Harmonic Oscillator Position Eigenstates Via Application of an Operator on the Vacuum

EDUCATION Revista Mexicana de F´ısica E 59 (2013) 122–127 JULY–DECEMBER 2013

Harmonic oscillator position eigenstates via application of an operator on the vacuum

Francisco Soto-Eguibar and Hector´ Manuel Moya-Cessa Instituto Nacional de Astrof´ısica, Optica´ y Electronica,´ Luis Enrique Erro 1, Santa Mar´ıa Tonantzintla, San Andres´ Cholula, Puebla, 72840 Mexico.´ Received 21 August 2013; accepted 17 October 2013

Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position.

Keywords: Position eigenstates; harmonic oscillator; squeezed states; minimum uncertainty states; squeeze operator.

PACS: 03.65.-w

1. Introduction duce the uncertainties either of the position or momentum, while still keeping the uncertainty principle to its minimum. Quantum states of the harmonic oscillator are very important Because of this, they belong to a special class of states named in the study of the quantum theory of the electromagnetic minimum uncertainty states. Once produced, for instance as field because of the fact that they may be used to describe electromagnetic fields in cavities, they may be monitored via a quantized field [1–3], each harmonic oscillator represent- two level atoms in order to check, or measure, that such states ing a mode of such electromagnetic field. In fact, this was have been indeed generated [10, 11]. the concept that Dirac used to build the first quantum theory Based on the above properties, we can think about eigen- of the electromagnetic field [2]. The easiest to understand states of position as limiting cases of squeezed states. As and to manipulate, and the most natural states of the quantum squeezed states are minimum uncertainty states, we can re- harmonic oscillator are number states |ni. Number states are duce to zero the uncertainty in the position, while the uncer- eigenstates of the harmonic oscillator Hamiltonian and, of tainty in the momentum goes to infinity, so that we keep the course, are also eigenstates of the number operator nˆ =a ˆ†aˆ, uncertainty principle to its minimum. Of course, there is also where aˆ† and aˆ are the well known creation and annihilation the option to reduce to zero the uncertainty in the momen- operators, respectively. However, for any n, no matter how tum, while the position gets completely undefined, obtain- big, the mean field is zero; i.e., hn|Eˆx|ni = 0, and we know ing that way the possibility to define momentum eigenstates. that a classical field changes sinusoidally in time in each point In Secs. 1 and 2, we analyze the possibility of define the of space; thus, these states can not be associated with classi- position eigenstates as the limit of extreme squeezing of the cal fields [3, 4]. squeezed states. In what follows, we will use a unit system such that ~ = m = ω = 1. In the first years of the sixties of the past century, Glauber [5] and Sudarshan [6] introduced the coherent states, There are two equivalent forms to define the squeezed and it has been shown that these states are the most clas- states. In the first one, introduced by Yuen [12], squeezed sical ones. Coherent states are denoted as |αi, and one states are obtained from the vacuum as way to define them is as eigenstates of the annihilation |α; ri = Sˆ(r)Dˆ(α)|0i = Sˆ(r)|αi, (1) operator; that is, aˆ|αi = α|αi. An equivalent definition is obtained¡ applying the¢ Glauber displacement operator † ∗ where h³ ´ i Dˆ(α) = exp αaˆ − α aˆ to the vacuum: |αi = Dˆ(α)|0i; 2 Sˆ(r) = exp aˆ2 − aˆ† r/2 (2) we see then coherent states as vacuum displaced states. Co- herent states also have the very important property that they is the so-called squeeze operator. In this view, squeezed states minimize the uncertainty relation for the two orthogonal field are created displacing the vacuum, and after, squeezing it. quadratures with equal uncertainties in each quadrature [3,4]. Note that when the squeeze parameter r is zero, the squeezed Since then, other states have been introduced. In partic- states reduce to the coherent states. In this work, we will con- ular, squeezed states [3, 4, 7–9] have attracted a great deal of sider only real squeeze parameters, as that is enough for our attention over the years because their properties allow to re- intentions. HARMONIC OSCILLATOR POSITION EIGENSTATES VIA APPLICATION OF AN OPERATOR ON THE VACUUM 123

