General phase spaces: from discrete variables to rotor and continuum limits

General phase spaces: from discrete variables to rotor and continuum limits

Victor V. Albert valbert4@gmail.com Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA    Saverio Pascazio Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy 
Istituto Nazionale di Ottica (INO-CNR), I-50125 Firenze, Italy 
INFN, Sezione di Bari, I-70126 Bari, Italy
   Michel H. Devoret michel.devoret@yale.edu Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
Abstract

We provide a basic introduction to discrete-variable, rotor, and continuous-variable quantum phase spaces, explaining how the latter two can be understood as limiting cases of the first. We extend the limit-taking procedures used to travel between phase spaces to a general class of Hamiltonians (including many local stabilizer codes) and provide six examples: the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the Kitaev honeycomb model. We obtain continuous-variable generalizations of all models, some of which are novel. The Baxter model is mapped to a chain of coupled oscillators and the Rabi model to the optomechanical radiation pressure Hamiltonian. The procedures also yield rotor versions of all models, five of which are novel many-body extensions of the almost Mathieu equation. The toric and cubic codes are mapped to lattice models of rotors, with the toric code case related to lattice gauge theory.

Rabi model, Baxter model, Harper equation, toric code, honeycomb model, Haah code
pacs:
03.65.-w, 02.30.Px

I Introduction

Continuous-variable (cv) limits of discrete-variable (dv) systems, if done carefully, can yield new models which are both interesting and helpful in illuminating low-energy properties of the original dv systems. A physical example comes from spin-wave theory, where the Hamiltonian of interacting spins is expanded in the limit of small quantum fluctuations () using the Holstein-Primakoff mapping Auerbach (1994). Another type of limit involves thinking of each dv system not as a spin, but as a finite quantum system Vourdas (2017) of dimension (in quantum information, a quit) whose conjugate variables, position and momentum, are bounded and discrete. This limit then involves making both variables continuous and unbounded. A less-used version makes one of the variables continuous and periodic (i.e., an angle) and the other an integer, resulting in the phase space of a rotor (; see Table 1 for all three phase spaces). While these limits deal with the underlying phase space of a dv system, it is not always clear when and how to apply them to specific dv Hamiltonians, particularly in composite scenarios (e.g., the Jaynes Cummings model) consisting of both dv and cv components. We attempt to address these issues by outlining general and straightforward limit-taking procedures and applying them to obtain known and new dv, rot, and cv extensions of six well-known models from condensed-matter physics and quantum computation (see Table 2).

In Sec. II, we provide a basic introduction of the three aforementioned phase spaces (dv, rot, and cv) such that the latter two can be thought of as limits of the former. In turn, dv phase space can be understood as a discretization of cv phase space in terms of fixed position and momentum increments and (similar to a computer approximating differential equations with difference equations). Section III outlines all limit-taking procedures, with comments on when can be a valid low-energy approximation of a dv Hamiltonian. We warm up with a known and exactly-solvable example of all limit-taking procedures — the harmonic oscillator AKA the Harper equation — in Sec. IV. In Sec. V, we study a many-body coupled-oscillator example — the Baxter parafermionic spin chain. In Sec. VI, we introduce the Rabi model, show that its -state extension has a dihedral symmetry, and provide its analogues in all three phase spaces. We continue with deriving the cv toric code model from the dv one while introducing novel rotor toric code extensions in Sec. VII. In Sec. VIII, we develop dv, rot, and cv extensions of the Haah cubic code. The Kitaev honeycomb model is generalized in Sec. IX. A final discussion is given in Sec. X.

Ii Classical and quantum phase spaces

dv rot cv
Sketch
Phase space
Weyl relation    
CCR  ” “  ”
Fourier Discrete Fourier transform Fourier series Fourier transform
“Bases”
Translations
Character
Relationship between bases
Completeness        
Orthonormality
Table 1: Comparison and summary of relations between conjugate variables in dv-, rot-, and cv-type phase spaces. Most physically relevant phase spaces consist of combinations of these three. An extension of the present analysis can be performed for any (locally compact) Abelian group Werner (2016) and for (compact) non-Abelian groups Lenz (1990) (if one is willing to allow commutators to be non-trivial operators, thereby dropping the notion of a phase space). The (Abelian) groups involved here are , the circle group , the integers , and the reals . The variables , , and . In order to respect the domains of their respective conjugate variables, the functionals and are respectively - and -periodic in their arguments. The canonical commutation relations (CCR’s) in quotes are incorrect due to not preserving domains, but nevertheless give the right physical intuition. The label “Bases” is in quotes because , , and are not normalizable. See related tables in Refs. Cotfas and Dragoman (2012); Werner (2016).

