Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t

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1 8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes as a bass n Hlbert space. Also, we ntroduce the so-called squeezed states that mnmze the uncertanty product qp. These states can b e vewed as a generalzaton of the coherent states and, lke the latter, are closely related to the propertes of the operators a and a. The coherent states dscussed above are members of a wder class of states havng the property that the product of the dspersons of q^ and p^ s a mnmum. Such states are called `squeezed states'. The possblty of reducng the uncertanty n a p h yscal varable, e.g., coordnate or momentum of a mechancal oscllator, or the duraton of an optcal pulse, provded by the squeezed states, s often useful n applcatons.. Uncertanty product Recall the proof of the uncertanty relaton qp h. For any two hermtan operators A, B, and any state, consder the quantty F () = h(a ^ B) ^ j(a ^ B) ^ = ha^ h[a ^ B] ^ hb^ () where h::: stands for the expectaton value h j:::j. Snce F () s nothng but the norm jj(a ^ B) ^ jj, t s non-negatve. The quadratc polynomal F () thus does not have real roots, whch gves ^ ^ ^ ^ ha hb h[a B] () Applyng ths to A ^ = q, ^ B ^ = p ^ = ; h@ q, obtan F () = h(q^ p^) j(q^ p^) = hq^ h[ q^ p^] hp^ 0 (3) leadng to hp^hq^ h. For a state wth q = hq^ = 0 and p = hp^ = 0, the coordnate and momentum uncertanty s gves whch hq = h( q^ ; q) = hq^ hp = h( p^ ; p) = hp^ () hq hp h (5) The uncertanty relaton for a more general state (q) wth nonzero q, p can be obtaned from the above. Let us shft the coordnate as and momentum (q) = e pq ~ (q ; q) (6)

2 Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze the uncertanty product and have q = p = 0 satsfy (q^ p^) = q h 0 = 0 (8) where s a complex parameter. Integratng ths equaton, wth h absorbed n, obtan (q) = ;= ( 0 ) = exp(;q =) = 0 (9) (normalzablty requres 0 > 0). More general states wth mnmal uncertanty can b e obtaned by a dsplacement of p and q, as n Eq.(6). The general states that mnmze hq hp are called squeezed states. As we shall see, a squeezed state may be constructed to have and arbtrarly small wdth hq =.. Squeezed states and the operators a and a. Suppose we prepared a squeezed state (9) of a harmonc oscllator H = h!(a a ) w th q = p = 0, and ask how wll t evolve n tme. The smplest way to obtan the tme evoluton s to rewrte the mnmal uncertanty condton q h 0 = 0 through the canoncal creaton and annhlaton operators, snce the latter evolve n tme n a very ;Ht ;!t Ht ;Ht smple way: a(t) = e Ht a e = e a, a (t) = e a e = e!t a. h Usng the relatons q = p (a a ), p = p (a ; a) (Lecture ), we obtan (q^ p^) = ( ; )a ( ) a 00 = 0 (0) The tme dependent state (t) = e ;Ht satses e ;Ht ( ; )a ( Ht ) a e (t) = ( ; )a(;t) ( ) a (;t) (t) = 0 (!t ; )e a ( ) e ;!t a (t) = 0 () Ths equaton has the form of Eq.(0) wth a tme-dependent gven by (t) ; ;!t cos!t ; sn!t = e (t) = () (t) (cos!t ; sn!t) The wavepacket remans gaussan at all tmes, whle ts wdth oscllates. For the ground state of the oscllator, (t) = at all tmes. If the ntal state s squeezed,, the wavepacket wdth reaches maxmum at the tmes when cos!t s close to zero, and collapses to the mnmal value when sn!t s near zero.

3 3 The Wgner functon of the squeezed state (9) evolvng n tme accordng to () s a gaussan dstrbuton n the phase space, centered at the orgn and rotatng wth the frequency! wthout changng shape (see Problem 3, PS#). In contrast wth the sotropc phase-space dstrbuton of the ground state (as well as any other coherent state), the squeezed states produce gaussans elongated n one drecton and squashed n a perpendcular drecton. The major axes rotate accordng to the classcal oscllator phase ow. There s another, more formal, denton of squeezed states based on the so-called `squeeze operators'. These are untary operators whch, when appled to the oscllator vacuum state, produce a squeezed state. The smplest example of a squeeze operator s U () = exp a^ ^a ; a^ a ^ = (3) Ths operator has the followng propertes (see Problem, PS#) U ()^au() = cosh ^ a ; snh ^ a U ()a U () = cosh ^ a ; snh ^ a () from whch t follows that U () q^ U () = U () p a ^ a^ U () = e ; q^ (5) h U () pu ^ () = U () p a ^ ; a^ U () = e p^ (6) Usng these results, one can show that the state U ()j0 s a mnmum uncertanty state (9) wth the parameter = e. Hence the wdth of the wavepacket hq = s e tmes smaller than that of oscllator vacuum. Other squeezed states can be obtaned n a smlar way usng untary operators U (z) = exp (z (^a^ a ; a^ a ^ ) =) wth complex parameter z. Because of the form of the squeeze operator, the squeezed states are also sometmes called `two-photon coherent states'..3 Squeezed states from tme evoluton How can one obtan a squeezed state, startng from, e.g., the vacuum state? It turns out that the squeezed states arse qute naturally from oscllator dynamcs, provded that the parameters of the oscllator Hamltonan, the frequency! and the mass m, are functons of tme. Let us consder the Schrodnger evoluton p m(t)! (t) h@ t = H(t) H(t) = q m(t) (7) Integratng formally the evoluton equaton n the tme nterval [0 t ], have Y N (t) = S(t) ^ t=0 = lm e ; N! j= h H(t j )t t=0 (8) wth t j = jt, t = t=n.