In the definition introduced by Caves [13], the vacuum is Proceeding in exactly the same way for the quadrature oper- squeezed and the resulting state is then displaced; that means, ator Yˆ , we obtain q that in this approach squeezed states are given as er ∆ Y ≡ hα; r|Yˆ 2|α; ri − hα; r|Yˆ |α; ri2 = . (15) 2 |α0; r0i = Dˆ (α0) Sˆ (r0) |0i. (3) As we already said, we can then think in the position eigen- Both definitions of the squeezed states agree when the states and in the momentum eigenstates as limit cases of squeeze factor is the same, r0 = r, and when the modified squeezed states. Indeed, when the squeeze parameter r goes amplitude α0 of the Caves approach is given by to infinity, the uncertainty in the position goes to zero, and the momentum is completely undetermined. Of course, when the 0 ∗ α = µα − να , (4) squeeze parameter goes to minus infinity, we have the inverse situation, and we can think in define that way the momentum being eigenstates. In the two following sections, we use the Yuen µ = cosh r (5) and the Caves definitions of the squeezed states to test this and hypothesis. ν = sinh r. (6) To analyze the uncertainties in the position and in the 2. A first attempt a` la Yuen momentum of the squeezed states, we introduce, following Loudon and Knight [7], the quadrature operators From Eq. (14) above, we can see that in the limit r → ∞ the uncertainty for position vanishes and so a position eigenstate aˆ +a ˆ† xˆ Xˆ = = √ (7) should be obtained (from now on, we consider α real), 2 2 x lim |√ ; ri → |xip. (16) and r→∞ 2 aˆ − aˆ† pˆ Yˆ = = √ (8) We have written a sub index p in the position eigenstate in 2i 2 order to emphasis that fact. Following the Yuen definition where xˆ is the position operator and pˆ the momentum opera- |α; ri = Sˆ(r)Dˆ(α)|0i = Sˆ(r)|αi, so µ ¶ tor. Note that the quadrature operators are essentially the po- x x x |√ ; ri = Sˆ(r)Dˆ √ |0i = Sˆ(r)|√ i. (17) sition and momentum operators; this definition just provides 2 2 2 us with two operators that have the same dimensions. We now write the squeeze operator as [16] In order to show that really the squeezed states are min- 1 ν †2 1 ν 2 imum uncertainty states, we need to calculate the expected ˆ − 2µ aˆ 2µ aˆ S(r) = √ e † e , (18) values in the squeezed state (1) of the quadrature operators µ µaˆ aˆ (7) and (8), and its squares. Using (7) and (1), we get where, as we already said, µ = cosh r and ν = sinh r. So,

2 † x 1 − ν aˆ† 1 ν aˆ2 x 1 † aˆ +a ˆ |√ ; ri = e 2µ e 2µ |√ i. (19) hα; r|Xˆ|α; ri = hα|Sˆ (r) Sˆ(r)|αi. (9) √ aˆ†aˆ 2 2 2 µ µ 2 The action of the squeeze operator on the creation and Now, we develop the first operator (from right to left) in annihilation operators is obtained using the Hadamard’s power series, we use the definition of the coherent states, lemma [14, 15], aˆ|αi = α|αi, and¡ the action of the number¢ operator over the number states aˆ†aˆ|ni =n ˆ|ni = n|ni , to obtain ˆ† ˆ † ˆ† † ˆ † S (r)âS(r) = µaˆ − νaˆ , S (r)â S(r) = µaˆ − νa,ˆ (10) µ ¶aˆ†aˆ ∞ µ ¶n 2 X x 1 − ν aˆ† 1 x 1 |√ ; ri = √ e 2µ √ √ |ni such that µ µ † 2 n=0 2 n! ˆ† aˆ +a ˆ ˆ −r ˆ S (r) S(r) = e X. (11) ∞ µ ¶n µ ¶n 2 2 X 1 − ν aˆ† x 1 1 † ∗ = √ e 2µ √ √ |ni. (20) Therefore, as aˆ|αi = α|αi and hα|aˆ = hα|α , it is easy to µ 2 n! µ see that n=0 ∗ 1 1 −r α + α r → ∞ = → 0 hα; r|Xˆ|α; ri = e , (12) As , µ cosh r , which means that the only term 2 that survives from the sum is n = 0, and then and that ν †2 − 2µ aˆ 2 |xip ∝ e |0i (21) 1 + 2 |α|2 + α2 + α∗ hα; r|Xˆ 2|α; ri = e−2r . (13) that would give an approximation for how to obtain a po- 4 sition eigenstate from the vacuum. However, note that the So, we obtain for the uncertainty in the quadrature operator above expression does not depend on x and therefore can not ˆ X, be correct. From all this analysis, we must conclude that the q e−r Yuen definition of the squeezed states, whatever representa- ∆ X ≡ hα; r|Xˆ 2|α; ri − hα; r|Xˆ|α; ri2 = . (14) 2 tion used, does not give the correct asymptotic states.