ii.1 Classical phase space

In classical physics, the phase space of a physical system with one degree of freedom is a two-dimensional manifold spanned by two infinitesimal translation generators, and , acting on the conjugate variables position and momentum, respectively. These translation operators commute,

(1)

Starting with the system located at an origin point, it is possible to reach a unique state in which the system is located at a well-defined phase space point using a sequence of elementary translations. The infinitesimal circuit associated with the sequence of translations from Eq. (1) defines a surface element of phase space (see Fig. 1). Any observable over phase space evolves in time according to Hamilton’s equation, written here as

(2)

where the Poisson bracket of two functions and of phase space is given by the exterior product

(3)

and is the Hamiltonian function characterizing the dynamics of system. Note that the Hamiltonian does not set the nature of the degree of freedom itself, which is set by the topology of phase space. For instance, the phase space of a 1-D massive particle evolving in an potential (ideal harmonic oscillator) is a flat plane whereas the phase space of a rigid pendulum or a rotor is an infinite cylinder. While various phase spaces are used in classical mechanics (depending on constraints one puts on a particle’s motion), there are only a few canonical topologies in quantum mechanics that meaningfully extend the classical notion of conjugate variables.

In order to be sufficiently general in our quantum mechanical treatment, instead of considering from the beginning a continuous phase space with infinitesimal generators and , we are going to introduce finite translation operators and and consider a topology of phase space where

(4)

where is a positive integer. Eventually, we will take the limits , and in varying ways. We thus consider that our 1-D degree of freedom, instead of evolving continuously, hops from site to site, the set of sites forming a ring graph shown in Fig. 2(a). The position variable denotes the site index and is thus an integer modulo , the total number of sites along the ring. If hopping between two sites takes the same universal amount of time, phase space is fully discrete, and the momentum also belongs to the set of integers modulo . Because of periodic boundary conditions for both position and momentum, as indicated by the set of black and white arrows in Fig. 2(b), phase space has the topology of a torus.

Figure 1: In phase space, four elementary translations along position and momentum define a closed circuit, here oriented clockwise, and its corresponding enclosed area, represented here in grey. Quantum physics associates a phase factor to an area in phase space, measured in units of Planck’s constant.

ii.2 Quantum phase space

In quantum physics, two conjugate translation generators do not commute, and one can categorize all possible relations into three cases:

(5)

The first case is obviously classical phase space. The second corresponds to a quantum phase space associated with a pair of Abelian groups of conjugate translations, typical of oscillators and rotors. The third case corresponds to spaces associated with non-Abelian groups, such as the group associated with a spin (we discuss later how the representation does produce the second case for particular ). In the following, we focus on the second case, that of a circuit in phase space producing a phase factor proportional to the area enclosed by the circuit, considering different limits for the step sizes and and the period . While we adhere to the definition of quantum phase space which corresponds only to the second case Werner (2016), we note that some of the properties we mention can also be extended to the third case Lenz (1990).

Let us explore in detail the second case. Using Eqs. (4-5) to simplify , we can see that

(6)

Since commutes with every operator in the algebra associated with the translations, we can represent it as

(7)

In our limit-taking procedures, the parameters , and do not vary independently, and we impose

(8)

In other words, , which corresponds to an elementary circuit in phase space accumulating a phase shift given by the encircled area divided by Planck’s constant, in similarity with Bohr’s old trajectory quantification rule.

Representing finite translations by exponentiation of translation generators (which we denote in bold),

(9a)
(9b)

we can rewrite the Weyl relation as

(10)

By naive expansion, we can recover the well-known result that the Weyl relation for the is equivalent to the canonical commutation relation (CCR) for the corresponding generating operators

(11)

The generators and and the commutation relation above define the algebra Gilmore (1974) (sometimes referred to as the Heisenberg-Weyl algebra Klein and Marshalek (1991)), which has only infinite-dimensional representations. We will review how the generators of motion for all common quantum-mechanical phase spaces, some of which are finite-dimensional, nevertheless emulate (see Table 1, fourth row).

