that the for 0 = 0 (ir/2). It is clear 2) from the product of the uncertainties (i~a 1)(~a2)is always fig.
1, we show (~a
391
Volume 89A, number 8
PHYSICS LETTERS
7 June 1982
8 (Aa~
0
5
10
15
At 2,and for times 0
Fig. 2. Same as fig. 1, but for (~a
~ At
~ 20. For ~
=
7r/2, this becomes (~aj)2.
2)
larger than 1/4. Note also the relative lack of temporal structure in (~a 2((~a 2),fig. 2, as compared to 2) 1) the conjugate component shown in fig. 1. In conclusion, we have shown that even the simplest mcxdel of quantum optics follows a dynamics leading to squeezed states of the electromagnetic field. However, this squeezing is rather weak, (~a 2~ 0.2. Fur1)academic thermore, this model is obviously only of interest. References [1] C.M. Caves, Phys. Rev. D23 (1981) 1693. [2] For a recent review of this problem, see: P. Meystre and M.D. Scully, eds., Quantum optics, experimental gravitation, and measurement theory (Plenum, New York, 1982), to be published. [3] D. Stoler, Phys. Rev. Lett. 33 (1974) 1397.
392
[4] G. Milburn and D.F. Walls, Opt. Commun. 39 (1981) 401. [5] D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (1981) 709. [6] H.P. Yuen and J.H. Shapiro, Opt. Lett. 4 (1979) 334. [7] L.A. Lugiato and G. Strini, to be published. [8] W. Becker, M.O. Scully and M.S. Zubairy, Phys. Rev. Lett. 48 (1982) 475. [9] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [10] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51(1963) 89. [111S. Stenholm, Phys. Rep. 6C (1973) 1. [12J T. von Foerster, J. Phys. A8 (1975) 95. [13] J.H. Eberly, N.B. Narozhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44 (1980) 1323. [14] S. Singh, Phys. Rev. A, to be published. [15] P.L. Knight and P.M. Radmore, Phys. Rev. A, to be published. [16] P. Meystre, E. Geneux, A. Quattropani and A. Faist, Nuovo Cimento 25 (1975) 521.