Rev. Mex. Fis. E 59 (2013) 122–127 124 F. SOTO-EGUIBAR AND H. M. MOYA-CESSA

ν 3. A second attempt a` la Caves We take now the limit when r → ∞, or µ → 1, so Ã ! µ ¶ 2 We now squeeze the vacuum and after we displace it. Thus, x2 aˆ† √ |xi ∝ exp − exp − + 2xaˆ† |0i. (29) in this case, p 2 2 µ ¶ x x We get an expression that gives us the position eigenstates as |xip = lim |√ ; ri = lim Dˆ √ Sˆ(r)|0i. (22) r→∞ 2 r→∞ 2 an operator applied to the vacuum. Unlike the Yuen case, expression (21), now we have an x dependence and it looks like ³ ³ ´ ³ ´ 1 2 nˆ+ 2 We use again expression Sˆ(r) = exp − ν aˆ† 1 a better candidate to be the position eigenstate. In fact, in the ³ ´ ´ 2µ µ next Section, we will show that this really is an eigenstate of ν 2 exp 2µ aˆ for the squeeze operator [12], where µ and ν the position. are deﬁned in (5) and (6),µ and¶ we write the displacement op- 2 ¡ ¢ ˆ |α| ∗ † erator as D(α) = exp 2 exp (−α aˆ) exp αaˆ [16], 4. Leaving squeezed states aside to obtain µ ¶ µ ¶ µ ¶ We will try now an alternative approach to the eigenstates of x x2 x x the position. We can write a position eigenstate, simply by |√ ; ri = exp exp −√ aˆ exp √ aˆ† 2 4 2 2 multiplying it by a proper unit operator ∞ µ ¶ µ ¶nˆ+ 1 µ ¶ X ν 2 1 2 ν × exp − aˆ† exp a2 |0i. (23) |xip = |ni hn|xip (30) 2µ µ 2µ n=0 Therefore the position eigenstate |xi may be written as [17] As aˆ|0i = 0 and aˆ†aˆ|0i =n ˆ|0i = 0, we cast the previous p formula as X∞ |xip = ψn(x)|ni (31) µ 2 ¶ µ ¶ x 1 x x n=0 |√ ; ri = √ exp exp −√ aˆ 2 µ 4 2 1 −x2/2 with ψn(x) = √ √ e Hn(x); such that |xip may µ ¶ µ ¶ 2n πn! x ν 2 × exp √ aˆ† exp − aˆ† |0i. (24) be re-written as 2µ 2 2 −x /2 ∞ e X 1 n |xi = H (x)ˆa† |0i, (32) Inserting twice the identity operator, written as p π1/4 2n/2n! n ³ ´ ³ ´ n=0 Iˆ = exp √x aˆ exp − √x aˆ , we obtain 2 2 that may be added via using the generating function for Her-