Figure 2: (a) The dv phase space can be thought of as consisting of a degree of freedom hopping from site to site. The set of sites forms a ring. The position variable is a site index and is thus an integer modulo , the number of sites along the ring. (b) If hopping between two sites takes the universal same amount of time, phase space is fully discrete, with the momentum belonging also to the set of integers modulo . Both position and momentum have periodic boundary conditions, as indicated by the set of black and white arrows, and thus phase space has the topology of a torus.

ii.3 Toroidal doubly-discrete quantum phase space (dv)

In the algebra representing the translation operators, the relations (5) and (7) have important consequences. Let us introduce the projectors and over respective regions of phase space and , where and are some small intervals which can be thought of as discretizations of ordinary continuous phase space and

(12)

We later vary the intervals and in a way which converts the dv phase space (and its associated Hilbert space) into the rot and cv phase spaces. We have made the ranges of “two-sided”, i.e., defined them such that both their maximum and minimum values are functions of , in order to properly perform said procedures. The projectors (by definition, and ) satisfy

(13a)
(13b)
(13c)
(13d)

Quantum-mechanically, due to the non-commutation of conjugate translation operators, the product of projectors on position and momentum can no longer be a projector (as in the classical case). Thus, since the projectors do not commute and it is not possible to identify a phase space point state with both definite position and momentum . As a consequence of projections on conflicting with projections on , we have the discrete Fourier relations

(14)

where the position state vectors and momentum state vectors are defined as the eigenvectors of the projectors with eigenvalues 0 and 1:

(15)

The projectors in the position basis and in the momentum basis both resolve the identity,

(16)

and the overlap amplitude of the basis vectors is a constant,111A pair of bases for which the norm of the overlap between their constituents is independent of the basis labels is called mutually unbiased Durt et al. (2010).

(17)

There are thus only independent projectors whereas there are orthogonal observables (counting the identity as the constant observable). Classically, there would be also orthogonal observables, but there would be also as much as projectors. Thus, in quantum mechanics, the number of independent pure states is much less than the number of properties that can be acquired from them!

Let us now introduce the conjugate variables that label states of fixed position and momentum,

(18)

and quantify their conjugate nature. These operators label the columns and the rows of Fig. 2b, but again the non-commutation of these operators prevents simultaneously assigning a fixed position and momentum to the points in the corresponding classical space. The quantum or discrete Fourier transform operator going from the momentum basis to the position basis is

(19)

The position and momentum operators are related by this transform via

(20)

Performing the discrete Fourier transform twice yields the dv parity operator

(21)

This important operator takes to modulo .

We can readily make contact with standard Fourier analysis by linking a quantum pure state to a function of a discrete periodic variable . Namely, writing using the position basis and looking at the overlap of with a momentum eigenstate produces the discrete Fourier series of :

(22)

The same holds for the dual momentum basis.

In the position state vector basis, translations in position and in momentum are explicitly given by222These were introduced first by Sylvester in the 19th century Sylvester (1909) and applied to quantum mechanics by von Neumann von Neumann (1931), Weyl Weyl (1950), and Schwinger Schwinger (2000). They have been called Schwinger bases, Weyl operators Werner (2016), Pauli operators Gottesman et al. (2001), generalized spin Pittenger and Rubin (2004) or Pauli Kibler (2009) matrices, and ’t Hooft generators or clock-and-shift matrices Sachse (2006).

(23a)
(23b)

and such operators naturally perform displacements along the ring of sites in either position or momentum cross-sections of phase space:333The set forms a group, called the Generalized Pauli Group Kibler (2009), and an algebra, sometimes called the non-commutative torus Landi et al. (2001).

(24)

Strictly speaking, we cannot relate the Weyl relation

(25)

to a CCR since expanding to first order will violate our imposed domains on Hall (2013). In other words, can only be inside functions that are periodic in . However, we can depart from mathematical rigor and represent the above Weyl relation as the CCR

(26)

which is analogous to the continuous case (11). Moreover, if we express everything in terms of

(27)

and use our imposed phase space area constraint (8), the translations can be expressed as and and Eq. (26) recovers the ordinary CCR (11).

We briefly describe a Wigner function representation for dv for being odd Vourdas (2017). Recall that cv Wigner functions can be expressed in terms of the trace of a density matrix with a certain displaced parity operator; we provide the expression later in Eq. (40). In dv, an analogous expression is

(28)

where , is an density matrix, (21) is the dv parity operator, and is the dv displacement operator. The Wigner function conveniently takes real values over phase space and thus shares some of the properties of classical probability distributions, despite not always having positive values.