4 We now show that the evolvng squeezed state satses where P (t) and Q(t) are functons of tme to be found. Indeed, let us dene the quantty Then, at all tmes, (P (t) q^; Q(t) p^) (t) = 0 (9) C(t) ^ = S(t) ^ ( P (0)^ q ; Q(0)^ p) S ^ ; (t) (0) C(t) ^ (t) = S(t) ^ ( P (0)^ q ; Q(0)^ p) S ^ ; (t)s(t) ^ = 0 () The quantty C(t) ^ obeys equaton of moton of the form ^ H(t)t ^ H(t)t h@ t C(t) = lm e ; h C(t)e h ; C(t) ^ = [ C(t) ^ H(t)] () t!0 t We also note that, f C ^ s a polynomal n q, p of a degree n, then the commutator [C ^ H] s also a polynomal of exactly the same degree. Snce [C ^ H] determnes the rate of change of C ^ wth tme, and the C ^ starts as a lnear functon n q and p at t = 0, t s natural to suppose that C ^ remans lnear n n q and p at all tmes, C(t) ^ = P (t) q^; Q(t) p^. Usng the evoluton equaton, we nd h! p^ m! P h@ t C(t) = [C(t) H(t)] = (P q^ ; Qp^) ; ^ ^ q^ = h p^ m! Qq^ (3) m m ^ t C(t) = P (t) q^ ; Q(t) p^, obtan Q _ = P=m P _ = ;m! Q () whch concdes wth the classcal Hamltonan equatons for the oscllator (7). The ntal condtons, e.g., correspondng to the state (9), are generally speakng complex valued. For nstance, for the oscllator vacuum state, have P t=0 = Q t=0 = ; (5) Thus the evolvng state has the form (9) at all tmes wth (t) = ;P (t)=q(t). Let us consder a smple example, when the oscllator frequency jumps from! 0 to!. After that, the former ground state, whch s not a ground state any more, starts to evolve n tme. To analyze the dynamcs of Q and P, t s convenent to ntroduce the varables z = Q P =m! whch obey z_ = ;!z z_ ; =!z ; (6) Solvng for the tme evoluton z (t), obtan z (t) = e ;!t z (t = 0) = e ;!t m! ; m! 0 (7)

5 5 Smlarly, for z ; (t), obtan!t!t z~ ; (t) = e z~ ; (t = 0) = ;e (8) m! m! 0 whch gves sn!t cos!t sn!t cos!t Q(t) = (z z ; ) = ; P (t) = ; (9) m! m! 0 m! 0 m! The resultng tme dependence (t) = ;P (t)=q(t) agrees wth Eq.(). As ths example demonstrates, any tme dependence of the Hamltonan can be used to generate squeezed states. The underlyng physcal reason for ths general behavor s that the certanty product, proportonal to the phase space area of the Wgner densty peak, remans constant n tme due to the phase volume conservaton n Hamltonan dynamcs. However, n practce t s often desrable to get a hghly squeezed state by employng only small varatons of the oscllator parameters. The above example suggests that ths may b e dcult, snce for the value! close to! 0 the wavepacket wdth vares very lttle as a functon of tme. However, large squeezng by small perturbaton can b e acheved by usng the phenomenon of parametrc resonance. It s well known (to any chld on a swng) that, when the frequency of the oscllator s modulated,! (t) =! 0 cos t (30) and the external frequency s close to! 0, the classcal oscllator becomes unstable, wth the oscllatons of small ampltude growng larger as a functon of tme. A hghly squeezed state s formed near the moments correspondng to the extremal ponts of classcal moton, when P (t) s large and Q(t) small (beng complex, Q(t) never completely vanshes, but can become really small at large tmes). For further dscusson of the detals of parametrc resonance we refer to Problem, PS# (see also Landau & Lfshts, \Classcal Mechancs").

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