2 mite polynomials [18] x 1 x − √x aˆ √x aˆ† |√ ; ri = √ e 4 e 2 e 2 X∞ k 2 µ −t2+2t x t e = Hk(x) , (33) 2 √x aˆ − √x aˆ − ν aˆ† √x aˆ − √x aˆ k! × e 2 e 2 e 2µ e 2 e 2 |0i. (25) n=0 to give ³ ´ 2 x −x /2 †2 √ It is clear that exp − √ aˆ |0i = |0i, and using the e − aˆ + 2xaˆ† 2 |xip = e 2 |0i. (34) Hadamard´s lemma [14], it is easy to prove that π1/4 ¡ ¢ ¡ ¢ The above expression allows us to write the position eigen- exp (−γ aˆ) η aˆ† exp (γ aˆ) = η aˆ† − γ , (26) state as an operator applied to the vacuum. Note that this ¡ ¢ expression is the same as the one obtained using the Caves for any well behaved function η aˆ† ; thus deﬁnition for the squeezed states, formula (28). We prove µ ¶ · µ ¶¸ now that indeed (32) is an eigenvector of the position oper- 2 † x 1 x x † x ator; for that, we write the position operator as xˆ = aˆ√+ˆa , |√ ; ri = √ exp exp √ aˆ − √ 2 2 µ 4 2 2 thus " # 2 µ ¶2 −x 2 2 / ¡ ¢ † √ † ν † x e † − aˆ + 2xaˆ × exp − aˆ − √ |0i. (27) xˆ|xip = √ aˆ +a ˆ e 2 |0i. (35) 2µ 2 π1/4 2 Inserting the identity operator in the above expression as After some algebra, †2 √ √ †2 − aˆ 2xaˆ† − 2xaˆ† aˆ Iˆ = e 2 e e e 2 , we get · 2 µ ¶¸ x 1 x ν 2 −x /2 †2 √ √ |√ ; ri = √ exp − 1 + e − aˆ 2xaˆ† − 2xaˆ† 2 µ 4 µ xˆ|xip = √ e 2 e e · µ ¶ ¸ π1/4 2 ν 2 x ν † † †2 †2 √ × exp − aˆ + √ 1 + aˆ |0i. (28) aˆ ¡ †¢ − aˆ 2xaˆ† 2µ 2 µ × e 2 aˆ +a ˆ e 2 e |0i; (36)

Rev. Mex. Fis. E 59 (2013) 122–127 HARMONIC OSCILLATOR POSITION EIGENSTATES VIA APPLICATION OF AN OPERATOR ON THE VACUUM 125

†2 ¡ ¢ †2 1−e2r 2 √ 1−e2r 2 √ aˆ † − aˆ † † aˆ x as e 2 aˆ +a ˆ e 2 =a ˆ − aˆ +a ˆ =a ˆ, It is very easy to see¯ that®e 4 | 2xi = e 2 | 2xi, †2 ¡ ¢ †2 iγnˆ ¯ iγ aˆ † − aˆ † † and that e |αi = e α , thus e 2 aˆ +a ˆ e 2 =a ˆ − aˆ +a ˆ =a ˆ, and aˆ|0i = 0, we obtain ½ ¾ 2 h¡ ¢ i −x /2 †2 √ 1 1 2r 2 ∗2 e − aˆ 2xaˆ† hα; r|xi = exp 2 − e x − α − r xˆ|xi = x e 2 e |0i = x|xi , (37) p 1/4 p π1/4 p π 2 D √ E −r as we wanted to show. × α| 2e x . (44) We can write (32) in terms of coherent states. We have 1 2 2 ∗ − 2 (|δ| +|²| −2δ ²) √ ∞ ³ ´ Finally, as hδ|²i = e , we have † X 1 √ k k e 2xaˆ |0i = 2x aˆ† |0i k! ( k=0 h 1 1 ¡ 2r −2r¢ 2 ¡√ ¢ hα; r|xip = exp 2 − e − 2e x X∞ k √ π1/4 2 2x x2 = √ |ki = e | 2xi, (38) ) k! i k=0 √ 2 + 2 2α∗e−rx − α∗ − |α|2 − r , (45) thus 2 x /2 †2 √ e − aˆ |xi = e 2 | 2xi. (39) p π1/4 as we wanted to show. With the expressions obtained, it is easy to show that the squeezed states have the form of a Gaussian wave packet. To conﬁrm this, we use the above expression to state that 5. The Husimi Q-function

† hα; r|xip = hα|Sˆ (r)|xip We can now ﬁnd the wave function of a coherent state as a

2 function of the position [19]. We use equation (32), that ex- x /2 †2 √ e † − aˆ press the eigenstates of the position as an operator acting on = hα|Sˆ (r)e 2 | 2xi. (40) π1/4 the vacuum, and get that

r aˆ2−raˆ†aˆ that after some algebra, can be re-written as In the Appendix, we disentangle the operator e 2 as 2r −raˆ†aˆ 1−e aˆ2 e e 4 , and we get 1 Q(β)= h ³ ´i 3/2 1 2 ∗2 π exp 2 x − α − r h √ i 2 2 ∗2 hα; r|xip = × exp −x − |β| −Re(β )+2 2Re(β)x . (48) π1/4 2r √ −raˆ†aˆ 1−e aˆ2 × hα|e e 4 | 2xi. (43) In the ﬁgures, we plot the Husimi Q-function for different values of x.