Properties of this dv fully discrete phase space for general are summarized in the first column of Table 1. We have only introduced the bare-bones framework, and there are many more quantities that can be defined in dv, including coherent states Galetti and Marchiolli (1996); Ruzzi et al. (2005), squeezed states Marchiolli et al. (2007), and quantum codes (Ketkar et al. (2006); Gottesman et al. (2001), Sec. II). There are also plenty of other ways to visualize states Galetti and de Toledo Piza (1988); Ruzzi and Galetti (2000); Gibbons et al. (2004); Marchiolli et al. (2005); Ferrie (2011); Marchiolli and Ruzzi (2012); Tilma et al. (2016); Ligabò (2016). We refer the reader to Refs. Miranowicz and Imoto (2001); de la Torre and Goyeneche (2003); Bengtsson and Zyczkowski (); Vourdas (2017) for further introductory reading.

The case — a spin one-half system

It is important to realize that the case is that of the ubiquitous spin-. In that case,

(29a)
(29b)
(29c)

Therefore, and . Since and same for , we have (modulo 2)

The Fourier transform corresponds to the well-known Hadamard transform and the parity operator (21) is trivial. We thus see that at the particular angle , the expression does produce a constant not equal to one — the second case in Eq. (5). This does not occur for any other values of . Recalling that the generators of the Lie algebra can be realized in a space of dimension given a spin Schiff (1968), the case at is the only time that spin rotations and dv translation operators coincide. Therefore, procedures involving representations of for , such as spin-coherent states Puri (2001), the Holstein-Primakoff transformation, and its associated Lie-algebraic contraction (; see, e.g., Arecchi et al. (1972); Atakishiyev et al. (2003); Gilmore (1974)), are not directly connected to the phase space analysis discussed here for .

ii.4 Cylindrical singly-discrete quantum phase space (rot)

We now take the limit , first considering the case where

(30)

invoking a universal constant . Then we can introduce conjugate variables

(31)

The operators and take their eigenvalues in the set of angles (compact set of reals modulo ) and the set of all integers, respectively:

(32)

hence the renaming of into .

In terms of wavefunctions, the limit as the number of points in the discretization is equivalent to the standard limit in which the discrete Fourier series of a discrete periodic wavefunction is transformed into the ordinary Fourier series of a continuous periodic function (see Wolf (1979), Sec. 3.4.5). In terms of the new conjugate variables, Eq. (22) becomes

(33)

where we have rescaled the coefficients as . In the large limit, this position-basis expansion of becomes an integral over the angle . In this infinite ladder or limit (), the circle labeled by eigenvalues of is essentially “cut open” and turns into the unbounded integer-valued variable , while at the same time is absorbed into the dense and bounded variable conjugate to . Of course, one could have instead done , which we call the the infinitely dense circle or limit. How the remaining properties of dv transform in this limit are listed in Table 1. The concepts discussed for dv in the text, such as coherent states Kowalski et al. (1996) and Wigner functions Berry (1977); Mukunda (1979), also naturally carry over to rot (see also, e.g., Zhang and Vourdas (2003); Ruzzi et al. (2006)).

In the rot1 limit, dv position and momentum eigenstates (14) become

(34)

respectively. Here, we encounter states which are normalizable only in the “Dirac” or “continuous” sense as well as the technicality that the orthonormality relation has to be -periodic:

(35)

Above, we define the -periodic -function in order to make sure that we can use any values of Raynal et al. (2010). However, if we restrict ourselves to using only , as in Table 1, the -function reduces to the ordinary -function. Since are not normalizable, they technically do not belong to the function space associated with rot, i.e., the space of functions such that (Hall (2013), Sec. 6.6).

Another consequence of domains and similar to the dv case, the Weyl relation between translations in and does not imply a proper CCR (see Ref. Hall (2013), Sec. 12.2). Assuming that restrict ourselves to using only , functions of must be -periodic in order to preserve its domain. Therefore, and its powers cannot act on states alone. If we ignore this fact and calculate the variances of states in and , then we will see that the former yields a finite number while the latter is zero. This violates Heisenberg’s uncertainty relation and thus the conventional CCR (we list what the CCR would have been if we did not worry about domains in Table 1).

Application for the rot phase space include i) the quantum rotor Raynal et al. (2010); Wen (2004), where () labels the angular momentum (position) of the rotor, ii) the motion of an electronic excitation in the periodic potential of crystal, where is the site index, assuming the crystal to be infinite, which makes analogous to the pseudo momentum in band-theory Ashcroft and Mermin (1976), or iii) the dynamics of a Josephson junction between two isolated islands, like in the Cooper pair box Girvin (2015); Devoret (1997), where is the phase difference between the two superconductors on either side of the junction and the number of Cooper pairs having tunneled across the junction.

ii.5 Flat-plane fully continuous quantum phase space (cv)

We again take the limit , but now consider the case where both

(36)

approach zero. Thus, while the whole of phase space has a number of points growing as , an area of order will harbor of order points and can still be considered continuous. Note that we did not have to split into two identical factors; any splitting and for is sufficient Ruzzi (2002); Ruzzi and Galetti (2002). Keeping with an even splitting, we introduce new conjugate variables

(37)

which become ordinary position and momentum in the large limit. We had already seen from Eq. (27) that this type of redefinition recovers the original commutation relation (11). In terms of wavefunctions, this is equivalent to the standard limit in which the discrete Fourier series of a periodic wavefunction is transformed into the continuum Fourier series as the functions period (see Wolf (1979), Sec. 3.4.5). In terms of the new conjugate variables, Eq. (22) becomes

(38)

where . Since for large , the above sum over (38) becomes an integral over . This completes the limit-taking procedure . The properties of this continuous flat phase space are summarized in the last column of Table 1. Just like (19), we can write the Fourier transform as a standalone operator:

(39)

One can easily confirm that is the parity operator taking . Note that eigenfunctions of position and momentum are, like (34), not normalizable and therefore not in the space of physical quantum states (Hall (2013), Sec. 6.6).

The cv Wigner function can then defined, analogous to (28), in terms of a cv displacement operator and the parity operator . One can easily confirm that takes . Letting and following Appx. A.2.1 of Ref. Haroche and Raimond (2006) yields

(40)

Now that we have performed the and limit-taking procedures, all that is left to complete the connections between them is the limit. Recall that the rot variables are the angular and integer . To perform the limit, we introduce a length scale which rescales the periodicity of and take this scale to infinity. The new variables this time are

(41)

The first redefinition transforms the already continuous variable into an unbounded variable while the second transforms the already unbounded variable into a continuous one (since its intervals go to zero). In terms of the new conjugate variables, the component of expanded in the basis becomes

(42)

where we define the rescaled coefficients . This completes the last limit , which is based on the well-known conversion of a Fourier series of a periodic function into a Fourier transform by taking the function’s periodicity to infinity.

Since the periodicity of both position and momentum goes to infinity, these and limit-taking procedures are well adapted to studies of harmonic and weakly anharmonic oscillators and to expansions of periodic functions of operators. Letting and be standard deviations of the zero point fluctuations of the oscillator, we can introduce the operators and such that

(43)

Then, the number of action quanta in the system is the operator

(44)

In general, the Hamiltonian is not a simple function of , but remains a balanced function of and . Using this notation, the Fourier transform and parity operator are simply

(45)

Note that in contrast with the situation with the pair and , there is no conjugate quantum operator for satisfying all of the properties in Table 1 (although there is an operator satisfying some of the properties Garrison and Wong (1970)). This is due to the fact that the polar representation of even a flat plane is singular when the radius is zero (equivalently, the eigenvalues of are bounded from below). This effect also obstructs us from creating an orthonormal basis of phase states for quantum optical applications (see Ref. Lynch (1995) or Ref. Schleich (2001), Problem 8.4).

dv rot cv
Sec. IV Harper equation Azbel (1964); Hofstadter (1976)   Almost Mathieu equation Avila and Jitomirskaya (2009) Harmonic oscillator
Sec. V   Baxter parafermionic spin chain Baxter (1989) Rotor Baxter chain Coupled-oscillator chain
Sec. VI -state Rabi model Rabi (1936, 1937); Albert (2012) Rotor-oscillator Rabi model Optomechanical Hamiltonian
Sec. VII Kitaev toric code Yu. Kitaev (2003); Wen (2003); Bullock and Brennen (2007) Rotor toric code   CV toric code Zhang et al. (2008)
Sec. VIII Haah cubic code Haah (2011) Rotor Haah code CV honeycomb model
Sec. IX Kitaev honeycomb model Yu. Kitaev (2006); Barkeshli et al. (2015); Fendley (2012) Rotor honeycomb model CV Haah code
Table 2: The first column lists models to which we apply the limit-taking procedures of Sec. III, yielding the rotor (rot) and continuous-variable (cv) models in the last two columns.