Rev. Mex. Fis. E 59 (2013) 122–127 126 F. SOTO-EGUIBAR AND H. M. MOYA-CESSA

FIGURE 4. The Husimi Q-function for x = 6. FIGURE 1. The Husimi Q-function for x = −3.

6. Conclusions

We have found an operator that applied to the vacuum gives us the eigenstates of the position. We did that by two different ways; first, using the Caves definition of the squeezed states, we took the limit of extreme squeezing in the position side, to get the position eigenstate. Second, we used the expansion of an arbitrary wave function in the base of the harmonic oscillator; i.e., we wrote an arbitrary wave function in terms of Hermite polynomials. The expressions obtained allows us to show certain properties of squeezed states, and also allow us to write in a very easy way the Husimi Q-function of the position eigenstates. The same procedure can be followed to find the eigenstates of the momentum, but taken the limit when the squeeze parameters goes to −∞. We can also conclude that from the point of view of this work, the Caves approach to define squeezed states is more adequate, because it gives the correct eigenstates of the posi- FIGURE 2. The Husimi Q-function for x = 0. tion; while the Yuen definition, formula (1), gives an expression that is incorrect. So, we must first squeeze the vacuum, and after, displace it.

A Appendix

In this appendix, we show how to disentangle the operator − r aˆ2+raˆ†aˆ e 2 . We deﬁne

− r aˆ2+raˆ†aˆ Fˆ(r) ≡ e 2 , (49)

and we suppose that (48) can be rewritten as £ ¤ £ ¤ Fˆ(r) = exp f(r)ˆa†aˆ exp g(r)ˆa2 , (50)

where f(r) and g(r) are two unknown well behaved functions; as Fˆ(0) = Iˆ, being Iˆ the identity operator, these functions most satisfy the conditions f(0) = g(0) = 0. At ﬁrst sight, one can thinkh that ini the proposal (45) should be a 2 FIGURE 3. The Husimi Q-function for x = 3 term of the form exp h(r)ˆa† ; however, this is not the case

Rev. Mex. Fis. E 59 (2013) 122–127 HARMONIC OSCILLATOR POSITION EIGENSTATES VIA APPLICATION OF AN OPERATOR ON THE VACUUM 127 £ ¤ because aˆ2, aˆ†aˆ = 2â2. We differentiate with respect to r, so µ ¶ to find dFˆ df dg = aˆ†aˆ + e−2f aˆ2 F.ˆ (54) dFˆ df £ ¤ £ ¤ dr dr dr = aˆ†aˆ exp faˆ†aˆ exp gaˆ2 dr dr Equating this equation to the one obtained differentiating the dg £ ¤ £ ¤ + exp faˆ†aˆ aˆ2 exp gaˆ2 , (51) original formula for Fˆ(r), equation (44), we get the follow- dr ing system of first order ordinary differential equations where for simplicity in the notation,£ we¤ have£ dropped¤ all r- dependency; we write Iˆ = exp −faˆ†aˆ exp faˆ†aˆ for the df dg 1 = 1, e−2f = − . (55) identity operator in the second term, to obtain dr dr 2 dFˆ df £ ¤ £ ¤ dg £ ¤ = aˆ†aˆ exp faˆ†aˆ exp gaˆ2 + exp faˆ†aˆ aˆ2 The solution of the first equation, that satisfies the initial con- dr dr dr £ ¤ £ ¤ £ ¤ dition f(0) = 0, is the function f(r) = r. Substituting this × exp −faˆ†aˆ exp faˆ†aˆ exp gaˆ2 . (52) solution in the second equation and solving it with the initial 2r condition g(0) = 0, we obtain g(r) = 1−e . Thus, finally Using the Hadamard´s lemma [14,15], it is very easy to prove 4 we write that 2r £ † ¤ 2 £ † ¤ −2f 2 − r aˆ2+raˆ†aˆ raˆ†aˆ 1−e aˆ2 exp faˆ aˆ aˆ exp −faˆ aˆ = e aˆ , (53) e 2 = e e 4 . (56)

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