Iii Continuum and rotor limit-taking procedures

The above formulations of rot and cv from dv are only done on the level of the Hilbert space. When it comes to applying them to Hamiltonians, there are some additional subtleties which have to be dealt with. In an attempt to resolve such subtleties, let us demonstrate our slightly generalized limit-taking procedures on a general dv-type Hamiltonian. Since we saw that there are two ways to take the limit in Sec. II.4, the limits we consider below are summarized in the following diagram:

(46)

We start with a dv Hamiltonian that can be written as

(47)

where modulate the hopping length scales for the respective variables and are analytic -periodic functions of (18). Simpler versions of the limit, which are applicable for all but one of the models we consider, can be performed with . However, these scales are necessary to be able to obtain cv via rot, so we keep them for now.

Following Sec. II, we first write new conjugate variables in terms of ,

(48a)
(48b)
(48c)

where we have set for the first case because we do not need different length scales there. We then take the limit and write new Hamiltonians which serve as extensions of into cv and rot; the remainders of the respective limit-taking procedures are discussed in the next two subsections.

Previous efforts have rigorously studied similar embeddings in the past, in particular Barker Barker (2001a, b, c, 2003) and Digernes et al. Digernes et al. (1994). However, the former requires exact knowledge of the eigenstructure of both and (see Barker (2001c), Prop. 3.7) and is thus rigorously applicable to only simple examples. The latter constrains the Hamiltonian to be of a different form than . Here, we extend the procedure in the Supplement of Ref. Massar and Spindel (2008) to any (in the language of Barker (2001a), by “dead reckoning”) with the goal of creating a meaningful extension of the dv system into cv and rot. We apply these procedures to six models (see Table 2), obtaining continuum and rotor generalizations that in some cases have not been known before. We sometime keep track of the symmetries of the model to demonstrate that our limits are symmetry-preserving.

iii.1 Continuum limit

For this case, we take and expand around the center of -phase space. We could in principle expand around a generic point , but we can always redefine that to be the origin. This procedure can also be generalized to dv systems ; we stick to one for simplicity. First, let us take

(49)

We perform this limit in order to remove any factors of occurring in the expansion of some later on, noting that it is not necessary if such factors occur for all . The case when this step is necessary is only for the Rabi model in Sec. VI. We pick in order to have the limits conveniently produce the same result as we are about to produce, but this is done for convention since a sequence of rational numbers can yield any real. As a sanity check, we see that that , i.e., the hopping between sites (determined by ) does not increase faster than the total number of sites (proportional to ).

Recalling that we have redefined variables as in Eq. (48a), let us approximate and with their expansions around the zero eigenvalue of , respectively. Such an expansion can be done in a similar way as operator exponentiation, i.e., by working in a basis for which the two functions are diagonal and then expanding each of their eigenvalues. Such an expansion will not hold for arbitrarily high eigenvalues of since they are not always much less than one (e.g., near the maximal values of ). This means that, as , we have to keep projecting ourselves to the intersection of the subspaces of small eigenvalues of and . Consequently, the eigenstates of which remain in such a limit are only those which are centered around and have small variance in either variable. Expanding, we obtain (apart from a constant shift in energy)

(50)

with the coefficients obvious functions of and their derivatives (evaluated at zero). Thus, such a limit always yields a Hamiltonian consisting of linear and bilinear terms.

Let us discuss when the limit corresponds to a physically meaningful low-energy expansion of . A trivial sufficient condition is that the ground state subspace of is localized around . That way, expansion around encapsulates the ground state subspace and the low-energy excited states. If is a global maximum for each , then the minus sign in front of the sum (47) guarantees that the lowest-energy states will be centered around . [If there is another maximum at say for all , then expansion will of course ignore the ground state centered at .] However, being centered at the origin still does not guarantee that the ground-state subspace is localized to the same degree in as it is in . Examples of systems whose ground states are centered but not equally localized around are and for . In both cases, the second term gives a higher energy penalty for states near the origin than the first term, so expanding only the second term is more appropriate. A similar example of such an expansion (albeit of type) is the expansion of the cosine term in the Josephson junction Hamiltonian,

(51)

(where are real and